Where do new math ideas come from and how are they explored?

[Greg Bernhardt]

Disclaimer: I am totally ignorant on all facets of this subject.

I have an airbnb guest today who is a PhD student in math theory. I've only had very limited communication because my Mandarin is severely lacking. However, what I could understand sparked a general question of how math research works. I feel it must be different than how it comes about with the sciences because for one thing math research doesn't use the scientific method, right? So I guess the question is, where does the inspiration come for what to research and then by what method is it explored (and ultimately verified)?

@[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] @lavinia @fresh_42

 

[fresh_42]

Disclaimer: I am totally ignorant on all facets of this subject.

I have an airbnb guest today who is a PhD student in math theory. I've only had very limited communication because my Mandarin is severely lacking. However, what I could understand sparked a general question of how math research works. I feel it must be different than how it comes about with the sciences because for one thing math research doesn't use the scientific method, right?

For centuries mathematics counted as closer related to philosophy than to natural sciences. It is a fairly new point of view two consider math as a language of physics, despite the fact that it has been used (and developed) to solve physical equation systems. There are no experiments which can be done to falsify concepts, only logical rigor. This is why the revolutionary work of Cantor, Russel et alii and later Gödel has been so fundamentally, which took place in the shadow of the physical revolution by quantum mechanics. It is also the reason for Hilbert's program. It would be an interesting discussion on how mathematicians deal with the provable gaps in their system in comparison to how physicists deal with theirs.

So I guess the question is, where does the inspiration come for what to research and then by what method is it explored (and ultimately verified)?

The easiest answer would be: from the need to solve equations. But this is as short sighted as it is probably wrong. Surely those needs have been and are an essential part of development: Descartes, Bernoulli, Liouville, Gauß, Graßmann, Cauchy, Riemann and many others all had to solve real world problems. However, I still believe one of my favorite metaphors isn't completely wrong. I like to compare mathematics with model railroading: Usually male persons flee into their basements and start playing in an idealized copy of the real world for hours without recognizing anything outside of it. Mathematics can be a playground where fantasy and imagination is far more important than physical problems. The latter often come afterwards to justify their achievements to the rest of the world, which don't understand the childish part of their motivation (which I find is mainly responsible for the gender gap). Mathematical concepts frequently are used later on outside their original context: Grothendiek, Fock etc. As there will be no experiments, there won't be any restrictions beside the requirement to have no logical contradictions. Therefore we can invent whatever we want and play with it. I regularly read Terence's blog (T. Tao) and he's a master in finding interesting problems where others never look at. He also demonstrates, how important a mathematical background is. His solutions are very often a crossover of various mathematical disciplines. Personally I even think that this freedom is a necessary condition to do math. Many ideas might not have been found, if mathematicians started with a specific problem in mind. It would restrict their possibilities too much.

The verification process is finally similar to other sciences: publications, reviews and eventually additional papers by others. On the other hand, there are proofs, which are hard to verify: the four-color theorem took a computer (plus correction) IIRC and many refused to consider it as a proof, the last Fermat (plus correction) is actually only understood by a few mathematicians worldwide, Perelmann's proof of the Poincaré conjecture took years before it had been accepted, and Mochizuki's proof of the abc-conjecture apparently isn't understood by anyone. But, hey, we could also talk about the sense and nonsense of string theories. Research at the extremes seem to require extreme methods and sometimes even extreme personalities. The only difference in mathematics is, that the number of restrictions at the beginning are fewer.

 

[Greg Bernhardt]

Is new math mostly about expanding existing an theory, tying together existing theories or creating a completely new theory? I suppose I am most confused over completely new theory. I suppose that is where the laws of philosophy and logic come into play? Was Newton's calculus completely new theory to the world?

The verification process is finally similar to other sciences: publications, reviews and eventually additional papers by others. On the other hand, there are proofs, which are hard to verify: the four-color theorem took a computer (plus correction) IIRC and many refused to consider it as a proof, the last Fermat (plus correction) is actually only understood by a few mathematicians worldwide, Perelmann's proof of the Poincaré conjecture took years before it had been accepted, and Mochizuki's proof of the abc-conjecture apparently isn't understood by anyone. But, hey, we could also talk about the sense and nonsense of string theories. Research at the extremes seem to require extreme methods and sometimes even extreme personalities. The only difference in mathematics is, that the number of restrictions at the beginning are fewer.

