4 ms·

I am baffled by this and feeling stupid

jordigh 12y ago
There's really no reason to feel stupid just because you don't know what the terminology means, if that is indeed the problem. Would you feel stupid if you were to walk into a Scientology meeting and you didn't know what SP, Xenu, or fair game meant? I wouldn't. It's just a bunch of jargon that I would have to pick up if I were really interested in knowing the underlying concepts being discussed (and I'm not, which makes me feel even less stupid for not wanting to learn this). And even if you know what all the words mean, it can take some time and effort to put them together, even for practitioners.
- westoncb 12y ago
  The situation is even trickier than that because you can keep on looking up these terms, one after another, but it's going to take a lot more than their particular definitions before things begin 'coming together' for someone. I think this largely comes down to the fact that there's no place people can go to learn what it is mathematics is, what it's about, how it works, etc.—and consequently, what mathematicians are interested in and why they have their particular quirks of formulation, and why certain points (which can seem utterly uninteresting to an outsider) are emphasized. Of course you can get this by doing a math degree (and maybe graduate work), but I'm just trying to describe the difficulty in teaching oneself. At least in the U.S., our exposure to mathematics in formal education (pre-university), gives exactly all of the wrong answers to those questions above, so that when you try to start teaching yourself you never know what direction to search in in order to resolve ambiguities.
gjulianm 12y ago
I wouldn't worry about that. I'm studying mathematics and I also feel stupid while reading new concepts on Wikipedia. The problem with mathematics is that, if you don't know the idea behind the concept or which kind of problem is it trying to solve, it's almost impossible to really understand the definitions. And even when you understand the concepts and ideas, it can still be difficult to understand the definitions. Few months ago, a professor told us the 100% formal definition of a function: it wasn't easy to understand and we all had this "wtf" face.
- jordigh 12y ago
  Yeah, the thing with formal definitions is that they're like little programs you have to compile before you can see their output. Trying to understand what a function is from a statement such as "a function from A to B is a subset F of the cartesian product AxB such that for all (a1,b1), and (a2,b2) in F, b1=b2 whenever a1=a1" really obscures the notion this is all about "mapping" or "doing" something. This definition is more like something that you would feed to a computer who doesn't "understand" what really is going on. Or to a human, who would work through it piecemeal, and try to build a few functions following this definition until the a-ha moment would come and this human would form a different intuitive understanding of what a function is or does. But the reason we write definitions as little programs to be compiled is to make sure that there can be no room for ambiguity. You compile the definition and arrive at the same mathematical object as anyone else. If we define a function as a mapping or a procedure, then we just shifted one word for another and still have to elucidate what a mapping or a procedure is.
  - ihm 12y ago
    Well, the set theoretic attempt to describe functions isn't a very nice one. Type theory gives a formal definition closer to the intuitive notion: a function from A to B is a program of type A -> B. I.e., a rule for turning things of type A into things of type B.
    - jordigh 12y ago
      I admit I don't know much about type theory. I assume "program" is a term of art? Also, does this allow for functions that have literally undescribable rules, such as a function arising from the application of the axiom of choice? Wanting to allow such undescribable functions is the reason for the set theoretic definition. I assume "rule" isn't a term of art? What's the book to read for this stuff?
      - ihm 12y ago
        Yes, program is a term of art. It means a "term" in the type theory under consideration. A type theory is essentially a programming language with a fancy type system, and there are many type theories, just as there are many programming languages. It does allow for functions without computable rules. In the case of the axiom of choice, one postulates as an axiom a program with a type that says something like "give me a type A and an A-indexed family of types, P : A -> Type, such that for all a:A, the type P a is non-empty, and I'll give you a function f : (a : A) -> P a". As far as I know, there is still no one comprehensive book on type theory. I think the type theory parts of the homotopy type theory book (https://hottheory.files.wordpress.com/2013/03/hott-online-611-ga1a258c.pdf https://hottheory.files.wordpress.com/2013/03/hott-online-61...) are quite good. Chapter 1 gives the basics of dependent type theory. Chapter 2 is about the groupoid interpretation of type theory and is quite nice. Chapter 3 is about "logical things" in type theory (things like axiom of choice). Chapter 5 is about inductive types which are quite central to type theory (these are generalizations of algebraic data types like trees and lists from functional programming languages).

