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Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". I just don't find them convincing; but more like using au

by hyp0 12y ago

Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". I just don't find them convincing; but more like using authorised forms o argument within an artificially stylised tradition (like English Literature). And I wonder if alien mathematics will reveal our mathematics as embarassingly parochial - and not the universal common ground usually assumed. So, instead of proof, I have to fall back on intuition and working code, with their severe limitations. However... studying mathematical proof has at times informed and grown my intuition, by revealing new ways to see a problem and new (bizarre and unintuitive) ways to decompose it. I might have been better off never having seen this story.

RevRal 12y ago
That was/is such a dilemma.
lutusp 12y ago
> Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". That's too bad, because the anecdote doesn't challenge the basis for mathematical proofs or logical reasoning, in fact it requires it as a precondition for the anecdote to move forward. Remember that Gödel's incompleteness theorems don't argue that there are no true statements, only that some of them cannot be proven true. > So, instead of proof, I have to fall back on intuition ... You might be better off reviewing the structure of logic and mathematical proof. Start here: http://en.wikipedia.org/wiki/Euclid's_theorem http://en.wikipedia.org/wiki/Euclid's_theorem My reasoning is that, if there's one proof sufficiently transparent to win acceptance from a skeptic of logic, then there might be two ... ad infinitum.
- hyp0 12y ago
  Thanks for the link; I now feel a little more convinced by Euclid's Theorem than last time I looked at it. Though I still don't feel fully convinced by it; I don't fully see it. It's entirely possible my obstacle is not so much my skepticism as my stupidity :-)
  - lutusp 12y ago
    To see the power of a given proof, try to imagine what would be required to refute it, falsify it. This is by no means the only avenue of attack, but it's instructive. Also, it resembles the approach used by scientists with respect to falsifiable scientific theories (which aren't the same thing as mathematical proofs).
  - phaemon 12y ago
    I'd be interested in hearing why you're not fully convinced. There seem to be 2 parts to the proof. If you have a list of primes: You can generate another number from that list You can always get a prime from that number to add to the list I'm guessing it's the second part that isn't clicking with you, but perhaps I'm wrong. As for 'stupidity', I wouldn't worry about it. The only people I've ever had call me a moron or question my intelligence in any way have always been people who were less intelligent than I am. And that's not because I'm a genius ;-)
    - hyp0 12y ago
      I can follow the steps, but not see it. Like turn-by-turn directions, but no map. Perhaps also because I couldn't come up with it on my own - I don't see the family of which it is an instance (partly, this is the magic open-endedness of mathematics, it's not predictable). But I'm seeing more: start with some primes. They needn't be consective or ordered, just some primes. Any old primes will do. eg 2 and 5 are OK (skipping 3). Now multiply them all to get p. Obviously, p is divisible by all the primes we started with, because we just created it by multiplying them. eg 2 * 5 = 10 Note that p will generally be quite a bit bigger than the primes. Typically, you'll have the primes bunched up near the left of the number line, perhaps with some primes skipped between them, then a big gap to p, and continuing to infinity on the right. Now we add one to p. This is just to the right of p on the number line. This p+1 is either prime or it isn't. 1. If it's prime, then there is a prime other than the ones we started with. eg 10 + 1 = 11 2. If it's not prime, it has divisors. This proof claims it must include divisors that are prime but are not among those we started with. <-- THIS IS THE BIT I DON'T GET You can keep doing this, including that new prime (ie either p+1 itself or a prime divisor of it), showing there are infinitely many primes. So, yes, it's the second part.
- tjgq 12y ago
  In my humble opinion, proofs by contradiction are not the best examples of mathematical reasoning to be presented to the uninitiated. It has been my experience that people untrained in mathematics find it difficult to (intuitively) accept them as valid.
  - phaemon 12y ago
    Euclid's proof isn't a proof by contradiction. As the wikipedia page says: >"Euclid is often erroneously reported to have proved this result by contradiction" It simply says that if you are constructing a list of primes, you can always add one more to the list, therefore there are infinitely many.
    - tjgq 12y ago
      The overall structure of the proof is not by contradiction, but one of the steps is. The Wikipedia article calls this out, right after the sentence you quoted.
