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.
Interpretation
This quote emphasizes that mathematics, like science, seeks to find similarities and connections between seemingly different concepts.
Gian-Carlo Rota highlights that mathematics is essentially about discovering relationships, drawing parallels between different analogies, much like the broader scientific endeavor. This pursuit reflects a deep-seated psychological drive to find unity in diversity, suggesting that true understanding comes from recognizing that disparate elements can share fundamental principles.
In practice
During a lecture on mathematical theories, a professor could use this quote to illustrate the nature of scientific exploration.
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.
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.
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.
Most Jupiter-sized planets orbit the mother star in a highly elliptical orbit. This means they will often cross the orbit of any Earth-like planet and fling it into outer space, making life impossible. But our Jupiter travels in a near-perfect circular orbit, preventing a collision with any Earth-like planet, making life possible.
A great power imposes the obligation of exercising restraint, and we did not live up to this obligation. I think this affected many of the scientists in a subtle sense, and it diminished their desire to continue to work on the bomb.
In the last analysis the entire field of psychology may reduce to biological electrochemistry.
I always say, 'Let your experiment speak to you.' What I mean by that is I - actually, we, or, at least, I'm not smart enough, actually, to guess how nature is working, but by looking and doing the right experiments and paying close attention to the subtleties of it, you start to catch on.
Name the greatest of all inventors. Accident.
Darwin gives courage to the rest of science that we shall end up understanding literally everything, springing from almost nothing - a thought extremely hard to comprehend and believe.
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