Axiomatic Thinking (Hilbert, 1918)

Wednesday September 23, 2020

I had difficulty finding this online, so I'm posting it here for reference.

Just as in the life of nations the individual nation can only thrive when all neighbouring nations are in good health; and just as the interest of states demands, not only that order prevail within every individual state, but also that the relationships of the states among themselves be in good order; so it is in the life of the sciences. In due recognition of this fact the most important bearers of mathematical thought have always evinced great interest in the laws and the structure of the neighbouring sciences; above all for the benefit of mathematics itself they have always cultivated the relations to the neighbouring sciences, especially to the great empires of physics and epistemology. I believe that the essence of these relations, and the reason for their fruitfulness, will appear most clearly if I describe for you the general method of research which seems to be coming more and more into its own in modern mathematics: I mean the axiomatic method.

When we assemble the facts of a definite, more-or-less comprehensive field of knowledge, we soon notice that these facts are capable of being ordered. This ordering always comes about with the help of a certain framework of concepts [Fachwerk von Begriffen] in the following way: a concept of this framework corresponds to each individual object of the field of knowledge, and a logical relation between concepts corresponds to every fact within the field of knowledge. The framework of concepts is nothing other than the theory of the field of knowledge.

Thus the facts of geometry order themselves into a geometry, the facts of arithmetic into a theory of numbers, the facts of statics, mechanics, electrodynamics into a theory of statics, mechanics, electrodynamics, or the facts from the physics of gases into a theory of gases. It is precisely the same with the fields of knowledge of thermodynamics, geometrical optics, elementary radiation-theory, the conduction of heat, or also with the calculus of probabilities or the theory of sets. It even holds of special fields of knowledge in pure mathematics, such as the theory of surfaces, the theory of Galois equations, and the theory of prime numbers, no less than for several fields of knowledge that lie far from mathematics, such as certain parts of psychophysics or the theory of money.

If we consider a particular theory more closely, we always see that a few distinguished propositions of the field of knowledge underlie the construction of the framework of concepts, and these propositions then suffice by themselves for the construction, in accordance with logical principles, of the entire framework.

Thus in geometry the proposition of the linearity of the equation of the plane and of the orthogonal transformation of point-coordinates is completely adequate to produce the whole broad science of spatial Euclidean geometry purely by means of analysis. Moreover, the laws of calculation and the rules for integers suffice for the construction of number theory. In statics the same role is played by the proposition of the parallelogram of forces; in mechanics, say, by the Lagrangian differential equations of motion; and in electrodynamics by the Maxwell equations together with the requirement of the rigidity and charge of the electron. Thermodynamics can be completely built up from the concept of energy function and the definition of temperature and pressure as derivatives of its variables, entropy and volume. At the heart of the elementary theory of radiation is Kirchhoff's theorem on the relationships between emission and absorption; in the calculus of probabilities the Gaussian law of errors is the fundamental proposition; in the theory of gases, the proposition that entropy is the negative logarithm of the probability of the state; in the theory of surfaces, the representation of the element of arc by the quadratic differential form; in the theory of equations, the proposition concerning the existence of roots; in the theory of prime numbers, the proposition concerning the reality and frequency of Rieman's function 𝜁(s).

These fundamental propositions can be regarded from an initial standpoint as the axioms of the individual fields of knowledge: the progressive development of the individual field of knowledge then lies solely in the further logical construction of the already mentioned framework of concepts. This standpoint is especially predominant in pure mathematics, and to the corresponding manner of working we owe the mighty development of geometry, of arithmetic, of the theory of functions, and of the whole of analysis.

Thus in the cases mentioned above the problem of grounding the individual field of knowledge had found a solution; but this solution was only temporary. In fact, in the individual fields of knowledge the need arose to ground the fundamental axiomatic propositions themselves. So one acquired 'proofs' of the linearity of the equation of the plane and the orthogonality of the transformation expressing a movement, of the laws of arithmetical calculation, of the parallelogram of forces, of the Lagrangian equations of motion, of Kirchhoff's law regarding emission and absorption, of the law of entropy, and of the proposition concerning the existence of roots of an equation.

