Re: Prime numbers
John, On 26 May 2013, at 00:54, John Mikes wrote: Bruno and others: did you read http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.single.html the information about prof. Zhang's discovery (U of New Hampshire)? It is still in the conjecture of mathematical proof and 'truth' with a position of primes are greater than 1 - with the interesting conclusion that 'primes' are the ATOMS of the number world. Any thoughts? Primes (1 is usually not considered as a prime number) are atoms of the numbers when conceived multiplicatively, because all numbers can be described uniquely as a product of primes. That is the existence and unicity of decomposition of numbers into prime factors (without taking the order of the multiplication into account). This is the so called fundamental theorem of arithmetic. It is easy to prove the existence of the decomposition into primes, but less easy to prove the uniqueness. For the twin conjecture, (it exists an infinity of pair of primes p and q with p - q = 2) it looks like an important step has been proved, (the case with p - q just bounded) but we are still far from proving the twin one. Most mathematician believe that the twin conjecture is true (like most believe that the Riemann conjecture is true). If they were false, the distribution of primes would not be statistically random, and that would mean something very special is at play, a bit like a number conspiracy! Why not, of course. We just don't know, but a non random behavior of the primes is a bit like the UFO of number theory. Well, except that for the UFO, there are (at least) some evidences (from time to time, most are eventually explained in general), but there is no evidence at all that the primes behave non- randomly (in the statistical sense, not in Chaitin-Kolmogorov sense as we can generate mechanically the distribution of primes). Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list?hl=en. For more options, visit https://groups.google.com/groups/opt_out.
Re: Prime numbers
http://www.newscientist.com/article/dn23595-weinsteins-theory-of-everything-is-probably-nothing.html Brent On 5/25/2013 3:54 PM, John Mikes wrote: Bruno and others: did you read http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.single.html the information about prof. Zhang's discovery (U of New Hampshire)? It is still in the conjecture of mathematical proof and 'truth' with a position of primes are greater than 1 - with the interesting conclusion that 'primes' are the ATOMS of the number world. Any thoughts? JM -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list?hl=en. For more options, visit https://groups.google.com/groups/opt_out. No virus found in this message. Checked by AVG - www.avg.com http://www.avg.com Version: 2013.0.3343 / Virus Database: 3184/6356 - Release Date: 05/25/13 -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list?hl=en. For more options, visit https://groups.google.com/groups/opt_out.
Re: Prime Numbers
On 27 Sep 2012, at 18:46, meekerdb wrote: On 9/27/2012 1:19 AM, Bruno Marchal wrote: On 26 Sep 2012, at 19:29, meekerdb wrote: On 9/25/2012 9:51 PM, Jason Resch wrote: On Sep 25, 2012, at 11:05 PM, meekerdb meeke...@verizon.net wrote: snip So you mean if some mathematical object implies a contradiction it doesn't exist, e.g. the largest prime number. But then of course the proof of contradiction is relative to the axioms and rules of inference. Well there is always some theory we have to assume, some model we operate under. This is needed just to communicate or to think. The contradiction proof is relevant to some theory, but so is the existence proof. You can't even define an object without using some agreed upon theory. Sure you can. You point and say, That! That's how you learned the meaning of words, by abstracting from a lot of instances of your mother pointing and saying, That. But this uses implicit theories selected by evolution. A brain *is* essentially a theory of the local universe already. At least that's your theory. :-) Hmm... If by brain you mean the material object, then a brain is not a theory, but the 3-I, the body description at the right comp- substitution level, is the theory. It is a word (finite object) interpreted by a universal system (physical forces, QM, bosons and fermions). The *material* brain, unfortunately perhaps, is not a word, it is an infinity of words interpreted by an infinity of competing universal numbers. We have to explain, with comp, why little numbers seems to win, because we can't prevent all the numbers to add their grains of salt, hopefully not their buggy grains of sand generating noise and/or white rabbits. Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 26 Sep 2012, at 19:29, meekerdb wrote: On 9/25/2012 9:51 PM, Jason Resch wrote: On Sep 25, 2012, at 11:05 PM, meekerdb meeke...@verizon.net wrote: snip So you mean if some mathematical object implies a contradiction it doesn't exist, e.g. the largest prime number. But then of course the proof of contradiction is relative to the axioms and rules of inference. Well there is always some theory we have to assume, some model we operate under. This is needed just to communicate or to think. The contradiction proof is relevant to some theory, but so is the existence proof. You can't even define an object without using some agreed upon theory. Sure you can. You point and say, That! That's how you learned the meaning of words, by abstracting from a lot of instances of your mother pointing and saying, That. But this uses implicit theories selected by evolution. A brain *is* essentially a theory of the local universe already. Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/27/2012 1:19 AM, Bruno Marchal wrote: On 26 Sep 2012, at 19:29, meekerdb wrote: On 9/25/2012 9:51 PM, Jason Resch wrote: On Sep 25, 2012, at 11:05 PM, meekerdb meeke...@verizon.net wrote: snip So you mean if some mathematical object implies a contradiction it doesn't exist, e.g. the largest prime number. But then of course the proof of contradiction is relative to the axioms and rules of inference. Well there is always some theory we have to assume, some model we operate under. This is needed just to communicate or to think. The contradiction proof is relevant to some theory, but so is the existence proof. You can't even define an object without using some agreed upon theory. Sure you can. You point and say, That! That's how you learned the meaning of words, by abstracting from a lot of instances of your mother pointing and saying, That. But this uses implicit theories selected by evolution. A brain *is* essentially a theory of the local universe already. At least that's your theory. :-) Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/25/2012 9:51 PM, Jason Resch wrote: On Sep 25, 2012, at 11:05 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 8:54 PM, Jason Resch wrote: On Sep 25, 2012, at 10:27 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 4:07 PM, Jason Resch wrote: Yes. If we cannot prove that their existence is self-contradictory Propositions can be self contradictory, but how can existence of something be self-contradictory? Brent Brent, it was roger, not I, who wrote the above. But in any case I interpreted his statement to mean if some theoretical object is found to have contradictory properties, then it does not exist. Sorry. No worries. So you mean if some mathematical object implies a contradiction it doesn't exist, e.g. the largest prime number. But then of course the proof of contradiction is relative to the axioms and rules of inference. Well there is always some theory we have to assume, some model we operate under. This is needed just to communicate or to think. The contradiction proof is relevant to some theory, but so is the existence proof. You can't even define an object without using some agreed upon theory. Sure you can. You point and say, That! That's how you learned the meaning of words, by abstracting from a lot of instances of your mother pointing and saying, That. Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Sep 26, 2012, at 12:29 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 9:51 PM, Jason Resch wrote: On Sep 25, 2012, at 11:05 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 8:54 PM, Jason Resch wrote: On Sep 25, 2012, at 10:27 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 4:07 PM, Jason Resch wrote: Yes. If we cannot prove that their existence is self- contradictory Propositions can be self contradictory, but how can existence of something be self-contradictory? Brent Brent, it was roger, not I, who wrote the above. But in any case I interpreted his statement to mean if some theoretical object is found to have contradictory properties, then it does not exist. Sorry. No worries. So you mean if some mathematical object implies a contradiction it doesn't exist, e.g. the largest prime number. But then of course the proof of contradiction is relative to the axioms and rules of inference. Well there is always some theory we have to assume, some model we operate under. This is needed just to communicate or to think. The contradiction proof is relevant to some theory, but so is the existence proof. You can't even define an object without using some agreed upon theory. Sure you can. You point and say, That! That's how you learned the meaning of words, by abstracting from a lot of instances of your mother pointing and saying, That. Brent There is still an implicitly assumed model that the two people are operating under (if they agree on what is meant by the chair they see). Or they may use different models and define the chair differently. For example, a solipist believes the chair is only his idea, a physicalist thinks it is a collection of primitive matter, a computationalist a dream of numbers. Then while they might all agree on the existence of something, that thing is different for each person because they are defining it under different models. Jason -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en . -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/26/2012 12:11 PM, Jason Resch wrote: On Sep 26, 2012, at 12:29 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 9:51 PM, Jason Resch wrote: On Sep 25, 2012, at 11:05 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 8:54 PM, Jason Resch wrote: On Sep 25, 2012, at 10:27 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 4:07 PM, Jason Resch wrote: Yes. If we cannot prove that their existence is self-contradictory Propositions can be self contradictory, but how can existence of something be self-contradictory? Brent Brent, it was roger, not I, who wrote the above. But in any case I interpreted his statement to mean if some theoretical object is found to have contradictory properties, then it does not exist. Sorry. No worries. So you mean if some mathematical object implies a contradiction it doesn't exist, e.g. the largest prime number. But then of course the proof of contradiction is relative to the axioms and rules of inference. Well there is always some theory we have to assume, some model we operate under. This is needed just to communicate or to think. The contradiction proof is relevant to some theory, but so is the existence proof. You can't even define an object without using some agreed upon theory. Sure you can. You point and say, That! That's how you learned the meaning of words, by abstracting from a lot of instances of your mother pointing and saying, That. Brent There is still an implicitly assumed model that the two people are operating under (if they agree on what is meant by the chair they see). Or they may use different models and define the chair differently. For example, a solipist believes the chair is only his idea, a physicalist thinks it is a collection of primitive matter, a computationalist a dream of numbers. Then while they might all agree on the existence of something, that thing is different for each person because they are defining it under different models. But if they are different then what sense does it make to say there is a contradiction in *the* model and hence something doesn't exist. That's why it makes no sense to talk about a contradiction disproving the existence of something you can define ostensively. It is only in the Platonia of statements that you can derive contradictions from axioms and rules of inference. If you can point to the thing whose non-existence is proven, then it just means you've made an error in translating between reality and Platonia. Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Wed, Sep 26, 2012 at 2:33 PM, meekerdb meeke...@verizon.net wrote: On 9/26/2012 12:11 PM, Jason Resch wrote: On Sep 26, 2012, at 12:29 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 9:51 PM, Jason Resch wrote: On Sep 25, 2012, at 11:05 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 8:54 PM, Jason Resch wrote: On Sep 25, 2012, at 10:27 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 4:07 PM, Jason Resch wrote: Yes. If we cannot prove that their existence is self-contradictory Propositions can be self contradictory, but how can existence of something be self-contradictory? Brent Brent, it was roger, not I, who wrote the above. But in any case I interpreted his statement to mean if some theoretical object is found to have contradictory properties, then it does not exist. Sorry. No worries. So you mean if some mathematical object implies a contradiction it doesn't exist, e.g. the largest prime number. But then of course the proof of contradiction is relative to the axioms and rules of inference. Well there is always some theory we have to assume, some model we operate under. This is needed just to communicate or to think. The contradiction proof is relevant to some theory, but so is the existence proof. You can't even define an object without using some agreed upon theory. Sure you can. You point and say, That! That's how you learned the meaning of words, by abstracting from a lot of instances of your mother pointing and saying, That. Brent There is still an implicitly assumed model that the two people are operating under (if they agree on what is meant by the chair they see). Or they may use different models and define the chair differently. For example, a solipist believes the chair is only his idea, a physicalist thinks it is a collection of primitive matter, a computationalist a dream of numbers. Then while they might all agree on the existence of something, that thing is different for each person because they are defining it under different models. But if they are different then what sense does it make to say there is a contradiction in *the* model and hence something doesn't exist. It means a certain object (which is defined in a model) does not exist in that model. A model in one object is not the same as another object in a different model, even if they might have the same name, symbol, or appearance. 2 in a finite field, is a different thing from 2 in the natural numbers. The chair in the solipist model is different from the chair in the materialist model. A chair made out of primitively real matter is non-existent in the solipist model. I don't see how you can escape having to work within a model when you make assertions, like X exists, or Y does not exist. What is X or Y outside of the model from which they are defined and exist within? Jason That's why it makes no sense to talk about a contradiction disproving the existence of something you can define ostensively. It is only in the Platonia of statements that you can derive contradictions from axioms and rules of inference. If you can point to the thing whose non-existence is proven, then it just means you've made an error in translating between reality and Platonia. Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.**comeverything-list@googlegroups.com . To unsubscribe from this group, send email to everything-list+unsubscribe@ **googlegroups.com everything-list%2bunsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/** group/everything-list?hl=enhttp://groups.google.com/group/everything-list?hl=en . -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/26/2012 2:53 PM, Jason Resch wrote: On Wed, Sep 26, 2012 at 2:33 PM, meekerdb meeke...@verizon.net mailto:meeke...@verizon.net wrote: On 9/26/2012 12:11 PM, Jason Resch wrote: On Sep 26, 2012, at 12:29 PM, meekerdb meeke...@verizon.net mailto:meeke...@verizon.net wrote: On 9/25/2012 9:51 PM, Jason Resch wrote: On Sep 25, 2012, at 11:05 PM, meekerdb meeke...@verizon.net mailto:meeke...@verizon.net wrote: On 9/25/2012 8:54 PM, Jason Resch wrote: On Sep 25, 2012, at 10:27 PM, meekerdb meeke...@verizon.net mailto:meeke...@verizon.net wrote: On 9/25/2012 4:07 PM, Jason Resch wrote: Yes. If we cannot prove that their existence is self-contradictory Propositions can be self contradictory, but how can existence of something be self-contradictory? Brent Brent, it was roger, not I, who wrote the above. But in any case I interpreted his statement to mean if some theoretical object is found to have contradictory properties, then it does not exist. Sorry. No worries. So you mean if some mathematical object implies a contradiction it doesn't exist, e.g. the largest prime number. But then of course the proof of contradiction is relative to the axioms and rules of inference. Well there is always some theory we have to assume, some model we operate under. This is needed just to communicate or to think. The contradiction proof is relevant to some theory, but so is the existence proof. You can't even define an object without using some agreed upon theory. Sure you can. You point and say, That! That's how you learned the meaning of words, by abstracting from a lot of instances of your mother pointing and saying, That. Brent There is still an implicitly assumed model that the two people are operating under (if they agree on what is meant by the chair they see). Or they may use different models and define the chair differently. For example, a solipist believes the chair is only his idea, a physicalist thinks it is a collection of primitive matter, a computationalist a dream of numbers. Then while they might all agree on the existence of something, that thing is different for each person because they are defining it under different models. But if they are different then what sense does it make to say there is a contradiction in *the* model and hence something doesn't exist. It means a certain object (which is defined in a model) does not exist in that model. A model in one object is not the same as another object in a different model, even if they might have the same name, symbol, or appearance. 