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.
The greatest service which can be rendered any country is to add a useful plant to its culture. --The Fruit Hunters
The only useful function of a statistician is to make predictions, and thus to provide a basis for action.
If Watson and I had not discovered the [DNA] structure, instead of being revealed with a flourish it would have trickled out and that its impact would have been far less. For this sort of reason Stent had argued that a scientific discovery is more akin to a work of art than is generally admitted. Style, he argues, is as important as content. I am not completely convinced by this argument, at least in this case.
My ambition was to bring to bear on medicine a chemical approach. I did that by chemical manipulation of viruses and chemical ways of thinking in biomedical research.
I was born on January 8, 1942, exactly three hundred years after the death of Galileo. I estimate, however, that about two hundred thousand other babies were also born that day. I don't know whether any of them was later interested in astronomy.
Almost everywhere, climate change denial now looks as stupid and as unacceptable as Holocaust denial.
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