But critical examination of these 'proofs' shows that they are not in themselves proofs, but basically only make it possible to trace things back to certain deeper propositions, which in turn are now to be regarded as new axioms instead of the propositions to be proved. The actual so-called axioms of geometry, arithmetic, statics, mechanics, radiation theory, or thermodynamics arose in this way. These axioms form a layer of axioms which lies deeper than the axiom-layer given by the recently-mentioned fundamental theorems of the individual field of knowledge. The procedure of the axiomatic method, as it is expressed here, amounts to a deepening of the foundations of the individual domains of knowledge—a deepening that is necessary for every edifice that one wishes to expand and to build higher while preserving its stability.

If the theory of a field of knowledge—that is, the framework of concepts that represents it—is to serve its purpose of orienting and ordering, then it must satisfy two requirements above all: first it should give us an overview of the independence and dependence of the propositions of the theory; second, it should give us a guarantee of the consistency of all the propositions of the theory. In particular, the axioms of each theory are to be examined from these two points of view.

Let us first consider the independence or dependence of the axioms.

The axiom of parallels in geometry is the classical example of the independence of an axiom. When he placed the parallel postulate among the axioms, Euclid thereby denied that the proposition of parallels is implied by the other axioms. Euclid's method of investigation became the paradigm for axiomatic research, and since Euclid geometry has been the prime example of an axiomatic science.

Classical mechanics furnishes another example of an investigation of the independence of axioms. The Lagrangian equations of motion were temporarily able to count as axioms of mechanics—for mechanics can of course be entirely based on these equations when they are generally formulated for arbitrary forces and arbitrary side-constraints. But further investigation shows that it is not necessary in the construction of mechanics to presuppose arbitrary forces or arbitrary side-constraints; thus the system of presuppositions can be reduced. This piece of knowledge leads, on the one hand, to the axiom system of Boltzmann, who assumes only forces (and indeed special central forces) but no side-constraints, and the axiom system of Hertz, who discards forces and makes do with side-constraints (and indeed special side-constraints with rigid connections). These two axiom systems form a deeper layer in the progressive axiomatization of mechanics.

If in establishing the theory of Galois equations we assume as an axiom the existence of roots of an equation, then this is certainly a dependent axiom; for, as Gauss was the first to show, that existence theorem can be proved from the axioms of arithmetic.

Something similar would happen if we were to assume as an axiom in the theory of prime numbers the proposition about the reality of the zeroes of the Riemann 𝜁(s)-function: as we progress to a deeper layer of purely arithmetical axioms the proof of this reality-proposition would become necessary, and only this proof would guarantee the reliability of the important conclusions which we have already achieved for the theory of prime numbers by taking it as a postulate.

A particularly interesting question for axiomatics concerns the independence of the propositions of a field of knowledge from the axiom of continuity.

In the theory of real numbers it is shown that the axiom of measurement—the so-called Archimedean axiom—is independent of all the other arithmetical axioms. As everybody knows, this information is of great significance for geometry; but it seems to me to be of capital interest for physics as well, for it leads to the following result: the fact that by adjoining terrestrial distances to one another we can achieve the dimensions and distances of bodies in outer space (that is, that we can measure heavenly distances with an earthly yardstick) and the fact that the distances within an atom can all be expressed in terms of metres—these facts are not at all a mere logical consequence of propositions about the congruence of triangles or about geometric configurations, but are a result of empirical research. The validity of the Archimedean axiom in nature stands in just as much need of confirmation by experiment as does the familiar proposition about the sum of the angles of a triangle.

In general, I should like to formulate the axiom of continuity in physics as follows: 'If for the validity of a proposition of physics we prescribe any degree of accuracy whatsoever, then it is possible to indicate small regions within which the presuppositions that have been made for the proposition may vary freely, without the deviation of the proposition exceeding the prescribed degree of accuracy.' This axiom basically does nothing more than express something that already lies in the essence of experiment; it is constantly presupposed by the physicists, although it has not previously been formulated.

For example, if one follows Planck and derives the second law of thermodynamics from the axiom of the impossibility of a perpetuum mobile of the second sort, then this axiom of continuity must be used in the derivation.

By invoking the theorem that the continuum can be well-ordered, Hamel has shown in a most interesting manner that, in the foundations of statics, the axiom of continuity is necessary for the proof of the theorem concerning the parallelogram of forces—at any rate, given the most obvious choice of other axioms.

