Ramanujan, math intuition and the voice of the goddess

Intuition takes many forms, but what is it exactly? Why do genius scientists talk about mysterious forces? Do we really need the supernatural or there are more plausible explanations? Can we train it?

The story of the goddess

Ramanujan was one of the greatest mathematicians of the 20th century. He wrote down several astonishing results which he derived intuitively, before dying of pneumonia at a very young age. Ramanujan was a self-taught genius…

I quote:

The goddess Namagiri, worshipped at the Temple of Namakkal, was held in veneration by the Ramanujan family. Prior to the arrival of Ramanujan, Komalatammal had been childless. The family prayed to Namagiri in the hope that she would bless Komalatammal with children. Ramanujan was born shortly thereafter.

Ramanujan’s mathematical talents were evident from an early age. As a boy, he often woke in the middle of the night and wrote mathematical formulas on a piece of slate he kept beside his bed. These were subsequently recorded in his notebooks. Legend has it that it was Namagiri, appearing in Ramanujan’s dreams, who gave him the formulas. No proofs appear in his notebooks.

Scientific intuition as a sense

Our senses are much more than organs. A huge part of our brain processes sensory information. Tapping into that brainpower is an immense advantage. We can occasionally do it during the REM stage of dreaming. And occasionally our senses get mixed up in synesthesia.

In fact, various types of synesthesia (and the selection is wide) and various types of lucid dreaming (again, more than one kind can be uses) are trainable. I personally acquired some of these skills out of curiosity and then lost them out of insufficient practice. Curiously, some people are born this way, and certain individuals can utilize these skills more than others.

Colored letters are not helping

Computerized synesthesia training usually includes seeing letters, numbers, and notes in color. The training is straight-forward and takes a couple of months. At the end of the training every letter, number, and  every note is colored. However, unlike code highlighter in software IDE, the colors are plenty and hardly meaningful. They help find small typos and improve accuracy, but we have computerized tools for that.

Lucid dreaming is also not very helpful in the form it is usually practiced. After several weeks of waking up at the 20th minute of the REM cycle (you will probably need an app and a fitness bracelet), you are likely to gain control of your dream. If you have a bad dream, you can wake up. If you have a good dream, you are likely to remember it later. Now what?

Another scientific way to zap the brain is by using electrodes. The current is very low, the placement does not have to be super-accurate, and the effect is mild, for approximately 20 min. You can get slightly better focus, spatial awareness, muscle control, or pain reduction. Say a 15% improvement for 20 min. Now what? Which real task takes 20 min and can be revolutionized by a small improvement? The skills we teach in keytostudy provide a permanent x3 improvement, and some people still cannot find usage for them.

Back to scientific intuition

How does scientific intuition work? In math and physics the answer is clear. Typically it includes some sorts of mental experiments and geometrical metaphors. Both processes are highly visual, subconscious, and hard to formalize. Then the end results are reprocessed, and the logical brain processes try to catch up and fill in the gaps. People who do this sort of magic often tend to have a very active connection between the brain hemispheres.

We have some notes of another great mathematician, Euler. He used to solve complex mathematical problems from young age, finding patterns where others tried brute force. This approach is especially visible in music. Euler introduced Tonnetz graph to describe musical harmonies. The harmonies which people tend to hear, somehow projected as geometric shapes in Euler’s brain. This synesthesia is absolutely brilliant, and not something you can acquire by watching colored letters.

Another great 20th century mathematician, Grothendieck, envisioned algebraic sequences as geometric projections and revolutionized algebra around this abstraction. One of my friends, Alexander Shapiro (funny, but this was the original surname of Grothendieck) solved a previously unsolvable math problem in a similar way. I tried to ask him about the process, but the answer focused on some trivia about Renaissance painting and music. Things beyond reason…

Is AI intuitive?

Artificial intelligence often provides clues regarding the way we think. Chess as a game is not yet solved by AI, but computers outplay humans the same way the world champion outplays me. A human does not stand a chance against the best AI out there.

A human chess player starting from the level of master uses two skills. He uses visualization and computation to judge possible scenarios, and he uses pattern recognition to evaluate each position. The pattern recognition reuses the brain’s face-recognition areas. We are highly unlikely to act outside of recognizable features, and sometimes train chess puzzles to think in wider parameters.

The computer often makes steps we as humans cannot predict. In his training, AI plays tens of millions of games very fast. The results of these games are encoded in neurons or weights, which are almost impossible to understand. And this allows the computer to recognize situations we as humans never even imagined. As a result, AI is highly intuitive, but it is also likely to make very stupid mistakes very rarely. This feels like a very human trait.

Can intuition be trained?

A person can be trained to ace the IQ tests and get very high admission scores. Professional football players tend to score very high or very low in IQ tests. They see the field and act intuitively, implementing strategies very fast. Some players can take the skills to the math tests, others cannot. Listening to complex music, solving spatial puzzles definite and athletic activities tend to improve IQ scores in ways we only partially understand.

I do not know how to train another Ramanujan, Euler, or Grothendieck. As a mathematician (my PhD has very little to do with psychology…), I wish I could acquire this level of skill. I have several friends vastly smarter than me, yet none of them has this sort of genius synesthesia. Typically they can reduce very complex problems to very simple subproblems, and then the solution is easy. I have several friends with unique perception and equally unique results, yet they cannot fully describe the process.

I do know how to design and train an AI network to get results significantly superior to any human.

Divide and conquer

A fundamental concept in training is to use several simple procedures to develop networks with specific skills, and then combine these networks to tackle more complex problems. By introducing mutations to the network, we can progressively improve the results.

We apply a similar approach in speedreading training, where we focus on developing several small skills and then integrate them to create a powerful overall ability. A comparable method can be used for technical analysis of stock data. While financial experts recognize patterns in stocks, I have trained myself to spot these patterns quickly. However, AI can outperform humans in this regard, so don’t expect to earn substantial profits solely through technical analysis.

Most of what we do involves training small networks for specific problems and then combining these skills to address more complex challenges. Ideally, we could leverage our entire range of experiences—senses, family roles, artistic endeavors, professional training, and philosophical insights. Typically, this integration only occurs when we are in a flow state and exceptionally fortunate.

Analogical thinking

There is a scientific discipline dealing with analogical thinking. Usually, this involves some math practice. If a certain trick solves a math puzzle, then a similar trick can solve another math puzzle. We do not really vocalize the logic of the trick, but we can verify its mathematical accuracy. The more tricks we see, the more likely we are to come up with a similar solution to a new problem.

Only there is a caveat in this logic. The most prolific mathematical geniuses made their greatest discovery in a very young age. Way before they could see all the relevant tricks.

I think it is the “goddess” effect. Some profoundly different ways to analyze mathematical problems intuitively. My thesis advisor told me a strange story. As a young boy, he was a gymnast and not very good in math. When he was 10 years old he fell on his head. He could not walk or talk for 3 days, but afterward, he had some uncanny math intuition. For him, the math was very simple, but translating the understanding back to the human language and logical formulas was a daily struggle.