Hermann Weyl Quotes

Besides language and music mathematics is one of the primary manifestations of the free creative power of the human mind.

The objective world is, it does not happen. Only to the gaze of my consciousness, crawling along the lifeline of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time.

The goal of mathematics is the symbolic comprehension of the infinite with human, that is finite, means.

My work has always tried to unite the True with the Beautiful and when I had to choose one or the other, I usually chose the Beautiful.

Mathematics has the inhuman quality of starlight, brilliant and sharp, but cold.

The Greeks made Space the subject-matter of a science of supreme simplicity and certainty. Out of it grew, in the mind of classical antiquity, the idea of pure science. Geometry became one of the most powerful expressions of that sovereignty of the intellect that inspired the thought of those times. At a later epoch, when the intellectual despotism of the Church, which had been maintained through the Middle Ages, had crumbled, and a wave of scepticism threatened to sweep away all that had seemed most fixed, those who believed in Truth clung to Geometry as to a rock, and it was the highest ideal of every scientist to carry on his science 'more geometrico.

By refraining from reducing multiplication to addition we are enabled through these axioms to banish continuity, which is so difficult to fix precisely, from the logical structure of geometry.

Our generation is witness to a development of physical knowledge such as has not been seen since the days of Kepler, Galileo and Newton, and mathematics has scarcely ever experienced such a stormy epoch. Mathematical thought removes the spirit from its worldly haunts to solitude and renounces the unveiling of the secrets of Nature. But as recompense, mathematics is less bound to the course of worldly events than physics.

If Hilbert's view prevails over intuitionism, as appears to be the case, then I see in this a decisive defeat of the philosophical attitude of pure phenomenology, which thus proves to be insufficient for the understanding of creative science even in the area of cognition that is most primal and most readily open to evidence – mathematics.

Or, we may use Cartesian co-ordinate systems from the outset:

It is a tragic and strange fact , a superb malice of the creator , that man's mind is so immensely better suited for handling what is irrelevant than what is relevant to him.

With mathematics we stand precisely at that intersection of bondage and freedom that is the essence of the human itself.

Brouwer's remark is simple but deep: we have here the creation of the “continuum,” which, although containing individual real numbers, does not dissolve into a set of real numbers as finished beings; we rather have a medium of free Becoming. We found ourselves in the domain of an ageold problem of thought, the problem of continuity, of change, and of Becoming.

Now, if on the one hand it is very satisfactory to be able to give a common ground in the theory of knowledge for the many varieties of statements concerning space, spatial configurations, and spatial relations which, taken together, constitute geometry, it must on the other hand be emphasised that this demonstrates very clearly with what little right mathematics may claim to expose the intuitional nature of space. Geometry contains no trace of that which makes the space of intuition what it is in virtue of its own entirely distinctive qualities which are not shared by “states of addition-machines” and “gas-mixtures” and “systems of solutions of linear equations”. It is left to metaphysics to make this “comprehensible” or indeed to show why and in what sense it is incomprehensible. We as mathematicians have reason to be proud of the wonderful insight into the knowledge of space which we gain, but, at the same time, we must recognise with humility that our conceptual theories enable us to grasp only one aspect of the nature of space, that which, moreover, is most formal and superficial.

Two possibilities present themselves for the analytical treatment of metrical geometry.

the problem is then to develop a theory of invariance with respect to arbitrary linear transformations, in which, however, in contra-distinction to the case of affine geometry, we have a definite invariant quadratic form, viz. the metrical groundform once and for all as an absolute datum.

While Brouwer has made clear to us to what extent the intuitively certain falls short of the mathematically provable, Gödel shows conversely to what extent the intuitively certain goes beyond what (in an arbitrary but fixed formalism) is capable of mathematical proof.

We must here follow the first course so as to be able to pass on later to generalisations which extend beyond the limits of Euclidean geometry.

We now come to the decisive step of mathematical abstraction: we forget about what the symbols stand for. ...[The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for. Hermann Weyl, The Mathematical Way of Thinking

The question of the ultimate foundations and the ultimate meaning of mathematics remains open; we do not know in which direction it will find its final solution nor even whether a final objective answer can be expected at all. "Mathematizing" may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization.

