stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2009-09-28 02:34 pm
stoutfellow) wrote2009-09-28 02:34 pm
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Ramble, Part 75: Exclude This!
In our last episode, a couple of months ago, we discussed the Intuitionists, and in particular their view that mathematics takes place in the human mind and, therefore, must be founded and justified in mental terms. This has enormous consequences.
"What is truth?" is one of the oldest philosophical questions. For Intuitionists, here is the answer: a mathematical statement is true if, and only if, it has been proven. It is false if, and only if, it has been proven false. (There are some subtleties here that I'm not going to go into, at least in part because I'm not sure I understand them; this is, still, a reasonable approximation of the Intuitionist stance.)
Consider any of the great open questions of mathematics - say, Goldbach's Conjecture. A Logicist or a Formalist, asked whether Goldbach's Conjecture is true or false, would say, perhaps, "I don't know", or maybe "it's probably true". An Intuitionist, asked, "Is it true or false?", would answer, "No". Until a proof or disproof is discovered, the Conjecture is neither true nor false.
That deserves thought. One of the oldest principles of logic is the Law of Excluded Middle: for any statement P, the statement P OR (NOT-P) is true. In more ordinary English: any meaningful statement is either true or false. The Intuitionists took the radical step of rejecting this principle.
There are back-door ways of obtaining Excluded Middle. Perhaps the most familiar is the Law of Double Negation: any statement P is logically equivalent to NOT-NOT-P. The Intuitionists reject this as well, and the consequences for the practicing mathematician are enormous.
One of the first sophisticated proof techniques the budding mathematician encounters is proof by contradiction: if the assumption that P is true leads to a contradiction, then P must be false; that is, NOT-P is true. It is a powerful and seductive tool. (Indeed, young mathematicians often overuse it, and need to learn when it is appropriate and when it can be dispensed with.) In particular, mainstream mathematicians - that is, Logicists and Formalists - use it in the following form: to prove P, assume that P is false, and deduce a contradiction; conclude P. The Intuitionists object: your assumption is that NOT-P is true; your contradiction shows that NOT-NOT-P is true - but that is not the same as showing that P is true!
It is surprisingly difficult for a mathematician trained in the mainstream tradition to adapt to this. I have a book which discusses calculus from an Intuitionist viewpoint. Working the exercises, I had to constantly stop myself and restart, because I was using the disallowed form of proof by contradiction. I eventually, more or less, got the hang of it, but it was tough!
The point is this: despite the name of the school, Intuitionism is more, not less, demanding in its search for mathematical rigor. Discarding Excluded Middle is not their only demand; their notion of existence proof is much more tightly constrained than that of the mainstream. (It is not surprising that they also reject the Axiom of Choice, although there is a much-weakened version which is acceptable to some Intuitionists.) Any mathematical statement that meets Intuitionist standards also meets those of the other two schools, but the converse is not true.
Intuitionism is hard. It has some distinct advantages over the other schools, which will be mentioned later, but the sheer difficulty of its concept of proof is discouraging, and - for the foreseeable future - it is likely to remain a minority position.
Perhaps.
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Ramble Contents
"What is truth?" is one of the oldest philosophical questions. For Intuitionists, here is the answer: a mathematical statement is true if, and only if, it has been proven. It is false if, and only if, it has been proven false. (There are some subtleties here that I'm not going to go into, at least in part because I'm not sure I understand them; this is, still, a reasonable approximation of the Intuitionist stance.)
Consider any of the great open questions of mathematics - say, Goldbach's Conjecture. A Logicist or a Formalist, asked whether Goldbach's Conjecture is true or false, would say, perhaps, "I don't know", or maybe "it's probably true". An Intuitionist, asked, "Is it true or false?", would answer, "No". Until a proof or disproof is discovered, the Conjecture is neither true nor false.
That deserves thought. One of the oldest principles of logic is the Law of Excluded Middle: for any statement P, the statement P OR (NOT-P) is true. In more ordinary English: any meaningful statement is either true or false. The Intuitionists took the radical step of rejecting this principle.
There are back-door ways of obtaining Excluded Middle. Perhaps the most familiar is the Law of Double Negation: any statement P is logically equivalent to NOT-NOT-P. The Intuitionists reject this as well, and the consequences for the practicing mathematician are enormous.
One of the first sophisticated proof techniques the budding mathematician encounters is proof by contradiction: if the assumption that P is true leads to a contradiction, then P must be false; that is, NOT-P is true. It is a powerful and seductive tool. (Indeed, young mathematicians often overuse it, and need to learn when it is appropriate and when it can be dispensed with.) In particular, mainstream mathematicians - that is, Logicists and Formalists - use it in the following form: to prove P, assume that P is false, and deduce a contradiction; conclude P. The Intuitionists object: your assumption is that NOT-P is true; your contradiction shows that NOT-NOT-P is true - but that is not the same as showing that P is true!
It is surprisingly difficult for a mathematician trained in the mainstream tradition to adapt to this. I have a book which discusses calculus from an Intuitionist viewpoint. Working the exercises, I had to constantly stop myself and restart, because I was using the disallowed form of proof by contradiction. I eventually, more or less, got the hang of it, but it was tough!
The point is this: despite the name of the school, Intuitionism is more, not less, demanding in its search for mathematical rigor. Discarding Excluded Middle is not their only demand; their notion of existence proof is much more tightly constrained than that of the mainstream. (It is not surprising that they also reject the Axiom of Choice, although there is a much-weakened version which is acceptable to some Intuitionists.) Any mathematical statement that meets Intuitionist standards also meets those of the other two schools, but the converse is not true.
Intuitionism is hard. It has some distinct advantages over the other schools, which will be mentioned later, but the sheer difficulty of its concept of proof is discouraging, and - for the foreseeable future - it is likely to remain a minority position.
Perhaps.
Previous
Ramble Contents