Quotes Post:

Ongoing collection of quotes attributed (mod my lapses in memory) to mathematicians and physicists or about their work:

d’Alembert:

Go forward, faith will follow! 

Laurent Schwartz:

I have always thought that morality in politics was something essential, just like feelings and affinities.

To discover something in mathematics is to overcome an inhibition and a tradition. You cannot move forward if you are not subversive.

Jean Baptiste Joseph Fourier (1768–1830)

Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.

Richard Feyman

Physics is imagination in a straitjacket.

When in doubt, integrate by parts.

We leave the operators, as Jeans said,hungry for something to differentiate.”

Sophia Kovalevskaya (1850–1891)

It is not possible to be a mathematician without being a poet at heart.

Anonymous quote (in an anecdote, probably by Gian-Carlo Rota)

I do discrete, not continuous.

Anonymous, paraphrased

Schwartz’s theory of distributions is an example of the French propensity to turn an operation into a theory–in this case, integration by parts.

Hermann Weyl

In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.

Banach

Mathematics is the study of analogies between analogies.

Andre Weil

to add a little bit, whatever it is, to Riemann, that would already be, as they say in Greek, to do something {[faire quelque chose]}

Andre Weil

Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur, whether he participates in it, or even if he is a historian contemplating it retrospectively, accompanied, nevertheless, by a touch of melancholy. The pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time; at least in the Gita there is a slew of prayers (slokas) on the subject, each one more final than the previous ones.

Gian-Carlo Rota

Although the notation of Hopf algebra satisfied the most ardent advocate of spic-and-span rigor, the translation of “classical” umbral calculus into the newly found rigorous language made the method altogether unwieldy and unmanageable. Not only was the eerie feeling of witchcraft lost in the translation, but, after such a translation, the use of calculus to simplify computation and sharpen our intuition was lost by the wayside…

Alain Lascoux

… it does not suffice to highlight algebraic structures to give them life, one has furthermore to market them. This entails showing that the energy spent for learning them is compensated by a new light shed on classical domains, connections between different fields and the creation of new mathematical objects–one cannot always grind the same grain, one also has to sow!

Benoit Mandelbrot (on tribalism in math)

They (Bourbaki) were a militant bunch, with strong biases against geometry and against every science, and ready to scorn and even to humiliate those who did not follow their lead. It was presented to us students as the best there was. And if we didn’t like it, we were advised to move out of math.

Arnaud Denjoy says of Bourbaki

I fear your absolutism, your certainty of holding the true faith in mathematics, your mechanical gesture of drawing the sword to exterminate the infidel to the Bourbaki Qur’an. […] We are many to judge you as despotic, capricious, sectarian.

Reid

I consider it regrettable and unhealthy that the algebra seminar seems to form a ghetto with its own internal language, attitudes, criterions for success and mechanisms for reproduction, and no visible interest in what the rest of the world is doing.

I. G. Petrovskii

Genuine mathematicians do not gang up, but the weak need gangs in order to survive. They can unite on various grounds (it could be super-abstractness, anti-Semitism or “applied and industrial” problems), but the essence is always a solution of the social problem – survival in conditions of more literate surroundings.

Arnold

Jacobi noted, as mathematics’ most fascinating property, that one and the same function controls both the presentations of a whole number as a sum of four squares and the real movement of a pendulum.

These discoveries of connections between heterogeneous mathematical objects can be compared with the discovery of the connection between electricity and magnetism in physics or with the discovery of the similarity between the east coast of America and the west coast of Africa in geology.

The emotional significance of such discoveries for teaching is difficult to overestimate–they teach us to search and find such wonderful phenomena of harmony of the Universe.

Hermann Weyl (I would say two extremes of mathematicians. For a simple, dramatic narrative, seems always a dichotomy is drawn.)

There are two Classes of mathematicians:


• ALGEBRAISTS: as Leibniz, Weierstrass
• GEOMETERS-PHYSICISTS: as Newton, Riemann, Klein


and people belonging to different classes may tend to be in conflict with each other.

The tools of the algebraists are logical argumentation, formulae and their clever manipulation, algorithms.

The other class relies more on intuition, and graphical and visual impressions. For them, it is more important to find a new truth than an elegant new proof.


The concept of rigour is the battlefield where the opposite parties confront themselves, and the conflicts which hence derived were sometimes harsh and longlasting.

John Maynard Keynes

Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind that looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago.

Jacques Hadamard

It is important for him who wants to discover not to confine himself to one chapter of science, but to keep in touch with various others.

John Adams


I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain.

W. S. Anglin


Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

Chesterton, G. K.


You can only find truth with logic if you have already found truth without it.

Mittag-Leffler

The best works of Abel are truly lyric poems of sublime beauty … rising farther above life’s commonplace and emanating more directly from the soul than those of any poet.

(This is more astounding once you understand the deprivations and miseries that Abel suffered. Fortunately, some mentors, such as Bernt Michael Holmboe, played key roles in nurturing his intellect. )

Clifford Truesdell on the approach to mathematical discovery/invention of Euler and the Bernoulli’s

(1) Always attack a special problem. If possible solve the special problem in a way that leads to a general method.
(2) Read and digest every earlier attempt at a theory of the phenomenon in question.
(3) Let a key problem solved be a father to a key problem posed. The new problem finds its place on the structure provided by the solution of the old; its solution in turn will provide further structure.
(4) If two special problems solved seem cognate, try to unite them in a general scheme. To do so, set aside the differences, and try to build a structure on the common features.
(5) Never rest content with an imperfect or incomplete argument. If you cannot complete and perfect it yourself, lay bare its flaws for others to see.
(6) Never abandon a problem you have solved. There are always better ways. Keep searching for them, for they lead to a fuller understanding. While broadening, deepen and simplify.


