"The modern development of the foundations of mathematics in the light of philosophy" by Kurt Godel,1961

Kurt Gödel,
Collected Works,
Volume III (1961.
Oxford University Press, 1981.

The Complete lecture reproduced here :
I would like to attempt here to describe, in terms of philosophical concepts, the development of foundational research in mathematics since around the turn of the century, and to fit it into a general schema of possible philosophical world-views [Weltanschauungen]. For this, it is necessary first of all to become clear about the schema itself. I believe that the most fruitful principle for gaining an overall view of the possible world-views will be to divide them up according to the degree and the manner of their affinity to or, respectively, turning away from metaphysics (or religion). In this way we immediately obtain a division into two groups: scepticism, materialism and positivism stand on one side, spiritualism, idealism and theology on the other. We also at once see degrees of difference in this sequence, in that scepticism stands even farther away from theology than does materialism, while on the other hand idealism, e.g., in its pantheistic form, is a weakened form of theology in the proper sense.

The schema also proves fruitful, however, for the analysis of philosophical doctrines admissible in special contexts, in that one either arranges them in this manner or, in mixed cases, seeks out their materialistic and spiritualistic elements. Thus one would, for example, say that apriorism belongs in principle on the right and empiricism on the left side. On the other hand, however, there are also such mixed forms as an empiristically grounded theology. Furthermore one sees also that optimism belongs in principle toward the right and pessimism toward the left. For scepticism is certainly a pessimism with regard to knowledge. Moreover, materialism is inclined to regard the world as an unordered and therefore meaningless heap of atoms. In addition, death appears to it to be final and complete annihilation, while, on the other hand, theology and idealism see sense, purpose and reason in everything. On the other hand, Schopenhauer's pessimism is a mixed form, namely a pessimistic idealism. Another example of a theory evidently on the right is that of an objective right and objective aesthetic values, whereas the interpretation of ethics and aesthetics on the basis of custom, upbringing, etc., belongs toward the left.

Now it is a familiar fact, even a platitude, that the development of philosophy since the Renaissance has by and large gone from right to left - not in a straight line, but with reverses, yet still, on the whole. Particularly in physics, this development has reached a peak in our own time, in that, to a large extent, the possibility of knowledge of the objectivisable states of affairs is denied, and it is asserted that we must be content to predict results of observations. This is really the end of all theoretical science in the usual sense (although this predicting can be completely sufficient for practical purposes such as making television sets or atom bombs).

It would truly be a miracle if this (I would like to say rabid) development had not also begun to make itself felt in the conception of mathematics. Actually, mathematics, by its nature as an a priori science, always has, in and of itself, an inclination toward the right, and, for this reason, has long withstood the spirit of the time [Zeitgeist] that has ruled since the Renaissance; i.e., the empiricist theory of mathematics, such as the one set forth by Mill, did not find much support. Indeed, mathematics has evolved into ever higher abstractions, away from matter and to ever greater clarity in its foundations (e.g., by giving an exact foundation of the infinitesimal calculus and the complex numbers) - thus, away from scepticism.

Finally, however, around the turn of the century, its hour struck: in particular, it was the antinomies of set theory, contradictions that allegedly appeared within mathematics, whose significance was exaggerated by sceptics and empiricists and which were employed as a pretext for the leftward upheaval. I say "allegedly" and "exaggerated" because, in the first place, these contradictions did not appear within mathematics but near its outermost boundary toward philosophy, and secondly, they have been resolved in a manner that is completely satisfactory and, for everyone who understands the theory, nearly obvious. Such arguments are, however, of no use against the spirit of the time, and so the result was that many or most mathematicians denied that mathematics, as it had developed previously, represents a system of truths; rather, they acknowledged this only for a part of mathematics (larger or smaller, according to their temperament) and retained the rest at best in a hypothetical sense namely, one in which the theory properly asserts only that from certain assumptions (not themselves to be justified), we can justifiably draw certain conclusions. They thereby flattered themselves that everything essential had really been retained. Since, after all, what interests the mathematician, in addition to drawing consequences from these assumptions, is what can be carried out. In truth, however, mathematics becomes in this way an empirical science. For if I somehow prove from the arbitrarily postulated axioms that every natural number is the sum of four squares, it does not at all follow with certainty that I will never find a counter-example to this theorem, for my axioms could after all be inconsistent, and I can at most say that it follows with a certain probability, because in spite of many deductions no contradiction has so far been discovered. In addition, through this hypothetical conception of mathematics, many questions lose the form "Does the proposition A hold or not?" For, from assumptions construed as completely arbitrary, I can of course not expect that they have the peculiar property of implying, in every case, exactly either A or ~A.

Although these nihilistic consequences are very well in accord with the spirit of the time, here a reaction set in obviously not on the part of philosophy, but rather on that of mathematics, which, by its nature, as I have already said, is very recalcitrant in the face of the Zeitgeist. And thus came into being that curious hermaphroditic thing that Hilbert's formalism represents, which sought to do justice both to the spirit of the time and to the nature of mathematics. It consists in the following: on the one hand, in conformity with the ideas prevailing in today's philosophy, it is acknowledged that the truth of the axioms from which mathematics starts out cannot be justified or recognised in any way, and therefore the drawing of consequences from them has meaning only in a hypothetical sense, whereby this drawing of consequences itself (in order to satisfy even further the spirit of the time) is construed as a mere game with symbols according to certain rules, likewise not supported by insight.

But, on the other hand, one clung to the belief, corresponding to the earlier "rightward" philosophy of mathematics and to the mathematician's instinct, that a proof for the correctness of such a proposition as the representability of every number as a sum of four squares must provide a secure grounding for that proposition - and furthermore, also that every precisely formulated yes-or-no question in mathematics must have a clear-cut answer. I.e., one thus aims to prove, for inherently unfounded rules of the game with symbols, as a property that attaches to them so to speak by accident, that of two sentences A and ~A, exactly one can always be derived. That not both can be derived constitutes consistency, and that one can always actually be derived means that the mathematical question expressed by A can be unambiguously answered. Of course, if one wishes to justify these two assertions with mathematical certainty, a certain part of mathematics must be acknowledged as true in the sense of the old rightward philosophy. But that is a part that is much less opposed to the spirit of the time than the high abstractions of set theory. For it refers only to concrete and finite objects in space, namely the combinations of symbols.

What I have said so far are really only obvious things, which I wanted to recall merely because they are important for what follows. But the next step in the development is now this: it turns out that it is impossible to rescue the old rightward aspects of mathematics in such a manner as to be more or less in accord with the spirit of the time. Even if we restrict ourselves to the theory of natural numbers, it is impossible to find a system of axioms and formal rules from which, for every number-theoretic proposition A, either A or ~A would always be derivable. And furthermore, for reasonably comprehensive axioms of mathematics, it is impossible to carry out a proof of consistency merely by reflecting on the concrete combinations of symbols, without introducing more abstract elements. The Hilbertian combination of materialism and aspects of classical mathematics thus proves to be impossible.

Hence, only two possibilities remain open. One must either give up the old rightward aspects of mathematics or attempt to uphold them in contradiction to the spirit of the time. Obviously the first course is the only one that suits our time and is therefore also the one usually adopted. One should, however, keep in mind that this is a purely negative attitude. One simply gives up aspects whose fulfilment would in any case be very desirable and which have much to recommend themselves: namely, on the one hand, to safeguard for mathematics the certainty of its knowledge, and on the other, to uphold the belief that for clear questions posed by reason, reason can also find clear answers. And as should be noted, one gives up these aspects not because the mathematical results achieved compel one to do so but because that is the only possible way, despite these results, to remain in agreement with the prevailing philosophy.

Now one can of course by no means close one's eyes to the great advances which our time exhibits in many respects, and one can with a certain justice assert that these advances are due just to this leftward spirit in philosophy and world-view. But, on the other hand, if one considers the matter in proper historical perspective, one must say that the fruitfulness of materialism is based in part only on the excesses and the wrong direction of the preceding rightward philosophy. As far as the rightness and wrongness, or, respectively, truth and falsity, of these two directions is concerned, the correct attitude appears to me to be that the truth lies in the middle or consists of a combination of the two conceptions.

Now, in the case of mathematics, Hilbert had of course attempted just such a combination, but one obviously too primitive and tending too strongly in one direction. In any case there is no reason to trust blindly in the spirit of the time, and it is therefore undoubtedly worth the effort at least once to try the other of the alternatives mentioned above, which the results cited leave open - in the hope of obtaining in this way a workable combination. Obviously, this means that the certainty of mathematics is to be secured not by proving certain properties by a projection onto material systems - namely, the manipulation of physical symbols but rather by cultivating (deepening) knowledge of the abstract concepts themselves which lead to the setting up of these mechanical systems, and further by seeking, according to the same procedures, to gain insights into the solvability, and the actual methods for the solution, of all meaningful mathematical problems.

In what manner, however, is it possible to extend our knowledge of these abstract concepts, i.e., to make these concepts themselves precise and to gain comprehensive and secure insight into the fundamental relations that subsist among them, i.e., into the axioms that hold for them? Obviously not, or in any case not exclusively, by trying to give explicit definitions for concepts and proofs for axioms, since for that one obviously needs other undefinable abstract concepts and axioms holding for them. Otherwise one would have nothing from which one could define or prove. The procedure must thus consist, at least to a large extent, in a clarification of meaning that does not consist in giving definitions.

