#### by Colin McLarty

Most mathematicians cannot define the spaces of algebraic geometry, called
*schemes*, off the top of their heads, nor the *morphisms*
__\( f:S \to S^{\prime} \)__ mapping one scheme algebro-geometrically to another.
But they can define the cartesian product __\( S \times S^{\prime} \)__
and the coproduct, or union, __\( S \cup S^{\prime} \)__ of schemes (up to
isomorphism) by familiar diagrams of morphisms,
as in the figure.

Mac Lane wrote two autobiographies. Obviously there is his *Mathematical Autobiography* [38].
Less obvious is *Mathematics: Form and Function*
[23],
aimed at philosophers.
It surveys the mathematics
Mac Lane wanted philosophers to know. Naturally, that is the mathematics
Mac Lane wanted to know over the years, so this sweeping survey
is little more sweeping than his own career. Having attended lectures by
David Hilbert
and
Emmy Noether,
and studied with
Paul Bernays
and
Hermann Weyl,
Mac Lane made noted contributions to algebra, topology
and group theory. He was influential in logic. He published on geometry,
mechanics and on teaching calculus. He looked over the whole scope
of mathematics as vice president of the National Academy of Sciences and
the American Philosophical Society.2
He was president of the Mathematical
Association of America and the American Mathematical Society, and
chair of Mathematics at Chicago from 1952 to 1958, where the mathematics
department was competing to be the strongest in the world.

His central achievement was the co-creation, with
Samuel Eilenberg,
of *category theory*, which became the way mathematicians organize each
branch of mathematics and especially the relations between branches.
*Mathematics: Form and Function* comes surprisingly close to
repeating,
in chronological order, his own mathematical interests. It develops a philosophy
already recognizable in his *Journal of Symbolic Logic* reviews in the
1930s and 1940s, yet which finally relied on
William Lawvere’s
categorical
foundations
([e18],
[e8]).

Mac Lane was distinctively American. His outlook was cosmopolitan and progressive. His trademark Mac Lane tartan suit showed his exuberant pride in his immigrant family past. He traced his intellectual roots to a Presbyterian grandfather “officially charged with heresy for preaching on the writings of Charles Darwin” in Ohio in 1883 ([38], p. 3).3 In many ways his career followed that of Eliakim Hastings Moore (1862–1932), who set out to make America a world force in mathematics.

Retired but still teaching, Professor Moore made a tremendous impression on the beginning graduate student ([38], p. 36). Both had studied at Yale. Moore studied in Germany and headed the mathematics department at Chicago from the time the university first opened in 1892. Mac Lane went to Göttingen for his doctorate in 1931 and later headed the department at Chicago. Both were leaders in the research-oriented American Mathematical Society. Moore’s interest in mathematics education made him a co-founder of the Mathematical Association of America to complement the AMS. As president of the MAA in the 1950s, Mac Lane worked to update mathematics teaching. Moore was the doctoral teacher of G. D. Birkhoff, father of Garrett Birkhoff who co-authored [9] and [16]. And Moore had a principle:

The existence of analogies between central features of various theories implies the existence of a general abstract theory which underlies the particular theories and unifies them with respect to those central features. ([e2], p. 98)

Category theory was created to find the unity behind deep analogies
between topology and algebra. Today it is a standard format for giving
such “general abstract” theories. Consideration of
the relation of Moore to Mac Lane would repay far more attention than it
has gotten.4
Studying logic in Göttingen from 1931 to 1933, right after Gödel’s
incompleteness theorem, Mac Lane developed a strong feeling for
Hilbert’s
“formalism.” He had an exchange in the *New York Review of Books* with
Freeman Dyson
on the point. Dyson claimed Hilbert had gone “into a blind
alley of reductionism … reducing mathematics to a set of marks written
on paper, and deliberately ignoring the context of ideas and applications
that give meaning to the marks.” Mac Lane said to the contrary that Hilbert
meant quite concrete mathematical knowledge in his famous slogan
*Wir müssen wissen; wir werden wissen* (We must know, we will know): “This
was the tone of his general lecture course on the philosophy of science
which I heard him give in late 1931.” Hilbert used formalization in its own
proper domain:

Hilbert himself called this “metamathematics.” He used this for a specific limited purpose, to show mathematics consistent. Without this reduction, no Gödel’s theorem, no definition of computability, no Turing machine, and hence no computers.

Dyson simply does not understand reductionism and the deep purposes it can serve. Hilbert was not “sterile.”

