How to create new mathematics?

Question

How do scientists and mathematicians create new mathematics for describing concepts? What is new mathematics? Is it necessarily in format of previous mathematics? Can one person make (invent or discover) a mathematics such that it isn't in format of geometry or algebra or analysis that we know?

Of course I know we can define a new meter and have a new geometry, or define a new n-ary operation and have a new algebra, etc. But can we create a new mathematics that is not like that?

In fact my question is this: when a scientist (in particular, a physicist), tries to create a mathematical theory for a concept, how does he/she do it? May he/she invent a new mathematics that is not in format of previous available mathematics at all? (i.e it wasn't geometry or algebra or analysis, etc.) Thanks a lot.

Answer 1

I think that the answer has to do with solving problems. Solving the problem may lead to new methods of solution being created. This is the origin of all new mathematics. The process of creation of new ideas is a mystery of the mind. We do not know exactly how this is done. Sorry.

Answer 2

This answer is explicitly a personal opinion.

My feeling is that new mathematics is created mostly for the purpose of illuminating some previously unseen (or, perhaps, hazily seen) connection between two or more things. The things may be in either mathematics or the natural world (including the human world), or both. We see the inklings of some model that bridges these things, and we create the scaffolding we need in order to explore them. Over time, this model permits us to understand those things, in terms of each other, better than we would have otherwise. Eventually, the model itself may become an object of study.

A related thought: We don't really understand why (if the question makes any sense to be asked) mathematics is as effective as it is in elucidating the natural world. One possibility, akin to the weak anthropic principle, is that if it weren't, there probably wouldn't have evolved beings sufficiently advanced to wonder why mathematics is as effective as it is.

Nonetheless, it is a remarkably effective way to explore the world. Combined with experimentation and observation, it permits us to articulate, in a more or less precise way, our understanding of how the world works. Even on its own, it exhibits powers of introspection: We have devised ways of having mathematics inspect and make assertions about itself. In that way, we've been able to make statements about what we believe to be the nature of truth and demonstration that would stun people from a mere millennium ago.

Answer 3

I agree with the previous answers but I want to add another point, most of the times, there is no big singular discovery of a big new theory.

Instead what usually happens, is that for example researcher A solves some problem using a combination of well known techniques, then researcher B applies a similar combination to a related problem, with slight improvements communicated by researcher C and so on.

This continues until 20 years later, when somebody else entirely looks through the whole stack of papers, unifies the notation, removes the dead ends that lead to nothing and publishes a book about the new theory that gradually has emerged.

Answer 4

I have a particular few ideas when it comes to this question. I am quite confident that the drive behind Mathematics is the drive to generalization.

A certain type of 'new' Mathematics bases itself on ideas which are quite rigorous and well known, and builds upon them; at every stage building a new platform where fresh results are able to be proven in a quite general landscape.

What's the motivation behind generalization? Well if you're proving facts in a general landscape, they easily apply to the particular case where you were proving facts before. You would like to make these results consistent and build definitions accordingly. This is general trend in Mathematics.

• Real Line Analysis $\subseteq$ Metric Spaces $\subseteq$ Topological spaces
• Riemann Integration $\subseteq$ Lebesgue Integration
• etc.

Another type of 'new' Mathematics comes from the interlacing of 2 different fields of Maths. Mathematics is surprisingly very interconnected with itself. This can be seen especially in the field of combinatorics where 'tricks' are used from other fields to prove results. One such example is Erdos's Probabilistic Combinatorics Method, where he rewords Graph Theoretic problems in a Probability framework to achieve previously never possible bounds.

In Graph theory there are also the realms of Algebraic Graph Theory and Topological Graph Theory where the subjects intermingle to see problems in a new light and prove results more generally using tools previously used to prove results in Topology and Linear Algebra...

In the realm of modelling...motivation to model a particular problem can lead to particular advancements...and even a whole new explosion of results such as Brownian motion and its origins to Wiener Processes etc Statistics...Stochastic Processes....these all are fueled by real life applications and have a basis in Functional Analysis.

So the moral of the story is what exactly? (Warning: My opinion)

1. See if anyone has done any work on the problem you're interested in
2. If not, use previous results to build on more problems which aim in the direction of the problem of interest
3. If not, see the problem in a different light. Use techniques and ideas from other areas and truly understand the deep essence of what they're doing and also pour in a lot of creativity to make the ideas rigorous and consistent
4. If all else fails, invent a whole new field of Mathematics

Step 4 might seem quite easy for the naive who bring up Galois who essentially invented Group Theory or Euler who accidentally gave birth to Graph Theory but in light of Step 1 given so much work that has been done with regards to Mathematics and so many smart people that have worked in such various fields of Mathematics...the probability of stumbling over a brand new field of Mathematics that everyone has overlooked is quite unlikely.

TLDR; A lot of creativity, imagination and hard work!

Answer 5

(Much of the following contains my own opinions.)

It really depends on the mathematics beeing created, and this is much more of a creative process than I think most people (at least non-mathematicians) realize. It is therefore hard (and somewhat disrespectful) to reduce to a simply cut explaination.