The axioms of classical mechanics can be deepened if, using the axiom of continuity, one imagines continuous motion to be decomposed into small straight-line movements caused by discrete impulses and following one another in rapid succession. One then applies Bertrand's maximum principle as the essential axiom of mechanics, according to which the motion that actually occurs after each impulse is that which maximizes the kinetic energy of the system with respect to all motions that are compatible with the law of the conservation of energy.

The most recent ways of laying the foundations of physics—of electrodynamics in particular—are all theories of the continuum, and therefore raise the demand for continuity in the most extreme fashion. But I should prefer not to discuss them because the investigations are not yet completed.

We shall now examine the second of the two points of view mentioned above, namely, the question concerning the consistency of the axioms. This question is obviously of the greatest importance, for the presence of a contradiction in a theory manifestly threatens the contents of the entire theory.

Even for successful theories that have long been accepted, it is difficult to know that they are internally consistent: I remind you of the reversibility and recurrence paradox in the kinetic theory of gases.

It often happens that the internal consistency of a theory is regarded as obvious, while in reality the proof requires deep mathematical developments. For example, consider a problem from the elementary theory of the conduction of heat—namely, the distribution of temperatures within a homogeneous body whose surfaces are maintained at a definite temperature that varies from place to place: then in fact the requirement that there be an equilibrium of temperatures involves no internal theoretical contradiction. But to know this it is necessary to prove that the familiar boundary-value problem of potential theory is always solvable; for only this proof shows that a temperature distribution satisfying the equations of the conduction of heat is at all possible.

But particularly in physics it is not sufficient that the propositions of a theory be in harmony with each other; there remains the requirement that they not contradict the propositions of a neighbouring field of knowledge.

Thus, as I showed earlier, the axioms of the elementary theory of radiation can be used to prove not only Kirchhoff's law of emission and absorption, but also a special law about the reflection and refraction of individual beams of light, namely, the law: If two beams of natural light and of the same energy each fall on the surface separating two media from different sides in such a way that one beam after its reflection, and the other after its passage, each have the same direction, then the beam that arises from uniting the two is also of natural light and of the same energy. This theorem is, as the facts show, not at all in contradiction with optics, but can be derived as a conclusion from the electromagnetic theory of light.

As is well known, the results of the kinetic theory of gases are in full harmony with thermodynamics.

Similarly, electrodynamic inertia and Einsteinian gravitation are compatible with the corresponding concepts of the classical theories, since the classical concepts can be conceived as limiting cases of the more general concepts in the new theories.

In contrast, modern quantum theory and our developing knowledge of the internal structure of the atom have led to laws which virtually contradict the earlier electrodynamics, which was essentially built on the Maxwell equations; modern electrodynamics therefore needs—as everybody acknowledges—a new foundation and essential reformulation.

As one can see from what has already been said, the contradictions that arise in physical theories are always eliminated by changing the selection of the axioms; the difficulty is to make the selection so that all the observed physical laws are logical consequences of the chosen axioms.

But matters are different when contradictions appear in purely theoretical fields of knowledge. Set theory contains the classic example of such an occurrence, namely, in the paradox of the set of all sets, which goes back to Cantor. This paradox is so serious that distinguished mathematicians, for example, Kronecker and Poincaré, felt compelled by it to deny that set theory—one of the most fruitful and powerful branches of knowledge anywhere in mathematics—has any justification for existing.

But in this precarious state of affairs as well, the axiomatic method came to the rescue. By setting up appropriate axioms which in a precise way restricted both the arbitrariness of the definitions of sets and the admissibility of statements about their elements, Zermelo succeeded in developing set theory in such a way that the contradictions disappear, but the scope and applicability of set theory remain the same.

In all previous cases it was a matter of contradictions that had emerged in the course of the development of a theory and that needed to be eliminated by a reformulation of the axiom system. But if we wish to restore the reputation of mathematics as the exemplar of the most rigorous science it is not enough merely to avoid the existing contradictions. The chief requirement of the theory of axioms must go farther, namely, to show that within every field of knowledge contradictions based on the underlying axiom-system are absolutely impossible.

In accordance with this requirement I have proved the consistency of the axioms laid down in the Grundlagen der Geometrie by showing that any contradiction in the consequences of the geometrical axioms must necessarily apppear in the arithmetic of the system of real numbers as well.

For the fields of physical knowledge too, it is clearly sufficient to reduce the problem of internal consistency to the consistency of the arithmetical axioms. Thus I showed the consistency of the axioms of the elementary theory of radiation by constructing its axiom system out of analytically independent pieces—presupposing in the process the consistency of analysis.

