Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Sunday, August 9, 2015 at 2:57:20 AM UTC+5:30, Marko Rauhamaa wrote:
Marko Rauhamaa :
Steven D'Aprano :
The contemporary standard approach is from Zermelo-Fraenkel set
theory: define 0 as the empty set, and the successor to n as the
union of n and the set containing n:
0 = {} (the empty set)
n + 1 = n ∪ {n}
That definition barely captures the essence of what a number *is*. In
fact, there have been different formulations of natural numbers.
Rehashing this old discussion. I ran into this wonderful website:
URL: http://at.metamath.org/mpeuni/mmset.html
Attention you Hilbertian!
Gödelian here — http://blog.languager.org/2015/07/cs-history-2.html
:-)
Thanks for that link. Need to study it carefully
--
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Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On 2015-07-24, Paul Rubin no.email@nospam.invalid wrote: Grant Edwards invalid@invalid.invalid writes: You can always pick out the topologist at a conference: he's the one trying to dunk his coffee cup in his doughnut. Did you hear about the idiot topologist? He couldn't tell his butt from a hole in the ground, but he *could* tell his butt from two holes in the ground. Wow. Now I know _two_ topologist jokes. The girls are going to be impressed! -- Grant Edwards grant.b.edwardsYow! I just got my PRINCE at bumper sticker ... But now gmail.comI can't remember WHO he is ... -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On 24/07/2015 15:13, Grant Edwards wrote: On 2015-07-24, Paul Rubin no.email@nospam.invalid wrote: Grant Edwards invalid@invalid.invalid writes: You can always pick out the topologist at a conference: he's the one trying to dunk his coffee cup in his doughnut. Did you hear about the idiot topologist? He couldn't tell his butt from a hole in the ground, but he *could* tell his butt from two holes in the ground. Wow. Now I know _two_ topologist jokes. The girls are going to be impressed! Here comes the third. Q: How many topologists does it take to change a light bulb? A: It really doesn't matter, since they'd rather knot. https://www.ocf.berkeley.edu/~mbarrien/jokes/lightblb.txt -- My fellow Pythonistas, ask not what our language can do for you, ask what you can do for our language. Mark Lawrence -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Grant Edwards invalid@invalid.invalid writes: Did you hear about the idiot topologist? He couldn't tell his butt from a hole in the ground, but he *could* tell his butt from two holes in the ground. Wow. Now I know _two_ topologist jokes. The girls are going to be impressed! I got it from here: http://mathoverflow.net/questions/1083/do-good-math-jokes-exist -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Chris Angelico ros...@gmail.com: Fortunately, we don't need to completely understand it. New Horizons reached Pluto right on time after a decade of flight that involved taking a left turn at Jupiter... we can predict exactly what angle to fire the rockets at in order to get where we want to go, even without knowing how that gravity yank works. Practicality beats purity? Engineer! At the time I was in college I heard topology was very fashionable among mathematicians. That was because it was one of the last remaining research topics that didn't yet have an application. Marko -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On 2015-07-23, Marko Rauhamaa ma...@pacujo.net wrote: Chris Angelico ros...@gmail.com: Fortunately, we don't need to completely understand it. New Horizons reached Pluto right on time after a decade of flight that involved taking a left turn at Jupiter... we can predict exactly what angle to fire the rockets at in order to get where we want to go, even without knowing how that gravity yank works. Practicality beats purity? Engineer! At the time I was in college I heard topology was very fashionable among mathematicians. That was because it was one of the last remaining research topics that didn't yet have an application. You can always pick out the topologist at a conference: he's the one trying to dunk his coffee cup in his doughnut. [Hey, how often do you get to use a topology joke.] -- Grant -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Fri, Jul 24, 2015 at 7:59 AM, Marko Rauhamaa ma...@pacujo.net wrote: Laura Creighton l...@openend.se: In a message of Fri, 24 Jul 2015 00:29:28 +0300, Marko Rauhamaa writes: At the time I was in college I heard topology was very fashionable among mathematicians. That was because it was one of the last remaining research topics that didn't yet have an application. I have a very good freind who is a knot-theorist. (Chad Musick, who may have proven something wonderful.) see: http://chadmusick.wikidot.com/knots He says there are lots of applications for this in the field of circuit board layouts. And most mathematicians accept knot-theory as part of topology. So topology, too, is lost. You remind me of the hipster mathematician cook, who burned himself by calculating pie before it was cool. ChrisA -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On 2015-07-23 22:50, Mark Lawrence wrote: On 23/07/2015 22:29, Marko Rauhamaa wrote: Chris Angelico ros...@gmail.com: Fortunately, we don't need to completely understand it. New Horizons reached Pluto right on time after a decade of flight that involved taking a left turn at Jupiter... we can predict exactly what angle to fire the rockets at in order to get where we want to go, even without knowing how that gravity yank works. Practicality beats purity? Engineer! Heard the one about the three engineers in the car that breaks down? The chemical engineer suggests that they could have contaminated fuel. They should try and get a sample and get someone to take it to a lab for analysis. The electrical engineer suggests that they check the battery and the leads for any problems. The Microsoft engineer suggests that they close all the windows, get out of the car, get back in the car, open all the windows and see what happens. And an Apple engineer would suggest buying a new car that runs only on its manufacturer's brand of fuel. :-) -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
