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roland comments on Rationality Quotes December 2011 - Less Wrong
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How can you be sure? Every mathematical structure has to be represented in a physical brain. So the mathematical structures are constrained by the physicality of this universe.
I'm not sure I understand what you're saying. I can imagine a lot of things that don't exist in our universe, from magic flying ponies to Cthulhu. Some of those things are physically impossible; and yet, this imagination still takes place in my physical brain... Doesn't it ?
That wasn't my point. Whatever you are imagining has to be represented in your brain even if it is a phantasy. Can you imagine an infinite set in your brain? I'm not talking about the concept of an infinite set, but an actual infinite set?
Your question doesn't make sense. Can you represent an actual elephant in your head?
I'm not entirely clear on the categorical difference between the concept of an infinite set, and an actual infinite set. Aren't sets concepts to begin with, even finite ones ?
Would you say that mathematics is invented? Or that it is discovered?
Good question. I don't know and honestly I don't care. It is one of those deep philosophical question that can be debated ad nauseum.
But it is relevant to this discussion:
If it is invented, then any particular piece of mathematics doesn't exist until someone thinks it up (i.e. requires a physical brain).
If it is discovered, then all mathematics exists (in some sense), but humanity can only see a small portion of the whole (and it being in a physical brain or not is irrelevant).