Re: Math Question
On 07 Aug 2011, at 21:41, Craig Weinberg wrote: On Aug 1, 2:29 pm, Bruno Marchal marc...@ulb.ac.be wrote: Bruno Stephen, Isn't there a concept of imprecision in absolute physical measurement and drift in cosmological constants? Are atoms and molecules all infinitesimally different in size or are they absolutely the same size? Certainly individual cells of the same type vary in all of their measurements, do they not? If so, that would seem to suggest my view - that arithmetic is an approximation of feeling, and not the other way around. Cosmos is a feeling of order, or of wanting to manifest order, but it is not primitively precise. Make sense? Not really. The size of a molecule can be considered infinite, if you describe the molecule by its quantum wave. I don't see why arithmetic would approximate feeling, nor what that could mean. I don't see what you mean by cosmos, etc. Biological processes then, could be conceived as a 'levelling up' of molecular arithmetic having been formally actualized, I don't understand. What do you mean by molecular arithmetic, etc. a more significant challenge is attempted on top of the completed molecular canvas - with more elasticity and unpredictibility, and a host of newer, richer feelings which expand upon the molecular range, becoming at once more tangible and concrete, more real, and more unreal and abstract. The increased potential for unreality in the subjective interiority of the cells is what creates the perspective necessary to conceive of the molecular world as objectively real by contrast. The nervous system does the same trick one level higher. I see the words, but fail to see any precise meaning. It seems to me that you postulate all the notions that I think we should explain from simpler notions we agree on. 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: Math Question
On Aug 8, 12:03 pm, Bruno Marchal marc...@ulb.ac.be wrote: On 07 Aug 2011, at 21:41, Craig Weinberg wrote: On Aug 1, 2:29 pm, Bruno Marchal marc...@ulb.ac.be wrote: Bruno Stephen, Isn't there a concept of imprecision in absolute physical measurement and drift in cosmological constants? Are atoms and molecules all infinitesimally different in size or are they absolutely the same size? Certainly individual cells of the same type vary in all of their measurements, do they not? If so, that would seem to suggest my view - that arithmetic is an approximation of feeling, and not the other way around. Cosmos is a feeling of order, or of wanting to manifest order, but it is not primitively precise. Make sense? Not really. The size of a molecule can be considered infinite, if you describe the molecule by its quantum wave. Wouldn't the quantum wave describe the character of groups of the molecule rather than an actual instance of the molecule? Don't individual molecules have measurable finite sizes? For instance, here http://www.quantum.at/research/molecule-interferometry-applications/molecular-quantum-lithography.html we can see C60 molecules are in the range of 2nm each. I don't see why arithmetic would approximate feeling, nor what that could mean. I don't see what you mean by cosmos, etc. For instance, a chef might make a meal by adding informal quantities of the ingredients and procedures according to how she feels. A pinch of salt, a chunk of butter, mix well, heat until crispy, etc. If she wants to publish this as a recipe, she might want to get more quantitatively precise with ingredient amounts, time and temp, etc. If however, the quantities were arithmetically precise to begin with, there would not be any need to blur them into informal terms. If the recipe for the universe is a book of numbers, there would be no need for blurry feelings to arise to mask them. Biological processes then, could be conceived as a 'levelling up' of molecular arithmetic having been formally actualized, I don't understand. What do you mean by molecular arithmetic, etc. I'm characterizing the mechanics of molecules as being more arithmetic and deterministic than that of organisms. Saying that molecular mechanics represent one level of feeling actualized into form, and that the next level is form actualizing a more powerful experience of feeling. a more significant challenge is attempted on top of the completed molecular canvas - with more elasticity and unpredictibility, and a host of newer, richer feelings which expand upon the molecular range, becoming at once more tangible and concrete, more real, and more unreal and abstract. The increased potential for unreality in the subjective interiority of the cells is what creates the perspective necessary to conceive of the molecular world as objectively real by contrast. The nervous system does the same trick one level higher. I see the words, but fail to see any precise meaning. I'm saying that it's the difference between feeling and it's opposite - arithmetic, which gives rise to the experience of 'reality'. It seems to me that you postulate all the notions that I think we should explain from simpler notions we agree on. Not sure what you mean. If you're saying that I postulate that feeling is not reducible but that you think we should reduce it to arithmetic, I agree. I think the idea that feeling seems like it should be reduced to something else is a consequence of the fact that our thoughts of reduction are themselves a feeling. Craig -- 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: Math Question
