1The understanding of nature and life is our noblest task. All human striving and desire culminate in this goal, and we have experienced ever-increasing success in the process. In recent decades, we have gained a richer and deeper understanding of nature than were previously gained in as many centuries. Today we want to take advantage of this favourable state of things to discuss an old philosophical problem related to our subject—namely, the much-debated question about the respective contributions of thought on one side and experience on the other to our knowledge. This ancient question is justified, for to answer it is essentially to determine in what sense all the knowledge that we gather in our scientific enterprises is true knowledge [Wahrheit].

Without presumption toward the old philosophers and researchers, we can now more confidently expect a correct resolution of the question than before – for two reasons: the first is the aforementioned rapid pace at which our sciences are developing today. The significant discoveries of earlier times—from Copernicus, Kepler, Galileo, and Newton to Maxwell—were spread out at greater intervals over nearly four centuries. The modern era begins with the discovery of Hertzian waves. Now, discovery follows discovery: Röntgen discovers his rays, Curie discovers radioactivity, and Planck establishes quantum theory. And in the most recent times, discoveries of new phenomena and surprising interrelations have rushed one after another, so that the abundance of faces appears almost disconcerting: Rutherford’s theory of radioactivity, Einstein’s hν-law, Bohr’s explanation of spectra, Moseley’s numbering of the elements, Einstein’s theory of relativity, Rutherford’s disintegration of nitrogen, Bohr’s construction of the elements, Aston’s isotope theory.

Thus, in physics alone, we have witnessed an unbroken series of discoveries—and what discoveries they are! In magnitude, not a single one of them falls short of the achievements of earlier times; moreover, they are more closely packed in time, yet internally just as multifaceted as the past ones. And in this, theory and practice, thought and experience, are consistently revealed as continually and intimately intertwined. At times, theory races ahead; at other times, experiment takes the lead—always mutually confirming, complementing, and stimulating one another. The same applies to chemistry, astronomy, and the biological disciplines.

Thus, we have the advantage over the older philosophers of having witnessed a great number of such discoveries and of having observed the resulting shifts in perspective as they emerged. Among these new discoveries were many that altered or entirely eliminated old, deeply rooted conceptions and ideas. Consider, for instance, the new concept of time in the theory of relativity or the decay of chemical elements, and how these have dispelled prejudices that no one would have ever thought of challenging before.

But today, a second circumstance helps us in resolution of our old philosophical problem. Not only has the technique of experimentation and the art of constructing theoretical-physical frameworks reached unprecedented heights, but its counterpart—namely the science of logic—has likewise made significant progress. Today, there exists a general method for the theoretical treatment of scientific questions, which in all cases facilitates the precise formulation of problems and helps to prepare their solution—namely, the axiomatic method.

What, then, is the significance of this much-discussed axiomatics? The basic idea rests on the fact that, even in broad areas of knowledge, a few statements—called axioms—are usually sufficient to construct the entire theoretical framework purely through logic. But this remark does not exhaust its significance. The best way to grasp the axiomatic method is through examples. The oldest and most well-known example of the axiomatic method is Euclidean geometry. However, I would prefer to briefly illustrate the axiomatic method with a striking example from modern biology.

Drosophila is a small fly, but our interest in it is great; it has been the subject of the most extensive, careful, and successful breeding experiments. This fly is usually gray, red-eyed, spotless, round-winged, long-winged. But flies with deviating special characteristics also occur: instead of gray they are yellow, instead of red-eyed they are white-eyed, etc. Usually, these five special characteristics are linked, i.e., if a fly is yellow, then it is also white-eyed and spotted, has split wings and club wings. And if it has club wings, then it is also yellow and white-eyed, etc. From this usually occurring linkage, however, in suitable crosses, fewer deviations appear among the offspring, specifically in a quantitatively constant percentage. The numbers found experimentally in this way agree with the linear Euclidean axioms of congruence and the axioms concerning the geometric concept of 'between', and thus, as an application of the linear congruence axioms, i.e., the elementary geometric propositions about marking off segments, the laws of inheritance emerge; so simple and precise – and at the same time so wonderful, as no imagination, however bold, could have devised them.