Now in fact, there exists today the beginning of a science which claims to possess a systematic method for such a clarification of meaning, and that is the phenomenology founded by Husserl. Here clarification of meaning consists in focusing more sharply on the concepts concerned by directing our attention in a certain way, namely, onto our own acts in the use of these concepts, onto our powers in carrying out our acts, etc. But one must keep clearly in mind that this phenomenology is not a science in the same sense as the other sciences. Rather it is or in any case should be a procedure or technique that should produce in us a new state of consciousness in which we describe in detail the basic concepts we use in our thought, or grasp other basic concepts hitherto unknown to us. I believe there is no reason at all to reject such a procedure at the outset as hopeless. Empiricists, of course, have the least reason of all to do so, for that would mean that their empiricism is, in truth, an apriorism with its sign reversed.

But not only is there no objective reason for the rejection of phenomenology, but on the contrary one can present reasons in its favour. If one considers the development of a child, one notices that it proceeds in two directions: it consists on the one hand in experimenting with the objects of the external world and with its own sensory and motor organs, on the other hand in coming to a better and better understanding of language, and that means - as soon - as the child is beyond the most primitive designating of objects - of the basic concepts on which it rests. With respect to the development in this second direction, one can justifiably say that the child passes through states of consciousness of various heights, e.g., one can say that a higher state of consciousness is attained when the child first learns the use of words, and similarly at the moment when for the first time it understands a logical inference.

Now one may view the whole development of empirical science as a systematic and conscious extension of what the child does when it develops in the first direction. The success of this procedure is indeed astonishing and far greater than one would expect a priori: after all, it leads to the entire technological development of recent times. That makes it thus seem quite possible that a systematic and conscious advance in the second direction will also far exceed the expectations one may have a priori.

In fact, one has examples where, even without the application of a systematic and conscious procedure, but entirely by itself, a considerable further development takes place in the second direction, one that transcends "common sense". Namely, it turns out that in the systematic establishment of the axioms of mathematics, new axioms, which do not follow by formal logic from those previously established, again and again become evident. It is not at all excluded by the negative results mentioned earlier that nevertheless every clearly posed mathematical yes-or-no question is solvable in this way. For it is just this becoming evident of more and more new axioms on the basis of the meaning of the primitive notions that a machine cannot imitate.

I would like to point out that this intuitive grasping of ever newer axioms that are logically independent from the earlier ones, which is necessary for the solvability of all problems even within a very limited domain, agrees in principle with the Kantian conception of mathematics. The relevant utterances by Kant are, it is true, incorrect if taken literally, since Kant asserts that in the derivation of geometrical theorems we always need new geometrical intuitions, and that therefore a purely logical derivation from a finite number of axioms is impossible. That is demonstrably false. However, if in this proposition we replace the term "geometrical" - by "mathematical" or "set-theoretical", then it becomes a demonstrably true proposition. I believe it to be a general feature of many of Kant's assertions that literally understood they are false but in a broader sense contain deep truths. In particular, the whole phenomenological method, as I sketched it above, goes back in its central idea to Kant, and what Husserl did was merely that he first formulated it more precisely, made it fully conscious and actually carried it out for particular domains. Indeed, just from the terminology used by Husserl, one sees how positively he himself values his relation to Kant.

I believe that precisely because in the last analysis the Kantian philosophy rests on the idea of phenomenology, albeit in a not entirely clear way, and has just thereby introduced into our thought something completely new, and indeed characteristic of every genuine philosophy - it is precisely on that, I believe, that the enormous influence which Kant has exercised over the entire subsequent development of philosophy rests. Indeed, there is hardly any later direction that is not somehow related to Kant's ideas. On the other hand, however, just because of the lack of clarity and the literal incorrectness of many of Kant's formulations, quite divergent directions have developed out of Kant's thought - none of which, however, really did justice to the core of Kant's thought. This requirement seems to me to be met for the first time by phenomenology, which, entirely as intended by Kant, avoids both the death-defying leaps of idealism into a new metaphysics as well as the positivistic rejection of all metaphysics. But now, if the misunderstood Kant has already led to so much that is interesting in philosophy, and also indirectly in science, how much more can we expect it from Kant understood correctly?

"Category Theory" by Samuel Eilenberg and Saunders Mac Lane,

Category theory, alongside set theory, serves as a universal language of modern mathematics. Categories, functors, and natural transformations are widely used in all areas of mathematics, allowing us to look uniformly and consistently on various constructions and formulate the general properties of diverse structures. The impact of category theory is irreducible to the narrow frameworks of its great expressive conveniences. This theory has drastically changed our general outlook on the foundations of mathematics and widened the room of free thinking in mathematics.
.
Set theory, a great and ingenious creation of Georg Cantor, occupies in the common opinion of the 20th century the place of the sole solid base of modern mathematics. Mathematics becomes sinking into a section of the Cantorian set theory. Most active mathematicians, teachers, and philosophers consider as obvious and undisputable the thesis that mathematics cannot be grounded on anything but set theory. The set-theoretic stance transforms paradoxically into an ironclad dogma, a clear-cut forbiddance of thinking (as L. Feuerbach once put it wittily). Such an indoctrinated view of the foundations of mathematics is false and conspicuously contradicts the leitmotif, nature, and pathos of the essence of all creative contribution of Cantor who wrote as far back as in 1883 that “denn das Wesen der Mathematik liegt gerade in ihrer Freiheit.”
====================================
Samuel Eilenberg (September 30, 1913—January 30, 1998) was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire (now in Poland) and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.

He earned his Ph.D. from Warsaw University in 1936. His thesis advisor was Karol Borsuk. His main interest was algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (whose names the Eilenberg-Steenrod axioms bear), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory.

Eilenberg took part in the Bourbaki group meetings, and, with Henri Cartan, wrote the 1956 book Homological Algebra, which became a classic.

Later in life he worked mainly in pure category theory, being one of the founders of the field. The Eilenberg swindle (or telescope) is a construction applying the telescoping cancellation idea to projective modules.

Eilenberg also wrote an important book on automata theory. The X-machine, a form of automaton, was introduced by Eilenberg in 1974.

Eilenberg was also a prominent collector of Asian art. His collection mainly consisted of small sculptures and other artifacts from India, Indonesia, Pakistan, Nepal, Thailand, Cambodia, Sri Lanka and Central Asia. In 1991-1992, the Metropolitan Museum of Art in New York staged an exhibition from more than 400 items that Eilenberg had donated to the museum, entitled The Lotus Transcendent: Indian and Southeast Asian Art From the Samuel Eilenberg Collection".
=====
The X-machine (XM) is a theoretical model of computation introduced by Samuel Eilenberg in 1974.

The X in "X-machine" represents the fundamental data type on which the machine operates; for example, a machine that operates on databases (objects of type database) would be a database-machine.

The X-machine model is structurally the same as the finite state machine, except that the symbols used to label the machine's transitions denote relations of type X→X. Crossing a transition is equivalent to applying the relation that labels it (computing a set of changes to the data type X), and traversing a path in the machine corresponds to applying all the associated relations, one after the other.

Interest in the X-machine was revived in the late 1980s by Mike Holcombe[2], who noticed that the model was ideal for software formal specification purposes, because it cleanly separates control flow from processing. Provided one works at a sufficiently abstract level, the control flows in a computation can usually be represented as a finite state machine, so to complete the X-machine specification all that remains is to specify the processing associated with each of the machine's transitions. The structural simplicity of the model makes it extremely flexible; other early illustrations of the idea included Holcombe's specification of human-computer interfaces,his modelling of processes in cell biochemistry, and Stannett's modelling of decision-making in military command systems.

X-machines have received renewed attention since the mid-1990s, when Gilbert Laycock's deterministic Stream X-Machine was found to serve as the basis for specifying large software systems that are completely testable. Another variant, the Communicating Stream X-Machine offers a useful testable model for biological processes[8] and future swarm-based satellite systems.

"Saunders Mac Lane (1909–2005): His Mathematical Life and Philosophical Works"

Cofounded category theory with Samuel Eilenberg.
.
Category theory, alongside set theory, serves as a universal language of modern mathematics. Categories, functors, and natural transformations are widely used in all areas of mathematics, allowing us to look uniformly and consistently on various constructions and formulate the general properties of diverse structures. The impact of category theory is irreducible to the narrow frameworks of its great expressive conveniences. This theory has drastically changed our general outlook on the foundations of mathematics and widened the room of free thinking in mathematics.
.
Set theory, a great and ingenious creation of Georg Cantor, occupies in the common opinion of the 20th century the place of the sole solid base of modern mathematics. Mathematics becomes sinking into a section of the Cantorian set theory. Most active mathematicians, teachers, and philosophers consider as obvious and undisputable the thesis that mathematics cannot be grounded on anything but set theory. The set-theoretic stance transforms paradoxically into an ironclad dogma, a clear-cut forbiddance of thinking (as L. Feuerbach once put it wittily). Such an indoctrinated view of the foundations of mathematics is false and conspicuously contradicts the leitmotif, nature, and pathos of the essence of all creative contribution of Cantor who wrote as far back as in 1883 that “denn das Wesen der Mathematik liegt gerade in ihrer Freiheit.”
======================================
Contributions :
After a thesis in mathematical logic, his early work was in field theory and valuation theory. He wrote on valuation rings and Witt vectors, and separability in infinite field extensions. He started writing on group extensions in 1942, and began his epochal collaboration with Samuel Eilenberg in 1943, resulting in what are now called Eilenberg–Mac Lane spaces K(G,n), having a single non-trivial homotopy group G in dimension n. This work opened the way to group cohomology in general.

