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Brit. J. Phil. Sci. page 1 of 36

The Last Mathematician from

Hilbert’s G¨

ottingen: Saunders

Mac Lane as Philosopher of

Mathematics

Colin McLarty

ABSTRACT

While Saunders Mac Lane studied for his D.Phil in G ¨

ottingen, he heard David Hilbert’s

weekly lectures on philosophy, talked philosophy with Hermann Weyl, and studied it

with Moritz Geiger. Their philosophies and Emmy Noether’s algebra all inﬂuenced

his conception of category theory, which has become the working structure theory of

mathematics. His practice has constantly afﬁrmed that a proper large-scale organization

for mathematics is the most efﬁcient path to valuable speciﬁc results— while he sees

that the question of which results are valuable has an ineliminable philosophic aspect.

His philosophy relies on the ideas of truth and existence he studied in G¨

ottingen. His

career is a case study relating naturalism in philosophy of mathematics to philosophy

as it naturally arises in mathematics.

1Introduction

2Structures and Morphisms

3Varieties of Structuralism

4G¨

ottingen

5Logic: Mac Lane’s Dissertation

6Emmy Noether

7Natural Transformations

8Grothendieck: Toposes and Universes

9Lawvere and Foundations

10 Truth and Existence

11 Naturalism

12 Austere Forms of Beauty

1 Introduction

Science concedes to idealism that its objective reality is not given but posed

as a problem. (Weyl [1927], p. 83)

The Author (2007). Published by Oxford University Press on behalf of British Society for the Philosophyof Science. All rights reserv ed.

doi:10.1093/bjps/axl030 For Permissions, please email: journals.permissions@oxfordjournals.org

2Colin McLarty

Speciﬁc mathematically conceived forms, forms of motion, relations, etc.

are fundamental to physical reality, and are real themselves [for the natural

sciences]. For mathematics, on the other hand, these same mathematical

forms etc. are not real but are special cases of an ideal objective world.

(Geiger [1930], p. 87)

When Bourbaki wanted to base their encyclopedic Elements of Mathematics

on a suitable idea of structures, they listened to a G ¨

ottingen trained algebraist

and logician who had also studied philosophy there:

As you know, my honourable colleague Mac Lane maintains every notion

of structure necessarily brings with it a notion of homomorphism, which

consists of indicating, for each of the data that make up the structure,

which ones behave covariantly and which contravariantly [...]whatdo

you think we can gain from this kind of consideration? (Andr´

eWeilto

Claude Chevalley, Oct. 15, 1951, quoted in Corry [1996], p. 380)

Saunders Mac Lane’s idea was not an axiom nor a deﬁnition nor a theorem.

It was not yet widely accepted and indeed Weil misunderstood it, as we will see.

Mathematically it was a huge extrapolation from Mac Lane’s collaboration

with Samuel Eilenberg on topology and algebra reﬂecting Emmy Noether’s

inﬂuence. Philosophically it reﬂected Mac Lane’s interest in foundations and

his studies with Hermann Weyl and Moritz Geiger. On the largest scale it

expressed Mac Lane’s view of the nature and value of mathematics.

Weil would not have cared for Mac Lane’s philosophy although it,

like his own, grew from the German scientiﬁc tradition. Weil saw this

tradition in Hilbert and in Bertrand Russell. He took from it a formalist

or instrumentalist view of mathematics, which he regarded as avoiding any

philosophic stand.1Alexander Grothendieck, to the contrary, found Weil’s

approach ‘an extraordinary Verﬂachung, a ‘‘ﬂattening,’’ a ‘‘narrowing’’ of

mathematical thought’ ([1987], p. 970).2

Mac Lane wrote his dissertation on logic, formalization, and practice. He

and Eilenberg addressed speciﬁc questions of mathematical existence, which

are still debated today (Eilenberg and Mac Lane [1945], p. 246). Mathematics

and philosophy were inextricable throughout his career and they crystallized

in his advocacy of categorical mathematics. He makes a case study for

naturalism in the philosophy of mathematics versus philosophy naturally

arising in mathematics. Weyl inﬂuenced him mathematically and by explicit

philosophy. Noether decisively affected his philosophy by her mathematics.

1See (Cartier [1998b], p. 11) and (Patras [2001], pp. 127– 67).

2Grothendieck took ‘Verﬂachung’ from number theorist Carl Ludwig Siegel’s attackon Bourbaki

which even compared Bourbaki to Hitler’s Brownshirts Lang ([1995]). Yet Siegel probably

considered Grothendieck a typical Bourbakiste.

Mac Lane as Philosopher 3

He notably inﬂuenced Grothendieck and William Lawvere. Lawvere and he

exchanged philosophic as well as mathematical ideas. Mac Lane’s inﬂuence on

Grothendieck was all mathematical but produced a philosophical convergence.

Each phase of his career faced him with ‘philosophical questions as to

Mathematical truth and beauty’ (Mac Lane [1986], p. 409). He urges in

philosophy the values that guided his research. The category theory he

needed for topology and algebra, which is now textbook material, makes up

his foundation.

2 Structures and Morphisms

Weil misunderstood Mac Lane and under-estimated the resources of set

theoretic mathematics. Weil supposed that if structures are sets then morphisms

must be functions. Plenty of examples ﬁt that model. A group Gis a set with

multiplication and a group morphism f:G→His a function that preserves

multiplication. In other words multiplication is covariant for group morphisms.

A topological space Sis a set with speciﬁed open subsets and the morphisms

are continuous functions or maps, meaning functions f:S→Tthat reﬂect

open subsets. The inverse image f−1(U ) of any open subset of Tis open in

S.Opensetsarecontravariant for continuous functions. However, Mac Lane

knew that morphisms had to be more general in practice.

First, there were transformations more elaborate than functions such as the

measurable functions, prominent in quantum mechanics. A textbook would

initially deﬁne a measurable function as a function that reﬂects measurable

subsets the way continuous functions reﬂect open subsets. But the theory

rested on a fact that philosophers today still mention: ‘the space L2of square-

integrable functions from Rto C[with a certain integral as inner product] is

a ‘concrete example’ of a Hilbert space’ (Hellman [2005], p. 536).3This claim

requires ‘two functions fand gof this class being considered as identical

if and only if f(x)=g(x) almost everywhere’ (Stone [1932], p. 23).4This

implies that fand gneed not agree everywhere, but everywhere outside some

subset of measure 0. Hilbert space techniques were central. This notion of

identity of functions is entirely natural in the context. So mathematicians

naturally thought of measurable functions this way—what the set theorists

would call equivalence classes of functions. Stone’s book complies with this,

and Mac Lane bought it in 1936.

3A measurable function fis square integrable if the squared absolute value has a well deﬁned

integral R|f(x)|2dx over the real line.

4Introductory texts today deﬁne measurable functions as functions but ‘for the sake of simplicity’

use the term to mean equivalence classes (Rudin [1966], p. 69). It is ‘usual’ for advanced texts

‘to identify two [measurable functions] if they coincide almost everywhere’ (Farkas and Kra

[1992], p. 29).

4Colin McLarty

Weil’s own algebraic geometry gave a far more elaborate example. He

would introduce a morphism f:X→Yfrom a space Xto a space Yas a list

of compatible ring morphisms f∗

i,j :RY,j →RX,i in reverse, from coordinate

rings RY,j on patches of Yto the rings RX,i on corresponding patches of X.

Actually, one algebraic space morphism was one equivalence class of such lists,

under a suitable equivalence relation. Special cases are scattered through (Weil

[1946]). A recent textbook notes that morphisms in algebraic geometry are not

functions and says ‘Students who disapprove are recommended to give up at

once and take a reading course in category theory instead’ (Reid [1990], p. 4).

Further, Eilenberg and Mac Lane used ‘morphisms’ not even based on

functions. For example, the real numbers form a category with inequalities

as morphisms. So √3≤πis a morphism from √3toπthough hardly a

function (Eilenberg and Mac Lane [1945], pp. 272ff).

All these non-function morphisms are easy to handle in set theory.5But

they are not functions. Weil’s strategy would need an impossible number of

extensions. It would have to handle partial functions, equivalence classes of

partial functions, lists of functions in the reverse direction...This was and

is entirely infeasible. There is no assignable limit to the devices that serve as

morphisms in practice.

Taking the opposite strategy, Eilenberg and Mac Lane called anything a

morphism, whether it was a function or built from functions or unrelated to

functions, if it satisﬁed the category theory axioms:

•Each morphism fhas an object Acalled domain andanobjectB

called codomain, written f:A→B.

•Morphisms f:A→Band g:B→Cwith matched codomain and

domain have a composite gf :A→C. Composition is associative so

that h(gf ) =(hg)f for any h:C→D. In a diagram:

•Each object Ahas an identity morphism 1A:A→Adeﬁned by its

composites f1A=1Bf=ffor every f:A→B.

5The set theory may be Zermelo Fraenkel, ZF, or the Elementary Theory of the Category of

Sets, ETCS. They work identically for our purposes (McLarty [2004]).

Mac Lane as Philosopher 5

The category Set has sets as objects and functions f:A→Bas morphisms.

Weil’s algebraic varieties are the objects of a category with quite complicated

morphisms. The real numbers Rform a category with real numbers as objects

and inequalities x≤yas morphisms. The identity morphism for any x∈Ris

just x≤x, and morphisms x≤yand y≤zcompose to x≤z. More examples

and explanations are in Mac Lane ([1986], pp. 386 –9).