In a message of Thu, 23 Jul 2015 23:01:51 +0100, MRAB writes: And an Apple engineer would suggest buying a new car that runs only on its manufacturer's brand of fuel. :-) Before you do that, read this: http://teslaclubsweden.se/test-drive-of-a-petrol-car/ (ps, if you can read Swedish, the Swedish version is a little more fun.) Laura -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On 23/07/2015 22:29, Marko Rauhamaa wrote: Chris Angelico ros...@gmail.com: Fortunately, we don't need to completely understand it. New Horizons reached Pluto right on time after a decade of flight that involved taking a left turn at Jupiter... we can predict exactly what angle to fire the rockets at in order to get where we want to go, even without knowing how that gravity yank works. Practicality beats purity? Engineer! Heard the one about the three engineers in the car that breaks down? The chemical engineer suggests that they could have contaminated fuel. They should try and get a sample and get someone to take it to a lab for analysis. The electrical engineer suggests that they check the battery and the leads for any problems. The Microsoft engineer suggests that they close all the windows, get out of the car, get back in the car, open all the windows and see what happens. -- My fellow Pythonistas, ask not what our language can do for you, ask what you can do for our language. Mark Lawrence -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Ian Kelly ian.g.ke...@gmail.com: Gravity existed before Newton, but the *theory* of gravity did not, so he composed the theory? Ironically, gravity is maybe the least well understood phenomenon in modern physics. Marko -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On 23/07/2015 23:01, MRAB wrote: On 2015-07-23 22:50, Mark Lawrence wrote: On 23/07/2015 22:29, Marko Rauhamaa wrote: Chris Angelico ros...@gmail.com: Fortunately, we don't need to completely understand it. New Horizons reached Pluto right on time after a decade of flight that involved taking a left turn at Jupiter... we can predict exactly what angle to fire the rockets at in order to get where we want to go, even without knowing how that gravity yank works. Practicality beats purity? Engineer! Heard the one about the three engineers in the car that breaks down? The chemical engineer suggests that they could have contaminated fuel. They should try and get a sample and get someone to take it to a lab for analysis. The electrical engineer suggests that they check the battery and the leads for any problems. The Microsoft engineer suggests that they close all the windows, get out of the car, get back in the car, open all the windows and see what happens. And an Apple engineer would suggest buying a new car that runs only on its manufacturer's brand of fuel. :-) Like it, marks out of 10, 15 :) -- My fellow Pythonistas, ask not what our language can do for you, ask what you can do for our language. Mark Lawrence -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
In a message of Fri, 24 Jul 2015 00:29:28 +0300, Marko Rauhamaa writes: Chris Angelico ros...@gmail.com: Fortunately, we don't need to completely understand it. New Horizons reached Pluto right on time after a decade of flight that involved taking a left turn at Jupiter... we can predict exactly what angle to fire the rockets at in order to get where we want to go, even without knowing how that gravity yank works. Practicality beats purity? Engineer! At the time I was in college I heard topology was very fashionable among mathematicians. That was because it was one of the last remaining research topics that didn't yet have an application. Marko I have a very good freind who is a knot-theorist. (Chad Musick, who may have proven something wonderful.) see: http://chadmusick.wikidot.com/knots He says there are lots of applications for this in the field of circuit board layouts. And most mathematicians accept knot-theory as part of topology. Laura -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Laura Creighton l...@openend.se: In a message of Fri, 24 Jul 2015 00:29:28 +0300, Marko Rauhamaa writes: At the time I was in college I heard topology was very fashionable among mathematicians. That was because it was one of the last remaining research topics that didn't yet have an application. I have a very good freind who is a knot-theorist. (Chad Musick, who may have proven something wonderful.) see: http://chadmusick.wikidot.com/knots He says there are lots of applications for this in the field of circuit board layouts. And most mathematicians accept knot-theory as part of topology. So topology, too, is lost. Marko -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Fri, Jul 24, 2015 at 7:50 AM, Mark Lawrence breamore...@yahoo.co.uk wrote: Heard the one about the three engineers in the car that breaks down? The chemical engineer suggests that they could have contaminated fuel. They should try and get a sample and get someone to take it to a lab for analysis. The electrical engineer suggests that they check the battery and the leads for any problems. The Microsoft engineer suggests that they close all the windows, get out of the car, get back in the car, open all the windows and see what happens. And the data scientist proved that the phenomenon is not wholly unlikely, given the predicted average reliability of cars; further studies would require improved sample size. ChrisA -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Thursday, July 23, 2015 at 12:28:19 PM UTC+5:30, Gregory Ewing wrote: Rustom Mody wrote: Ive known good ones) most practicing-mathematicians proceed on the assumption that they *discover* math and not that they *invent* it. For something purely abstract like mathematics, I don't see how there's any distinction between discovering and inventing. They're two words for the same thing. I don't know what kind of -ist that makes me... By some strange coincidence, a colleague just sent me