2 in a finite field, is a different thing from 2 in the natural numbers. The chair in the solipist model is different from the chair in the materialist model. A chair made out of primitively real matter is non-existent in the solipist model. I don't see how you can escape having to work within a model when you make assertions, like X exists, or Y does not exist. I don't try to escape that. What is X or Y outside of the model from which they are defined and exist within? The whole point of having a model is that X and Y refer to something outside the model. The model is a model *of* reality, not reality itself. So when you prove X and ~X in the model you may have proved X doesn't exist or you may have shown your model doesn't correspond to reality. Brent Jason That's why it makes no sense to talk about a contradiction disproving the existence of something you can define ostensively. It is only in the Platonia of statements that you can derive contradictions from axioms and rules of inference. If you can point to the thing whose non-existence is proven, then it just means you've made an error in translating between reality and Platonia. Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com mailto:everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com mailto:everything-list%2bunsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Wed, Sep 26, 2012 at 5:01 PM, meekerdb meeke...@verizon.net wrote: On 9/26/2012 2:53 PM, Jason Resch wrote: On Wed, Sep 26, 2012 at 2:33 PM, meekerdb meeke...@verizon.net wrote: On 9/26/2012 12:11 PM, Jason Resch wrote: On Sep 26, 2012, at 12:29 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 9:51 PM, Jason Resch wrote: On Sep 25, 2012, at 11:05 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 8:54 PM, Jason Resch wrote: On Sep 25, 2012, at 10:27 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 4:07 PM, Jason Resch wrote: Yes. If we cannot prove that their existence is self-contradictory Propositions can be self contradictory, but how can existence of something be self-contradictory? Brent Brent, it was roger, not I, who wrote the above. But in any case I interpreted his statement to mean if some theoretical object is found to have contradictory properties, then it does not exist. Sorry. No worries. So you mean if some mathematical object implies a contradiction it doesn't exist, e.g. the largest prime number. But then of course the proof of contradiction is relative to the axioms and rules of inference. Well there is always some theory we have to assume, some model we operate under. This is needed just to communicate or to think. The contradiction proof is relevant to some theory, but so is the existence proof. You can't even define an object without using some agreed upon theory. Sure you can. You point and say, That! That's how you learned the meaning of words, by abstracting from a lot of instances of your mother pointing and saying, That. Brent There is still an implicitly assumed model that the two people are operating under (if they agree on what is meant by the chair they see). Or they may use different models and define the chair differently. For example, a solipist believes the chair is only his idea, a physicalist thinks it is a collection of primitive matter, a computationalist a dream of numbers. Then while they might all agree on the existence of something, that thing is different for each person because they are defining it under different models. But if they are different then what sense does it make to say there is a contradiction in *the* model and hence something doesn't exist. It means a certain object (which is defined in a model) does not exist in that model. A model in one object is not the same as another object in a different model, even if they might have the same name, symbol, or appearance. 2 in a finite field, is a different thing from 2 in the natural numbers. The chair in the solipist model is different from the chair in the materialist model. A chair made out of primitively real matter is non-existent in the solipist model. I don't see how you can escape having to work within a model when you make assertions, like X exists, or Y does not exist. I don't try to escape that. What is X or Y outside of the model from which they are defined and exist within? The whole point of having a model is that X and Y refer to something outside the model. The model is a model *of* reality, not reality itself. So when you prove X and ~X in the model you may have proved X doesn't exist or you may have shown your model doesn't correspond to reality. Okay. I think we are in agreement then. The main idea is to make a model of reality and test it by seeing how well the model's predictions for observations match our observations. Jason -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Re: Prime Numbers
Hi Stephen P. King Yes, I think that the structures and attributes of matter are provided by a creator (the All, the supreme monad, or God). Plato used the analogy of geometrical shapes for his structures. Roger Clough, rclo...@verizon.net 9/25/2012 Forever is a long time, especially near the end. -Woody Allen - Receiving the following content - From: Stephen P. King Receiver: everything-list Time: 2012-09-24, 10:42:12 Subject: Re: Prime Numbers On 9/24/2012 9:46 AM, Roger Clough wrote: God's ideas is fine. The numbers and arithmetic etc. can inhere in some mind. The numbers are (idealistically) real, as I think all arithmetic must be. For it is true whether known or not. At least as you stay with common numbers and arithmetic. Pretty sure. Hi Roger, One question I have to pose: How do the properties of entities become discriminated from each other and collected together? Are the properties on a object inherent or is there some other active system of property attribution in Nature? Does God play a role in this? -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Re: Prime Numbers
On Tue, Sep 25, 2012 at 7:35 AM, Roger Clough rclo...@verizon.net wrote: Hi Stephen P. King Yes, I think that the structures and attributes of matter are provided by a creator (the All, the supreme monad, or God). Plato used the analogy of geometrical shapes for his structures. But if you believe in the All do you also believe there are other types of matter, other universes, other planets with intelligent beings, etc? Jason -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/25/2012 10:24 AM, Jason Resch wrote: On Tue, Sep 25, 2012 at 7:35 AM, Roger Clough rclo...@verizon.net mailto:rclo...@verizon.net wrote: Hi Stephen P. King Yes, I think that the structures and attributes of matter are provided by a creator (the All, the supreme monad, or God). Plato used the analogy of geometrical shapes for his structures. But if you believe in the All do you also believe there are other types of matter, other universes, other planets with intelligent beings, etc? Jason Hi Jason, Yes. If we cannot prove that their existence is self-contradictory then we should consider them as possible. Just because I cannot experience or imagine something is not a proof of impossibility. -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Tue, Sep 25, 2012 at 5:37 PM, Stephen P. King stephe...@charter.netwrote: On 9/25/2012 10:24 AM, Jason Resch wrote: On Tue, Sep 25, 2012 at 7:35 AM, Roger Clough rclo...@verizon.net wrote: Hi Stephen P. King Yes, I think that the structures and attributes of matter are provided by a creator (the All, the supreme monad, or God). Plato used the analogy of geometrical shapes for his structures. But if you believe in the All do you also believe there are other types of matter, other universes, other planets with intelligent beings, etc? Jason Hi Jason, Yes. If we cannot prove that their existence is self-contradictory then we should consider them as possible. Just because I cannot experience or imagine something is not a proof of impossibility. Roger, I agree with you here. But then this seems to contradict the notion that *this* world is the best of all possible worlds, unless by this world you mean the All. After all Leibniz said Everything that is possible demands to exist. Jason -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/25/2012 7:07 PM, Jason Resch wrote: On Tue, Sep 25, 2012 at 5:37 PM, Stephen P. King stephe...@charter.net mailto:stephe...@charter.net wrote: On 9/25/2012 10:24 AM, Jason Resch wrote: On Tue, Sep 25, 2012 at 7:35 AM, Roger Clough rclo...@verizon.net mailto:rclo...@verizon.net wrote: Hi Stephen P. King Yes, I think that the structures and attributes of matter are provided by a creator (the All, the supreme monad, or God). Plato used the analogy of geometrical shapes for his structures. But if you believe in the All do you also believe there are other types of matter, other universes, other planets with intelligent beings, etc? Jason Hi Jason, Yes. If we cannot prove that their existence is self-contradictory then we should consider them as possible. Just because I cannot experience or imagine something is not a proof of impossibility. Roger, I agree with you here. But then this seems to contradict the notion that *this* world is the best of all possible worlds, unless by this world you mean the All. After all Leibniz said Everything that is possible demands to exist. Jason Hi Jason, Well said! I think that Leibniz' idea that *this* world is the best of all possible worlds has a stipulation that was not stated! It only seems to make sense that Leibniz was defining this world as the world that we observe *and* communicate about with each other. It is the best possible by necessity as it is impossible for us to experience any other lesser version. We have least action rules in physics that are nice demonstration of this... -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/25/2012 4:07 PM, Jason Resch wrote: Yes. If we cannot prove that their existence is self-contradictory Propositions can be self contradictory, but how can existence of something be self-contradictory? Brent then we should consider them as possible. Just because I cannot experience or imagine something is not a proof of impossibility. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Sep 25, 2012, at 10:27 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 4:07 PM, Jason Resch wrote: Yes. If we cannot prove that their existence is self-contradictory Propositions can be self contradictory, but how can existence of something be self-contradictory? Brent Brent, it was roger, not I, who wrote the above. But in any case I interpreted his statement to mean if some theoretical object is found to have contradictory properties, then it does not exist. Jason then we should consider them as possible. Just because I cannot experience or imagine something is not a proof of impossibility. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en . -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/25/2012 8:54 PM, Jason Resch wrote: On Sep 25, 2012, at 10:27 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 4:07 PM, Jason Resch wrote: Yes. If we cannot prove that their existence is self-contradictory Propositions can be self contradictory, but how can existence of something be self-contradictory? Brent Brent, it was roger, not I, who wrote the above. But in any case I interpreted his statement to mean if some theoretical object is found to have contradictory properties, then it does not exist. Sorry. So you mean if some mathematical object implies a contradiction it doesn't exist, e.g. the largest prime number. But then of course the proof of contradiction is relative to the axioms and rules of inference. Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Sep 25, 2012, at 11:05 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 8:54 PM, Jason Resch wrote: On Sep 25, 2012, at 10:27 PM, meekerdb meeke...@verizon.net wrote: On 9/25/2012 4:07 PM, Jason Resch wrote: Yes. If we cannot prove that their existence is self-contradictory Propositions can be self contradictory, but how can existence of something be self-contradictory? Brent Brent, it was roger, not I, who wrote the above. But in any case I interpreted his statement to mean if some theoretical object is found to have contradictory properties, then it does not exist. Sorry. No worries. So you mean if some mathematical object implies a contradiction it doesn't exist, e.g. the largest prime number. But then of course the proof of contradiction is relative to the axioms and rules of inference. Well there is always some theory we have to assume, some model we operate under. This is needed just to communicate or to think. The contradiction proof is relevant to some theory, but so is the existence proof. You can't even define an object without using some agreed upon theory. Jason Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en . -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Re: Prime Numbers
Hi Bruno Marchal Numbers are not in spacetime, that is, are not at location r at time t. So they are ideas, they are not physical. To be physical you have to have a specific location at a specific time. This is not my view, it is that of Descartes. The same with arithmetic. Numbers and arithmetic statements are not at (r,t). Which is not to say that they are not real, if by real I mean true or as is without an observer. Like in a textbook. Roger Clough, rclo...@verizon.net 9/24/2012 Forever is a long time, especially near the end. -Woody Allen - Receiving the following content - From: Bruno Marchal Receiver: everything-list Time: 2012-09-23, 03:42:03 Subject: Re: Prime Numbers On 22 Sep 2012, at 22:10, Stephen P. King wrote: On 9/22/2012 7:32 AM, Roger Clough wrote: How could mathematics be fiction ? If so, then we could simply say that 2+2=5 because it's saturday. How could we have a world we many minds can, on rare occasions, come to complete agreement if that where the case? Perhaps it is true that 2+2=4 because we all agree, at some level, that it is true. (I am not just considering humans here with the word we!) How will you define we without accepting 2+2=4, given that IF we assume comp, we are defined by (L?ian) universal number and their relations with other universal numbers? Why do you keep an idealist conception of numbers, which contradicts your references to papers which use, as most texts in science, the independence and primitivity of elementary arithmetic? Or you remark was ironic? Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Re: Prime Numbers
Hi Bruno Marchal I believe that there are at least three attributes of numbers: 1) Are they true or false as in a numerical equation ? Does 2+ 2 = 4 ? True. 2) Do they physically exist or do they mentally inhere ? They inhere. You can't touch them. 3) Are they real or not ? Numbers are always real (in the philosophical sense). Roger Clough, rclo...@verizon.net 9/24/2012 Forever is a long time, especially near the end. -Woody Allen - Receiving the following content - From: Bruno Marchal Receiver: everything-list Time: 2012-09-23, 03:42:03 Subject: Re: Prime Numbers On 22 Sep 2012, at 22:10, Stephen P. King wrote: On 9/22/2012 7:32 AM, Roger Clough wrote: How could mathematics be fiction ? If so, then we could simply say that 2+2=5 because it's saturday. How could we have a world we many minds can, on rare occasions, come to complete agreement if that where the case? Perhaps it is true that 2+2=4 because we all agree, at some level, that it is true. (I am not just considering humans here with the word we!) How will you define we without accepting 2+2=4, given that IF we assume comp, we are defined by (L?ian) universal number and their relations with other universal numbers? Why do you keep an idealist conception of numbers, which contradicts your references to papers which use, as most texts in science, the independence and primitivity of elementary arithmetic? Or you remark was ironic? Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Re: Prime Numbers
Hi Stephen P. King That's what Peirce gave as a pragmatic definition of truth, something that we would all agree to, given time enough. But fiction can be true (as true fiction, a narrative woven about actual events) or not be true. Arithmetic isn't, it's either always true or always false. Roger Clough, rclo...@verizon.net 9/24/2012 Forever is a long time, especially near the end. -Woody Allen - Receiving the following content - From: Stephen P. King Receiver: everything-list Time: 2012-09-22, 16:10:38 Subject: Re: Prime Numbers On 9/22/2012 7:32 AM, Roger Clough wrote: How could mathematics be fiction ? If so, then we could simply say that 2+2=5 because it's saturday. How could we have a world we many minds can, on rare occasions, come to complete agreement if that where the case? Perhaps it is true that 2+2=4 because we all agree, at some level, that it is true. (I am not just considering humans here with the word we!) -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 24 Sep 2012, at 12:39, Roger Clough wrote: Hi Bruno Marchal Numbers are not in spacetime, that is, are not at location r at time t. So they are ideas, God's ideas? Then I am OK. The comp God is arithmetical truth, so this works. they are not physical. OK. To be physical you have to have a specific location at a specific time. I am OK with this, but note that it makes the Universe into a non physical object. The Universe cannot belong to a location r at time t, as it is the gauge making such position and time consistent in the picture. This is not my view, it is that of Descartes. The same with arithmetic. Numbers and arithmetic statements are not at (r,t). OK. Which is not to say that they are not real, if by real I mean true or as is without an observer. Like in a textbook. OK. So you can understand how comp is interesting, as it explains (partially but more than any other theory) how the physical beliefs appears and why they come in two sort of shapes (quanta and qualia), and this without assuming anything more than elementary arithmetic and the invariance of consciousness for some digital transformations. Then the big picture happens to be closer to the neoplatonists one than the aristotelian one, which I think you should appreciate. Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Re: Prime Numbers
