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Not sure about this. I understand the proofs that there is no largest prime number, and that the integers can't be put into a one-to-one correspondence with the

Not sure about this. I understand the proofs that there is no largest prime number, and that the integers can't be put into a one-to-one correspondence with the reals, but I wouldn't say either is "obvious".

daniel-cussen 15y ago
The second proof you mention is the most controversial proof in the history of mathematics. Some mathematicians refuse to believe it.
- impendia 15y ago
  I am a mathematics postdoc at Stanford, constantly going to conferences and meeting other mathematicians, and I have not met anyone who refuses to believe this. Indeed, I have met no one who refuses to believe any proof that is widely accepted by the mathematical community. You could (presumably) construct axiom systems under which this is not true, and I believe that some logicians do this sort of thing, but this is rather far removed from mainstream research mathematics.
  - xyzzyz 15y ago
    You could apply the Lowenheim-Skolem theorem to the ZFC theory to obtain a countable model of set theory, but the object representing the set of real numbers in this model would still be "uncountable" inside of the model, altough clearly countable for us outside of it.
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  - jeffcoat 15y ago
    Diagonalization is a poor example -- though I don't think Cantor's logic was widely accepted until the early 20th century -- but I think there are examples of proofs where mathematicians will accept the logic, but argue with it none the less: The Banach-Tarski paradox comes to mind first. I don't think anyone argues that the proof is wrong, but you'll easily find people to argue that that very fact means that the full axiom of choice should be held in deep suspicion. The 4-color map coloring theorem is more interesting in this thread, though; I think it's the best known example of a proof that offers little to no insight into why the theorem is true. I don't think any solution that requires breaking a problem into 1,936 special cases, and then mechanically checking each one of those cases, will ever lead to an understanding that makes the theorem "obvious".
    - xyzzyz 15y ago
      I think once you understand the proof of 5 color theorem and believe the fact that the proof of 4 color theorem is more or less the same, but more hardcore, then it is as much insight as you can get. Looking for insight is what many people do in mathematics, but nonetheless believing that every true fact is true for a reason is naive, just as naive as believing that every problem has to have a solution. So far, most of the time we succeed in explaining why true things are true, but it may be the case that some facts are true merely by combinatorics, and not for human-imaginable reason. Sure, I would like to believe that there is a crucial observation we haven't made yet that makes 4 color theorem easy, just as it is for 6 color theorem. Nevertheless, I accept the fact that there may be no such thing and 4 color theorem holds just because the constraints force it to be.
      - jeffcoat 15y ago
        I didn't mean to defend the idea that every fact is true for a reason. I remember reading something of Chaitin's a couple of years ago that drove home the point that most mathematical facts are, in some fairly-well defined sense, true for no reason at all. (Alas, I don't remember the argument well enough to re-cap it here.)
    - Dn_Ab 15y ago
      The axiom of choice also allows you to prove the existence of a strategy that allows near perfect predictive accuracy given a function and a set of all past points. The reals themselves admit many far out properties such as virtually all reals being "inaccessible" or the existence of a know it all number. No, the problem that those who dislike the axiom of choice have is not inherent with the axiom but more the law of excluded middle. Constructivism has no rooms for such imaginings hence those who follow it are not happy with existential proofs. This more practical viewpoint will prove more profound IMHO just as bayesianism is winning the day. Just too many things connected.. types and terms in programs, intuitionistic logic, parts of physics, topoi, CCC.
pash 15y ago
Funny you picked those examples, because I recall them both seeming quite obvious to me when I learned them. And I'm no great mathematician. It's, um, obviously quite hard to explain why you find something obvious, but I think in the first case it's because you can just think about rescaling the number line (using a bigger unit) or shifting the origin, which to me makes it clear that bigness is completely arbitrary and implies that there can't possibly be any biggest number in this sense; in the second case, there clearly is no "next" number in the reals as there is in the naturals, and that insight gets right at the heart of the concept of countability.
- xyzzyz 15y ago
  in the second case, there clearly is no "next" number in the reals as there is in the naturals, and that insight gets right at the heart of the concept of countability. I think you missed the point. There is no "next number" concept in the rationals either, and yet they're countable. There is "next number" concept in the first uncountable ordinal, and, as the name suggests, it is uncountable.
  - pash 15y ago
    You're right. The "next number" test (under the natural ordering) makes the concept of density obvious, not countability. So I think I've given credence to the notion that those who claim to find something obvious might not fully understand it. Or at least not at four in the morning. (Oh, and a point that Stephen Abbott made in his wonderful intro analysis book now springs to mind: proofs are useful also as a check on our intuition.)