However, often it comes down to a combination of solving a specific problem, simplifying/generalizing things and formalizing them. If you have a specific problem that you want to solve, say in physics, it often helps to reformulate and/or generalize things to get a better understanding of what you are actually doing.

Take the "simple" example of measuring areas and volumes. I really think that this video does a better job than me, in explaining the reformulation of the problem of calculating the area of a circle. It is not hard to understand that these were crucial problems, that needed an exact solution, and by dealing with these sort of problems, calculus was pretty much independently invented/discovered by Gottfried Leibniz and Isaac Newton. Similar problems and related solutions have even been found from the ancient greeks, where for example the, so called, Method of Exhaustion, is a method for finding the area of a shape in a very similar way as one would do using limits and calculus later.

The previous paragraph also sheds some light on another realization, that these developments often take alot of time. Usually, a whole area of mathematics, such as calculus, can take hundereds of years to actually develop. Especially if you count the time people have spent on developing methods to solve the type of problems that led up to the actual development of the field in question. For calculus this was perhaps measuring areas and volumes, and for algebra it was perhaps trying to generalize arithmetic of numbers, for some, perhaps in order to solve equations. Group theory is also a more modern (c. 1800 or so) example of a field, within algebra it self, that more or less emerged from trying to generalize things when solving equations. Many would probably agree that even such an abstract area as logic itself came out of trying to generalize and/or formalize reason and language.

One should also note that even though an area, such as algebra say, may have come out of trying to solve a physical problem, development of the area is also very much due to pure curiosity, and creativity. Even though solving equations may perhaps have started with a physical problem, people who got obsessed with solving equations that others did not managed to solve, were probably among the ones pushing the field forwards. For a thorough read about the development of these fields I suggest either John Stillwell's Mathematics and its History, or (for a somewhat easier read i.m.o.) Victor J. Katz's A History of Mathematics: An Introduction. I honestly think that there are few better ways to get a solid answer to your question, than reading about the history of the development.

Of course fields of mathematics pop up quicker than over a period of hundereds of years. Chaos theory, is perhaps a field closer to physics than many, but by many still considered a field in its own right. Even this perhaps has its root in earlier problems, for example in the studies of the three-body problem by Poincare [9], but as a field most agree it really first developed during the sixtiees and seventiees, by Edward Lorentz, among others. It, more or less, emerged from the fact that deterministic, sometimes seemingly simple dynamical systems, could have unpredictable and very complicated behaviour. Chaos theory was pretty much what came out of trying to understand how and why this happens.

To conclude, I think solving a specific problem (regardless of what area you are dealing with) precisely, often takes generalizing things and this often results in doing mathematics in one way or the other. The idea of formalizing and generalizing things serves many purposes i.m.o., but perhaps foremost it often makes it more clear to your self, and it makes the problem more accessible to others (at least mathematicians), not familiar with your specific problem.

I hope this gives you an ok grasp of what it might mean to develop mathematics, in as few words as possible. Often new areas does not pop up out of nowhere, but are rather more changes/developments of old ones. A better understanding takes reading alot about other's creations, for example via reading about the history of mathematics. Perhaps someone else can add more concrete examples from, perhaps, more modern developments, where a whole area has more or less popped up from nothing.

[9]: History of Mathematics, Volume: 11, AMS, (1997), pp.272.

Answer 6

Creating new types of math is easy*

*for certain values of easy

To create new kinds of math. Just make up some rules. Or take an existing set of mathematical rules, and add a new rule. That's the easy part.

The hard part is creating new math that is consistent, interesting, and useful.

After you've made up your rules, you probably want to know if your system is consistent. Suppose I wanted to add the rule that there is a biggest natural number n. But then what is n+1? Well, either n+1 = n because it's the biggest number, or I have that n+1 isn't a natural number. In either case I have problems because now. I could decide that the rule is that $ n+1 \neq n $ doesn't hold for all natural numbers anymore or addition on the natural numbers doesn't always result in a natural number. But either of these choices makes my new natural numbers less nice. And there are all sorts of contradictions that can be reached from either of those choices. And if a contradiction can be reached, in some sense, what I'm playing with is non-sense.

That's what happened to set theory. At one point, they realized that given the current set of rules they were using for sets, you could define a set which both must contain itself and must not contain itself. But instead of giving up on sets, they just added some more rules to their system that prevented such a set from being defined and thereby left them with a set theory that didn't contain such contradictions.

But even if you choose a new rule that is consistent, it might not be very interesting. It might turn out to be something that some one already thought of before but just described in different terms.

And even if it's new and interesting, it might not be useful. But don't let that dissuade you. There are many branches of math that upon their creation were not very useful, but people found uses for them later. Knot theory was once considered completely useless, but it was used to prove the structure of DNA.

Even if it's not new, discovering something on your own that had previously been discovered by others can be very rewarding. It can give you a much deeper understanding of a topic, and it can be gratifying to realize you independently came up with the same thing as some clever person in the past. And it can be just fun.