Hilbert Quotes & Sayings

Every mathematical discipline goes through three periods of development: the naive, the formal, and the critical. — David Hilbert

Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof. — David Hilbert

In mathematics ... we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials ... being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations. — David Hilbert

No mathematician of equal stature has risen from our generation ... Hilbert was singularly free from national and racial prejudices; in all public questions, be they political, social or spiritual, he stood forever on the side of freedom. — Hermann Weyl

I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, the Senate is not a bathhouse. Objecting to sex discrimination being the reason for rejection of Emmy Noether's application to join the faculty at the University of Gottingen. — David Hilbert

Some people have got a mental horizon of radius zero and call it their point of view. — David Hilbert

The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas. — David Hilbert

[On Cantor's work:] The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity. — David Hilbert

I didn't work especially hard at mathematics at school, because I knew that's what I'd be doing later. — David Hilbert

So the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Nothing else has a baggage-free description. In other words, we all live in a gigantic mathematical object-one that's more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today's most advanced physics theories. Everything in our world is purely mathematical-including you. — Max Tegmark

How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments. — David Hilbert

Mathematics is a game played according to certain simple rules with meaningless marks on paper. — David Hilbert

We must know. We will know. — David Hilbert

In Hilbert space no one can hear you scream. — Yakir Aharonov

We do not master a scientific theory until we have shelled and completely prised free its mathematical kernel. — David Hilbert

Is mathematics doomed to suffer the same fate as other sciences that have split into separate branches? ... Mathematics is, in my opinion, an indivisible whole ... May the new century bring with it ingenious champions and many zealous and enthusiastic disciples. — David Hilbert

In Oppenheimer's view, it would be a waste of his precious time, or of mine, to concern ourselves with the details of particular solutions. This was how the philosophy of reductionism led Oppenheimer and Einstein astray. Since the only purpose of physics was to reduce the world of physical phenomena to a finite set of fundamental equations, the study of particular solutions such as black holes was an undesirable distraction from the general goal. Like Hilbert, they were not content to solve particular problems one at a time. They were entranced by the dream of solving all the basic problems at once. And as a result, they failed in their later years to solve any problems at all. — Freeman Dyson

Before beginning [to try to prove Fermat's Last Theorem] I should have to put in three years of intensive study, and I haven't that much time to squander on a probable failure. — David Hilbert

Galileo was no idiot. Only an idiot could believe that science requires martyrdom - that may be necessary in religion, but in time a scientific result will establish itself. — David Hilbert

No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite. — David Hilbert

Hilbert once had a student in mathematics who stopped coming to his lectures, and he was finally told the young man had gone off to become a poet. Hilbert is reported to have remarked: 'I never thought he had enough imagination to be a mathematician.' — George Polya

Dr. Jules Hilbert: Hell Harold, you could just eat nothing but pancakes if you wanted.
Harold Crick: What is wrong with you? Hey, I don't want to eat nothing but pancakes, I want to live! I mean, who in their right mind in a choice between pancakes and living chooses pancakes?
Dr. Jules Hilbert: Harold, if you pause to think, you'd realize that that answer is inextricably contingent upon the type of life being led ... and, of course, the quality of the pancakes. — Zach Helm

The infinite! No other question has ever moved so profoundly the spirit of man. — David Hilbert

Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics. — David Hilbert

The total subject of mathematics is clearly too broad for any of us. I do not think that any mathematician since Gauss has covered it uniformly and fully; even Hilbert did not and all of us are of considerably lesser width quite apart from the question of depth than Hilbert. — John Von Neumann

Physics is becoming too difficult for the physicists. — David Hilbert

I would like to make a confession which may seem immoral: I do not believe in Hilbert space anymore. — John Von Neumann

If indeed, as Hilbert asserted, mathematics is a meaningless game played with meaningless marks on paper, the only mathematical experience to which we can refer is the making of marks on paper. — E. T. Bell

One must be able to say at all times
instead of points, straight lines, and planes
tables, chairs, and beer mugs — David Hilbert

