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I disagree, and I believe you have made a category error: you are mistaking the description of a thing for the thing itself. I believe your error is the same as

I disagree, and I believe you have made a category error: you are mistaking the description of a thing for the thing itself. I believe your error is the same as that of the one who asserts "the natural numbers are the finite transitive sets". In fact, the finite ordinals are only one possible implementation of the natural numbers in set theory, and what the natural numbers really are is "the abstract notion of the smallest structure you can do induction on". Similarly, any given symbolic representation of a thing is an implementation of that thing in mathematics, but it need not be the thing itself. Perhaps our best (or even only) hope of automatic mathematics verification is to find a symbolic representation of a thing, and there are some areas where we have in some sense "definitely found the right symbols" which cleave very tightly to the thing they represent; and it's probably always true that if you can phrase something in the symbolic language, then you will be more rigorously constrained to say correct things by construction. But when an actual human actually does mathematics, they may very well not be manipulating symbols at all until they come to the point where they must transmit their thoughts to someone else. In fact, in some cases it may even be possible to transmit nontrivial mathematical thoughts without symbols; one possible method is simply "analogy". To revisit the example at the start, by the way: If you choose the language of category theory, of course, you can formalise "the abstract notion of the smallest structure you can do induction on" into the symbolic representation whose name is "natural numbers object", and thereby claim that once more we are in the world of rules for manipulating symbols. But I do not believe the implied claim that "unless you think of the naturals as their category-theoretic representation, then you are not doing mathematics"!

lisper 5y ago
> I disagree With what? > you are mistaking the description of a thing for the thing itself How? > what the natural numbers really are is "the abstract notion of the smallest structure you can do induction on". And who granted you the authority to define what the natural numbers "really are"? The last time I checked, numbers were not actually part of physical reality, they were a kind of concept or idea. People are free to attach whatever labels they like to concepts and ideas. They can even overload terms to mean different things in different contexts. So just because I use the word "number" in some colloquial context to stand for a different idea than you do in some other context doesn't make it wrong.
- 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 ago
      Certainly! 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 disagree With what? > you are mistaking the description of a thing for the thing itself How? > what the natural numbers really are is "the abstract notion of the smallest structure you can do induction on". And who granted you the authority to define what the natural numbers "really are"? The last time I checked, numbers were not actually part of physical reality, they were a kind of concept or idea. People are free to attach whatever labels they like to concepts and ideas. They can even overload terms to mean different things in different contexts. So just because I use the word "number" in some colloquial context to stand for a different idea than you do in some other context doesn't make it wrong.

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 ago
    Certainly! 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".