Dyson finally agreed with him about Hilbert and said “Hilbert himself was, of course, a master of both [formal and concrete] kinds of mathematics,” but oddly Dyson would not extend the same concession to Mac Lane. He insisted Mac Lane “is by temperament a reductionist.” In fact, Mac Lane was no kind of reductionist. He saw what formalization could achieve while he was constantly involved with the ideas and applications that give it meaning.5 The specific project of his Göttingen dissertation was to give formal means of abbreviating formal proofs. But the larger goal was to bring formal logic closer to practice and make it express more directly how

A proof is not just a series of individual steps, but a group of steps, brought together according to a definite plan or purpose …. So we affirm that every mathematical proof has a leading idea, which determines all the individual steps, and can be given as a plan of the proof. ([1], p. 60)

This larger project became a philosophical article in the *Monist* the year
that Mac Lane started publishing
[2].
It is reaffirmed, for
example, in
[35].

Mac Lane’s most penetrating work in mathematics, philosophy and
history of mathematics came from his encounter with
Emmy Noether
in Göttingen. He went there for Hilbert’s logic and because
Øystein Ore
at Yale had told him of Noether, who was yet little known in the U.S.
He had trouble absorbing her ideas on *factor sets* and
*crossed products*,
which replaced massive amounts of calculation in number theory with
sharp conceptual arguments, although he notes that “they still involve heavy
computation.”6
Mac Lane always said that a mathematician needs to keep practiced
at long calculation. Over time, Mac Lane clarified factor sets and
crossed products by a yet tighter conceptual framework: *group cohomology*.
Noether’s unification of arithmetic and abstract algebra became a
unification of group theory and topology in a thoroughly categorical way.
He later wrote historical expository articles with philosophic investigation
of creation and conceptualization, algebra and geometry.7
These ideas
are omnipresent in advanced number theory today, in the form of their
descendants *Galois cohomology, étale cohomology* and others; see, e.g.,
[e15].

Mac Lane’s most sustained pedagogical effort began with Noether as
well. That was the 25 year evolution of his algebra textbook with
Birkhoff.
First
[9]
was an American take on the Noether
school of algebra. Eventually
[16]
became many
students’ first look at categorical mathematics.8
Göttingen is also the reason, though, that philosophers today have
trouble reading Mac Lane, especially his *Mathematics: Form and Function*.
He pursued philosophy at length with Hermann Weyl and worked on
revising
[e3].
As part of his D. Phil. degree, he studied philosophy of
mathematics with phenomenologist and Hilbert disciple Moritz Geiger.9
For Göttingen mathematicians at that time, philosophy was phenomenology.
This phenomenology came from Edmund Husserl, who had trained
in mathematics with
Karl Weierstrass
and
Leopold Kronecker
and whose
doctoral dissertation was on calculus of variations. Hilbert got him hired
in philosophy at Göttingen.10
At Harvard, from the mid-1930s to 1947,
Mac Lane spoke often with Willard Quine but rejected Quine’s “undue
concern with logic, as such.”
([23], p. 443)
Mac Lane would absorb a
good dose of Karl Popper. But his base note would remain 1930s Göttingen
phenomenology — the philosophy of mathematicians.

Phenomenologists accept many different attitudes towards being.
Geiger especially describes the “naturalistic attitude,” which reduces
everything to physical existence, the “immediate
attitude,” which recognizes
social existence (as of a poem or a law), psychic existence (as of a
feeling or perception) and more. None of these is *right* or *wrong* for Geiger;
rather each has its role in life. At times they compete for a role. He says
some people take a naturalistic attitude towards logic and thus reduce it to
a branch of psychology, which he finds mistaken. For Geiger, mathematics
can only be understood in the immediate attitude. Its objects exist as *forms*
(*Gestalten, Gebilde*), which may be forms of physical objects but are not
themselves physically real
([e4],
esp. pp. 82, 86–88, 115).