In Riemann, Hilbert or in Banach space Let superscripts and subscripts go their ways. Our symptotes no longer out of phase, We shall encounter, counting, face to face. — Stanislaw Lem

The tool which serves as intermediary between theory and practice, between thought and observation, is mathematics; it is mathematics which builds the linking bridges and gives the ever more reliable forms. — David Hilbert

People who roll their eyes are stupid, — Andrew Hilbert

One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it. — David Hilbert

However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes. — David Hilbert

Indignant reply to the blatent sex discrimination expressed in a colleague's opposition when Hilbert proposed appointing Emmy Noether as the first woman professor at their university. — David Hilbert

In 1931, Kurt Godel proved in his famous second incompleteness theorem that there could be no finitary proof of the consistency of arithmetic. He had killed Hilbert's program with a single stroke.

So should you be worried that all of mathematics might collapse tomorrow afternoon? For what it's worth, I'm not. I do believe in infinite sets, and I find the proofs of consistency that use infinite sets to be convincing enough to let me sleep at night. — Jordan Ellenberg

Geometry, like arithmetic, requires for its logical development only a small number of simple, fundamental principles. These fundamental principles are called the axioms of geometry. — David Hilbert

Von Neumann, by contrast, wore a three-piece suit at almost all times, including on a donkey ride down the Grand Canyon; even as a student he was so well dressed that, upon first meeting him, the mathematician David Hilbert reportedly had but one question: Who is his tailor?45 — Walter Isaacson

Science in its everyday practice is much closer to art than to philosophy. When I look at Godel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as Chartres Cathedral. Godel took Hilbert's formalized axioms of mathematics as his building blocks and built out of them a lofty structure of ideas into which he could finally insert his undecidable arithmetical statement as the keystone of the arch. The proof is a great work of art. It is a construction, not a reduction. It destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas. Godel proved that in mathematics the whole is always greater than the sum of the parts. Every formalization of mathematics raises questions that reach beyond the limits of the formalism into unexplored territory. — Freeman Dyson

As long as a branch of science offers an abundance of problems, so long it is alive; a lack of problems foreshadows extinction or the cessation of independent development. — David Hilbert

Plenty of mathematicians, Hardy knew, could follow a step-by-step discursus unflaggingly-yet counted for nothing beside Ramanujan. Years later, he would contrive an informal scale of natural mathematical ability on which he assigned himself a 25 and Littlewood a 30. To David Hilbert, the most eminent mathematician of the day, he assigned an 80. To Ramanujan he gave 100. — Robert Kanigel

When people asked hilbert why he didn't prove Fermat's Last Theorem and win the Wolfskehl Prize, he said, "Why should I kill the goose that lays the golden egg? — Constance Bowman Reid

An old French mathematician said: A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us. — David Hilbert

If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven? — David Hilbert

I do not want to presuppose anything as known. I see in my explanation in section 1 the definition of the concepts point, straight line and plane, if one adds to these all the axioms of groups i-v as characteristics. If one is looking for other definitions of point, perhaps by means of paraphrase in terms of extensionless, etc., then, of course, I would most decidedly have to oppose such an enterprise. One is then looking for something that can never be found, for there is nothing there, and everything gets lost, becomes confused and vague, and degenerates into a game of hide and seek. — David Hilbert

This is not mathematics, it is theology.
(On being exposed to Hilbert's work in invariant theory.) — Paul Gordan

Physics is much too hard for physicists. — David Hilbert

Every town has a psychopath or two. Not just the everyday crazy person, either. Not like Crazy Larry, the paint huffing weirdo peddling around town on a child-sized Huffy ranting about the end of the world, or the old lady dressed in rags who hands out filthy doll clothes to the kiddies. I'm talking about the cold, never remorseful lunatic, who may never have seemed insane up until the day he hacked apart his mother and shoved her stinking corpse into the attic. This town is overflowing with them; bloodcurdling murderers like Kenny Wayne Hilbert, Charlie Fender ... Orland Winthro. And Al, the crazy had to come from somewhere. — Nikki Ferguson

The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science. — David Hilbert

The entire world is understood as a living interrelated and interactive system. Thus, if a tree outside of the zendo is damaged, I am also damaged. Just so, if I am damaged, so too, the Universe. — Harvey Daiho Hilbert-roshi

No one shall expel us from the paradise which Cantor has created for us.

