Why don't mathematicians introduce intuition behind concepts as physicists do?

Question

First of all please don't be angry - if anyone might be - and thoughtlessly downvote this post. I'll make it clear that I'm not here to criticise mathematicians - but rather to understand.

I understand the importance and significance of accurate statements of definitions and theorems and their respective proofs.

I read some mathematical methods in physics books and felt that they lack systematic approaches to concepts. They are more like well-written novels with a good plot.

So I always refer to some (relatively) more rigorous books that discuss each area of mathematical methods in more detail. These are usually books with titles subject name + "with applications" or something like that.

But at the same time, I felt that all of these proper maths books only contain, from the beginning till the end, definition - theorem - example - proof - remark kind of approach. Honestly, I find them very boring yet solidly written.

I saw maths department professors take the identical approach in their classes. Here comes what I think is a more serious problem - such ways of teaching have no intuition.

I'll compare two different ways of explaining a single concept of differentiation.

The first way I would do is to introduce the $\varepsilon-\delta$ definition of a derivative and provide proofs of various theorems about it - like linearity, product rule, quotient rule, chain rule and so on.

The second way I would do is to initially very informally introduce the notion of tangent vector, differential forms $dx, dy$, linear approximation and Taylor expansion. And then talk about how it is historically originated from Newton's study of mechanics, and how it is contemporarily applied to various optimisation problems. And then start dealing with the basic proofs of facts, perhaps by exploiting $\varepsilon-\delta$.

I don't see how the first way of teaching has any pedagogical advantage which is how mathematicians are doing mathematics. It definitely trains you to have a rigorous understanding. However, it does not only de-motivate students but also stops them from taking the intuitions out of it.

So, my questions come.

1. How come mathematicians have intuitions though? I felt that I would never be able to have an intuitive understanding if this was the only way in which I was taught mathematics.

2. Why do mathematicians hate to include intuitions? To phrase it a bit more offensively, what will then be the point of countless repetitions of theorems and proofs if there's no motivation? Why not nicely mix intuition and rigorousness?

Answer 1

The question is a reasonable one. I think part of the issue is that there are secretly a few types of math texts and typically it's not clear which book is of which type.

1. Some are written for people who "already know the material" but need to either fill in minor gaps or have a reference for some proofs. These will typically be written concisely with not so much motivation, because the target audience would not need these things as much.
2. Some of them are written for the newcomer. These texts will typically have more filler text and background information. Often, they have sections discussing motivation and historical context. More often than not, these texts are directed at undergraduates.

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Opinion: My personal opinion is that motivations are best learned from asking friends working in the respective fields, though I do think it is important to have these motivations written down in some places. However, it does seem that sometimes a little too much emphasis on carrying out the details of proofs is present in some math courses. I feel that the most precious information that a lecturer can impart is the context and motivation of a field - after all, the proofs are usually in many different books.

It can be quite important to understand historical context in learning math. For instance, it's easier to understand the definition of a "scheme" if you understand at least a little bit the classical theories that preceded schemes and what problems the definition was meant to address.

Answer 2

Given that physics is all about converting physical problems into mathematical problems and the latter's solutions back into physical solutions, what you propose isn't in concert with physicists' true motive for introducing such intuitions. It's not about augmenting the opportunities for understanding that proofs provide. It's about several factors (which pure mathematics doesn't succumb to due to not being empirical). I'll present a by no means complete list:

• Helping us decide what the axioms should be in the first place, by reminding us of the real-world explanatory power we seek;
• Reminding us of the older physical theory we must improve upon;
• Identifying where the theory connects to something we can test against observation;
• Critiquing the current state of our understanding, either as the scientific community or students who haven't learned all the community knows, so we can direct future efforts, which will often involve theoretical refinements;
• Comparing the practical benefits of specific notations, calculation techniques etc.;
• Knowing which approximations our models should take in which regimes, especially where this explains qualitative phase changes in response to continuous changes around values either observation or calculation shows to be critical.