Hi Bruno Marchal God's ideas is fine. The numbers and arithmetic etc. can inhere in some mind. The numbers are (idealistically) real, as I think all arithmetic must be. For it is true whether known or not. At least as you stay with common numbers and arithmetic. Pretty sure. Roger Clough, rclo...@verizon.net 9/24/2012 Forever is a long time, especially near the end. -Woody Allen - Receiving the following content - From: Bruno Marchal Receiver: everything-list Time: 2012-09-24, 09:12:29 Subject: Re: Prime Numbers On 24 Sep 2012, at 12:39, Roger Clough wrote: Hi Bruno Marchal Numbers are not in spacetime, that is, are not at location r at time t. So they are ideas, God's ideas? Then I am OK. The comp God is arithmetical truth, so this works. they are not physical. OK. To be physical you have to have a specific location at a specific time. I am OK with this, but note that it makes the Universe into a non physical object. The Universe cannot belong to a location r at time t, as it is the gauge making such position and time consistent in the picture. This is not my view, it is that of Descartes. The same with arithmetic. Numbers and arithmetic statements are not at (r,t). OK. Which is not to say that they are not real, if by real I mean true or as is without an observer. Like in a textbook. OK. So you can understand how comp is interesting, as it explains (partially but more than any other theory) how the physical beliefs appears and why they come in two sort of shapes (quanta and qualia), and this without assuming anything more than elementary arithmetic and the invariance of consciousness for some digital transformations. Then the big picture happens to be closer to the neoplatonists one than the aristotelian one, which I think you should appreciate. Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/24/2012 9:46 AM, Roger Clough wrote: God's ideas is fine. The numbers and arithmetic etc. can inhere in some mind. The numbers are (idealistically) real, as I think all arithmetic must be. For it is true whether known or not. At least as you stay with common numbers and arithmetic. Pretty sure. Hi Roger, One question I have to pose: How do the properties of entities become discriminated from each other and collected together? Are the properties on a object inherent or is there some other active system of property attribution in Nature? Does God play a role in this? -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 22 Sep 2012, at 22:10, Stephen P. King wrote: On 9/22/2012 7:32 AM, Roger Clough wrote: How could mathematics be fiction ? If so, then we could simply say that 2+2=5 because it's saturday. How could we have a world we many minds can, on rare occasions, come to complete agreement if that where the case? Perhaps it is true that 2+2=4 because we all agree, at some level, that it is true. (I am not just considering humans here with the word we!) How will you define we without accepting 2+2=4, given that IF we assume comp, we are defined by (Löbian) universal number and their relations with other universal numbers? Why do you keep an idealist conception of numbers, which contradicts your references to papers which use, as most texts in science, the independence and primitivity of elementary arithmetic? Or you remark was ironic? Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/23/2012 3:42 AM, Bruno Marchal wrote: On 22 Sep 2012, at 22:10, Stephen P. King wrote: On 9/22/2012 7:32 AM, Roger Clough wrote: How could mathematics be fiction ? If so, then we could simply say that 2+2=5 because it's saturday. How could we have a world we many minds can, on rare occasions, come to complete agreement if that where the case? Perhaps it is true that 2+2=4 because we all agree, at some level, that it is true. (I am not just considering humans here with the word we!) How will you define we without accepting 2+2=4, given that IF we assume comp, we are defined by (Löbian) universal number and their relations with other universal numbers? Why do you keep an idealist conception of numbers, which contradicts your references to papers which use, as most texts in science, the independence and primitivity of elementary arithmetic? Or you remark was ironic? Bruno http://iridia.ulb.ac.be/~marchal/ The continued confusion of the symbols and what they represent makes this entire conversation an exercise in futility. -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Re: Prime Numbers
Hi Rex Allen How could mathematics be fiction ? If so, then we could simply say that 2+2=5 because it's saturday. Roger Clough, rclo...@verizon.net 9/22/2012 Forever is a long time, especially near the end. -Woody Allen - Receiving the following content - From: Rex Allen Receiver: everything-list Time: 2012-09-21, 09:20:41 Subject: Re: Prime Numbers Just to avoid confusion, this sentence: I would say that mathematics is just very tightly plotted fiction where so many details of the story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the reader.? Should probably be: I would say that mathematics is just very tightly plotted fiction where so many details of the back-story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the reader.? On Fri, Sep 21, 2012 at 8:40 AM, Rex Allen wrote: On Tue, Sep 18, 2012 at 11:50 PM, Terren Suydam wrote: On Tue, Sep 18, 2012 at 10:19 PM, Rex Allen wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. I'm curious about what a plausible fictionalist account of the Mandelbrot set could be. Is fictionalism the same as constructivism, or the idea that knowledge doesn't exist outside of a mind? I lean towards a strong form of fictionalism - which says that there are few important differences between mathematics and literary fiction. So - I could give a detailed answer - but I think I'd rather give a sketchy answer at this point. I would say that mathematics is just very tightly plotted fiction where so many details of the story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the reader. Mathematics is a kind of world building. ?n the?maginative?ense. ? But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? How is it that we are able to reliably know things about Platonia? I think just doing logic and math - starting from axioms and proving things from them - is interacting with the Platonic realm. But how is it that we humans do that? ?his is my main question. ?hat exactly are we doing when we start from axioms and prove things from them? ?here does this ability come from? ?hat does it consist of? I would have thought that quarks and electrons from which we appear to be constituted would be indifferent to truth. Which would fit with the fact that I seem to make a lot of mistakes. But you think otherwise? I didn't understand the above... what do quarks and electrons have to do with arithmetical platonism? Are we not composed from quarks and electrons? ?f so - then how do mere collections of quarks and electrons connect with platonic truths? By chance? ?re we just fortunate that the initial conditions and causal laws of the universe are such that our quarks and electrons take forms that mirror Platonic Truths? ? How can you make sense of that in terms of the constructivist point of view that you are (I think) compelled to take if you argue against arithmetical platonism? ?t seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. I will agree with you that all intelligences that start from the same premises as you, and follow the same rules as inference as you, will also draw the same conclusions about the Mandelbrot set as you do. However - I do not agree with you that this amenable group exhausts the set of all *possible* intelligences. I only meant that all possible intelligences that start from a mathematics that includes addition, multiplication, and complex numbers will find that if they iterate the function z' = z^2 + c, they will find that some orbits become periodic or settle on a point, and some escape to infinity. If they draw a graph of which orbits don't escape, they will draw the Mandelbrot Set. All possible intelligences that undertake that procedure will draw the same shape... and this seems like discovery, not creation. It seems like a tautology to me. ?f you do what I do and believe what I believe then you will be a lot like me...? Is there anything to mathematics other than belief? What are beliefs? ?hy do we have the beliefs that we have? ?ow do we form beliefs - what lies behind belief? Can *our* mathematical abilities be reduced to something
Re: Re: Prime Numbers
Hi Terren Suydam I don't see that mathematics and fiction have anything in common. With fiction, anything can happen. A would of could be, or should be. With mathematics you've got that nasty equals sign. A world of is. Hume pointed out that there's no way to get from is to ought or vice versa. Roger Clough, rclo...@verizon.net 9/22/2012 Forever is a long time, especially near the end. -Woody Allen - Receiving the following content - From: Terren Suydam Receiver: everything-list Time: 2012-09-21, 12:29:56 Subject: Re: Prime Numbers On Fri, Sep 21, 2012 at 8:40 AM, Rex Allen rexallen31...@gmail.com wrote: On Tue, Sep 18, 2012 at 11:50 PM, Terren Suydam terren.suy...@gmail.com wrote: I'm curious about what a plausible fictionalist account of the Mandelbrot set could be. Is fictionalism the same as constructivism, or the idea that knowledge doesn't exist outside of a mind? I lean towards a strong form of fictionalism - which says that there are few important differences between mathematics and literary fiction. Can you articulate any important differences between them? So - I could give a detailed answer - but I think I'd rather give a sketchy answer at this point. I would say that mathematics is just very tightly plotted fiction where so many details of the story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the reader. Mathematics is a kind of world building. In the imaginative sense. I am not unsympathetic with this view, given the creativity that goes into mathematical proofs. However, it falls apart for me when I consider that an alien civilization is constrained to build the same worlds if they start from the same logical axioms. I think just doing logic and math - starting from axioms and proving things from them - is interacting with the Platonic realm. But how is it that we humans do that? This is my main question. What exactly are we doing when we start from axioms and prove things from them? Where does this ability come from? What does it consist of? We're using our intelligence and creativity to search a space of propositions (given a set of axioms) that are either provably true or false. I would say our intelligence and creativity comes from our animal nature, evolved as it is to make sense of the world (and each other) and draw useful inferences that help us survive. I'm not sure how to answer the question what does it consist of. Are you asking how we can act intelligently, how creativity works? I didn't understand the above... what do quarks and electrons have to do with arithmetical platonism? Are we not composed from quarks and electrons? If so - then how do mere collections of quarks and electrons connect with platonic truths? By chance? Are we just fortunate that the initial conditions and causal laws of the universe are such that our quarks and electrons take forms that mirror Platonic Truths? I see. Assuming comp, we are some infinite subset of the trace of the UD (universal dovetailer), which is a platonic entity. Quarks and electrons are a part of the physics that emerges from that (the numbers' dreams)... that's the reversal, where physics emerges from computer science. The question of how we, as mere collections of quarks etc. connect back with Platonia, is answered by CT (Church-Turing Thesis). As we are universal machines, we can emulate any computation, including the universal dovetailer (for instance). I only meant that all possible intelligences that start from a mathematics that includes addition, multiplication, and complex numbers will find that if they iterate the function z' = z^2 + c, they will find that some orbits become periodic or settle on a point, and some escape to infinity. If they draw a graph of which orbits don't escape, they will draw the Mandelbrot Set. All possible intelligences that undertake that procedure will draw the same shape... and this seems like discovery, not creation. It seems like a tautology to me. If you do what I do and believe what I believe then you will be a lot like me...? Is there anything to mathematics other than belief? The point is that you are constrained in what you can prove starting from a given set of axioms. You are not constrained in which axioms you start with - that's where the belief comes in since there is no way to prove that your axioms are True, except within a more encompassing logical framework with its own axioms. What are beliefs? Why do we have the beliefs that we have? How do we form beliefs - what lies behind belief? Beliefs in the everyday sense are inferences about our experience that we hold to be true. They help us navigate the world as we experience it, and make sense of it. Mostly our beliefs are formed by suggestion from our parents and peers when we are young, and as we learn and grow we complicate our worldview with new beliefs
Re: Re: Prime Numbers
Hi meekerdb Mathematical objects such as proofs ansd new theorems are found by intuition. Penrose suggests that intuition is a peep into Platonia. So these come from Platonia. Roger Clough, rclo...@verizon.net 9/22/2012 Forever is a long time, especially near the end. -Woody Allen - Receiving the following content - From: meekerdb Receiver: everything-list Time: 2012-09-21, 13:30:03 Subject: Re: Prime Numbers On 9/21/2012 5:40 AM, Rex Allen wrote: On Tue, Sep 18, 2012 at 11:50 PM, Terren Suydam terren.suy...@gmail.com wrote: On Tue, Sep 18, 2012 at 10:19 PM, Rex Allen rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.com wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. I'm curious about what a plausible fictionalist account of the Mandelbrot set could be. Is fictionalism the same as constructivism, or the idea that knowledge doesn't exist outside of a mind? I lean towards a strong form of fictionalism - which says that there are few important differences between mathematics and literary fiction. So - I could give a detailed answer - but I think I'd rather give a sketchy answer at this point. I would say that mathematics is just very tightly plotted fiction where so many details of the story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the reader. Mathematics is a kind of world building. In the imaginative sense. But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? How is it that we are able to reliably know things about Platonia? I think just doing logic and math - starting from axioms and proving things from them - is interacting with the Platonic realm. But how is it that we humans do that? This is my main question. What exactly are we doing when we start from axioms and prove things from them? Where does this ability come from? What does it consist of? I would have thought that quarks and electrons from which we appear to be constituted would be indifferent to truth. Which would fit with the fact that I seem to make a lot of mistakes. But you think otherwise? I didn't understand the above... what do quarks and electrons have to do with arithmetical platonism? Are we not composed from quarks and electrons? If so - then how do mere collections of quarks and electrons connect with platonic truths? By chance? Are we just fortunate that the initial conditions and causal laws of the universe are such that our quarks and electrons take forms that mirror Platonic Truths? How can you make sense of that in terms of the constructivist point of view that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. I will agree with you that all intelligences that start from the same premises as you, and follow the same rules as inference as you, will also draw the same conclusions about the Mandelbrot set as you do. However - I do not agree with you that this amenable group exhausts the set of all *possible* intelligences. I only meant that all possible intelligences that start from a mathematics that includes addition, multiplication, and complex numbers will find that if they iterate the function z' = z^2 + c, they will find that some orbits become periodic or settle on a point, and some escape to infinity. If they draw a graph of which orbits don't escape, they will draw the Mandelbrot Set. All possible intelligences that undertake that procedure will draw the same shape... and this seems like discovery, not creation. It seems like a tautology to me. If you do what I do and believe what I believe then you will be a lot like me...? Is there anything to mathematics other than belief? What are beliefs? Why do we have the beliefs that we have? How do we form beliefs - what lies behind belief? Can *our* mathematical abilities be reduced to something that is indifferent to mathematical truth? Could there be intelligences who start from vastly difference premises, and use vastly different rules of inference, and draw vastly different conclusions? Of course, but then what they are doing doesn't relate to the Mandelbrot Set. However - they might *believe* their creations to be just as significant and universal as you consider the Mandelbrot Set to be - mightened they? What would make them wrong in their belief but you right in yours
Re: Prime Numbers
On 21 Sep 2012, at 19:17, meekerdb wrote:
On 9/21/2012 1:22 AM, Bruno Marchal wrote:
On 20 Sep 2012, at 20:14, meekerdb wrote:
On 9/20/2012 10:31 AM, Bruno Marchal wrote:
On 20 Sep 2012, at 18:14, meekerdb wrote:
On 9/20/2012 2:05 AM, Bruno Marchal wrote:
A modal logic of probability is given by the behavior of the
probability one. In Kripke terms, P(x) = 1 in world alpha
means that x is realized in all worlds accessible from alpha,
and (key point) that we are not in a cul-de-sac world.
