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.
I cannot stress often enough that what science is all about is not proving things to be true but proving them to be false.
I have harnessed the cosmic rays and caused them to operate a motive device.
Evolution advances, not by a priori design, but by the selection of what works best out of whatever choices offer. We are the products of editing, rather than of authorship.
There is fuel in every bit of vegetable matter that can be fermented. There's enough alcohol in one year's yield of an acre of potatoes to drive the machinery necessary to cultivate the fields for a hundred years.
It is not always the magnitude of the differences observed between species that must determine specific distinctions, but the constant preservation of those differences in reproduction.
One can say, looking at the papers in this symposium, that the elucidation of the genetic code is indeed a great achievement. It is, in a sense, the key to molecular biology because it shows how the great polymer languages, the nucleic acid language and the protein language, are linked together.
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