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Ah, it is fun to be belligerent on the Internet. But I shall answer in good faith. I disagree with… well, what could I possibly be disagreeing with except the
by Smaug123 5y ago
Ah, it is fun to be belligerent on the Internet. But I shall answer in good faith.
I disagree with… well, what could I possibly be disagreeing with except the primary thesis of the comment to which I was replying, namely that mathematics is solely symbolic reasoning. More precisely, "Math is the act of inventing sets of rules for manipulating symbols that produce interesting or useful results, or exploring the behavior of sets of rules invented by others.".
I believe I have demonstrated both with analogies and specific examples what descriptions you were mistaking for the things themselves. I believe you are blinded by the fact that we communicate about maths using well-defined formalisms (drawings of graphs, logical formulae, definitions, theorems), and are missing the fact that we don't necessarily think in those formalisms; and I assert that thinking about the objects referred to by those formalisms is still performing mathematics.
In fact I believe it is you who is claiming the authority over what the natural numbers "really are", because you assert that one cannot think about the natural numbers without adhering to formalisms! I gave both an informal notion and one possible casting of that informality into a formal system. But my point stands precisely as stated if you simply change "what they really are" to "a more general way of thinking of them, outside the ordinals formalism"; you have certainly done an excellent job of quibbling away the actual points. I did assert clumsily that I had the one true way of thinking about the natural numbers, and I shouldn't have done so; my argument certainly doesn't rely on that assertion.
- lisper 5y ago> I assert that thinking about the objects referred to by those formalisms is still performing mathematics. OK. So can you give me an example of thinking about something such that that act of thinking isn't "performing mathematics"?
- Smaug123 5y agoCertainly! I am not performing mathematics when I am reading a fiction book and (unconsciously or consciously) mentally simulating the actions and internal states of the characters. I never said anything like "all thought is mathematics", merely "not all mathematics is symbolic thought".
- lisper 5y ago> I am not performing mathematics when I am reading a fiction book How do you know? You said that thinking about the objects referred to by mathematical formalisms is performing mathematics. How can you be sure that your fiction book can't be described by some mathematical formalism, but you are simply as yet unaware of it?
- Smaug123 5y agoMaybe you'll come along in ten years and convince me that I was performing mathematics all along! Maths has expanded to embrace other fields in the past. Archimedes probably would have called work on the nature of truth "philosophy" and very much not "mathematics". I, for one, do not wish to adopt any stance that rules out future progress.
- lisper 5y agoI don't have to wait ten years. Unless you reject the Church-Turing thesis, then on your definition everything is math, which makes your definition vacuous.
- Smaug123 5y agoI shall quote myself: > I never said anything like "all thought is mathematics", merely "not all mathematics is symbolic thought".
- lisper 5y agoWe're going around in circles. If not all mathematics is symbolic thought, then, unless you're a dualist, there is nothing that can be meaningfully distinguished as "not mathematics" (because Turing completeness) and the whole concept of mathematics becomes vacuous and useless.
- Smaug123 5y ago