{Expressing the importance of Georg Cantor's set theory in the development of mathematics.} — David Hilbert

A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution. — David Hilbert

To the average mathematician who merely wants to know his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency ... The Realist position is probably the one which most mathematicians would prefer to take. It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism, while his normal position will be somewhere between the two, trying to enjoy the best of two worlds. — Paul Cohen

Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of
our science and at the secrets of its development during future centuries? What particular goals will there be toward
which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the
wide and rich field of mathematical thought will the new centuries disclose? — David Hilbert

In other words, the idea is the there's a fourth level of parallel universes that's vastly larger than the three we've encountered so far, corresponding to different mathematical structures. The first three levels correspond to noncommunicating parallel universes within the same mathematical structure: Level I simply means distant regions from which light hasn't yet had time to reach us, Level II covers regions that are forever unreachable because of the cosmological inflation of intervening space, and Level III, Everett's "Many Worlds," involves noncommunicating parts of the Hilbert space of quantum mechanics. Whereas all the parallel universes at Levels I, II and III obey the same fundamental mathematical equations (describing quantum mechanics, inflation, etc.), Level IV parallel universes dance to the tunes of different equations, corresponding to different mathematical structures. Figure 12.2 illustrates this four-level multiverse hierarchy, one of the core ideas of this book. — Max Tegmark

Quantum phenomena do not occur in a Hilbert space. They occur in a laboratory. — Asher Peres

Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country. — David Hilbert

Sometimes it happens that a man's circle of horizon becomes smaller and smaller, and as the radius approaches zero it concentrates on one point. And then that becomes his point of view. — David Hilbert

Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. — David Hilbert

A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. — David Hilbert

We ought not to believe those who today, adopting a philosophical air and with a tone of superiority, prophesy the decline of culter and are content with the unknowable in a self-satisfied way. For us there is no unknowable, and in my opinion there is also non whatsoever for the natural sciences. In place of this foolish unknowable, let our watchword on the contrary be: we must know - we shall know. — David Hilbert

Keep computations to the lowest level of the multiplication table. — David Hilbert

But in the present century, thanks in good part to the influence of Hilbert, we have come to see that the unproved postulates with which we start are purely arbitrary. They must be consistent, they had better lead to something interesting. — Julian Coolidge

I tried reading Hilbert. Only his papers published in mathematical periodicals were available at the time. Anybody who has tried those knows they are very hard reading. — Alonzo Church

The art of doing mathematics consists in finding that special case which contains all the germs of generality. — David Hilbert

Begin with the simplest examples. — David Hilbert

I have tried to avoid long numerical computations, thereby following Riemann's postulate that proofs should be given through ideas and not voluminous computations. — David Hilbert

Later Michel went up to the priest as he was packing away the tools of the trade. "I was very interested in what you were saying earlier ... " The man of God smiled urbanely, then Michel began to talk about the Aspect experiments and the EPR paradox: how two particle, once united, are forever and inseparable whole, "which seems pretty much in keeping with what you were saying about one flesh." The priest's smile froze slightly. "What I'm trying to say, "Michel went on enthusiastically, "is that from an ontological point of view, the pair can be assigned a single vector in Hilbert space. Do you see what I mean? — Michel Houellebecq

Geometry is the most complete science. — David Hilbert

One hears a lot of talk about the hostility between scientists and engineers. I don't believe in any such thing. In fact I am quite certain it is untrue ... There cannot possibly be anything in it because neither side has anything to do with the other. — David Hilbert

He who seeks for methods without having a definite problem in mind seeks in the most part in vain. — David Hilbert

If one were to bring ten of the wisest men in the world together and ask them what was the most stupid thing in existence, they would not be able to discover anything so stupid as astrology. — David Hilbert