[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.
Dissections daily convince us of our ignorance of the seats of diseases, and cause us to blush at our prescriptions. How often are we disappointed in our expectation from the most certain and powerful of our remedies, by the negligence or obstinacy of our patients! What mischief have we done under the belief of false facts and false theories! We have assisted in multiplying diseases. We have done more — we have increased their mortality.
Every existence above a certain rank has its singular points; the higher the rank the more of them. At these points, influences whose physical magnitude is too small to be taken account of by a finite being may produce results of the greatest importance.
[M]y work, which I've done for a long time, was not pursued in order to gain the praise I now enjoy, but chiefly from a craving after knowledge, which I notice resides in me more than in most other men. And therewithal, whenever I found out anything remarkable, I have thought it my duty to put down my discovery on paper, so that all ingenious people might be informed thereof.
The future belongs to Science. More and more she will control the destinies of the nations. Already she has them in her crucible and on her balances.
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?
The separation of science and non-science is not only artificial but also detrimental to the advancement of knowledge. If we want to understand nature, if we want to master our physical surroundings, then we must use all ideas, all methods, and not just a small selection of them.
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