"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.
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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.”
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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".
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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.
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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.”
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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.
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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.
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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.
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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

Saunders Mac Lane,"His Mathematical Life and Philosophical Works"1960

The University of Chicago’s Saunders Mac Lane, one of the most influential American mathematicians of the 20th century and a recipient of the National Medal of Science, died Thursday, April 14, in San Francisco after a long illness. He was 95.

“He was one of the most important figures in the University of Chicago Mathematics Department, or indeed in American mathematics,” David Eisenbud wrote of Mac Lane in the preface of the latter’s autobiography, which will be published in late May by A K Peters Ltd. Eisenbud, who received his Ph.D. in Mathematics from the University of Chicago under Mac Lane’s supervision in 1970, is the president of the American Mathematical Society and director of the Mathematical Sciences Research Institute in Berkeley, Calif.

Peter May, Professor in Mathematics at the University of Chicago, described the multifaceted Mac Lane as “extremely energetic, dynamic, clear-headed, opinionated, a raconteur.” In his research, “he was extraordinarily perceptive and original, and he was especially strong as a philosopher of mathematics,” May said.

“With Sammy Eilenberg he created a new way of thinking about mathematics. In a landmark 1945 paper, they introduced and named the concepts of ‘categories,’ ‘functors’ and ‘natural transformations.’ The language they introduced there transformed modern mathematics,” he said. “In fact, a very great deal of mathematics since then would quite literally have been unthinkable without that language.”

F. William Lawvere, Emeritus professor of mathematics at the State University of New York at Buffalo, said, “Category theory is still exploding in its influence after 60 years, illuminating and guiding the development of practically every one of the many varied fields of mathematics.”

Category theory was first developed as a language to describe transformations from one area of mathematics into another, although it later developed into a field of study in its own right. Said May, “Mac Lane was one of the pioneers of algebraic topology, a subject in which one transforms, or describes, spatial structures, which one first sees in terms of shapes—spaces with holes, like a doughnut, say—into algebraic structures, with addition and multiplication, in which one can do calculations.”

The development of category theory and algebraic topology was accompanied by the development of another subject, homological algebra, a kind of algebra that plays a prominent role in algebraic topology and other branches of mathematics. “Mac Lane played a major role in establishing the foundations of these and related areas of modern mathematics, and he wrote famous texts that are some of the most readable accounts of these fields,” May said.

Category theory was at first perceived by some mathematicians as too abstract for practical mathematics, May said. Consequently, Mac Lane titled his introductory book to the field Categories for the Working Mathematician.

Peter Johnstone, professor of the foundations of mathematics at the University of Cambridge, also lauded Mac Lane for his legacy as a mathematical historian. “In the many papers he wrote on historical topics in his later years, he has left a unique body of material for future historians of 20th-century mathematics, written by someone who was there at the time and who knew what it was like to be working at the cutting edge of mathematical research,” Johnstone said.

Mac Lane also steered national science and mathematics policy through his work on numerous boards, and guided dozens of students to mathematical careers.

In the support of scientific research, Mac Lane served as vice president of the National Academy of Sciences and the American Philosophical Society, and as president of the Mathematical Association of America and the American Mathematical Society. As president of the Mathematical Association of America in the 1950s, he began the first activity of that organization toward improvement in the teaching of modern mathematics.

He was a member of the National Science Board from 1974 to 1980, providing science policy advice to the U.S. government. In 1976, he led a delegation of mathematicians to the People’s Republic of China to examine the conditions affecting the development of mathematics in that nation.

“Saunders’ principal characteristic was his immensely strong sense of duty and service,” Johnstone said. “He felt it was his duty to the profession to which he was privileged to belong, even though I’m sure he found it less congenial work than actually doing mathematics.”

For a time there were three members of the University of Chicago Mathematics Department who had received their Ph.D.s under Mac Lane’s supervision. The first of the three was Alfred Putnam, who studied under Mac Lane at Harvard University in 1942. The other two were John Thompson, who received his Ph.D. in 1959, and Arunas Liulevicius, who did likewise in 1960, both from the University of Chicago. Thompson received the Fields Medal, sometimes referred to as the Nobel Prize of mathematics, in 1970.

Another prominent student of Mac Lane’s, Irving Kaplansky, also served on the University of Chicago faculty. Kaplansky was Mac Lane’s first Ph.D. student, receiving his degree at Harvard in 1941. Kaplansky served as Chairman of Chicago’s Department of Mathematics and as director of Berkeley’s Mathematical Sciences Research Institute, and is a member of the National Academy of Sciences.