3 Varieties of Structuralism

Today one may speak of three varieties of mathematical structuralism:

Bourbaki’s theory of structures, category theory, and the family of recent

philosophical structuralisms based on ‘the central framework of model

theory’.6The ﬁrst two were created as working mathematics although the

ﬁrst was never actually used even by Bourbaki (Corry [1996], Chap. 7). The

third has philosophical motives discussed below. Of course, these approaches

need not be judged only by their adequacy to describe mathematical practice,

let alone their inﬂuence on practice. But Mac Lane judges every view of

mathematics that way.

Bourbaki’s preliminary account describes a structure as a structured set,

that is a set plus some higher-order data ([1939]). It mentions no morphisms

except isomorphisms, which are 1-1 onto functions preserving and reﬂecting

all structure. It was not yet a working theory but merely a fascicule de r´

esultats,

a booklet of theorems without proofs. The project was interrupted by World

War II.

After the war, Bourbaki hotly debated how to make a working theory. All

agreed it must include morphisms. Members Cartier, Chevalley, Eilenberg,

and Grothendieck championed categories, as did their visitor Mac Lane. But

Weil was a majority of one in the group, so they created a theory with structure

preserving functions as morphisms (Bourbaki [1958]). They never used it, and

not for lack of trying. They could not make it work on the actual mathematics

they wanted to cover. The planned unity of the Elements gave way to various

6Quoting (Shapiro [1997], p. 93). Other examples are (Hellman [1989], Resnik [1997]). The

structuralist style of mathematics goes back to Dedekind and really to Riemann (Corry [1996],

Ferreir´

os [1999], Laugwitz [1999]).

6Colin McLarty

methods for various subjects. Most members themselves used categories and

indeed invented much of the theory as it is today. Some have expressed bitter

disappointment over Bourbaki choosing an obviously inadequate tool.7

In sum: Bourbaki’s structure theory follows category theory in using

morphisms to handle structures. It was developed by largely the same

people who developed the category theory. It failed. Bourbaki stipulates what

morphisms are: they are suitable functions. The category axioms merely say

how morphisms relate to each other: they compose associatively, with identity

elements. Even if we suppose everything is a set, categorical morphisms need

not be functions.

Mac Lane praises Bourbaki’s ‘magniﬁcent multi-volume monster’ for its

sweeping coverage ([1986], p. 5). On their theory of structures he says:

Categorical ideas might well have ﬁtted in with the general program

of Nicolas Bourbaki [...]. However, his ﬁrst volume on the notion

of mathematical structure was prepared in 1939 before the advent of

categories. It chanced to use instead an elaborate notion of an ´

echelle de

structure which has proved too complex to be useful. Apparently as a

result, Bourbaki never took to category theory. At one time, in 1954, I

was invited to attend one of the private meetings of Bourbaki, perhaps in

the expectation that I might advocate such matters. However, my facility

in the French language was not sufﬁcient to categorize Bourbaki. (Mac

Lane [1996a], p. 132)

More sharply, he considered Bourbaki’s deﬁnition ‘a cumbersome piece of

pedantry’ (Mac Lane [1996b], p. 181).

In technical respects the philosophical structuralisms are close to Bourbaki’s

preliminary account. Their structures are structured sets, or sui generis objects

very much like sets in Shapiro ([1997]). They consider no morphisms except

isomorphisms, and these are suitable functions. They differ from Bourbaki in

their philosophic motives, which go back to Benacerraf and Putnam.

Benacerraf noted that we do not normally assign set theoretic properties to

numbers—we normally assign them only arithmetic relations to each other.

He called for a theory of abstract structures, which differ from ZF sets in that

‘the ‘‘elements’’ of the structure have no properties other than those relating

them to other ‘‘elements’’ of the same structure’ (Benacerraf [1965], p. 70).

These elements may really have no individuating properties.8Putnam sought

to avoid Platonism by making mathematics deal with possibilities. Rather than

7See (Grothendieck [1985–87], p. P62), (Cartier [1998a], pp. 22– 7), and Chevalley in (Mashaal

[2000], p. 54). The debate was reported in detail in Bourbaki’s internal newsletter. See Corry

([1996], pp. 376–87) and many of the jokes in Beaulieu ([1998]).

8In arithmetic each number is individuated by arithmetic relations: it is the unique ﬁrst natural

number in its structure, or the unique ninety ﬁfth...In structures with more symmetry an

element may not be individuated at all.

Mac Lane as Philosopher 7

say Fermat’s Last Theorem is true (of the, existing natural numbers), we will

say it necessarily holds for every possible system of objects related to each

other the way natural numbers are supposed to relate (Putnam [1967]). This

is compatible with supposing that the elements of each particular structure

have individuating properties (e.g., as on ZF set theoretic foundations), but

the identities in any particular example are irrelevant as we never refer to any

particular example.

Philosophically, then, we face three dichotomies: Should structure theory

posit actual structures or only possible ones? Should it posit elements without

individuating properties, or is it only that individuation is irrelevant? Should it

follow Bourbaki’s theory of structured sets with structure preserving functions,

or category theory with its more general morphisms? Any combination is

logically possible. Different combinations achieve different things. Here we

can only survey the issues as they relate to Mac Lane.

The question of possible versus actual objects has never mattered to

Mac Lane, whose own quite different ontology goes back to the 1930s in

G¨

ottingen, as we will see. Yet, Hellman puts an interesting question mark

in his table of virtues of various structuralist theories, on the matter of

whether category theoretic structuralism uses any notion of modality as

a primitive ([2005], p. 560). Section 10 will note that Mac Lane’s notion of

‘correct’ mathematics is close to what modalists express as necessary inferences

drawn from possible premises. Without speciﬁcally taking correctness as

primitive, Mac Lane leaves it open to further explication. Perhaps it could be

construed modally, though for now the question mark must stand. Certainly,

as Hellman says, category theoretic structuralism admits modal variants that

no one has yet given.

Mac Lane has never focused on individuation of elements beyond what

is implicit in his advocacy of Lawvere’s Elementary Theory of the Category

of Sets (ETCS) as a foundation, as described in more detail in Section 9.

Here ETCS is asserted as describing the category of sets and not as just an

axiomatic theory. In ETCS the elements of any one set are distinct but have

no distinguishing properties. Each function between sets f:A→Bestablishes

a relation between the elements x∈Aand the elements fx ∈B,andthese

relations are the only properties that the elements have. So the ETCS axioms

meet Benacerraf’s requirement for a theory of abstract structures, unless

Benacerraf is taken to rule out ETCS by requiring a ZF foundation. See the

commentary to the reprint of Lawvere ([1965]).

An alternative categorical structuralism expresses mathematics entirely in

categorical terms, but takes this category theory as an axiomatic theory with

no intended referent (Awodey [1996], Awodey [2004]). If the huge resulting

axiomatic theory is interpreted in ZF as foundation, then the objects and

8Colin McLarty

morphisms do in fact have set theoretic individuating properties. But those

properties are irrelevant as they are never invoked in the axiomatic theory.

This brings us to our third dichotomy, between Bourbaki’s structure theory

and category theory as structure theory. Mac Lane wrote one article on

‘structures’ in the Bourbaki or model-theoretic sense.9His small interest in the

idea is clear. He mentions a 1933 abstract in which he stated, without proof, a

theorem on ‘structures’, which may have used the term in something like this

sense:10

To give a proof of such a theorem, I must have had some speciﬁc deﬁnition

of ‘structure’. I no longer recall that deﬁnition. (Mac Lane [1996b], p. 179)

More positively, he emphasizes morphisms. On the last page he remarks that

‘there can be quite different views of structure —as something arising in set

theory and then formulated in Bourbaki’s typical structures, or as something

located in some ethereal category’ (Mac Lane [1996b], p. 183). Today the

ethereal occurs all across mathematics. Few people have ever learned more of

Bourbaki’s approach than the name.

Category theory became a standard tool through decades of decisions

by thousands of mathematicians. But Mac Lane had tremendous personal

inﬂuence as he pushed it very hard in his research, exposition, and

popularization. The ways he did this and his reasons for it go back to

his student days in G¨

ottingen.

4G

¨

ottingen

Mac Lane worked for his doctorate in G¨

ottingen from 1931 to 1933. In 1931

he went to Hilbert’s weekly lectures on ‘Introduction to Philosophy on the

Basis of Modern Science’. There, Hilbert urged that mathematics can meet no

limits: Wir m¨

ussen wissen; wir werden wissen—We must know, we will know.11

The philosophy Mac Lane most studied was not Hilbert’s directly, though. It

was from two of Hilbert’s prot´

eg´

es and phenomenologists, Geiger and Weyl.

These two drew on their friend Edmund Husserl, who was a regular in

Hilbert’s circle.12 They both practiced a philosophy of ﬁne observation and

sweeping intellectual ambition expertly informed on the latest mathematics

and physics. Of course Weyl was personally prominent in both ﬁelds. Neither

Geiger nor Weyl gave long or detailed arguments for theses. Faced with

9Compare Mac Lane ([1986], p. 33) where he calls these sets-with-structure and gives them as

important examples but not the only kind of structure.