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Another example of the axiomatic method in a completely different field is the following:
In our theoretical sciences, we are accustomed to the application of formal thought processes and abstract methods. The axiomatic method belongs to logic. When hearing the word logic, in many circles people think of something very tedious and difficult. Today, the logical science [die logische Wissenschaft] has become easy to understand and very interesting. For example, it has been realised that even in daily life, we use methods and conceptual constructions that require a high degree of abstraction and can only be understood through the unconscious application of axiomatic methods. For instance, the general process of negation and, in particular, the concept of 'infinite'. Regarding the concept of 'infinite', we must be clear that 'infinite' has no intuitive meaning and, without closer examination, no sense at all. For there are only finite things everywhere. There is no infinite speed and no infinitely fast propagating force or effect. Moreover, the action [Wirkung] itself is of a discrete nature and exists only in quanta. There is nothing continuous at all that could be divided infinitely often. Even light has an atomistic structure, just as magnitude of action [Wirkungsgröße]. Even the universe, I firmly believe, is only of finite extent, and someday astronomers will be able to tell us how many kilometers long, high, and wide the cosmos is. Although we often encounter cases of very large numbers in reality, e.g., the distances of stars in kilometers or the number of essentially different chess games possible, nevertheless endlessness or infinity is a monstrous abstraction – achievable only through the conscious or unconscious application of the axiomatic method – precisely because it is the negation of a universally prevailing state. This conception of the infinite, which I have substantiated through detailed investigations, resolves a series of fundamental questions; in particular, Kant's antinomies concerning space and unlimited divisibility thereby become unfounded/without object [gegestandlos], and thus the difficulties arising from them are solved.

Now, when we turn to our problem itself, how nature and thought are connected, we want to express three main viewpoints here. The first connects to the problem of infinity just discussed. We saw that the infinite is nowhere realised; it is neither present in nature nor permissible as a foundation in our thinking without special precautions. In this already I perceive an important parallelism between nature and thought, a fundamental agreement between experience and theory.

We see yet another parallelism: our thinking proceeds from unity and seeks to form unity; we observe the unity of substance in matter, and everywhere we confirm the unity of natural laws. In doing so, nature actually very much comes forth to us in our research, as if she were willing to reveal her secrets. The sparse distribution of mass in celestial space enabled the discovery and precise confirmation of Newton's law. Michelson, despite the great speed of light, was still able to establish with certainty the invalidity of the law of addition of velocities, because our Earth completes its orbit around the Sun just rapidly enough. And Mercury happens to do us the favour of executing the perihelion precession, so that we can test Einstein's theory on it. And the light rays from fixed stars pass just close enough to the sun for their deflection to be observed.

But even more striking is a phenomenon, which we, in a different sense than Leibniz, shall call the pre-established harmony, which is virtually an embodiment and realisation of mathematical thoughts. The older examples of this are the conic sections, which were studied long before it was suspected that our planets or even electrons moved in such paths. But the grandest and most wonderful case of the pre-established harmony is the famous Einsteinian theory of relativity. Here, solely through the general demand for invariance combined with the principle of greatest simplicity, the differential equations for the gravitational potential are mathematically uniquely established; and this formulation would have been impossible without the profound and difficult mathematical investigations of Riemann, which existed long before. In recent times, more and more instances have accumulated where the most important mathematical theories standing at the center of mathematical interest are at the same time precisely those needed in physics. I had developed the theory of infinitely many variables out of purely mathematical interest and had even used the term spectral analysis, without ever having suspected that these would later be realised in the actual spectrum of physics.

