Wednesday, 14 April 2010

The Epistemological Argument against Mathematical Platonism

An amended version of my "epistemological argument against mathematical platonism" (forthcoming in Bruce & Barbone, Just the Argument : 100 of the most important Arguments in Western Philosophy, Blackwell) can be read here : draft.


Anonymous said...

Hi, Jonathan Speke laudly here,
Seems to me a realist vs nominalist distinction---which is a version of the mind/ body, material/ mental split;
Is it a notion or is it a real thing? One could ask, is the real vs notion split==itself a real distinction, a real actual thing or just a notion? Round we go.
This is the context, it seems to me in which you present your argument. I am skeptical of all such distinctions if they are taken as indubitable. After all, one could argue that the objectivity of a thing is determined subjectively-- such a distinction is a mental construct and that subjectivity does exist objectively.
Or one could say that, no, there is an intuition of such a mathematical object, sans concept, that such a mathematical is an independent entity--and that this can be state using concepts. But the reply to this would be that such a claim is just a mental, word, conceptual thing. But the reply to this could be that words are separate from things. But the reply to this is that this too is mental construct. And round we go.
Mental versus physical, real versus imagined, seem to me akin to one claiming one can see that form is entity existing independent of color in the vase-- and another denying--- saying form is just an abstraction and not independently real or material or actual at all. Take your pick. How do you decide such a thing?
Ultimately, if someone says "I have an intuition that there are independent mathematical objects, that, though we cannot touch them, are just as real as any physical object" --or some such.
What can you say? He has such an intuition and you do not. Are you going to deny the existence or validity of his intuition? He will deny your denial and round we go.
And one can argue that there is no clear deviding line between mental and physical even if such a split is presumed.
One could say these are two different points of view of the same thing, or could say they are completely different things altogether ---and tere seems no necessity to label either as entity nor a thought-----just say it is a showing up that receives assent.
Hard for me to agree that a thought is not as real as a taste or a touch. In any case, all this stuff--whether you categorize it as mental or physical or avoid categorization--- --- is the world. What other world do we have? Of what else could the world
be formed?

Mikolka said...

Thank you for your message. And sorry for the delay.

1) “ After all, one could argue that the objectivity of a thing is determined subjectively-- such a distinction is a mental construct and that subjectivity does exist objectively. “ Indeed, one can ask that question. One can also ask which one is the longer word between “realist” and “nominalist”… OK I am answering seriously now. But I see to much of that kind of strategy in Philosophy : one is asking a direct question and is waiting for an answer to that precise question, when everybody else is trying to avoid a direct answer to that question.

2) I am not fond of the subjective/objective distinction. For many reasons that I will not state here because they are irrelevant.

3) “Round we go”. Yeah. This is precisely why skepticism is what we call in French an “impasse”, a dead-end, and why I am not a skepticist. Used without extreme delicacy, it is an unweilding theoretical tool.

4) What can I say ? I will say : prove it ! We are here talking about mathematical concepts. If somebody, even Pascal or Gödel, says to you : “I believe that the solution of Fermat problem is Z, because I just I had this intuition” are you going to accept Z ? On what ground are you accepting this result ? Is the fact that he just had this intuition a sufficient ground for you ? Well, from my point of view, it is not ! What I mean is that the belief that one has a mathematical "intuition" of an abstract is not an admissible proof to hold that a certain mathematical proposition is true. Here, we are talking about intuition or revelation in general, but intuition in the case of mathematics. Do not make the mistake of mixing problems of intuition in general and problems of mathematical intuition of abstract objects, because the latter is very specific.

Brenton said...

Dear Mikolka,

I hope this will be a more helpful comment than the previous post.

I liked your note on the epistemic argument, I found some parts tough going though. In particular, I kept coming back to the thought that we might satisfy our realist intuitions without transgressing epistemic limits.

As you point out the intuition is that truths, even the mathematical truths, require truthmakers.

The epistemic argument (which I am used to hearing as the "Eleatic principle") is mounted against abstract objects conceived of in an extreme sense; that is, outside space and time. Fair enough.

What about truthmakers within space and time? These might also be abstract objects in a sense; they might be relations.

A few years ago I was briefly apprenticed at a maintenance site in isolated Pt Hedland. I learnt something there from a craftsman in the art of bricolage. That is, creatively using unlikely, or stand in, materials and tools at hand rather than ideal "bought in" materials.

I think some sort of "philosophical bricolage" is required here.

If truths of mathematics can be furnished truthmakers within space and time then they it seems these truthmakers will stand in causal or law-like relations to us, and so avert the difficulties of the epistemic argument.

Before I forget! This view will still be worthy of the name 'Platonism'. Why order in epistemically expensive transcendent objects when ordinary material objects (and relations between them) might do just as well. No worries that its metaphysical posits rest in the realm below Plato's heaven.

Looking forward to your response.

Brenton Welford