Artigo sugerido por Antonio Augusto P. Videira
About five years ago, Andrew Robinson, who has written quite a bit about Tagore, sent
me the Einstein-Tagore discussion and asked me for my comments on it. It seemed to me, as has been said previously, that they were rather talking past each other. Einstein was not
understanding what Tagore was saying. The point at issue was whether we have an objective world or a human world. Einstein thought that there was objective reality and that Tagore’s position was nonsense. I think my comment at the time was that Einstein was not appreciating how much the processes of construction, engaged in by our senses and minds, affect what we see and what our science consists of. I’ll come back to that point later. But anyway I think it’s because of that vague comment that I found myself invited here and trying to make sense of this issue. I’ve been explaining at various times that I don’t really understand Tagore and I find what is written on these subjects pretty confusing. But I have been attempting to make sense of it. So what I’m going to do is talk a bit about the relationships between what Tagore seems to be saying and the kind of approach or the kind of problems I’ve been working on myself.
As I said, the question is whether we have an objective world or a human world, and
Tagore’s position was that essentially everything is human, everything we know about is human. I’d like to start with something which perhaps not much attention has been given to—the mathematical side. It just so happens that on my way here I looked in the airport bookshop at Heathrow to find something to read, and lo and behold there was a book there called The Mathematical Experience, by Davis and Hersh, evidently put there to assist me with my coming lecture. But let me first, before I say what’s in that book, quote from the Tagore-Einstein Dialogue.
Tagore was saying that “truth is the perfect comprehension of the universal mind” and
that’s the thing I should perhaps just say a bit about. Tagore talks about the individual minds and the universal mind. The universal mind is like a perfected version of the human mind. It’s the ideal version of it, I believe. So “individuals approach” the abilities of the universal mind “through our own mistakes and blunders,” our “experience,” and so on. Einstein objected to this in the case of say mathematics. He said, “I cannot prove that scientific truth must be conceived as a truth that is valid independent of humanity, but I believe it firmly. I believe, for instance, that the Pythagorean theorem in geometry,” about the square on the hypotenuse, “states something that is approximately true independent of the existence of man.” So the question was raised, how much mathematical truth is independent of man, and how much is human construction.
Going back to The Mathematical Experience, what a lot of the book discusses is about
various happenings in the history of mathematics, what happens when new ideas emerge and people work them through. They also discuss attempts to understand what mathematics is. Because the curious thing is that, once one is trained in mathematics, one does the mathematics fairly automatically. Well, sometimes it’s hard work trying to understand something, but one simply feeds the problem into one’s mind and then the understanding emerges. But if you ask what is going on, what is mathematics, then this is a very difficult problem. Every approach seems to have some difficulties and to be in some ways not reflecting reality. Davis and Hersch discuss three approaches, or three theories about what mathematics is: Platonism, constructivism, and formalism.
Formalism is the idea that Hilbert introduced in trying to get a proper foundation of
mathematics—philosophers like Frege as well. The idea is that mathematics is proving theorems from axioms. So you might think you would simply state what your axioms are, then formalise your processes of deduction. Then you could go through mechanically, or a computer could go through verifying the proof, and that is mathematics. Well the difficulty of this is that, for a start, that we don’t really do things that way. There are lots of gaps in the arguments. Then there’s the question of where do the axioms come from, because we don’t actually play mathematics as a game like chess. The general belief is that we adopt axioms because they are true in some sense.
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