A mathematician's work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.
Interpretation
Mathematicians rely on intuition and creativity as much as on formal proof, which serves to validate their thoughts.
Gian-Carlo Rota highlights the complex nature of mathematical discovery, emphasizing that the journey involves a mixture of guesswork, analogy, and creativity. Proofs, often considered the essence of mathematics, are depicted as tools for confirming our intuitions and ideas, rather than the foundational elements of the discovery process itself.
In practice
In a lecture on mathematical creativity, this quote can illustrate the importance of intuition in problem-solving.
The apex of mathematical achievement occurs when two or more fields which were thought to be entirely unrelated turn out to be closely intertwined. Mathematicians have never decided whether they should feel excited or upset by such events.
Every lecture should state one main point and repeat it over and over, like a theme with variations. An audience is like a herd of cows, moving slowly in the direction they are being driven towards. If we make one point, we have a good chance that the audience will take the right direction; if we make several points, then the cows will scatter all over the field. The audience will lose interest and everyone will go back to the thoughts they interrupted in order to come to our lecture.
God created infinity, and man, unable to understand infinity, had to invent finite sets.
[In mathematics] There are two kinds of mistakes. There are fatal mistakes that destroy a theory, but there are also contingent ones, which are useful in testing the stability of a theory.
Mathematics is the study of analogies between analogies. All science is. Scientists want to show that things that don't look alike are really the same. That is one of their innermost Freudian motivations. In fact, that is what we mean by understanding.
It was like a new world opened to me, the world of science, which I was at last permitted to know in all liberty.
We look for medicine to be an orderly field of knowledge and procedure. But it is not. It is an imperfect science, an enterprise of constantly changing knowledge, uncertain information, fallible individuals, and at the same time lives on the line. There is science in what we do, yes, but also habit, intuition, and sometimes plain old guessing. The gap between what we know and what we aim for persists. And this gap complicates everything we do.
I dug things up. I was curious. I liked to draw what I found.
Arguably, my student status and perhaps my gender were also my downfall with respect to the Nobel Prize, which was awarded to Professor Antony Hewish and Professor Martin Ryle. At the time, science was still perceived as being carried out by distinguished men.
But just as astronomy succeeded astrology, following Kepler's discovery of planetary regularities, the discoveries of these many principles in empirical explorations of intellectual processes in machines should lead to a science, eventually.
People credit me for making the universe interesting when in fact the universe is inherently interesting, and I'm merely revealing that fact. I don't think I'm anything special for this to happen.
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