Intuitionistic Proof Versus Classical Truth

When I was a PhD student at UCSD in the 1980s I got hold of a Dutch magazine article titled, “De eenzaamheid van het gelijk.” Being a play on words, this phrase is hard to translate into idiomatically acceptable English, but the thrust of the pun is the use of the word “gelijk,” which means, at one and the same time, right, correct, equal, and simultaneous. So the author of the article evidently meant to suggest not only that his subject’s loneliness or isolation was due to his being correct in the face of his opposition, but also that he had it all right in the mathematical sense while others had it wrong: the article was a paean to L. E. J. (well, as long as I am returning to my native language, here it is: Luitzen Egbertus Johannes) Brouwer.

By the way, the article was to be found in a magazine that was available as part of the La Jolla Nachlass of the then recently deceased Errett Bishop, known for his contributions to constructivist mathematics, and, given my fluency in Dutch, making me an isolated singularity, I was the natural designate for this keepsake. I also had more than a passing interest in mathematical logic, so I went for it: I still have the magazine.

The article itself was very informative and interesting, particularly regarding Brouwer’s life, with his famous pugnacity and well-known bizarre life-style front and center. His pugnacity is abundantly illustrated by what Einstein, as a spectator, called “the frog and mouse battle” among the mathematicians, pitting Brouwer and his intuitionists against the mathematical establishment led by none other than David Hilbert, who would have none of what Brouwer was pushing (see e.g. the account in Constance Reid’s wonderful biography Hilbert). As for Brouwer’s life-style, especially in older age, it was, to say the least, unusual, given that it included his holding forth as the rooster in a hen-house on the “Veluwe”: he lived on a proto-commune in an agrarian area of Holland, surrounded by an arrangement of women.

So we are certainly dealing with something very different from the usual state of affairs. Indeed, Brouwer should be counted perhaps equally as a philosopher as a mathematician. In the latter area, he is of course remembered for his marvelous fixed point theorem and his work on the theory of dimension: a true pioneer in topology. But he moved away from this field into mathematical logic when he became the fons et origo of the assault on mainstream mathematics that has come to be known as intuitionism: it’s still around (certainly in Holland), although it’s not unfair to say that it did not succeed in bringing about the Götterdämmerung he had intended, given the war waged on him by the Wotan early 20th century mathematics, Hilbert.

It is perhaps not overly simplistic to characterize intuitionism as a philosophy of mathematics that insists on having the mathematician, as a human mind functioning in time and space, be an inalienable part of the process of mathematizing, as well as of the product itself, i.e. of mathematics itself. So, a certain inalienable subjectivism is posited, and the discipline itself is denied its Platonic status, i.e. its independence as an edifice of absolutely true (because provable) statements.

Many mainstays of mathematics are ipso facto challenged or rejected, since in Brouwer’s view they are out of reach for the inspecting human mind. Constructivism is the order of the day (on most days), and many proofs need to be redone in constructivist terms, if the intuitionists should have their way. In particular, the all-important tertium non datur, the law of the excluded middle, is out, and that is what spurred Hilbert into full attack: he likened what Brouwer was suggesting to taking his boxing gloves away from the boxer (cf. Reid’s book again).

Well, when the echoes of all the shouting died down, Hilbert had won the day, at least in practical terms: the debate was not really settled (especially if you ask the remaining intuitionists), but the rank and file returned to their normal activity, quietly but unmistakably closing ranks behind the status quo. And it’s still \(\neg[P\wedge (\neg P)]\) for just about all of us, and somewhere Hilbert is smiling.

All right, then, what’s the present book about? The author, Enrico Martino, tells us in his Preface: “I was stimulated … to collect some of my old papers on Intuitionism … In those days [the 1980s and ‘90s] there was a lively philosophical debate between classical and intuitionistic logicians and mathematicians … A crucial question discussed … is the following: to what extent [does the] intuitionistic perspective [succeed in avoiding] the classical realistic notion of truth?”

Most emphatically, then, Martino is concerned with a theme that is alien to most working mathematicians, but is obviously the sort of thing that attracts epistemologists and metaphysicians like bees to honey. Martino continues: “My answer is that a form of realism is hidden in the idealization of Brouwer’s creative subject [the aforementioned human mind functioning in space and time] … we need to think of him as if he were a real being: the mere image of him in our mind would not be able to perform the actions required by his role.”

All this having been said, the material in the book under review is also not at all like what one would encounter in philosophy as distinct from mathematics, for instance on my building’s third floor as opposed to (our) second floor: philosophy, as generally understood (at least popularly — whatever that means), does not deal with epistemology and theories of knowledge in this way. But mathematics is of course part of human knowledge, and so we are de facto dealing with an academic hybrid. In my undergraduate days at UCLA, we were graced with the presence on our campus of none other than Alonzo Church, sometimes counted as the greatest logician of his age, save for Gödel, and Church’s office was in the philosophy department across campus instead of in the mathematics department. The nature of the beast, I guess.

Back to the book, then: suffice to say it is a necessary amalgam of philosophical prose and mathematicalese, and the audience should be accordingly disposed and prepared. Just check out Chapter 2, suggestively titled, “Creative subject and the Bar Theorem.” The Bar Theorem is a mainstay of intuitionism, developed by Brouwer early on, its thrust being that it is a sort-of induction method on steroids with such things as decidability thrown into the mix. Brouwer famously used it to prove that a real function defined at every point of an interval is uniformly continuous. And there we are: we are not in Kansas anymore.

And this is the context of what Martino addresses in this second chapter, it being the text of a paper he gave at the 1981 L. E. J. Brouwer Centenary Symposium in Noordwijkerhout, Holland. Specifically, in this article Martino proposes “a reasonably precise description of Brouwer’s notion of ‘creative subject’ and an axiom … which is conceptually equivalent to the bar theorem.”

The preceding example should suffice to give an indication of just how intuitionistic mathematics differs in flavor from what we are used to in the mainstream, especially, of course, due to an admixture of philosophical and even psychological elements. One need only consult the titles of the other papers that appear as chapters in the book under review: to say that intuitionism is a profound departure from what most of us take for granted and are used to is a gross understatement.

Thus, the book under review is a collection of scholarly articles in and about intuitionistic mathematics, and as such should appeal to the corresponding specialists. The articles, being research pieces, are pitched at a high level: one needs a serious background in mathematical logic, and particularly intuitionism, to handle it all, as well as a decent dose of the kind of philosophy Martino and intuitionists generally practice. To say it departs dramatically from Aristotle doesn’t even begin to describe what’s afoot: it departs dramatically from Hilbert and Gödel, too. So: caveat.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.