10 For other things he might have meant, see Mac Lane ([1939a], p. 18).

11 (Mac Lane [1995a], Mac Lane [1995b]).

12 See Reid ([1986], index) and Tieszen ([2000]); van Atten, van Dalen and Tieszen ([2002]).

Mac Lane as Philosopher 9

competing ideas they chose the best in each and gave short shrift to what they

rejected. They would brieﬂy contrast their ideas to others but spent no time

arguing with contemporaries. Theirs was little like Carnap’s philosophy and

less like Quine’s (both later colleagues of Mac Lane).13 Among the issues of

today’s philosophy of mathematics, G ¨

ottingen’s mathematician philosophers

were little concerned with analytic epistemology and not at all with modal

logic. Under Husserl’s inﬂuence they adopted nuanced ontologies of the kind

Quine would entirely reject.

It would be natural to think they shared today’s interest in minimizing

ontological commitments. Hilbert’s formalism treated mathematics as dealing

with formulas, ﬁnite strings of symbols, and not with inﬁnite sets or other

ideal objects. This is a sharply minimal ontology. But Geiger and Weyl had

no interest in this ontology, either according to their publications around

Mac Lane’s time in Germany or according to Mac Lane’s recollections.

Hilbert’s great work of that time was Geometry and the Imagination.His

preface to the book denounces

the superstition that mathematics is but a continuation, a further

development, of the ﬁne art of juggling with numbers. Our book aims

to combat that superstition, by offering, instead of formulas, ﬁgures that

may be looked at and that may be easily supplemented by models which

the reader can construct. (Hilbert and Cohn-Vossen [1932], p. iv).

Hilbert’s weekly philosophy lectures showed Mac Lane that Hilbert was

not just juggling with formulas either. Formalism was a strategy for certain

purposes. When Freeman Dyson complained of Hilbert ‘reducing mathematics

to a set of marks written on paper’, Mac Lane gave a sharp reply:

Hilbert himself called this ‘‘metamathematics’’. He used this for a speciﬁc

limited purpose, to show mathematics consistent. Without this reduction,

no G¨

odel’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. (Mac Lane [1995b])

It serves speciﬁc deep purposes. Mac Lane never took it for the actual

ontology of mathematics or thought that Hilbert did. The philosopher

mathematicians around him in G ¨

ottingen showed no interest in any ontological

minimalism.

13 When G¨

ottingers were less interested in logic than he hoped, Mac Lane thought of going to

Carnap in Vienna (Mac Lane [1979], p. 64). But he had little to do with Carnap when both

taught at Chicago.

10 Colin McLarty

Mac Lane’s doctoral study included Geiger’s lectures on philosophy of

mathematics and an examination by Geiger on Reality in the Sciences and

Metaphysics (Geiger [1930]).14 Geiger wrote much on mathematical sciences

though he is best known today for aesthetics. He wrote a Systematic Axiomatics

for Euclidean Geometry to improve Hilbert’s axioms by drawing out the real

connections of ideas.15 Compare Mac Lane later citing various proofs for

a theorem, then singling out one as ‘the reason’ for it.16 The question of

whether proofs do give ‘reasons’ or not, and whether different valid proofs

give different reasons, remains open today. Geiger and Mac Lane have tried

to apply the idea in detail. For Geiger, systematic axiomatics is philosophy but

he says ‘I have tried to exclude all that is philosophical in the narrow sense so

that philosophic foundations and philosophic evaluation are left for another

occasion’ ([1924], p. XVIII). That later occasion was the book Mac Lane

studied and the key to much of his later philosophy.

Geiger describes ‘the relation between reality (Wirklichkeit), as deﬁned in

science, and reality as metaphysics strives to know it’ ([1930], p. 1). It opens

with a Kantian perspective but without accepting Kant’s critical solution: The

sciences advance by secure methods while each metaphysician begins anew, yet

we inevitably seek metaphysical clarity and unity, while any attempt to take

the assumptions of science as metaphysical absolutes leads to contradiction.

Geiger distinguishes two attitudes, which he calls naturalistic and immediate.

The naturalistic attitude assumes a world existing-in-itself and grounded-in-

itself, and is so little interested in the conscious observer that it does not even

bother to say this world is independent of the observer. This attitude takes

physicalistic reduction for granted:

Psychic and physical are in contradictory opposition for the naturalistic

attitude. What is not physical is psychic, and what is not psychic is

physical —this is their methodological axiom [...]. The physical is the

real (Reale) in space and time, the ‘‘objective’’; the psychic in contrast is

the non-spatial and non-objective. Whatever is not objective is ‘‘merely’’

subjective, is psychic. (Geiger [1930], p. 18)

On the other hand ‘the unreﬂected stance of ordinary life is not the

naturalistic, but the immediate attitude’ which starts with an observing subject

in an object world ([1930], p. 20). The immediate attitude takes the psychic

as real along with many kinds of being beyond the physical. The psychic is

‘what the subject experiences as belonging immediately to the subject’ such

14 See Alexanderson and Mac Lane ([1989], p. 15), Mac Lane ([1995a], p. 1136), Mac Lane

([2005], p. 55).

15 Weyl cites it (Weyl [1927], p. 24).

16 For example (Mac Lane [1986], pp. 145, 189, 427, 455).

Mac Lane as Philosopher 11

as wishes, passions, and acts of will ([1930], p. 21). Other objects are neither

physical nor psychic. Examples are a poem, or a language, or the Congress

of ‘the United States when they declared the slaves free’ ([1930], p. 25). That

example would please the staunch New Englander Mac Lane.

The whole point of Geiger’s discussion of mathematics is to say

‘Consideration of the structure of mathematics shows that the adequate

attitude for it is the immediate’:

The naturalistic attitude knows only psychic and physical forms (Gebilde).

If Mathematics were a science in the naturalistic attitude, it would have

to be either a science of physical objects, thus a kind of applied physics,

or a science of psychic objects, thus a kind of applied psychology. Yet

Mathematics is neither the one nor the other.17 (Geiger [1930], p. 82)

He blames the naturalistic attitude for promoting psychologism in logic but

ﬁnds it has little inﬂuence in Mathematics ([1930], pp. 115, 88).

The philosophic problem for Geiger is to clarify ‘the structure of

mathematical forms (Gebilde).’ The structure analysis would explain how

the non-naturalist mathematical objects can apply in naturalistic sciences: ‘as

ideal objects, mathematical objects are in fact accessible only to the immediate

attitude, but as forms (Gestalten) of real objects they are indifferent to the

attitude’ ([1930], pp. 86–7).18 He never got to it though.

Around the same time Mac Lane lived in Weyl’s house, helped him practice

English, and regularly spoke of philosophy with him. They worked on revising

Weyl’s Philosophie der Mathematik und Naturwissenschaft (Weyl [1927]). As

Mac Lane later recalled it, their effort was not much like the eventual

revision (Weyl [1949]).

In his fast-paced booklet Weyl recounts

important philosophical results and viewpoints given primarily by work

in mathematics and natural science. I point out the connection with great

philosophers of the past wherever I have been sensitive to it (sie mir f ¨

uhlbar

geworden ist). ([1927], p. 3)

He was very sensitive. He cites Fichte, Schelling, and Hegel. He quotes

Heraclitus and Euclid in Greek. He goes from logic and axiomatics to non-

Euclidean and projective geometry. He describes how Helmholtz and Lie

made transformation groups basic to geometry. This ﬁrst, mathematical part

takes just 60 pages to reach Riemann on metrics and topology. The last 100

17 I capitalize Mathematics here because Mac Lane does in ([1986]). I will do this whenever I

mean to invoke his ideas.

18 Geiger uses ‘Gebilde’and‘Gestalten’ interchangeably and I argue that both appear as ‘forms’

in Mac Lane ([1986]).

12 Colin McLarty

pages treat space, time, matter, and causality from the ground up to arrive

at relativity and quantum theory. He draws on sources from Pythagoras and

Proclus through Galileo and Hobbes, Leibniz and Euler, much on Kant,

plus Maxwell and Helmholtz and Mach. Weyl writes as a colleague of his

contemporaries in physics. Substantial mathematics is assumed throughout.

The inﬂuence on Mac Lane was broad and deep though Mac Lane never

shared Weyl’s ﬂuency with philosophical and historical references. Mac Lane

was strongly marked by Weyl’s encyclopedic breadth and clear style in

mathematics and by his certainty that the best philosophic insights on science

would depend on detailed mastery of the best science. Weyl, like Geiger, spoke

of mathematical Gebilde with a different order of being than actual things:

To the Greeks we owe the recognition that the structure of space,

manifest in the relations between spatial forms (Gebilde) and their lawful

dependence on one another, is something completely rational. This is

unlike the case of an actual particular where we must ever build from new

input of intuition. (Weyl [1927], p. 3)

Ontological theory is far less developed in Weyl than in Geiger, while Weyl

does more to locate it in mathematical practice. He quotes Hermann Hankel’s

textbook on complex function theory saying modern pure mathematics is

a purely intellectual mathematics freed from all intuition, a pure theory

of forms (Formenlehre) dealing with neither quanta nor their images the

numbers, but intellectual objects which may correspond to actual objects

or their relations but need not.

