[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.
It was unthinkable not long ago that a biologist or paleontologist would be at the same conference as an astrophysicist. Now we have accumulated so much data in each of these branches of science as it relates to origins that we have learned that no one discipline can answer questions of origins alone.
Scientific and technological progress themselves are value-neutral. They are just very good at doing what they do. If you want to do selfish, greedy, intolerant and violent things, scientific technology will provide you with by far the most efficient way of doing so. But if you want to do good, to solve the world's problems, to progress in the best value-laden sense, once again, there is no better means to those ends than the scientific way.
Mathematics is one of the deepest and most powerful expressions of pure human reason, and, at the same time, the most fundamental resource for description and analysis of the experiential world.
We can invent as many theories we like, and any one of them can be made to fit the facts. But that theory is always preferred which makes the fewest number of assumptions.
The three-pound organ in your skull - with its pink consistency of Jell-o - is an alien kind of computational material. It is composed of miniaturized, self-configuring parts, and it vastly outstrips anything we've dreamt of building.
Economics has never been a science - and it is even less now than a few years ago.
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