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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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P or not-P
The past few days has seen an interesting discussion emerge on the Foundations of Mathematics listserv (FOM), which grew out of link to a recent article by Frank Quinn in Notices of the AMS, “A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today.”
Quinn’s article highlighted the role of the Law of the Excluded Middle (also known as tertium non datur) in mathematical reasoning. The Law of the Excluded Middle is a logical law that is usually stated, “P or not-P.” Though this sounds simple enough, it has been a sore spot in mathematics because it allows one to “prove” the existence of something that one can neither construct or exhibit — once all the alternatives are eliminated (and with classical negation, there is only one alternative), one is left with the mathematical equivalent of the last man standing.
Dutch philosopher of mathematics L. E. J. Brouwer formulated one of the most influential schools of constructivist thought, intuitionism, by jettisoning the Law of the Excluded Middle and simply doing without it.
Frank Waaldijk (like Brouwer, from the Netherlands) wrote on the FOM list:
The revolution in mathematics spreads much further than `just´ methodology. The revolution is about the concepts underlying all of our thinking about math, science and reality.
And my preliminary conclusion is: we still know nothing for sure. We are stumbling in the dark.I therefore cannot take Quinn’s stance on the role of `excluded middle´ in mathematics very seriously. Classical mathematics, in its full-fledged embrace of excluded middle, can be compared to science fiction…or dreamland if you would like a stronger metaphor. It’s nice to dream, and nice to be able to conjure battlestars and time travel and black hole mining and…But it is also important to return to reality from time to time. This is where constructive mathematics comes in. Constructive mathematics and classical mathematics are not always at odds per se…it is `just’ a major difference of focus and perspective. But I am personally convinced that we need constructive mathematics for a better understanding of our physical world and physical reality. And constructive views on excluded middle should already be taught in high school, not exclusively but at least for comparison.
Panu Raatikainen responded to Frank Waaldijk writing:
These are strong claims, and we’ve heard them now and then before, but it would be nice to hear some convincing arguments supporting them… I’ve honestly tried hard to find one for some time, but have so far failed…
Today I wrote the following (though with additional material added below) to the FOM list:
Perhaps a more charitable conception of the relationship between classical eclecticism and its tolerance of non-constructive modes of reasoning on the one hand, and on the other hand the many species of constructivism that have been proposed in order to place limits on classical eclecticism, is to be found in an image proposed by the mathematician Alain Connes:
“Constructivism may be compared to mountain climbers who proudly scale a peak with their bare hands, and formalists to climbers who permit themselves the luxury of hiring a helicopter to fly over the summit.”
(Conversations on Mind, Matter, and Mathematics, Changeux and Connes, Princeton, 1995, p. 42)
On the next page Connes says, continuing the image,
“…the uncountable axiom of choice gives an aerial view of mathematical reality — inevitably, therefore, a simplified view.”
If we think of the constructivist perspective very roughly as a “bottom up” approach, like a mountain climber who starts at the base and clambers over every cliff and every ledge on the way up, and non-constructive methods as a “top down” approach, an aerial view of mathematics, perhaps lacking in definite detail, but giving the big picture of the scene, then the two approaches are complementary.
An adequate conception of mathematical reality must include both constructive and non-constructive approaches, rather than dismiss classical mathematics as science fiction or dreamland.
I suggest that the top-down perspective of classical mathematics and the bottom-up perspective of constructivism meet in the middle, and that middle is constituted by macroscopic mathematical intuitions — the familiar instances of mathematical knowledge and experience like counting with cardinal numbers.
The classical foundationalist project plunges down from the heights and seeks to immerse itself in the details of mathematical knowledge from above. The relationship of the foundationalist to foundations established regressively (in Russell’s sense) from macroscopic mathematical intuitions is analogous the relationship believed by the classical mathematician to hold between macroscopic mathematical intuitions and the mathematical reality from which they are thought to descend. Thus the foundationalist project is a project to bring down the truth of macroscopic mathematics down its foundations.
The constructivist is no believer in the truth or truth-giving properties of macroscopic mathematics, which he regards as riddled with errors. The constructivist seeks to build from below only what can be built step-by-step, content to neglect the big picture and therefore blind to the landscape in which he patiently builds. To the constructivist, and foundationalist is falling off a cliff when he plunges downward; to the classical mathematician, the constructivist is so absorbed in this life on firm ground, with his feet in the mud, that he has ceased to dream and no longer looks up at the stars.
To put the distinction between the two in a quasi-scientific idiom, constructivism “explains” macroscopic mathematical intuitions as being constructed from ur-intuitions (as, for example, from Brouwer’s first and second acts of intuitionism), while classical eclecticism “explains” macroscopic mathematics intuitions from the top down, with reference to the abstract mathematical entities, from which flow macroscopic mathematical intuitions when the mind directly “perceives” mathematical reality (as in Gödelian mathematical realism, where the mind possesses a special faculty for “perceiving” mathematical reality).
The constructivist, who is a mathematical fox and knows many little things, many details of mathematics, and the classical mathematician, who is tolerant of non-constructive methods and as a mathematical hedgehog knows one big thing, need each other.
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Sir Michael Dummett obituary →
Sir Michael Dummett has died at the age of 86 on 27 December 2011. He was one of the most renowned philosophers of his generation. He sometimes characterized late-twentieth century analytical philosophy as Post-Fregean philosophy, and although he wrote voluminously about Frege and acknowledged the obvious influence, Dummett’s own philosophy diverged considerably from Frege’s unapologetic realism in regard to abstract objects.
Dummett wrote Elements of Intuitionism, which is a classic textbook on basic intuitionistic logic and mathematics, and then went on to apply his generally constructivistic and specifically intuitionistic views to wider vistas of philosophy in The Logical Basis of Metaphysics. These works might be informally characterized as “Frege meets Brouwer.”
Although Dummett’s work came after the great heyday of mid-century linguistic philosophy, his work embodied the “linguistic turn” even when writing about logic, mathematics, and metaphysics. Dummett emphasized the central role of language and of meaning in philosophical theories, and here, too, Frege was a touchstone and a crucial influence. His first major publication was Frege: Philosophy of Language.
Putting this linguistic turn into practice, in The Logical Basis of Metaphysics he wrote:
“…we must attain a clear conception of what a meaning-theory can be expected to do. Such a conception will form a base camp for an assault on the metaphysical peaks: I have no greater ambition in this book than to set up a base camp.”
