What motivates a mathematical definition?

Question

Take the dot product for example. It is defined as:

Why is that definition more significant than any other definitions (e.g. )? I know that this one is one important starting point in the development of functional analysis, which is amazing to learn, but it's still bugging me on why it seems to be the only interesting product. What motivate authors to define it like that? My hypothesis is that many authors had tried other definitions, but none of them yields other interesting results or generalizations as this, so this one attracts more and more citations, until one day a textbook author decides to just define it upfront and ignore other variations.

Is it correct? If yes, is it also true for all other mathematical definitions? How about axioms? Is there any theory behind this? I suppose that theory would need to also explain what makes a concept interesting or not.

Answer 1

There are a lot of different motivations, they depend on the mathematical context:

• The motivation is to abstract several concrete mathematical examples to an abstract mathematical structure: To define the abstract concept “group” or “vector space” or “topological space”.
• The motivation is to generalize a well-known concrete structure to cover different, but related cases: To generalize the concept of a metric from the Euclidean metric to general non-degenerate bilinear forms of different signature.
• The motivation is to create complete new objects which replace well-known objects in a new context: To introduce the concept of ideals in a number ring to substitute the concept of numbers in the ring of integers.
• etc.

Answer 2

You ask:

What motivates a mathematical definition?

For starters, in the general sense, if one examines math definitions by category, the same motivations of any definition in any domain of discourse apply. Let's take a look at Robinson's Definition (GB) to guide us. His work is involved, but we can take a look at four basic types of definitions: real, lexical, stipulative, and precising definitions.

First, he talks about real definitions which he takes as not a definition at all, but rather in line with traditional notions about a thing's essence. Mathematicians have to decide and describe the basic abstract objects (SEP) with which they work. Thus, points, lines, real numbers, lattices, and groups are examples of abstract objects that form the ontology of mathematical discourse; one cannot work with the ontology unless one has clear criteria as to what those objects are.

Secondly, lexical definitions are necessary to clarify communication. It's a property of human thought that retaining many details in mind is a difficult thing to do, even with strategies like chunking. By writing down definitions and opening them to peer review as well as publishing them, it clarifies communication so mathematicians can be sure they are talking about the same thing even if they are in radically different places, cultures, and times.

Third, sometimes mathematicians want to propose new ideas and objects, and since those objects are novel, or are modifications to current ideas, they use stipulative definitions. Remember that a field like category theory does not spring up from the minds of a single person whole. When Elienberg and Mac Lane proposed their ideas initially, they had to get buy-in from others. This in turn leads to discussions and debates, and as new ideas are proposed, definitions are stipulated to see if they make sense or are useful.

Lastly, a mathematician may seek to take an intuitive concept and make it more formal with a precising definition. Here, the goal of the mathematician is to take a term that is already in use, but affix more notation to it to help prevent any confusion or to dispel vagueness. Natural language and naive mathematics has for a couple of hundred years now, been subjected to a process of formalization, and analytic philosophy is in some regards the a byproduct of Frege, Hilbert, and others' attempts to use symbols to clarify mathematics and establish its foundations using metaphysical grounding (SEP).

What's important to know is that in real-world mathematical discourse, these categories may be somewhat blurred. When does a precising definition create a new abstract object? Isn't recording the conventional use of the term a form of formalization? If a definition that is stipulated becomes adopted by a language community, isn't that really the discovery a mathematical object's essence? So on, and so forth.

Answer 3

A good notation reflects the underlying mathematics. Early mathematical problems were written as text. See for example the Moscow Mathematical Papyrus.. A problem is given as...

A number taken 1 and 1/2 times and added to 4 to make 10. What is the number?

The method of solution is given as...

Calculate the excess of this 10 over 4. The result is 6. You operate on 1 and 1/2 to find 1. The result is 2/3. You take 2/3 of this 6. The result is 4.

If we represent the unknown as X then we have the equation...

X + X/2 + 4 = 10

The power of using the equals sign is that the left and right arguments are treated equally. When we first learned how to simplify equations, we would have subtracted 4 from both sides. We then learned a typographical rule that lets us take the 4 from the left, change its sign, and put it on the right...

X + X/2 = 6

We can then multiply everything by two to eliminate the fraction...

2X + X = 12

...and so on. We cannot just rely on typographical rules. If X was the divisor, then we could have to remember to treat the special case where X is zero. But it makes things a lot easier.

There are lots of ways of writing two times X. We can write 2X, and some mathematicians do, but we cannot write two times two this way. A multiplication sign (dot, cross, or star) represents the multiplication by a symbol, and makes the notation the same for numbers and symbols.

Sometimes we get stuck with the second-best choice. The ancients took the pi symbol to represent the ratio of a circle's circumference to its diameter. It might have been better to choose the ratio of the circumference to the radius, but it is probably too late to change things now.

Let's pretend we are inventing a new notation. I never liked the notation for numeric intervals where...

[A,B)

...represents the numeric range between A and B, including A, but not including B. The unbalanced brackets bugs me, and will bug most text editors. I would like to write this as...

[A:]B

I have replaced the comma with a colon, which is used in some programming notations as a vertical ellipsis. [A:B] is used in some computer languages to represent the range from A to B. I have taken B out of the range by putting it outside the square brackets denoting the range.

If I wrote this in a paper, people would reject it as non-standard notation. However, if that had been suggested early in the evolution of set theory, it might have stuck. I think this is what happens with notation in general: there is a short period when everything is new, and you can rework the notation. There may be special fields where alternative notations flourish, such as suffix notation for matrices and tensors. But the opportunity for radical change evaporates once the notation is in general use.