Question regarding proofs and theorems

[res3210]

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

 

[chiro]

Hey res3210.

I think you should be aware that there is a lot of intuition that has to be built up in mathematics before you can grasp the whole theorem/proof presentation. It's the same reason why when you start learning, you do it slowly with examples and applications rather than just going right into graduate analysis in your first year.

By the time you are in graduate school you are usually expected to have the intuition to understand the theorems and proofs in different ways including visually, symbolically, and mathematically. It is basically as compact as it gets but that compactness is expected to have some sort of redundancy with what you have done in the rest of undergraduate studies.

What you would probably find if you actually spoke with a lot of mathematicians are a lot of notes with drawings, ideas, and things that help lead to a proof and when something is published, then all of that is essentially stripped away in favor of the kinds of things you read in textbooks and journal articles.

In terms of proving and solving mathematical problems there are three main things that are always involved and they are transformation, representation, and constraints and you use these to go from one situation to another.

The representation deals with the structure of the information you are using. The transformations deal with how you go from expression (or set of expressions or relationships) to something else. The constraints define the state space involved.

If you can understand how these work in harmony across all of mathematics, then it will help you with solving mathematical problems and proving things.

For proving things though, you should remember that having experience in proving will be necessary to get better in this. Proving things mathematically requires a specificity that is not found in other endeavors and not even in the applied sciences or mathematics. Being specific is actually really hard and it requires practice to become proficient at it like say riding a bike or playing a sport.

What I would recommend is that if you do want to understand the theorem/proof style, then get other resources that go into the intuition of the subject and keep some separate diary or notebook that covers the intuition behind the ideas presented.

Always keep in mind that this is the most compact form of presenting the idea and when it is actually unpacked (as it were) then the book would probably be about four or five times bigger than it is. A lot of information is crammed into a small space and it does require a lot of unpacking to really make sense of it.

 

[Stephen Tashi]

So I am wondering, how exactly do mathematicians construct theorems?

A question that takes priority to that is how they construct definitions. They need the definitions before they construct any assumptions or theorems. Mathematics is a cultural activity. The history of how standard definitions arise has a sociological aspect. It reveals how some human beings have agreed to cooperate. The modern definitions of things like "limits", "vector spaces", "norms", "groups" evolved over many years. The modern definitions didn't come out of the brains of persons who weren't aware of previous efforts.

I suppose a mathematician that lives at a time when a particular field of mathematics has been sorted out by his predecessors has the definitions and assumptions at hand. There might be a few individuals that can function as you imagine - that is, their thought processes are completely formal and abstract. (A commentator on John Von Neumann says there there are "visual" mathematicians and "acoustic" mathematicians. Von Neumann was an "acoustic" mathematician. He didn't rely on a geometric pictures to think. It's reported that when he concentrated on a question, he often stood in a corner of a room and whispered to himself, as if language itself was his main tool.)

The formal presentation of mathematics has much to do with human cooperation. Mathematical papers need to be peer-reviewed. ; textbook writers need to have their books proof read. It is simpler for experts to check mathematical work that is concise and formal. People starting out in math may prefer articles that are chatty and informal, but it wouldn't be possible to peer-review informal mathematical papers because they would have some ambiguity about them. The formal "face" of mathematics is a portrait of how most people think about mathematics. It shows people's thinking after it has been reduced down to a concise and precise minimum.