He quotes Husserl that ‘without this viewpoint [...] one cannot speak of

understanding the mathematical method’.19

Over time, Mac Lane would agree and disagree with various of Weyl’s

claims. He heartily agrees ‘as Weyl once remarked, [set theory] contains far

too much sand’ (Mac Lane [1986], p. 407). It posits a huge universe with just

an inﬁnitesimal sliver of any conceivable interest. This means categorical set

theory as well as Zermelo–Fraenkel. Mac Lane prefers the categorical but

has to say: ‘We conclude that there is yet no simple and adequate way of

conceptually organizing all of Mathematics’ ([1986], p. 407).

By 1927 Weyl stressed the indispensability of formal mathematics and

Hilbert’s use of the inﬁnite. Mac Lane evidently agreed. Yet he was unmoved by

Weyl’s two main philosophic concerns beyond that: Brouwerian intuitionism,

and the relation to physics. Weyl had famously torn allegiance:

19 (Hankel [1867], p. 10) and (Husserl [1922], p. 250) quoted at (Weyl [1927], p. 23).

Mac Lane as Philosopher 13

Mathematics with Brouwer achieves the highest intuitive clarity. He is able

to develop the beginnings of analysis more naturally, and in closer contact

with intuition, than before. But one cannot deny that, in progressing to

higher and more general theories, the unavailability of the simple axioms

of classical logic ﬁnally leads to nearly insupportable difﬁculties. (Weyl

[1927], p. 44)

However, Mac Lane found Brouwer ‘often pontiﬁcal and obscure’ and

eventually found formally intuitionistic logic convenient precisely for higher

theories.20 As to physics, while Mac Lane always appreciates applications of

Mathematics, he would never agree that: ‘Mathematics must stand in the

service of natural science’ (Weyl [1927], p. 49).

5 Logic: Mac Lane’s Dissertation

Mac Lane proposed to read Principia Mathematica as an undergraduate at

Yale. His teacher talked him into the more practical Set Theory (Hausdorff

[1914]). This was primarily on point set topology, as we would say today, but

paid some attention to foundations. ‘This was the ﬁrst serious mathematical

text that I read and it made a big impression on me’ (Alexanderson and Mac

Lane [1989], p. 6). Mac Lane has ever since urged that logic should not merely

study inference in principle, but the inferences made daily by mathematicians.

He went on to active involvement in the Association for Symbolic Logic, and

teaching logicians, as described below. But he ﬁnds ‘Mathematical logic is a

lively, but unusually specialized ﬁeld of research’ (Mac Lane [2005], p. 198).

He ﬁnds that too much research in set theory has only tenuous links to any

other part of Mathematics.21 He insists that theoretical study of logic could

do much more to address practical issues:

There remains the real question of the actual structure of mathematical

proofs and their strategy. It is a topic long given up by mathematical

logicians, but one which still—properly handled —might give us some

real insight. (Mac Lane [1979], p. 66)

His dissertation says: ‘the task of logic is to draw proofs from given

premisses’ (Mac Lane [1934], p. 5), meaning that logic aims to study and

improve the means of inference as actually practiced. In particular, logic

should study more than the correctness of single inferences, and it need not

only address symbolic reasoning:

20 On Brouwer see Mac Lane ([1939b], p. 292). Forcing arguments appear as simple r intuitionistic

set theory, and classical theorems on real valued functions appear as simpler intuitionistic

theorems on real numbers (Mac Lane and Moerdijk [1992], pp. 277– 84 and 318–31).

21 See the debate (Mathias [1992], Mac Lane [1992], Mathias [2000], Mac Lane [2000]).

14 Colin McLarty

A proof is not just a series of individual steps, but a group of steps,

brought together according to a deﬁnite plan or purpose....Soweafﬁrm

that every mathematical proof has a leading idea (leitende Idee), which

determines all the individual steps, and can be given as a plan of the proof

(Beweisplan).

...Many fundamentally different styles can be used to give any one

proof—the precise, symbolic, detailed style, which is used in Principia

and many other parts of Mathematics, which requires rigorous exposition

of proof steps at the cost of the underlying ideas—and the intuitive,

conceptual style, which always displays the main ideas and methods of

a proof, so as to understand the individual manipulations in the light of

these ideas. This style is particularly practiced in the books and lectures of

H. Weyl. (Mac Lane [1934], pp. 60– 1)

The dissertation was part of a projected ‘structure theory for Mathematics

based on the principle of leading ideas’ to bring intuitive proof closer to formal

logic (Mac Lane [1934], p. 61). The dissertation would shorten formal proofs

by abbreviating routine sequences of steps. Mac Lane aimed to organize proof

and the discovery of proofs: ‘one can construct broader and deeper methods

of abbreviation based on the concept of a plan of a proof ...which efﬁciently

(zweckm¨

aßig) determines the individual steps of the proof ’ (p. 6).

Mac Lane has always felt that right logical foundations would mesh well

with practice. In 1948 he advanced Emmy Noether’s algebra by a categorical

study of homomorphism and isomorphism theorems.22 This led to Mac Lane’s

Abelian categories described below. But he paused on a foundational detail.

Integers x,y are said to be congruent modulo 3, written

x≡3y

if the difference x−yis divisible by 3. So 1 ≡37 and so on. Arithmetic with

various moduli, such as modulus 3, was important to number theory in the

1930s, and still is today. Mathematicians then recognized two ways to deﬁne

the factor group Z/3 of integers modulo 3.23 Many textbooks favoured the

way still common today: Deﬁne the coset modulo 3 of any integer x∈Zto be

the equivalence class of xfor this relation. Writing xfor the coset of xthat

says:

x={y∈Z|x≡3y}

Then Z/3 has exactly three elements, namely, the cosets 0,1,2, since every

integer x∈Zbelongs to exactly one of these. Another approach was to say the

elements of Z/3 are the usual integers, but with ≡3taken as the new equality

22 (Mac Lane [1948]) For Noether’s reliance on these theorems see Alexandroff ([1981], p. 108

and passim). For Mac Lane on her school, see Mac Lane ([1997]).

23 Today the name quotient group is more common.

Mac Lane as Philosopher 15

relation. Then Z/3 still has exactly three elements. The elements are integers

and not sets of integers, but there are exactly 3 different integers for this new

equality relation since every integer xsatisﬁes exactly one of

x≡30x≡31x≡32

Noether constantly used factor groups not only of Zbut also of any group

G. Mac Lane paused over a detail. Take any group Gand factor group of it

G/N , and then form a factor group of that: (G/N)/M .24 Intuitively, (G/N)/M

is a coarser factor group of Gand mathematicians would work with it that

way. But, for factor groups deﬁned using cosets, it is not strictly so. The

elements of (G/N)/M are cosets of cosets of elements of G, not cosets of

elements of G. The group (G/N)/M is only isomorphic to a factor group of

G. Mac Lane wrote:

This apparent difﬁculty can be surmounted by an attention to

fundamentals. A factor group G/N may be described either as a group

in which the elements are cosets of N, and the equality of elements is

the equality of sets, or as a group in which the elements are the elements

of Gand the ‘‘equality’’ is congruence modulo N. Both approaches are

rigorous and can be applied (with approximately equal inconvenience!)

throughout group theory. The difﬁculties cited disappear when we adopt

the second point of view, and regard a group Gas a system of elements

Gwith a reﬂexive symmetric and transitive ‘‘equality’’ relation such that

logically identical elements are equal (but not necessarily conversely) and

such that products of equal elements are equal.25 ([1948], pp. 265 –7)

On the ‘equality approach,’ a factor group of a factor group of Gis quite

strictly, and not only up to isomorphism, a factor group of G.

Mac Lane later dropped that problem as he pioneered more practical,

powerful, rigorous ways to work with isomorphisms. But he never lost faith

that the right foundations will give the right working methods. He chose

algebra as a career over logic only because it was easier to get a job (Mac

Lane [2005], p. 62). He joined the Association for Symbolic Logic and was

on the Council from 1944 to 1948. He encouraged Stephen Kleene to write

Introduction to Metamathematics and critiqued drafts (Kleene [1952], p. vi).

His doctoral students include logicians William Howard, Michael Morley,

Anil Nerode, Robert Solovay, and recently Steven Awodey.

In practice, though, Mac Lane found that the way to radically shorter

proofs—and to previously infeasible proofs —is not through abbreviation or

apt details. It is through new concepts. His dissertation had introduced the

concept of the ‘leading idea’ of a proof, which was itself meant to be a leading

24 E.g. take Zand Z/12. Then (Z/12)/3 is isomorphic to Z/3.

25 Mac Lane cites Haupt ([1929]) for the equality approach.

16 Colin McLarty

idea for further work in logic. He soon found leading ideas that still guide

work in algebra and topology today. They grew from where he did not expect

them.

6 Emmy Noether

Bernays and I both took a course of Noether’s. The course was based on

an article on the structure of algebras that she subsequently published. She

was a rather confused and hurried-up lecturer because she was working

it out as she went. I found the subject interesting, but I wasn’t anxious

to pursue it... I can recall walking up and down the corridors with

Bernays during the 20 minute break, pumping him about things in logic.

(Alexanderson and Mac Lane [1989], p. 14)

Yet the two projects of his most productive mathematical decade came from

Noether.

The ﬁrst was how to organize algebraic topology. By 1930, each (suitable)

topological space Xwas assigned a series of cohomology groups:

H0(X), H 1(X), H 2(X) ...

