Tuesday, April 7, 2009

The beauty of mathematics

Consider the claims:

  1. There is exquisite beauty in mathematics.
  2. Mathematics is grounded in the existence of a maximally great being.
Then the following argument might be offered:
  1. P(B|G) > P(B|~G).
  2. Therefore, B is evidence for G.
It does not, however, follow from (2) alone that B is evidence for the existence of God.

7 comments:

Chad said...

Dr. Pruss,

What would you adduce, if anything, for G?

Alexander R Pruss said...

You mean, what else, besides B?

Maybe that G has the resources to solve the epistemological problems plaguing acausal Platonic theories.

Vlastimil Vohánka said...

"It does not ... follow from (2) alone that B is evidence for the existence of God."

Why?

Alexander R Pruss said...

It's possible that P is evidence for Q, Q entails R, but P isn't evidence for R. More work is needed to rule out this possibility.

Alexander R Pruss said...

Here is a case of this phenomenon. There are ten balls, numbered 1-10, in an urn. Balls 1-5 are black; 6-10 are white. I pull out a ball from the urn, but you don't see me do it. I then tell you that it's black.

p = the ball is black
q = the ball is one of 2-5
r = the ball is not 1

Then, p is evidence for q. In fact, P(q|p) = 4/5, while P(q), not given p, is only 4/10.

Moreover, q entails r.

However, p is actually evidence against r. For P(r)=9/10, and P(r|q)=4/5, so P(r|q)>P(r).

I am no good with numbers, so I may have screwed up somewhere.

Vlastimil Vohánka said...

Thanks.

I see you similarly wrote in a paper on skeptical theism that "evidence for an entailing proposition need not be evidence for an entailed proposition. A standard case to show this is as follows. Let’s suppose I throw two dice, a red die and a green die. Let p be the proposition that the total I throw is 12. That the red die came up 6 is evidence in favor of p, and p entails that the green die came up 6. But that the red die came up 6 is no evidence for the green die’s coming up 6."

Here the evidence (for the entailing proposition) and the entailed proposition are independent.

As for your new scenario, you have there two typos. You say: "p is ... evidence against r. For P(r)=9/10, and P(r|q)=4/5, so P(r|q) greater than P(r)." There should have been: "p is ... evidence against r. For P(r)=9/10, and P(r|p)=4/5, so P(r|p) lower than P(r)".

Here, the evidence (for the entailing proposition) and the entailed proposition are not independent: the evidence is also an evidence against the entailed proposition.

1. Are there some other options (for it being the case the evidence for the entailing proposition is not an evidence for the entailed proposition) except the two just mentioned?

2. If not, are the beauty of math (the evidence for the entailing proposition) and theism (the entailed proposition) independent or is the beauty an evidence against theism?

3. Some would opt for the former alternative, claiming that math is beautiful necessarily, similarly to the alleged necessity of some moral propositions (like, you should not torture children just for fun), and thus the beauty provides no support for theism, like logical truths provide no evidence for anything. They are necessary, maybe even analytically.

4. But does maximally great being entail God (i.e., "an all-powerful, all-knowing perfectly good immaterial person who has created the world, has created human beings 'in his own image,' and to whom we owe worship, obedience and allegiance," as Plantinga has it in his SEP entry on science and religion)?

5. By the way, does God entail maximally great being? In other words, is God maximally great? If so, in which sense? In the sense that He possesses "the maximally valuable consistent conjunction of great­ making properties", as some analytical philosophers of religion put it? If so, is there some great making property (i.e., a property which makes the thing instantiating it better) or perfection which is not instantiated by a maximally great being? This does not seem to be (straightforwardly) ruled out by the just given definition of maximally great being. But the original, Anselmian, concept of maximally great being does rule it out. So, there seem to be different standard meanings of "maximally great being." Further, some more basic, ontological, differences lurk behind. Cf. Klima's www.fordham.edu/gsas/phil/klima/FILES/DZP-GK.pdf , sections 7 and 8.

Matthew said...

I think I once read a paper "Does mathematical beauty pose a problem to naturalism". I think it can be found online.