One may and should in some circumstances proceed similarly in the construction of a mathematical theory. For example, if in the development of the theory of Galois groups we have taken the proposition of the existence of roots as an axiom, or if in the theory of prime numbers we have taken the hypothesis concerning the reality of the zeros of the Riemann 𝜁(s)-function as an axiom, then in each case the proof of the consistency of the axiom system comes down to a proof, using the means of analysis, of the proposition of the existence of roots or of the Riemann hypothesis concerning 𝜁(s)—and only then has the theory been securely completed.

The problem of the consistency of the axiom system for the real numbers can likewise be reduced by the use of set-theoretic concepts to the same problem for the integers: this is the merit of the theories of the irrational numbers developed by Weierstrass and Dedekind.

In only two cases is this method of reduction to another special domain of knowledge clearly not available, namely, when it is a matter of the axioms for the integers themselves, and when it is a matter of the foundation of set theory; for here there is no other discipline besides logic which it would be possi-ble to invoke.

But since the examination of consistency is a task that cannot be avoided, it appears necessary to axiomatize logic itself and to prove that number theory and set theory are only parts of logic.

This method was prepared long ago (not least by Frege's profound investigations); it has been most successfully explained by the acute mathematician and logician Russell. One could regard the completion of this magnificent Russellian enterprise of the axiomatization of logic as the crowning achievement of the work of axiomatization as a whole.

But this completion will require further work. When we consider the matter more closely we soon recognize that the question of the consistency of the integers and of sets is not one that stands alone, but that it belongs to a vast domain of difficult epistemological questions which have a specifically mathematical tint: for example (to characterize this domain of questions briefly) the problem of the solvability in principle of every mathematical question, the problem of the subsequent checkability of the results of a mathematical investigation, the question of a criterion of simplicity for mathematical proofs, the question of the relationship between content and formalism in mathematics and logic, and finally the problem of the decidability of a mathematical question in a finite number of operations.

We cannot rest content with the axiomatization of logic until all questions of this sort and their interconnections have been understood and cleared up.

Among the mentioned questions, the last—namely, the one concerning decidability in a finite number of operations—is the best-known and the most discussed; for it goes to the essence of mathematical thought.

I should like to increase the interest in this question by indicating several particular mathematical problems in which it plays a role.

In the theory of algebraic invariants we have the fundamental theorem that there is always a finite number of whole rational invariants by means of which all other such invariants can be represented. In my opinion, my first general proof of this theorem completely satisfied our requirements of simplicity and perspicuity; but it is impossible to reformulate this proof so that we can obtain from it a statable bound for the number of the finitely many invariants of the full system, let alone obtain an actual listing of them. Instead, new principles and considerations of a completely different sort were necessary in order to show that the construction of the full system of invariants requires only a finite number of operations, and that this number is less than a bound that can be stated before the calculation.

We see the same thing happening in an example from the theory of surfaces. It is a fundamental question in the geometry of surfaces of the fourth order to determine the maximum number of separate sheets it takes to make up such a surface.

The first step towards an answer to this question is the proof that the number of sheets of a curved surface must be finite. This can easily be shown function-theoretically as follows. One assumes the existence of infinitely many sheets, and selects a point inside each spatial region bounded by a sheet. A point of accumulation for these infinitely many chosen points would then be a point of a singularity that is excluded for an algebraic surface.

This function-theoretic path does not at all lead to an upper bound for the number of surface-sheets. For that, we need instead certain observations on the number of cut-points, which then show that the number of sheets certainly cannot be greater than 12.

The second method, entirely different from the first, in turn cannot be applied or transformed to decide whether a surface of the fourth order with twelve sheets actually exists.

Since a quaternary form of the fourth order possesses 35 homogeneous coefficients, we can conceive of a given surface of the fourth order as a point in 34-dimensional space. The discriminant of the quaternary form of fourth order is of degree 108 in its coefficients; if it is set equal to zero, it accordingly represents in 34-dimensional space a surface of order 108. Since the coefficients of the discriminant are themselves determinate integers, the topological character of the discriminant surface can be precisely determined by the rules that are familiar to us from 2- and 3-dimensional space; so we can obtain precise information about the nature and significance of the individual subdomains into which the discriminant surface partitions the 34-dimensional space. Now, the surfaces of fourth order represented by points of these subdomains all certainly possess the same sheet-number; and it is accordingly possible to establish, by a long and wearying but finite calculation, whether we have a surface of fourth order with n ≤ 12 sheets or not.