What does 'accessible' mean?
In modal logic semantic, it is a technical world for any element
in set + a binary relation on it.
A mapping of the set onto itself?
?
A relation is not a map. A world can access more than one world.
For example {a, b} with the relation {(a, a), (a, b)}, or aRa, aRb.
When applied to probability, the idea is to interpret the worlds
by the realization of some random experience, like throwing a
coin would lead to two worlds accessible, one with head, the
other with tail. In that modal (tail or head) is a certainty as
(tail or head) is realized everywhere in the accessible worlds.
Then accessible means nomologically possible.
Accessible means only that some binary relation exists on a set.
But in some concrete model of a multi-world or multi-situation
context, nomological possibility is not excluded.
Then I don't understand what other kinds of possibility are
allowed? I don't see how logical possibility could be considered an
accessibility relation (at least not an interesting one) because it
would allow Rxy where y was anything except not-x.
But in the worlds of the UD there is no nomological constraint, so
there's no probability measure?
I am not sure why there is no nomological constraints in the UD.
UD* is a highly structured entity. You might elaborate on this.
A nomological constraint is one of physics.
Why? Define perhaps nomological.
But physics is derivative from part of the UD. The UD is structured
only by arithmetic.
Why would this be not enough, given that physics will supervene on
arithmetical relations (computations)?
Bruno
Generally speaking a different world is defined as not
accessible. If you can go there, it's part of your same world.
Yes. OK. Sorry. Logician used the term world in a technical
sense, and the worlds can be anything, depending of which modal
logic is used, for what purpose, etc. Kripke semantic main used
is in showing the independence of formula in different systems.
Bruno
Brent
This gives KD modal logics, with K: = [](p - q)-([]p -
[]q), and D: []p - p. Of course with [] for Gödel's
beweisbar we don't have that D is a theorem, so we ensure the D
property by defining a new box, Bp = []p t.
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Re: Prime Numbers
On 9/22/2012 7:32 AM, Roger Clough wrote: How could mathematics be fiction ? If so, then we could simply say that 2+2=5 because it's saturday. How could we have a world we many minds can, on rare occasions, come to complete agreement if that where the case? Perhaps it is true that 2+2=4 because we all agree, at some level, that it is true. (I am not just considering humans here with the word we!) -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 20 Sep 2012, at 20:14, meekerdb wrote: On 9/20/2012 10:31 AM, Bruno Marchal wrote: On 20 Sep 2012, at 18:14, meekerdb wrote: On 9/20/2012 2:05 AM, Bruno Marchal wrote: A modal logic of probability is given by the behavior of the probability one. In Kripke terms, P(x) = 1 in world alpha means that x is realized in all worlds accessible from alpha, and (key point) that we are not in a cul-de-sac world. What does 'accessible' mean? In modal logic semantic, it is a technical world for any element in set + a binary relation on it. When applied to probability, the idea is to interpret the worlds by the realization of some random experience, like throwing a coin would lead to two worlds accessible, one with head, the other with tail. In that modal (tail or head) is a certainty as (tail or head) is realized everywhere in the accessible worlds. Then accessible means nomologically possible. Accessible means only that some binary relation exists on a set. But in some concrete model of a multi-world or multi-situation context, nomological possibility is not excluded. But in the worlds of the UD there is no nomological constraint, so there's no probability measure? I am not sure why there is no nomological constraints in the UD. UD* is a highly structured entity. You might elaborate on this. Bruno Generally speaking a different world is defined as not accessible. If you can go there, it's part of your same world. Yes. OK. Sorry. Logician used the term world in a technical sense, and the worlds can be anything, depending of which modal logic is used, for what purpose, etc. Kripke semantic main used is in showing the independence of formula in different systems. Bruno Brent This gives KD modal logics, with K: = [](p - q)-([]p - []q), and D: []p - p. Of course with [] for Gödel's beweisbar we don't have that D is a theorem, so we ensure the D property by defining a new box, Bp = []p t. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com . To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en . http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en . http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Tue, Sep 18, 2012 at 11:50 PM, Terren Suydam terren.suy...@gmail.comwrote: On Tue, Sep 18, 2012 at 10:19 PM, Rex Allen rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.com wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. I'm curious about what a plausible fictionalist account of the Mandelbrot set could be. Is fictionalism the same as constructivism, or the idea that knowledge doesn't exist outside of a mind? I lean towards a strong form of fictionalism - which says that there are few important differences between mathematics and literary fiction. So - I could give a detailed answer - but I think I'd rather give a sketchy answer at this point. I would say that mathematics is just very tightly plotted fiction where so many details of the story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the reader. Mathematics is a kind of world building. In the imaginative sense. But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? How is it that we are able to reliably know things about Platonia? I think just doing logic and math - starting from axioms and proving things from them - is interacting with the Platonic realm. But how is it that we humans do that? This is my main question. What exactly are we doing when we start from axioms and prove things from them? Where does this ability come from? What does it consist of? I would have thought that quarks and electrons from which we appear to be constituted would be indifferent to truth. Which would fit with the fact that I seem to make a lot of mistakes. But you think otherwise? I didn't understand the above... what do quarks and electrons have to do with arithmetical platonism? Are we not composed from quarks and electrons? If so - then how do mere collections of quarks and electrons connect with platonic truths? By chance? Are we just fortunate that the initial conditions and causal laws of the universe are such that our quarks and electrons take forms that mirror Platonic Truths? How can you make sense of that in terms of the constructivist point of view that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. I will agree with you that all intelligences that start from the same premises as you, and follow the same rules as inference as you, will also draw the same conclusions about the Mandelbrot set as you do. However - I do not agree with you that this amenable group exhausts the set of all *possible* intelligences. I only meant that all possible intelligences that start from a mathematics that includes addition, multiplication, and complex numbers will find that if they iterate the function z' = z^2 + c, they will find that some orbits become periodic or settle on a point, and some escape to infinity. If they draw a graph of which orbits don't escape, they will draw the Mandelbrot Set. All possible intelligences that undertake that procedure will draw the same shape... and this seems like discovery, not creation. It seems like a tautology to me. If you do what I do and believe what I believe then you will be a lot like me...? Is there anything to mathematics other than belief? What are beliefs? Why do we have the beliefs that we have? How do we form beliefs - what lies behind belief? Can *our* mathematical abilities be reduced to something that is indifferent to mathematical truth? Could there be intelligences who start from vastly difference premises, and use vastly different rules of inference, and draw vastly different conclusions? Of course, but then what they are doing doesn't relate to the Mandelbrot Set. However - they might *believe* their creations to be just as significant and universal as you consider the Mandelbrot Set to be - mightened they? What would make them wrong in their belief but you right in yours? What are the limits of belief, do you think? Is there any belief that is so preposterous that even the maddest of the mad could not believe such a thing? I don't think so... based on my understanding of how mad maddest of the mad can get. And if there is no such belief - then is it conceivable that quarks and electrons could configure themselves in such a way as to *cause* a being who holds such beliefs to come into existence?
Re: Prime Numbers
On Wed, Sep 19, 2012 at 12:27 AM, Jason Resch jasonre...@gmail.com wrote: On Sep 18, 2012, at 9:19 PM, Rex Allen rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.com terren.suy...@gmail.com wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? We study and create theories about objects in the mathematical realm just as we study and create theories about objects in the physical realm. So in the physical realm, we start from our senses - what we see, hear, feel, etc. From this, we infer the existence of electrons and wavefunctions and strings and whatnot. Or some of us do. Others take a more instrumental view of scientific theories. So you're saying that thought is another kind of sense? And that what occurs to us in thought can also be used as a basis to infer the existence of objects which help explain those thoughts? But we believe that electrons interact causally with us because we are made from similar stuff - and by doing so make themselves known to us...right? How do Platonic objects interact causally with us? Via a Platonic Field? PFT - Platonic Field Theory? It's not much different from how we develop theories about other things we cannot interact with: the early universe, the cores of stars, the insides of black holes, etc. We test these theories by following their implications and seeing if they lead to contridictions with other, more established, facts. Just as with physical theories, we ocasionally find that we need to throw out the old set of theories (or axioms) for a new set which has greater explanatory power. So you think our current mathematical theories are not true in any metaphysical sense - but rather are approximations of what exists in Platonia? Is there an equivalent of the idea of domains of validity that holds in some circles in physics? I'm not sure any of this counts as being evidence in favor of Platonism... How is it that we are able to reliably know things about Platonia? The very idea of knowing implies a differentiation between true and false. Nearly any intelligent civilization that notices a partition between true and false will eventyally get here. True in what sense? A coherentist conception of truth? A correspondence conception of truth? How do we know truth? Do we have an innate truth sense? Does the ability to know truth require free will? For instance: If we say a statement is true because it is true, that is different than saying it is true because our neurons fired in a way that determined our response. If all our decisions were predetermined from the moment of the big bang then rational discussion is meaningless. Whether or not anyone agrees with you has nothing to do with the truth of your claim. Their beliefs were hardwired from the beginning of time. It follows then that your own beliefs are not based on their truth value. You believe what you believe because your neurons have determined that you will believe in this rather than that. SO - what is this truth stuff, really? I would have thought that quarks and electrons from which we appear to be constituted would be indifferent to truth. The unreasonable effectiveness of math in the physical sciences is yet further support if Platonism. If this, and seemingly infinite physical universes exist, and they are mathematical structures, why can't others exist? Which would fit with the fact that I seem to make a lot of mistakes. But you think otherwise? We are imperfect beings. What is the source of imperfection? Where does it come from? What explains it? Objectively, intrinsically, absolutely imperfect? Have you heard the term Works as coded, with respect to software development? So I can write a program that has a bug in it - and the computer will run it perfectly. The computer will do exactly what I told it to do. The program works as coded. When running my program, the computer is perfectly imperfect. I am the source of its imperfection. However, in a functionalist theory of mind - I am actually just executing my own program right? Given the initial conditions of the universe and the causal laws that govern it - I could not do other than I did when I wrote that buggy code. I also work as coded. I also am perfectly imperfect. And since in this view I am not the source of my own imperfection - the universe's initial conditions and causal laws must be that source. But what explains that imperfection? But - maybe there really is no such thing as imperfection? It's all just made up...like mathematical
Re: Prime Numbers
On Sep 21, 2012, at 8:13 AM, Rex Allen rexallen31...@gmail.com wrote: On Wed, Sep 19, 2012 at 12:27 AM, Jason Resch jasonre...@gmail.com wrote: On Sep 18, 2012, at 9:19 PM, Rex Allen rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.com wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? We study and create theories about objects in the mathematical realm just as we study and create theories about objects in the physical realm. So in the physical realm, we start from our senses - what we see, hear, feel, etc. From this, we infer the existence of electrons and wavefunctions and strings and whatnot. Or some of us do. Others take a more instrumental view of scientific theories. Right, and we have similarly inferred the existence of primes, fractals, non-computable functions, etc. So you're saying that thought is another kind of sense? Thought is needed for inference and building theories, equally in the physical sciences and math. And that what occurs to us in thought can also be used as a basis to infer the existence of objects which help explain those thoughts? Right, like you might think up genesis and dualism, or big bang and materialism, or platonic truth and computationalism. These are ontological theories for what exists, and why we are here experiencing it. If you say math is fiction and only exists only as a story in our brains, then obviously you can't use platonic truth and computationalism as one if your theories of existence. I think the fact that mathematics can serve as a theory for our existence shows absolutely that mathematical theories and physical theories are on equal footing. We can gather evidence for them and build cases for them, find out we were wrong about them, and so on. Why do we believe in quarks, electrons, strings, etc.? Because they can explain our observations. Why do I believe in the platonic realm? For the same reasons. But we believe that electrons interact causally with us because we are made from similar stuff - and by doing so make themselves known to us...right? How do Platonic objects interact causally with us? Via a Platonic Field? PFT - Platonic Field Theory? How did the warping of space and time cause Einsteins brain to figure out relativity? I think you are looking at it in the wrong way. Our brains seek good explanations. They sometimes find one. That's all that is going on. Now you say our explainations when it comes to mathematics are fiction, but if that is so, why not say the same of the physical theories? Why not say the big bang is fiction, or matter is fiction? I think this leads to declaring everything but one's current thought is fiction, which does not seem very useful. It's not much different from how we develop theories about other things we cannot interact with: the early universe, the cores of stars, the insides of black holes, etc. We test these theories by following their implications and seeing if they lead to contridictions with other, more established, facts. Just as with physical theories, we ocasionally find that we need to throw out the old set of theories (or axioms) for a new set which has greater explanatory power. So you think our current mathematical theories are not true in any metaphysical sense - but rather are approximations of what exists in Platonia? They may or may not be true, but they are certainly incomplete. Just like our physical theories may or not be true descriptions of the universe, and are certainly incomplete. Is there an equivalent of the idea of domains of validity that holds in some circles in physics? I don't know what this concept means well enough to say. I'm not sure any of this counts as being evidence in favor of Platonism... How is it that we are able to reliably know things about Platonia? The very idea of knowing implies a differentiation between true and false. Nearly any intelligent civilization that notices a partition between true and false will eventyally get here. True in what sense? A coherentist conception of truth? A correspondence conception of truth? In the sense of the notion that a proposition is either true or false. How do we know truth? Do we have an innate truth sense? How do we know anything? Do we know anything? Does the ability to know truth require free will? Comparabalist or incompatibalist? For instance: If we say a statement is true because it is true, What we say or