The group Hn(X) counts the n-dimensional holes and twists in X.AtorusT,

or ‘doughnut surface,’ has no twists but two 1-dimensional holes: one inside

the surface is encircled by the dotted line on the left, and one through the

centre is encircled by the dotted line on the right:

The 1-dimensional cohomology group H1(T ) of the torus assigns one integer

coefﬁcient, say a, to the ﬁrst hole and one, say b, to the second. It is the group

N2of pairs of integers a, bwith coordinatewise addition26

a, b+c, d =a+c, b +d

A map of topological spaces f:X→Yinduces group homomorphisms in the

other direction

Hn(f ) :Hn(Y ) →Hn(X)

26 There are also cohomologies with other coefﬁcients than integers.

Mac Lane as Philosopher 17

for each n∈N. A great deal of information about maps to the torus from any

space Xis captured in the simple form of homomorphisms

H1(T ) ∼

=N2−→ H1(X)

between the 1-dimensional cohomology groups.

Contrary to legend, Noether did not introduce these groups in topology.

They were long known but unused. Rather, she organized all of algebra

around morphisms, speciﬁcally the homomorphism and isomorphism theorems.

She also got topologists to use the groups by showing how interrelations

of group morphisms with topological maps can give radically more efﬁcient

proofs (McLarty [2006]).

New theorems and methods poured in faster than anyone could follow.

Topologist Awould use theorems proved by topologist Band vice

versa—when in fact the two topologists used completely different deﬁnitions.

Topologists felt that the many algebraic approaches were ‘naturally equivalent’

so they should all agree in effect. But no one could precisely deﬁne this idea,

let alone prove it. It was hard to know exactly what, if anything, anyone had

proved. Even Noether’s pure algebra was expanding explosively when she died

in 1935. How could it all be organized?

The other problem Mac Lane took from Noether was in those lectures he

attended with Bernays. Noether invented factor sets to replace huge number

theoretic calculations by conceptual arguments. Mac Lane writes:

I personally did not understand factor sets well at the time of Noether’s

lectures, but later Eilenberg and I used factor sets to invent the cohomology

of groups. (Green, LaDuke, Mac Lane and Merzbach [1998], p. 870)

Group cohomology is described below. Calculation remains the basis of

number theory, but each step radically reduced the calculations for any given

problem. In other words, ever larger problems became feasible.

A series of philosophical and historical works on creation and conceptu-

alization, algebra, and geometry grew from Mac Lane’s confrontation with

Noether.27 No doctrinal philosophy seems to have passed between them. Yet

they share a single-minded devotion to Mathematics (which we will return

to in connection with naturalism), and a sense of humour, and both are

peripatetics:

One day, at her lecture, Professor Noether observed with distaste that the

Mathematical Institute would be closed at her next lecture, in honour of

some holiday. To save mathematical research from this sorry interruption,

she proposed an excursion to the coffee house of Kerstlingeroden Feld, up

27 (Mac Lane [1976a], [1978], [1981], [1988a], [1988b], [1989]).

18 Colin McLarty

in the hills. So on that day we all met at the doors of the Institute —Noether,

Paul Bernays, Ernst Witt, etc. After a good hike we consumed coffee,

talked algebra, and hiked back, to our general proﬁt. (Mac Lane [1995a],

p. 1137)

7 Natural Transformations

At least since his dissertation, Mac Lane has been interested in the ‘leading

ideas’ that structure any proof or any branch of mathematics. The great

example in his career was the collaboration with Eilenberg. On the face

of it they made an arcane calculation of the cohomology of a certain

inﬁnitely tangled topological space (Mac Lane [1976b]). Yet Eilenberg and

Mac Lane emphasized the key to their calculations: natural equivalence,or

natural isomorphism (Mac Lane [1986], p. 195).

Two constructions might start with a group Gand give different results, but

always isomorphic results, where the isomorphism is deﬁned the same way for

all groups G. Then the isomorphism

is considered ‘‘natural,’’ because it furnishes for each Ga unique

isomorphism, not dependent on any choice [of how to describe G].

(Eilenberg and Mac Lane [1942], p. 538)

[It] is ‘‘natural’’ in the sense that it is given simultaneously for all [groups]

(Eilenberg and Mac Lane [1945], p.232)

They stress capturing the common notion of naturalness. They frequently

put ‘natural’ in quotes to emphasize that it gives ‘a clear mathematical meaning’

to a colloquial idea ([1942], p. 538).

They illustrate their sense of naturality not only in group theory and

topology but all over mathematics, and they make a sweeping claim far

beyond their actual proofs:

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. In

particular it provides opportunities for the comparison of constructions

and of the isomorphism occurring in different branches of mathematics;

in this way it may occasionally suggest new results by analogy. ([1945],

p. 236)

They note that the category of all groups or the category of all sets are

illegitimate objects in set theory. However, they say this matters little:

The difﬁculties and antinomies here involved are exactly those of ordinary

intuitive Mengenlehre [set theory]; no essentially new paradoxes are

Mac Lane as Philosopher 19

apparently involved. Any rigorous foundation capable of supporting the

ordinary theory of classes would equally well support our theory. Hence

we have chosen to adopt the intuitive standpoint, leaving the reader free

to insert whatever type of logical foundation (or absence thereof) he may

prefer. ([1945], p. 246)

They sketch foundations based on circumlocution, type theory, and G ¨

odel-

Bernays set theory. But foundations were not the leading idea.

From naturality the lead quickly shifted towards functoriality. Eilenberg

and Steenrod axiomatized cohomology as a series of functors from a suitable

category of topological spaces to that of Abelian groups. The axioms

became standard among topologists even before they were announced in

print (Eilenberg and Steenrod [1945]). As Mac Lane expected for leading ideas,

the axioms went a long way to routinize proofs in topology. Functoriality

organized the general theorems and worked quietly in the background to let

geometric ideas lead in speciﬁc results.

In fact, Eilenberg and Mac Lane had a sweeping analogy in mind between

group theory and topology. Each topological space Xalso has a fundamental

group π1Xmeasuring the ways a curve can get tangled in X.28 Topologists

using ideas from Emmy Noether had found that for many spaces Xall of

the cohomology groups Hn(X) can be calculated by pure algebra from the

one group π1X. This link between topology and group theory was seriously

puzzling. Eilenberg and Mac Lane set out to explain it and use it.

Within a few years the analogy was formalized as a new mathematical

subject. Each group Ggot its own cohomology groups:

H0(G), H 1(G), H 2(G)...

Each group homomorphism f:G→Ginduces homomorphisms in the other

direction

Hn(f ) :Hn(G)→Hn(G)

It is harder to say what these groups Hn(G) count compared to the topological

case. Mac Lane explains them by deriving them from topology ([1988b]).

They are extremely useful in group theory per se and in applications of it.

Henri Cartan’s Paris seminar spent 1950–51 exploring the parallel between

groups and topological spaces with Eilenberg. From that came a profusion of

cohomology theories in complex analysis, algebraic geometry, number theory,

and more.

Cartan’s seminar deﬁned a cohomology theory as a suitable sequence of

functors Hn:X→Awhere Xis a category based on a geometric or algebraic

28 See many topology textbooks or Mac Lane ([1986], pp. 322– 8).

20 Colin McLarty

object which ‘has’ cohomology, and Aa category of ‘values’ of cohomology. So

X=XTmight be based on a topological space T.29 If A=Ab is the category of

Abelian groups, then the functors Hn:XT→Ab give the classical cohomology

of T.OrX=XGcould be based on a group Gto give the cohomology of G.

Other categories would be used for the category of values A, say the category

of real vector spaces, to reveal somewhat different information.

At ﬁrst, the categories Xand Awere deﬁned by whatever nuts and bolts

would work. Then, Mac Lane gave purely categorical axioms on a category

Asufﬁcient to make it work as a category of values for cohomology. He

called such a category an Abelian category. He gave the ﬁrst purely categorical

deﬁnitions of many simple constructions, which he says ‘would have pleased

Emmy Noether’ (Mac Lane [2005], p. 210).30 In 1945 he and Eilenberg

apparently considered these constructions too simple to need categorical

treatment. By 1950, Mac Lane saw them as so simple they must have categorical

deﬁnitions.

8 Grothendieck: Toposes and Universes

Grothendieck simpliﬁed and strengthened Mac Lane’s Abelian category

axioms into the standard textbook foundation for cohomology. 31 Then he

went to the categories Xwhich have cohomology.

Cohomology used the category ShTof sheaves on any topological space

T,whereasheaf is a kind of set varying continuously over T. Grothendieck

saw how to do mathematics inside ShTalmost the way it is done in sets.32

Constructions familiar for sets lift into ShTbut with the brilliant difference

that each construction itself ‘varies continuously’ over T. Grothendieck saw

how the cohomology of Texpresses a simple relation between the varying

Abelian groups in ShTand ordinary constant groups.33 The same relation

gives the cohomology of any group Gin terms of a category ShGof sets acted

on by the group G. Grothendieck deﬁned a new kind of category called a topos,

with sheaf categories ShTand group action categories ShGas examples, such

that each topos has a natural cohomology theory. He uniﬁed the cohomology

of the known cases and obviously opened the way to cohomology theories as

yet unknown.

29 This is the category of sheaves of Abelian groups on T. For this and related terms see Mac

Lane and Moerdijk ([1992]).