We can only understand this agreement [Übereinstimmung] between nature and thought, between experiment and theory, if we take into account the formal element, and the related mechanisms on both sides - that of nature and that of our mind [Verstand]. The mathematical process of elimination provides, it seems, the resting points and stations where bodies in the real world, as well as thoughts in the world of the mind, rest [verweilen] and thus offer themselves to inspection [Kontrolle] and comparison.

However, even this pre-established harmony does not yet exhaust the relations between nature and thought, nor does it reveal the deepest secrets of our problem. To arrive at this, let us examine the entire complex of physical-astronomical knowledge. We then notice in today's science a viewpoint that goes far beyond the older questions and goals of our science: it is the fact that today's science not only teaches how to predict future movements and expected phenomena from present data, in the sense of classical mechanics, but also shows that the present concrete states [gegenwärtigen tatsächlichen Zustände] of matter on Earth and in the universe are not accidental or arbitrary, but themselves follow from the physical laws.

The most important evidence for this are Bohr's atomic models, the structure of the stellar world [Sternenwelt], and finally the entire developmental history of organic life. The pursuit of these methods would then have to lead, it seems, to a system of natural laws that fits reality in its entirety, and then indeed only thought, i.e., conceptual deduction, would suffice to gain all physical knowledge; as if Hegel would have been right in his claim to be able to deduce all natural phenomena [Naturgeschehen] from concepts. But this conclusion is incorrect/inaccurate [unzutreffend]. For what about the origins of the laws of the world? How do we obtain such laws? And who teaches us that they fit reality? The answer is that experience alone makes this possible for us. In contrast to Hegel, we recognise that the laws of the world can be obtained in no other way but from experience. Although various speculative viewpoints may contribute to the construction of the framework of physical concepts: whether the established laws and the logical framework of concepts built from them are correct, only experience can decide. Sometimes an idea had its first origin in pure thought, such as Democritus's idea of atomism, whereas the existence of atoms was proven only two millennia later by experimental physics. Sometimes experience leads the way and forces the speculative viewpoint upon the mind. Thus, we owe it to the powerful impetus of the Michelson experiment that the deep-rooted prejudice of absolute time was cleared away and finally the idea of general relativity could be grasped by Einstein.

Whoever nevertheless wishes to deny that the laws of the world [Weltgesetze] originate from experience must assert that besides deduction and besides experience, there exists yet a third source of knowledge.
Indeed, philosophers – and Kant is the classic representative of this standpoint – have asserted that besides logic and experience, we have a certain a priori knowledge about reality. Now, I admit that certain a priori insights are necessary even for the construction of theoretical frameworks and that such insights always underlie the formation of our knowledge. I believe that even mathematical knowledge ultimately rests on a kind of such intuitive insight. And that we require a priori a certain intuitive stance [Einstellung] even for the construction of number theory. Thus, the most general fundamental idea of Kantian epistemology retains its significance: namely, the philosophical problem of establishing that intuitive stance a priori and thereby investigating the condition of the possibility of all conceptual knowledge and, at the same time, of all experience. I maintain that this has essentially been done in my investigations into the principles of mathematics. The a priori is thereby nothing more and nothing less than a fundamental stance or the expression for certain indispensable preconditions of thought and experience.

But the boundary between, on the one hand, what we possess a priori, and on the other hand, that for which experience is necessary, we must draw differently than Kant; Kant greatly overestimated the role and scope of the a priori.

In Kant's time, one could think that the conceptions of space and time that one had were applicable to reality just as generally and immediately as, for example, our conceptions of number, order [Reihenfolge], and magnitude, which we constantly use in mathematical and physical theories in the familiar way. Then, indeed, the doctrine of space and time, and thus geometry in particular, would be something that precedes all knowledge of nature, just as arithmetic does. But this viewpoint of Kant's was already abandoned – quite rightly – before the development of physics compelled it, notably by Riemann and Helmholtz; for geometry is nothing other than that part of the entire framework of physical concepts which depicts the possible positional relations of rigid bodies to one another in the world of real things. That movably rigid bodies exist at all, and what their positional relations are, is purely a matter of experience. The proposition that the sum of angles in a triangle equals two right angles and that the parallel axiom holds, is precisely, as Gauss already recognised, something to be established or refuted solely by experiment. If, for example, all the facts expressed by the congruence theorems proved to be in agreement with experience, but, in contrast, the sum of the angles in a triangle constructed from rigid rods turned out to be smaller than two right angles, no one would lapse into thinking that the parallel axiom was valid in the space of real bodies.

