- #1
res3210
- 47
- 0
Hey guys,
I have been interested in formalistic mathematics for a while, about a year now. Every time I read a formalistic book on math (Principles of Mathematical Analysis by Rudin is a great example) I never understand how mathematicians develop the structure they present in the books. And since the books are presented very structurally, e.g. posit a definition, the definition is followed by a theorem, which is followed by a proof, I am struck with the impression that mathematicians run around stating definitions, whereby theorems magically fall out, upon which they are struck by some stroke of genius with a proof. This seems very romantic and very implausible to me. So I am wondering, how exactly do mathematicians construct theorems? Do they lay down rules for some algebra (or any other fomalistic method of logical thinking) and play with it until they can deduce some insightful conclusions? Or does it indeed happen as above, where they are given some natural gift which allows them insight into the inner workings of human though processes? Does the theorem come first, from which they can deduce definitions and explanations as to why it is so? Or does it come about because someone says "hey, this would be useful, maybe I can prove this. What definitions would be required and how can I show that this follows from them?" Any kind of insight would be greatly appreciated, and while you are at it, feel free to share how you think about constructing formalistic, logical thought, if it is to your fancy.
RS
I have been interested in formalistic mathematics for a while, about a year now. Every time I read a formalistic book on math (Principles of Mathematical Analysis by Rudin is a great example) I never understand how mathematicians develop the structure they present in the books. And since the books are presented very structurally, e.g. posit a definition, the definition is followed by a theorem, which is followed by a proof, I am struck with the impression that mathematicians run around stating definitions, whereby theorems magically fall out, upon which they are struck by some stroke of genius with a proof. This seems very romantic and very implausible to me. So I am wondering, how exactly do mathematicians construct theorems? Do they lay down rules for some algebra (or any other fomalistic method of logical thinking) and play with it until they can deduce some insightful conclusions? Or does it indeed happen as above, where they are given some natural gift which allows them insight into the inner workings of human though processes? Does the theorem come first, from which they can deduce definitions and explanations as to why it is so? Or does it come about because someone says "hey, this would be useful, maybe I can prove this. What definitions would be required and how can I show that this follows from them?" Any kind of insight would be greatly appreciated, and while you are at it, feel free to share how you think about constructing formalistic, logical thought, if it is to your fancy.
RS