daniel-cussen 15y ago

The second proof you mention is the most controversial proof in the history of mathematics. Some mathematicians refuse to believe it.

impendia 15y ago
I am a mathematics postdoc at Stanford, constantly going to conferences and meeting other mathematicians, and I have not met anyone who refuses to believe this. Indeed, I have met no one who refuses to believe any proof that is widely accepted by the mathematical community. You could (presumably) construct axiom systems under which this is not true, and I believe that some logicians do this sort of thing, but this is rather far removed from mainstream research mathematics.
- xyzzyz 15y ago
  You could apply the Lowenheim-Skolem theorem to the ZFC theory to obtain a countable model of set theory, but the object representing the set of real numbers in this model would still be "uncountable" inside of the model, altough clearly countable for us outside of it.
- deleted 15y ago
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- deleted 15y ago
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- jeffcoat 15y ago
  Diagonalization is a poor example -- though I don't think Cantor's logic was widely accepted until the early 20th century -- but I think there are examples of proofs where mathematicians will accept the logic, but argue with it none the less: The Banach-Tarski paradox comes to mind first. I don't think anyone argues that the proof is wrong, but you'll easily find people to argue that that very fact means that the full axiom of choice should be held in deep suspicion. The 4-color map coloring theorem is more interesting in this thread, though; I think it's the best known example of a proof that offers little to no insight into why the theorem is true. I don't think any solution that requires breaking a problem into 1,936 special cases, and then mechanically checking each one of those cases, will ever lead to an understanding that makes the theorem "obvious".
  - xyzzyz 15y ago
    I think once you understand the proof of 5 color theorem and believe the fact that the proof of 4 color theorem is more or less the same, but more hardcore, then it is as much insight as you can get. Looking for insight is what many people do in mathematics, but nonetheless believing that every true fact is true for a reason is naive, just as naive as believing that every problem has to have a solution. So far, most of the time we succeed in explaining why true things are true, but it may be the case that some facts are true merely by combinatorics, and not for human-imaginable reason. Sure, I would like to believe that there is a crucial observation we haven't made yet that makes 4 color theorem easy, just as it is for 6 color theorem. Nevertheless, I accept the fact that there may be no such thing and 4 color theorem holds just because the constraints force it to be.
    - jeffcoat 15y ago
      I didn't mean to defend the idea that every fact is true for a reason. I remember reading something of Chaitin's a couple of years ago that drove home the point that most mathematical facts are, in some fairly-well defined sense, true for no reason at all. (Alas, I don't remember the argument well enough to re-cap it here.)
  - Dn_Ab 15y ago
    The axiom of choice also allows you to prove the existence of a strategy that allows near perfect predictive accuracy given a function and a set of all past points. The reals themselves admit many far out properties such as virtually all reals being "inaccessible" or the existence of a know it all number. No, the problem that those who dislike the axiom of choice have is not inherent with the axiom but more the law of excluded middle. Constructivism has no rooms for such imaginings hence those who follow it are not happy with existential proofs. This more practical viewpoint will prove more profound IMHO just as bayesianism is winning the day. Just too many things connected.. types and terms in programs, intuitionistic logic, parts of physics, topoi, CCC.

pash 15y ago

Funny you picked those examples, because I recall them both seeming quite obvious to me when I learned them. And I'm no great mathematician. It's, um, obviously quite hard to explain why you find something obvious, but I think in the first case it's because you can just think about rescaling the number line (using a bigger unit) or shifting the origin, which to me makes it clear that bigness is completely arbitrary and implies that there can't possibly be any biggest number in this sense; in the second case, there clearly is no "next" number in the reals as there is in the naturals, and that insight gets right at the heart of the concept of countability.

xyzzyz 15y ago
in the second case, there clearly is no "next" number in the reals as there is in the naturals, and that insight gets right at the heart of the concept of countability. I think you missed the point. There is no "next number" concept in the rationals either, and yet they're countable. There is "next number" concept in the first uncountable ordinal, and, as the name suggests, it is uncountable.
- pash 15y ago
  You're right. The "next number" test (under the natural ordering) makes the concept of density obvious, not countability. So I think I've given credence to the notion that those who claim to find something obvious might not fully understand it. Or at least not at four in the morning. (Oh, and a point that Stephen Abbott made in his wonderful intro analysis book now springs to mind: proofs are useful also as a check on our intuition.)