After introducing, via the Eilenberg-Steenrod axioms, the abstract approach to homology theory, he and Eilenberg originated category theory in 1945. He is especially known for his work on coherence theorems. A recurring feature of category theory, abstract algebra, and of some other mathematics as well, is the use of diagrams, consisting of arrows (morphisms) linking objects, such as products and coproducts. According to McLarty (2005), this diagrammatic approach to contemporary mathematics largely stems from Mac Lane (1948).

Mac Lane had an exemplary devotion to writing approachable texts, starting with his very influential A Survey of Modern Algebra, coauthored in 1941 with Garrett Birkhoff. From then on, it was possible to teach elementary modern algebra to undergraduates using an English text. His Categories for the Working Mathematician remains the definitive introduction to category theory.

Mac Lane supervised the Ph.Ds of, among many others, David Eisenbud, William Howard, Irving Kaplansky, Michael Morley, Anil Nerode, Robert Solovay, and John G. Thompson.

In addition to reviewing a fair bit of his mathematical output, the obituary articles McLarty (2005, 2007) clarify Mac Lane's contributions to the philosophy of mathematics. Mac Lane (1986) is an approachable introduction to his views on this subject.
=====================================
Most mathematicians cannot define the spaces of algebraic geometry, called schemes, off the top of their heads, nor the morphisms mapping one scheme algebro-geometrically to another f:SS'. But they can define the cartesian product S x S' and the coproduct or union S S' of schemes (up to isomorphism) by familiar diagrams of morphisms.
.
Products and coproducts of any kind of structure are defined this way today1. This rigorous ‘structuralism’, where structures are defined up to isomorphism by their morphisms to and from other structures, has been textbook mathematics since Lang [1965] and Mac Lane and Birkhoff [1967]. It is in large part due to Saunders Mac Lane. He first gave these definitions of product and coproduct in Mac Lane [1948], where he emphasized that they are more fundamental than set-theoretic definitions for much work with structures like Abelian groups.
========================================
SAUNDERS MAC LANE,
THE KNIGHT OF MATHEMATICS

San Francisco and April 14, 2005 form the terminal place and date of the marvellous almost centennial life of the prominent American mathematician Saunders Mac Lane who shared with Samuel Eilenberg (1913–1998) the honor of creation of category theory which ranks among the most brilliant, controversial, ambitious, and heroic mathematical achievements of the 20th century.

It is category theory that one of the most ambitious projects of the 20th century mathematics was realized within in the 1960s, the project of socializing set theory. This led to topos theory providing a profusion of categories of which classical set theory is an ordinary member. Mathematics has acquired infinitely many new degrees of freedom. All these rest on category theory originated with the article by Mac Lane and Eilenberg “General Theory of Natural Equivalences,” which was presented to the American Mathematical Society on September 8, 1942 and published in 1945 in the Transactions of the AMS.

Mac Lane authored or coauthored more than 100 research papers and 6 books:
A SURVEY OF MODERN ALGEBRA (1941, 1997 ; with G. Birkhoff);
HOMOLOGY (1963);
ALGEBRA (1967; with G. Birkhoff);
CATEGORIES FOR THE WORKING MATHEMATICIAN (1971, 1998);
MATHEMATICS, FORM AND FUNCTION (1985);
SHEAVES IN GEOMETRY AND LOGIC: A FIRST INTRODUCTION TO TOPOS THEORY (1992; with Ieke Moerdijk).

Mac Lane was the advisor of 39 Ph.D. theses. Alfred Putman, John Thompson, Irving Kaplansky, Robert Solovay, and many other distinguished scientists are listed as his students. He was elected to the National Academy of Sciences of the USA and received the National Medal of Science, the highest scientific award of the USA in 1989. Mac Lane served as vice-president of the National Academy of Sciences and the American Philosophical Society and the Mathematical Association of America. He contributed greatly to modernization of the teaching programs in mathematics. Mac Lane received many signs of honor from the leading universities of the world and possessed an impressive collection of mathematical awards and prizes. Mac Lane became a living legend of the science of the USA.

Mac Lane was born on August 4, 1909 in Norwich near Taftville, Connecticut in the family of a Congregationalist minister and was christened as Leslie Saunders MacLane. The name Leslie was suggested by his nurse, but his mother disliked the name. A month later, his father put a hand on the head of the son, looked up to the God, and said: “Leslie forget.” His father and uncles changed the spelling of their surname and began to write MacLane instead of MacLean in order to avoid sounding Irish. The space in Mac Lane was added by Saunders himself at request of his first wife Dorothy. That is how Mac Lane narrated about his name in A Mathematical Biography which was published soon after his death.

Saunders's father passed away when the boy was 15 and it was Uncle John who supported the boy and paid for his education in Yale. Saunders was firstly fond of chemistry but everything changed after acquaintance with differential and integral calculus by the textbook of Longley and Wilson (which reminds of the later book by Granville, Smith, and Longley). The university years revealed Mac Lane's attraction to philosophy and foundations of mathematics. He was greatly impressed by the brand-new three volumes by Whitehead and Russell, the celebrated PRINCIPIA MATHEMATICA. The mathematical tastes of Mac Lane were strongly influenced by the lectures of a young assistant professor Oystein Ore, a Norwegian mathematician from the Emmy Noether's school. After graduation from Yale, Mac Lane continued education in the University of Chicago. At that time he was very much influenced by the personalities and research of Eliakim Moore, Leonard Dickson, Gilbert Bliss, Edmund Landau, Marston Morse, and many others. Mac Lane was inclined to wrote a Ph.D. thesis in logic but this was impossible in Chicago and so Saunders decided to continue education in Göttingen.

The stay in Germany in 1931–1933 was decisive for the maturity of Mac Lane's gift and personality. Although David Hilbert had retired, he still delivered weekly lectures on philosophy and relevant general issues. The successor of Hilbert was Hermann Weyl who had recently arrived from Zürich and was in the prime of his years and talents. Weyl advised Saunders to attend the lectures on linear associative algebras by Emmy Noether whom Weyl called “the equal of each of us.” In the Mathematical Institute Mac Lane met and boiled with Edmund Landau, Richard Courant, Gustav Herglotz, Otto Neugebauer, Oswald Teichmüller, and many others. Paul Bernays became the advisor of Mac Lane's Ph.D. thesis “Abbreviated Proofs in Logic Calculus.”

The Nazis gained power in Germany in February 1933. The feast of antisemitism started immediately and one of the first and fiercest strokes fell upon the Mathematical Institute. The young persons are welcome to read as an antidote Mac Lane's masterpiece “Mathematics at Göttingen under the Nazis” in the Notices of the AMS, 42:10, 1134–1138 (1995).

In the fall of 1933 Mac Lane returned to the States with Dorothy Jones Mac Lane whom he had married recently in Germany. The further academic career of Mac Lane was mainly tied with Harvard and since 1947 with Chicago.

To evaluate the contribution of Mac Lane to mathematics is an easy and pleasant task. It suffices to cite the words A. G. Kurosh, a renowned Russian professor of Lomonosov State University. In the translator's preface to the Russian edition of the classical HOMOLOGY book, Kurosh wrote:

The author of this book, a professor of Chicago University, is one of the most prominent American algebraists and topologists. His role in homological algebra as well as category theory is the role of one of the founders of this area.

Homological algebra implements a marvelous project of algebraization of topological spaces by assigning to such a space X the sequence of (abelian) homology groups Hn(X). Moreover, each continuous map f: X→ Y from X to Y induces a family of homomorphisms of the homology groups fn: Hn(X) → Hn(Y). The aim of homological algebra consists in calculation of homologies.

In his research into homological algebra and category theory Mac Lane cooperated with Eilenberg whom he met in 1940. Eilenberg had arrived from Poland two years earlier. He saw the affinity of the algebraic calculations of Mac Lane with those he encountered in algebraic topology. Eilenberg offered cooperation to Mac Lane. The union of Eilenberg and Mac Lane lasted for 14 years and resulted in 15 joint papers which noticeably changed the mathematical appearance of the 20th century.

The pearl of this cooperation was category theory. Mac Lane always considered category theory “a natural and perhaps inevitable aspect of the 20th century mathematical emphasis on axiomatic and abstract methods—especially as those methods when involved in abstract algebra and functional analysis.” He stressed that even if Eilenberg and he did not propose this theory it will necessarily appear in the works of other mathematicians. Among these potential inventors of the new conceptions Mac Lane listed Claude Chevalley, Heinz Hopf, Norman Steenrod, Henri Cartan, Charles Ehresmann, and John von Neumann.

In Mac Lane's opinion, the conceptions of category theory were close to the methodological principles of the project of Nicholas Bourbaki. Mac Lane was sympathetic with the project and was very close to joining in but this never happened (the main obstacles were in linguistic facilities). However, even the later membership of Eilenberg in the Bourbaki group could not overcome a shade of slight disinclination and repulsion. It turned out impossible to “categorize Bourbaki” with a theory of non-French origin as Mac Lane had once phrased the matter shrewdly and elegantly. It is worth noting in this respect that the term “category theory” had roots in the mutual interest of its authors in philosophy and, in particular, in the works of Immanuel Kant.

Set theory rules in the present-day mathematics. The buffoon's role of “abstract nonsense” is assigned in mathematics to category theory. History and literature demonstrate to us that the relations between the ruler and the jester may be totally intricate and unpredictable. Something very similar transpires in the interrelations of set theory and category theory and the dependency of one of them on the other.

