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On the Nature of Mathematics

For some reason I have been thinking of and talking over lunch with friends about the nature of mathematics. So here you go.

I think that most intelligent species in the universe (and we are obviously one of them), at least an instance of the class of beings we belong to (which I sometimes refer to as technological species), must have tools to address two fundamental problems: first is tools to find out what the truth is, and second is tools to find out how things work (in fact tools to describe correctly how they work), and as importantly if not more, tools to describe how to build them. (For instance "what is a square root ?" is about truth --here about definition--, but "how to compute a square root ?" is about how to build things).

Us, Humans, we have mathematics to solve the first problem. And yes, I do define mathematics as a language and set of methods to find out what the truth is. Not necessary the truth in absolute (if such thing as the absolute actually exists), but the truth in specific, sometimes artificial, contexts (we call them axiomatic theories). Mathematics started for us at least 5000 years ago (at least historical icons of mathematical activity are that old), but I think that mathematical thinking in the human mind must have origins as old as few hundred thousand years ago).

I think that we have just started to address the second problem by inventing computers. But the important thing is not the machines themselves. I think that computers are for Computing (the art and study of idealised processes) what telescopes are for cosmology. I think that in this very moment (beginning of the 21st century), we are playing around with programs, programming languages, networks and the like, the same way that tax collectors and farmers thousands of years ago used compasses and straight edged to rebuild lands around the Nile after the yearly floodings. To them mathematics (did they even already have a name for it?) was the very fact of using those tools, but what was really happening was that their minds were embarking on a fabulous journey to which we (mathematicians) are the latest children. I think that similarly, in the future, the computing equivalent of somebody like Euclid will give to Computing its true, native, conceptual framework.

...and I think that when this day comes (note that the following could actually perfectly happen before that day comes) people will also understand something else: the mathematical skills of humans beings, the feeling that we have of manipulating "abstract thoughts" is just a shadow of something deeper: computing is the true basis of way the mind works (and the correct way of describing its inner workings).

Note that I am not saying that the "human mind is a computer". I am saying that the fundamental concepts that we currently barely touch by studying computer science, are what is needed to correctly describe how mathematics is born in the human mind.