[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.
Interpretation
Mistakes in mathematics can either ruin a theory or help test its stability.
Gian-Carlo Rota highlights the dual nature of mistakes in mathematics: while some mistakes can be catastrophic and invalidate a theory, others can serve a constructive purpose by providing insights into the theory's robustness. This suggests that errors can be instrumental in the process of refining and strengthening mathematical concepts.
In practice
This quote could be shared during a mathematics lecture to encourage students to embrace their errors.
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.
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.
There are no miracles in agricultural production.
More stars in the north are seen not to set, while in the south certain stars are no longer seen to rise.
The progress of science is strewn, like an ancient desert trail, with the bleached skeleton of discarded theories which once seemed to possess eternal life.
We send messages all the time, free of charge. There's a big shell out there now, 80 light-years around us. A civilization only a little more advanced than we are can pick those things up.
Civilization begins with distillation
The world has changed far more in the past 100 years than in any other century in history. The reason is not political or economic but technological-technologies that flowed directly from advances in basic science. Clearly, no scientist better represents those advances than Albert Einstein: TIME's Person of the Century.
Subscribe for the occasional hand-picked quote. No noise.