# Classroom platonism

I'd like to know whether any form a certain hypothesis about the learning of higher mathematics has entered the mathematical or educational literature. I'll frame the hypothesis here but not defend it since this is not a blog-in-disguise; likewise I'm not soliciting debate.

This hypothesis opposes to some degree the common shibboleth which holds that "mastering abstraction" constitutes the single major plateau which undergraduate mathematics students must, but often do not, scale.

For the sake of making the distinction, I'll first flesh out what I mean by "mastering abstraction." Generally speaking, abstraction means reducing to essentials. So mastering mathematical abstraction breaks into two major challenges: learning modelling and learning formal work. Modelers must first know how to decide what they may safely ignore and then how to select or construct formal systems that adequately capture what remains. With a formal system in hand, getting answers requires skill with its internal operation, sometimes despite the loss of intuition that arises from distance to the original situation. Of course a feedback cycle often arises -- "answers" from formal work can demand systemic revision of the formalism.

On the the current hypothesis, namely that something else constitutes the major glass ceiling for advance mathematics students. I'll call that something else cognitive platonization. (If someone else has already coined a better name I'd like to know!) So cognitive platonization occurs when mathematicians confer objecthood on the collection of some or all configurations of a known object. Examples abound: taking all solutions of certain differential equations as elements of a vector space, forming (iterated) power sets and cumulative hierarchies in set theory, studying state spaces in dynamical systems, moduli spaces in geometry, homology and cohomology groups or Stone-Cech compactifications in topology. Like abstraction, cognitive platonization often induces a loss of intuition due to distance from the original situation, but I contend a different sort of distance. Abstraction involves reasoning away from a picture you may feel afraid to lose; cognitive platonization involves reasoning on the way to a picture you may fear will never congeal.

As an aside, I chose the name because some radical philosophers challenge the very "existence" of just these sort of things I see students struggling to comprehend.

I'd like to know several things:

1) Does the challenge of teaching cognitive platonization (known by whatever name) have a theoretical literature?

2) Does cognitive platonization have a practical literature, meaning materials aimed directly at students, perhaps at the (American) college sophomore level?

3) Do any books from the popular science genre frame this issue and do a good job at communicating its essentials to a wide-audience?

4) What testable implications of the hypothesis can anyone suggest? Might success or failure with, say, abstract algebra or measure theory correlate with a student's response to tasks, otherwise unrelated to that subject matter, that indicate their ability or willingness to embrace this process of conferring objecthood? If so, what sort of tasks?

Final note: I'm asking here because most mathematics education research looks at K-12 teaching and learning, or perhaps calculus. Almost all writing about teaching higher mathematics comes from practicing mathematicians.

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I used scare quotes because I'm philosophically skeptical about any single unified concept of existence. For example, a radical philosopher might contend that $10^{10^{10^{10}}}$ does not exist (something like this comes up in Edward Nelson's Radically Elementary Probability Theory). And we might all agree that unicorns do not exist. But I don't find it clear that "not exist" means the same thing in both claims. –  David Feldman Jun 18 '12 at 20:23
Not an answer, just a comment about cognitive platonization: I've seen this problem at a much lower level of mathematics. Children who know perfectly well how to add 3 dogs + 2 dogs, or 3 books + 2 books, can still fail to see how to add fractions like 3 tenths + 2 tenths. The missing step, treating the abstract tenths like the concrete dogs and books, seems to be an instance of what you call cognitive platonization. –  Andreas Blass Jun 18 '12 at 20:33
Trying to understand your terminology better: Would you say that Descartes's use of the term "imaginary number" was symptomatic of a resistance to cognitive platonism? Was the historical lateness of the discovery of non-Euclidean geometries perhaps due in part to a discomfort with cognitive platonism? Would you characterize Emmy Noether as a master of cognitive platonization? If so, then it seems professional mathematicians have also had trouble with cognitive platonization historically, and maybe you're witnessing "ontogeny recapitulating phylogeny" in the classroom? –  Timothy Chow Jun 18 '12 at 20:51
To platonize X, don't you necessarily abstract Y? If you want to view the class of all X as really a certain kind of Y, you have to reduce your picture of Y to its essential components and formally check that the class of all X satisfies those conditions? –  Will Sawin Jun 18 '12 at 20:57
I would have thought that these conceptual difficulties arise because of the way our brains function. In which perhaps we should try to translate the issues into psychology rather than philosophy? –  Chris Godsil Jun 18 '12 at 20:57

The platonization you are looking at seems related to the idea of reification that appears in mathematics education literature (roughly the compression of a mathematical process to a mathematical object). Note: This is clearly distinct from what you are asking about, but shares the feature of bestowing objecthood on something in a way that requires the shift to (what is at first) a radically different viewpoint.

A few things on this that may be helpful can be found here.

Particularly, here are a couple of papers on reification.

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My sole experience with the term reification (before encountering Jon Bannon's usage) was in a book of Stephen Jay Gould (The Mismeasure of Man, if my memory serves me), where he mentioned its misuse to create things which weren't really there to help indicate intelligence. A weak but worrying parallel is the creation of derivatives as investment vehicles. I recommend Gould's take on the subject of reification. Gerhard "Ask Me About System Design" Paseman, 2012.06.18 –  Gerhard Paseman Jun 19 '12 at 0:13
@Gerard: It is interesting that reification is also a logical fallacy: en.wikipedia.org/wiki/Reification_(fallacy) –  Jon Bannon Jun 19 '12 at 1:15
Oddly, I thought to call my concept reification. I'll respond further once I've had a chance to read what you cite. –  David Feldman Jun 19 '12 at 4:10