On Aug 1, 2:29 pm, Bruno Marchal marc...@ulb.ac.be wrote: Bruno Stephen, Isn't there a concept of imprecision in absolute physical measurement and drift in cosmological constants? Are atoms and molecules all infinitesimally different in size or are they absolutely the same size? Certainly individual cells of the same type vary in all of their measurements, do they not? If so, that would seem to suggest my view - that arithmetic is an approximation of feeling, and not the other way around. Cosmos is a feeling of order, or of wanting to manifest order, but it is not primitively precise. Make sense? Biological processes then, could be conceived as a 'levelling up' of molecular arithmetic having been formally actualized, a more significant challenge is attempted on top of the completed molecular canvas - with more elasticity and unpredictibility, and a host of newer, richer feelings which expand upon the molecular range, becoming at once more tangible and concrete, more real, and more unreal and abstract. The increased potential for unreality in the subjective interiority of the cells is what creates the perspective necessary to conceive of the molecular world as objectively real by contrast. The nervous system does the same trick one level higher. Craig -- 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: Math Question
On 7/31/2011 7:40 PM, Pzomby wrote: The following quote is from the book “What is Mathematics Really?” by Reuben Hersh “0 (zero) is particularly nice. It is the class of sets equivalent to the set of all objects unequal to themselves! No object is unequal to itself, so 0 is the class of all empty sets. But all empty sets have the same members….none! So they’re not merely equivalent to each other…they are all the same set. There’s only one empty set! (A set is characterized by its membership list. There’s no way to tell one empty membership list from another. Therefore all empty sets are the same thing!) Once I have the empty sets, I can use a trick of Von Neumann as an alternative way to construct the number 1. Consider the class of all empty sets. This class has exactly one member: the unique empty set. It’s a singleton. ‘Out of nothing’ I have made a singleton set…a “canonical representative” for the cardinal number 1. 1 is the class of all singletons…all sets but with a single element. To avoid circularity: 1 is the class of all sets equivalent to the set whose only element is the empty set. Continuing, you get pairs, triplets, and so on. Von Neumann recursively constructs the whole set of natural numbers out of sets of nothing. ….The idea of set…any collection of distinct objects…was so simple and fundamental; it looked like a brick out of which all mathematics could be constructed. Even arithmetic could be downgraded (or upgraded) from primary to secondary rank, for the natural numbers could be constructed, as we have just seen, from nothing…ie., the empty set…by operations of set theory.” Any comments or opinions on whether this theory is the basis for the natural numbers and their relations as is described in the quote above? Thanks Hi Pzomby, Nice post, but I need to point out that that von Neumann's construction depends on the ability to bracket the singleton an arbitrary number of times to generate the pairs, triplets, etc. which implies that more exists than just the singleton. What is the source of the bracketing? I have long considered that this bracketing is a primitive form of 'making distinctions' which is one of the necessary (but not sufficient) properties of consciousness. Onward! Stephen -- 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: Math Question
On Aug 1, 5:24 am, Stephen P. King stephe...@charter.net wrote: On 7/31/2011 7:40 PM, Pzomby wrote: The following quote is from the book What is Mathematics Really? by Reuben Hersh 0 (zero) is particularly nice. It is the class of sets equivalent to the set of all objects unequal to themselves! No object is unequal to itself, so 0 is the class of all empty sets. But all empty sets have the same members .none! So they re not merely equivalent to each other they are all the same set. There s only one empty set! (A set is characterized by its membership list. There s no way to tell one empty membership list from another. Therefore all empty sets are the same thing!) Once I have the empty sets, I can use a trick of Von Neumann as an alternative way to construct the number 1. Consider the class of all empty sets. This class has exactly one member: the unique empty set. It s a singleton. Out of nothing I have made a singleton set a canonical representative for the cardinal number 1. 