Thus for Mac Lane, mathematics studies *forms* or
*aspects*,
which can be applied to study physically real things. All of the Euclidean
and
non-Euclidean geometries can be applied to physical space by suitable
definitions of “distance.” Which one best suits physics is not a mathematical
question. Mathematical statements may be proved or not, illuminating
or not, but they cannot be *true* or *false*, because they are not empirically
falsifiable. So mathematics makes no ontological commitments. Mac Lane
never became a Quinean, far less a post-Hilary Putnam Quinean.11

Back in the 1930s, returned from Germany, Mac Lane focussed on algebra because it was hard to get a job in logic ([38], p. 62). He joined the Association for Symbolic Logic and wrote a survey of topics in logic for discussion in math clubs [5]. His logic and philosophy are scattered, often in reviews. He shows far more knowledge of proof theory in [3] than in his 1934 dissertation. He reviews many ideas on the foundations of mathematics in his [8] and [11].12 He endorses Church:

This is an up-to-date description of the central problem in the foundations of mathematics. This problem is taken to be “the construction of a symbolic system within which the body of extant mathematics may be derived in accordance with sharply stated and immediately applicable formal rules.” Other problems, connected with the ultimate nature of mathematics or with the possible characterization of mathematics as a branch of formal logic, must wait on this central problem for solution. [8]

Mac Lane, too, was unsatisfied with existing set theories. He would much later repeat “there is yet no simple and adequate way of conceptually organizing all of Mathematics” though he considered categorical foundations the best prospect and he no longer saw this issue as prior to all others ([23], p. 407).

He treated themes that would reappear after he met Lawvere. Marshall Stone “shows surprising geometric content in many apparently purely algebraic or logical ideas” in ways topos theory would later extend [6], [28]. Work by Tarski does the same [4]. J-L. Destouches proposes a “new logic” for physics but:

This calculus is symbolized only incompletely, and tends to confuse system with meta-system. It is not free from technical errors …. Any such “new logic” for Physics must remain a gratuitous speculation until someone produces such a logic and uses it in Physics in some way not possible for the classical logic. [7]

Later he would find that [e11] and related works succeeded at this.

Mac Lane had great fun reviewing Eric Temple Bell’s
*Development of Mathematics*.
His review is worth reading today for discussion of many
mathematical advances — and for Mac Lane’s defence of the philosophers:
“Is Plato as vicious as Bell’s everywhere dense cracks would indicate? …
The philosopher will disagree with the jabs at Kant.” It discusses ambiguities
in the idea of “structure.” And it conveys Mac Lane’s approach to
every account of mathematics and its philosophy: “don’t wonder about it,
but go, read, and disagree for yourself”
[13].

From 1944 to 1948, Mac Lane was on the Council of the Association
for Symbolic Logic. Around that time, as editor of the Carus Mathematical
Monographs, he urged Stephen Kleene and Barkley Rosser to write on
logic. “In Kleene’s case this led to the preparation of his magnificant volume
*Introduction to Metamathematics*.”
([31], p. 4)13
He became the
Ph.D. advisor to logicians William Howard, Michael Morley, Anil Nerode,
Robert Solovay and Steven Awodey.

His mathematics stayed in touch with foundations. The founding paper of category theory made a startling claim.14 While giving trivial examples from linear algebra, and deep applications to extremely advanced topology, Eilenberg and Mac Lane say:

In a metamathematical sense our theory provides general concepts applicable to all branches of mathematics, and so contributes to the current trend towards uniform treatment of different mathematical disciplines. ([12], p. 236)

The paper also gives several strategies for the set-theoretic problems posed by large categories ([12], p. 246).

The great mathematical debate over category theory took place in private
over the next few years, indeed in closed meetings of the Bourbaki group,
with Eilenberg a member and Mac Lane sometimes visiting. He was photographed
at one
([e17],
p. 85).
Category theory lost in debate
and won in practice. Bourbaki formulated a set-theoretic notion of
“structure”
as basis for their encyclopedic *Elements of Mathematics*
[e6],
yet even their own later volumes never use this theory. Members
who relied on (and invented much) category theory in their work were
bitterly disappointed.15
Bourbaki structures are something like Tarski models.16
Mac Lane
made at least one effort to deal with “structures” in this sense since it
is familiar to philosophers
[32].
But compare
[23], p. 33,
where
these are called *sets-with-structure* and treated as important examples but
not the only kind of “structure.” He could never really engage with versions
of “structuralism” which neglect the tools mathematicians use. To
say the least, mathematicians handle structures by the *morphisms* between
them which are neglected by philosophical “
structuralists.” To say the truth,
mathematicians handle morphisms by category theory.

His encounter with Lawvere brought Mac Lane back to a focus on logic,
foundations and philosophy of mathematics.17
When he first heard of
Lawvere’s ideas on set theory, which became the
*Elementary theory of the category of sets*
or ETCS for short, he found them entirely implausible. He
found the axioms entirely persuasive when he learned them. Unfortunately,
in one of those misplaced jokes that so complicate the history of thought,
Mac Lane was fond of calling ETCS “set theory without elements”
([38], at length on p. 192).
Set theory is no joking matter. A good number of
logicians have taken him literally, notwithstanding that elements are defined
early in any account of ETCS
([e8],
p. 1507).
Mac Lane has
consistently backed ETCS as the best working foundation for mathematics
([37], Appendix).

