Let's try another exercise in philosophical imagination. Suppose Platonism and dualism are true. Then consider a theory on which our souls actually inhabit a purely mathematical universe. All the things we ever observe—dust, brains, bodies, stars and the like—are just mathematical entities. As our souls go through life, they become "attached" to different bits and pieces of the mathematical universe. This may happen according to a deterministic schedule, but it could also happen an indeterministic way: today you're attached to part of a mathematical object A1, and tomorrow you might be attached to B2 or C2, instead. You might even have free will. One model for this is the traveling minds story, but with mathematical reality in the place of physical reality.
This is a realist idealism. The physical reality around us on this story is really real. It's just not intrinsically different from other bits of Platonic mathematical reality. The only difference between our universe and some imaginary 17-dimensional toroidal universe is that the mathematical entities constituting our universe are connected with souls, while those constituting that one are not.
One might wonder if this is really a form of idealism. After all, it really does posit physical reality. But physical reality ends up being nothing but Platonic reality.
The view is akin to Tegmark's ultimate ensemble picture, supplemented with dualism.
Given Platonism and dualism, this story is an attractive consequence of Ockham's Razor. Why have two kinds of things—the physical universe and the mathematical entities that represent the physical universe? Why not suppose they are the same thing? And, look, how neatly we solve the problem of how we have mathematical knowledge—we are acquainted with mathematical objects much as we are with tables and chairs.
"But we can just see that chairs and tables aren't made of mathematical entities?" you may ask. This, I think, confuses not seeing that chairs and tables are made of mathematical entities with seeing that they are not made of them. Likewise, we do not see that chairs and tables are made of fundamental particles, but neither do we see that they are not made of them. The fundamental structure of much of physical reality is hidden from our senses.
So what do we learn from this exercise? The view is, surely, absurd. Yet given Platonism and dualism, Ockham's razor strongly pulls to it. Does this give us reason to reject Platonism or dualism? Quite possibly.
Dr. Pruss,
ReplyDeleteI've been trying to select one of two mathematical realisms: Platonism and Scholastic Realism. The Platonism is not unlike the view expressed in this post, except I'm not especially hung up on having a mathematical universe. Properties are also fine, though probably lead to similar weirdness to the Tegmarkian view. In contrast, the Scholastic Realism is either Thomistic or grounds mathematical truths in the Divine Nature, instead of Intellect. I'm still researching the latter, but why prefer it to the former?
There are definitely problems with Platonism, but Divine Exemplarism seems to compound Benacerrafan worries. To rephrase Benacerraf's argument, human beings exist entirely within space-time. The God of classical theism exists outside space-time. Therefore, if Divine Exemplarism about mathematical truths is correct, human beings could not attain mathematical knowledge.
The God of classical theism is also Divinely Simple. As I understand, by Divine Simplicity, God is in principle too simple for human minds to comprehend except by analogy to more comprehensible entities. But there is something weird about saying mathematical truths can only be known by analogies.
Inferring mathematical truths from instantiations in nature also doesn't seem to match how mathematicians actually work. What am I misunderstanding?
Well, one argument against Platonic Realism is:
ReplyDelete1. Everything that exists is God or freely created by God.
2. If there are Platonic forms, they aren't freely created by God.
3. So, there are no Platonic forms.
By the way, it may be weird to say that mathematical truths can only be known by analogy, but it's not so weird to say that the *grounds* of mathematical truths can only be known by analogy.
Dr. Pruss,
ReplyDelete2. If there are Platonic forms, they aren't freely created by God.
Could the Forms not be necessary by virtue of God having created them in every possible world?
By the way, it may be weird to say that mathematical truths can only be known by analogy, but it's not so weird to say that the *grounds* of mathematical truths can only be known by analogy.
That's true. I suppose I'm getting caught up in another analogy—the analogy of truths as "reflections" of God's Nature under certain limitations.
Could the Forms not be necessary by virtue of God having created them in every possible world?
ReplyDeleteOn further reflection, this was a stupid question. Never mind. Thank you for the help.
I don't believe that Benacerraf's problem (in many its forms) is insolvable for Platonist. Mark Balauger offer one interesting solution in ''Platonism and Anti-Platonism in Mathematics'' (see very useful summary in Zalta and Colyvan critical study: https://mally.stanford.edu/Papers/logic-metaphysics.pdf).
ReplyDeleteProf Pruss, if I remember properly in your recent article on Divine creative freedom you wrote that you disagree with philosophers who hold that abstract objects can not be created by God. What is your view on this subject? Could theist be Platonist?
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ReplyDeleteMilos,
ReplyDeleteThanks for the recommendation.
I agree, and have read Balaguer's work. My concern was just that Divine Exemplarism might compound Benacerrafan epistemological problems by burying mathematics not just outside of time and space, but also in the mind or nature of a Divinely Simple being.
If you are still searching for the proof of God in a mathematical equation, simplify the equation to the absolute and behold the Divine truth. When All is equal, All is truly One.
ReplyDeleteBe One, =
...oh and as for the proof: have you ever doubted the equal sign in an equation? Surely One can and should doubt the symbols on either side of an equation, but never =, = simply is. Truth is.
Godel my dear friends was incomplete. =