this article on the mathematician John Horton Conway: http://www.theguardian.com/science/2015/jul/23/john-horton-conway-the-most-charismatic-mathematician-in-the-world In which is this paragraph: -- Conway is the rare sort of mathematician whose ability to connect his pet mathematical interests makes one wonder if he isn't, at some level, shaping mathematical reality and not just exploring it, James Propp, a professor of mathematics at the University of Massachusetts Lowell, once told me. The example of this that I know best is a connection he discovered between sphere packing and games. These were two separate areas of study that Conway had arrived at by two different paths. So there's no reason for them to be linked. But somehow, through the force of his personality, and the intensity of his passion, he bent the mathematical universe to his will. -- -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Friday, July 24, 2015 at 2:59:41 AM UTC+5:30, Marko Rauhamaa wrote: Chris : Fortunately, we don't need to completely understand it. New Horizons reached Pluto right on time after a decade of flight that involved taking a left turn at Jupiter... we can predict exactly what angle to fire the rockets at in order to get where we want to go, even without knowing how that gravity yank works. Practicality beats purity? Engineer! At the time I was in college I heard topology was very fashionable among mathematicians. That was because it was one of the last remaining research topics that didn't yet have an application. Probably shows more than anything else how siloed university depts are: http://www.amazon.in/Topology-Cambridge-Theoretical-Computer-Science/dp/0521576512 -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Thursday, July 23, 2015 at 7:08:10 PM UTC-5, Grant Edwards wrote: You can always pick out the topologist at a conference: he's the one trying to dunk his coffee cup in his doughnut. [Hey, how often do you get to use a topology joke.] Don't sale yourself short Grant. You get extra bonus points here: (1) told a rare joke and (2) perpetuated the off topic ramblings in an effort to deflect from main subject. Nice multitasking dude! -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Grant Edwards invalid@invalid.invalid writes: You can always pick out the topologist at a conference: he's the one trying to dunk his coffee cup in his doughnut. [Hey, how often do you get to use a topology joke.] Did you hear about the idiot topologist? He couldn't tell his butt from a hole in the ground, but he *could* tell his butt from two holes in the ground. -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Thursday, July 23, 2015 at 9:03:15 PM UTC-5, Paul Rubin wrote: Did you hear about the idiot topologist? He couldn't tell his butt from a hole in the ground, but he *could* tell his butt from two holes in the ground. This sounds more like a riddle than a joke. So in other words: the message passing requires himself in one hole and a clone of himself in another hole speaking the message simultaneously? -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Jul 22, 2015 9:46 PM, Steven D'Aprano steve+comp.lang.pyt...@pearwood.info wrote: On Thursday 23 July 2015 04:09, Rustom Mody wrote: tl;dr To me (as unprofessional a musician as mathematician) I find it arbitrary that Newton *discovered* gravity whereas Beethoven *composed* the 9th symphony. Newton didn't precisely *discover* gravity. I'm pretty sure that people before him didn't think that they were floating through the air weightless... *wink* Did gravity exist before Newton? Then he discovered it (in some sense). Did the 9th Symphony exist before Beethoven? No? Then he composed it. Gravity existed before Newton, but the *theory* of gravity did not, so he composed the theory? -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Fri, Jul 24, 2015 at 6:59 AM, Marko Rauhamaa ma...@pacujo.net wrote: Ian Kelly ian.g.ke...@gmail.com: Gravity existed before Newton, but the *theory* of gravity did not, so he composed the theory? Ironically, gravity is maybe the least well understood phenomenon in modern physics. Fortunately, we don't need to completely understand it. New Horizons reached Pluto right on time after a decade of flight that involved taking a left turn at Jupiter... we can predict exactly what angle to fire the rockets at in order to get where we want to go, even without knowing how that gravity yank works. Practicality beats purity? ChrisA -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Rustom Mody wrote: Ive known good ones) most practicing-mathematicians proceed on the assumption that they *discover* math and not that they *invent* it. For something purely abstract like mathematics, I don't see how there's any distinction between discovering and inventing. They're two words for the same thing. I don't know what kind of -ist that makes me... -- Greg -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Thursday, July 23, 2015 at 12:28:19 PM UTC+5:30, Gregory Ewing wrote: Rustom Mody wrote: Ive known good ones) most practicing-mathematicians proceed on the assumption that they *discover* math and not that they *invent* it. For something purely abstract like mathematics, I don't see how there's any distinction between discovering and inventing. They're two words for the same thing. I don't know what kind of -ist that makes me... Ummm... Clever! You give few clues... except for 'purely abstract'. Does that mean entirely in your consciousness? Or does it mean completely independent of the (physical) world and even time. Latter is more or less definition of platonist Former is some kind of combo of intuitionist and formalist (I guess!). JFTR: I believe that post-Cantor 'platonism' is an abuse of Plato's original https://en.wikipedia.org/wiki/Allegory_of_the_Cave -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Steven D'Aprano st...@pearwood.info:
I think that we can equally choose the natural numbers to be
axiomatic, or sets to be axiomatic and derive natural numbers from
them. Neither is more correct than the other.