To philosophers, though, he says there is no single “foundation” for mathematics and the purported foundations are better seen as “proposals for the organization of Mathematics” ([23], p. 406). They cannot be “starting points” or “justifications” for mathematics which is far older and more securely known than they are.

Mac Lane had used category theory to simplify many aspects of mathematics. Lawvere brought a radical idea that even the simplest notions — such as the natural numbers or sets of functions — have simple categorical versions which could bring foundations closer to normal practice. Foundations could use the very tools that mathematicians use daily. Mac Lane jumped onto this program. He regarded it as marking a deep shift in the direction of category theory and part of a larger shift away from generality and towards focussed themes in abstract algebra. See especially his [21], [24].

This brings us up to Mac Lane’s first autobiography, written “as a background
for the philosophy of Mathematics”
([23], back cover).18
Like
his career it begins with the foundations of algebra plus the basic topology
Mac Lane learned from Hausdorff’s *Mengenlehre*: “the first serious
mathematical text that I read”
([27], p. 6).
It
moves to axiomatic geometry, group theory and Galois theory, all close to
themes of his 1930s research; then calculus and linear algebra which he
taught in the 1940s; and then complex analysis and mechanics which he
began to teach in the 1950s and 1960s. (I find no trace of his war work in
applied mathematics.) It ends with foundations and philosophy. Of course
that puts philosophy out of order. He studied it early and late. The book
is not meant to be chronological. It is just such a powerfully, personally
informed, view of mathematics that it naturally follows his course. The
later autobiography
[38]
has great personal detail and great
mathematics. Indeed it has more advanced mathematics than
*Mathematics: Form and Function*.19
But *Mathematics: Form and Function* is what Saunders
had to say to the philosophers.

Let him have the last word, from a talk he gave at mid-career, fifty-one years ago:

It cannot be too often reiterated that the aim of collegiate mathematics is the understanding of mathematical ideas per se. The applications support the understanding, and not vice versa…

We must contrive ever anew to expose our students — be they general students or specialized students — to the beauty and excitement and relevance of mathematical ideas. We must set forth the extraordinary way in which mathematics, springing from the soil of basic human experience with numbers and data and space and motion, builds up a far-flung architectural structure composed of theorems which reveal insights into the reasons behind appearances and of concepts which relate totally disparate concrete ideas…

Mathematics as it is today … can no longer be presented by piecemeal courses, for it is simply no longer true that advanced mathematics can be split neatly into compartments labelled “algebra,” “analysis,” “geometry,” and “applied mathematics”…

You may protest that I am talking about research and not about education… I answer, first: The character and direction of current research is the best indication of the ideas which we ought to be teaching. I answer, second: One of our main responsibilities is that of training the research mathematicians of the future. American Mathematics has made tremendous strides [by] the infusion of European mathematical talent. In the decades to come, we must produce a similar infusion on our own and from our own students…

The proper treatment of calculus for functions of several variables requires vector ideas; the budding statistician and the coming physicist need them; modern analysis is unthinkable without the notion of linear dependence and all that flows from it. Throughout these courses the infusion of a geometrical point of view is of paramount importance. A vector is geometrical; it is an element of a vector space, defined by suitable axioms — whether the scalars be real numbers or elements of a general field. A vector is not an

\( n \)-tuple of numbers until a coordinate system has been chosen. Any teacher and any text book which starts with the idea that vectors are\( n \)-tuples is committing a crime for which the proper punishment is ridicule. The\( n \)-tuple idea is not “easier,” it is harder; it is not clearer, it is more misleading. By the same token, linear transformations are basic and matrices are their representations… ([15], pp. 152–154)

#### Bibliographic note

A bibliography of Mac Lane’s works through 1978 is in [19], pp. 545–553. The present bibliography lists works of philosophic interest. Mac Lane’s reviews are listed only when cited above. Mac Lane also made three 60 minute videotapes for the American Mathematical Society, Providence, Rhode Island: “Algebra as a means of understanding mathematics” (a joint AMS-MAA lecture, Columbus, Ohio, August 1990), “Fifty years of Mathematical Reviews” (a joint AMS-MAA lecture, Louisville, Kentucky, January 1990), “Some major research departments of mathematics” (a joint AMS-MAA invited address, Providence, Rhode Island, August 1988).