To gaze up from the ruins of the oppressive present towards the stars is to recognise the indestructible world of laws, to strengthen faith in reason, to realise the "harmonia mundi" that transfuses all phenomena, and that never has been, nor will be, disturbed.

In Brouwer’s analysis, the individual place in the continuum, the real number, is to be defined not by a set but by a sequence of natural numbers, namely, by a law which correlates with every natural number n a natural number φ(n). . . How then do assertions arise which concern. . . all real numbers, i.e., all values of a real variable? Brouwer shows that frequently statements of this form in traditional analysis, when correctly interpreted, simply concern the totality of natural numbers. In cases where they do not, the notion of sequence changes its meaning: it no longer signifies a sequence determined by some law or other, but rather one that is created step by step by free acts of choice, and thus necessarily remains in statu nascendi. This “becoming” selective sequence (werdende Wahlfolge) represents the continuum, or the variable, while the sequence determined ad infinitum by a law represents the individual real number in the continuum. The continuum no longer appears, to use Leibniz’s language, as an aggregate of fixed elements but as a medium of free “becoming”. Of a selective sequence in statu nascendi, naturally only those properties can be meaningfully asserted which already admit of a yes-or-no decision (as to whether or not the property applies to the sequence) when the sequence has been carried to a certain point; while the continuation of the sequence beyond this point, no matter how it turns out, is incapable of overthrowing that decision.

Mathematics is the science of the infinite. The great achievement of the Greeks was to have made the tension between the finite and the infinite fruitful for the knowledge of reality. The feeling of the calm and unquestioning acknowledgement of the infinite belongs to the Orient, but for the East it remained a mere abstract awareness that left the concrete manifold of existence lying indifferently to one side, unshaped and impervious. Coming out of the Orient, the religious feeling of the infinite, the apeiron, took possession of the Greek soul in the Dionysian-Orphic epoch that preceded the Persian Wars. Here the Persian Wars also mark the release of the Occident from the Orient. That tension and its overcoming became for the Greeks the driving motive of knowledge. But every synthesis, as soon as it was achieved, permitted the old antithesis to break out anew in deepened form. Thus, it determined the history of theoretical knowledge into our time. Indeed, today we are compelled everywhere in the foundations of mathematics to return directly to the Greeks.

Mathematics is the science of the infinite. The great achievement of the Greeks was to have made the tension between the finite and the infinite fruitful for the knowledge of reality. The feeling of the calm and unquestioning acknowledgement of the infinite belongs to the Orient, but for the East it remained a mere abstract awareness that left the concrete manifold of existence lying indifferently to one side, unshaped and impervious. Coming out of the Orient, the religious feeling of the infinite, the apeiron, took possession of the Greek soul in the DionysianOrphic epoch that preceded the Persian Wars. Here the Persian Wars also mark the release of the Occident from the Orient. That tension and its overcoming became for the Greeks the driving motive of knowledge. But every synthesis, as soon as it was achieved, permitted the old antithesis to break out anew in deepened form. Thus, it determined the history of theoretical knowledge into our time. Indeed, today we are compelled everywhere in the foundations of mathematics to return directly to the Greeks.

Brouwer's remark is simple but deep: we have here the creation of the "continuum," which, although containing individual real numbers, does not dissolve into a set of real numbers as finished beings; we rather have a medium of free Becoming. We found ourselves in the domain of an ageold problem of thought, the problem of continuity, of change, and of Becoming.

Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the contact with intuition much more closely than had been done before. It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the larger part of his towering edifice which he believes to be built of concrete blocks dissolve into mist before his eyes.

However the power of science rests on the combination of experiment, i.e., observation under freely chosen conditions, with symbolic construction, and the latter is its mathematical aspect. Thus if science is found guilty, mathematics cannot evade the verdict.

The introduction of numbers and coordinates is an act of violence.

when placed in two positions successively, realises this idea of the equality of two portions of space ; by a rigid body we mean one which, however it be moved or treated, can always be made to appear the same to us as before, if we take up the appropriate position with respect to it. I shall