How do we create new mathematics symbols/concepts?

Question

In philosophy, you can just create a new word or give existing meaning to a new word, and often the philosophers don't even define what these concepts mean. For example, Hegel in his book Phenomenology of Spirit keeps using words that don't exist and words that already exist without defining them such that it's impossible to understand him without looking up the specific definitions of the words he uses in a reference book. Is it similar in mathematics, or are there rules that force people to use a set procedure to define and create new concepts and symbols? Could you give an example?

Answer 1

Philosophy deals in particular with concepts that are occasionally indeed vague (or perceived as such) or unsufficiently defined. But the names associated to these concepts may - by themselves - introduce a bias, be ambiguous, in resonance (sometimes unfortunate) with other names for not-so-close concepts, etc.

In mathematics, in principle, concepts are supposed to be un-ambiguously defined. By comparison, the name given to these concepts is rather secondary or frankly arbitrary. The name "group" associated to a set with an internal law such that such and such... is unimportant by itself ; it could have been "horse" as well, without harm for mathematical developments.

Besides, you question the "set procedure" ; if you mean by that the fact to settle definitions in the framework of set notations (I refrain from saying "set theory" !), it is largely true.

As you ask for examples, two of them come to my mind :

1. "Analysis situ" was a word coined by Henri Poincaré in the 1890s to designate the "non-especialy-metric" properties of "some" spaces (this word was abandoned later on for the name "topology"). He had a word, his genius was certainly encompassing in a right way what the concepts were behind ; for sure, he could explain to people by examples, periphrases, counterexamples, ... what he meant by this name, but it is only in the 1930s (Kolmogorov formulation as a set of the so-called open sets of the topology) that "topology" became accepted by the mathematical community.

2. The concept of graph (a naive description being "points" and "straight lines" or "arrows" relating some of these points) has taken a long time to be defined as a couple $(V,E)$ (a set of "Vertices", instead of points, and a set of "Edges" which are oriented and in this case, they modelize the arrows, or not). In order to have an idea of the "non-mathematical" way (according to our present-day standards), see this 1869 paper of Camille Jordan, a great mathematician, in the process of discovering this mathematical concept and some of its properties.

The axiomatisation of probability has undergone a very parallel "evolution".

Here comes the word "axiomatisation" which is at the very heart of modern mathematics... Entering into the subject would lead us very very far : I leave there the field to professional historians of mathematics.

Nevertheless, there is a question : has this necessarily-set-notational-framework the drawback of corseting the thought ? When you think to the huge advances made by Grothendieck in many cases associated with drawings, forging many many new words ("drawings" for example...) the meaning of which were understood at most by a handful of followers...

Answer 2

I would say there are some informal rules. To be accepted by the community, a new mathematical term or concept has to be useful, novel, and clearly and unambiguously defined. For an author/proposer to get a new concept accepted it generally has to pass expert peer review in a major journal. With but few exceptions, expert peer review is quite good at catching and rejecting errant attempts to add concepts erroneously. One need only consider the expansion of the classes of number to see this: natural number, integer, rational, complex, ... hyper-real... Each got accepted when they passed the criteria mentioned above.

Frankly, I think this vigilance is better in mathematics than in nearly every other scientific and technical discipline. Offhand, I can't think of any "goofball" or wildly inaccurate mathematical concepts that have gotten any traction, while in physics (say), the fringe elements have come up with terms and "concepts" that don't pass muster. (Just look at the ridiculous "physics" terms bandied about by Deepak Chopra.)

Even if some errant attempt slips through, its products will be short lived because ideas, concepts, and notations "live" by being taken up by the broader community, and if a concept doesn't have the properties listed above, it will die out.