Ask HN: I want to learn what mathematics is really about
11 comments
> I want answers to all these questions in a comprehensive way.
IMO what you need is an overview of the philosophy of mathematics/logic (as someone pointed out)
Being a philosophy, you should not expect definite answers. Mathematics itself is a historical process, so there are always trends and fashions to be aware of.
To get a quick comprehensive map, I personally always start with online articles (before books) on encyclopaedias like Plato [0] or even Wikipedia [1].
>Why certain things can be proved and others not?
One of the most captivating events of the 20th century is how Kurt Gödel's incompleteness theorems "destroyed" Hilbert’s dream of creating pure formalized foundations of all mathematics (without paradoxes and inconsistencies). I don’t think we ever fully comprehended the implications of this.
> how did Computer science or computation emerge as a branch of mathematics
Besides Gödel’s incompleteness, there is Turing's completeness. Turing Machine is the associated model of computation. These models are mathematical abstraction of computers (see "Theory of computation")
[0] https://plato.stanford.edu/entries/philosophy-mathematics/
[1] https://en.wikipedia.org/wiki/Category:Philosophy_of_mathema...
IMO what you need is an overview of the philosophy of mathematics/logic (as someone pointed out)
Being a philosophy, you should not expect definite answers. Mathematics itself is a historical process, so there are always trends and fashions to be aware of.
To get a quick comprehensive map, I personally always start with online articles (before books) on encyclopaedias like Plato [0] or even Wikipedia [1].
>Why certain things can be proved and others not?
One of the most captivating events of the 20th century is how Kurt Gödel's incompleteness theorems "destroyed" Hilbert’s dream of creating pure formalized foundations of all mathematics (without paradoxes and inconsistencies). I don’t think we ever fully comprehended the implications of this.
> how did Computer science or computation emerge as a branch of mathematics
Besides Gödel’s incompleteness, there is Turing's completeness. Turing Machine is the associated model of computation. These models are mathematical abstraction of computers (see "Theory of computation")
[0] https://plato.stanford.edu/entries/philosophy-mathematics/
[1] https://en.wikipedia.org/wiki/Category:Philosophy_of_mathema...
Don't overthink it. Mathematics is just a conceptual tool we've invented to help us think about and solve problems. Nothing more. Language is another such tool.
Now, we could invent a conceptual tool to have arbitrary meaning, but without rigid definitions, symbols have no meaning, and therefore they cease to be useful for solving/comprehending anything.
I mean, say the word "XUNS" could mean anything, be it "cat", "sovereign", "hallelujah" or whatever. And I just say "XUNS XUNS XUNS XUNS XUNS". Would you know exactly what I said? No it is nonsense, because, without out giving a strict definition it could mean anything.
So by ascribing rigid definitions/meaning to things, we can now reason about them in sensible ways. Which is what mathematics does, and why it is useful tool for proving/comprehending something.
As for the axiomatic method... it produces truths, mainly because it's built beforehand upon pre-defined truths. By this I mean, if I define an inch to be so long, and measure an object to be 3 inches, then the statement that "the object is 3 inches in length" is true, because we defined it so. If instead, you changed the definition of an inch to be any arbitrary length, then it ceases to be useful, and we can no longer prove exactly how many inches the object is in length.
Now, we could invent a conceptual tool to have arbitrary meaning, but without rigid definitions, symbols have no meaning, and therefore they cease to be useful for solving/comprehending anything.
I mean, say the word "XUNS" could mean anything, be it "cat", "sovereign", "hallelujah" or whatever. And I just say "XUNS XUNS XUNS XUNS XUNS". Would you know exactly what I said? No it is nonsense, because, without out giving a strict definition it could mean anything.
So by ascribing rigid definitions/meaning to things, we can now reason about them in sensible ways. Which is what mathematics does, and why it is useful tool for proving/comprehending something.
As for the axiomatic method... it produces truths, mainly because it's built beforehand upon pre-defined truths. By this I mean, if I define an inch to be so long, and measure an object to be 3 inches, then the statement that "the object is 3 inches in length" is true, because we defined it so. If instead, you changed the definition of an inch to be any arbitrary length, then it ceases to be useful, and we can no longer prove exactly how many inches the object is in length.
I recommend reading histories of mathematics including more recent histories like Morris Kline's "Mathematics: the loss of certainty." According to other books by Reuben Hersh, such as "the Mathematical Experience" and "What is mathematics really?", mathematics can be seen as a social activity where what gets studied and what gets accepted as proven has a definite relationship to human existence and human social life. If you agree with this, even in part, history of math becomes important.
A few words on your questions:
Mathematics is no different from any other field except that its objects of interest are all abstractions. As an engineer you use it, so it appears to be about number and symbol pushing, but the theory of how and where to push the numbers and symbols and why it works was discovered by a mathematician at some point. The proofs are not ultimate but hopefully convincing given certain usually acceptable assumptions or axioms. We like the axiomatic method because axioms are usually not controversial and everything proceeds from them using strict logical reasoning.