When accepting items into the a priori inventory, extreme caution is required; for indeed many insights formerly considered a priori have today even been recognised as incorrect. The most striking example of this is the notion of the absolute present [Gegenwart]. An absolute present does not exist, however much we may be accustomed from childhood to assuming it, since in daily life we are only dealing with short distances and slow movements. Were this otherwise, no one would have hit upon the idea of introducing absolute time. As it was, however, even such profound thinkers as Newton and Kant did not even think of doubting the absoluteness of time. The cautious Newton even formulated this requirement as starkly as possible: absolute, true time flows in itself and by virtue of its nature uniformly and without relation to any object whatsoever. Newton thereby frankly cut off any retreat or compromise, and Kant, the critical philosopher, proved here not critical at all, by accepting Newton without further ado. Only Einstein definitively freed us from this prejudice – this will always remain one of the mightiest deeds of the human spirit – and the overly extensive a priori theory could not have been more strikingly reduced to absurdity than by this advance in physics. For the assumption of absolute time entails, among other things, the proposition of the addition of velocities when composing two velocities – incidentally, a proposition that in itself could scarcely be surpassed in apparent evidence and popular comprehensibility – and yet it emerged compellingly from the most diverse experiments in the fields of optics, astronomy, and electricity theory that this proposition of the addition of velocities is not correct; in fact, another, more complicated law holds for the composition of two velocities.

We can say: in recent times the view represented by Gauss and Helmholtz concerning the empirical nature of geometry has become a firmly established result of science. It must today serve as a firm reference point for all philosophical speculations concerning space and time. For Einstein's theory of gravitation makes it obvious: geometry is nothing but a branch of physics; geometric truths are in no single respect fundamentally different in status or kind from physical ones. Thus, for example, the Pythagorean theorem and Newton's law of gravitation are intrinsically related, insofar as they are governed by the same fundamental physical concept, that of potential. But even more is certain for anyone familiar with Einstein's theory of gravitation: these two laws, so dissimilar and hitherto seemingly separated by vast distances—the one a theorem of elementary geometry known since antiquity and since taught everywhere in schools, the other a law concerning the effect of masses upon one another—are not merely of the same character, but are only parts of one and the same general law.

The fundamental similarity of geometric and physical facts could hardly have come to light more drastically. Of course, in the usual logical construction and in our ordinary daily experiences, familiar from childhood, the geometric and kinematic propositions precede the dynamic ones, and this circumstance explains why it was forgotten that they are experiences at all. Thus we see: Kant's a priori theory still contains anthropomorphic residues, from which it must be freed, and after whose removal only that a priori stance remains which also underlies purely mathematical knowledge: it is essentially the finite stance characterised by me in various treatises2.

The instrument that enables the mediation between theory and practice, between thought and observation, is mathematics; it builds the connecting bridge and makes it ever more robust. Hence it comes about that our entire present culture, insofar as it rests on the intellectual penetration and harnessing of nature, finds its foundation in mathematics. Already Galileo said: Nature can only be understood by those who have learned its language and the signs in which it speaks to us; but this language is mathematics, and its signs are mathematical figures. Kant made the statement: "I maintain that in any particular natural science, only so much genuine science can be found as there is mathematics contained therein." Indeed: We do not master a scientific theory until we have peeled out its mathematical core and completely unveiled it. Without mathematics, today's astronomy and physics are impossible; these sciences, in their theoretical parts, virtually dissolve into mathematics. It is to these and the numerous further applications that mathematics owes its prestige, to the extent that it enjoys such among the wider public.