      - lutusp 12y ago
        Also, even though it's correct that Euclid didn't pose it as a proof by contradiction, it can certainly be posed that way, and I often use that form when presenting it to nonmathematicians.
        tjgq 12y ago
        Not everyone thinks the same way. There are certainly lay persons out there who do not find proofs by contradiction jarring when they come across them for the first time. I remember I was one of them. But the impression I get from my (possibly biased) sample is that most non-trained people intuitively see proof by contradiction (or any form of nonconstructive proof, really) as a way of "cheating", because it asserts something does or does not exist without actually producing a (counter)example. YMMV, of course.
rspeer 12y ago
Why would you be better off? I don't see any advantage in having the overly optimistic belief that there's some universal, correct set of axioms. You can still do all the math you could do before -- and if Carroll or GEB gets you more interested in the fundamentals of math, you can do even more. Yes, you have to accept some basis of mathematics, and you now understand that some true things will be unprovable in the basis you just accepted. But that doesn't stop you from proving things. I think you might have just transferred your optimism about math to code instead. How do you know your programming language is doing what you asked it to? That you asked it to do the right thing at all? That the compiled code has the correct behavior? That your hardware works as advertised and is not failing at the moment? In both code and math, you have to accept some abstractions that you're not going to worry about, but the things you do with math are certainly more verifiable.
- lutusp 12y ago
  > I think you might have just transferred your optimism about math to code instead. To add to your point, if someone begins to doubt the utility of mathematical logic and responds by refocusing his attention from logic to code, he's somehow overlooking the fact that code is built on a foundation of mathematical logic.
- hyp0 12y ago
  Better off, because I could have learnt the conventional axioms as a skill, like the perceptual and motor skills comprising reading, writing, arithmetic. And only later question them. My "optimism" is more for my intuition; working code is experimental confirmation. It's easier to be optimistic about code than proofs. Firstly, working code only needs to work in the specific cases you're using (that you test for); but a proof must work in every possible case. Thus, working code is simpler and easier to check, because it's aiming at less. My code almost always confirms my intuition. Yes, it's also helpful to have the automatic, mechanical check of executable code; and as you say, this relies on compilers, OSes, silicon, hardware. (Though, anecdotally, I have noticed subtle problems that I eventually diagnosed and confirmed to be compiler and hardware bugs.) BTW, yes I have tried COQ (proof assistant; somewhat mechanical proof checking), but simple ideas become very complex to prove, and the problem of bugs in COQ itself etc is of greater concern, for the next reason: Secondly, and relatedly, is that the standard is much lower for code. It just needs to work. Whereas a mathematical proof is supposed to be absolutely true. In other words, I don't ask as much from code. If there turns out to be a bug, it's just learning more about the problem; about the world. It's an engineering flaw. But if my proof is wrong, the game is lost. An argument against my intuition is probably more telling. Though my faith in it has turned out to be justified many many times, I certainly can be wrong. My only real excuse is that, as a human being, I have nothing else to fall back on but my sense of reality and reason. That's my hardware; if it's wrong, I really am lost. So I might as well trust it. Fortunately, it's almost always right; probably because I try to see things from many angles and check them in many ways before my intutive sense is fully formed.
anigbrowl 12y ago
Wait wait wait, proofs are still possible! The lesson of Godel's theorem is that you can have correctness or completeness, but not both within the same formal system. So you can have a system that yields only true statements, it just won't be able to encompass all true statements; or you can have a system that does encompass all true statements, but from which it is impossible to exclude some falsehoods.
PhasmaFelis 12y ago
> Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". I'm not sure I understand why. If the tortoise had insisted that one plus one is three, would it have destroyed your ability to accept arithmetic? I'm not sure what Carroll intended by this piece, but what I take from it is that there's no sense arguing with irrational people. One can certainly claim to accept A, and accept if-A-then-B, but deny B; one can also claim that up is down and the sky is candy-striped.

