Timeline for Classroom platonism
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 4, 2017 at 8:21 | review | Close votes | |||
| Dec 4, 2017 at 15:02 | |||||
| Oct 25, 2017 at 13:34 | review | Close votes | |||
| Oct 25, 2017 at 17:01 | |||||
| Jun 23, 2012 at 3:59 | comment | added | Will Sawin | @David: In all your examples, Y is something concrete. A teacher would be insane to teach function spaces before "we live in R^3", power sets before sets of letters, numbers, etc., and state spaces and moduli spaces before donuts and coffee cups. So the student must abstract away their original concrete concept for Y. They must replace their concept with its formal properties, and see that X satisfies its formal properties. I am not saying that abstraction is the fundamental problem in platonization. I am just saying that abstraction is a key component of platonization. | |
| Jun 19, 2012 at 4:08 | comment | added | David Feldman | @Chris I'm not sure I read you. I will say, having tried to teach some philosophy of mathematics to undergraduates, that I don't think I would have served them well if my course induced them to align as formalists, Platonist, intuitionists or whatever. All the major philosophical positions capture vital aspects of mathematical practice and I believe that the sense of conflict arises because we overload words like "existence" and "identity" and then forget that we did. | |
| Jun 19, 2012 at 3:55 | comment | added | David Feldman | @Will Maybe I don't understand, but you haven't convinced me. I would agree that after platonizing configurations of $X$ to get $Y$ one then has the option of abstracting $Y$ away from its association with $X$. For example this happens when we forget the coordinates. And of course the elements of $X$ to start with may represent abstractions of some sort. Abstracting and platonizing work together very well, but I see a distinction. And it does seem to me that the latter represents the most persistent difficulty for my students. | |
| Jun 19, 2012 at 3:50 | comment | added | David Feldman | ...Agree about Emmy Noether...and also Poincaré. You can read Poincaré selling platonization right at the beginning of Analysis Situs. | |
| Jun 19, 2012 at 3:49 | comment | added | David Feldman | @Timothy The trouble with "imaginary numbers" strikes me as analogous to civil unions and gay marriage...an struggle about essential attributes and privilege. The old idea that full-fledged numbers should measure or count seems reasonable. Had the originators not called $\sqrt{-1}$ a "number" the whole historical chapter might have unfolded without a fuss. Agree about non-Euclidean geometry - the whole geometry becomes the object of study rather than the circles and triangles... | |
| Jun 19, 2012 at 3:42 | comment | added | David Feldman | @Andreas It strikes me that teachers of small children may either choose to concretize tenths (say via slices of pie) or, say, ask how to add 3 wizzlewazzles + 2 wizzlewazzles to get across the idea that the answer that it doesn't even matter (for this) what tenths are. I don't see treating abstract tenths like concrete dogs and books as cognitive platonization, but I do see cognitive platonization in the grammatical transition from "5 tenths are..." to "5/10 is..." | |
| Jun 18, 2012 at 23:55 | answer | added | Jon Bannon | timeline score: 6 | |
| Jun 18, 2012 at 23:43 | comment | added | Jon Bannon | The closest I've heard to cognitive platonization is the reification mentioned (if I remember right) in William Byers's "How mathematicians think". The idea there was that a process (or processes) is (are) considered as an object. As for the abstraction obstruction, I think I still suffer from that one...as I'm not yet able to think like Grothendieck :) | |
| Jun 18, 2012 at 20:57 | comment | added | Chris Godsil | 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? | |
| Jun 18, 2012 at 20:57 | comment | added | Will Sawin | 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? | |
| Jun 18, 2012 at 20:51 | comment | added | Timothy Chow | 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? | |
| Jun 18, 2012 at 20:33 | comment | added | Andreas Blass | 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. | |
| Jun 18, 2012 at 20:23 | comment | added | David Feldman | 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. | |
| Jun 18, 2012 at 20:08 | comment | added | Nik Weaver | I'd just like to point out that you characterize as radical those philosophers who challenge something you put in scare quotes. | |
| Jun 18, 2012 at 19:58 | history | asked | David Feldman | CC BY-SA 3.0 |