Mathematicians quit trying to define what natural numbers mean and just
chose a standard enumerable sequence as *the* set of natural numbers.
That's analogous to physicists defining the meter as a particular rod in
a vault in Paris. So not even the length of the rod but the rod itself.
To modern mathematicians, the concept three simply means the fourth
element in the standard enumeration. When mathematicians need to use
natural numbers to count, they have to escape to predicate logic. For
example, to express that a natural number n has precisely two divisors,
you have to say,
∃x∈N ∃y∈N ¬x=y ∧ x|n ∧ y|n
(which avoids counting) or:
∃B∈P({x∈N : x|n}×2)
(∀x∈{x∈N : x|n} ∃y∈2 ((x,y)∈B ∧ ∀z∈2 (x,z)∈B→y=z) ∧
∀x∈2 ∃y∈{x∈N : x|n} ((y,x)∈B ∧ ∀z∈{x∈N : x|n} (z,x)∈B→y=z))
This latter clumsy expression captures the notion of counting. However,
in programming terms, you could say counting is not first-class in
mathematics. Counting is done as a macro if you will.
If the logicians had managed to define natural numbers the way they
wanted, counting would be first class and simple:
{x∈N : x|n}∈2
Marko
--
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Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
One way to look at this is to see that arithmetic is _behaviour_. Like all behaviours, it is subject to reification: see: https://en.wikipedia.org/wiki/Reification and especially as it is done in the German language, reification has this nasty habit of turning behaviours (i.e. things that are most like a verb) into nouns, or things that require nouns. Even the word _behaviour_ is suspect, as it is a noun. This noun-making can be contagious if we thought of the world, not as a thing, but happening-now (and see how hard it is to not have a noun like 'process' there) would we come to the question of 'Who made it?' For there would be no 'it' there to point at. It is not too surprising that the mathematicians have run into the limits of reification. There is only so much 'pretend this is a thing' you can do under relentless questioning before the 'thing-ness' just goes away ... Laura -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Thu, 23 Jul 2015 02:58 am, Steven D'Aprano wrote: 1. We have reason to expect that the natural numbers are absolutely fundamental and irreducible That's wrong. If we had such a reason, we could state it: the reason we expect natural numbers are irreducible is ... and fill in the blank. But I don't believe that such a reason exists (or at least, as far as we know). Sorry, that's not as clear as I intended. By a reason, I mean a direct reason for that choice, rather than some reason for the opposite choice. I hope that's clear? Perhaps an example will help. Are tomatoes red? We can have direct reasons for believing that tomatoes are red, e.g. the light reflecting off these tomatoes peaks at frequency X, which is within the range of red light. Alternatively, we might not have any such reason, and be reduced to arguing against the alternatives. Only if all the alternatives are false could we then believe that tomatoes are red. If tomatoes were blue, they would appear green when viewed under yellow light; since these tomatoes fail to appear green, they might be red. I'm suggesting that we have no direct reason for believing that the natural numbers are irreducible concepts, only indirect ones, namely that all the attempts to reduce them are unsatisfactory in some fashion or another. But neither do we have direct reasons for expecting them to be reducible. -- Steven -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Wednesday, July 22, 2015 at 11:22:57 PM UTC+5:30, Oscar Benjamin wrote: On Wed, 22 Jul 2015 at 18:01 Steven D'Aprano st...@pearwood.info wrote: I think that the critical factor there is that it is all in the past tense. Today, I believe, the vast majority of mathematicians fall into two camps: (1) Those who just use numbers without worrying about defining them in some deep or fundamental sense; Probably. I'd say that worrying too much about the true essence of numbers is just Platonism. Numbers are a construct (a very useful one). There are many other constructs used within mathematics and there are numerous ways to connect them or define them in terms of each other. Usually these are referred to as connections or sometimes more formally as isomorphisms and they can be useful but don't need to have any metaphysical meaning. Philosophers-of-mathematics decry platonism. However from my experience (I am not a professional mathematician, though Ive known good ones) most practicing-mathematicians proceed on the assumption that they *discover* math and not that they *invent* it. To me this says that though they may not know the meaning or spelling of platonism, they all layman-adhere to it. tl;dr To me (as unprofessional a musician as mathematician) I find it arbitrary that Newton *discovered* gravity whereas Beethoven *composed* the 9th symphony. Maybe Beethoven was sent my God to write Ode-to-Joy and reunite Europe? -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Nice Thanks for that Laura! I am reminded of | The toughest job Indians ever had was explaining to the whiteman who their | noun-god is. Repeat. That's because God isn't a noun in Native America. | God is a verb! From