Hilbert Quotes

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

Geometry is the most complete science. ~ 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

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 Hilbert space no one can hear you scream. ~ Yakir Aharonov

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

In Hilbert space, the classical states are just in one corner of the room, quantum attention functions are in the middle, and they are the matrix of cosmic functions that can collapse any other quantum wave function and transform the non-classical states into classical states in the Hilbert space. ~ Amit Ray

The essence of Hilbert's program was to find a decision process that would operate on symbols in a purely mechanical fashion, without requiring any understanding of their meaning. Since mathematics was reduced to a collection of marks on paper, the decision process should concern itself only with the marks and not with the fallible human intuitions out of which the marks were reduced. In spite of the prolonged efforts of Hilbert and his disciples, the Entscheidungsproblem was never solved. Success was achieved only in highly restricted domains of mathematics, excluding all the deeper and more interesting concepts. Hilbert never gave up hope, but as the years went by his program became an exercise in formal logic having little connection with real mathematics. Finally, when Hilbert was seventy years old, Kurt Godel proved by a brilliant analysis that the Entscheindungsproblem as Hilbert formulated it cannot be solved.

Godel proved that in any formulation of mathematics, including the rules of ordinary arithmetic, a formal process for separating statements into true and false cannot exist. He proved the stronger result which is now known as Godel's theorem, that in any formalization of mathematics including the rules of ordinary arithmetic there are meaningful arithmetical statements that cannot be proved true or false. Godel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not ~ Freeman Dyson

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

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

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

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

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

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

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

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 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

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

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

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

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

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

Good, he did not have enough imagination to become a mathematician.

[Upon hearing that one of his students had dropped out to study poetry] ~ David Hilbert

Keep computations to the lowest level of the multiplication table. ~ 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

Imagine what would have happened had the logicist endeavor been entirely successful. This would have implied that mathematics stems fully from logic-literally from the laws of thought. But how could such a deductive science so marvelously fit natural phenomena? What is the relation between formal logic (maybe we should even say human formal logic) and the cosmos? The answer did not become any clearer after Hilbert and Godel. Now all that existed was an incomplete formal "game," expressed in mathematical language. How could models based on such an "unreliable" system produce deep insights about the universe and its workings? ~ Mario Livio

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

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

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

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

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

Relativity theory applies to macroscopic bodies, such as stars. The event of coincidence, that is, in ultimate analysis of collision, is the primitive event in the theory of relativity and defines a point in space-time, or at least would define a point if the colliding panicles were infinitely small. Quantum theory has its roots in the microscopic world and, from its point of view, the event of coincidence, or of collision, even if it takes place between particles of no spatial extent, is not primitive and not at all sharply isolated in space-time. The two theories operate with different mathematical conceptsãthe four dimensional Riemann space and the infinite dimensional Hilbert space, respectively. So far, the two theories could not be united, that is, no mathematical formulation exists to which both of these theories are approximations. All physicists believe that a union of the two theories is inherently possible and that we shall find it. Nevertheless, it is possible also to imagine that no union of the two theories can be found. This example illustrates the two possibilities, of union and of conflict, mentioned before, both of which are conceivable. ~ Eugene Paul Wigner

We must know. We will know. ~ David Hilbert

But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results. ~ David Hilbert

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

Physics is becoming too difficult for the physicists. ~ 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

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

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

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

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

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

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

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

Physics is much too hard for physicists. ~ 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

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

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

Certainly not! I didn't build a machine to solve ridiculous crossword puzzles! That's hack work, not Great Art! Just give it a topic, any topic, as difficult as you like..."
Klapaucius thought, and thought some more. Finally he nodded and said:
"Very well. Let's have a love poem, lyrical, pastoral, and expressed in the language of pure mathematics. Tensor algebra mainly, with a little topology and higher calculus, if need be. But with feeling, you understand, and in the cybernetic spirit."
"Love and tensor algebra?" Have you taken leave of your senses?" Trurl began, but stopped, for his electronic bard was already declaiming:

Come, let us hasten to a higher plane,
Where dyads tread the fairy fields of Venn,
Their indices bedecked from one to n,
Commingled in an endless Markov chain!

Come, every frustum longs to be a cone,
And every vector dreams of matrices.
Hark to the gentle gradient of the breeze:
It whispers of a more ergodic zone.

In Reimann, Hilbert or in Banach space
Let superscripts and subscripts go their ways.
Our asymptotes no longer out of phase,
We shall encounter, counting, face to face.

I'll grant thee random access to my heart,
Thou'lt tell me all the constants of thy love;
And so we two shall all love's lemmas prove,
And in bound partition never part.

For what did Cauchy know, or Christoffel,
Or Fourier, or any Boole or Euler, ~ Stanisław Lem

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

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

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

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