1 is the class of all singletons all sets but with a single element. To avoid circularity: 1 is the class of all sets equivalent to the set whose only element is the empty set. Continuing, you get pairs, triplets, and so on. Von Neumann recursively constructs the whole set of natural numbers out of sets of nothing. .The idea of set any collection of distinct objects was so simple and fundamental; it looked like a brick out of which all mathematics could be constructed. Even arithmetic could be downgraded (or upgraded) from primary to secondary rank, for the natural numbers could be constructed, as we have just seen, from nothing ie., the empty set by operations of set theory. Any comments or opinions on whether this theory is the basis for the natural numbers and their relations as is described in the quote above? Thanks Hi Pzomby, Nice post, but I need to point out that that von Neumann's construction depends on the ability to bracket the singleton an arbitrary number of times to generate the pairs, triplets, etc. which implies that more exists than just the singleton. What is the source of the bracketing? I have long considered that this bracketing is a primitive form of 'making distinctions' which is one of the necessary (but not sufficient) properties of consciousness. Onward! Stephen- Hide quoted text - - Stephen: The full three paragraphs are from the book. The sentence ‘Once I have the empty sets, I can use a trick of Von Neumann as an alternative way to construct the number 1.’ is Hersh’s words. I was looking for opinions, as you have given, on Hersh’s conclusions. Your comment on ‘making distinctions’ is the direction I was heading in understanding the role of primitive mathematics (sets, numbers) underlying human consciousness. Thanks Pzomby -- 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: Math Question
On 01 Aug 2011, at 01:40, Pzomby wrote: The following quote is from the book “What is Mathematics Really?” by Reuben Hersh “0 (zero) is particularly nice. It is the class of sets equivalent to the set of all objects unequal to themselves! No object is unequal to itself, so 0 is the class of all empty sets. But all empty sets have the same members….none! So they’re not merely equivalent to each other…they are all the same set. There’s only one empty set! (A set is characterized by its membership list. There’s no way to tell one empty membership list from another. Therefore all empty sets are the same thing!) Once I have the empty sets, I can use a trick of Von Neumann as an alternative way to construct the number 1. Consider the class of all empty sets. This class has exactly one member: the unique empty set. It’s a singleton. ‘Out of nothing’ I have made a singleton set…a “canonical representative” for the cardinal number 1. 1 is the class of all singletons…all sets but with a single element. To avoid circularity: 1 is the class of all sets equivalent to the set whose only element is the empty set. Continuing, you get pairs, triplets, and so on. Von Neumann recursively constructs the whole set of natural numbers out of sets of nothing. ….The idea of set…any collection of distinct objects…was so simple and fundamental; it looked like a brick out of which all mathematics could be constructed. Even arithmetic could be downgraded (or upgraded) from primary to secondary rank, for the natural numbers could be constructed, as we have just seen, from nothing…ie., the empty set…by operations of set theory.” Any comments or opinions on whether this theory is the basis for the natural numbers and their relations as is described in the quote above? To use set theory for studying the numbers is like taking an airbus 380 to go to the grocery. Set theory is too big, and it flatten the concepts (unlike categories which sharpen them, when used carefully). Now, ZF, the Zermelo-Fraenkel formal set theory, is a cute example of (arithmetical) little Löbian Universal Machine, and is handy as an example of a very imaginative machine capable of handling most of PA's theology. PA is much weaker than ZF, but like the guy in the chinese room which can simulate a chinese talking person, PA can simulate (emulate, even) ZF. Well, even RA can do that. But set theories and most toposes give too much larger ontology, when you assume comp. They do have epistemological roles, to be sure, and they do prove *much* more arithmetical truth than PA. But then many other theories do. There is no real problem if you prefer to adopt set theoretical realism, instead of arithmetical realism, when assuming comp. This will not change anything in the extraction of theology and physics from comp, except you will meet even more people criticizing your ontology (as being too much big!). If you like set, you can take the theory of hereditary finite sets, which can be shown equivalent with PA. Well, to be sure, putting infinite sets in the ontology can inadvertently leads to treachery in the explanation of why machines (finite beings) can believe in infinite sets. 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: Math Question
