Book on how to define ideas rigorously

[CGandC]

I know the logic of proving/disproving mathematical statements, I learned it by reading books, texts regarding to the matter, lots of exercises ( in the subject of how to do mathematical proofs and in the subject of proving/disproving statements in my math courses [ e.g. Linear Algebra, Real Analysis, Discrete Mathematics... ] ) , lots of mistakes and lots of questions as to where I've been wrong.
But besides the Proof side of mathematics there's the art of defining ideas in a mathematical fashion/rigorous fashion ( like the definition of limit of a sequence ) which feels to me as being doable by someone who is trained in doing mathematical proofs but also requires some experience of taking ideas and attempting to formulate them/define them ( this is an aspect of some people who are researching mathematics would try to gain experience in).

One book that I consider the holy grail of learning to do mathematical proofs ( which helped me tremendously ) is How to prove it by Velleman. ( Another helpful book I've read some parts of it is Polya's book )
I learned all the topics in the book, did almost all of the exercises, I've been doing math courses which are very proof-oriented like Real Analysis, Set Theory..., all of which are close hand to the mathematician's framework of proofs, rigour and language. But sometimes when I want to take a concept and formulate it, I feel stumbled ( I did learn about formulation with regards to learning predicate logic. But I'm talking about deeper stuff, how to be able to formulate constructions/ideas either for the sake of themselves or for the sake of solving a problem, such as the one provided in the answer in this question: https://cs.stackexchange.com/questions/54990/filling-bins-with-pairs-of-balls)

I don't know of any book that teaches you how to take abstract ideas and define/formulate them in a rigorous/mathematical fashion, and I'm interested in finding such a book to learn from, do you know of any such book that you can recommend about? or do you have a recommendation as to how to get better at this skill? ( because just doing proofs doesn't help )

 

[Office_Shredder]

The answer seems like a pretty standard combinatorial proof. You should just study some graph theory if you want to see more examples of problems with similar solutions.

You're not going to find a book that teaches you to turn this question into a graph theory question, that just comes from experience. But this was a combinatorial question and a combinatorial answer, so nothing magical here happened other than the person answering the question is smart and experienced in the area.

 

[CGandC]

But how one would come up with definitions of for example, limit and cardinality before having experience in Real Analysis? how would one come up to define the fruit Banana in terms of its shape? how one would formulate a process of me doing A then B then C... . Is there a book on this subject?

It's like one of those cases when the math professor explains what the intuition is behind some proof he's about to do, and then he goes on and formulates his proof ideas and writes a nice, elegant proof.

Basically, I would like to be like my professor and reach to such level of formulating my intuitive ideas rigorously - as much as I can, without having any a-priori knowledge on the subject ( or maybe even If I do have a knowledge in the subject and I want to come up with something new that relates to the current field of research/interest ), I wondered if there was some source of knowledge that helps gaining some level of grasp of how such processes can be carried out.

I don't think there's a strict answer to my question because from what I've experienced myself, only If I have a solid, thorough understanding and control of a mathematical/physical theory, only then I can formulate ( or at-least reach to half formulation ) some ideas that have a relation to the subject.
But it seems to me like a problem that physicists and mathematicians would face whenever they are trying to come up with new theories/definitions/lemmas, but what is the process of coming up with these ideas?

 

[Mark44]

But how one would come up with definitions of for example, limit and cardinality before having experience in Real Analysis?

The ideas and rigorous definitions that underlie limits took a long time to develop, culminating in the definitions involving $\delta$ and $\epsilon$ as well as sequences, refining the rather vague ideas about numbers being "close" to some other number. In short, as real analysis was developed, better definitions came out of that work.

how would one come up to define the fruit Banana in terms of its shape?

One would need a good understanding of geometry/analytic geometry as well as knowledge of a variety of graphs of functions.

how one would formulate a process of me doing A then B then C... . Is there a book on this subject?

AFAIK, there isn't such a book, but if you can break down a complex process into a sequence of steps, then the task is to figure out what needs to be done for task A, then task B, and so on.

don't think there's a strict answer to my question because from what I've experienced myself, only If I have a solid, thorough understanding and control of a mathematical/physical theory, only then I can formulate ( or at-least reach to half formulation ) some ideas that have a relation to the subject.

That's what I think, as well.

But it seems to me like a problem that physicists and mathematicians would face whenever they are trying to come up with new theories/definitions/lemmas, but what is the process of coming up with these ideas?

There really are two things here: theories/theorems and definitions. Anyone can make a definition however they want, but only those that are useful in some sense will stand the test of time.

Lemmas can be thought of as subtasks or steps that lead up to a theorem that uses them. In a way, this is like the A, B, C, ... that you mentioned before.

 

[robphy]

possibly useful?
https://en.wikipedia.org/wiki/Abstract_nonsense
https://en.wikipedia.org/wiki/Category_theory

 

[fishturtle1]

Topology through Inquiry by Starbird and Su.

 

[romsofia]

Basically, I would like to be like my professor and reach to such level of formulating my intuitive ideas rigorously

This is called getting a doctorate. Getting a doctorate isn't "more coursework" or "reading papers", you are being trained by a professional to do exactly this: formulate ideas rigorously in the attempts to advance your field in a given direction. So no, don't expect a book to spell out ABC, boom now you can research. Each field has a different methodology and expectations. You learn your craft from your advisors. All these subtle issues, and doubts, get addressed in the doctoral program. You learn to be confident in research methodology, that is formulating intuitive ideas rigorously.

If you are not at the point in your life where you can commit to getting one, then you can try your best on your own. Which in a nutshell is: observe something, read papers in the area to see what has been done, ask rigorous questions that you see aren't addressed in the papers (hard part, this is why you have to be trained because not all questions are fruitful in research, you can spend weeks on something that doesn't matter so that's where the advisor steps in), answer said questions (again, formulating the answer is also hard, which is where an advisor comes into play), look to your peers to see if they agree. But who are your peers? Is the question your addressing just an underlying assumption your field makes? How do you submit a paper to a journal, and which journal do you choose?

And oh boy, what happens if you can't find ANY papers on it? Is that because it's truly a novel idea, or do you just not know how to find niche papers correctly? Are you looking in the right journals? Do you know the bigger names in the field, maybe they have some ideas, or seen a paper addressing it... you see the point? Lot's of subtle issues on just starting!

If you truly believe in your base, and I'm not sure if you're still an undergraduate, self studying, etc. But just ask different groups at your university. If you're an undergraduate, ask if you can sit in on their seminars and get involved. Ask your professors if they know of any research opportunities. That's truly the best way to learn any craft, just start doing it under the guidance of those who can.

If you're self studying, reach out to former professors, check your university groups, see if there is a group you're interested in, and contact them asking to sit in on their seminars (always ask for permission, because you could just go, but it'd be weird seeing a new face and no one knows who they are).

I get that is a long drawn out answer for a book recommendation, and I didn't recommend a book, but hopefully you get the point. Your professor can do this with ease because it's his career and ,usually, passion. He was trained to do it, so if you want to be like him, follow his path!