From a logic standpoint, set theory and category theory are instances of a first order theory. The former deals with sets and the membership relation between them. The latter speaks of objects and morphisms (or arrows). Of course, there is no principle difference between the atomic formulas a∈b and a→b. However, the precipice in meaning is abysmal between the two concepts that are formalized by the two atomic formulas. The stationary universe of Zermelo–Fraenkel, cluttered up with uncountably many copies of equipollent sets confronts the free world of categories, ensembles of arbitrary nature that are determined by the dynamics of their transformations.

The individual dualities of set theory, dependent on the choice of particular realizations of the pairs of objects under study, give up their places to the universal natural transformations of category theory. One of the most brilliant achievements of category theory was the development of axiomatic homology theory. Instead of the homological diversity for topological spaces (the simplicial homology for a polyhedron, singular and Čech homology, Vietoris homology, etc.) Eilenberg and Steenrod suggested as far back as in 1952 the new understanding of each homology or cohomology theory as a functor from the category of spaces under consideration to the category of groups. The axiomatic approach to defining such a functor radically changed the manner of further progress in homological algebra and algebraic topology. The study of the homology of Eilenberg–Mac Lane spaces and the method of acyclic models demonstrated the strength of the ideas of category theory and led to universal use of simplicial sets in K-theory and sheaves.

In 1948 Mac Lane proposed the concept of abelian category abstracting the categories of abelian groups and vector spaces which played key roles in the first papers on axiomatic homology theory. The abelian categories were rediscovered in 1953 and became a major tool in research into homological algebra by Cartan, Eilenberg, and their followers.

Outstanding advances in category theory are connected with the names of Alexander Grothendieck and F. William Lawvere. Topos theory, their aesthetic creation, appeared in the course of “point elimination” called upon by the challenge of invariance of the objects we study in mathematics. It is on this road that we met the conception of variable sets which led to the notion of topos and the understanding of the social medium of set-theoretic models.

A category is called an elementary topos provided that it is cartesian closed and has a suboject classifier. The sources of toposes lie in the theory of sheaves and Grothendieck topology. Further progress of the concept of topos is due to search for some category-theoretic axiomatization of set theory as well as study into forcing and the nonstandard set-theoretic models of Dana Scott, Robert Solovay, and Petr Vopěnka. The new frameworks provide a natural place for the Boolean valued models that are viewed now as the toposes with Aristotle logic and which pave king's ways to the solution of the problem of the continuum by Kurt Gödel and Paul Cohen. These toposes are the main arena of Boolean valued analysis.

Bidding farewell to Mac Lane, reading his sincere and openhearted autobiography, enjoying his vehement polemics with Freeman J. Dyson, and perusing his deep last articles on general mathematics, anyone cannot help but share his juvenile devotion and love of mathematics and its creators. His brilliant essays “Despite Physicists, Proof Is Essential in Mathematics” and “Proof, Truth, and Confusion” form an anthem of mathematics which is only possible by proof.

Let me summarize where we have come. As with any branch of learning, the real substance of mathematics resides in the ideas. The ideas of mathematics are those which can be formalized and which have been developed to fit issues arising in science or in human activity. Truth in mathematics is approached by way of proof in formalized systems. However, because of the paradoxical kinds of self-reference exhibited by the barn door and Kurt Gödel, there can be no single formal system which subsumes all mathematical proof. To boot, the older dogmas that “everything is logic” or “everything is a set” now have competition—“everything is a function.” However, such questions of foundation are but a very small part of mathematical activity, which continues to try to combine the right ideas to attack substantive problems. Of these I have touched on only a few examples: Finding all simple groups, putting groups together by extension, and characterizing spheres by their connectivity. In such cases, subtle ideas, fitted by hand to the problem, can lead to triumph.
Numerical and mathematical methods can be used for practical problems. However, because of political pressures, the desire for compromise, or the simple desire for more publication, formal ideas may be applied in practical cases where the ideas simply do not fit. Then confusion arises—whether from misleading formulation of questions in opinion surveys, from nebulous calculations of airy benefits, by regression, by extrapolation, or otherwise. As the case of fuzzy sets indicates, such confusion is not fundamentally a trouble caused by the organizations issuing reports, but is occasioned by academicians making careless use of good ideas where they do not fit.
As Francis Bacon once said, “Truth ariseth more readily from error than from confusion.” There remains to us, then, the pursuit of truth, by way of proof, the concatenation of those ideas which fit, and the beauty which results when they do fit.

So wrote Saunders Mac Lane, a great genius, creator, master, and servant of mathematics. His unswerving devotion to the ideals of truth and free thinking of our ancient science made him the eternal and tragicomical mathematical Knight of the Sorrowful Figure and Category...

S. Kutateladze

"Lectures on Intuitionism" by Lej Brouwer,1951

The gradual transformation of the mechanism of mathematical thought is a consequence of the modifications which, in the course of history, have come about in the prevailing philosophical ideas, firstly concerning the origin of mathematical certainty, secondly concerning the delimitation of the object of mathematical science. In this respect we can remark that in spite of the continual trend from object to subject of the place ascribed by philosophers to time and space in the subject-object medium, the belief in the existence of immutable properties of time and space, properties independent of experience and of language, remained well-nigh intact far into the nineteenth century. To obtain exact knowledge of these properties, called mathematics, the following means were usually tried: some very familiar regularities of outer or inner experience of time and space were postulated to be invariable, either exactly, or at any rate with any attainable degree of approximation. They were called axioms and put into language. Thereupon systems of more complicated properties were developed from the linguistic substratum of the axioms by means of reasoning guided by experience, but linguistically following and using the principles of classical logic. We will call the standpoint governing this mode of thinking and working the observational standpoint, and the long period characterised by this standpoint the observational period. It considered logic as autonomous, and mathematics as (if not existentially, yet functionally) dependent on logic.

For space the observational standpoint became untenable when, in the course of the nineteenth and the beginning of the twentieth centuries, at the hand of a series of discoveries with which the names of Lobatchefsky, Bolyai, Riemann, Cayley, Klein, Hilbert, Einstein, Levi-Cività and Hahn are associated, mathematics was gradually transformed into a mere science of numbers; and when besides observational space a great number of other spaces, sometimes exclusively originating from logical speculations, with properties distinct from the traditional, but no less beautiful, had found their arithmetical realisation. Consequently the science of classical (Euclidean, three-dimensional) space had to continue its existence as a chapter without priority, on the one hand of the aforesaid (exact) science of numbers, on the other hand (as applied mathematics) of (naturally approximative) descriptive natural science.

In this process of extending the domain of geometry, an important part had been played by the logico-linguistic method, which operated on words by means of logical rules, sometimes without any guidance from experience and sometimes even starting from axioms framed independently of experience. Encouraged by this the Old Formalist School (Dedekind, Cantor, Peano, Hilbert, Russell, Zermelo, Couturat), for the purpose of a rigorous treatment of mathematics and logic (though not for the purpose of furnishing objects of investigation to these sciences), finally rejected any elements extraneous to language, thus divesting logic and mathematics of their essential difference in character, as well as of their autonomy. However, the hope originally fostered by this school that mathematical science erected according to these principles would be crowned one day with a proof of its non-contradictority was never fulfilled, and nowadays, after the logical investigations performed in the last few decades, we may assume that this hope has been relinquished universally.

Of a totally different orientation was the Pre-intuitionist School, mainly led by Poincaré, Borel and Lebesgue. These thinkers seem to have maintained a modified observational standpoint for the introduction of natural numbers, for the principle of complete induction, and for all mathematical entities springing from this source without the intervention of axioms of existence, hence for what might be called the 'separable' parts of arithmetic and of algebra. For these, even for such theorems as were deduced by means of classical logic, they postulated an existence and exactness independent of language and logic and regarded its non-contradictority as certain, even without logical proof. For the continuum, however, they seem not to have sought an origin strictly extraneous to language and logic. On some occasions they seem to have contented themselves with an ever-unfinished and ever-denumerable species of 'real numbers' generated by an ever-unfinished and ever-denumerable species of laws defining convergent infinite sequences of rational numbers. However, such an ever-unfinished and ever-denumerable species of 'real numbers' is incapable of fulfilling the mathematical function of the continuum for the simple reason that it cannot have a positive measure. On other occasions they seem to have introduced the continuum by having recourse to some logical axiom of existence, such as the 'axiom of ordinal connectedness', or the 'axiom of completeness', without either sensory or epistemological evidence. In both cases in their further development of mathematics they continued to apply classical logic, including the principium tertii exclusi, without reserve and independently of experience. This was done regardless of the fact that the noncontradictority of systems thus constructed had become doubtful by the discovery of the well-known logico-mathematical antonomies.

In point of fact, pre-intuitionism seems to have maintained on the one hand the essential difference in character between logic and mathematics, and on the other hand the autonomy of logic, and of a part of mathematics. The rest of mathematics became dependent on these two.

Meanwhile, under the pressure of well-founded criticism exerted upon old formalism, Hilbert founded the New Formalist School, which postulated existence and exactness independent of language not for proper mathematics but for meta-mathematics, which is the scientific consideration of the symbols occurring in perfected mathematical language, and of the rules of manipulation of these symbols. On this basis new formalism, in contrast to old formalism, in confesso made primordial practical use of the intuition of natural numbers and of complete induction. It is true that only for a small part of mathematics (much smaller than in pre-intuitionism) was autonomy postulated in this way. New formalism was not deterred from its procedure by the objection that between the perfection of mathematical language and the perfection of mathematics itself no clear connection could be seen.

