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dbaupp comments on Rationality Quotes December 2011 - Less Wrong
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The "controversy" about infinite sets is about their existence/usefulness as physical objects, not their mathematical existence (you'll note that I was careful to say that they were not physical objects in the grandparent). From that article:
Thus, infinite sets are a perfect example of a mathematical object disconnected from physical reality/practical experience.
We can construct the natural numbers by starting with two symbols "0" and "1" that are naturals, and saying that if n is a natural, then n+1 is too i.e. adding 1 over and over again. Part of the definition is each time we add 1, we get a number we haven't seen before; and so we have an infinite set by construction. And we can make bigger ones by taking the power set (the power set always has a larger cardinality then the set it comes from).
So infinite sets are definitely mathematical objects because we can (and just have) construct them.
Watch Eliezers response to this question, http://www.youtube.com/watch?v=3dufqGC8X8c
scroll to 4:40 I like his one argument: if we have finite neurons and thus cannot construct an infinite set in our "map" what makes you think that you can make it correspond to a (hypothetical) infinity in the territory?
I don't really see what this argument comes to. The map-territory metaphor is a metaphor; neural structures do not have to literally resemble the structures they have beliefs about. In fact, if they did, then the objection would work for any finite structure that had more members than there are synapses (or whatever) in the brain.
If he is saying that infinite sets are a mathematical impossibility then he is wrong.
But I'm fairly sure that he is saying they are a physical impossibility. Which is not at all unreasonable. (this is the "territory" I think he is talking about)
I have a feeling we are working with different definitions of the "mathematics". I think your definition of "mathematics" might be "symbols that occur in physics and can be manipulated to give answers about the universe".
My definition is something like "set of axioms => conclusions about the structure of the object generated by the axioms" (which includes things like the real numbers, which gives calculus, so the first version of "mathematics" is included the second).
No. You have a rule that hypothetically would produce an infinite set if applied ad infinitum. This may seem like nitpicking but there is a difference between the concept of an infinite set and an actual infinite set, the latter can't be represented in a finite brain(I suppose).
I can write down the rules of a turing machine, but this doesn't produce a working computer to spring to life if you get my point.
Yep, exactly; no problem with that, that's how mathematics works. There is only a problem if someone wants to write down every element of an infinite set.
This is mathematics. The concept of a mathematical object is the object, because the "concept" version satisfies all the same rules (axioms) as any "actual" version, and these rules completely describe its structure, and (broadly) mathematics is the study of structure/patterns.
One does not need a physical basis for these rules, and so one does not need a physical basis for structures generated by such rules.