by hyp0 12y ago

RevRal 12y ago

That was/is such a dilemma.

lutusp 12y ago

> Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". That's too bad, because the anecdote doesn't challenge the basis for mathematical proofs or logical reasoning, in fact it requires it as a precondition for the anecdote to move forward. Remember that Gödel's incompleteness theorems don't argue that there are no true statements, only that some of them cannot be proven true. > So, instead of proof, I have to fall back on intuition ... You might be better off reviewing the structure of logic and mathematical proof. Start here: http://en.wikipedia.org/wiki/Euclid's_theorem http://en.wikipedia.org/wiki/Euclid's_theorem My reasoning is that, if there's one proof sufficiently transparent to win acceptance from a skeptic of logic, then there might be two ... ad infinitum.

hyp0 12y ago
Thanks for the link; I now feel a little more convinced by Euclid's Theorem than last time I looked at it. Though I still don't feel fully convinced by it; I don't fully see it. It's entirely possible my obstacle is not so much my skepticism as my stupidity :-)
- lutusp 12y ago
  To see the power of a given proof, try to imagine what would be required to refute it, falsify it. This is by no means the only avenue of attack, but it's instructive. Also, it resembles the approach used by scientists with respect to falsifiable scientific theories (which aren't the same thing as mathematical proofs).
- phaemon 12y ago
  I'd be interested in hearing why you're not fully convinced. There seem to be 2 parts to the proof. If you have a list of primes: You can generate another number from that list You can always get a prime from that number to add to the list I'm guessing it's the second part that isn't clicking with you, but perhaps I'm wrong. As for 'stupidity', I wouldn't worry about it. The only people I've ever had call me a moron or question my intelligence in any way have always been people who were less intelligent than I am. And that's not because I'm a genius ;-)
  - hyp0 12y ago
    I can follow the steps, but not see it. Like turn-by-turn directions, but no map. Perhaps also because I couldn't come up with it on my own - I don't see the family of which it is an instance (partly, this is the magic open-endedness of mathematics, it's not predictable). But I'm seeing more: start with some primes. They needn't be consective or ordered, just some primes. Any old primes will do. eg 2 and 5 are OK (skipping 3). Now multiply them all to get p. Obviously, p is divisible by all the primes we started with, because we just created it by multiplying them. eg 2 * 5 = 10 Note that p will generally be quite a bit bigger than the primes. Typically, you'll have the primes bunched up near the left of the number line, perhaps with some primes skipped between them, then a big gap to p, and continuing to infinity on the right. Now we add one to p. This is just to the right of p on the number line. This p+1 is either prime or it isn't. 1. If it's prime, then there is a prime other than the ones we started with. eg 10 + 1 = 11 2. If it's not prime, it has divisors. This proof claims it must include divisors that are prime but are not among those we started with. <-- THIS IS THE BIT I DON'T GET You can keep doing this, including that new prime (ie either p+1 itself or a prime divisor of it), showing there are infinitely many primes. So, yes, it's the second part.
tjgq 12y ago
In my humble opinion, proofs by contradiction are not the best examples of mathematical reasoning to be presented to the uninitiated. It has been my experience that people untrained in mathematics find it difficult to (intuitively) accept them as valid.
- phaemon 12y ago
  Euclid's proof isn't a proof by contradiction. As the wikipedia page says: >"Euclid is often erroneously reported to have proved this result by contradiction" It simply says that if you are constructing a list of primes, you can always add one more to the list, therefore there are infinitely many.
  - tjgq 12y ago
    The overall structure of the proof is not by contradiction, but one of the steps is. The Wikipedia article calls this out, right after the sentence you quoted.
    - lutusp 12y ago
      Also, even though it's correct that Euclid didn't pose it as a proof by contradiction, it can certainly be posed that way, and I often use that form when presenting it to nonmathematicians.
      - tjgq 12y ago
        Not everyone thinks the same way. There are certainly lay persons out there who do not find proofs by contradiction jarring when they come across them for the first time. I remember I was one of them. But the impression I get from my (possibly biased) sample is that most non-trained people intuitively see proof by contradiction (or any form of nonconstructive proof, really) as a way of "cheating", because it asserts something does or does not exist without actually producing a (counter)example. YMMV, of course.

