Nothing of value in human experience is intuitive. All that comes naturally to the human animal is eating, defecating and killing. People aren’t “gifted” at activities. Babies don't even know how to sleep peacefully through the night. As any parent will tell you, they must learn how to sleep.
We are creatures of noise, madness and chaos. Everywhere in the world where people live together, and the inhabitants are uneducated and idle, regardless of culture, race, or time, all of them share a common characteristic – they are noisy. By contrast, two places people congregate in large numbers are quiet: churches and libraries. That’s where people get on with the work of engaging with the unknown. Understanding is a function of work. And the first point on that function is (0,0).
For instance, there is nothing intuitive in mathematics beyond the addition of natural numbers less than 10. Subtraction is not intuitive, nor is the concept of zero. The postulates of Euclid are not intuitive. If they were, Euclid wouldn't get credit for them. Fractions? Forget it. And don't get me started on calculus.
As far as I can remember, most maths before calculus is computation. To master computation takes work. Not a little work, not 15 minutes a day – it takes a lot of work. Over and over and over again, like running beep-test drills in basketball drills. Or endless scales in piano practice. Most of it is practice for the real interesting maths later. If you’re willing to put in the practice, calculus teaches you composition, the system and patterns which not only produce order but define it.
To understand something, you must recreate it. If you can do this, you stare for hours at what you just did. The sun goes down outside, and you don't see it because all you see is how the starting point so obviously contained the brilliant insight at the end. How did anyone not see it? The world you know, the same world as you lived in as an infant – all wood and metal and separate pieces – starts to look thin and you start to see the fields and flows that have always been there, and you wondered why you ignored them, and what else you are ignoring.
This is why it doesn't matter what you are studying, fluid dynamics, electromagnetism, topology, physics, aerodynamics. You are listening to the different music the universe plays, but even though it isn't expected, you know how to listen, how to hear it. You know what to expect before you hear it. You look at planetary charts and you expect the mathematical model to be something like Bach – some periodicity, synchronicity, some counterpoint within the unity.
You look at the data coming from an atom smasher, and you don't expect to hear Bach or Mozart. Maybe Stravinsky. Probably more like Xenakis. Discord and unpredictability in an irresistible force. An energy. Charlie Parker. Ornette Coleman. A force with its own internal order invisible from the outside. A fixed beginning with a very definite and different ending.
And it’s not memorisation that makes a good mathematician or a thinker, although it does add an important factor. People who make educational music with a hip-hop voice and cadence reciting multiplication tables or national capitals over a drumbeat send the message that kids can't/won't memorise pieces of information unless its put in music. But they should still memorise them if for no other reason than to train their minds in the process of memorising abstract information.
Funny how kids don't have any problem memorising the characters on their TV shows or video games but can't be expected to remember the names of nine (or eleven) planets or the identity of half a dozen species of tree in their backyard. The idea that kids don't have to memorise multiplication tables because they have iPhones is beyond ignorant. People have had abacuses for thousands of years, then soroban, slide rules, calculators, etc. and it was still worthwhile to memorise.
I was never very interested or good at maths, but I’ve since discovered the reason to memorise multiplication tables is so the patterns in the tables, for example, the relationships between the 2, 3, and 6's, and the 4's will become apparent over time. Just like it's useful for a child to see a picture of a farm to understand farms, it is worthwhile to see the landscape of multiplication to understand what happens to numbers when you multiply them.
Likewise, when you visualise a memorised table in your mind, you start to notice things that are missing from it, like 11, 19, and all the other primes. You start to notice that some numbers show up a lot, like 24, and others infrequently, like 21. When the student thinks about the table, these patterns all become part of the memory of the multiplication table. He begins to learn things about mathematics, completely independently, from the simple act of repeatedly recalling the dumb multiplication table.
If it were religion, we’d say you take your passage from the bible or zen kōan or whatever and pray or meditate on it. Maths isn't a religion, but the iterative process of thought is the same. You think about a thing, you see it in your mind, and your mind becomes so familiar with this conjured image that it can detect new patterns in it.
It’s this same reason kids are taught to memorise the postulates of geometry, formulae for conic sections, aphorisms, the names of common birds and trees, various types of insects, the continents, oceans, planets and national capitals.
We memorise datasets so we can develop insights and new ideas about the data that haven’t existed before. Memorisation is the basis of inquiry, investigation and answer. It allows you to formulate and answer your own intelligent, probing questions. Otherwise, education is nothing more than instructions for using tools.
But the emphasis here is on the work. There is no intuition or natural talent. The child who sits at the piano for hours after the other children get bored does not have a natural talent for music. He has a natural talent for work. Same with reading. The joy is in figuring out, in trying.
There is only comfort in work because all that is great comes from inhuman amounts of work. The only way to create order out of disorder, to create a pattern out of noise, is work. Work fights entropy. Bach and Beethoven worked. They suffered. They were possessed of Plato's demon, driven to study and study and study and slowly to learn to see the patterns, and until they learn to recreate the patterns, change them, and finally dare to wrangle the forces underlying those patterns, and set to work on those forces themselves, creating new patterns surprising and unexpected even to accomplished musicians.
This is what mathematicians and thinkers do. The field attracts introverts and people comfortable with solitude. Maths – not computation, but maths – requires concentration, devotion and humility. A mathematician is a good listener. Maybe he thinks he's listening to God, or the Universe, or the white noise of bombinating quarks. He isn’t working his systems of equations by mechanically jostling symbols around the page. The good mathematician and thinker listens and wonders where he's heard it before.