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The Last Mathematician from Hilbert's Gottingen: Saunders Mac Lane as Philosopher of Mathematics



While Saunders Mac Lane studied for his D.Phil in Göttingen, 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 Göttingen. His career is a case study relating naturalism in philosophy of mathematics to philosophy as it naturally arises in mathematics. Introduction Structures and Morphisms Varieties of Structuralism Göttingen Logic: Mac Lane's Dissertation Emmy Noether Natural Transformations Grothendieck: Toposes and Universes Lawvere and Foundations Truth and Existence Naturalism Austere Forms of Beauty
Brit. J. Phil. Sci. page 1 of 36
The Last Mathematician from
Hilbert’s G¨
ottingen: Saunders
Mac Lane as Philosopher of
Colin McLarty
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.
2Structures and Morphisms
3Varieties of Structuralism
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:
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´
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:GHis 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:STthat reflect
open subsets. The inverse image f1(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:XYfrom 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 3πis 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:AB.
Morphisms f:ABand g:BCwith matched codomain and
domain have a composite gf :AC. Composition is associative so
that h(gf ) =(hg)f for any h:CD. In a diagram:
Each object Ahas an identity morphism 1A:AAdefined by its
composites f1A=1Bf=ffor every f:AB.
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:ABas 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 xyas morphisms. The identity morphism for any xRis
just xx, and morphisms xyand yzcompose to xz. 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´
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],
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
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:ABestablishes
a relation between the elements xAand 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
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¨
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´
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
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 ¨
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....Soweafrm
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
...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
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
if the difference xyis 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 xZto be
the equivalence class of xfor this relation. Writing xfor the coset of xthat
Then Z/3 has exactly three elements, namely, the cosets 0,1,2, since every
integer xZbelongs 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
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
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
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:XYinduces 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 nN. 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
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 ¨
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:GGinduces homomorphisms in the other
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:XAwhere 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:XTAb 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
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 <>.
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
EEfrom 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.
{xA|fx =gx}>−→ Af
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 xAof a set Acorrespond exactly
to the functions x:1Afrom 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:1Aand function f:AB,
the composite fx:1Bis an element of B. The key axiom is extensionality
applied to functions: given parallel arrows f, g :A
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
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
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.
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,
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... Is it possible that mathematical development has passed something by, just because MacLane did not find resonance for this work and, back in the U.S., was soon successful, as the strong mathematician he was, in other fields. His work on the conceptual structure of mathematics, category theory and its pervading influence throughout mathematics, is well known, (1). ...
... In this essay we attempt to revive the idea of an algebra of proofs and place MacLane's thesis work and its vision in a new framework and try to address his original vision. He believed that the mathematical enterprise as a whole should be cast into a grand algebraic 1 As Bernays was only Dozent (and was soon to be dismissed as foreign, Swiss and of jewish ancestry), Weyl was official thesis advisor, and of course took personal interest until he also left (for Princeton, Bernays for ETH.) framework. If completed, it would embody axiomatic, methodology of inventing and proving ,and would underly mathematical exposition; a youthful dream. ...
Looking at MacLane's thesis on proof theory in the light of combinatory logic
... The moral is that every type of structure is equipped with a special type of map between structures of the given type. In current mathematical argo it is said that such maps "respect" or "preserve" the corresponding structure (even if what is functors, and in 1961 to produce a separate volume on the Abelian Categories [48]; see also [195] for further historical details. These plans were never realised, is also at odds with the existing practice of "categorical reasoning", which makes the Bourbaki-style set-theoretic underpinning wholly redundant (at least from the viewpoint of a working mathematician). ...
The received concepts of axiomatic theory and axiomatic method, which stem from David Hilbert, need a systematic revision in view of more recent mathematical and scientific axiomatic practices, which do not fully follow in Hilbert’s steps and re-establish some older historical patterns of axiomatic thinking in unexpected new forms. In this work I motivate, formulate and justify such a revised concept of axiomatic theory, which for a variety of reasons I call constructive, and then argue that it can better serve as a formal representational tool in mathematics and science than the received concept.
... Their first printed use of category is in (1945) giving the general definition of functors. For more relating Mac Lane to Noether see Koreuber (2015); Krömer (2007); Mac Lane (1981Lane ( , 1997bMcLarty (2006McLarty ( , 2007a. ...