So the situation left by formalism and pre-intuitionism can be summarised as follows: for the elementary theory of natural numbers, the principle of complete induction and more or less considerable parts of arithmetic and of algebra, exact existence, absolute reliability and non-contradictority were universally acknowledged, independently of language and without proof. As for the continuum, the question of its languageless existence was neglected, its establishment as a set of real numbers with positive measure was attempted by logical means and no proof of its non-contradictory existence appeared. For the whole of mathematics the four principles of classical logic were accepted as means of deducing exact truths.

In this situation intuitionism intervened with two acts, of which the first seems to lead to destructive and sterilising consequences, but then the second yields ample possibilities for new developments.

FIRST ACT OF INTUITIONISM

Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognising that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities. And it is this common substratum, this empty form, which is the basic intuition of mathematics.

Inner experience reveals how, by unlimited unfolding of the basic intuition, much of 'separable' mathematics can be rebuilt in a suitably modified form. In the edifice of mathematical thought thus erected, language plays no part other than that of an efficient, but never infallible or exact, technique for memorising mathematical constructions, and for communicating them to others, so that mathematical language by itself can never create new mathematical systems. But because of the highly logical character of this mathematical language the following question naturally presents itself. Suppose that, in mathematical language, trying to deal with an intuitionist mathematical operation, the figure of an application of one of the principles of classical logic is, for once, blindly formulated. Does this figure of language then accompany an actual languageless mathematical procedure in the actual mathematical system concerned?

A careful examination reveals that, briefly expressed, the answer is in the affirmative, as far as the principles of contradiction and syllogism are concerned,' if one allows for the inevitable inadequacy of language as a mode of description and communication. But with regard to the principle of the excluded third, except in special cases, the answer is in the negative, so that this principle cannot in general serve as an instrument for discovering new mathematical truths.

Indeed, if each application of the principium tertii exclusi in mathematics accompanied some actual mathematical procedure, this would mean that each mathematical assertion (i.e. an assignment of a property to a mathematical entity) could be judged, that is to say could either be proved or be reduced to absurdity.

Now every construction of a bounded finite nature in a finite mathematical system can only be attempted in a finite number of ways, and each attempt proves to be successful or abortive in a finite number of steps. We conclude that every assertion of possibility of a construction of a bounded finite nature in a finite mathematical system can be judged, so that in these circumstances applications of the Principle of the Excluded Third are legitimate.

But now let us pass to infinite systems and ask for instance if there exists a natural number n such that in the decimal expansion of pi the nth, (n+1)th, ..., (n+8)th and (n+9)th digits form a sequence 0123456789. This question, relating as it does to a so far not judgeable assertion, can be answered neither affirmatively nor negatively. But then, from the intuitionist point of view, because outside human thought there are no mathematical truths, the assertion that in the decimal expansion of pi a sequence 0123456789 either does or does not occur is devoid of sense.

The aforesaid property, suppositionally assigned to the number n, is an example of a fleeing property, by which we understand a property f, which satisfies the following three requirements:

(i) for each natural number n it can be decided whether or not n possesses the property f,

(ii) no way of calculating a natural number n possessing f is known;

(iii) the assumption that at least one natural number possesses f is not known to be an absurdity.

Obviously the fleeing nature of a property is not necessarily permanent, for a natural number possessing f might at some time be found, or the absurdity of the existence of such a natural number might at some time be proved.

...

The belief in the universal validity of the principle of the excluded third in mathematics is considered by the intuitionists as a phenomenon of the history of civilization of the same kind as the former belief in the rationality of pi, or in the rotation of the firmament about the earth. The intuitionist tries to explain the long duration of the reign of this dogma by two facts: firstly that within an arbitrarily given domain of mathematical entities the non-contradictority of the principle for a single assertion is easily recognized; secondly that in studying an extensive group of simple every-day phenomena of the exterior world, careful application of the whole of classical logic was never found to lead to error. [This means de facto that common objects and mechanisms subjected to familiar manipulations behave as if the system of states they can assume formed part of a finite discrete set, whose elements are connected by a finite number of relations.]

The mathematical activity made possible by the first act of intuitionism seems at first sight, because mathematical creation by means of logical axioms is rejected, to be confined to 'separable' mathematics, mentioned above; while, because also the principle of the excluded third is rejected, it would seem that even within 'separable' mathematics the field of activity would have to be considerably curtailed. In particular, since the continuum appears to remain outside its scope, one might fear at this stage that in intuitionism there would be no place for analysis. But this fear would have assumed that infinite sequences generated by the intuitionistic unfolding of the basic intuition would have to be fundamental sequences, i.e. predeterminate infinite sequences proceeding, like classical ones, in such a way that from the beginning the nth term is fixed for each n. Such however is not the case; on the contrary, a much woder field of development, including analysis and often exceeding the frontiers of classical mathematics, is opened by the second act of intuitionism.

SECOND ACT OF INTUITIONISM

Admitting two ways of creating new mathematical entities: firstly in the shape of more or less freely proceeding infinite sequences of mathematical entities previously acquired (so that, for decimal fractions having neither exact values, not any guarantee of ever getting exact values admitted); secondly in the shape of mathematical species, i.e. properties supposable for mathematical entities previously acquired, satisfying the condition that if they hold for a certain mathematical entity, they also hold for all mathematical entities which have been defined to be 'equal' to it, definitions of equality having to satisfy the conditions of symmetry, relfexivity and transitivity. ...

Theorems holding in intuitionistic, but not in classical, mathematics often originate from the circumstance that for mathematical entities belonging to a certain species the inculcation of a certain property imposes a special character on their way of development from the basic intuition; and that from this compulsory special character properties ensue which for classical mathematics are false. Striking examples are the modern theorems that the continuum does not split, and that a full function of the unit continuum is necessarily uniformly continuous.

Notes

Introvert science, directed at beauty, does not carry risks for consequences.

The stock of mathematical entities is a real thing, for each person, and for humanity.

The inner experience (roughly sketched):

twoity;
twoity stored and preserved aseptically by memory;
twoity giving rise to the conception of invariable unity;
twoity and unity giving rise to the conception of unity plus unity;
threeity as twoity plus unity, and the sequence of natural numbers;
mathematical systems conceived in such a way that a unity is a mathematical system and that two mathematical systems, stored and aseptically preserved by memory, apart from each other, can be added;
etc.

Fragments from a lecture 'Changes in the relation between classical logic and mathematics. (The influence of intuitionistic mathematics on logic)', given in November 1951

Classical logic presupposed that independently of human thought there is a truth, part of which is expressible by means of sentences called 'true assertions', mainly assigning certain properties to certain objects or stating that objects possessing certain properties exist or that certain phenomena behave according to certain laws. Furthermore classical logic assumed the existence of general linguistic rules allowing an automatic deduction of new true assertions from old ones, so that starting from a limited stock of 'evidently' true assertions, mainly founded on experience and called axioms, an extensive supplement to existing human knowledge would theoretically be accessible by means of linguistic operations independently of experience. Finally, using the term 'false' for the 'converse of true', classical logic assumed that in virtue of the so-called 'principle of the excluded third' each assertion, in particular each existence assertion and each assignment of a property to an object or of a behaviour to a phenomenon, is either true or false independently of human beings knowing about this falsehood or truth, so that, for example, contradictorily of falsehood would imply truth whilst an assertion a which is true if the assertion b is either true or false would be universally true. The principle holds if 'true' is replaced by 'known and registered to be true', but then this classification is variable, so that to the wording of the principle we should add 'at a certain moment'.

As long as mathematics was considered as the science of space and time, it was a beloved field of activity of this classical logic, not only in the days when space and time were believed to exist independently of human experience, but still after they had been taken for innate forms of conscious exterior human experience. There continued to reign some conviction that a mathematical assertion is either false or true, whether we know it or not, and that after the extinction of humanity mathematical truths, just as laws of nature, will survive. About half a century ago this was expressed by the great French mathematician Charles Hermite in the following words: 'Il existe, si je ne me trompe, tout un monde qui est l'ensemble des vérités mathématiques, dans lequel nous n'avons d'accés que par l'intelligence, comme existe le monde des réalités physiques; l'un et l'autre indépendant de nous, tous deux de création divine et qui ne semblent distincts quà cause de la faiblesse de notre esprit, par contre ne sont pour une pensée puissante qu'une seule et même chose, et dont la synthèse se rélève partiellement dans cette merveilleuse correspondence entre les Mathématiques abstraites d'une part, I'Astronomie, et toutes les branches de la Physique de I'autre'.

Only after mathematics had been recognized as an autonomous interior constructional activity which, although it can be applied to an exterior world, neither in its origin nor in its methods depends on an exterior world, firstly all axioms became illusory, and secondly the criterion of truth or falsehood of a mathematical assertion was confined to mathematical activity itself, without appeal to logic or to hypothetical omniscient beings. An immediate consequence was that for a mathematical assertion a the two cases of truth and falsehood, formerly exclusively admitted, were replaced by the following three:

(1) a has been proved to be true;

(2) a has been proved to be absurd;

(3) a has neither been proved to be true nor to be absurd, nor do we know a finite algorithm leading to the statement either that a is true or that a is absurd. [The case that a has neither been proved to be true nor to be absurd, but that we know a finite algorithm leading to the statement either that a is true, or that a is absurd, obviously is reducible to the first and second cases. This applies in particular to assertions of possibility of a construction of bounded finite character in a finite mathematical system, because such a construction can be attempted only in a finite number of particular ways, and each attempt proves successful or abortive in a finite number of steps.]