30 Examples are products and quotients (Mac Lane [1950], pp. 489– 91).

31 (Lang [1995], Hartshorne [1977]). On Grothendiec k see McLarty ([forthcoming]) and resources

on the Grothendieck Circle website at <www.grothendieck-circle.org>.

32 Again, for details see Mac Lane and Moerdijk ([1992]).

33 These varying groups form the category called XTabove.

Mac Lane as Philosopher 21

He was fascinated with these new worlds, which on one hand support new

interpretations of mathematics and on the other hand have cohomology. But

each topos Eis a proper class, as large as the universe of all sets, and indeed

contains that universe. For example, take any topological space T. The objects

of ShTare sets ‘varying continuously’ over Tto any degree, and Set ShT

appears as the subcategory of sets with 0 variation or in other words the sets

constant over T.

Grothendieck’s approach quantiﬁes freely over toposes. This is natural

since they represent spaces, groups etc. He constructs the ‘set’ of all functors

E→Efrom one topos to another just as he would the set of all maps from

one space to another. But these topos moves are illegitimate in ordinary set

theory, whether ZF or categorical. They quantify over proper classes, form

the superclass of all functions from one proper class to another, and raise all

of this to ever higher levels. Grothendieck tested the limits of Eilenberg and

Mac Lane’s claim:

Any rigorous foundation capable of supporting the ordinary theory of

classes would equally well support our theory. (Eilenberg and Mac Lane

[1942], p. 246)

It is not entirely true since the simplest versions of many important theorems

use superclasses of classes and so on.

So Grothendieck posited his universes.34 A universe is a set of sets which

itself models the basic set theory axioms so that you can do essentially

ordinary mathematics inside any universe. The basic axioms do not imply that

any universes exist. Grothendieck posited that every set is a member of some

universe, implying that each universe is a member of inﬁnitely many larger

universes. He could deﬁne a U-topos within any universe Uso that it looks like

a proper class from the viewpoint of Ubut is merely a set from the viewpoint

of any larger universe U. He could rise through any number of levels by

invoking as many universes. This did not entirely preserve the naive simplicity

of his ideas, though, since it meant keeping track of universes.

Another popular solution in practice is circumlocution. Instead of toposes

this uses much smaller Grothendieck topologies. It works for technical purposes

in number theory and algebraic geometry and one is free to use topos language

as a convenient but technically illegitimate fac¸on de parler. But that fac¸on de

parler remains common and compelling. According to Grothendieck the real

insights occur at that level (Artin, Grothendieck and Verdier [1972], Preface

and passim). To put his viewpoint into terms familiar in the philosophy of

mathematics: reducing toposes to Grothendieck topologies is like reducing

34 See Artin, Grothendieck and Verdier ([1972], Appendix to Exp. 1).

22 Colin McLarty

full set theoretic real analysis to second order Peano arithmetic. It sufﬁces for

many purposes but it has strictly lower logical strength, and doing it rigorously

would require lengthy circumlocutions that obscure geometric intuition.

As a philosophical matter neither Mac Lane nor Grothendieck is interested

in fac¸ons de parler. The only thing either one wants from a foundation is that

it be correct and illuminating. Goals such as ontological or proof theoretic

parsimony have no appeal. A practically useful way of thinking ought to

ﬁnd natural, legitimate expression in a rigorous foundation. Like Mac Lane,

Grothendieck is unconcerned with whether universes ‘really exist.’ He knows

the general consensus that universes are consistent. So long as they give

the easiest formal foundation for cohomology he will use them. He developed

explicit interests in philosophy later and in a very different style (Grothendieck

[1985–87]). But on ontology, foundations, and the roles of conceptualization

and formalization, his practice led in the same direction as Mac Lane.

9 Lawvere and Foundations

Small theorems had a large impact when Mac Lane put simple features of

Abelian groups into categorical terms ([1948]). Categories not only captured

overarching ideas like ‘natural equivalence’ and reduced huge arguments to

a feasible scope, but also proved new theorems by directly addressing simple

ideas. Grothendieck’s extension of this into Abelian categories became bread

and butter for algebraists and topologists and one of the founding topics of

category theory as a subject in its own right.35

Mac Lane met Lawvere as a graduate student with a program to unify all

mathematics from the simplest to the most advanced in categorical terms. This

included purely categorical axioms for the set theory. Mac Lane found the set

theory absurdly implausible— until he saw the axioms—and then he sent it

to the Proceedings of the National Academy of Sciences as Lawvere ([1964]).

The axioms used Mac Lane’s categorical deﬁnitions of cartesian products and

equalizers. This last is a categorical deﬁnition of solution sets to equations.

{x∈A|fx =gx}>−→ Af

−→

−→

gB

The axioms also used Lawvere’s original categorical accounts of the natural

numbers, power sets, and more (Mac Lane [1986], § XI.12.).

On any account of sets, the elements x∈Aof a set Acorrespond exactly

to the functions x:1→Afrom a singleton 1 to A. Lawvere’s axioms deﬁne an

element as such a function. So elements are not sets themselves and in fact

35 The ﬁrst verbatim reference to ‘category theory’ in Mathematical Reviews was in 1962 #B419

reviewing a work on systems biology (Rosen [1961]).

Mac Lane as Philosopher 23

the elements of a set Ahave no properties except that they are elements of A.

Rather the functions to and from Aestablish relations between elements of A

and those of other sets. Given any element x:1→Aand function f:A→B,

the composite fx:1→Bis an element of B. The key axiom is extensionality

applied to functions: given parallel arrows f, g :A→

→B,ifeveryx∈Agives

equal values fx =gx then f=g.

Lawvere had found no new facts about sets. His axioms are familiar truths to

all mathematicians. He found how to say them rigorously without the aspects

of ZF unfamiliar to mathematicians: the transﬁnite cumulative hierarchy, and

specifying every number or geometric point or whatever as a set. The familiar

truths sufﬁce.

These axioms, and Lawvere’s vision of the scope of category theory, widely

extended Mac Lane’s own ideas and became the technical core of Mac Lane’s

philosophy, although he has never entirely agreed with Lawvere on them. One

striking difference is that Lawvere always stresses many different things that

‘foundations’ can mean in a formal-logical sense or an ontological sense or a

working sense and he offers several alternative formal-logical ‘foundations.’

For Mac Lane, a ‘foundation’ is always a formal-logical theory in which

to interpret Mathematics. Mac Lane insists foundations are only ‘proposals

for the organization of Mathematics’ and taking one as the actual basis

of Mathematics ‘would preclude the novelty which might result from the

discovery of new form’ (Mac Lane [1986], pp. 406, 455). So to urge one

is in no way to deny the others. Yet he does consistently urge one, namely,

Lawvere’s Elementary Theory of the Category of Sets (Lawvere [1964], Lawvere

[1965]).36

He offers two extensions of the axioms. When talking about foundations

for category theory he often adds an axiom positing one universe (Mac

Lane [1998], pp. 21–2). Other times he has said his ‘categorical foundation

takes functors and their composition as the basic notions’ as if he sees the

ETCS axioms being stated for one category in a category of categories (Mac

Lane [2000], p. 527). 37 That would be one reasonably conservative take on

Lawvere’s Category of Categories as Foundation (Lawvere [1966]). These are

two closely analogous ways to strengthen the ETCS axioms. The ﬁrst posits

a world of sets in which one set models ETCS. The second posits a world

of categories in which one category models ETCS. Again, Mac Lane offers

36 See Mac Lane ([1986], chap. XI), Mac Lane ([1998], Appendix), Mac Lane and Moerdijk

([1992], VI.10), Mac Lane ([1992]), and Mac Lane ([2000]).

37 In an unpublished note ‘The categorical foundations of mathematics,’ circulated in 1998,

Mac Lane says axioms for the category of categories are ‘Lawvere’s second version’ of axioms

for the category of sets. This shows how closely he relates them though it gets the order

backwards (Lawvere [1963], Lawvere [1964]).

24 Colin McLarty

neither of these as an explanation of what math is really all about, nor as

restraints on Mathematics in practice, but as proposals for organization.

Throughout his work Mac Lane uses ‘the usual category of all sets’ which

we can formalize by Zermelo-Fraenkel set theory, but he prefers to formalize

it by ETCS (Mac Lane [1998], pp. 290–1). The category is prior to any

formalization. Both ETCS and ZF, and stronger variants of either one,

describe this category. They say different things about it. He does insist there

is no use asking if one or the other axiom system is true or false. Each is correct

in the sense of consistent and adequate to interpret ordinary Mathematics. So

each ‘can serve as a foundation for mathematics’ (Mac Lane and Moerdijk

[1992], p. 331). We can ask how illuminating, or promising, or relevant each

one is for mathematical practice. Mac Lane excludes the question of truth for

reasons taken from Weyl, Geiger, and Karl Popper whose book appeared just

after Mac Lane left G ¨

ottingen (Popper [1935]).

10 Truth and Existence

In a book section titled ‘Is Geometry a Science?’ Mac Lane says each of many

geometries can be applied in the physical world by suitable ‘deﬁnitions

used in the measurement of distance’ so that ‘in the language of Karl

Popper, statements of a science should be falsiﬁable; those of geometry are

not’ (Mac Lane [1986], p. 91). Notice he is writing precisely of the geometry of

physical space, which some philosophers might say is an empirical question.

Mac Lane follows many others in saying it is not, because we can always deﬁne

measurements to support any desired physical geometry.