On 31 Jul 2011, at 14:15, Craig Weinberg wrote: Reblogging myself here, but curious to see what you think of the idea that 1 cannot be proven greater than 0. In which theory? The notion of proof is theory and definition dependent. (contrary to computability, which is absolute, by Church thesis). If you agree to define x y by Ez(z+x = y)E = It exists. I assume classical logic + the axioms: x+0 = x x+s(y) = s(x+y) 0 denotes the number zero, and s(x) denotes the successor of x, often noted as x+1. Cf the whole theory I gave last week. I use only a subset of that theory here. So we have to prove that 0 s(0). By the definition of above, we have to prove that Ez(z + 0 = s(0)) But s(0) + 0 = s(0) by the axiom x + 0 = x given above. So Ez(0 + z = s(0)) is true, with z = s(0). (This is the usual use of the existence rule of classical logic). Of course we could have taken the theory with the unique axiom 1 is greater than 0. For all proposition we can always find a theory which proves it. The interesting thing consists in proving new fact in some fixed theory, and change only a theory when it fails to prove a fact for which we have compelling evidences. Bruno Someone’s comment on the previous chart mentioned the difficulty (impossibility?) of proving that 1 0. It’s an interesting kernel there, and it reminds me of the whole “time does not physically exist” realization. On one level, I can think of zero as having no different relation to 1 than it has with any other number. Zero does the same thing to any number as it does to one and should be thought of more properly as the hub of the decimal spiral. I’m no mathematician, but I suppose that 0 is also formally defined as an integer between 1 and -1 or something. Still it exposes the question of whether the elemental underpinnings of our ability to count is really anchored in anything at all other than our own anthropological conventions of counting. Beyond numbers themselves, it appears that the whole quantitative notion - of greater than or less than, and of ‘equal’ is nothing but a figment of our feelings about order. There may not be any inherent moreness to something than the absence of something. If it’s the same thing, it actually seems more palatable to see the absence of something being a condition predicated upon the things’ a priori presence, no? Even if we want to get into quantum atopoietic craziness where things come out of nothing, rendering such a possibility discretely seems to threaten the whole notion of mathematical coherence. If any or all quantities, variables, and formulas can be generated arbitrarily from 0, then 0 would seem to be the same thing as ∞, and greater than 1 or any other arithmetic expression. Anthrodeximal Numberline Maybe it’s time to create a new numberline, without all of the repetitive decimal numerals. Instead there could be a Wiki of new quantitative symbols and names which anyone can add to and own as a permanent vector in the schema. It would be easy to translate them to and from Arabic numerals online and some interesting possibilities for informal encryption and unanticipated mathematic-linguistic synchronicity. By removing the aspect of repetition, we would unmask the semantic bias of the math logos and arrive at a pure generic linear calibration defined only in it’s own idiosyncratic a-signifying terms. Sort of like breaking the mantra of math, it’s trance-like rhythms that disguise it’s human neurological origin from us. By adding more unique qualitative sense to the thing, the quality-flattening power drains out and the system seems to disqualify itself. -- 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: Math Question
On Jul 31, 9:49 am, Bruno Marchal marc...@ulb.ac.be wrote: In which theory? The notion of proof is theory and definition dependent. (contrary to computability, which is absolute, by Church thesis). If you agree to define x y by Ez(z+x = y) E = It exists. I assume classical logic + the axioms: x+0 = x x+s(y) = s(x+y) 0 denotes the number zero, and s(x) denotes the successor of x, often noted as x+1. Cf the whole theory I gave last week. I use only a subset of that theory here. So we have to prove that 0 s(0). By the definition of above, we have to prove that Ez(z + 0 = s(0)) But s(0) + 0 = s(0) by the axiom x + 0 = x given above. So Ez(0 + z = s(0)) is true, with z = s(0). (This is the usual use of the existence rule of classical logic). Of course we could have taken the theory with the unique axiom 1 is greater than 0. For all proposition we can always find a theory which proves it. The interesting thing consists in proving new fact in some fixed theory, and change only a theory when it fails to prove a fact for which we have compelling evidences. How do we know that 0 has a successor though? If 0 x = x and x -0 = x then maybe s(0)=0 or Ezs(0)... Can we disprove the idea that a successor to zero does not exist? Sorry, I'm probably not at the minimum level of competence to understand this. -- 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: Math Question