In contrast to the perpetual character of cases (1) and (2), an assertion of type (3) may at some time pass into another case, not only because further thinking may generate an algorithm accomplishing this passage, but also because in modern or intuitionistic mathematics, as we shall see presently, a mathematical entity is not necessarily predeterminate, and may, in its state of free growth, at some time acquire a property which it did not possess before.

See lecture above on fleeing property

One of the reasons [incorrect, the extension is an immediate consequence of the self-unfolding; so here only the utility of the extension is explained.] that led intuitionistic mathematics to this extension was the failure of classical mathematics to compose the continuum out of points without the help of logic. For, of real numbers determined by predeterminate convergent infinite sequences of rational numbers, only an ever-unfinished denumerable species can actually be generated. This ever-unfinished denumerable species being condemned never to exceed the measure zero, classical mathematics, in order to compose a continuum of positive measure out of points, has recourse to some logical process starting from at least an axiom. A rather common method of this kind is due to Hilbert who, starting from a set of properties of order and calculation, including the Archimedean property, holding for the arithmetic of the field of rational numbers, and considering successive extensions of this field and arithmetic to the extended fields and arithmetics conserving the foresaid properties, including the preceding fields and arithmetics, postulates the existence of an ultimate such extended field and arithmetic incapable of further extension, i.e. he applies the so-called axiom of completeness. From the intuitionistic point of view the continuum created in this way has a merely linguistic, and no mathematical, existence. It is only by means of the admission of freely proceeding infinite sequences that intuitionistic mathematics has succeeded to replace this linguistic continuum by a genuine mathematical continuum of positive measure, and the linguistic truths of classical analysis by genuine mathematical truths.

However, notwithstanding its rejection of classical logic as an instrument to discover mathematical truths, intuitionistic mathematics has its general introspective theory of mathematical assertions. This theory, which with some right may be called intuitionistic mathematical logic, we shall illustrate by the following remarks.

"Epistemology & Modern Physics" by Moritz Schlick,1925

There is no longer any doubt nowadays, that theoretical philosophy has standing only in close connection with the sciences, whether it seeks in them a basis on which it attempts to build further, or whether they form for it merely the subject-matter of its own analyses, whereby it then makes individual inquiry into the first principles of knowledge. This is very much the case if, as I believe, philosophy can be nothing else whatever but the activity whereby we clarify all our concepts. And it is also beyond doubt that, of all the sciences, physics here stands at the forefront. Physics, that is, occupies an exceptional position, because in it two elements are united, which are only found separately in the other sciences: in the first place its exactness, the quantitative determinacy of its laws, whereby it differs from all other factual sciences, more particularly the historical sciences; and secondly the fact that it has as its subject-matter the real, and in this respect differs from mathematics.

Even a person who did not follow Kant, in permitting only absolutely certain, exact knowledge to count as knowledge at all, would yet be convinced that at any rate it represents the high point of knowledge, so that a philosophy which could do complete justice to exact knowledge would thereby at the same time have solved the entire problem of knowledge. But this it can do, however, only if it deals not merely with strict knowledge, but at the same time with knowledge of the real, since merely imagined or contrived objects are of little interest to the philosopher; it is the world of reality which yields him the major problems.

Hence the physical sciences are assured of having a unique significance for philosophy, though this has not always been apparent in equal measure to philosophers of different periods. After the making, in our own day, of some attempts that were already methodically defective, to couple the historical with the exact sciences from a philosophical standpoint, the modern development of physics, which has taken on a highly philosophical character, has brought out the peculiar position of this science a great deal more clearly than ever before. So clearly, indeed, that, given the present state of research, some altogether crucial questions about the mutual relationship of physics and epistemology can perhaps be brought to a decision.

The most important of these questions seems to me this: when, through alliance with exact empirical science, philosophy emancipates itself from the speculative procedure, does it acquire a better criterion of its own truth? Given a proposition of physics, we know how its truth can in principle be established: it has to be confirmed by experience. But the question of how we actually recognise the truth of a philosophical system is so far from having found a generally satisfying answer, that it has often been put forward only for the purpose of deriding philosophy.

Today, however, after the insight gained into the thorough interpenetration of philosophy and the sciences, we can and must say of epistemology, at least, that the correct theory is that which prevails in course of the advance of physical research.

But this formulation of the criterion of truth is initially so indeterminate and general, that we still need very accurate elucidations in order to understand its meaning aright. And here it is contemporary physics alone which provides us with the instances of cognition that are needed to specify and explain matters in full.

Before examining individual cases, we shall ask in what sense we can really expect beforehand to find epistemological statements confirmed in physics. Can philosophy predict any experimental finding of the empirical sciences? We certainly have no right to assume this, for if so, philosophy would be dabbling in the trade of physics, and nobody believes any more nowadays that physical results can be obtained by purely philosophical methods. The task of epistemology is not to predict what will be observed in nature. It merely tells us beforehand how science will react, if this or that is observed. What it prophesies, therefore, is not the results of experiments, but the impact of experimental results on the system of physics.

By far the most important limiting case of such statements occurs when it lays down specific principles with the claim that science will always adhere to them, whatever sort of observations may be made. In short, epistemology makes statements about the dependence of physics on, and in limiting cases its independence of possible observations. The statements are correct when, on the occurrence of these observations, physical science actually takes the form predicted.

Here now is the weightiest example from modern physics. The epistemology pursued by the great mathematicians of the 19th century (Gauss, Riemann, Helmholtz) had maintained that a specific course of processes in nature (a specific mode of behaviour on the part of light-rays and measuring rods) was conceivable, on observation of which physics would turn over to employing non-Euclidean geometries. This prediction, as we know, has been most brilliantly confirmed by the general theory of relativity, and the premises on whose basis this prophecy was made, have thereby demonstrated their truth-value. But what role did these premises play in the epistemology of the said mathematicians? Do they form the inmost heart of their philosophy, determining the character of the whole edifice of thought, or are they of a less essential kind, so that they might perhaps equally find a place in an altogether different theory of knowledge? This question has to be answered in order to know in what degree and what aspect modern physics is actually to be seen as confirmation of that particular epistemology, which was notoriously that of empiricism.

An important step towards deciding the matter is taken if we establish whether, or in what degree, the opposite theory to empiricism, that of Kantian apriorism, would be equally capable of validating the principles of modern physics. This apriorism teaches, of course, that natural science will always adhere to certain general principles, whatever any given experimenter may happen to observe. These principles are said to be synthetic, that is, not to express mere tautologies, and they are also said to be a priori. The latter has a double significance in the Kantian system. First, that is, that they represent logical presuppositions of science, so that without them we could erect no structure of connected truths about nature at all; but secondly, too, that these principles are self-evident for us, so that we simply cannot imagine their invalidity, and hence that our ideational consciousness is inexorably linked to them. Of these two aspects the so-called logical interpretation of Kant (the Marburg school) emphasises the first, while the psychological view stresses the second. The conflict between the two attitudes is strange, since both interpretations are quite indubitably combined together in Kant: synthetic a priori propositions are for him both the logically necessary presuppositions of science, and also imbued with the psychological compulsion of self-evidence.

Now which, according to the doctrine of apriorism, are the basic synthetic judgements of all science? For Kant, they include the axioms of Euclidean geometry, of which, as we have just seen, modern physics demonstrates that they are not a priori in the first sense, after it had already been made clear before this that they are not so in the second (psychological) sense. For in that sense apriorism with regard to Euclidean geometry has already been refuted by psychological considerations, which many philosophers still seem to overlook.

In this one connection, concerning some (and hence not yet all) geometrical axioms, modern physics therefore opts decisively in favour of empiricism. But apriorism can take a variety of forms; its principle is elastic, and does not have to be defended precisely in Kant's version. It would be quite generally refuted only if it turned out that science contains no synthetic a priori propositions whatsoever. Anyone who maintains their existence must of course be able to produce them. An apriorism that cannot really enumerate a single synthetic a priori principle, has thereby pronounced its own death-sentence. For this reason I raised the question some years ago' , as to which judgements about nature a modern apriorisrn would now be able to propose, in the light of contemporary physics, as absolutely inescapable presuppositions of all science, independent of any possible observations.

And to this question modern scientific research appears to give an answer of the same kind as that given in the case of Euclidean geometry; for it shows that physical science refuses to regard any one of the principles which might come into question here as the sole possible basis. In order to convince ourselves of this, let us go through the particular proposals which have been made for keeping apriorism on its feet!

In the first place, now that a portion of the Euclidean axioms has had to be dropped, the attempt has been made to extract a complex from the remaining axioms of geometry, and to proclaim it as the unshakeable foundation of all scientific accounts of space. Reverting to an older belief, it has been sought to ascribe this rank to the axioms of analysis situs, to those principles, that is, which describe the purely qualitative inter-relationships of space, without reference to 'metrical' relations of magnitude - in short, to the axioms of 'topological' space'. But there are indications in modern physics that it has no wish to allow itself to be fettered for ever by such axioms. Hermann Weyl has already outlined a peculiar theory of matter according to which electrons, the ultimate constituents of matter, are as it were outside space. The latter would have such peculiar topological properties that it would be impossible, for example, to imagine a spherical volume of space containing electrons to contract by steady shrinkage into a point. Still bolder constructions are scientifically possible, and there is simply no predicting the assumptions to which we may be driven by the astonishing physical facts disclosed by modern research. Hence the appearance of contemporary physics gives us clear warning against the attempt to view the topological axioms, say, as a noli me tangere [touch me not].