Johannes Kepler

Ubi materia, ibi geometria.

Heaviside

There was a time indeed in my life when I was something like old Teufelsdröckh in his garret, and was in some measure satisfied with a mere subsistence. But that was when I was making discoveries. It matters not what others think of their importance. They were meat and drink and company to me

From the The New York Times book review “Two Brilliant Siblings and the Curious Consolations of Math” (a variation on Boethius) by Paul Sehgal of “The Weil Conjectures” by Karen Olsson. The siblings are Andre and Simone Weil:

At the 1994 reception for the prestigious Kyoto Prize, awarded for achievements that contribute to humanity, the French mathematician André Weil turned to his fellow honoree, the film director Akira Kurosawa, and said: “I have a great advantage over you. I can love and admire your work, but you cannot love and admire my work.”

This was a lament, not a boast.

The book unfurls effortlessly, loose and legato. There are no real revelations — the subjects are well known and long dead. There are no stakes; there is no suspense. I was riveted. Olsson is evocative on curiosity as an appetite of the mind, on the pleasure of glutting oneself on knowledge. André “gorges himself” on mathematics and Sanskrit. Simone crawls between her books arrayed on the floor, “leans over Descartes like an animal drinking.”

Paul Seghal quotes Olsson about her college days studying math:

We were a small band of students giddily, exhaustedly trekking through an abstract moonscape,” she writes. “The egos, the insecurity, the unabashed nerdiness! I miss it still.”

Why do we represent the unknown with x? Credit René Descartes’s printer, who was running out of letters while producing copies of the treatise “La Géométrie.” X, y and z remained, and the printer settled on x, the least used letter in French.

The mathematician Henri Poincaré imagined that thoughts lived in the mind like static particles, “as if hooked to a wall,” Olsson writes. Thinking liberated them and allowed them to crash into and attach to one another. Her book is full of such moments of connection, combustion and surprise. And if x goes unsolved, there is something apt and beautiful there, too. For all the riddles of mathematics, there is also the ordinary and eternal mystery of other people’s minds.

Edwin Abbott (on loan as an apt description in general of the work of passionate mathematicians):

A romance of many dimensions.

Keynes

The difficulty lies not so much in developing new ideas as in escaping from old ones.

Maryam Mirazhkani

<When I come across a nice idea in mathematics,> I feel  really facinated. It’s like listening to music or seeing an amazing drawing. It’s really like art.

Eden Philpotts (from A Shadow Passes):

In the marshes the buckbean has lifted its feathery mist of flower spikes above the bed of trefoil leaves. The fimbriated flowers are a miracle of workmanship and every blossom exhibits an exquisite disorder of ragged petals finer than lace. But one needs a lens to judge of their beauty: it lies hidden from the power of our eyes, and menyanthes must have bloomed and passed a million times before there came any to perceive and salute her loveliness. The universe is full of magical things patiently waiting for our wits to grow sharper.

Yau

May charm and beauty always be the guiding principle of geometry.

Chern

… while analysis maps a whole mine, geometry looks out for the beautiful stones. 

Atiyah

The history of mathematics is full of instances of happy inspiration triumphing over a lack of rigour. Euler’s use of wildly divergent series or Ramanujan’s insights are among the more obvious […] The marvelous formulae emerging at present from heuristic physical arguments are the modern counterparts of Euler and Ramanujan, and they should be accepted in the same spirit of gratitude tempered with caution.

Yau

Linear or non-linear analysis is developed to understand the underlying geometric or combinatorial structure. In the process, geometry will provide deeper insight of analysis. An important guideline is that space of special functions defined by the structure of the space can be used to define the structure of this space itself.

von Neumann

… it is hardly possible to believe in the existence of an absolute, immutable conceptof mathematical rigour, dissociated from all human experience. I am trying to take a very low-brow attitude on this matter. Whatever philosophical or epistemological preferences anyone may have in this respect, the mathematical fraternities’ actual experiences with its subject give little support to the assumption of the existence of an a priori concept of mathematical rigour.

The most vitally characteristic fact about mathematics is, in my opinion, its quite peculiar relationship to the natural sciences, or, more generally, to any science which interprets experience on a higher than purely descriptive level. […] It is undeniable that some of the best inspirations in mathematics—in those parts of it which are as pure mathematics as one can imagine—have come from the natural sciences.

… ?

I think it is very important in studying intellectual history not to indulge in hero worship. History is not just an account of the great triumphs and successes of the past but also of the false leads and errors and mistakes that our heroes made. Only if we learn about these can we truly appreciate their triumphs. Only by studying the false paths they sometimes followed can we begin to appreciate them as real human beings and not as gods.

Klein (about a book by Salmon):

… like refreshing and instructive walks through wood and field and cultivated gardens, where the guide draws attention now to this beauty, now to that strange appearance, without forcing everything into a rigid system of faultless perfection . . . In this flower garden we have all grown up, here we have gathered the foundation of knowledge on which we have to build.

Truesdell

I will tell you these stories, not in the fashion of those textbook writerswho manufacture historical notices so as to bear out their own views of how science ought have developed, but instead as they really did occur.

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1 Response to Quotes Post:

  1. Poetry is imagination in a straitjacket that you can stretch as far as you want.

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