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I believe you're mistaken. There is value in axioms and axiomatic proofs: two different people will most definitively have a different notion of "obvious", and

I believe you're mistaken. There is value in axioms and axiomatic proofs: two different people will most definitively have a different notion of "obvious", and even have a different understanding of a mathematical problem. So a proof may be accepted by one person and rejected by another. Given a set of axioms and proofs it's possible to mechanically check a proof. It's not quite possible to reliably check proofs otherwise.

hugh4 10y ago
But are people more likely to accept the axioms of Principia Mathematica (and the soundness of every logical step from page 1 to 300) than they are to accept the notion that 2 + 2 = 4 based on intuitive notions of what twoness, fourness and plusness are?
- CarolineW 10y ago
  Of course they are more likely to believe their intuitions. They also believe that it makes no difference whether or not you swap doors in the Monty Hall problem, and don't believe that with only 23 people the odds of a shared birthday are more than 50%. To some extent, there is the problem. People trust their intuitions, and their intuitions are often wrong. That's why for some things we need proper proofs.
  - hugh4 10y ago
    But proofs always come back to axioms, and on what basis do we accept axioms? That they sound intuitively right. So we've just kicked the problem upstairs a bit, we can't avoid using our intuition. Personally I'm more likely to believe 2 + 2 = 4, something I can easily check to my own satisfaction using four objects, than I am to believe the Axiom of Choice.
    - kosievdmerwe 10y ago
      We still use our intuitions, but now everyone knows the starting set of assumptions. As for the axioms in use, I think the big reasons they were chosen is: They lead to results we already wanted/proved to be true. Another thing to keep in mind, not everyone works with the same sets of axioms. Which, as someone with a formalist[2] view on mathematics, I find interesting. For example, not everyone studying logic assumes the principle of the excluded middle[1]. One of the consequences of this is that you can no longer do proofs by contradiction. The axiom of choice is another example of this where two groups of mathematicians accept it or not. I'm a formalist, so I don't have issues with this (as long as both sets of axioms are interesting and "intuitive"), other philosophies of maths might. [1] https://en.wikipedia.org/wiki/Law_of_excluded_middle https://en.wikipedia.org/wiki/Law_of_excluded_middle [2] https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...
    - mnx 10y ago
      Well, the whole idea is to pick simple axioms, so it's harder to get wrong. And also, every successful prediction that math makes based on the axioms, is in a sense a verification of them.
    - fmap 10y ago
      It comes down to levels of assurance. The rules of inference you are using are part of your design space. Mathematics at its core is about clever problem solving. When you encounter a problem you have to decide what you would accept as a proof, and this forces you to accept certain reasoning principles. The caveat is that this is non-trivial and it's very easy to make people accept assumptions which are completely wrong. That's really the main reason to accept the axioms of set theory: People have been trying to poke holes in them for a hundred years and nobody has managed it yet. If you can use set theory (or something equiconsistent) to solve your problem, chances are that nobody will be able to call you out on a mistake.
pavelrub 10y ago
I'm not saying anything about the ability to check proofs, or the value of axiomatic proofs in general, only that 2+2=4 specifically doesn't require an axiomatic proof in order to convince anybody that it is true. This is like saying that we need a rigorous theory of color in order to be convinced that black is darker than red. Mathematicians didn't axiomatize natural numbers in order to show that 1+1=2 or 2+2=4, or any other trivial arithmetical fact. They have never doubted it, and I don't know what "doubting 2+2=4" even means. In fact the entire process is reversed: they invented axioms that can form a formal basis for what we already know to be true. If Peano axioms proved that 2+2 = 6 - they wouldn't be a valid axiomatization of the natural numbers. One cannot axiomatize the natural numbers without already assuming that all the basic arithmetic facts we know about them are true (or else he wouldn't be axiomatizing the natural numbers, but something else). Somebody who rejects 2+2=4 has a problem understanding human language, not proofs.
- thaumasiotes 10y ago
  > This is like saying that we need a rigorous theory of color in order to be convinced that black is darker than red. You do, if you want to be right. The fact that you can get people to agree with you doesn't make you right, and red is frequently darker than black by some pretty normal definitions of "darker". Red and black are differentiated by the shape of their reflective spectrum, not the amplitude.
  - jessriedel 10y ago
    You guys are basically arguing over Moore's here-is-one-hand problem. https://en.wikipedia.org/wiki/Here_is_one_hand https://en.wikipedia.org/wiki/Here_is_one_hand pavelrub's point is that you sometimes have less reason to believe the axioms of your formalization than their derived consequences. We have better reason to believe the intuitive idea that 2+2=4 than we do any putative axioms of arithmetic. If we derived that 2+2=5 from some particular axioms of arithmetic, we would conclude those axioms were wrong (or rather, were not the proper system for formalizing 2-plus-2-ness) rather than conclude that 2+2=5.
    - deleted 10y ago

hugh4 10y ago

But are people more likely to accept the axioms of Principia Mathematica (and the soundness of every logical step from page 1 to 300) than they are to accept the notion that 2 + 2 = 4 based on intuitive notions of what twoness, fourness and plusness are?