There's really no reason to feel stupid just because you don't know what the terminology means, if that is indeed the problem. Would you feel stupid if you were to walk into a Scientology meeting and you didn't know what SP, Xenu, or fair game meant? I wouldn't. It's just a bunch of jargon that I would have to pick up if I were really interested in knowing the underlying concepts being discussed (and I'm not, which makes me feel even less stupid for not wanting to learn this). And even if you know what all the words mean, it can take some time and effort to put them together, even for practitioners.

westoncb 12y ago
The situation is even trickier than that because you can keep on looking up these terms, one after another, but it's going to take a lot more than their particular definitions before things begin 'coming together' for someone. I think this largely comes down to the fact that there's no place people can go to learn what it is mathematics is, what it's about, how it works, etc.—and consequently, what mathematicians are interested in and why they have their particular quirks of formulation, and why certain points (which can seem utterly uninteresting to an outsider) are emphasized. Of course you can get this by doing a math degree (and maybe graduate work), but I'm just trying to describe the difficulty in teaching oneself. At least in the U.S., our exposure to mathematics in formal education (pre-university), gives exactly all of the wrong answers to those questions above, so that when you try to start teaching yourself you never know what direction to search in in order to resolve ambiguities.

gjulianm 12y ago

I wouldn't worry about that. I'm studying mathematics and I also feel stupid while reading new concepts on Wikipedia. The problem with mathematics is that, if you don't know the idea behind the concept or which kind of problem is it trying to solve, it's almost impossible to really understand the definitions. And even when you understand the concepts and ideas, it can still be difficult to understand the definitions. Few months ago, a professor told us the 100% formal definition of a function: it wasn't easy to understand and we all had this "wtf" face.

jordigh 12y ago
Yeah, the thing with formal definitions is that they're like little programs you have to compile before you can see their output. Trying to understand what a function is from a statement such as "a function from A to B is a subset F of the cartesian product AxB such that for all (a1,b1), and (a2,b2) in F, b1=b2 whenever a1=a1" really obscures the notion this is all about "mapping" or "doing" something. This definition is more like something that you would feed to a computer who doesn't "understand" what really is going on. Or to a human, who would work through it piecemeal, and try to build a few functions following this definition until the a-ha moment would come and this human would form a different intuitive understanding of what a function is or does. But the reason we write definitions as little programs to be compiled is to make sure that there can be no room for ambiguity. You compile the definition and arrive at the same mathematical object as anyone else. If we define a function as a mapping or a procedure, then we just shifted one word for another and still have to elucidate what a mapping or a procedure is.
- ihm 12y ago
  Well, the set theoretic attempt to describe functions isn't a very nice one. Type theory gives a formal definition closer to the intuitive notion: a function from A to B is a program of type A -> B. I.e., a rule for turning things of type A into things of type B.
  - jordigh 12y ago
    I admit I don't know much about type theory. I assume "program" is a term of art? Also, does this allow for functions that have literally undescribable rules, such as a function arising from the application of the axiom of choice? Wanting to allow such undescribable functions is the reason for the set theoretic definition. I assume "rule" isn't a term of art? What's the book to read for this stuff?
    - ihm 12y ago
      Yes, program is a term of art. It means a "term" in the type theory under consideration. A type theory is essentially a programming language with a fancy type system, and there are many type theories, just as there are many programming languages. It does allow for functions without computable rules. In the case of the axiom of choice, one postulates as an axiom a program with a type that says something like "give me a type A and an A-indexed family of types, P : A -> Type, such that for all a:A, the type P a is non-empty, and I'll give you a function f : (a : A) -> P a". As far as I know, there is still no one comprehensive book on type theory. I think the type theory parts of the homotopy type theory book (https://hottheory.files.wordpress.com/2013/03/hott-online-611-ga1a258c.pdf https://hottheory.files.wordpress.com/2013/03/hott-online-61...) are quite good. Chapter 1 gives the basics of dependent type theory. Chapter 2 is about the groupoid interpretation of type theory and is quite nice. Chapter 3 is about "logical things" in type theory (things like axiom of choice). Chapter 5 is about inductive types which are quite central to type theory (these are generalizations of algebraic data types like trees and lists from functional programming languages).