In the second place, the language of the new physics pronounces more clearly still against the endeavour to cling, say, to the continuity of nature as a necessary and invariably satisfied condition, which now finds expression in certain synthetic a priori propositions. For since Riemann, some decades ago, examined the physical possibility of a discontinuous space composed of discrete points, Planck's quantum theory, in our own day, has so domesticated the idea of jumpiness and discontinuity in our view of nature, that our physics is nowhere prepared to contest in principle the possibility of discontinuities. Here too, therefore, apriorism finds no resting-place.

Third and lastly, let us examine the attitude of present-day physics to that principle which appears in Kant as the most important of synthetic a priori propositions, and is also not infrequently declared to be such even today: I mean, of course, the causal principle. If, appropriately enough, we mean by causality the existence of regularity in nature, it certainly represents a necessary presupposition of science; without causality, a knowledge of nature would be impossible, for such knowledge consists, in fact, of discovering laws. From this simple fact many have already sought to conclude that the causal law is to be regarded as an a priori principle in the fullest sense. But this is undoubtedly quite mistaken, or at least a misuse of terminology. For this does nothing to establish an epistemological apriorism. The latter only comes about if we add the claim that we should continue to uphold the validity of the causal principle for all natural processes, whatever science may disclose to us in. the way of facts in nature. In other words, we should have to possess an unshakeable conviction of the factual validity of the causal principle. We see here how the logical a priori is inseparable from the psychological, if it is to characterise a particular epistemological position, namely the Kantian notion that our understanding prescribes laws to nature. So when Ernst Cassirer expresses the opinion that the idea of universal regularity in nature, as such, continues to hold good as a synthetic a priori principle, or when J. Winternitz , among others, describes the causal law as a constitutive principle of science in Kant's sense, the view of these exponents of a modified apriorism can only be understood to mean that they regard the possibility of science as absolutely assured, and consider a nature that would furnish no laws to man an absurdity.

As against this, it can be read off from the present state of physics, that science does not recognise a priori constraints of this kind, and opposes to the view in question the healthy scepticism of the empiricist. The pursuit of processes within the atom by the methods of quantum theory has led many physicists to conclude that, within certain limits, processes that are strictly causeless occur there; to these, therefore, the causal law could find no applications.

Even if - like the author - one fails to perceive in the facts available any sufficient basis for this conclusion, it could still become perfectly legitimate if further facts were to hand, and so this case has the following lesson to teach: Although physics is well aware that the causal principle, the reciprocal dependence of natural processes on each other, is a presupposition for its own existence, it still by no means assumes this presupposition to be satisfied a priori everywhere, — or even in a particular area; it ascertains for itself, rather, using its own methods (and with the exactitude of these methods), whether and to what extent this is the case. It establishes for itself, that is, the boundaries of its own kingdom. That the methods of science are able to make such an examination, can be confirmed by a subsequent analysis of its procedure. All this in contradiction to apriorism, according to which the causal principle is supposed not to be an empirically testable proposition.

The empiricist, of course, is well aware that it would always be possible in principle to sustain the causal law by suitable hypotheses -just as he knows that Euclidean geometry could be held valid without exception, if we really wanted this; but he denies that the human mind is unconditionally obliged to do this, and denies also that the application of scientific methods could always lead only to a confirmation of the causal principle. On the contrary, it is quite easy to imagine observations which would make it possible to sustain the causal law only by an infraction of these methods: namely, by a continual introduction of new hypotheses constructed ad hoc. And the modern physicist confirms the empiricist's prediction the moment he thinks himself actually confronted with observations of that kind.

Thus a survey of the state of modern physics indicates that it presents us in surprising sequence with a series of cases, in which the empiricist and apriorist views of natural knowledge may contend with one another; that without exception it pursues the course recommended by empiricism; and that not one of its principles is accorded those properties which a synthetic a priori judgement of the Kantian type would have to possess.

We may say, therefore, that modern physics shows us, that even for epistemology there is a sort of confirmation by experience, an objective criterion of truth, and that this criterion decides in favour of the empiricist theory of knowledge.

A remark needs to be added, to guard against erroneous conclusions from what we have said.

The relation outlined between modern physics and philosophy could occasion regret that epistemology should cast the anchor of its criterion of truth into empirical science, and thereby partake of its uncertainty and mutability. But if the hope of grounding philosophy on a firmer soil than that of experience and logic must be abandoned (and it has never been more than a hope anyway), this would have to be set off in the bargain against the advantage of having obtained any objective criterion at all. It is very notable that an actual exponent of apriorism, Elsbach (in his book Kant und Einstein ), expresses the view that epistemology can be expected only to vindicate the mutable state of science at any time, but not science as such. This position is no longer that of Kantianism (Einstein, in his critique of Elsbach's book, says of him that he is in agreement neither with Mohammed nor with the prophet); it is more empiricist than empiricism. For the empiricist is unable to join in the lament of many onlookers, that physics is constantly changing, that its theories are short-lived and that hitherto supposedly correct laws are liable to be overthrown at any moment by new discoveries. He knows, rather, that no law till now, in the sense and with the exactitude whereby it has once been confirmed, has ever again had to be abandoned. The changeable elements in physics are not the relations of dependency, which once established, continue to find repeated confirmation, but rather the intuitive ideas which serve for interpretation and interpolation. The split between the purely conceptual and empirically confirmed content of a science, and the intuitive images which illustrate the content without themselves belonging thereto - this split is one of the most important achievements of modern epistemology. A philosophy that knows how to achieve it tidily everywhere may justifiably regard a confirmation by modern physics in the sense outlined above as a confirmation by science as such.

"The modern development of the foundations of mathematics in the light of philosophy" by Kurt Gödel,1961

Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
He also showed that the continuum hypothesis cannot be disproved from the accepted axioms of set theory, if those axioms are consistent. He made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.
-------------------------------------------------------------------------------
I would like to attempt here to describe, in terms of philosophical concepts, the development of foundational research in mathematics since around the turn of the century, and to fit it into a general schema of possible philosophical world-views [Weltanschauungen]. For this, it is necessary first of all to become clear about the schema itself. I believe that the most fruitful principle for gaining an overall view of the possible world-views will be to divide them up according to the degree and the manner of their affinity to or, respectively, turning away from metaphysics (or religion). In this way we immediately obtain a division into two groups: scepticism, materialism and positivism stand on one side, spiritualism, idealism and theology on the other. We also at once see degrees of difference in this sequence, in that scepticism stands even farther away from theology than does materialism, while on the other hand idealism, e.g., in its pantheistic form, is a weakened form of theology in the proper sense.

The schema also proves fruitful, however, for the analysis of philosophical doctrines admissible in special contexts, in that one either arranges them in this manner or, in mixed cases, seeks out their materialistic and spiritualistic elements. Thus one would, for example, say that apriorism belongs in principle on the right and empiricism on the left side. On the other hand, however, there are also such mixed forms as an empiristically grounded theology. Furthermore one sees also that optimism belongs in principle toward the right and pessimism toward the left. For scepticism is certainly a pessimism with regard to knowledge. Moreover, materialism is inclined to regard the world as an unordered and therefore meaningless heap of atoms. In addition, death appears to it to be final and complete annihilation, while, on the other hand, theology and idealism see sense, purpose and reason in everything. On the other hand, Schopenhauer's pessimism is a mixed form, namely a pessimistic idealism. Another example of a theory evidently on the right is that of an objective right and objective aesthetic values, whereas the interpretation of ethics and aesthetics on the basis of custom, upbringing, etc., belongs toward the left.

Now it is a familiar fact, even a platitude, that the development of philosophy since the Renaissance has by and large gone from right to left - not in a straight line, but with reverses, yet still, on the whole. Particularly in physics, this development has reached a peak in our own time, in that, to a large extent, the possibility of knowledge of the objectivisable states of affairs is denied, and it is asserted that we must be content to predict results of observations. This is really the end of all theoretical science in the usual sense (although this predicting can be completely sufficient for practical purposes such as making television sets or atom bombs).

It would truly be a miracle if this (I would like to say rabid) development had not also begun to make itself felt in the conception of mathematics. Actually, mathematics, by its nature as an a priori science, always has, in and of itself, an inclination toward the right, and, for this reason, has long withstood the spirit of the time [Zeitgeist] that has ruled since the Renaissance; i.e., the empiricist theory of mathematics, such as the one set forth by Mill, did not find much support. Indeed, mathematics has evolved into ever higher abstractions, away from matter and to ever greater clarity in its foundations (e.g., by giving an exact foundation of the infinitesimal calculus and the complex numbers) - thus, away from scepticism.

Finally, however, around the turn of the century, its hour struck: in particular, it was the antinomies of set theory, contradictions that allegedly appeared within mathematics, whose significance was exaggerated by sceptics and empiricists and which were employed as a pretext for the leftward upheaval. I say "allegedly" and "exaggerated" because, in the first place, these contradictions did not appear within mathematics but near its outermost boundary toward philosophy, and secondly, they have been resolved in a manner that is completely satisfactory and, for everyone who understands the theory, nearly obvious. Such arguments are, however, of no use against the spirit of the time, and so the result was that many or most mathematicians denied that mathematics, as it had developed previously, represents a system of truths; rather, they acknowledged this only for a part of mathematics (larger or smaller, according to their temperament) and retained the rest at best in a hypothetical sense namely, one in which the theory properly asserts only that from certain assumptions (not themselves to be justified), we can justifiably draw certain conclusions. They thereby flattered themselves that everything essential had really been retained. Since, after all, what interests the mathematician, in addition to drawing consequences from these assumptions, is what can be carried out. In truth, however, mathematics becomes in this way an empirical science. For if I somehow prove from the arbitrarily postulated axioms that every natural number is the sum of four squares, it does not at all follow with certainty that I will never find a counter-example to this theorem, for my axioms could after all be inconsistent, and I can at most say that it follows with a certain probability, because in spite of many deductions no contradiction has so far been discovered. In addition, through this hypothetical conception of mathematics, many questions lose the form "Does the proposition A hold or not?" For, from assumptions construed as completely arbitrary, I can of course not expect that they have the peculiar property of implying, in every case, exactly either A or ~A.