http://hilgart.org/enformy/dma-god.htm On Wednesday, July 22, 2015 at 10:48:38 PM UTC+5:30, Laura Creighton wrote: One way to look at this is to see that arithmetic is _behaviour_. Like all behaviours, it is subject to reification: see: https://en.wikipedia.org/wiki/Reification This is just a pointer to various disciplines/definitions... Which did you intend? By and large (for me, a CSist) I regard reification as philosophicalese for what programmers call first-classness. As someone brought up on Lisp and FP, was trained to regard reification/firstclassness as wonderful. However after seeing the overwhelming stupidity of OOP-treated-as-a-philosophy, Ive become suspect of this. If http://steve-yegge.blogspot.in/2006/03/execution-in-kingdom-of-nouns.html was just a joke it would be a laugh. I believe it is an accurate description of the brain-pickling it does to its religious adherents. And so now I am suspect of firstclassness in FP as well: http://blog.languager.org/2012/08/functional-programming-philosophical.html (last point) and especially as it is done in the German language, reification has this nasty habit of turning behaviours (i.e. things that are most like a verb) into nouns, or things that require nouns. Even the word _behaviour_ is suspect, as it is a noun. This noun-making can be contagious if we thought of the world, not as a thing, but happening-now (and see how hard it is to not have a noun like 'process' there) would we come to the question of 'Who made it?' For there would be no 'it' there to point at. It is not too surprising that the mathematicians have run into the limits of reification. There is only so much 'pretend this is a thing' you can do under relentless questioning before the 'thing-ness' just goes away ... Yes but one person's threshold where thing-ness can be far away from another's. Newton used thingness of ∞ (infinitesimals) with impunity and invented calculus. Gauss found this very improper and re-invented calculus without 'completed infinity'. Yet mathematicians habitually find that, for example generating functions that are obviously divergent (∴ semantically meaningless) are perfectly serviceable to solve recurrences; solutions which can subsequently be verified without the generating functions. Which side should be embarrassed? -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
In a message of Wed, 22 Jul 2015 10:49:13 -0700, Rustom Mody writes: Nice Thanks for that Laura! I am reminded of | The toughest job Indians ever had was explaining to the whiteman who their | noun-god is. Repeat. That's because God isn't a noun in Native America. | God is a verb! From http://hilgart.org/enformy/dma-god.htm On Wednesday, July 22, 2015 at 10:48:38 PM UTC+5:30, Laura Creighton wrote: One way to look at this is to see that arithmetic is _behaviour_. Like all behaviours, it is subject to reification: see: https://en.wikipedia.org/wiki/Reification This is just a pointer to various disciplines/definitions... Which did you intend? I meant -- depending on your background -- go find a meaning for reification that makes sense to you. And then extend this to some other areas. By and large (for me, a CSist) I regard reification as philosophicalese for what programmers call first-classness. Me too. But there are more people out there who know something about reification than there are that know about first classness. As someone brought up on Lisp and FP, was trained to regard reification/firstclassness as wonderful. However after seeing the overwhelming stupidity of OOP-treated-as-a-philosophy, Ive become suspect of this. If http://steve-yegge.blogspot.in/2006/03/execution-in-kingdom-of-nouns.html was just a joke it would be a laugh. I believe it is an accurate description of the brain-pickling it does to its religious adherents. And so now I am suspect of firstclassness in FP as well: http://blog.languager.org/2012/08/functional-programming-philosophical.html (last point) and especially as it is done in the German language, reification has this nasty habit of turning behaviours (i.e. things that are most like a verb) into nouns, or things that require nouns. Even the word _behaviour_ is suspect, as it is a noun. This noun-making can be contagious if we thought of the world, not as a thing, but happening-now (and see how hard it is to not have a noun like 'process' there) would we come to the question of 'Who made it?' For there would be no 'it' there to point at. It is not too surprising that the mathematicians have run into the limits of reification. There is only so much 'pretend this is a thing' you can do under relentless questioning before the 'thing-ness' just goes away ... Yes but one person's threshold where thing-ness can be far away from another's. Newton used thingness of ∞ (infinitesimals) with impunity and invented calculus. Gauss found this very improper and re-invented calculus without 'completed infinity'. Yet mathematicians habitually find that, for example generating functions that are obviously divergent (∴ semantically meaningless) are perfectly serviceable to solve recurrences; solutions which can subsequently be verified without the generating functions. Which side should be embarrassed? Embarassment is a function of the ego. The ego is _another_ one of those nouns where if you try to stalk it, it falls apart because it was produced by behaviour, rather than the cause of behaviour. Descartes: I think, therefore I am. (Because there must be an I that is doing the thinking.) Modern Day Western Neurologist: Thinking is going on. Therefore an I is produced. Laura -- https://mail.python.org/mailman/listinfo/python-list
OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Wed, 22 Jul 2015 11:34 pm, Rustom Mody wrote: On Tuesday, July 21, 2015 at 4:09:56 PM UTC+5:30, Steven D'Aprano wrote: We have no reason to expect that the natural numbers are anything less than absolutely fundamental and irreducible (as the Wikipedia article above puts it). It's remarkable that we can reduce all of mathematics to essentially a single axiom: the concept of the set. These two statements above contradict each other. With the double-negatives and other lint removed they read: I meant what I said. 1. We have reason to expect that the natural numbers are absolutely fundamental and irreducible That's wrong. If we had such a reason, we could state it: the reason we expect natural numbers are irreducible is ... and fill in the blank. But I don't believe that such a reason exists (or at least, as far as we know). However, neither do we have any reason to think that they are *not* irreducible. Hence, we have no reason to think that they are anything but irreducible. 2. We can reduce all of mathematics to essentially a single axiom: the concept of the set. Heh, yes, that is a bit funny. But I don't think it's really a contradiction, because I don't think that numbers really are sets. The set-theoretic definition of the natural numbers is quite nice, but is it really simpler than the old fashioned idea of natural numbers as counting numbers? Certainly it shows that we have a one-to-one correspondence from the natural numbers to sets, building from the empty set alone, and in some sense sets seem more primitive. But I think that proving that sets actually are more primitive is another story. I think that we can equally choose the natural numbers to be axiomatic, or sets to be axiomatic and derive natural numbers from them. Neither is more correct than the other. By the way, my comment about essentially a single axiom should be read informally. Formally, you need a whole bunch of axioms: http://math.stackexchange.com/questions/68659/set-theoretic-construction-of-the-natural-numbers I stated that the set-theoretic definition was generally accepted, but that's quite far from saying it is logically proven. The definition comes from Zermelo–Fraenkel set theory (ZF) plus the Axiom of Choice (ZFC), but the Axiom of Choice is not uncontroversial. E.g. the Banach-Tarski paradox follows from AC. https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox Like any other Axiom, the Axiom of Choice is unprovable. You can either accept it or reject it, and there is also ZF¬C. Jerry Bona jokes: The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma? the joke being that all three are mathematically equivalent. So are you on the number-side -- Poincare, Brouwer, Heyting... Or the set-side -- Cantor, Russel, Hilbert... ?? Yes. I think both points of view are useful, even if only useful in understanding what the limits of that view are. On Tuesday 21 July 2015 19:10, Marko Rauhamaa wrote: Our ancestors defined the fingers (or digits) as the set of numbers. Modern mathematicians have managed to enhance the definition quantitatively but not qualitatively. So what? This is not a problem for the use of numbers in science, engineering or mathematics (including computer science, which may be considered a branch of all three). There may be still a few holdouts who hope that Gödel is wrong and Russell's dream of being able to define all of mathematics in terms of logic can be resurrected, but everyone else has moved on, and don't consider it to be an embarrassment any more than it is an embarrassment that all of philosophy collapses in a heap when faced with solipsism. That's a bizarre view. Really? Which part(s) do you consider bizarre? (1) That the lack of any fundamental definition of numbers in terms is not a problem in practice? (2) That comp sci can be considered a branch of science, engineering and mathematics? (3) That there may still be a few people who hope Gödel is wrong? (4) That everyone else (well, at least everyone else in mathematics, for some definition of everyone) has moved on? (5) That philosophy is unable to cope with the problem of solipsism? I don't think any of those statements are *bizarre*, i.e. conspicuously or grossly unconventional or unusual; eccentric; freakish; gonzo; outlandish; outre; grotesque. In hindsight, my claim of everyone else having moved on is too strong -- there are millions of mathematicians in the world, and I'm sure that you can find one or two who don't fall into either of the two categories I gave. That is, they neither wish to dispute Gödel nor do they accept the irreducibility of numbers as something matter-of-fact and not embarrassing. So let's just quietly insert an almost before everyone and move on, shall we? As a subjective view I dont feel embarrassed by... its not arguable other than to say embarrassment is like
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