Re: Prime Numbers
On Fri, Sep 21, 2012 at 8:40 AM, Rex Allen rexallen31...@gmail.com wrote: On Tue, Sep 18, 2012 at 11:50 PM, Terren Suydam terren.suy...@gmail.com wrote: I'm curious about what a plausible fictionalist account of the Mandelbrot set could be. Is fictionalism the same as constructivism, or the idea that knowledge doesn't exist outside of a mind? I lean towards a strong form of fictionalism - which says that there are few important differences between mathematics and literary fiction. Can you articulate any important differences between them? So - I could give a detailed answer - but I think I'd rather give a sketchy answer at this point. I would say that mathematics is just very tightly plotted fiction where so many details of the story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the reader. Mathematics is a kind of world building. In the imaginative sense. I am not unsympathetic with this view, given the creativity that goes into mathematical proofs. However, it falls apart for me when I consider that an alien civilization is constrained to build the same worlds if they start from the same logical axioms. I think just doing logic and math - starting from axioms and proving things from them - is interacting with the Platonic realm. But how is it that we humans do that? This is my main question. What exactly are we doing when we start from axioms and prove things from them? Where does this ability come from? What does it consist of? We're using our intelligence and creativity to search a space of propositions (given a set of axioms) that are either provably true or false. I would say our intelligence and creativity comes from our animal nature, evolved as it is to make sense of the world (and each other) and draw useful inferences that help us survive. I'm not sure how to answer the question what does it consist of. Are you asking how we can act intelligently, how creativity works? I didn't understand the above... what do quarks and electrons have to do with arithmetical platonism? Are we not composed from quarks and electrons? If so - then how do mere collections of quarks and electrons connect with platonic truths? By chance? Are we just fortunate that the initial conditions and causal laws of the universe are such that our quarks and electrons take forms that mirror Platonic Truths? I see. Assuming comp, we are some infinite subset of the trace of the UD (universal dovetailer), which is a platonic entity. Quarks and electrons are a part of the physics that emerges from that (the numbers' dreams)... that's the reversal, where physics emerges from computer science. The question of how we, as mere collections of quarks etc. connect back with Platonia, is answered by CT (Church-Turing Thesis). As we are universal machines, we can emulate any computation, including the universal dovetailer (for instance). I only meant that all possible intelligences that start from a mathematics that includes addition, multiplication, and complex numbers will find that if they iterate the function z' = z^2 + c, they will find that some orbits become periodic or settle on a point, and some escape to infinity. If they draw a graph of which orbits don't escape, they will draw the Mandelbrot Set. All possible intelligences that undertake that procedure will draw the same shape... and this seems like discovery, not creation. It seems like a tautology to me. If you do what I do and believe what I believe then you will be a lot like me...? Is there anything to mathematics other than belief? The point is that you are constrained in what you can prove starting from a given set of axioms. You are not constrained in which axioms you start with - that's where the belief comes in since there is no way to prove that your axioms are True, except within a more encompassing logical framework with its own axioms. What are beliefs? Why do we have the beliefs that we have? How do we form beliefs - what lies behind belief? Beliefs in the everyday sense are inferences about our experience that we hold to be true. They help us navigate the world as we experience it, and make sense of it. Mostly our beliefs are formed by suggestion from our parents and peers when we are young, and as we learn and grow we complicate our worldview with new beliefs. There isn't much behind belief except habituation. Certainly most of us hold onto some beliefs that are contradicted by facts (particularly the beliefs we hold of ourselves). Can *our* mathematical abilities be reduced to something that is indifferent to mathematical truth? I think if you were doing math in a way that was indifferent to mathematical truth, you wouldn't be very good at math. Of course, but then what they are doing doesn't relate to the Mandelbrot Set. However - they might *believe* their creations to be just as significant and universal
Re: Prime Numbers
On 9/21/2012 1:22 AM, Bruno Marchal wrote: On 20 Sep 2012, at 20:14, meekerdb wrote: On 9/20/2012 10:31 AM, Bruno Marchal wrote: On 20 Sep 2012, at 18:14, meekerdb wrote: On 9/20/2012 2:05 AM, Bruno Marchal wrote: A modal logic of probability is given by the behavior of the probability one. In Kripke terms, P(x) = 1 in world alpha means that x is realized in all worlds accessible from alpha, and (key point) that we are not in a cul-de-sac world. What does 'accessible' mean? In modal logic semantic, it is a technical world for any element in set + a binary relation on it. A mapping of the set onto itself? When applied to probability, the idea is to interpret the worlds by the realization of some random experience, like throwing a coin would lead to two worlds accessible, one with head, the other with tail. In that modal (tail or head) is a certainty as (tail or head) is realized everywhere in the accessible worlds. Then accessible means nomologically possible. Accessible means only that some binary relation exists on a set. But in some concrete model of a multi-world or multi-situation context, nomological possibility is not excluded. Then I don't understand what other kinds of possibility are allowed? I don't see how logical possibility could be considered an accessibility relation (at least not an interesting one) because it would allow Rxy where y was anything except not-x. But in the worlds of the UD there is no nomological constraint, so there's no probability measure? I am not sure why there is no nomological constraints in the UD. UD* is a highly structured entity. You might elaborate on this. A nomological constraint is one of physics. But physics is derivative from part of the UD. The UD is structured only by arithmetic. Brent Bruno Generally speaking a different world is defined as not accessible. If you can go there, it's part of your same world. Yes. OK. Sorry. Logician used the term world in a technical sense, and the worlds can be anything, depending of which modal logic is used, for what purpose, etc. Kripke semantic main used is in showing the independence of formula in different systems. Bruno Brent This gives KD modal logics, with K: = [](p - q)-([]p - []q), and D: []p - p. Of course with [] for Gödel's beweisbar we don't have that D is a theorem, so we ensure the D property by defining a new box, Bp = []p t. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/21/2012 5:40 AM, Rex Allen wrote: On Tue, Sep 18, 2012 at 11:50 PM, Terren Suydam terren.suy...@gmail.com mailto:terren.suy...@gmail.com wrote: On Tue, Sep 18, 2012 at 10:19 PM, Rex Allen rexallen31...@gmail.com mailto:rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.com mailto:terren.suy...@gmail.com wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. I'm curious about what a plausible fictionalist account of the Mandelbrot set could be. Is fictionalism the same as constructivism, or the idea that knowledge doesn't exist outside of a mind? I lean towards a strong form of fictionalism - which says that there are few important differences between mathematics and literary fiction. So - I could give a detailed answer - but I think I'd rather give a sketchy answer at this point. I would say that mathematics is just very tightly plotted fiction where so many details of the story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the reader. Mathematics is a kind of world building. In the imaginative sense. But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? How is it that we are able to reliably know things about Platonia? I think just doing logic and math - starting from axioms and proving things from them - is interacting with the Platonic realm. But how is it that we humans do that? This is my main question. What exactly are we doing when we start from axioms and prove things from them? Where does this ability come from? What does it consist of? I would have thought that quarks and electrons from which we appear to be constituted would be indifferent to truth. Which would fit with the fact that I seem to make a lot of mistakes. But you think otherwise? I didn't understand the above... what do quarks and electrons have to do with arithmetical platonism? Are we not composed from quarks and electrons? If so - then how do mere collections of quarks and electrons connect with platonic truths? By chance? Are we just fortunate that the initial conditions and causal laws of the universe are such that our quarks and electrons take forms that mirror Platonic Truths? How can you make sense of that in terms of the constructivist point of view that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. I will agree with you that all intelligences that start from the same premises as you, and follow the same rules as inference as you, will also draw the same conclusions about the Mandelbrot set as you do. However - I do not agree with you that this amenable group exhausts the set of all *possible* intelligences. I only meant that all possible intelligences that start from a mathematics that includes addition, multiplication, and complex numbers will find that if they iterate the function z' = z^2 + c, they will find that some orbits become periodic or settle on a point, and some escape to infinity. If they draw a graph of which orbits don't escape, they will draw the Mandelbrot Set. All possible intelligences that undertake that procedure will draw the same shape... and this seems like discovery, not creation. It seems like a tautology to me. If you do what I do and believe what I believe then you will be a lot like me...? Is there anything to mathematics other than belief? What are beliefs? Why do we have the beliefs that we have? How do we form beliefs - what lies behind belief? Can *our* mathematical abilities be reduced to something that is indifferent to mathematical truth? Could there be intelligences who start from vastly difference premises, and use vastly different rules of inference, and draw vastly different conclusions? Of course, but then what they are doing doesn't relate to the Mandelbrot Set. However - they might *believe* their creations to be just as significant and universal as you consider the Mandelbrot Set to be - mightened they? What would make them wrong in their belief but you right in yours? What are the limits of belief, do you think? Is there any belief that is so preposterous that even the
Re: Prime Numbers
On 9/21/2012 8:59 AM, Jason Resch wrote: On Sep 21, 2012, at 8:13 AM, Rex Allen rexallen31...@gmail.com mailto:rexallen31...@gmail.com wrote: On Wed, Sep 19, 2012 at 12:27 AM, Jason Resch jasonre...@gmail.com mailto:jasonre...@gmail.com wrote: On Sep 18, 2012, at 9:19 PM, Rex Allen rexallen31...@gmail.com mailto:rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.com mailto:terren.suy...@gmail.com wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? We study and create theories about objects in the mathematical realm just as we study and create theories about objects in the physical realm. So in the physical realm, we start from our senses - what we see, hear, feel, etc. From this, we infer the existence of electrons and wavefunctions and strings and whatnot. Or some of us do. Others take a more instrumental view of scientific theories. Right, and we have similarly inferred the existence of primes, fractals, non-computable functions, etc. We invented counting, addition, etc and found it implied true propositions about primes, fractals, etc. To say they exist in the same way tables and chairs exist is going much further. So you're saying that thought is another kind of sense? Thought is needed for inference and building theories, equally in the physical sciences and math. And that what occurs to us in thought can also be used as a basis to infer the existence of objects which help explain those thoughts? Right, like you might think up genesis and dualism, or big bang and materialism, or platonic truth and computationalism. These are ontological theories for what exists, and why we are here experiencing it. If you say math is fiction and only exists only as a story in our brains, then obviously you can't use platonic truth and computationalism as one if your theories of existence. I think the fact that mathematics can serve as a theory for our existence shows absolutely that mathematical theories and physical theories are on equal footing. We can gather evidence for them and build cases for them, find out we were wrong about them, and so on. Why do we believe in quarks, electrons, strings, etc.? Because they can explain our observations. Why do I believe in the platonic realm? For the same reasons. But we believe that electrons interact causally with us because we are made from similar stuff - and by doing so make themselves known to us...right? How do Platonic objects interact causally with us? Via a Platonic Field? PFT - Platonic Field Theory? How did the warping of space and time cause Einsteins brain to figure out relativity? I think you are looking at it in the wrong way. Our brains seek good explanations. They sometimes find one. That's all that is going on. Now you say our explainations when it comes to mathematics are fiction, but if that is so, why not say the same of the physical theories? Why not say the big bang is fiction, or matter is fiction? They are stories which we intend to have referents independent of the stories (theories). Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