Distinguishing mathematical geometry from physical would not affect this

point. But the passage also denies that distinction:

We are more concerned with the positive aspects of the question: What,

then, is geometry? It is a sophisticated intellectual structure, rooted in

questions about the experience of motion, of construction, of shaping. It

leads to propositions and insights which form the necessary backdrop for

any science of motion or of engineering practices of construction [...].

Geometry is a variety of intellectual structures, closely related to each

other and to the original experiences of space and motion [...]. Geometry

is indeed an elaborate web of perception, deduction, ﬁgures, and ideas.

(Mac Lane [1986], pp. 91–2)

This is easier to understand in comparison with Weyl.

Weyl focusses on physical geometry in a section titled ‘Subject and object

(the scientiﬁc consequences of epistemology).’ He cites Kant among other

precedents for his view that the geometry of the ‘objective world’ itself is a

construction of our reason. It is: ‘ﬁnally a symbolic construction in exactly

Mac Lane as Philosopher 25

the way it is carried out in Hilbert’s mathematics.’ This is where he says

‘science concedes to idealism that its objective reality is not given but posed

as a problem.’ For Weyl as for Kant there are no physical geometric ‘data’

until our reason constructs space, and it does that by the same means as it

constructs pure mathematics. Unlike Kant, Weyl knows that our reason can

construct and apply many different geometries.38

Geiger and Mac Lane agree that mathematics is not a body of formal

truths, to be applied to another body of physical facts. In Mac Lane’s terms

quoted above, ‘geometry is a variety of intellectual structures, closely related

to each other and to the original experiences of space and motion.’ The same

intellectual faculty that sees curves in the world sees curves in differential

geometry. Recall Geiger quoted in the epigraph on how the mathematical

forms are ‘fundamental to physical reality, and are real themselves’ for the

physical sciences while for mathematics they are ‘not real but are special cases

of an ideal object world’ ([1930], p. 87).

Mac Lane somewhat combines Geiger and Popper. Geiger’s naturalistic

attitude merges with Popperian empirical science. Falsiﬁability becomes the

criterion of both. Mac Lane reserves truth for this naturalistic domain. He

concludes that Mathematics is not true and this is central to his philosophy.

A section title in the concluding chapter to Mac Lane’s philosophy book

asks ‘Is Mathematics True?’ He says ‘The whole thrust of our exhibition

and analysis of Mathematics indicates that this issue of truth is a mistaken

question.’ The right questions to ask of a given piece of math are: is it correct

by the rules and axioms, is it responsive to some problem or open question, is it

illuminating, promising, relevant? He says ‘To be sure, it is easy and common to

think that Mathematics is true’ but that is a mistake: ‘Mathematics is ‘‘correct’’

but not ‘‘true’’.’39

One may object that theorems correctly proved from true axioms are also

true. Or one may adopt ‘if-thenism’ and claim that mathematics studies

true conditionals of the form ‘IF (some axioms) THEN (some theorem).’

Mac Lane has the same response to both: These axioms and conditionals are

alike immune to empirical falsiﬁcation and so are neither true nor false. They

are, if properly given, correct.

What is this correctness? Mac Lane could take the usual position of

structuralists since Putnam ([1967]). They posit ‘logically possible’ structures

where: ‘logical possibility is taken as primitive’ (Hellman [1989], p. 8). Like

Putnam, Hellman offers no deﬁnition of the ‘possible’ but claims we have

reasonable intuitions on what is possible. Shapiro writes of coherence rather

than logical possibility, but he similarly takes coherence to be ‘a primitive,

38 Quotes are (Weyl [1927], pp. 80, 83).

39 Direct and indirect quotes from Mac Lane ([1986], pp. 440– 3).

26 Colin McLarty

intuitive notion, not reduced to something formal, and so [he does] not venture

a rigorous deﬁnition’ (Shapiro [1997], pp. 133, 135). Mac Lane’s ‘correctness’

has the same role as ‘logical possibility’ or ‘coherence’ and can as well be

declared primitive. Mac Lane has not said this himself, though. Probably he

takes a full account of correctness as one of the ‘hard problems’ yet to be

solved. It might fall under either of:

Question II. How does a Mathematical form arise from human activity

or scientiﬁc questions? What is it that makes a Mathematical formulation

possible?

Question IV. What is the boundary between Mathematics and (say)

Physical Science? (Mac Lane [1986], p. 444– 5)

Certainly he agrees with Hellman’s penultimate sentence, that we are ‘far

from a ﬁnal resolution of deep philosophical issues in this corner of the

foundations of mathematics’ (Hellman [1989], p. 144).

Compare Mac Lane’s part in the Bulletin of the American Mathematical

Society debate in 1994 over proof versus speculation in mathematics.

Mathematicians Arthur Jaffe and Frank Quinn had pointed to large and

increasing numbers of mathematical claims being published, especially on

the internet, and especially in mathematical physics, with no clear indication

of whether they are proven, conjectured, wished for, or mere scattershot

guesses.40 They say ‘Modern mathematics is nearly characterized by the use of

rigorous proofs’ but it has not always been so ([1993], p. 1). To put their case

in 18 words: They urge measuring degrees of speculation to keep its beneﬁts

without blurring the boundary around what is proved. The Bulletin editors

solicited replies from prominent mathematicians and printed 17 of them plus

a rejoinder from Jaffe and Quinn.41

Mac Lane’s response talks of ‘inspiration, insight, and the hard work of

completing proof.’ He says:

The sequence for the understanding of mathematics may be: intuition,

trial, error, speculation, conjecture, proof. The mixture and sequence of

these events may differ widely in different domains, but there is general

agreement that the end product is rigorous proof—which we know and

can recognize, without the formal advice of the logicians. (Atiyah et al.

[1994], p. 14).

Referring to some proofs published years after the results were announced,

he says: ‘the old saying applies ‘‘better late than never,’’ while in this case

40 They mean a trend inﬂuenced by Fields Medalist Edward Witten. See Louis Kaufmann’s

perceptive review Mathematical Reviews (94h:00007).

41 (Atiyah et al. [1994], Jaffe and Quinn [1994], Thurston [1994]).

Mac Lane as Philosopher 27

‘‘never’’ would have meant that it was not mathematics.’ For him no conjecture

is true or false, rather it is proved or not, and ‘It is not mathematics until it

is ﬁnally proved.’ He never speaks of mathematical truth nor of speculation

as a possible source of truth or falsity nor of proof as guarantor of truth. He

simply says ‘Mathematics rests on proof—and proof is eternal’ (Atiyah et al.

[1994] pp. 14–5).

He criticizes ‘False and advertised claims’ about Mathematics, notable

claims that various results have been proved, and blames The New York Times

for ‘recent ﬂamboyant cases’ (Atiyah et al. [1994], p. 14). He never speaks of

true or false claims in Mathematics. Truth comes up exactly once. He says

his mathematical research works by ‘getting and understanding the needed

deﬁnitions, working with them to see what could be calculated and what might

be true’ (Atiyah et al. [1994], p. 13). That is, he ﬁnds what can be calculated

using the deﬁnitions and what is true of them. To read ‘true’ here as referring

to mathematical truth deduced from the deﬁnitions would be to ignore the

‘whole thrust’ of his philosophical book (Mac Lane [1986], p. 440).

In the book Mac Lane says:

The view that Mathematics is ‘‘correct’’ but not ‘‘true’’ has philosophical

consequences. First, it means that Mathematics makes no ontological

commitments [...]. Mathematical existence is not real existence. ([1986],

p. 443)

Neither does Mathematics study marks on article:

Mathematics aims to understand, to manipulate, to develop, and to apply

those aspects of the universe which are formal. ([1986], p. 456)

Formal aspects are not physical objects any more than they are ﬁnite strings

of symbols. Mathematics takes them as ideal objects and does not even care

whether they really are aspects of the physical. If future quantum theory ﬁnds

space-time is discrete it will change neither the mathematics of the continuum

nor the origin of that idea in our experience of space. Mac Lane calls these

aspects forms where Geiger and Weyl spoke of Gebilde and Gestalten.Heraises

numerous philosophic issues about these forms. Some challenge his own ideas

while others relate them to speciﬁc Mathematics (Mac Lane [1986], pp. 444ff).

11 Naturalism

Maddy describes Quine as the founding ﬁgure in current naturalism and she

deﬁnes naturalism in his words as ‘the recognition that it is within science

itself, and not in some prior philosophy, that reality is to be identiﬁed and

described’ (Quine [1981b], p. 21). Quine requires ‘abandonment of the goal of

28 Colin McLarty

a ﬁrst philosophy’ and urges that we begin our reasoning ‘within the inherited

world theory [given by science] as a going concern’ ([1981a], p. 72).42 This is

congenial both to Mac Lane’s practice and to his explicit philosophy.

