Don't think there is one single Less Wrongist philosophy of math, but the infinite set atheism tag might be of interest, it links to 2009!Yudkowsky's guess that "I suspect that an AI does not have to reason about large infinities, or possibly any infinities at all, in order to deal with reality".
Where does mathematics go? How is it that human mental mathematical abilities and the related "manipulations of symbols" can be used to achieve things in the world? Cf The Unreasonable Effectiveness of Mathematics.
What do you think of Eric S. Raymond's take?
Where does mathematics come from? On the surface, human mathematical activity seems quite different from what I imagine apes would be doing on the savanna.
Same question, what do you think of this take?
It suffices for me if there is one philosophy of mathematics that exhibits Less Wrongist virtues 🙂
The Yud view seems not very complete from this. I'm saddened to see I can't access the Q&A videos, but I found a transcript of #14 about infinite atheism here. Note that the most infinite thing in my example suspicious statement is quantification over natural numbers.
I like both your posts, thanks for summarizing these views. I think in combination it somewhat dissolved the philosophical mystery for me, but there is still some normal empirical mystery left. The Mac Lane stuff reminded me of something I read at Terence Tao's, about how mathematicians use different intuitions like geometric, economic.
I don't understand the comment relating the equation to the continuum hypothesis so apologies if I'm missing a fundamental part of your question, but you do say:
> I have an intuition that arithmetical statements are true or false independently of my ability to prove or disprove them.
When I ask myself the question about the meaning/truth of mathematical statements, I usually end up convinced that mathematical statements are all tautologies. Or in other words, when I say "1 + 1 = 2", what I mean to say is that "Peano => (1 + 1 = 2)", the former just being a shorthand for the latter. Under this framing, all mathematical statements are tautologies and mathematics becomes the study of complicated tautologies. I would then say that math is so effective because we choose our axioms to resemble our experiences of the real world (the natural numbers resemble the concept of counting things, the real numbers locally resemble physical space). And if I we omit the assumption of the Peano axioms, then "1 + 1 = 2" really is true in a literal sense. It's not even a truth-claim about reality, rather it's a statement that evaluates to "true" (in context).
I consider myself to subscribe to "LessWrongist" philosophy: Bayesian epistemology, words are clusters in thingspace, materialist/reductionist view of consciousness and metaethics. These views seem to me to solve the typical philosophical nonsense you can often hear.
But there is one topic on which I don't know what to think without a little nonsense. I have an intuition that arithmetical statements are true or false independently of my ability to prove or disprove them. I have seen a similar view from famous rat-adjacent philosopher Scott Aaronson. But this intuition seems not very reductionist.
Consider the statement
On the one hand, every Turing machine either halts or doesn't. If we apply this to all the
On the other hand, if I say that this statement is either true or false, I'm asserting an apparent question of objective fact that will never be answered by a being in our universe, and there isn't even in principle a way to know the answer. Which is very "philosophical nonsense"-coded.
But if I bite the formalist bullet and say that sometimes I have a proof or disproof and sometimes I don't and that's all there is to it, I'm not making a distinction between the above statement and, say, the Continuum Hypothesis, which feels to me like an important distinction.
A somewhat related, less philosophical question is the Cotton Eyed Joe problem of mathematics, ie:
Seems a bit mysterious to me.
I thought to try the classic Yudkowskian trick of checking what, if anything, an AI would have to answer to a philosophical problem to be able to do stuff in the world. An AI would probably want to do some math to do physics, engineering, design encryption schemes etc.
I can imagine an AI getting all what it needs from some formal system like ZFC and not bothering with deeper questions. But, what if the AIs maker neglected to hardcode the most suitable system into the AI? When I consider which mathematical axioms to apply to my daily life, I use my mysterious philosophical abilities that I don't know how to write down in code. So, I'm not sure what AI would think of this.