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What is the relationship between constructive and non-constructive mathematics?
A few days ago in P or not-P I mentioned a discussion concerning constructivism on the FOM listserv. In that post I quoted Frank Waaldijk, and gave my response to this. Since then I have been honored with a response, which follows herein:
I think that to compare classical mathematics to science fiction is not the same as to dismiss it. Actually I’m quite appreciative of good science fiction. And I also think it is important to dream. I just stated that it is also important to return to reality from time to time.
Connes’ image above suggests that constructive math and classical math actually study the same mathematical earth, from a different point of view. For very important parts of mathematics, I can more or less live with this image although I also consider it too simplified and too charitable. However, the unrestricted axiom of choice gives rise to whole galaxies which are not studied in detail by constructive mathematicians (because to them these galaxies are rather wild science fiction, with little realistic content). Perhaps a nice short paper to also read is `Reality and Virtual Reality in Mathematics’ by Douglas Bridges (http://www.math.canterbury.ac.nz/~d.bridges/files/real.pdf).Apart from that, my more important point was and is that there could be a better balance between the large number of classical mathematicians which swarm in the sky and the handful of constructive mathematicians who try to validate what has been glimpsed from above. So my intention was not to dismiss classical mathematics (since I value its helicopter view also) but to emphasize that for a better understanding of reality, we need constructive mathematics. And for this we need more researchers in this field, and we also should teach constructive views already in high school, not exclusively like I said, but for comparison. One should not forget that all constructive mathematicians have had a heavy training in classical math… but vice versa?This to me seems the deeper implication of the foundational crisis that Frank Quinn mentioned. Some progress can be noted, since nowadays acceptance of constructive mathematics seems much better than in Brouwer’s time. On the other hand, one still finds occurrences of Zorn’s lemma (equivalent to axiom of choice) being used in texts on number theory to prove the existence of a maximal ideal, where a simple constructive proof can also be given… This means that generally, mathematicians have little feel for the constructive level/content of what they are doing. Reverse mathematics then helps only so much.
For the record, I agree that constructivist methods should be taught in schools as early as it is reasonable to suppose that students can grasp the essential ideas behind constructivism. I think it would be great improvement simply to make students aware of the difference between constructive and non-constructive reasoning, and this distinction is pervasive throughout formal thought but is little known beyond mathematics and philosophy. The mathematical curriculum would be the perfect opportunity to inculcate this distinction.
Given, then the virtues of knowing both what classical, non-constructive methods are, and the constructivist reaction that they have precipitated, what exactly are the essential ideas of constructivism? While most mathematicians, like most scientists, eschew philosophical thought (perhaps this is a reason to classify mathematics among the sciences), this is a paradigmatically philosophical question, and it cannot be answered without some significant effort.
When I was corresponding with Torkel Franzén in 1997, I wrote to him:
“Do the intuitionists have a replacement for set theory, or is this wrongly construing the entire intuitionist project? What, in this context, does ‘replacement’ mean?”
And he responded to me that:
“It means, quite simply, that everybody stops doing, teaching, and using classical mathematics and instead does, teaches, and uses intuitionistic mathematics.”
There is a kind of mathematical positivism in this response, and I have encountered this attitude elsewhere. It is as though intuitionism here had no philosophical content at all, whereas in fact it is fundamentally the expression of a philosophical perspective on mathematics.
How then are we to understand constructivist mathematics in relation to classical mathematics? If the distinctive feature of constructivism is that it limits the forms of mathematical reasoning, may we conclude that constructivist thought is wholly contained within classical thought, as in the following Venn diagram?
Here I have colored constructivist mathematics is a bright Bolshevik red because Frank Ramsey spoke of the “Bolshevik menace” of Weyl and Brouwer — although I seem to recall that someone told me Ramsey was converted to intuitionism shortly before his untimely death at a early age. I’ve also used the exotic “aliens ate my mom” font to emphasize its nontraditional character.
I don’t think that many philosophers would say that constructivist thought is merely a subset of classical thought. There is an interesting quote from Michael Dummett (very recently deceased, rest his soul) from his “Concluding Remarks at the Cerisy conference” that, in its brief compass, touches on several important themes:
“…the new non-realist conception demands a change in the logic with which we operate. Brouwer of course developed intuitionistic mathematics not only by restricting proofs to the modified logic his conception validated, but by introducing new notions and new principles governing them. With these a beautiful new version of mathematics was born…” (One Hundred Years of Intuitionism, pp. 341-342)
Dummett here emphasizes Brouwer’s novel contributions to mathematics, which implies that intuitionist mathematics contains elements that are lacking in classical mathematics, which implies in turn that intuitionist mathematics is not wholly contained within classical mathematics. Of course, if classical mathematics took up the intuitionist ideas and made them its own, then, like Hinduism declaring the Buddha to be an Avatar of Vishnu, all would be classical mathematics again. But intuitionist ideas like bar induction and the fan theorem seem to be mostly confined to intuitionist thinkers.
