David Mumford - Collected Quotes

"Complex numbers are really not as complex as you might expect from their name, particularly if we think of them in terms of the underlying two dimensional geometry which they describe. Perhaps it would have been better to call them 'nature's numbers'. Behind complex numbers is a wonderful synthesis between two dimensional geometry and an elegant arithmetic in which every polynomial equation has a solution." (David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002)

"Logic has virtually nothing to do with the way we think." (David Mumford, [International Congress of Mathematics] 2002)

"Ordinary numbers have immediate connection to the world around us; they are used to count and measure every sort of thing. Adding, subtracting, multiplying and dividing all have simple interpretations in terms of the objects being counted and measured. When we pass to complex numbers, though, the arithmetic takes on a life of its own. Since -1 has no square root, we decided to create a new number game which supplies the missing piece. By adding in just this one new element √-1. we created a whole new world in which everything arithmetical, miraculously, works out just fine." (David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002)

"All of us mathematicians have discovered a sad truth about our passion: It is pretty hard to tell anyone outside your field what you are so excited about!" (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)

"But the drifting apart of pure and applied mathematics is not the whole story. The two worlds are tied more closely than you might imagine. Each contributes many ideas to the other, often in unexpected ways." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)

"It turns out that our knowledge is always too incomplete and our visual data is too noisy and cluttered to be interpreted by deduction. In this situation, the method of reasoning needed to parse a real-world scene must be statistical, not deductive. To implement this form of reasoning, our knowledge of the world must be encoded in a probabilistic form, known as an a priori probability distribution." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)

"If we learn to say things simply and build up slowly from the concrete to the abstract, we may be able to build many bridges among our various specialties." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)

"To mathematicians who study them, moduli schemes are just as real as the regular objects in the world. […] The key idea is that an ordinary object can be studied using the set of functions on the object. […] Secondly, you can do algebra with these functions - that is, you can add or multiply two such functions and get a third function. This step makes the set of these functions into a ring. […] Then the big leap comes: If you start with any ring - that is, any set of entities that can be added and multiplied subject to the usual rules, you simply and brashly declare that this creates a new kind of geometric object. The points of the object can be given by maps from the ring to the real numbers, as in the example of the pot. But they may also be given by maps to other fields. A field is a special sort of ring in which division is possible." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)

"To the average layperson, mathematics is a mass of abstruse formulae and bizarre technical terms (e.g., perverse sheaves, the monster group, barreled spaces, inaccessible cardinals), usually discussed by academics in white coats in front of a blackboard covered with peculiar symbols. The distinction between mathematics and physics is blurred and that between pure and applied mathematics is unknown. But to the professional, these are three different worlds, different sets of colleagues, with different goals, different standards, and different customs." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)