A few words on your questions:
Mathematics is no different from any other field except that its objects of interest are all abstractions. As an engineer you use it, so it appears to be about number and symbol pushing, but the theory of how and where to push the numbers and symbols and why it works was discovered by a mathematician at some point. The proofs are not ultimate but hopefully convincing given certain usually acceptable assumptions or axioms. We like the axiomatic method because axioms are usually not controversial and everything proceeds from them using strict logical reasoning.
My think about it is this, Math is a language that allows formal declaration of relationships between different spaces. When I say space I mean any sized dimensional system that has some regularity to it. So you can have scalar values like your age or descriptive values like the kind of things you might find collected from your Facebook page or maybe spatial things like where you are located or how your body is distributed about space. Math let's you say things like "X changes by A amount with respect to Y". It let's you define how the space exists and how it changes. It let's you transform the space from one domain to another. For example a picture is a collection of pixels with RGB values, but when you look at it you see a dog. It's likely there are bits and pieces of the picture that when taken together describe a dog that you could tell us a dog without the picture anymore (four legs, a tail, fur, etc).
I think, as a physicist and a scientist my view of what math is, is very data driven and concept oriented. Others probably have different ideas. I'm not even sure I like the description I've offered here haha. But I tried to keep it as simple as possible.
I think, as a physicist and a scientist my view of what math is, is very data driven and concept oriented. Others probably have different ideas. I'm not even sure I like the description I've offered here haha. But I tried to keep it as simple as possible.
The use of math in science comes with a larger emphasis on the semantic (sign-objects relations).
Interestingly pure math and applied math both have an interpretation focus (pragmatics or sign-object-interpreter relations) in that elegance and "interestingness" tends to play a role.
So I won't say pure math is only syntax but sometimes it get interpreted that way.
Interestingly pure math and applied math both have an interpretation focus (pragmatics or sign-object-interpreter relations) in that elegance and "interestingness" tends to play a role.
So I won't say pure math is only syntax but sometimes it get interpreted that way.
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Can you describe the level of your mathematical maturity?
I don't know of a single book that captures all that (there probably is one though).
I don't know of a single book that captures all that (there probably is one though).
I have done my bachelors and masters in electrical and communication engineering. I know calculus, matrix algebra and prob stats on a level of engineering. Not in much depth.
There are philosophy of math books like Joel David Hampkins. But given the breath of your question you might be better served by just a general philosophy text. Like something covering an introduction to metaphysics or even better 'philosophical logic'.
A more rigorous approach would be to study mathematics as mathematics (and not through traditional philosophy). In that case you would have to go down the path of mathematical logic, proof theory and model theory. Then after that fall into a pure theoretical CS / metamathematical rabbit hole.
A good intro to formal logic is "forallx". The key to truth in logic being the concept of tautology.
There is also the cognitive linguistics approach of focusing on the genesis of these abstractions from underlying metaphors. There is a famous book by George Lakoff on this subject.
More generally my favorite wider contextualization of the subject of reason, meaning and formalism (though it might seem distant and antiquated) is C S Peirce and his triadic approach to meaning and philosophy.
I have a very simple article on communication, reason and philosophical logic that gets across this triadic perspective to seeing these things: https://medium.com/the-sphinx/what-is-meant-by-a-reason-part...
A more rigorous approach would be to study mathematics as mathematics (and not through traditional philosophy). In that case you would have to go down the path of mathematical logic, proof theory and model theory. Then after that fall into a pure theoretical CS / metamathematical rabbit hole.
A good intro to formal logic is "forallx". The key to truth in logic being the concept of tautology.
There is also the cognitive linguistics approach of focusing on the genesis of these abstractions from underlying metaphors. There is a famous book by George Lakoff on this subject.
More generally my favorite wider contextualization of the subject of reason, meaning and formalism (though it might seem distant and antiquated) is C S Peirce and his triadic approach to meaning and philosophy.
I have a very simple article on communication, reason and philosophical logic that gets across this triadic perspective to seeing these things: https://medium.com/the-sphinx/what-is-meant-by-a-reason-part...
Godel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter is specifically about this. It's a big book with a lot of other ideas, just take your time and you'll get there.
Why certain things can be proved and others not? And in between all these how did Computer science or computation emerge as a branch of mathematics that had the confidence of a mountain that it can solve all issues of humanity even with the warning that all problems can't be proven.
I am myself confused what I am asking about. But, I want answers to all these questions in a comprehensive way.
I will love book suggestions that have answers to all my questions and maybe more.
Edit: I have done my bachelors and masters in electrical and communication engineering. I know calculus, matrix algebra and prob stats on a level of engineering. Not in much depth.