Is it ever unnecessary to mathematically formalize a concept?

Question

From my understanding, mathematics sometimes gives rise to new physical/tangible laws and the converse is also true. In particular, physical phenomena give rise to new mathematics.

In all of the cases that I have seen, the mathematics is usually formalized. That is, definitions, lemmas, theorems and their proofs are developed in either of the two cases mentioned above.

Are there ever cases where formally defining physical phenomena in mathematical language is unnecessary?

For example, deep learning has recently been formalized. Is this formalism necessary for developing new techniques in the field? I can see how it would make sense if the formal theory was developed first.

Answer 1

Your question is

Are there ever cases where formally defining physical phenomena in mathematical language is unnecessary?

It is never possible to define physical phenomena directly in mathematical language. Mathematics (and even sciences) can deal with real phenomena only through models.

To deal with a model mathematically, it must be a mathematical model, described in mathematical terms.

I think all mathematical models go back to real needs -- even if very indirectly, through many layers of abstraction and generalization.

So, at least in applications of mathematics, an important goal is to faithfully and efficiently model real phenomena of interest, so as to be able to make accurate enough predictions about the real phenomena.

The clarity and effectiveness of mathematical presentation are of course very important. Therefore, formalization can hardly ever hurt, unless it is taken to such an extent as to actually hamper readers' understanding. The power of formalization is to code very rich, deep, and complex mathematical content into very compact, unambiguously defined, and easy to grasp at once (after some practice) mathematical symbols and formulas.

It is usually helpful if a formal presentation is complemented by appropriate illustrations and imagery. In particular, if a new term is introduced, it helps to give it a descriptive name, easy to remember.

However, if the presentation is very informal and mainly consists of many images each with multiple possible interpretations, then it is easy for the uninitiated reader to quickly get lost (as happened with me many times).

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To me, the presentation of deep learning theory linked in your post does not look mathematical -- see e.g. formulas (2.30) and (2.32) on p. 51 there.

Answer 2

The post asks: "Are there ever cases where formally defining physical phenomena in mathematical language is unnecessary?"

The key clarification is: necessary for what?

• Mathematical rigor may require formal definitions -- but this may be the only goal where the formal definitions are necessary.

• A successful application does not require formal definitions.

• Motivating good mathematics does not require formal definitions.

• Even doing good mathematics does not require formal definitions.

First example. Consider the top-voted question in mathematical physics here on MathOverflow ("what is an integrable system?". The top-voted answer includes the telling phrase "I'm being notably vague here." So for doing research mathematics by the standards of MathOverflow, formalization is apparently not necessary.

Second example. The field of deep learning existed before the formalization that you describe, and some people continue in the field without using that formal approach. So the field does not seem like an example of necessary formalization.

Maybe ten years from now we will say that most good work in deep learning uses the formalization; in that case we might say it was necessary. More likely, we will say that someone proposed a formalization, and some people used it, but most contributors to the field did not use it.

Third example. Would Newton's definitions, as in Ian Bruce's translation of the Principia here, qualify as formal mathematical definitions?

• I: The quantity of matter is a measure of the same arising jointly from the density and magnitude.
• II: The quantity of motion is a measure of the same arising from the velocity and quantity of matter jointly.
• III: The innate force of matter is the resisting force, by which each individual body, however great it is in itself, persists in its state either of rest or of moving uniformly straight forwards.
• IV: The impressed force is the action exercised on the body, to changing the state either of rest or of motion uniform in direction.
• V: It is the centripetal force, by which bodies are drawn, impelled, or tend in some manner from all sides towards some point, as towards a centre.

I'd say that I, IV and V are or could be formal definitions in the usual sense (mass := density * volume, momentum := velocity * mass, centripetal force := force towards a fixed point), but that II and III are not. So formal definitions were apparently not a requirement for Newton’s work, the canonical example of analyzing physical phenomena in mathematical language.