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.
No doubt science cannot admit of compromises, and can only bring out the complete truth. Hence there must be controversy, and the strife may be, and sometimes must be, sharp. But must it even then be personal? Does it help science to attack the man as well as the statement? On the contrary, has not science the noble privilege of carrying on its controversies without personal quarrels?
We wish to find the truth, no matter where it lies. But to find the truth we need imagination and skepticism both. We will not be afraid to speculate, but we will be careful to distinguish speculation from fact.
In less than eight years "The Origin of Species" has produced conviction in the minds of a majority of the most eminent living men of science. New facts, new problems, new difficulties as they arise are accepted, solved, or removed by this theory; and its principles are illustrated by the progress and conclusions of every well established branch of human knowledge.
This investigation has yielded an unanticipated result that reaction of cyanic acid with ammonia gives urea, a noteworthy result in as much as it provides an example of the artificial production of an organic, indeed a so-called animal, substance from inorganic substances.
According to 'M' theory, ours is not the only universe. Instead, 'M' theory predicts that a great many universes were created out of nothing.
Even before string theory, especially as physics developed in the 20th century, it turned out that the equations that really work in describing nature with the most generality and the greatest simplicity are very elegant and subtle.
Subscribe for the occasional hand-picked quote. No noise.