The geometric method just described is thus a third way of treating our question about the maximum number of sheets of a surface of the fourth order. It proves the decidability of this question in a finite number of opera-tions. So in principle an important demand of our problem has been satisfied: it has been reduced to a problem of the level of difficulty of determining the \( 10^{(10^{10})} \)th numeral in the decimal expansion for 𝜋—a task which is clearly solvable, but which remains unsolved.

Rather, it took a profound and difficult algebraic-geometric investigation by Rohn to show that 11 sheets are not possible in a surface of the fourth order, while 10 sheets actually occur. Only this fourth method delivered the full solution of the problem.

These particular discussions show how a variety of methods of proof can be applied to the same problem, and they ought to suggest how necessary it is to study the essence of mathematical proof itself if one wishes to answer such questions as the one about decidability in a finite number of operations.

All such questions of principle, which I characterized above and of which the question just discussed—that is, the question about decidability in a finite number of operations—was only the last, seem to me to form an important new field of research which remains to be developed. To conquer this field we must, I am persuaded, make the concept of specifically mathematical proof itself into an object of investigation, just as the astronomer considers the movement of his position, the physicist studies the theory of his apparatus, and the philosopher criticizes reason itself.

To be sure, the execution of this programme is at present still an unsolved task.

In conclusion, I should like to sum up in a few sentences my general conception of the essence of the axiomatic method. I believe: anything at all that can be the object of scientific thought becomes dependent on the axiomatic method, and thereby indirectly on mathematics, as soon as it is ripe for the for-mation of a theory. By pushing ahead to ever deeper layers of axioms in the sense explained above we also win ever-deeper insights into the essence of scien-tific thought itself, and we become ever more conscious of the unity of our knowledge. In the sign of the axiomatic method, mathematics is summoned to a leading role in science.

From Ewald's introduction to the above, in From Kant to Hilbert, volume 2:

"Hilbert is persistently misconstrued as a 'formalist', i.e. as somebody who was so shaken by the paradoxes that he took up the theory that mathematics is merely a game played with meaningless symbols. But the intellectual background to Hilbert's proof theory was richer than this. To be sure, the paradoxes were an important goad; but he had more positive ambitions than the mere avoidance of paradox. (Hilbert, in fact, in his unpublished Göttingen lectures, repeatedly indicated that in his opinion Zermelo's work had successfully resolved the known paradoxes.) His study of the axiomatic foundations of geometry had yielded a rich harvest of mathematical results—non-Archimedean geometries, a new topological characterization of the plane, new theorems on the nature of continuity—and there was every reason to hope that the same powerful tool would prove equally useful in the other branches of mathematics and physics mentioned in this article.

"As for the term 'formalist', it is so misleading that it should be abandoned altogether as a label for Hilbert's philosophy of mathematics. On the face of it, 'formalism' is not a felicitous description for the style of reasoning one finds in Anschauliche Geometrie–'Intuitive geometry' (Hilbert and Cohn-Vossen 1932), a work which was written at the high-point of Hilbert's work in proof theory; nor does it accurately convey the convictions of the man who, in his Göttingen lectures, derided those who saw mathematics as a mere heaping-up of consequences mechanically derived from a given stock of axioms; nor the man who ended his essay, 'On the infinite', by saying, 'We gain a conviction that runs counter to the earlier endeavours of Frege and Dedekind, the conviction that, if scientific knowledge is to be possible, certain intuitive conceptions [Vorstellungen] and insights are indispensable; logic alone does not suffice' (Hilbert 1926). As the present selection makes clear, Hilbert viewed formal axiom systems instrumentally, as a powerful tool for mathematical research, a tool to be employed when a field had reached a point of sufficient ripeness. But he nowhere suggests that the whole of mathematics can simply be identified with the study of formal systems; and indeed in his proof-theoretical writings he took considerable pains to point out that the genuine mathematics–inhaltliche Mathematik–takes place, not in the formalism, but in the meta-language. For all these reasons, Hilbert himself rejected the label 'formalist' (see, for example, Hilbert 1931a, §35), and students of his thought would do well to follow his example."