rspeer 12y ago

Why would you be better off? I don't see any advantage in having the overly optimistic belief that there's some universal, correct set of axioms. You can still do all the math you could do before -- and if Carroll or GEB gets you more interested in the fundamentals of math, you can do even more. Yes, you have to accept some basis of mathematics, and you now understand that some true things will be unprovable in the basis you just accepted. But that doesn't stop you from proving things. I think you might have just transferred your optimism about math to code instead. How do you know your programming language is doing what you asked it to? That you asked it to do the right thing at all? That the compiled code has the correct behavior? That your hardware works as advertised and is not failing at the moment? In both code and math, you have to accept some abstractions that you're not going to worry about, but the things you do with math are certainly more verifiable.

lutusp 12y ago
> I think you might have just transferred your optimism about math to code instead. To add to your point, if someone begins to doubt the utility of mathematical logic and responds by refocusing his attention from logic to code, he's somehow overlooking the fact that code is built on a foundation of mathematical logic.
hyp0 12y ago
Better off, because I could have learnt the conventional axioms as a skill, like the perceptual and motor skills comprising reading, writing, arithmetic. And only later question them. My "optimism" is more for my intuition; working code is experimental confirmation. It's easier to be optimistic about code than proofs. Firstly, working code only needs to work in the specific cases you're using (that you test for); but a proof must work in every possible case. Thus, working code is simpler and easier to check, because it's aiming at less. My code almost always confirms my intuition. Yes, it's also helpful to have the automatic, mechanical check of executable code; and as you say, this relies on compilers, OSes, silicon, hardware. (Though, anecdotally, I have noticed subtle problems that I eventually diagnosed and confirmed to be compiler and hardware bugs.) BTW, yes I have tried COQ (proof assistant; somewhat mechanical proof checking), but simple ideas become very complex to prove, and the problem of bugs in COQ itself etc is of greater concern, for the next reason: Secondly, and relatedly, is that the standard is much lower for code. It just needs to work. Whereas a mathematical proof is supposed to be absolutely true. In other words, I don't ask as much from code. If there turns out to be a bug, it's just learning more about the problem; about the world. It's an engineering flaw. But if my proof is wrong, the game is lost. An argument against my intuition is probably more telling. Though my faith in it has turned out to be justified many many times, I certainly can be wrong. My only real excuse is that, as a human being, I have nothing else to fall back on but my sense of reality and reason. That's my hardware; if it's wrong, I really am lost. So I might as well trust it. Fortunately, it's almost always right; probably because I try to see things from many angles and check them in many ways before my intutive sense is fully formed.

anigbrowl 12y ago

Wait wait wait, proofs are still possible! The lesson of Godel's theorem is that you can have correctness or completeness, but not both within the same formal system. So you can have a system that yields only true statements, it just won't be able to encompass all true statements; or you can have a system that does encompass all true statements, but from which it is impossible to exclude some falsehoods.

PhasmaFelis 12y ago

> Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". I'm not sure I understand why. If the tortoise had insisted that one plus one is three, would it have destroyed your ability to accept arithmetic? I'm not sure what Carroll intended by this piece, but what I take from it is that there's no sense arguing with irrational people. One can certainly claim to accept A, and accept if-A-then-B, but deny B; one can also claim that up is down and the sky is candy-striped.