This is very interesting and perhaps ties into my other thread: https://www.physicsforums.com/threads/what-are-the-constraints-on-research-progress.935170/

If only a few people can understand the cutting edge of a field, how possibly can that cutting edge be adequately progressed and even applied? Over time does the amount of education needed to get to that point ultimately become a hindrance?

 

[fresh_42]

I suppose I am most confused over completely new theory. I suppose that is where the laws of philosophy and logic come into play?

I'm not sure I see this bridge. There are always new concepts and ideas. E.g. I haven't heard about nets (topology) in my study, or of several recent concepts in category theory, a field I never would have expected major developments. Apparently the rate of abstraction is increasing. This means on the other hand, that many new developments look invented rather than discovered. In this sense parallels to philosophy are still given. We tend to judge retrospectively and then inventions appear as discoveries, especially if they can be applied to modern physics as it is the case e.g. for algebraic topology.

Was Newton's calculus completely new theory to the world?

No. It has been as so often in science a product of time. Leibniz developed similar concepts in parallel, and I've even recently read that the work of both already based on someone else's previous achievements. Unfortunately, I have forgotten his name. That's a disadvantage of modern days: you never know on which website you've read something. O.k. at least I tend to forget it. The main difference to nowadays was, that they had to write letters to each other, or publish thick books, and the intermediate step of publications on single results has been a far more an insider issue (letters) than it is today (journals). I have an interesting book written by J. Dieudonné about the history of mathematics between 1700 and 1900 in which he has worked out the many influences and connections which finally evolved to something we nowadays call a theory, or a calculus. In this sense there is no difference. Even Newton's famous quote "If I have seen further it is by standing on ye sholders of Giants." dates back to the 12th century "Dicebat Bernardus Carnotensis nos esse quasi nanos gigantum umeris insidentes, ut possimus plura eis et remotiora videre, non utique proprii visus acumine, aut eminentia corporis, sed quia in altum subvehimur et extollimur magnitudine gigantea" (Bernard of Chartres).

Over time does the amount of education needed to get to that point ultimately become a hindrance?

Maybe. We piled up a lot more scientific results compared to former days. How often have you read Peter's (or someone else's) advice: "Read a book about quantum mechanics (or general relativity) first"? Yet we have the Perelmanns, Taos and Wiles who manged to succeed. I think research at the front line has always been a kind of elitist undertaking. On the other hand, we all remember the giants, the Newtons and Gauß, and tend to forget all the others. Just as I obviously did (see above).

 

[mathman]

The cutting edge in mathematics is advanced by the few who are there because these people are usually in universities and they have graduate students who work under them in these areas.

 

[fresh_42]

By the way. A real life anecdote about Mandarin.

I had a fellow student who did his thesis and had been interested in a paper he's read about. So he wrote a letter to this guy in China and asked for a copy. A few weeks later, he received a copy. I mean, actually a copy.
I forgot whether he managed to get it translated ...

 

[Greg Bernhardt]

There are always new concepts and ideas.

To my extreme ignorance I can't imagine anything short of day dreaming. How is math theory developed without experiments?

 

[fresh_42]

How is math theory developed without experiments?

Well, the experiments are how it fits into existing theories and how it applies to examples.

I remember that I once found Galois' paper in an old journal. To be honest, it didn't really manage to understand it. His theory nowadays is presented in a completely different way. And even Artin's book on the subject is very different. It is somewhere in between, as it doesn't use modern Bourbakiism. Wikipedia says it took more than a decade before people realized what Galois had achieved. So sometimes, simply time is needed to distinguish between valuable developments or just another paper.

Another example are some formulas I've found. I'm pretty sure, that they aren't well-known as they arose from a totally different field and lie aside the mainstream canon. They are not hard to verify and probably all of us could do. This is the easy part. My suspicion is, that they could be used to classify (or help to classify) some sort of algebras which still appear to be more of a jungle. Unfortunately I'm not smart enough to see the wider range or to proceed beyond an elementary level. Here's what real geniuses do: they work it out to at least an interesting stage and some others might take it further.

I remember a paper from V. Strassen in which he constituted a concept in algebraic geometry which appeared to be rather artificial at the beginning. What the heck should succinct tensors be? In the end, he could improve the matrix exponent. A result which has been intended, but didn't appear to be achievable by his methods. This demonstrates the fact, that creativity is part of the show. An essential part.