Although these nihilistic consequences are very well in accord with the spirit of the time, here a reaction set in obviously not on the part of philosophy, but rather on that of mathematics, which, by its nature, as I have already said, is very recalcitrant in the face of the Zeitgeist. And thus came into being that curious hermaphroditic thing that Hilbert's formalism represents, which sought to do justice both to the spirit of the time and to the nature of mathematics. It consists in the following: on the one hand, in conformity with the ideas prevailing in today's philosophy, it is acknowledged that the truth of the axioms from which mathematics starts out cannot be justified or recognised in any way, and therefore the drawing of consequences from them has meaning only in a hypothetical sense, whereby this drawing of consequences itself (in order to satisfy even further the spirit of the time) is construed as a mere game with symbols according to certain rules, likewise not supported by insight.

But, on the other hand, one clung to the belief, corresponding to the earlier "rightward" philosophy of mathematics and to the mathematician's instinct, that a proof for the correctness of such a proposition as the representability of every number as a sum of four squares must provide a secure grounding for that proposition - and furthermore, also that every precisely formulated yes-or-no question in mathematics must have a clear-cut answer. I.e., one thus aims to prove, for inherently unfounded rules of the game with symbols, as a property that attaches to them so to speak by accident, that of two sentences A and ~A, exactly one can always be derived. That not both can be derived constitutes consistency, and that one can always actually be derived means that the mathematical question expressed by A can be unambiguously answered. Of course, if one wishes to justify these two assertions with mathematical certainty, a certain part of mathematics must be acknowledged as true in the sense of the old rightward philosophy. But that is a part that is much less opposed to the spirit of the time than the high abstractions of set theory. For it refers only to concrete and finite objects in space, namely the combinations of symbols.

What I have said so far are really only obvious things, which I wanted to recall merely because they are important for what follows. But the next step in the development is now this: it turns out that it is impossible to rescue the old rightward aspects of mathematics in such a manner as to be more or less in accord with the spirit of the time. Even if we restrict ourselves to the theory of natural numbers, it is impossible to find a system of axioms and formal rules from which, for every number-theoretic proposition A, either A or ~A would always be derivable. And furthermore, for reasonably comprehensive axioms of mathematics, it is impossible to carry out a proof of consistency merely by reflecting on the concrete combinations of symbols, without introducing more abstract elements. The Hilbertian combination of materialism and aspects of classical mathematics thus proves to be impossible.

Hence, only two possibilities remain open. One must either give up the old rightward aspects of mathematics or attempt to uphold them in contradiction to the spirit of the time. Obviously the first course is the only one that suits our time and is therefore also the one usually adopted. One should, however, keep in mind that this is a purely negative attitude. One simply gives up aspects whose fulfilment would in any case be very desirable and which have much to recommend themselves: namely, on the one hand, to safeguard for mathematics the certainty of its knowledge, and on the other, to uphold the belief that for clear questions posed by reason, reason can also find clear answers. And as should be noted, one gives up these aspects not because the mathematical results achieved compel one to do so but because that is the only possible way, despite these results, to remain in agreement with the prevailing philosophy.

Now one can of course by no means close one's eyes to the great advances which our time exhibits in many respects, and one can with a certain justice assert that these advances are due just to this leftward spirit in philosophy and world-view. But, on the other hand, if one considers the matter in proper historical perspective, one must say that the fruitfulness of materialism is based in part only on the excesses and the wrong direction of the preceding rightward philosophy. As far as the rightness and wrongness, or, respectively, truth and falsity, of these two directions is concerned, the correct attitude appears to me to be that the truth lies in the middle or consists of a combination of the two conceptions.

Now, in the case of mathematics, Hilbert had of course attempted just such a combination, but one obviously too primitive and tending too strongly in one direction. In any case there is no reason to trust blindly in the spirit of the time, and it is therefore undoubtedly worth the effort at least once to try the other of the alternatives mentioned above, which the results cited leave open - in the hope of obtaining in this way a workable combination. Obviously, this means that the certainty of mathematics is to be secured not by proving certain properties by a projection onto material systems - namely, the manipulation of physical symbols but rather by cultivating (deepening) knowledge of the abstract concepts themselves which lead to the setting up of these mechanical systems, and further by seeking, according to the same procedures, to gain insights into the solvability, and the actual methods for the solution, of all meaningful mathematical problems.

In what manner, however, is it possible to extend our knowledge of these abstract concepts, i.e., to make these concepts themselves precise and to gain comprehensive and secure insight into the fundamental relations that subsist among them, i.e., into the axioms that hold for them? Obviously not, or in any case not exclusively, by trying to give explicit definitions for concepts and proofs for axioms, since for that one obviously needs other undefinable abstract concepts and axioms holding for them. Otherwise one would have nothing from which one could define or prove. The procedure must thus consist, at least to a large extent, in a clarification of meaning that does not consist in giving definitions.

Now in fact, there exists today the beginning of a science which claims to possess a systematic method for such a clarification of meaning, and that is the phenomenology founded by Husserl. Here clarification of meaning consists in focusing more sharply on the concepts concerned by directing our attention in a certain way, namely, onto our own acts in the use of these concepts, onto our powers in carrying out our acts, etc. But one must keep clearly in mind that this phenomenology is not a science in the same sense as the other sciences. Rather it is or in any case should be a procedure or technique that should produce in us a new state of consciousness in which we describe in detail the basic concepts we use in our thought, or grasp other basic concepts hitherto unknown to us. I believe there is no reason at all to reject such a procedure at the outset as hopeless. Empiricists, of course, have the least reason of all to do so, for that would mean that their empiricism is, in truth, an apriorism with its sign reversed.

But not only is there no objective reason for the rejection of phenomenology, but on the contrary one can present reasons in its favour. If one considers the development of a child, one notices that it proceeds in two directions: it consists on the one hand in experimenting with the objects of the external world and with its own sensory and motor organs, on the other hand in coming to a better and better understanding of language, and that means - as soon - as the child is beyond the most primitive designating of objects - of the basic concepts on which it rests. With respect to the development in this second direction, one can justifiably say that the child passes through states of consciousness of various heights, e.g., one can say that a higher state of consciousness is attained when the child first learns the use of words, and similarly at the moment when for the first time it understands a logical inference.

Now one may view the whole development of empirical science as a systematic and conscious extension of what the child does when it develops in the first direction. The success of this procedure is indeed astonishing and far greater than one would expect a priori: after all, it leads to the entire technological development of recent times. That makes it thus seem quite possible that a systematic and conscious advance in the second direction will also far exceed the expectations one may have a priori.

In fact, one has examples where, even without the application of a systematic and conscious procedure, but entirely by itself, a considerable further development takes place in the second direction, one that transcends "common sense". Namely, it turns out that in the systematic establishment of the axioms of mathematics, new axioms, which do not follow by formal logic from those previously established, again and again become evident. It is not at all excluded by the negative results mentioned earlier that nevertheless every clearly posed mathematical yes-or-no question is solvable in this way. For it is just this becoming evident of more and more new axioms on the basis of the meaning of the primitive notions that a machine cannot imitate.

I would like to point out that this intuitive grasping of ever newer axioms that are logically independent from the earlier ones, which is necessary for the solvability of all problems even within a very limited domain, agrees in principle with the Kantian conception of mathematics. The relevant utterances by Kant are, it is true, incorrect if taken literally, since Kant asserts that in the derivation of geometrical theorems we always need new geometrical intuitions, and that therefore a purely logical derivation from a finite number of axioms is impossible. That is demonstrably false. However, if in this proposition we replace the term "geometrical" - by "mathematical" or "set-theoretical", then it becomes a demonstrably true proposition. I believe it to be a general feature of many of Kant's assertions that literally understood they are false but in a broader sense contain deep truths. In particular, the whole phenomenological method, as I sketched it above, goes back in its central idea to Kant, and what Husserl did was merely that he first formulated it more precisely, made it fully conscious and actually carried it out for particular domains. Indeed, just from the terminology used by Husserl, one sees how positively he himself values his relation to Kant.

I believe that precisely because in the last analysis the Kantian philosophy rests on the idea of phenomenology, albeit in a not entirely clear way, and has just thereby introduced into our thought something completely new, and indeed characteristic of every genuine philosophy - it is precisely on that, I believe, that the enormous influence which Kant has exercised over the entire subsequent development of philosophy rests. Indeed, there is hardly any later direction that is not somehow related to Kant's ideas. On the other hand, however, just because of the lack of clarity and the literal incorrectness of many of Kant's formulations, quite divergent directions have developed out of Kant's thought - none of which, however, really did justice to the core of Kant's thought. This requirement seems to me to be met for the first time by phenomenology, which, entirely as intended by Kant, avoids both the death-defying leaps of idealism into a new metaphysics as well as the positivistic rejection of all metaphysics. But now, if the misunderstood Kant has already led to so much that is interesting in philosophy, and also indirectly in science, how much more can we expect it from Kant understood correctly?