What is math intuition

Günter Ziegler in an interview

brand eins: Mr. Ziegler, what is the mathematical formula for intuition?

Günter Ziegler: I don't think it will ever be found. Mathematical formulas allow precise calculations. Intuition describes a process that cannot be calculated. That would be a contradiction in terms.

In one of your books you wrote that you were proud of your intuition. Isn't that a contradiction in terms for a mathematician?

I'm a mathematician, but I don't let myself be reduced to mathematical formulas. When solving mathematical problems, we mathematicians may even rely particularly heavily on our intuition. We have to feel relatively early on whether a certain working hypothesis or a certain solution could be the right one. When this instinct is lacking, we get lost on the wrong paths, work on the wrong ideas, possibly for months or even years. At the end there is no proof or no solution on paper, but endless frustration.

Can you be more specific?

Much of my work is based on the fact that I construct examples or counterexamples for conjectures made by other mathematicians. Experience plays a major role in this. I've been doing geometry and polyhedron theory for decades. Polyhedra are geometric objects with straight faces, such as cubes or pyramids, to name the simplest examples. I know a lot of these little animals, also much more interesting and complicated than cubes or pyramids. They speak to me. I mean that almost literally. If you as a geometer have dealt with a question long enough, then the forms tell you where to go. Or to put it a little less esoterically: I have developed a great deal of familiarity with the material. This familiarity gives rise to the feeling of whether a certain geometric shape fits a question, which in turn indicates that the assumption is very likely to be correct. Or on the contrary: I feel when something does not fit, is not right and that I have to think and work in a different direction than the person who made the assumption.

Mathematical assumptions are theses that others then work on. Do you need a special instinct to make these assumptions?

It depends. Some of the conjectures and problems to be solved result from relatively simple number games, such as the famous Fermat conjecture by the mathematician Pierre de Fermat. However, this does not mean that these problems can then be easily solved. In the case of other tasks, it is indeed the case that the formulation of the question or the thesis requires a very good sense of what could be right at all.

Is Mathematical Intuition Innate? A partial aspect of the math talent?

Sure, intuition is an important part of talent, but it doesn't have to be innate. My guess is that mathematical intuition has a lot to do with experience and is no different from intuition in other fields, such as politics. An experienced politician senses what he has to say to make people go to the barricades or, if it helps, to keep calm. He feels it because he has already experienced similar situations umpteen times. For a mathematician, experience helps to be able to concretely imagine extremely abstract problems, which in turn simplifies the solution considerably, at least for mathematicians who work a lot with examples. In this respect, mathematical intuition is, above all, hard-earned. Even if we go through the notebooks of the grandees of our discipline, by Gauss, Euler or Riemann, we see that their flashes of inspiration did not just come out of nowhere. They were hardworking and did a lot of math before making their assumptions.

If good intuition is the result of years of work, older mathematicians should have an advantage with their experience. Most of the big prizes are won by young mathematicians, often under 30 years of age.

The cliché is: mathematicians over 30 are burned out. There are many counterexamples for this, so this cannot be true in general. It is correct: the big prices are there for solutions and proofs of big problems and conjectures. To crack it, you need enormous persistence and concentration and a strong focus. This is often easier for young mathematicians to find when they don't have a family, a professorship or committee meetings.

Conversely, older mathematicians often see large arcs or surprising connections that no one has seen before. This is very important for research, but there are usually no prices for it. In this respect, counting math prices and comparing them with age distorts the picture.

In addition, the most important mathematics prize, the Fields Medal, can only be awarded to mathematicians under 40 due to regulations. However, the Abel Prize for life's work has been there since 2003. In the long term, those who have achieved a high level of research performance over their entire career will prevail.

Mathematics is the most precise of all sciences. The essence of intuition is heuristics, i.e. assumptions based on less information. Heuristics are not precise. How does that work together?

Math is precise in the results. And wherever we have solved the big problems and provided the evidence for them. The part of mathematics that is more exciting for me is the one that we cannot yet overlook and therefore cannot yet describe precisely. We have to work with heuristics there, otherwise we won't get any further. In this context, I find it exciting that - as in everyday life - we can intuitively solve complex mathematical problems approximately.

We all? Or "we mathematicians"?

We all. For example, if you see a sports car, you can tell whether it is special or just very streamlined. To be able to make this distinction scientifically precise requires advanced mathematical skills. The calculation of wind resistance values ​​is extremely complex, so our intuitive feeling for streamlinedness is remarkable. The same applies to traffic flows. These can also be described and calculated in complex equations, and yet we often have a good sense of where a traffic jam could soon arise and that it would now be better to take a different route.

You also write in one of your books: “Evidence is made from ideas and intuition. Mathematicians rarely talk about their intuition. ”Why is that?

We mathematicians work intuition on a problem for years, and when we have found the solution, we write it down as formally as possible. From the published evidence it can no longer be recognized which role intuition played in finding the solution. I think that's a shame, because I'm convinced that we mathematicians could learn a lot from each other if we knew more precisely how a colleague came up with a particular solution. This cannot of course be described with numbers or formulas, but in explanatory prose. So we would have to tell how an intuition became a concrete idea and finally a formal proof - including the wrong paths we went by the wayside. Perhaps prose texts are not exactly the strength of many mathematicians.

Can computers derive mathematical proofs?

I have never seen a computer that provided interesting evidence. Also, as far as I know, there is not a single interesting conjecture that a computer program has made. The state of the art is that computers can help humans concretize solutions and ultimately formalize the solution or proof in such a way that computers can verify the evidence. The software remains a tool. Computers don't really help us with the intellectual creative achievement in mathematics today. Exciting evidence needs an original idea, the really exciting needs several ideas that are originally linked. Of course, the programs are also getting better with machine learning, but so far humans still have a very large lead.

In the summer, the cinema screened “The Poetry of Infinite”, a film about an Indian math genius who came from the poorest of backgrounds and had little formal education. The film portrays Srinivasa Ramanujan, who was discovered by British mathematicians and brought to Cambridge, as arguably the most intuitive mathematician of all time. Is that true?

Of course, one always has to be careful with such superlatives, and such biopic films always come to a head. But what was really exciting about the real Ramanujan was that he was technically so bad and still came up with incredible formulas, many of which in retrospect turned out to be correct. He probably had a special, more visual approach to the formulas, and that really shows how great a role intuition can play in mathematics. The bottom line is that there is no intuition formula that you initially asked for. But there are people with a lot of intuition for formulas.

How far would Ramanujan have gotten if he had also had a good mathematical school and university education?

This is of course pure speculation. But it is also conceivable that he would have made it much less far. Perhaps this formal education would have buried its great strength, intuition, under itself. If you have read and understood everything, there is a risk that the imagination will be stifled. With Ramanujan there was the ingenious situation that he was then sitting in Cambridge with two technically excellent mathematicians, Hardy and Littlewood, who came across number theory and quickly realized that there was actually something valuable in Ramanujan's insane formulas. The magic of the three working together was that the talents were so different. This incredible intuition on the one hand and technical mastery on the other. The connection brings success. ---