Just to avoid confusion, this sentence: *I would say that mathematics is just very tightly plotted fiction where so many details of the story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the reader.* Should probably be: *I would say that mathematics is just very tightly plotted fiction where so many details of the back-story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the reader. * On Fri, Sep 21, 2012 at 8:40 AM, Rex Allen rexallen31...@gmail.com wrote: On Tue, Sep 18, 2012 at 11:50 PM, Terren Suydam terren.suy...@gmail.comwrote: On Tue, Sep 18, 2012 at 10:19 PM, Rex Allen rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.com wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. I'm curious about what a plausible fictionalist account of the Mandelbrot set could be. Is fictionalism the same as constructivism, or the idea that knowledge doesn't exist outside of a mind? I lean towards a strong form of fictionalism - which says that there are few important differences between mathematics and literary fiction. So - I could give a detailed answer - but I think I'd rather give a sketchy answer at this point. I would say that mathematics is just very tightly plotted fiction where so many details of the story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the reader. Mathematics is a kind of world building. In the imaginative sense. But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? How is it that we are able to reliably know things about Platonia? I think just doing logic and math - starting from axioms and proving things from them - is interacting with the Platonic realm. But how is it that we humans do that? This is my main question. What exactly are we doing when we start from axioms and prove things from them? Where does this ability come from? What does it consist of? I would have thought that quarks and electrons from which we appear to be constituted would be indifferent to truth. Which would fit with the fact that I seem to make a lot of mistakes. But you think otherwise? I didn't understand the above... what do quarks and electrons have to do with arithmetical platonism? Are we not composed from quarks and electrons? If so - then how do mere collections of quarks and electrons connect with platonic truths? By chance? Are we just fortunate that the initial conditions and causal laws of the universe are such that our quarks and electrons take forms that mirror Platonic Truths? How can you make sense of that in terms of the constructivist point of view that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. I will agree with you that all intelligences that start from the same premises as you, and follow the same rules as inference as you, will also draw the same conclusions about the Mandelbrot set as you do. However - I do not agree with you that this amenable group exhausts the set of all *possible* intelligences. I only meant that all possible intelligences that start from a mathematics that includes addition, multiplication, and complex numbers will find that if they iterate the function z' = z^2 + c, they will find that some orbits become periodic or settle on a point, and some escape to infinity. If they draw a graph of which orbits don't escape, they will draw the Mandelbrot Set. All possible intelligences that undertake that procedure will draw the same shape... and this seems like discovery, not creation. It seems like a tautology to me. If you do what I do and believe what I believe then you will be a lot like me...? Is there anything to mathematics other than belief? What are beliefs? Why do we have the beliefs that we have? How do we form beliefs - what lies behind belief? Can *our* mathematical abilities be reduced to something that is indifferent to mathematical truth? Could there be intelligences who start from vastly difference premises, and use vastly different rules of inference, and draw vastly different conclusions? Of course, but then what they are doing doesn't relate to the Mandelbrot Set. However - they
Re: Prime Numbers
On Fri, Sep 21, 2012 at 1:55 PM, meekerdb meeke...@verizon.net wrote: On 9/21/2012 8:59 AM, Jason Resch wrote: On Sep 21, 2012, at 8:13 AM, Rex Allen rexallen31...@gmail.com wrote: On Wed, Sep 19, 2012 at 12:27 AM, Jason Resch jasonre...@gmail.comwrote: On Sep 18, 2012, at 9:19 PM, Rex Allen rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.comwrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? We study and create theories about objects in the mathematical realm just as we study and create theories about objects in the physical realm. So in the physical realm, we start from our senses - what we see, hear, feel, etc. From this, we infer the existence of electrons and wavefunctions and strings and whatnot. Or some of us do. Others take a more instrumental view of scientific theories. Right, and we have similarly inferred the existence of primes, fractals, non-computable functions, etc. We invented counting, addition, etc and found it implied true propositions about primes, fractals, etc. To say they exist in the same way tables and chairs exist is going much further. All of our scientific theories are inventions too. We can only hope they bear some resemblance to reality. So you're saying that thought is another kind of sense? Thought is needed for inference and building theories, equally in the physical sciences and math. And that what occurs to us in thought can also be used as a basis to infer the existence of objects which help explain those thoughts? Right, like you might think up genesis and dualism, or big bang and materialism, or platonic truth and computationalism. These are ontological theories for what exists, and why we are here experiencing it. If you say math is fiction and only exists only as a story in our brains, then obviously you can't use platonic truth and computationalism as one if your theories of existence. I think the fact that mathematics can serve as a theory for our existence shows absolutely that mathematical theories and physical theories are on equal footing. We can gather evidence for them and build cases for them, find out we were wrong about them, and so on. Why do we believe in quarks, electrons, strings, etc.? Because they can explain our observations. Why do I believe in the platonic realm? For the same reasons. But we believe that electrons interact causally with us because we are made from similar stuff - and by doing so make themselves known to us...right? How do Platonic objects interact causally with us? Via a Platonic Field? PFT - Platonic Field Theory? How did the warping of space and time cause Einsteins brain to figure out relativity? I think you are looking at it in the wrong way. Our brains seek good explanations. They sometimes find one. That's all that is going on. Now you say our explainations when it comes to mathematics are fiction, but if that is so, why not say the same of the physical theories? Why not say the big bang is fiction, or matter is fiction? They are stories which we intend to have referents independent of the stories (theories). I don't see how this is any different from our mathematical theories though. Jason -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/21/2012 12:56 PM, Jason Resch wrote: On Fri, Sep 21, 2012 at 1:55 PM, meekerdb meeke...@verizon.net mailto:meeke...@verizon.net wrote: On 9/21/2012 8:59 AM, Jason Resch wrote: On Sep 21, 2012, at 8:13 AM, Rex Allen rexallen31...@gmail.com mailto:rexallen31...@gmail.com wrote: On Wed, Sep 19, 2012 at 12:27 AM, Jason Resch jasonre...@gmail.com mailto:jasonre...@gmail.com wrote: On Sep 18, 2012, at 9:19 PM, Rex Allen rexallen31...@gmail.com mailto:rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.com mailto:terren.suy...@gmail.com wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? We study and create theories about objects in the mathematical realm just as we study and create theories about objects in the physical realm. So in the physical realm, we start from our senses - what we see, hear, feel, etc. From this, we infer the existence of electrons and wavefunctions and strings and whatnot. Or some of us do. Others take a more instrumental view of scientific theories. Right, and we have similarly inferred the existence of primes, fractals, non-computable functions, etc. We invented counting, addition, etc and found it implied true propositions about primes, fractals, etc. To say they exist in the same way tables and chairs exist is going much further. All of our scientific theories are inventions too. We can only hope they bear some resemblance to reality. So you're saying that thought is another kind of sense? Thought is needed for inference and building theories, equally in the physical sciences and math. And that what occurs to us in thought can also be used as a basis to infer the existence of objects which help explain those thoughts? Right, like you might think up genesis and dualism, or big bang and materialism, or platonic truth and computationalism. These are ontological theories for what exists, and why we are here experiencing it. If you say math is fiction and only exists only as a story in our brains, then obviously you can't use platonic truth and computationalism as one if your theories of existence. I think the fact that mathematics can serve as a theory for our existence shows absolutely that mathematical theories and physical theories are on equal footing. We can gather evidence for them and build cases for them, find out we were wrong about them, and so on. Why do we believe in quarks, electrons, strings, etc.? Because they can explain our observations. Why do I believe in the platonic realm? For the same reasons. But we believe that electrons interact causally with us because we are made from similar stuff - and by doing so make themselves known to us...right? How do Platonic objects interact causally with us? Via a Platonic Field? PFT - Platonic Field Theory? How did the warping of space and time cause Einsteins brain to figure out relativity? I think you are looking at it in the wrong way. Our brains seek good explanations. They sometimes find one. That's all that is going on. Now you say our explainations when it comes to mathematics are fiction, but if that is so, why not say the same of the physical theories? Why not say the big bang is fiction, or matter is fiction? They are stories which we intend to have referents independent of the stories (theories). I don't see how this is any different from our mathematical theories though. It is different. It's confusing because arithmetic (to take an example) is both a theory about discrete objects, 1apple + 1apple = 2apples, which requires a correct interpretation like any theory of physics, 1raindrop + 1raindrop = 1raindrop, but it's also a closed story without any external referents, s(0)+s(0)=s(s(0)). This is what makes mathematics (and logic and language) useful; you can abstract from the physical world to the Platonia story, manipulate it by some rules, and if you did it right interpret the result back in the physical world. But that doesn't mean language and logic and mathematics exist in the same sense. Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email
Re: Prime Numbers
On 19 Sep 2012, at 21:51, Stephen P. King wrote: On 9/19/2012 2:39 PM, Bruno Marchal wrote: Dear Bruno, Your remarks raise an interesting question: Could it be that both the object and the means to generate (or perceive) it are of equal importance ontologically? Yes. It comes from the embedding of the subject in the objects, that any monist theory has to do somehow. In computer science, the universal (in the sense of Turing) association i - phi_i, transforms N into an applicative algebra. The numbers are both perceivers and perceived according of their place x and y in the relation of phi_x(y). You can define the applicative operation by x # y = phi_x(y). The combinators are not far away from this, and provides intensional and extensional models. I remind you that phi_i represent the ith computable function in some effective universal enumeration of the partial computable functions. You can take LISP, or c++ to fix the things. Bruno Dear Bruno, You are highlighting of the key property of a number, that it can both represent itself and some other number. It is a key property of anything finite, not just number. Lists and strings do this even more easily and naturally. My question becomes, how does one track the difference between these representations? By quotations, like when using Gödel number, or quoted list in LISP. Those are computable operations. You speak of measures, but I have never seen how relative measures are discussed or defined in modal logic. ? A modal logic of probability is given by the behavior of the probability one. In Kripke terms, P(x) = 1 in world alpha means that x is realized in all worlds accessible from alpha, and (key point) that we are not in a cul-de-sac world. This gives KD modal logics, with K: = [](p - q)-([]p - []q), and D: []p - p. Of course with [] for Gödel's beweisbar we don't have that D is a theorem, so we ensure the D property by defining a new box, Bp = []p t. It seems to me that if we have the possibility of a Godel numbering scheme on the integers, then we lose the ability to define a global index set on subsets of those integers ? unless we can somehow call upon something that is not a number and thus not directly representable by a number.. ? Not clear. We appeal to something non representable by adding the p in the definition of the modal box, but this is for the qualia and first person notion. The Dt (and variant like DDt, DDBDt, etc.) should give the first person plural, normally. many possibility remains, as the quantum p - []p appears in the three main material variants of: S4Grz1, Z1*, and X1*, for p arithmetic sigma_1 proposition (the arithmetical UD). Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/20/2012 2:05 AM, Bruno Marchal wrote: A modal logic of probability is given by the behavior of the probability one. In Kripke terms, P(x) = 1 in world alpha means that x is realized in all worlds accessible from alpha, and (key point) that we are not in a cul-de-sac world. What does 'accessible' mean? Generally speaking a different world is defined as not accessible. If you can go there, it's part of your same world. Brent This gives KD modal logics, with K: = [](p - q)-([]p - []q), and D: []p - p. Of course with [] for Gödel's beweisbar we don't have that D is a theorem, so we ensure the D property by defining a new box, Bp = []p t. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 20 Sep 2012, at 18:14, meekerdb wrote: On 9/20/2012 2:05 AM, Bruno Marchal wrote: A modal logic of probability is given by the behavior of the probability one. In Kripke terms, P(x) = 1 in world alpha means that x is realized in all worlds accessible from alpha, and (key point) that we are not in a cul-de-sac world. What does 'accessible' mean? In modal logic semantic, it is a technical world for any element in set + a binary relation on it. When applied to probability, the idea is to interpret the worlds by the realization of some random experience, like throwing a coin would lead to two worlds accessible, one with head, the other with tail. In that modal (tail or head) is a certainty as (tail or head) is realized everywhere in the accessible worlds. Generally speaking a different world is defined as not accessible. If you can go there, it's part of your same world. Yes. OK. Sorry. Logician used the term world in a technical sense, and the worlds can be anything, depending of which modal logic is used, for what purpose, etc. Kripke semantic main used is in showing the independence of formula in different systems. Bruno Brent This gives KD modal logics, with K: = [](p - q)-([]p - []q), and D: []p - p. Of course with [] for Gödel's beweisbar we don't have that D is a theorem, so we ensure the D property by defining a new box, Bp = []p t. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en . http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 18 Sep 2012, at 18:02, meekerdb wrote: On 9/18/2012 8:13 AM, Bruno Marchal wrote: On 17 Sep 2012, at 22:25, meekerdb wrote: But did anybody think z' = z^2 + c was interesting before that? Yes. This was known by people like Fatou and Julia, in the early 1900. I knew they considered what are now called fractal sets, but not that particular one. I think Julia worked on the Mandelbrot's Julia sets, notably. The Mandelbrot set is a classifier of the Julia sets. You can define the Mandelbrot set by the the set of z such that z belongs to its Julia J(z). The point is that in math and physics such object are hard to miss, even if you need a computer to figure out what they looks like. Iterating analytical complex functions leads to the Mandelbrot fractal sets, or similar. The computer has made those objects famous, but the mathematicians know them both from logic (counterexamples to theorem in analysis, like finding a continuous function nowhere derivable), or from dynamic system and iteration. If you iterate the trigonometric cosec function on the Gauss plane C, you can't miss the Mandelbrot set. But this iteration is a tedious and impractical *construction* which in practice depends on computers. In practice, yes. But if I remember well, the point is that the M sets and alike are discovered, not fictitious human's construction. To see them, we need a computer, but to see a circle you need a compass, or a very massive object, like the sun or the moon, ... In nature too as the following video does not illustrate too much seriously :) http://www.youtube.com/watch?v=JGxbhdr3w2I In such beautiful imagery it is generally overlooked that it is not the Mandelbrot set you are looking at, but rather regions colored according how close they are to the set (which cannot be seen at all). Hmm, the inside mandelbrot set has dimension 2, as you can extrapolate from the big spot, and then the filament ar made of little mandelbrot set. So you can always see something. You are correct, for the filaments: usually we can see them, as the little Mandelbrot sets are too small. The coloring only makes them less thin and more easily observable, but you would see the same basic shape with a pure black and white picture. for example, everywhere on the main (straight) antenna, there is a little mandelbrot set, so even black and white resolution will make a thin line (with always too big pixels, of course). Of course, the line can become thinner and thinner, so with deeper zoom, you will have to darken the picture, and then light it up, etc. Of course this is true also for a circle, or a straight line, which are too thin to be seen, too, but we don't worry to draw them with chalks or pens, which approximates them quite well. You can see that phenomenon here: http://www.youtube.com/watch?v=QXzgrtntRTY Bruno Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en . http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/19/2012 8:39 AM, Bruno Marchal wrote: On 18 Sep 2012, at 18:02, meekerdb wrote: On 9/18/2012 8:13 AM, Bruno Marchal wrote: On 17 Sep 2012, at 22:25, meekerdb wrote: But did anybody think z' = z^2 + c was interesting before that? Yes. This was known by people like Fatou and Julia, in the early 1900. I knew they considered what are now called fractal sets, but not that particular one. I think Julia worked on the Mandelbrot's Julia sets, notably. The Mandelbrot set is a classifier of the Julia sets. You can define the Mandelbrot set by the the set of z such that z belongs to its Julia J(z). The point is that in math and physics such object are hard to miss, even if you need a computer to figure out what they looks like. Iterating analytical complex functions leads to the Mandelbrot fractal sets, or similar. The computer has made those objects famous, but the mathematicians know them both from logic (counterexamples to theorem in analysis, like finding a continuous function nowhere derivable), or from dynamic system and iteration. If you iterate the trigonometric cosec function on the Gauss plane C, you can't miss the Mandelbrot set. But this iteration is a tedious and impractical *construction* which in practice depends on computers. In practice, yes. But if I remember well, the point is that the M sets and alike are discovered, not fictitious human's construction. To see them, we need a computer, but to see a circle you need a compass, or a very massive object, like the sun or the moon, ... In nature too as the following video does not illustrate too much seriously :) http://www.youtube.com/watch?v=JGxbhdr3w2I In such beautiful imagery it is generally overlooked that it is not the Mandelbrot set you are looking at, but rather regions colored according how close they are to the set (which cannot be seen at all). Hmm, the inside mandelbrot set has dimension 2, as you can extrapolate from the big spot, and then the filament ar made of little mandelbrot set. So you can always see something. You are correct, for the filaments: usually we can see them, as the little Mandelbrot sets are too small. The coloring only makes them less thin and more easily observable, but you would see the same basic shape with a pure black and white picture. for example, everywhere on the main (straight) antenna, there is a little mandelbrot set, so even black and white resolution will make a thin line (with always too big pixels, of course). Of course, the line can become thinner and thinner, so with deeper zoom, you will have to darken the picture, and then light it up, etc. Of course this is true also for a circle, or a straight line, which are too thin to be seen, too, but we don't worry to draw them with chalks or pens, which approximates them quite well. You can see that phenomenon here: http://www.youtube.com/watch?v=QXzgrtntRTY Bruno Dear Bruno, Your remarks raise an interesting question: Could it be that both the object and the means to generate (or perceive) it are of equal importance ontologically? -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 19 Sep 2012, at 17:03, Stephen P. King wrote: On 9/19/2012 8:39 AM, Bruno Marchal wrote: On 18 Sep 2012, at 18:02, meekerdb wrote: On 9/18/2012 8:13 AM, Bruno Marchal wrote: On 17 Sep 2012, at 22:25, meekerdb wrote: But did anybody think z' = z^2 + c was interesting before that? Yes. This was known by people like Fatou and Julia, in the early 1900. I knew they considered what are now called fractal sets, but not that particular one. I think Julia worked on the Mandelbrot's Julia sets, notably. The Mandelbrot set is a classifier of the Julia sets. You can define the Mandelbrot set by the the set of z such that z belongs to its Julia J(z). The point is that in math and physics such object are hard to miss, even if you need a computer to figure out what they looks like. Iterating analytical complex functions leads to the Mandelbrot fractal sets, or similar. The computer has made those objects famous, but the mathematicians know them both from logic (counterexamples to theorem in analysis, like finding a continuous function nowhere derivable), or from dynamic system and iteration. If you iterate the trigonometric cosec function on the Gauss plane C, you can't miss the Mandelbrot set. But this iteration is a tedious and impractical *construction* which in practice depends on computers. In practice, yes. But if I remember well, the point is that the M sets and alike are discovered, not fictitious human's construction. To see them, we need a computer, but to see a circle you need a compass, or a very massive object, like the sun or the moon, ... In nature too as the following video does not illustrate too much seriously :) http://www.youtube.com/watch?v=JGxbhdr3w2I In such beautiful imagery it is generally overlooked that it is not the Mandelbrot set you are looking at, but rather regions colored according how close they are to the set (which cannot be seen at all). Hmm, the inside mandelbrot set has dimension 2, as you can extrapolate from the big spot, and then the filament ar made of little mandelbrot set. So you can always see something. You are correct, for the filaments: usually we can see them, as the little Mandelbrot sets are too small. The coloring only makes them less thin and more easily observable, but you would see the same basic shape with a pure black and white picture. for example, everywhere on the main (straight) antenna, there is a little mandelbrot set, so even black and white resolution will make a thin line (with always too big pixels, of course). Of course, the line can become thinner and thinner, so with deeper zoom, you will have to darken the picture, and then light it up, etc. Of course this is true also for a circle, or a straight line, which are too thin to be seen, too, but we don't worry to draw them with chalks or pens, which approximates them quite well. You can see that phenomenon here: http://www.youtube.com/watch?v=QXzgrtntRTY Bruno Dear Bruno, Your remarks raise an interesting question: Could it be that both the object and the means to generate (or perceive) it are of equal importance ontologically? Yes. It comes from the embedding of the subject in the objects, that any monist theory has to do somehow. In computer science, the universal (in the sense of Turing) association i - phi_i, transforms N into an applicative algebra. The numbers are both perceivers and perceived according of their place x and y in the relation of phi_x(y). You can define the applicative operation by x # y = phi_x(y). The combinators are not far away from this, and provides intensional and extensional models. I remind you that phi_i represent the ith computable function in some effective universal enumeration of the partial computable functions. You can take LISP, or c++ to fix the things. Bruno -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en . http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/19/2012 2:39 PM, Bruno Marchal wrote: Dear Bruno, Your remarks raise an interesting question: Could it be that both the object and the means to generate (or perceive) it are of equal importance ontologically? Yes. It comes from the embedding of the subject in the objects, that any monist theory has to do somehow. In computer science, the universal (in the sense of Turing) association i - phi_i, transforms N into an applicative algebra. The numbers are both perceivers and perceived according of their place x and y in the relation of phi_x(y). You can define the applicative operation by x # y = phi_x(y). The combinators are not far away from this, and provides intensional and extensional models. I remind you that phi_i represent the ith computable function in some effective universal enumeration of the partial computable functions. You can take LISP, or c++ to fix the things. Bruno Dear Bruno, You are highlighting of the key property of a number, that it can both represent itself and some other number. My question becomes, how does one track the difference between these representations? You speak of measures, but I have never seen how relative measures are discussed or defined in modal logic. It seems to me that if we have the possibility of a Godel numbering scheme on the integers, then we lose the ability to define a global index set on subsets of those integers unless we can somehow call upon something that is not a number and thus not directly representable by a number.. -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 17 Sep 2012, at 22:25, meekerdb wrote: But did anybody think z' = z^2 + c was interesting before that? Yes. This was known by people like Fatou and Julia, in the early 1900. Iterating analytical complex functions leads to the Mandelbrot fractal sets, or similar. The computer has made those objects famous, but the mathematicians know them both from logic (counterexamples to theorem in analysis, like finding a continuous function nowhere derivable), or from dynamic system and iteration. If you iterate the trigonometric cosec function on the Gauss plane C, you can't miss the Mandelbrot set. In nature too as the following video does not illustrate too much seriously :) http://www.youtube.com/watch?v=JGxbhdr3w2I Bruno Bretn On 9/17/2012 1:17 PM, Terren Suydam wrote: I would say computers were the tool that allowed us to see it, like a microscope allowed us to see bacteria, and a telescope stars. On Mon, Sep 17, 2012 at 3:14 PM, meekerdbmeeke...@verizon.net wrote: On 9/17/2012 10:36 AM, Terren Suydam wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? How can you make sense of that in terms of the constructivist point of view How can you make sense of it otherwise. The Mandelbrot set is only interesting because it became possible to construct it by use of computers. Brent that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. Terren -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com . To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en . http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/18/2012 8:13 AM, Bruno Marchal wrote: On 17 Sep 2012, at 22:25, meekerdb wrote: But did anybody think z' = z^2 + c was interesting before that? Yes. This was known by people like Fatou and Julia, in the early 1900. I knew they considered what are now called fractal sets, but not that particular one. Iterating analytical complex functions leads to the Mandelbrot fractal sets, or similar. The computer has made those objects famous, but the mathematicians know them both from logic (counterexamples to theorem in analysis, like finding a continuous function nowhere derivable), or from dynamic system and iteration. If you iterate the trigonometric cosec function on the Gauss plane C, you can't miss the Mandelbrot set. But this iteration is a tedious and impractical *construction* which in practice depends on computers. In nature too as the following video does not illustrate too much seriously :) http://www.youtube.com/watch?v=JGxbhdr3w2I In such beautiful imagery it is generally overlooked that it is not the Mandelbrot set you are looking at, but rather regions colored according how close they are to the set (which cannot be seen at all). Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.comwrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? How is it that we are able to reliably know things about Platonia? I would have thought that quarks and electrons from which we appear to be constituted would be indifferent to truth. Which would fit with the fact that I seem to make a lot of mistakes. But you think otherwise? How can you make sense of that in terms of the constructivist point of view that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. I will agree with you that all intelligences that start from the same premises as you, and follow the same rules as inference as you, will also draw the same conclusions about the Mandelbrot set as you do. However - I do not agree with you that this amenable group exhausts the set of all *possible* intelligences. Could there be intelligences who start from vastly difference premises, and use vastly different rules of inference, and draw vastly different conclusions? If not - what makes them impossible intelligences? =*= What are the limits of belief, do you think? Is there any belief that is so preposterous that even the maddest of the mad could not believe such a thing? And if there is no such belief - then is it conceivable that quarks and electrons could configure themselves in such a way as to *cause* a being who holds such beliefs to come into existence? And if this is beyond the capacity of quarks and electrons, does it seem possible that there might be some other form of matter with more exotic properties that might be up to the task? And if not - why not? Rex -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Tue, Sep 18, 2012 at 10:19 PM, Rex Allen rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.com wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. I'm curious about what a plausible fictionalist account of the Mandelbrot set could be. Is fictionalism the same as constructivism, or the idea that knowledge doesn't exist outside of a mind? But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? How is it that we are able to reliably know things about Platonia? I think just doing logic and math - starting from axioms and proving things from them - is interacting with the Platonic realm. It is reliable because such proofs are necessarily valid no matter what sort of computational agent is computing them. Bruno really takes it to the next level though when he talks of interviewing ideally correct machines and treating them as entities (strictly platonic, of course) that can talk about what they can prove (believe). I would have thought that quarks and electrons from which we appear to be constituted would be indifferent to truth. Which would fit with the fact that I seem to make a lot of mistakes. But you think otherwise? I didn't understand the above... what do quarks and electrons have to do with arithmetical platonism? How can you make sense of that in terms of the constructivist point of view that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. I will agree with you that all intelligences that start from the same premises as you, and follow the same rules as inference as you, will also draw the same conclusions about the Mandelbrot set as you do. However - I do not agree with you that this amenable group exhausts the set of all *possible* intelligences. I only meant that all possible intelligences that start from a mathematics that includes addition, multiplication, and complex numbers will find that if they iterate the function z' = z^2 + c, they will find that some orbits become periodic or settle on a point, and some escape to infinity. If they draw a graph of which orbits don't escape, they will draw the Mandelbrot Set. All possible intelligences that undertake that procedure will draw the same shape... and this seems like discovery, not creation. Could there be intelligences who start from vastly difference premises, and use vastly different rules of inference, and draw vastly different conclusions? Of course, but then what they are doing doesn't relate to the Mandelbrot Set. If not - what makes them impossible intelligences? =*= What are the limits of belief, do you think? Is there any belief that is so preposterous that even the maddest of the mad could not believe such a thing? I don't think so... based on my understanding of how mad maddest of the mad can get. And if there is no such belief - then is it conceivable that quarks and electrons could configure themselves in such a way as to *cause* a being who holds such beliefs to come into existence? I'm guessing you meant to say and if there is such a belief I'm having a tough time understanding where you're going with this... it seems like an interesting line of questions, but I have no idea how it relates to what we were discussing. Terren And if this is beyond the capacity of quarks and electrons, does it seem possible that there might be some other form of matter with more exotic properties that might be up to the task? And if not - why not? Rex -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Sep 18, 2012, at 9:19 PM, Rex Allen rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam terren.suy...