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.
Equipped with his five senses, man explores the universe around him and calls the adventure Science.
Even Johannes Kepler, Isaac Newton, Charles Darwin, Gregor Mendel, and Albert Einstein made serious mistakes. But the scientific enterprise arranges things so that teamwork prevails: What one of us, even the most brilliant among us, misses, another of us, even someone much less celebrated and capable, may detect and rectify.
A theory which is not refutable by any conceivable event is non-scientific. Irrefutability is not a virtue of a theory (as people often think) but a vice.
If we find life out there, and it's not us, we will deem it not intelligent. But what may be equally as likely is that we find life that's vastly more intelligent than we are. If that's the case, we are putty in their hands.
Any one who has studied the history of science knows that almost every great step therein has been made by the "anticipation of Nature," that is, by the invention of hypotheses, which, though verifiable, often had very little foundation to start with; and, not unfrequently, in spite of a long career of usefulness, turned out to be wholly erroneous in the long run.
Gravitational waves, because they are so imperturbable - they go through everything - they will tell you the most information you can get about the earliest instants that go on in the universe.
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