W. V. O. Quine made the argument that a change in logic entails a change in meaning, and therefore implies a change in subject. Quine wrote in his Philosophy of Logic:
They think they are talking about negation, ‘~’, ‘not’; but surely the notation ceased to be recognizable as negation when they took to regarding some conjunctions of the form ‘p & ~p’ as true, and stopped regarding such sentences as implying all others. Here, evidently, is the deviant logician’s predicament: when he tries to deny the doctrine he only changes the subject.
Quine is here discussing the principle of non-contradiction — “not (P and not-P)” — rather than the law of the excluded middle — “P or not-P” — but the Quinean objection is invariant across laws of logic. In fact, the whole point in Quine’s making this assertion is to insist upon the invariance of classical logic, which implies the invariance of classical mathematics if mathematics follows from logic.
If Quine is right about this (and I myself have never agreed with the “change in logic; change in subject” thesis), and if constructivist mathematics can be correctly defined as a body of formal truths derived from a uniquely intuitionistic logic (perhaps derived from the ur-intuitions posited by Brouwer, but using Brouwer’s logic), then Brouwer has simply changed the subject (when he thought he was changing mathematics, or, as Dummett puts it, creating a new version of mathematics) and classical mathematics and intuitionistic mathematics are incommensurable, as in the Venn diagram below:
Here classical mathematics and constructive mathematics wholly disjoint, and this seems a little extreme to me, and I don’t think that many philosophers of mathematics would argue for this position — though no doubt there are a few who would so argue.
The next most obvious relationship would be that classical mathematics and intuitionistic mathematics overlap and intersect (to invoke a classic Wittgensteinian formulation), and I have a ready answer for the field of their intersection: this would be what I previously called “macroscopic mathematical intuitions” as familiar as counting with cardinal numbers.
This is a good as far as it goes, but it can’t be quite right. This Venn diagram implies that classical mathematics and intuitionistic mathematics have nothing in common except macroscopic mathematical intuitions, and this seems clearly mistaken. There are finite inferences from macroscopic mathematical intuitions, made with as much logic as classical and intuitionist reasoning have in common (i.e., logic without tertium non datur), that would constitute a body of mathematical knowledge distinct from macroscopic mathematical intuitions but held in common by classical and intuitionist thought.
Beyond the problem of the relation of intuitionism or constructivism to classicism — which is, it must be admitted, a difficult problem — there is perhaps the yet more difficult problem of the relationship of intuitionism to constructivism. I have indirectly alluded to this already by the ambiguity of the previously language, sometimes speaking of “intuitionism” and sometimes speaking of “constructivism” without making a clear distinction or defining either one relative to the other. This is just as much a disputed philosophical question as the above question of the relationship between classicism and non-classical thought.
Intuitionism has been unproblematically associated with constructivism and it has been explicitly denied any association with constructivism. For my part, the denial the intuitionism is a species of constructivism is sort of like Heidegger and Jaspers denying that they were existentialists — but in so far as the Sartrean maxim is that a writer should not allow himself to be turned into an institution, they were justified in doing so. We could call this philosophizing under erasure.
Here we come to a parallel sequence of problematic relationships, similar to those mentioned above in trying to determine the relationship of classicism to its non-classicist other. Are constructivism and intuitionism incommensurable?
Is intuitionism wholly contained within constructivism, so that intuitionism is a particular species of constructivist thought, but not the whole of constructivist thought?
These positions on the interrelationship seem as unlikely as the parallel formulations above, which brings us to constructivism and intuitionism overlapping and intersecting. If this is the proper model of how constructivism relates on intuitionism, what exactly is the common core, why is it common to them, and how is it to be distinguished from classicism?
The next step ought to be an attempt to see classicism, constructivism, and intuitionism all together, and again I can appeal to macroscopic mathematical intuitions as the common ground of all three schools of thought. Beyond that, their interrelationships are less clear.
If this is correct — or even approximately correct, if you’d like to grant me some latitude — what, then, is mathematics?
Is mathematics this whole structure — classicism, constructivism, and intuitionism taken together — plus whatever else is done in the name of mathematics?
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