On 31 Jul 2011, at 17:08, Craig Weinberg wrote: On Jul 31, 9:49 am, Bruno Marchal marc...@ulb.ac.be wrote: In which theory? The notion of proof is theory and definition dependent. (contrary to computability, which is absolute, by Church thesis). If you agree to define x y by Ez(z+x = y)E = It exists. I assume classical logic + the axioms: x+0 = x x+s(y) = s(x+y) 0 denotes the number zero, and s(x) denotes the successor of x, often noted as x+1. Cf the whole theory I gave last week. I use only a subset of that theory here. So we have to prove that 0 s(0). By the definition of above, we have to prove that Ez(z + 0 = s(0)) But s(0) + 0 = s(0) by the axiom x + 0 = x given above. So Ez(0 + z = s(0)) is true, with z = s(0). (This is the usual use of the existence rule of classical logic). Of course we could have taken the theory with the unique axiom 1 is greater than 0. For all proposition we can always find a theory which proves it. The interesting thing consists in proving new fact in some fixed theory, and change only a theory when it fails to prove a fact for which we have compelling evidences. How do we know that 0 has a successor though? If 0 x = x and x -0 = x then maybe s(0)=0 or Ezs(0)... Can we disprove the idea that a successor to zero does not exist? No. 0 is primitive term, and the language allows the term s(t) for all term t, so you have the terms 0, s(0), s(s(0)), etc. The rest follows from the axioms For all x 0 ≠ s(x), s(x) = s(y) - x = y (so that all numbers have only one successor. So you can, prove, even without induction, that 0 has a unique successor, different from itself. Sorry, I'm probably not at the minimum level of competence to understand this. I look on the net, but I see errors (Wolfram's definition is Dedekind Arithmetic!)? On wiki, the definition of Peano arithmetic seems correct. You need to study some elementary textbook in mathematical logic. Most presentation assumes you know what is first order predicate logic. You can google on those terms. There are good books, but it is a bit involved subject and ask for some works. Peano Arithmetic is the simplest example of Löbian theory or machines or belief system. It is very powerful. You light take time to find an arithmetical proposition that you can prove to be true and that she can't, especially without using the technics for doing that. Most interesting theorem in usual (non Logic) mathematics can be prove in or by PA. And PA, like all Löbian machine, can prove its own Gödel theorem (if I am consistent then I cannot prove that I am consistent). The I is a 3-I. Bruno -- 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: Math Question
On 7/31/2011 8:15 AM, Craig Weinberg wrote: Reblogging myself here, but curious to see what you think of the idea that 1 cannot be proven greater than 0. Someone’s comment on the previous chart mentioned the difficulty (impossibility?) of proving that 1 0. It’s an interesting kernel there, and it reminds me of the whole “time does not physically exist” realization. On one level, I can think of zero as having no different relation to 1 than it has with any other number. Zero does the same thing to any number as it does to one and should be thought of more properly as the hub of the decimal spiral. I’m no mathematician, but I suppose that 0 is also formally defined as an integer between 1 and -1 or something. Still it exposes the question of whether the elemental underpinnings of our ability to count is really anchored in anything at all other than our own anthropological conventions of counting. Beyond numbers themselves, it appears that the whole quantitative notion - of greater than or less than, and of ‘equal’ is nothing but a figment of our feelings about order. There may not be any inherent moreness to something than the absence of something. If it’s the same thing, it actually seems more palatable to see the absence of something being a condition predicated upon the things’ a priori presence, no? Even if we want to get into quantum atopoietic craziness where things come out of nothing, rendering such a possibility discretely seems to threaten the whole notion of mathematical coherence. If any or all quantities, variables, and formulas can be generated arbitrarily from 0, then 0 would seem to be the same thing as ∞, and greater than 1 or any other arithmetic expression. Anthrodeximal Numberline Maybe it’s time to create a new numberline, without all of the repetitive decimal numerals. Instead there could be a Wiki of new quantitative symbols and names which anyone can add to and own as a permanent vector in the schema. It would be easy to translate them to and from Arabic numerals online and some interesting possibilities for informal encryption and unanticipated mathematic-linguistic synchronicity. By removing the aspect of repetition, we would unmask the semantic bias of the math logos and arrive at a pure generic linear calibration defined only in it’s own idiosyncratic a-signifying terms. Sort of like breaking the mantra of math, it’s trance-like rhythms that disguise it’s human neurological origin from us. By adding more unique qualitative sense to the thing, the quality-flattening power drains out and the system seems to disqualify itself. Hi Craig, Umm, what would be the point of coming up with yet another representation system for quantities? We already established that a description is not its referent even though for every referent there is at least one description and for every description there is at least one referent. Zero, 0, null, the empty set is an absence of sorts; a placeholder. So in that sense it is a referent and just as space is 'the place where referents could be but are not', so too is 0. Onward! Stephen -- 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: Math Question