CarolineW 10y ago
Of course they are more likely to believe their intuitions. They also believe that it makes no difference whether or not you swap doors in the Monty Hall problem, and don't believe that with only 23 people the odds of a shared birthday are more than 50%. To some extent, there is the problem. People trust their intuitions, and their intuitions are often wrong. That's why for some things we need proper proofs.
- hugh4 10y ago
  But proofs always come back to axioms, and on what basis do we accept axioms? That they sound intuitively right. So we've just kicked the problem upstairs a bit, we can't avoid using our intuition. Personally I'm more likely to believe 2 + 2 = 4, something I can easily check to my own satisfaction using four objects, than I am to believe the Axiom of Choice.
  - kosievdmerwe 10y ago
    We still use our intuitions, but now everyone knows the starting set of assumptions. As for the axioms in use, I think the big reasons they were chosen is: They lead to results we already wanted/proved to be true. Another thing to keep in mind, not everyone works with the same sets of axioms. Which, as someone with a formalist[2] view on mathematics, I find interesting. For example, not everyone studying logic assumes the principle of the excluded middle[1]. One of the consequences of this is that you can no longer do proofs by contradiction. The axiom of choice is another example of this where two groups of mathematicians accept it or not. I'm a formalist, so I don't have issues with this (as long as both sets of axioms are interesting and "intuitive"), other philosophies of maths might. [1] https://en.wikipedia.org/wiki/Law_of_excluded_middle https://en.wikipedia.org/wiki/Law_of_excluded_middle [2] https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...
  - mnx 10y ago
    Well, the whole idea is to pick simple axioms, so it's harder to get wrong. And also, every successful prediction that math makes based on the axioms, is in a sense a verification of them.
  - fmap 10y ago
    It comes down to levels of assurance. The rules of inference you are using are part of your design space. Mathematics at its core is about clever problem solving. When you encounter a problem you have to decide what you would accept as a proof, and this forces you to accept certain reasoning principles. The caveat is that this is non-trivial and it's very easy to make people accept assumptions which are completely wrong. That's really the main reason to accept the axioms of set theory: People have been trying to poke holes in them for a hundred years and nobody has managed it yet. If you can use set theory (or something equiconsistent) to solve your problem, chances are that nobody will be able to call you out on a mistake.

pavelrub 10y ago

I'm not saying anything about the ability to check proofs, or the value of axiomatic proofs in general, only that 2+2=4 specifically doesn't require an axiomatic proof in order to convince anybody that it is true. This is like saying that we need a rigorous theory of color in order to be convinced that black is darker than red. Mathematicians didn't axiomatize natural numbers in order to show that 1+1=2 or 2+2=4, or any other trivial arithmetical fact. They have never doubted it, and I don't know what "doubting 2+2=4" even means. In fact the entire process is reversed: they invented axioms that can form a formal basis for what we already know to be true. If Peano axioms proved that 2+2 = 6 - they wouldn't be a valid axiomatization of the natural numbers. One cannot axiomatize the natural numbers without already assuming that all the basic arithmetic facts we know about them are true (or else he wouldn't be axiomatizing the natural numbers, but something else). Somebody who rejects 2+2=4 has a problem understanding human language, not proofs.

thaumasiotes 10y ago
> This is like saying that we need a rigorous theory of color in order to be convinced that black is darker than red. You do, if you want to be right. The fact that you can get people to agree with you doesn't make you right, and red is frequently darker than black by some pretty normal definitions of "darker". Red and black are differentiated by the shape of their reflective spectrum, not the amplitude.
- jessriedel 10y ago
  You guys are basically arguing over Moore's here-is-one-hand problem. https://en.wikipedia.org/wiki/Here_is_one_hand https://en.wikipedia.org/wiki/Here_is_one_hand pavelrub's point is that you sometimes have less reason to believe the axioms of your formalization than their derived consequences. We have better reason to believe the intuitive idea that 2+2=4 than we do any putative axioms of arithmetic. If we derived that 2+2=5 from some particular axioms of arithmetic, we would conclude those axioms were wrong (or rather, were not the proper system for formalizing 2-plus-2-ness) rather than conclude that 2+2=5.
  - deleted 10y ago