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Intuitively Clear Slippery Concepts
Or, into the conceptual wild…
The above phrase, “intuitively clear slippery concepts” just appeared today on the FOM (foundations of mathematics) listserv, as used by Hendrik Boom.
Here’s some context:
“…well-founded is one of these intuitively clear slippery concepts that goes awry when things become too general.”
This appeared in the tread, “iterative conception/cumulative hierarchy,” so this phrase that I ripped out the context of a sentence is also ripped out the context of a long discussion on the foundations of set theory.
Philosophers have used a variety of terms to indicate imprecise concepts and their manipulation: open-textured concepts, loose concepts, fuzzy concepts, and the like. This is the first time I’d seen “slippery concepts” used in this kind of context, but of course we all know what “slippery” means when it is used in colloquial speech. A slippery concept is difficult to pin down.
At the same time, philosophers have used a variety of terms of indicate intuitively clear concepts. The above phrase, “intuitively clear slippery concepts,” has the virtue of combining both the wide recognition that some concepts are intuitively clear, some concepts are slippery, and some concepts are both intuitively clear and slippery at the same time. One might hope or expect intuitively clear concepts to not be slippery, but sad experience has shown us that some of the least slippery concepts are counter-intuitive.
Moreover, Boom adds the caveat that these concepts, “go awry when things become too general.” In other words, in a suitably restrictive context, we ought not to have much trouble with concepts both intuitively clear and slippery, but once let out into the conceptual wild, as it were, things can go wrong.
This is a constructivist theme. In a couple of posts here, P or not-P and What is the relationship between constructive and non-constructive mathematics? (also inspired by posts on FOM), I have discussed constructivist thought in a general context.
For several years I have been thinking about how to characterize constructivist thought is absolutely general terms, so that all the diverse forms of constructivism can be brought together in a principled unity under a single concept. One obvious approach (which also has obvious problems) is to characterize constructivistic reasoning in terms of the limitations that it places on classical formal reasoning.
For example, “P or not-P” — the law of the excluded middle — doesn’t cause many problems in finite contexts, but once you try to apply it in unlimited or transfinite contexts its seems to generate paradoxes. The more rigorously we limit our reasoning (and this is one of the functions of axiomatization) the more we can be certain about what we say and what it means.
This is especially the point that I tried to make in What is the relationship between constructive and non-constructive mathematics?, as I there suggested that finite mathematics is a common core shared by constructivist and non-constructive thought alike.
That is to say, despite profound philosophical differences, there is a little bit of logic and mathematics upon which all (or almost all) formal thought can agree. It is when we go further afield, “when things become too general” as Boom wrote, or we head out into the “conceptual wild” (as I put it above), that differences emerge. The farther afield we go the more these differences harden into ideological differences informed by a “party spirit.”
In so far as almost all philosophies of mathematics can recognize the intuitive clarity of finite mathematics, but differences emerge once we pass beyond this restrictive boundary, the whole of mathematics could be said to consist of intuitively clear slippery concepts.
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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.