Steven D'Aprano st...@pearwood.info writes: That's wrong. If we had such a reason, we could state it: the reason we expect natural numbers are irreducible is ... and fill in the blank. But I don't believe that such a reason exists (or at least, as far as we know). However, neither do we have any reason to think that they are *not* irreducible. Hence, we have no reason to think that they are anything but irreducible. But by the same reasoning, we have no reason to think they are anything but non-irreducible (reducible, I guess). What the heck does it mean for a natural number to be irreducible anyway? I know what it means for a polynomial to be irreducable, but the natural number analogy would be a composite number, and there are plenty of those. You might like this: https://web.archive.org/web/20110806055104/http://www.math.princeton.edu/~nelson/papers/hm.pdf Remember also that in ultrafinitism, Peano Arithmetic goes from 1 to 88 (due to Shachaf on irc #haskell). ;-) -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Wed, 22 Jul 2015 at 18:01 Steven D'Aprano st...@pearwood.info wrote: I think that the critical factor there is that it is all in the past tense. Today, I believe, the vast majority of mathematicians fall into two camps: (1) Those who just use numbers without worrying about defining them in some deep or fundamental sense; Probably. I'd say that worrying too much about the true essence of numbers is just Platonism. Numbers are a construct (a very useful one). There are many other constructs used within mathematics and there are numerous ways to connect them or define them in terms of each other. Usually these are referred to as connections or sometimes more formally as isomorphisms and they can be useful but don't need to have any metaphysical meaning. Conventional mathematics treats the natural numbers as subsets of the complex numbers and usually treats the complex numbers as the most basic type of numbers. Exactly how you construct this out of sets is not as important as the usefulness of this concept when actually trying to use numbers. (2) Those who understand Gödel and have given up or rejected Russell's program to define mathematics in terms of pure logic. snip It's that I think that Russell's program is a degenerate research program and irrelevant paradigm abandoned by nearly everyone. Not only does Gödel prove the impossibility of Russell's attempt to ground mathematics in pure logic, but mathematicians have by and large rejected Russell's paradigm as irrelevant, like quintessence or aether to physicists, or how many angels can dance on the head of a pin to theologians. In simple terms, hardly anyone cares how you define numbers, so long as the definition gives you arithmetic. Actually in the decades since the incompleteness theorems were published much of mathematics has simply ignored the problem. Hilbert's idea to construct everything out of formal systems of axioms and proof rules continues to be pushed to its limits. This is now a standard approach in the literature, in textbooks and published papers, and in undergraduate programs. In contrast Gödel's (Platonist IMO) intuitionist idea of mathematical proof is ignored. The thing is that it turns out that even if you can't prove everything then you can at least prove a lot: Gödel demonstrated the existence of at least one unprovable theorem. Since we know that there are loads of unproven theorems and that loads of them continue to be proven all the time we clearly haven't yet hit any kind of Gödel limit that would impede further progress. Quoting Wikipedia: In the years following Gödel's theorems, as it became clear that there is no hope of proving consistency of mathematics, and with development of axiomatic set theories such as Zermelo–Fraenkel set theory and the lack of any evidence against its consistency, most mathematicians lost interest in the topic. They lost interest in the topic of proving consistency, completeness etc. They didn't lose interest in creating an explosion of different sets of axioms and proof systems, studying the limits of each and trying to push as much of conventional mathematics as possible into grand frameworks. For a modern example of Hilbert's legacy take a look at these guys: http://us.metamath.org/mpegif/mmtheorems.html They've tried to construct all of mathematics out of set theory using a fully formal (computer verifiable) proof database. They define the natural numbers as a subset of the complex numbers: http://us.metamath.org/mpegif/mmtheorems87.html#mm8625s The complex numbers themselves are defined in terms of the ordinal numbers which are similar to the natural numbers but have a distinct definition in terms of sets: http://us.metamath.org/mpegif/mmtheorems77.html#mm7635s I think they did it that way because it's just too awkward if the complex number 1 isn't the same as the natural number 1. -- Oscar -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Wednesday, July 22, 2015 at 11:18:23 PM UTC+5:30, Paul Rubin wrote: Remember also that in ultrafinitism, Peano Arithmetic goes from 1 to 88 (due to Shachaf on irc #haskell). ;-) No No No Its 42; Dont you know? -- https://mail.python.org/mailman/listinfo/python-list
Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Tuesday, July 21, 2015 at 4:09:56 PM UTC+5:30, Steven D'Aprano wrote: We have no reason to expect that the natural numbers are anything less than absolutely fundamental and irreducible (as the Wikipedia article above puts it). It's remarkable that we can reduce all of mathematics to essentially a single axiom: the concept of the set. These two statements above contradict each other. With the double-negatives and other lint removed they read: 1. We have reason to expect that the natural numbers are absolutely fundamental and irreducible 2. We can reduce all of mathematics to essentially a single axiom: the concept of the set. So are you on the number-side -- Poincare, Brouwer, Heyting... Or the set-side -- Cantor, Russel, Hilbert... ?? On Tuesday 21 July 2015 19:10, Marko Rauhamaa wrote: Our ancestors defined the fingers (or digits) as the set of numbers. Modern mathematicians have managed to enhance the definition quantitatively but not qualitatively. So what? This is not a problem for the use of numbers in science, engineering or mathematics (including computer science, which may be considered a branch of all three). There may be still a few holdouts who hope that Gödel is wrong and Russell's dream of being able to define all of mathematics in terms of logic can be resurrected, but everyone else has moved on, and don't consider it to be an embarrassment any more than it is an embarrassment that all of philosophy collapses in a heap when faced with solipsism. That's a bizarre view. As a subjective view I dont feel embarrassed by... its not arguable other than to say embarrassment is like thick-skinnedness -- some have elephant-skins some have gossamer skins As an objective view its just wrong: Eminent mathematicians have disagreed so strongly with each other as to what putative math is kosher and what embarrassing that they've sent each other to mental institutions. And -- most important of all -- these arguments are at the root of why CS 'happened' : http://blog.languager.org/2015/03/cs-history-0.html The one reason why this view -- the embarrassments in math/logic foundations are no longer relevant as they were in the 1930s -- is because people think CS is mostly engineering, hardly math. So (the argument runs) just as general relativity is irrelevant to bridge-building, so also meta-mathematics is to pragmatic CS. The answer to this view -- unfortunately widely-held -- is the same as above: http://blog.languager.org/2015/03/cs-history-0.html A knowledge of the history will disabuse of the holder of these misunderstandings and misconceptions -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Thu, Jul 23, 2015 at 11:44 AM, Dennis Lee Bieber wlfr...@ix.netcom.com wrote: On Thu, 23 Jul 2015 03:21:24 +1000, Steven D'Aprano st...@pearwood.info declaimed the following: Are tomatoes red? In answer I offer a novel: /Fried Green Tomatoes at the Whistle Stop Cafe/ (and lots of recipes for such on Google) I think we can take it as red. *dives for cover* ChrisA -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Thursday 23 July 2015 03:48, Paul Rubin wrote: Steven D'Aprano st...@pearwood.info writes: That's wrong. If we had such a reason, we could state it: the reason we expect natural numbers are irreducible is ... and fill in the blank. But I don't believe that such a reason exists (or at least, as far as we know). However, neither do we have any reason to think that they are *not* irreducible. Hence, we have no reason to think that they are anything but irreducible. But by the same reasoning, we have no reason to think they are anything but non-irreducible (reducible, I guess). What the heck does it mean for a natural number to be irreducible anyway? Reducible in the sense that we can define the natural numbers in terms of a simpler concept. E.g. the rationals can be reduced to the quotient of integers. One might argue that defining them in terms of sets is precisely that, but then we get into an argument as to which is simpler, natural numbers or sets? -- Steve -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On Thursday 23 July 2015 04:09, Rustom Mody wrote: tl;dr To me (as unprofessional a musician as mathematician) I find it arbitrary that Newton *discovered* gravity whereas Beethoven *composed* the 9th symphony. Newton didn't precisely *discover* gravity. I'm pretty sure that people before him didn't think that they were floating through the air weightless... *wink* Did gravity exist before Newton? Then he discovered it (in some sense). Did the 9th Symphony exist before Beethoven? No? Then he composed it. -- Steve -- https://mail.python.org/mailman/listinfo/python-list
Re: OT Re: Math-embarrassment results in CS [was: Should non-security 2.7 bugs be fixed?]
On 22/07/2015 19:14, Laura Creighton wrote: I don't suppose anybody could spare a bit of time for something that is slightly more important IMHO, like getting the core workflow going? I'm fairly well convinced that the vast majority of people here aren't in the slightest bit interested in actually doing anything, but just in case you could try reading. https://www.python.org/dev/peps/pep-0462/ https://www.python.org/dev/peps/pep-0474/ -- My fellow Pythonistas, ask not what our language can do for you, ask what you can do for our language. Mark Lawrence -- https://mail.python.org/mailman/listinfo/python-list
