Classical Theism, Philosophy, and Religion Forum / Mathematical Platonism

**Posted by Etzelnik** 10/22/2015 7:05 pm

As I understand it, a fair number of you have extensive mathematical background. Might you be willing to provide a synopsis of the for and against arguments and a basic bibliography?

Thanks!

**Posted by Mysterious Brony** 10/22/2015 11:57 pm

The Quine-Putnam indispensability argument is an argument for the existence of mathematical entities. You can find the argument here: http://plato.stanford.edu/entries/mathphil-indis/. Also Mary Leng's Mathematics and Reality responds to the indispensability argument and proposes anti-realist/nominal arguments against mathematical Platonism.

**Posted by iwpoe** 10/23/2015 1:43 am

I'm more broadly Platonist than merely mathematically. I can think of no broad Platonism that wouldn't entail Mathematical Platonism (in some sense), but is mathematical Platonism usually thought to have broader implications beyond the domain of mathematics?

**Posted by iwpoe** 10/23/2015 1:45 am

Mysterious Brony wrote:
The Quine-Putnam indispensability argument is an argument for the existence of mathematical entities.

How does Quine square that with his whole 'web of belief' thing? If I recall correctly he thinks you could do away with the law of non-contradiction if you wanted, or are "webs of belief" ontologically indifferent to the things they refer?

**Posted by Etzelnik** 10/23/2015 6:01 am

I found a pretty good IEP article on Quine-Putnam which goes into more detail than does SEP.
http://www.iep.utm.edu/indimath/#H9

**Posted by Mysterious Brony** 10/23/2015 5:09 pm

@iwpoe Well, I'm not familiar with Quine's works. I just presented the argument.

**Posted by Tyrrell McAllister** 10/26/2015 1:48 am

For me, the most compelling case for mathematical platonism appeals to my strong intuition that (say) the twin prime conjecture must have a determinate truth value. The intuition is that this conjecture has a truth value even if

(1) it turns out not to be a logical consequence of any of our axioms,

(2) it turns out not to be a logical consequence of any axioms that would ever seem "intuitively obvious" to us, and

(3) its truth or falsity depends on the behavior of integers so large that no actual thing will ever realize them (i.e., be numerous enough to be counted by them).

I don't know how to make sense of this intuition without positing something like "platonic" integers that exist independently of human thought and even of the existence of numerable things.

The biggest problem with such platonic entities, for me, is that I don't see how we could have any knowledge of them, even if they do exist. They don't seem to do any work in accounting for actual mathematical knowledge or practice.

The upshot is that I've gradually lost confidence in my intuition that the twin prime conjecture would have a truth value even under the circumstances that I described above.

**Posted by iwpoe** 10/26/2015 1:52 am

Tyrrell McAllister wrote:
The biggest problem with such platonic entities, for me, is that I don't see how we could have any knowledge of them, even if they do exist.

νόησις

Empiricist and naturalist epistemologies are fundamentally wrong and should be thrown out given that they cannot even account for their own activity.

Last edited by iwpoe (10/26/2015 1:53 am)

**Posted by Tyrrell McAllister** 10/26/2015 2:05 am

iwpoe wrote:
Empiricist and naturalist epistemologies are fundamentally wrong and should be thrown out given that they cannot even account for their own activity.

Who said anything about empiricism or naturalism? I will settle for an epistemological epistemology: An account of a known thing must account for how I know the thing. This is where mathematical platonism fails, so far as I can see.

**Posted by iwpoe** 10/26/2015 2:12 am

Tyrrell McAllister wrote:
iwpoe wrote:
Empiricist and naturalist epistemologies are fundamentally wrong and should be thrown out given that they cannot even account for their own activity.
Who said anything about empiricism or naturalism? I will settle for an epistemological epistemology: An account of a known thing must account for how I know the thing. This is where mathematical platonism fails, so far as I can see.

While the story might be more complicated (Hegel has a complicated story but gets to the same place) one must ultimately claim that we have a faculty for intuiting inteligibles to answer this challenge. The only reason I see for denying it (since it seems phenomenologically fundamental that was do know such things) is on the basis of an overly strict story about how we come to know (e.g. as I noted most usually "empiricist" or "naturalist" assumptions in doing epistemology. Kantian epistemology *also* denies that we have a faculty for knowing separate intelligibles, but claims we know them anyway as structures of consciousness itself.)

Last edited by iwpoe (10/26/2015 2:13 am)