## Thursday, March 18, 2010

### Seeing Sets

I used to laugh about (early) Pen Maddy claiming that we could see sets. But now I think that's almost right- though not in the way that Maddy intended it.

1.
I can see that my program doesn't infinitely loop, or that the 1000th prime is 7919 by pressing enter, waiting a few seconds and then looking above the command line prompt on my computer. These are all claims about mathematical objects, yet (given suitable equipment and background knowledge, we would ordinarily say that I can see these things to be true).

This seems just as literally true as the claim that I can see that the electricity is on, when I look at the lit windows of the house next door.

In both cases I immediately form the belief, probably am justified, am depending on a lot of contingent assumptions about electronic wiring etc.

2.
But maybe we should distinguish seeing Xs from seeing that some fact about Xs obtains? Maybe there's something especially problematic about believing in objects which you can't see?

-If seeing x = seeing that x exists, then I can see that there is a 1000th prime in the above example (suppose I wrote the program but had never seen the proof that there are infinitely many primes)

-If we take a more intuitive approach to seeing xs (i.e. is it awkward to say I am now looking an X) then:
a) certainly it is awkward to say `I am now looking at a number'...hmm though we might say `I am now seeing the line of the program that causes the crash (and lines in programs are abstract objects, just like lines in poems),'.
b) it's also pretty awkward to say `I am now seeing a drought', or `I am now seeing North America' or 'I am now seeing a proton'.

3.
If you can see a drought when you look at a color map of precipitation, why can't you see a pair of twin primes by looking at a chart?

4.
Overall conclusion:

Seeing that P really means little more than having some visual experience which causes you to immediately believe that P, which you might cite as part of your justification for believing that p. So if you can know things (e.g. all the background mathematical beliefs involved in the program case) about numbers, then it's not too hard to arrange to see things about them.

Of course, the anti-platonist won't think that you can know things about numbers either - well that's where my thesis comes in. But if we can know some things about the numbers, its not hard to arrange things so that we can see further things about them ie rig up reliable methods for forming beliefs about them whose last step involves visual experience.

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2. This post represents a fundamental failure to grasp the nature of philosophy and repeats the widespread philosophical mistake of assuming that just because we have some nice word 'see' that it somehow tracks some principled distinction/concept rather than just being your normal Wittgenstein 'game' type word.

For starters let's look at what's going on when someone like Maddy says she can 'see' the sets. Perhaps she is playing some frustrating game where she codes up her ideas into technically correct but highly confusing statements to bring herself fame and fortune. Maybe she is just throughly lost on the see of philosophical confusion. However, the charitable assumption which also matches up with normal human language is that she chose an unusual word like 'see' to describe coming to know the sets existed preciscely because she wanted to strongly distingush her claim from the one your interpratation would give: Maddy is saying she feels a powerful intuition that sets exist and that justifies belief in them.

Basically your interpretation fails one of the central cannons of legal and literary interpretation. If the author went out of their way to do something they did it for a reason not just to fuck with you. Either Maddy is throwing her readers under the confusion coach to advance her career or she is trying to indicate that she really thinks we know about the sets in a way that is much closer to perceptual knowledge than the knowledge gleaned from 'seeing' how a proof works.

Moreover, my interpratation of her has the significant advantage that her supposed epistemological answers/benefits don't come out as total gibberish. On your view any argument Maddy made for us knowing the sets exist would be no less applicable to any other philosophy of math (nothing beyond your own thoughts enters into your kind of seeing). On the other hand it's obvious why Maddy would think she had a novel solution to the problem of numerical/set existence if she meant 'see' to entail close parallels with perception.

My story goes like this: Maddy is troubled by questions about how we know there are sets and numbers and wants to put mathematical knowledge on a foundation just as firm as that we have for knowledge about external physical objects so she sits down and thinks about how philosophers have tried to stear clear of the Scylla of extreme skepticism and Charybdis of totally permissive knowledge claims and remembers a common solution is to argue that perception is a different sort of thing than mere reflection or intuition and thus gets epistemic super powers that aren't accessible to mere pondering. She then naturally wonders if one should also understand our knowledge of mathematical objects in a similar fashion. Note that at it's core this picture entierly rests on the idea that there is a distingushed kind of learning: perception which has different epistemic status. If you have to interpret Maddy as rejecting this then her motivations for the whole project suddenly seem totally mystifying.