Cognitive control in mathematics
Jukka Keranen's interesting PhD thesis is available here. By a careful comparison of an 1826 proof of Abel and a 1940 proof by Artin concerning the solvability of polynomials by radicals, taken as characteristic of their respective centuries, Keranen articulates what has been gained epistemically as we pass from the nineteenth to the twentieth century.
As someone who adheres to the idea that the aim of mathematical practice is improved understanding, this thesis is very welcome. I think one should not take the first of these passages to be in conflict with my position, since I take it that there's more to understanding than the understanding of facts.
I was led to conclude that while many of the resources characteristic of 20th century mathematics do in fact allow us to understand some range of mathematical facts better than do the corresponding 19th century resources, this is typically not the fundamental difference between them. As I will argue in this essay, typically the fundamental difference is that the 20th century resources allow us to attain better cognitive control over our mathematical epistemic processes such as proving theorems and solving problems: they make it possible for us to attain a higher grade of a certain kind of rational mastery over what we do in mathematics, one I chose to call “cognitive control.” (p. 3)
I motivated my characterization of cognitive control by noting that there are three basic types of epistemic challenges we need to be able to negotiate in typical mathematical epistemic processes: identify a terrain of facts to be examined, find a theoretically productive way of representing that terrain, and examine the appropriate locations in that terrain so as to extract features thereof that are directly relevant to answering the question driving the process. The basic idea was that, depending on one’s epistemic resources, one may or may not be able to negotiate these challenges in a rationally orchestrated manner. (p. 224)Pages 225 to 229 then give a neat synopsis of his findings.
As someone who adheres to the idea that the aim of mathematical practice is improved understanding, this thesis is very welcome. I think one should not take the first of these passages to be in conflict with my position, since I take it that there's more to understanding than the understanding of facts.
Recalling MacIntyre's comments about the master craftsman, good mathematicians don’t just know facts like people at a pub quiz, they know how things behave, they sense promising directions. They communicate a vision of how things might be. This is surely why mathematics exam questions go a certain way. State a result, prove it, then apply it in a novel situation. What is being tested is fledgling understanding. (How Mathematicians May Fail to be Fully Rational: p. 10)I went on to describe MacIntyre's account of the Aristotelian-Thomist view of understanding as the adequacy of the mind to its objects (see p. 12). I would want to argue that Keranen has provided us with some conceptual resources with which to clarify this adequacy in the case of mathematics. I also recommend Colin McLarty's account of Grothedieck's philosophy of mathematics - The rising tide: Grothendieck on simplicity and generality - which will appear soon as an article. Grothendieck had cognitive control if anyone did. See also the quotation in the fifth comment to this post, and those in the September 30 post.
8 Comments:
If you are unable to link to the thesis above. Try this link
http://etd.library.pitt.edu/ETD/available/etd-10282005-060742/
An amateur mathematician
Thanks for pointing that out, Zelah. I've fixed the link now, though one of my pdf readers still isn't happy and would prefer your indirect route.
Thank you for an interesting link.
I find it interesting that if I want to look at how mathematicians define/describe the processes underlying mathematical thinking, I need to read philosophy.
It's worth pointing out that, from time to time, modern mathematicians find it extremely fruitful to read the classics from eras with inferior epistemic resources. Geometric invariant theory certainly owes a lot to Mumford reading Hilbert. Intersection homology owes a lot to MacPherson reading Poincare. Manjul Bhargava of course read Gauss.
Nobody had thought about how many degree 6 curves you can draw through 6 points, or whatever, for a 150 years until quantum cohomology came along. There's probably all kinds of great stuff from the 19th century covered in dust.
Gian-Carlo Rota once described reading nineteenth century mathematical texts as like entering a hothouse full of exotic plants whose existence you had never suspected.
The quaternions are another example of 19th century math finding application (including within pure math) a century or more later; they turn out to be useful to computer games designers in parsimonious representation of motion in 3D.
I think these examples are very important to keep in mind. It is easy for non-scientists, in our instrumental times, to imagine that all research should be "useful". The British EPSRC, for instance, requires applicants for research grants to describe the wider impacts of their research, with the clear message that pure, blue-skies, research will not be funded without such impacts. Yet, none of us can know what will be useful and what not to future science or technology.
Who would have thought Aristotle's study of dialogue games would help in the design of computer-to-computer interaction?
I think mathematicians ought not even slide down the slope of saying that their work is justified because their theories may eventually be useful. If they give the impression that the only good of mathematics is external (in the sense of Aristotle's Nicomachean Ethics) to mathematics, they will be judged in means-ends fashion.
It would be interesting to share some "great stuff from the 19th century covered in dust." as my historical consciousness doesn't go further back than Poincare. I think both Poincare and Grothendieck liked to create new fruitful fields of mathematics by giving interesting new concepts an interesting name. Would one ever have thought about non mathematical applications for singularity theory if it had not been baptised catastrophe theory? I find it particulary hard to find sources that talk about the cognitive and conceptual parts of mathematics. Papers by Poincare "Science and method", Connes "A view of mathematics", Grothendieck's "sketch of a program" and "Récoltes et Semailles", Cartier's "A mad days work", Corfield's book, Baez, Atiyah, Mac Lane's "Form and function" and perhaps Polya's "How to solve it" are the only sources I am aware of that deal with this "soft" part of mathematics. Perhaps work in neuroscience on the cognition of visual information in the brain will make it clearer how we informally think about spaces.
As a nice example of an unexpected old source the logician Graham Priest links his ideas about dialetheism (contradictory logic) back to the budhist philosopher Nagarjuna (100 AD).
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