Not all of Quine’s philosophy suits Mac Lane so well. The two often spoke

when Mac Lane taught at Harvard from the mid 1930s to 1947. During these

years Quine arrived at his famous slogan ‘to be is to be the value of a variable’

([1948], pp. 32, 34). Mac Lane took this ontology as a foreign intrusion into

Mathematics, reﬂecting Quine’s ‘undue concern with logic, as such’:

For Mathematics, the ‘‘laws’’ of logic are just those formal rules which it

is expedient to adopt in stating Mathematical proofs. They are (happily)

parallel to the laws of logic that philosophers or lawyers might use in

arguing about reality—but Mathematics itself is not concerned with

reality but with rule. (Mac Lane [1986], p. 443)

Maddy’s own naturalism in the philosophy of mathematics says ‘the goal

of philosophy of mathematics is to account for mathematics as it is practiced,

not to recommend reform’ ([1997], p. 161). This is much more than eschewing

ﬁrst philosophy. And it is hard to apply to Mac Lane because, as with many

leading mathematicians, much of his practice consisted of reforms. Mac Lane

links all kinds of mathematical progress with philosophy:

A thorough description or analysis of the form and function of

Mathematics should provide insights not only into the Philosophy

of Mathematics but also some guidance in the effective pursuit of

Mathematical research. (Mac Lane [1986], p. 449)

Of course he has never encouraged immodesty in anyone. No philosopher

or mathematician need overrate their importance in guiding or reforming

Mathematics! He does encourage anyone to get informed on Mathematics and

its Philosophy and make their own judgements.

Maddy’s naturalism allows such judgements only when they rely on ‘integral

parts of mathematical method’ and not ‘extramathematical philosophizing;’

but ‘this is a difﬁcult distinction to draw’ (Maddy [2005b], p. 453). Mac Lane

does not draw it. He judges any proposed reform in Mathematics by the criteria

already quoted: is it correct, responsive, illuminating, promising, relevant?(Mac

Lane [1986], pp. 441). No doubt, much of what Maddy calls extramathematical

philosophizing, Mac Lane would call irrelevant. But Mac Lane sees many

varieties of irrelevance besides ‘philosophizing.’

Compare a point Maddy makes, which a non-naturalist might consider

merely sociological since it concerns the way practitioners view their

42 These quotes of Quine are on Maddy ([2005], pp. 437f).

Mac Lane as Philosopher 29

practice. Maddy claims ‘mathematicians tend to shrink from the task’ of

relating their work to other mathematics ‘especially in conversation with

philosophers’ ([1997], p. 170). As a naturalist, her standard for what the

practice should be is to see what practitioners take it to be, and she makes this

an argument for a claim central to her project: ‘the choice of methods for set

theory is properly adjudicated within set theory itself’ and not in relation to

other mathematics, let alone philosophy ([2005a], p. 358).

Of course mathematicians in Hilbert’s G ¨

ottingen did not shrink from the

task or from philosophy. Today Maddy’s observation seems true in set theory

but not in other branches of mathematics. It is a clich´

e to say number theorists

praise their ﬁeld as the Queen of Mathematics reigning over it all —and it is

true. See topics from spherical geometry to coupled oscillators in McKean and

Moll ([1999]). No set theory text is at all like that. Research number theory too

makes connections all across mathematics, as in Waldschmidt, Moussa, Luck

and Itzykson ([1992]). Geometers show more tendency to address philosophers.

They heavily dominated the only sustained public philosophic discussion by

mathematicians in recent times, the B.A.M.S. debate over proof.43

Strong internalism has never been natural to Mac Lane, whether it means

each branch of mathematics should look primarily to itself, or that mathematics

need not address philosophy. His work pulled together algebra, number theory,

and topology. He has surveyed Mathematics as a whole, most thoroughly

in Mac Lane ([1980]). His sweeping claim on ‘general concepts applicable to

all branches of mathematics’ grew from one technical problem for inﬁnitely

tangled topological spaces (Eilenberg and Mac Lane [1945], p. 236). His

philosophy book emphasizes ‘the intimate interconnection of Mathematical

ideas which is striking’ and urges resisting ‘the increasing subdivision of

mathematics attendant upon specialization [...] a resultant lack of attention

to connections [...] and neglect of some of the original objectives’ (Mac Lane

[1986], pp. 418, 428). For him, each single result in Mathematics takes its

value from the whole and the value of the whole is as much philosophical as

technical. There are valuable, purely technical mathematical articles. There is

no valuable Mathematics without philosophy.

None of this attacks the heart of Maddy’s naturalism. Maddy has created a

naturalist character called the ‘Second Philosopher’ who rejects ﬁrst philosophy

but not all philosophy. This character ‘will ask traditional philosophical

questions about what there is and how we know it,’ just as Descartes does, but

unlike Descartes in the Meditations, she will approach them in terms of ‘physics,

chemistry, optics, geology...neuroscience, linguistics, and so on’ (Maddy

[2003], pp. 80f). So far the Second Philosopher is very like Mac Lane. The key

43 These include Atiyah, Borel, Mandelbrot, Thom, Thurston, Witten, Zeeman (Atiyah et al.

[1994], Thurston [1994]).

30 Colin McLarty

to this character’s naturalism is that ‘all the Second Philosopher’s impulses are

methodological, just the thing to generate good science [...] she doesn’t speak

the language of science ‘‘like a native’’; she is a native’ (Maddy [2003], p. 98).

This is Mac Lane to a tee. His entire philosophical impulse is methodological

and his philosophy aims single-mindedly at generating good Mathematics. In

this way he is much closer to Noether, who could not conceive a holiday from

Mathematics, than to the classical, literary, historical philosophy of Weyl.

The Second Philosopher is not very close to Mac Lane’s formal functionalism

but actually seems not to have considered it—though it comes from a fellow

‘native.’ Maddy agrees with Mac Lane that Quine imposed irrelevant logical-

ontological concerns on mathematics and so offered too narrow a methodology

for it.44

Maddy and Mac Lane agree: ‘if you want to answer a question of

mathematical methodology, look not to traditionally philosophical matters

about the nature of mathematical entities, but to the needs and goals of

mathematics itself’ (Maddy [1997], p. 191). But Mac Lane ﬁnds that the

traditional philosophies are wrong while Maddy ﬁnds them extramathematical.

Mac Lane wants better philosophy in Mathematics, not less. He argues from

his long and inﬂuential career that the needs and goals of Mathematics do not

show in isolated results or even in isolated branches of Mathematics. They

show in the larger form and function of Mathematics. He is concerned with

Mathematics per se, and so with the on-going reforms of it, and for this very

reason with the love of wisdom.

12 Austere Forms of Beauty

Mac Lane says his rejection of truth in mathematics ‘does not dispose of

the hard questions about the philosophy of Mathematics; they are merely

displaced.’ They include:

What are the characteristics of a Mathematical idea? How can an idea be

recognized? described? [...] How does a Mathematical form arise from

human activity or scientiﬁc questions? (Mac Lane [1986], p. 444– 5)

But displacing the problems is already a lot. It is just what Mac Lane has

done in mathematics to very great effect. Cohomology does not itself solve

hard problems in topology or algebra. It clears away tangled multitudes of

individually trivial problems. It puts the hard problems in clear relief and

makes their solution possible. The same holds for category theory in general.

44 See e.g. (Maddy [1997], p. 184) or (Maddy [2005b], p. 450).

Mac Lane as Philosopher 31

Mac Lane continued the line of Dedekind, Hilbert, and Noether and the

famous Moderne Algebra (van der Waerden [1930]). This did not prove more

theorems than the old algebra. Curmudgeons truthfully complained that van

der Waerden taught less about ﬁnding the Galois group or the roots of speciﬁc

low-degree polynomials than older textbooks, and less advanced calculations

with matrices. Noether’s school claimed their theorems were better because

they applied to more ﬁelds of mathematics. Traditionalists accepted the new

proofs but construed them otherwise: they claimed the modern theorems were

mere generalities obscuring the speciﬁc ‘substance’ of each ﬁeld (Weyl [1935],

esp. p. 438). The same arguments took place over category theory into the

1970s and still continues today in some quarters. These value questions are

not subject to mathematical proof.

We come back to ‘Mathematical beauty’ (Mac Lane [1986], p. 409). When

Mac Lane told Weil and many others that every notion of structure necessarily

brings with it a notion of morphism it was not true in any ordinary sense. It was

no theorem, axiom, or deﬁnition. The only foundational axioms Mac Lane

knew around 1950 were membership-based set theories, which did not rely

on morphisms. There was no standard deﬁnition of structure let alone of

morphism. Mac Lane’s claim was a postulate in Euclid’s Greek sense of

’α´ιτ η µα or ‘demand.’ Weyl cited and praised Euclid for using this word which,

according to Weyl, still expresses ‘the modern attitude’ in mathematics ([1927],

p. 23). Mac Lane never shrinks from it. His Mathematics is free to demand any

kind of ideal object including, for example, the proper-class sized categories

of all groups or all topological spaces.

He urged his demand for morphisms because it expressed what is valuable

in Mathematics far beyond solutions to equations: ‘Mathematics is in part a

search for austere forms of beauty’ (Mac Lane [1986], p. 456). His claim about

structures and morphisms was a vision of vast order within and among all

the branches of Mathematics, a vision of articulate global organization, of

categorical Mathematics. It was a vision of Mathematical beauty.

Acknowledgements

Most of this article was written during Saunders Mac Lane’s life (1909–2005).

I am hugely indebted to him for conversation and for his work in mathematics,

history, and philosophy. Thanks to Steven Awodey, William Lawvere, Barry

Mazur, and the anonymous referees for valuable comments.

32 Colin McLarty

Department of Philosophy,

Case Western Reserve University,

Cleveland, OH,

USA 44106

colin.mclarty@case.edu

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