I’m talking about math related subjects. Why do we need to learn to count? Why is not obvious to count up to 5, for example? Why is not obvious to add 2 with 3? Why do we need to learn to evaluate a boolean expression? Which is the part not obvious from the definition? Why do we need to exercise? If, in lambda calculus, the beta reduction is defined as then why is it not instantaneously obvious that if then for any term we can reduce to ?

If you don’t like examples from mathematics learning, then I am puzzled, because mathematician = learner. It’s an ancient greek branch of a pythagorean sect, the one of people dedicated to learn.

Learning is a process which needs time to unfold. What happens in our brains during this time? It must be that we rely too much, without acknowledging this in mathematical approaches, on a collection of more simple, not goal oriented, brain activities, like for example pattern recognition. Wait, what IS a pattern? And what means “recognition”?

Let’s take it another way. I shall make the dumbest hypothesis. When we learn a new (for us) mathematical definition, we need to transform it in “brain language”. We need to “compile” it into something which happens physically in the brain. (Btw, I am completely skeptical that whatever happens in the brain can be described by a language. Main argument: vision, which occupies large parts of brain activities of all creatures which possess one.) So, the mathematical definition (say, of the truth value of a boolean expression) has to be translated into something else, comprising a procedure to parse the expression, passing by all sort of patterns recognition and all kinds of high level human brain activity. Or even better, we write a program for solving this problem. The computer “learns”, i.e. compiles the program and then it works as nice as a mechanical clock. For humans, not so much, because we may forget how to do it, then recall some ideas “behind the definition”, we say, and reconstruct the procedure from scratches, like memories about the day we learned it first time, past mistakes and common sense reasoning.

Brains, let’s take this as part of the dumbest hypothesis, are just networks of neurons, which send one to another some signals and which modify themselves and their connectivity according to some simple rule, based on mechanical clock (physics) and chemistry rules, satisfied by any part of the physical universe, not only by the brain. So, whatever happens when we learn a mathematical concept, we transform it into some very simple thing in the brain. Forget about categories, truth values, group operations, definitions and theorems. No, there must be something which mimics a pattern of behaviours associated with the implicit expectations of people writing definitions, theorems, problems and exercises. Should we write a program for any of the items mentioned in the questions from the first paragraph, then we would see that we have to take care for a lot of things not mentioned in definitions, like well managing the variable names, again some mechanized form of pattern recognition, along with more or less precise knowledge about the programming language we use and it’s idiosyncrasies.

More we try to approach the realm of obvious, more abstract it becomes. And in our brains, learning to ride a bike or to evaluate boolean expressions is equally concrete, it’s physical.

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τὰ δὲ μοι παθήματα ἐόντα ἀχάριτα μαθήματα γέγονε

In other words —

No Pain ⇒ No Brain.No Brain ⇒ Mo Pain.I am not trying to argue against learning, nor dismiss conditioning as learning. Instead, I want to stress the fact that learning math from definitions is time (and probably energy) consumed for translating back from abstract concepts into a real, physical, concrete phenomenon in the brain. It follows that some concrete, physical phenomena are, in a sense, a higher

~~class~~level than abstract mathematical constructs.UPDATES:1. I shall give examples, 2. Gromov has, I think, a close viewpoint when he writes about playing and “ergo” stuff, non goal-oriented part of brain functioning, in his ergobrain article.