Mochizuki basically developed a new math to prove the abc conjecture, which is one of the reasons, nobody can follow him. Assuming he's right, it might be new math, but the conjecture is old.

I've read that Andrew Wiles' basement actually had been a room under the roof. It took him many years and wrong tracks before his persistency won. He didn't fail to thank his wife for her's.

I think theoretical physics is pretty similar in its initial processes. The final step, to actually observe the Venus transit during an eclipse wasn't the cause to develop relativity theory. The initial thoughts and principles are fairly simple. The calculations and applications are not.

 

[lavinia]

I think of mathematics as a science.

The various sciences study certain empirical objects such as living systems or charged plasmas. Mathematics studies mathematical objects - geometrical structures, numbers, function spaces.

The physical sciences examine specific cases and through experiment search for repeated phenomena. They then attempt to model these phenomena in such a way that the results of future experiments can be predicted.

Mathematics also examines specific cases and searches for properties that are repeated among many examples. It then attempts to find a theory from which these properties can be derived and which holds true for cases that have not yet been examined.

The big difference is that physical sciences do not have proof but only faith in the rationality of Nature. Mathematics has proof so faith is not required.

In practice, most scientists believe that the laws of Nature are immutable and that a well verified theory is essentially proved.

Physicists particularly are searching for axioms - which they call physical principles - from which the behavior of Nature can be predicted by logical deduction. This is much like a lot of mathematics. For instance the axiom of Euclidean geometry that two lines intersect in at most one point is analogous to a physical principle.

On Physics Forums everyone says that Mathematics is just a language of Physics. Nonsense. If anything, the empirical world is a model of mathematics. For instance, Riemann used heat flow to define the Dirichlet Principle.

I know that most would disagree with this. And maybe what I have said is overstated. But the parallels between mathematical research and physical research are often not appreciated in my opinion.

 

[Greg Bernhardt]

Well, the experiments are how it fits into existing theories and how it applies to examples.

hmmm my guest made it clear he was pretty well allergic to anything applied and focused on theory. I should still ask him if his work is somehow rooted in experiments.

 

[mfb]

If only a few people can understand the cutting edge of a field, how possibly can that cutting edge be adequately progressed and even applied? Over time does the amount of education needed to get to that point ultimately become a hindrance?

One person or a couple of persons develop a new field, if it is promising over time more mathematicians spend time trying to understand it, contribute to it, and they spend more time finding better ways to present this field.
This is similar to theoretical physics. If you learn General Relativity today, you don't start with Einstein's publications. You start with textbooks that were written decades later, when the theory was much better understood and better approaches to work with the theory were developed.

The amount of education you need before you can contribute yourself is increasing, sure. The fraction of time you spend just to keep up with the work of others increases as well. There is a natural limit where you do exclusively that, but we are not there yet.

 

[Stephen Tashi]

To my extreme ignorance I can't imagine anything short of day dreaming. How is math theory developed without experiments?

We can say "creative" math is developed by thought experiments! It often begins with less than precise concepts expressed in "natural language" such as "information" or "risk". Then follows the effort to invent precise mathematical definitions that represent those concepts and to invent axioms that describe relations among the things defined. Then comes the experiment of applying logic to see what the definitions and axioms imply.

Often the results of the thought experiments are disappointing. For example, an attempt to define and axiomatize "risk" might logically lead to the theorem that risk is always zero. So a mathematical system can utterly fail to mimic the behavior of the concepts it seeks to describe. When that happens, it's back to the drawing board.

 

[aheight]

So I guess the question is, where does the inspiration come for what to research and then by what method is it explored (and ultimately verified)?

To answer the first: I just wade in the mathematical waters and let the tide take me. To answer the second: In the words of a fellow PF member: "you don't just stare at the problem and wait for the answer to pop into you head. Rather you try things and if they don't work you try something else." Often the path to the answer is littered on both sides with the wrong ones and you must get through this junk in order to reach the right one. The successful have a high tolerance for failure -- to endure the assault on the mind of getting through the wrong ones.

 

[VijayJ]

They are developing the new mathematical ideas to tackle the previously stated unsolved problems. These new ideas emerge from the previous understanding of the subject and intuitions.