"Self-portrait of Mr. S. U.
"His expression is usually ironic and quizzical. In truth he is very much affected by all that is ridiculous. Perhaps he has some talent to recognize and feel it at once, so it is not surprising that this is reflected in his facial expression.
"His conversation is very uneven, sometimes serious, sometimes gay, but never tiring or pedantic. He only tries to amuse and distract the people he likes. With the exception of the exact sciences, there is nothing which appears so certain or obvious to him that he would not allow for differing opinions: on almost any subject one can say almost anything.
"He brought to the study of mathematics a certain talent and facility which allowed him to make a name for himself at an early age. Dedicated to work and solitude until he was twenty-five, he became more worldly rather late. Nevertheless he is never rude because he is neither coarse nor hard. If he sometimes offends it is through inattention or ignorance. In speech he is neither gallant nor graceful. When he says kind things it is because he means them. Therefore the essence of his character is a frankness and truthfulness which are sometimes a little strong but never really shocking.
"Impatient and choleric to the point of violence, everything that contradicts or wounds him affects him in an uncontrollable way, but this usually disappears when he has vented his feelings.
"He is easy to influence or govern provided he is unaware that this is intended.
"Some people think that he is malicious because he makes merciless fun of pretentious bores. His temperament is naturally sensitive and renders him subject to delicate moods. This makes him at once gay and melancholy.
"Mr. U. behaves according to this general rule: he says a lot of foolish things, seldom writes them and never does any.
When Françoise read this description she felt it agreed well with what she then knew of me but was very surprised at the quality of my French, until she came to the last paragraph:
"And now I shall change from my text which I came upon by chance yesterday. The above are verbatim extracts from a letter of d'Alembert to Mademoiselle de Lespinasse written some two hundred years ago!" (D'Alembert was a famous French mathematician and encyclopedist of the eighteenth century). Francoise was very amused.
Some thirty years have elapsed since I copied this little text. I will now add as a finishing touch that I don't think I have changed much, but that there is one trait which d'Alembert did not mention that I possess — all this merely si parva magnis comparare licet — it is a certain impatience. I have been afflicted with this all my life. It may be increasing with advancing years. (If Einstein or Cantor came to lecture here today I would have the split reaction of a schoolboy — wanting to learn on the one hand and to skip class on the other.) While I still feel quite happy giving lectures, talks, or discussions I am becoming less and less able to sit through hours of such given by others. I am, I told some colleagues, "like an old boxer who can still dish it out but can't take it any more." This amused them no end.
Chapter 15. Random Reflections on Mathematics and Science
This chapter will be somewhat different in content from the preceding account of my ''adventures" and of scientists I have known.
Here I have tried to gather, review, and sometimes amplify some of the general ideas I have touched upon so lightly throughout the book. I hope that in their randomness these reflections will give the reader an added glimpse into the manifold aspects of science and especially the relation of mathematics to other sciences. It is merely about the "gist of the gist." For greater detail, I can only refer the reader to some of my more general scientific publications.What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. Roughly speaking, people know that it deals with numbers and figures, with patterns, relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes. They also know that it purports to form the foundations of all rational thought.
Some could say it is the external world which has molded our thinking — that is, the operation of the human brain — into what is now called logic. Others — philosophers and scientists alike — say that our logical thought (thinking process?) is a creation of the internal workings of the mind as they developed through evolution "independently" of the action of the outside world. Obviously, mathematics is some of both. It seems to be a language both for the description of the external world, and possibly even more so for the analysis of ourselves. In its evolution from a more primitive nervous system, the brain, as an organ with ten or more billion neurons and many more connections between them must have changed and grown as a result of many accidents.
The very existence of mathematics is due to the fact that there exist statements or theorems, which are very simple to state but whose proofs demand pages of explanations. Nobody knows why this should be so. The simplicity of many of these statements has both aesthetic value and philosophical interest.
The aesthetic side of mathematics has been of overwhelming importance throughout its growth. It is not so much whether a theorem is useful that matters, but how elegant it is. Few non-mathematicians, even among other scientists, can fully appreciate the aesthetic value of mathematics, but for the practitioners it is undeniable. One can, however, look conversely at what might be called the homely side of mathematics. This homeliness has to do with having to be punctilious, of having to make sure of every step. In mathematics one cannot stop at drawing with a big, wide brush; all the details have to be filled in at some time.
"Mathematics is a language in which one cannot express unprecise or nebulous thoughts," said Poincaré, I believe in a speech on world science which he gave at the St. Louis fair many years ago. And he gave as an example of the influence of language on thought a description of how differently he felt using English instead of French.
I tend to agree with him. It is a truism to say that there is a clarity to French which is not there in other tongues, and I suppose this makes a difference in the mathematical and scientific literature. Thoughts are steered in different ways. In French generalizations come to my mind and stimulate me toward conciseness and simplification. In English one sees the practical sense; German tends to make one go for a depth which is not always there.
In Polish and Russian, the language lends itself to a sort of brewing, a development of thought like tea growing stronger and stronger. Slavic languages tend to be pensive, soulful, expansive, more psychological than philosophical, but not nebulous or carried by words as much as German, where words and syllables concatenate. They concatenate thoughts which sometimes do not go very well together. Latin is something else again. It is orderly; clarity is always there; words are separated; they do not glue together as in German; it is like well-cooked rice compared to overcooked.
Generally speaking, my own impressions of languages are the following: When I speak German everything I say seems overstated, in English on the contrary it feels like an understatement. Only in French does it seem just right, and in Polish, too, since it is my native language and feels so natural.