Mac Lane supervised 39 Ph.D. students at Chicago, a record surpassed by only three other professors in the history of the University’s Mathematics Department. His last Ph.D. student, Steve Awodey, completed his degree in logic in 1997. Then in his 80s, Mac Lane was still energetic, always insisting upon taking the stairs to his third-floor office in the Ryerson Physical Laboratory, Awodey recalled.

Mac Lane was “a link to a past world of mathematics that’s gone now,” said Awodey, an associate professor in philosophy at Carnegie Mellon University. Mac Lane studied logic and mathematics as a Ph.D. student under several world-renowned mathematicians in the early 1930s in Göttingen, Germany, including David Hilbert, Emmy Noether and Hermann Weyl. At the time Göttingen was the world’s center of mathematical research, and Mac Lane often told stories of the great figures he encountered there.

Mac Lane was born in Norwich, Conn., on Aug. 4, 1909. He earned his bachelor’s degree from Yale College in 1930, and his master’s from the University of Chicago in 1931. His first scientific paper was published during this period, in physics. It was co-authored by Irving Langmuir, who received the 1932 Nobel Prize in chemistry.

For his Ph.D., which Mac Lane received in 1934 from the Mathematisches Institut of Göttingen, Germany, he studied under two prominent mathematicians, Hermann Weyl and Paul Bernays.

Mac Lane worked as a mathematics instructor at Harvard and Cornell universities and at the University of Chicago from 1934 to 1938. He served again on the Harvard faculty from 1938 to 1947, and at Chicago beginning in 1947. In 1944 and 1945, he also directed Columbia University’s Applied Mathematics Group, which was involved in the war effort.

Mac Lane was Chairman of the Mathematics Department at Chicago from 1952 to 1958, taking the reins from Marshall Stone at a time many consider the high point in the department’s history. He was appointed the Max Mason Distinguished Service Professor in Mathematics in 1963, and became Professor Emeritus in 1982.

He was the author or co-author of more than 100 research papers and six books: A Survey of Modern Algebra (1941); Homology (1963); Algebra (1967); Categories for the Working Mathematician (1971); Mathematics, Form and Function (1985); and Sheaves in Geometry and Logic: A First Introduction to Topos Theory, with Ieke Moerdijk, (1992).

His A Survey of Modern Algebra, written with Garrett Birkhoff, was for years the leading textbook in its field and was republished by A K Peters Ltd. in 1997.

Mac Lane was elected to the National Academy of Sciences in 1949. He received the nation’s highest award for scientific achievement, the National Medal of Science, in 1989. Mac Lane received two Guggenheim Fellowships and visited Australia as a Fulbright Scholar. He also received honorary degrees from Purdue University, Yale University and the University of Glasgow, among others. Other honors include both the Chauvenet Prize and the Distinguished Service award of the Mathematical Association of America, the Steele Career Prize of the American Mathematical Society, and honorary fellowship in the Royal Society of Edinburgh.

The University of Chicago Alumni Association presented Mac Lane the Norman Maclean Faculty Award in 2003. The Maclean Award recognizes emeritus professors or senior faculty members who have made outstanding contributions to teaching and to the experience of student life on campus.

Mac Lane is survived by his widow, Osa, of San Francisco, and two daughters, Gretchen Mac Lane of New York City and Cynthia Hay of London; one grandson, William Hay, of London; three stepchildren, William Segal of Washington, D.C.; Andrew Segal of Santa Fe, N.M.; and Karen Segal of San Francisco;, and five step-grandchildren. Mac Lane’s first wife, Dorothy Jones Mac Lane, died in 1985.

"Introduction to Mathematical Philosophy" by Bertrand Russell,1919

..
Mathematics can be shown to be a logical development of certain basic ideas ;
mathematics can be reduced to logic.
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The number of a class is the class of all those classes which are similar to it,.
(Classes are similar when their members can be put into a one-to-one relation with each other.)
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A relation is symmetrical when if one thing has the relation to another,
the other has the same relation to it ;
a relation is transitive when if one thing has the relation to a second,
and the second has the same relation to a third,
the first has the relation to the third.
(Other relations are defined.)
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An infinite cardinal number satisfies the equation,n equals n plus l.
(An infinite collection has parts which have as many terms as the infinite collection itself.)
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By distinguishing between types of entities it is possible to avoid paradoxes which have perplexed philosophers for centuries.
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Mathematical truths are a priori and have nothing to do with facts about the world ;
they are logical tautologies.
..