Math teacher Ali Kayaspor has thought a lot about how mathematicians have come up with fundamental ideas about the nature of reality and he shares anecdotes that give us a glimpse. But first, the cold shower:

Unfortunately, there is no clear way to answer the question of how a mathematician thinks. But we can approach this question as follows; if you watched any chess tournament, the game’s analysis is shared in detail at the end of the match. When you examine the analysis, you will see a breaking point in each game. Similarly, mathematicians also experience a breaking point while working on a problem before finding a solution.

Ali Kayaspor, “How Does a Mathematician’s Brain Differ from Other Brains?” at Medium (September 1, 2021)

Sometimes, perhaps, they hardly notice the breaking point. Consider this anecdote from the life of Carl Friedrich Gauss (1777–1855):

When the children in an elementary school were very naughty, the teacher wrote a difficult question on the chalkboard to silence all the children. The teacher asked the children to add up all the numbers from 1 to 100.

On that day, the young German boy named Gauss, who would grow up to be one of the greatest mathematicians of the future, was in that class. While the teacher thought it would take a long time for the children to solve the question, Gauss resumed talking to his friends again in just a few minutes. When Gauss’s teacher asked why he was talking, he said he had already solved the question. That day, all the students in that class tried to add all the numbers one by one, but Gauss did something unusual. He saw that he would always get 101 if he added a number from the left of the sequence and a number from the right. For example, 1+100, 2+99, 3+98,…, 50+51; it was always 101, and there were 50 of them.

If we look carefully at this method, we will see that this is an observation that even a small child can see. However, it is a fact that not all young children solve this question like that.

Ali Kayaspor, “How Does a Mathematician’s Brain Differ from Other Brains?” at Medium (September 1, 2021)

It involves creative abstract thinking at a level unusual in a child. We are told that “He went on to publish seminal works in many fields of mathematics including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy, optics, etc. Number theory was Gauss’s favorite and he referred to number theory as the ‘queen of mathematics.’” – “Gauss: The Prince of Mathematics”

Surprisingly, great mathematicians are not always good with conventional basic math. Take Srinivasa Ramanujan (1887–1920), one of India’s greatest mathematical geniuses:

Despite solving infinite sums, he could not understand the most basic analysis technique. Ramanujan did not have the slightest idea of complex analysis, but he could work on zeta functions. So Ramanujan had a different mindset in his mind that only he knew, and one we will never understand.

One day, when [his friend and fellow mathematician G. H.] Hardy wondered about this situation and asked how he wrote all those formulas, Ramanujan told him that God gave him all the formulas, and he had just written them. To me, this was a very reasonable answer because Ramanujan was a math practitioner 24/7, and he often forgot to eat. His wife or mother reminded him that Ramanujan had to eat. The few times he did sleep, he continued doing math in his dreams.

Ali Kayaspor, “How Does a Mathematician’s Brain Differ from Other Brains?” at Medium (September 1, 2021)

A number of anecdotes capture Ramanujan’s affinity for pure mathematics. His health was fragile and during a bout of illness, he surprised even his friend and mentor Hardy:

Eventually Ramanujan was confined to a nursing home to await his return to India. Hardy paid frequent visits to his friend and colleague. Not surprisingly, the conversation usually turned to mathematics. On one such visit, 1,729 cropped up. This was the number of the taxi cab Hardy had taken to the clinic, and as befits two number theorists they discussed its significance. Hardy thought 1,729 to be a boring run-of-the-mill number, but Ramanujan disagreed. “That is a really, really interesting number,” he declared. How so? “It is the smallest number that can be expressed as the sum of two cubes in two different ways!”

Ramanujan could see immediately that: 12³ + 1³ = 10³ + 9³ = 1,729.

This amusing anecdote came to symbolise Ramanujan’s humble genius, and numbers that can be expressed as the sum of two cubes in two separate ways are known as “taxi numbers” in recognition. Other taxi numbers are 4,014, 13,832 and 20,638. But 1,729 is the smallest.

Paul Davies, “Ramanujan – a humble maths genius” at Cosmos Magazine (December 28, 2015)

And no one really knows how he did it.

Perhaps it requires a lot of imagination. Mathematics is full of unusual numbers. Consider, for example, the irrationals that burble on forever without forming a pattern (though some, like the Golden Ratio and pi, are critical). They are more numerous than rational numbers which eventually start to make sense.

But also, there are the imaginary numbers (they don’t make sense according to our number system but of course our computers and the entire modern world depend on them).

And the hyperreals: “Thinking about infinities is somewhat mind-bending, but it turns out that actually manipulating infinities with the hyperreal system is incredibly easy if you are familiar with basic algebra.”

Then there’s 1/137, which keeps turning up in physics and no one is sure why: “What’s special about alpha is that it’s regarded as the best example of a pure number, one that doesn’t need units. It actually combines three of nature’s fundamental constants – the speed of light, the electric charge carried by one electron, and the Planck’s constant, as explains physicist and astrobiologist Paul Davies to Cosmos magazine. Appearing at the intersection of such key areas of physics as relativity, electromagnetism and quantum mechanics is what gives 1/137 its allure.”

And Chaitin’s unknowable number, which is critical to computer function: “The number exists. If you write programs in C++, Python, or Matlab, your computer language has a Chaitin number. It’s a feature of your computer programming language. But we can prove that even though Chaitin’s number exists, we can also prove it is unknowable.”

One needs a lot of imagination combined with a lot of numerical discipline to keep track of it all. And that’s as far as we have got in understanding the way creative mathematicians think.

You may also wish to read: Why the unknowable number exists but is uncomputable. Sensing that a computer program is “elegant” requires discernment. Proving mathematically that it is elegant is, Chaitin shows, impossible. Gregory Chaitin walks readers through his proof of unknowability, which is based on the Law of Non-Contradiction.