the 3 philosophy of #math of 1900s are: ① logicism. Math is pure derivation of logic. ② Hilbert's formalism. Math is just bunch of fomulaic symbols. No meaning. ③ intuitionism. Math is mind's construction.

these 3 schools are the basis of foundations of math. Each has its problems. logic became formal (i.e. symbolic) logic, which is basically formalism. They in turn, became constructivism (a variation of intuitionism), so it can run by computer.

these 3 schools are the basis of foundations of math. Each has its problems. logic became formal (i.e. symbolic) logic, which is basically formalism. They in turn, became constructivism (a variation of intuitionism), so it can run by computer.

- People say "constructivism", not "constructionism". So, if you prove something exists, mathematicians may ask if you proof is "constructive", meaning that you actually exhibit an example.5w
- Xah Lee+1thanks for correction.

i've always wondered what's the difference of logicism and formalism. Back in 1990s, I was introduced to them by Russell. (my top 3 fav author) Tried to read about them now and then. In 2000s, wikipedia is still not great, but in past year, i read a lot math again, and i seem to get some understanding. Is my characterization roughly correct?

i never cared about intuitionism, until now, and now i very much appreciate constructivism. As i understand, it is what mechanical manipulation can actually build.

PS few days ago i tried to read wikipedia about category theory again. Again, came away as incomprehensible abstraction. Meant to write a rant about it... but here's a gist...

so i spend half hour thinking, what does abstraction mean? After all, numbers 1 2 3 ... are abstraction to begin with. But then we have equations, such as 2*x+3 = 4, which is abstraction of description of math problems. Then, abstract algebra and 1800s's math, are the 2nd stage of abstraction. e.g. abstract algebra came from the systematic formal maniputlation of equations. And likewise stuff in algebraic geometry e.g. variety, and differential geometry geometry e.g. manifold, and so on in other branches of math.

Then, what's the next level abstraction? I'm thinking, scheme and sheaf etc (which i have no idea what they are), or, the category theory stuff.

but anyway, so i was reading Wikipedia on category theory.

https://en.wikipedia.org/wiki/Category_theory

in other math articles, say, holomorphic function, homotopy, hilbert space, riemann sphere, homomorphism, etc i've recently read, there's a cold definition. Most of the time, i can at least understand the definition, and go on from there. (in the above examples, i can also appreciate what they mean, why they are there, etc, except hilbert space.)

But the category theory, it begin with pages of pages of meta description. And am at loss. But perhaps, to appreciate it, one must first have solid understanding of various branches of math at graduate level?

but then, my first exposure to it is

Conceptual Mathematics: A First Introduction to Categories

by F William Lawvere, Stephen Hoel Schanuel

https://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/0521478170/

in 1990s. Which claimed to be written for laymen, and useful even outside of math. I recall, after reading 1 chapter, it's going too slow, and i was feeling, “what's the point?”. I never continued after the 1 chapter. (i don't have the book now. must gave away or something long ago.)5w - +John Baez when i heard about category theory in the 1990s, i was thinking, it's so nice: the math of analogy!

and few days ago i came across:

"If one axiomatizes relations instead of functions, one obtains the theory of allegories."

allegories!

love those names. ^_^5w - +Xah Lee - logicism is the belief that all mathematics can be reduced to "logic". This is tricky because it raises the question of what counts as "logic". For example, is 1+1=2 a statement in "logic"? I don't really care about the answer to that question anymore. But thanks to the logicists, we know many nice ways of understanding the meaning of statements like 1+1=2: we can define the natural numbers and addition using set theory, and 1+1=2 becomes a theorem of set theory. People still argue about whether set theory is part of logic... but again, I have trouble caring about that.

Formalists believed that all mathematics is the study of the consequences of formal rules - that is, rules for manipulating symbols. Thanks to the formalists we have many wonderful theorems about what can be done with various formal rules. Goedel's theorem and computers are ultimately spinoffs of this research.

Constructivists believed that mathematics is a mental act of constructing entities and working with these entities. Constructivists denied the validity of proof by contradiction and the axiom of choice, which allow one to prove things exist without actually constructing them. Thanks to constructivists we have intuitionistic logic, which became topos theory, which has now expanded to include homotopy type theory. Constructivism is very well-suited to working with mathematics using computers.5w - +Xah Lee wrote: " But perhaps, to appreciate it, one must first have solid understanding of various branches of math at graduate level?"

I've never tried to explain category theory to people who don't have a solid understanding of several branches of mathematics - typically algebra, topology and algebraic topology. Then one can compare these different branches, notice how the same patterns show up in each branch, and notice how some problems in one branch of mathematics can be solved by transferring them to another branch. Category theory was invented by algebraic topologists to make this "transfer" easy.

Another road to category theory goes through computer science. Have you watched "Categories for the Working Hacker"?5w - Xah Lee+1+John Baez thanks. will watch that video this time.

Watched some of Philip Wadler's video before, but never finished one.5w

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