@gmail.com wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? I find fictionalism to be the most plausible view of mathematics, with all that implies for the Mandelbrot set. But ;et me turn the question around on you, if I can: Do you have an explanation for how we discover mathematical objects and otherwise interact with the Platonic realm? We study and create theories about objects in the mathematical realm just as we study and create theories about objects in the physical realm. It's not much different from how we develop theories about other things we cannot interact with: the early universe, the cores of stars, the insides of black holes, etc. We test these theories by following their implications and seeing if they lead to contridictions with other, more established, facts. Just as with physical theories, we ocasionally find that we need to throw out the old set of theories (or axioms) for a new set which has greater explanatory power. How is it that we are able to reliably know things about Platonia? The very idea of knowing implies a differentiation between true and false. This leads quite directly to boolean algebra. Boolean algebra leads to concepts of numbers. (e.g., even numbers of not operators cancel out, so counting them becomes an issue). Once you get counting and numbers, you get the uncapturable infinite truths concerning them, and infinite hierarchies if ever more powerful consistent theories. Nearly any intelligent civilization that notices a partition between true and false will eventyally get here. I would have thought that quarks and electrons from which we appear to be constituted would be indifferent to truth. The unreasonable effectiveness of math in the physical sciences is yet further support if Platonism. If this, and seemingly infinite physical universes exist, and they are mathematical structures, why can't others exist? Which would fit with the fact that I seem to make a lot of mistakes. But you think otherwise? We are imperfect beings. Jason How can you make sense of that in terms of the constructivist point of view that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. I will agree with you that all intelligences that start from the same premises as you, and follow the same rules as inference as you, will also draw the same conclusions about the Mandelbrot set as you do. However - I do not agree with you that this amenable group exhausts the set of all *possible* intelligences. Could there be intelligences who start from vastly difference premises, and use vastly different rules of inference, and draw vastly different conclusions? If not - what makes them impossible intelligences? =*= What are the limits of belief, do you think? Is there any belief that is so preposterous that even the maddest of the mad could not believe such a thing? And if there is no such belief - then is it conceivable that quarks and electrons could configure themselves in such a way as to *cause* a being who holds such beliefs to come into existence? And if this is beyond the capacity of quarks and electrons, does it seem possible that there might be some other form of matter with more exotic properties that might be up to the task? And if not - why not? Rex -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en . -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/18/2012 9:27 PM, Jason Resch wrote: The unreasonable effectiveness of math in the physical sciences is yet further support if Platonism. I don't see that this follows. If we invent language, including mathematics, to describe our theories of the world that explains their effectiveness. But it doesn't imply that every description refers. The mathematics of Maxwell's equations was (and is) very effective, but we now believe they only approximately describe what exists. Brent If this, and seemingly infinite physical universes exist, and they are mathematical structures, why can't others exist? -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Mon, Sep 17, 2012 at 12:27 AM, Rex Allen rexallen31...@gmail.com wrote: On Sun, Sep 16, 2012 at 6:10 PM, Stephen P. King stephe...@charter.netwrote: HI Rex, Nice post! Could you riff a bit on what the number PHI tells us about this characteristic. How is it that it seems that our perceptions of the world find anything that is close to a PHI valued relationship to be beautiful? Thanks Stephen! Actually my initial example of numeracy isn't quite right, but it's not important to the rest of the argument. My main point is that you can get to the concept of prime numbers just using relative magnitudes that we have an innate sense of. I think an easier way to intuit prime numbers that can't be represented as rectangles, only a 1-wide lines. While the concept of primes is straight forward, there is an unending set of not-so-obvious facts that we continue to discover about the Primes. For example: The average distance between primes of size N is approximately the natural log of N, yet we know of no way to predict where the next prime will exactly be. ( http://en.wikipedia.org/wiki/Prime_gap ) Between N and 2N, there will always be at least one prime. ( http://en.wikipedia.org/wiki/Bertrand's_postulate ) There is a one-to-one correspondence, and method to get one from the other, between perfect numbers and primes of the form ((2^p) - 1) ( http://en.wikipedia.org/wiki/Perfect_number#Even_perfect_numbers ) For any prime p, and any integer i where 0 i p, i^p divided by p has a remainder of i. This almost never works for composite numbers. ( http://en.wikipedia.org/wiki/Fermat's_little_theorem ) the exception for composite numbers where this does hold are known as Carmichael numbers ( http://en.wikipedia.org/wiki/Carmichael_number ) but they are rare. And there are an infinite number of other such patterns waiting to be discovered. Jason As for the significance of PHI - well - I guess there's probably some plausible sounding evolutionary story that could be told about that. Though how satisfying or useful an explanation like that is just depends on what you're after and what your interests are. An explanation that might be useful in one context might be useless in some other context. Explanations are observer dependent. Probably. Rex -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Mon, Sep 17, 2012 at 2:05 AM, Jason Resch jasonre...@gmail.com wrote: I think an easier way to intuit prime numbers that can't be represented as rectangles, only a 1-wide lines. While the concept of primes is straight forward, there is an unending set of not-so-obvious facts that we continue to discover about the Primes. Right. My proposal is that this entire infinite edifice is built on top of our innate sense of more, less, and equal. Which I am tentatively advancing as the basis of an argument against Platonism. Rex -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? How can you make sense of that in terms of the constructivist point of view that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. Terren On Mon, Sep 17, 2012 at 12:32 PM, Rex Allen rexallen31...@gmail.com wrote: On Mon, Sep 17, 2012 at 2:05 AM, Jason Resch jasonre...@gmail.com wrote: I think an easier way to intuit prime numbers that can't be represented as rectangles, only a 1-wide lines. While the concept of primes is straight forward, there is an unending set of not-so-obvious facts that we continue to discover about the Primes. Right. My proposal is that this entire infinite edifice is built on top of our innate sense of more, less, and equal. Which I am tentatively advancing as the basis of an argument against Platonism. Rex -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/17/2012 10:36 AM, Terren Suydam wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? How can you make sense of that in terms of the constructivist point of view How can you make sense of it otherwise. The Mandelbrot set is only interesting because it became possible to construct it by use of computers. Brent that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. Terren -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
I would say computers were the tool that allowed us to see it, like a microscope allowed us to see bacteria, and a telescope stars. On Mon, Sep 17, 2012 at 3:14 PM, meekerdb meeke...@verizon.net wrote: On 9/17/2012 10:36 AM, Terren Suydam wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? How can you make sense of that in terms of the constructivist point of view How can you make sense of it otherwise. The Mandelbrot set is only interesting because it became possible to construct it by use of computers. Brent that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. Terren -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
But did anybody think z' = z^2 + c was interesting before that? Bretn On 9/17/2012 1:17 PM, Terren Suydam wrote: I would say computers were the tool that allowed us to see it, like a microscope allowed us to see bacteria, and a telescope stars. On Mon, Sep 17, 2012 at 3:14 PM, meekerdbmeeke...@verizon.net wrote: On 9/17/2012 10:36 AM, Terren Suydam wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? How can you make sense of that in terms of the constructivist point of view How can you make sense of it otherwise. The Mandelbrot set is only interesting because it became possible to construct it by use of computers. Brent that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. Terren -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
Benoit Mandelbrot did. But what does interesting have to do with it? Did anyone think that empty patch of sky was interesting before Hubble turned it into one of the most amazing photos ever taken? On Mon, Sep 17, 2012 at 4:25 PM, meekerdb meeke...@verizon.net wrote: But did anybody think z' = z^2 + c was interesting before that? Bretn On 9/17/2012 1:17 PM, Terren Suydam wrote: I would say computers were the tool that allowed us to see it, like a microscope allowed us to see bacteria, and a telescope stars. On Mon, Sep 17, 2012 at 3:14 PM, meekerdbmeeke...@verizon.net wrote: On 9/17/2012 10:36 AM, Terren Suydam wrote: Rex, Do you have a non-platonist explanation for the discovery of the Mandelbrot set and the infinite complexity therein? How can you make sense of that in terms of the constructivist point of view How can you make sense of it otherwise. The Mandelbrot set is only interesting because it became possible to construct it by use of computers. Brent that you are (I think) compelled to take if you argue against arithmetical platonism? It seems obvious that all possible intelligences would discover the same forms of the Mandelbrot so long as they iterated on z' = z^2 + c, but maybe I am missing the point of your argument. Terren -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/17/2012 2:45 PM, Terren Suydam wrote: Benoit Mandelbrot did. I wasn't aware of that. Did he have a proof of the fractal nature of the set before he calculated it? Brent But what does interesting have to do with it? Did anyone think that empty patch of sky was interesting before Hubble turned it into one of the most amazing photos ever taken? -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Mon, Sep 17, 2012 at 6:52 PM, meekerdb meeke...@verizon.net wrote: On 9/17/2012 2:45 PM, Terren Suydam wrote: Benoit Mandelbrot did. I wasn't aware of that. Did he have a proof of the fractal nature of the set before he calculated it? Brent I don't know. I doubt it, I'm not even sure he had even coined the term 'fractal' yet. I would be willing to bet though that what made plotting z' = z^2 + c interesting to him was the same basic curiosity that led astronomers to point Hubble at an empty patch of space (despite the considerable cost of doing so): is there anything there to be discovered? Terren -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On 9/16/2012 3:43 PM, Rex Allen wrote: It seems to me that numbers are based on our ability to judge relative magnitudes: Which is bigger, which is closer, which is heavier, etc. Many animals have this ability - called numeracy. Humans differ only in the degree to which it is developed, and in our ability to build higher level abstractions on top of this fundamental skill. SO - prime numbers, I think, emerge from a peculiar characteristic of our ability to judge relative magnitudes, and the way this feeds into the abstractions we build on top of that ability. =*= Let’s say you take a board and divide it into 3 sections of equal length (say, by drawing a line on it at the section boundaries). Having done so – is there a way that you could have divided the board into fewer sections of equal length so that every endpoint of a long section can be matched to the end of a shorter section? In other words – take two boards of equal length. Divide one into 3 sections. Divide the other into two sections. The dividing point of the two-section-board will fall right into the middle of the middle section of the three-section-board. There is no way to divide the second board into fewer sections so that all of its dividing points are matched against a dividing point on the longer board. Because of this – three is a prime. (Notice that I do not say: “this is because 3 is prime” – instead I reverse the causal arrow). =*= Let’s take two boards and divide the first one into 10 equally sized sections. Now – there are two ways that we can divide the second board into a smaller number of equally sized sections so that the end-points of every section on this second board are matched to a sectional dividing point on the first board (though the opposite will not be true): We can divide the second board into either 2 sections (in which case the dividing point will align with the end of the 5th section on the first board), OR We can divide the second board into 5 sections – each of which is the same size as two sections on the first board. Because of this, the number 10 is not prime. =*= The entire field of Number Theory grows out of this peculiar characteristic of how we judge relative magnitudes. Do you think? HI Rex, Nice post! Could you riff a bit on what the number PHI tells us about this characteristic. How is it that it seems that our perceptions of the world find anything that is close to a PHI valued relationship to be beautiful? -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Prime Numbers
On Sun, Sep 16, 2012 at 6:10 PM, Stephen P. King stephe...@charter.netwrote: HI Rex, Nice post! Could you riff a bit on what the number PHI tells us about this characteristic. How is it that it seems that our perceptions of the world find anything that is close to a PHI valued relationship to be beautiful? Thanks Stephen! Actually my initial example of numeracy isn't quite right, but it's not important to the rest of the argument. My main point is that you can get to the concept of prime numbers just using relative magnitudes that we have an innate sense of. As for the significance of PHI - well - I guess there's probably some plausible sounding evolutionary story that could be told about that. Though how satisfying or useful an explanation like that is just depends on what you're after and what your interests are. An explanation that might be useful in one context might be useless in some other context. Explanations are observer dependent. Probably. Rex -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: prime numbers etc
Touche. But I don't believe (in?) it - I am agnostic. Nonbeliever. (SONG: I lost my turf in San Francisco) J On Thu, Sep 6, 2012 at 10:36 PM, Stathis Papaioannou stath...@gmail.comwrote: On Fri, Sep 7, 2012 at 8:07 AM, John Mikes jami...@gmail.com wrote: Stathis wrote (to Craig): But you believe that the neurochemicals do things contrary to what chemists would predict, for example an ion channel opening or closing without any cause such as a change in transmembrane potential or ligand concentration. We've talked about this before and it just isn't consistent with any scientific evidence. You interpret the existence spontaneous neural activity as meaning that something magical like this happens, but it doesn't mean that at all. Stathis, you know ... whatever we state as 'knowledge about mind etc.' is an explanation for the little we think we learned - with lots we have no idea about. Like: chemicals ... potentials ... scientific evidence ... even cause (meaning the part we alredy know about) and mauch much more. It is your turf, you must know about more we don't know only think we do. It's your turf too - you're a chemist. -- Stathis Papaioannou -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en. -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: prime numbers etc
On Fri, Sep 7, 2012 at 8:07 AM, John Mikes jami...@gmail.com wrote: Stathis wrote (to Craig): But you believe that the neurochemicals do things contrary to what chemists would predict, for example an ion channel opening or closing without any cause such as a change in transmembrane potential or ligand concentration. We've talked about this before and it just isn't consistent with any scientific evidence. You interpret the existence spontaneous neural activity as meaning that something magical like this happens, but it doesn't mean that at all. Stathis, you know ... whatever we state as 'knowledge about mind etc.' is an explanation for the little we think we learned - with lots we have no idea about. Like: chemicals ... potentials ... scientific evidence ... even cause (meaning the part we alredy know about) and mauch much more. It is your turf, you must know about more we don't know only think we do. It's your turf too - you're a chemist. -- Stathis Papaioannou -- You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