On Jul 31, 11:58 am, Bruno Marchal marc...@ulb.ac.be wrote: How do we know that 0 has a successor though? If 0 x = x and x -0 = x then maybe s(0)=0 or Ezs(0)... Can we disprove the idea that a successor to zero does not exist? No. 0 is primitive term, and the language allows the term s(t) for all term t, so you have the terms 0, s(0), s(s(0)), etc. It sounds like you're saying that it's a given that 0 has a successor and therefore doesn't need to be proved. The rest follows from the axioms For all x 0 ≠ s(x), s(x) = s(y) - x = y (so that all numbers have only one successor. So you can, prove, even without induction, that 0 has a unique successor, different from itself. Sorry, I'm probably not at the minimum level of competence to understand this. I look on the net, but I see errors (Wolfram's definition is Dedekind Arithmetic!)? On wiki, the definition of Peano arithmetic seems correct. You need to study some elementary textbook in mathematical logic. Most presentation assumes you know what is first order predicate logic. You can google on those terms. There are good books, but it is a bit involved subject and ask for some works. Peano Arithmetic is the simplest example of Löbian theory or machines or belief system. It is very powerful. You light take time to find an arithmetical proposition that you can prove to be true and that she can't, especially without using the technics for doing that. Most interesting theorem in usual (non Logic) mathematics can be prove in or by PA. And PA, like all Löbian machine, can prove its own Gödel theorem (if I am consistent then I cannot prove that I am consistent). The I is a 3-I. Thanks, I'll see if I can nibble on it sometime. Craig -- 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: Math Question
On Jul 31, 1:19 pm, Stephen P. King stephe...@charter.net wrote: Hi Craig, Umm, what would be the point of coming up with yet another representation system for quantities? We already established that a description is not its referent even though for every referent there is at least one description and for every description there is at least one referent. Zero, 0, null, the empty set is an absence of sorts; a placeholder. So in that sense it is a referent and just as space is 'the place where referents could be but are not', so too is 0. Right. I like that. My point in the alt numeracy idea is to bring out the true a-signifying potential of quantity - to take generic mechanism to it's reductio ad absurdum and reveal the implicit sentimentality of arithmetic which is hidden in base-10 rhyming. If it looked like this instead: 1 2 3 4 5 6 7 8 9 A-Z Every name in every phonebook in India Every word in every language various random squiggles, etc then we could truly expunge all remnants of beauty or symmetry in arithmetic and reveal itself in pure abstraction and marvel at how utterly devoid of usefulness that makes it. That way we could recover our orientation to the genuine by admitting that what we get out of arithmetic is a happy feeling of satisfaction - dopamine, oxytocin, serotonin, and endorphins. Our 1p experience of a human brain in communion with the 1p sense of those categories of molecules are the primitives of arithmetic. Craig -- 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: Math Question
The following quote is from the book “What is Mathematics Really?” by Reuben Hersh “0 (zero) is particularly nice. It is the class of sets equivalent to the set of all objects unequal to themselves! No object is unequal to itself, so 0 is the class of all empty sets. But all empty sets have the same members….none! So they’re not merely equivalent to each other…they are all the same set. There’s only one empty set! (A set is characterized by its membership list. There’s no way to tell one empty membership list from another. Therefore all empty sets are the same thing!) Once I have the empty sets, I can use a trick of Von Neumann as an alternative way to construct the number 1. Consider the class of all empty sets. This class has exactly one member: the unique empty set. It’s a singleton. ‘Out of nothing’ I have made a singleton set…a “canonical representative” for the cardinal number 1. 1 is the class of all singletons…all sets but with a single element. To avoid circularity: 1 is the class of all sets equivalent to the set whose only element is the empty set. Continuing, you get pairs, triplets, and so on. Von Neumann recursively constructs the whole set of natural numbers out of sets of nothing. ….The idea of set…any collection of distinct objects…was so simple and fundamental; it looked like a brick out of which all mathematics could be constructed. Even arithmetic could be downgraded (or upgraded) from primary to secondary rank, for the natural numbers could be constructed, as we have just seen, from nothing…ie., the empty set…by operations of set theory.” Any comments or opinions on whether this theory is the basis for the natural numbers and their relations as is described in the quote above? Thanks -- 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.
