Ramble, Part 74: All in Your Mind
Jul. 30th, 2009 03:39 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe Logicists attempted to establish the truth of mathematics by grounding it in indisputable logic. The Formalists avoided the question of mathematical truth altogether. The third major school, Intuitionism, led by the great Dutch topologist L. E. J. Brouwer, took another tack, by redefining "truth".
There are two sides to Intuitionism: on the one hand the foundation it gives to mathematics, and on the other the proof techniques it permits. I'll talk about the first here and the second in the next post.
The fundamental assertion of Intuitionism is that mathematics is an activity of the human mind. The foundations, the justifications, for mathematics are therefore to be sought in the nature of the mind itself. You may recall that Kant had devised a general theory of truth on this basis, and that, as far as mathematics is concerned, he greatly overreached himself. The Intuitionist claims were far more modest; they pertained only to mathematical truth, and they took only one concept as inherent in the mind. (This "intuitive" claim to truth explains the name of the movement.)
That concept is succession: we all know the meaning of "the next", [ed. - That is, the next natural number,] and there is no need to interpose logical constructs, or anything else, to justify it. This is certainly a plausible stance, and - thanks to the work of mathematicians stretching from Descartes and Fermat, through Cauchy and Dedekind, and culminating in Peano - gives access to the entirety of classical mathematics, as well as much of the new mathematics that had arisen during the nineteenth century.
Much; not all. The Intuitionists were very leery of anything which smacked of infinity. Indeed, some of them held to the old Greek position, accepting virtual infinities but not actual ones - i.e., infinite sets. Most, however, were willing to grant countably infinite sets; but none gave credence to uncountable sets. (Virtuality is still available here; it is permissible to admit that, given any list of real numbers, there is a real number not on that list - but not to accept that there is a set consisting of all the real numbers!)
This, then, is the foundation for mathematics given by the Intuitionists: we accept the notion of succession, without resting it on any more fundamental concept, because it is intuitively obvious - inherent in the structure of our minds. This - and the name of the movement - may lead you to think that the Intuitionists were less committed to rigor and logic than other mathematicians. Nothing could be further from the truth, as we shall see next time.
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Ramble Contents
There are two sides to Intuitionism: on the one hand the foundation it gives to mathematics, and on the other the proof techniques it permits. I'll talk about the first here and the second in the next post.
The fundamental assertion of Intuitionism is that mathematics is an activity of the human mind. The foundations, the justifications, for mathematics are therefore to be sought in the nature of the mind itself. You may recall that Kant had devised a general theory of truth on this basis, and that, as far as mathematics is concerned, he greatly overreached himself. The Intuitionist claims were far more modest; they pertained only to mathematical truth, and they took only one concept as inherent in the mind. (This "intuitive" claim to truth explains the name of the movement.)
That concept is succession: we all know the meaning of "the next", [ed. - That is, the next natural number,] and there is no need to interpose logical constructs, or anything else, to justify it. This is certainly a plausible stance, and - thanks to the work of mathematicians stretching from Descartes and Fermat, through Cauchy and Dedekind, and culminating in Peano - gives access to the entirety of classical mathematics, as well as much of the new mathematics that had arisen during the nineteenth century.
Much; not all. The Intuitionists were very leery of anything which smacked of infinity. Indeed, some of them held to the old Greek position, accepting virtual infinities but not actual ones - i.e., infinite sets. Most, however, were willing to grant countably infinite sets; but none gave credence to uncountable sets. (Virtuality is still available here; it is permissible to admit that, given any list of real numbers, there is a real number not on that list - but not to accept that there is a set consisting of all the real numbers!)
This, then, is the foundation for mathematics given by the Intuitionists: we accept the notion of succession, without resting it on any more fundamental concept, because it is intuitively obvious - inherent in the structure of our minds. This - and the name of the movement - may lead you to think that the Intuitionists were less committed to rigor and logic than other mathematicians. Nothing could be further from the truth, as we shall see next time.
Previous
Ramble Contents