Nevertheless, mathematicians have refused to let applications be the measure of the value for mathematics. The prince of mathematicians, Gauss, who was undoubtedly also an applied mathematician par excellence, who created anew entire sciences, such as error theory and geodesy, in order to let mathematics play the leading role therein, who, when the astronomers had lost the newly discovered planet Ceres – a particularly important and interesting planet – and could not find it again, devised a new mathematical theory based on which he correctly predicted the location of Ceres, who invented the telegraph and many other practical things, was nevertheless of the same opinion. Pure number theory is that field of mathematics which has hitherto never found application. But it is precisely number theory that is called the queen of mathematics by Gauss and is glorified by him and almost all great mathematicians. Gauss speaks of the magical charm that made number theory the favourite science of the first mathematicians, not to mention its inexhaustible richness, in which it so far surpasses all other parts of mathematics. Gauss describes how already in his early youth the charms of number-theoretic investigations so captivated him that he could no longer let them go. He praises Fermat, Euler, Lagrange, and Legendre as men of incomparable fame because they opened up access to the sanctuary of this divine science and showed how full of great riches it is. And mathematicians before and after Gauss, such as Lejeune, Dirichlet, Kummer, Hermite, Kronecker, and Minkowski, express themselves with quite similar enthusiasm – Kronecker compares number theorists to the Lotus-eaters, who, once they have partaken of this delight, can never let go of it again.

Poincaré too, the most brilliant mathematician of his generation, who was essentially also a physicist and astronomer, is of the same opinion. Poincaré once turned with striking sharpness against Tolstoy, who had declared that the demand "science for science's sake" was foolish. "Should we," Tolstoy had said, "allow ourselves to be guided in the choice of our occupation by the whim of our curiosity? Would it not be better to make the decision according to usefulness, i.e., according to our practical and moral needs?" It is peculiar that it is precisely Tolstoy whom we mathematicians must reject here as a flat realist and cold-hearted/narrow-minded [enghertzig] utilitarian. Poincaré argues against Tolstoy that if one had proceeded according to Tolstoy's recipe, science would never have arisen at all. One only needs to open one's eyes, Poincaré concludes, to see how, for example, the achievements of industry would never have seen the light of day if these practitioners alone had existed and if these achievements had not been fostered by disinterested fools who never thought of practical exploitation. We are all of the same opinion.

Our great Königsberg mathematician Jacobi¹ also thought this way—Jacobi, whose name stands beside Gauss's and is still mentioned with reverence today by every student of our subjects. When the famous Fourier had once said that the main purpose of mathematics lay in the explanation of natural phenomena, it was Jacobi who rebuked him with the full passion of his temperament. A philosopher, such as Fourier surely was, ought to have known, Jacobi exclaims, that the honour of the human spirit is the sole purpose of all science and that from this viewpoint, a problem of pure number theory is worth just as much as one that serves applications.

Whoever feels the truth of the generous way of thinking and worldview that shines forth from these words of Jacobi's will not fall prey to regressive and fruitless skepticism; such a person will not believe those who today, with a philosophical air and superior tone, prophesy the downfall of culture and content themselves with the Ignorabimus. For the mathematician, there is no Ignorabimus, and, in my opinion, not for natural science at all either. Once the philosopher Comte – intending to name a certainly insoluble problem – said that science would never succeed in fathoming the secret of the chemical composition of celestial bodies. A few years later, this problem was solved by the spectral analysis of Kirchhoff and Bunsen, and today we can say that we make use of the most distant stars as the most important physical and chemical laboratories, such as we cannot find anywhere on Earth. The true reason, in my opinion, why Comte did not succeed in finding an insoluble problem is that there is no such thing as an insoluble problem at all. On the contrary, instead of the foolish Ignorabimus, let our slogan be:
We must know,
We shall know.

¹ David Hilbert delivered this address in Königsberg, his and Jacobi's hometown.