Saunders Mac Lane heard David Hilbert’s weekly lectures on philosophy and utterly believed Hilbert’s declaration that mathematics will know no limits. He studied algebra with Emmy Noether, and both mathematics and philosophy with Hermann Weyl. As a young algebraist he created today’s standard working method for mathematical structure: category theory, with topologist Samuel Eilenberg. As one step, they created the now standard definition of “isomorphism.” They originally saw categories as just a working tool. But in the 1950s, Mac Lane saw Alexander Grothendieck and others radically extend the range of the theory, and in the 1960s, he took up William Lawvere’s idea of categorical foundations. The essay relates all of this to current philosophical structuralism, especially concerning isomorphisms and automorphisms of structures. It concludes by comparing Mac Lane’s motives for structuralist working mathematics with current philosophical motives for structuralist ontology.
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There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized.
A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.
Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowledge, a mathematical proof must be rigorous . What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view , a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely, the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it.
Today's number theory solves classical problems by structural tools that violate standard philosophical expectations in ontology. The far-reaching practical demands of this mathematics require on one hand that the tools be fully explicit and more rigorous than many philosophical theories are, and on the other hand that they relate as directly and as concisely as possible to guiding intuitions. The standard tools grew from a century-long trend of unifying algebra, topology, and arithmetic, notably in the Weil conjectures; and they rely on devices that Kronecker produced for his idea of a pure arithmetic. Very large functors serve to organize individually simple kinds of data that can themselves even be depicted in simple pictures. Mathematicians and philosophers have debated issues of individuation and identity raised by these tools.
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What is so special and mysterious about the Continuum, this an-cient, always topical, and alongside the concept of integers, most in-tuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathe-matical minds of all times committed to penetrate once and for all its set-theoretical enigma ? The double-edged purpose of the present study is to save from the transfinite deadlock of higher set theory the jewel of mathemati-cal Continuum – this genuine, even if mostly forgotten today raison d'être of all set-theoretical enterprises to Infinity and beyond, from Georg Cantor to W. Hugh Woodin to Buzz Lightyear, by simulta-neously exhibiting the limits and pitfalls of all old and new reduc-tionist foundational approaches to mathematical truth: be it Can-tor's or post-Cantorian Idealism, Brouwer's or post-Brouwerian Con-structivism, Hilbert's or post-Hilbertian Formalism, Gödel's or post-Gödelian Platonism. 1 In the spirit of Zeno's paradoxes, but with the enormous historical advantage of hindsight, we claim that Cantor's set-theoretical method-ology, powerful and reach in proof-theoretic and similar applications as it might be, is inherently limited by its epistemological framework of transfinite local causality, and neither can be held accountable for the properties of the Continuum already acquired through geomet-rical, analytical, and arithmetical studies, nor can it be used for an adequate, conceptually sensible, operationally workable, and axiomat-ically sustainable re-creation of the Continuum. From a strictly mathematical point of view, this intrinsic limita-tion of the constative and explicative power of higher set theory finds its explanation in the identified in this study ultimate phenomenologi-cal obstacle to Cantor's transfinite construction, similar to topological obstacles in homotopy theory and theoretical physics: the entangle-ment capacity of the mathematical Continuum.
Some central philosophical issues concern the use of mathematics in putatively non-mathematical endeavors. One such endeavor, of course, is philosophy, and the philosophy of mathematics is a key instance of that. The present article provides an idiosyncratic survey of the use of mathematical results to provide support or counter-support to various philosophical programs concerning the foundations of mathematics.
Current research in mathematics involves a wide variety of interlocking ideas, old and new. For example, results about the curves and surfaces defined by polynomial equations, as in algebraic geometry, appear in the study of solitary waves and also in the gauge theories in physics. Centuries-old problems in number theory have been solved, while others have been revealed as insoluble. The classification of all finite simple groups is nearly achieved (and the full treatment will be voluminous); the representation of groups aids in their application to the study of symmetry. These developments, and many others, attest to the vitality of mathematics.
In the last few decades, after the appearance of T. S. Kuhn’s The Structure of Scientific Revolutions (1962, 1970), it has become fashionable to discuss revolutions in mathematics. The book by D. Gillies 1992 (ed.) Revolutions in Mathematics, Clarendon Press, Oxford, to be referred to henceforth as G followed by a page number, is a collection of relevant papers. The opinions range from Michael Crowe’s thesis that there are no revolutions in mathematics to the opposite opinion, represented by J.W. Dauben besides others, that almost every situation in the history of mathematics that was sensed to have been critical, every situation followed by changes viewed as radical, should be regarded as indicating a revolution. Gillies goes so far as to speak of a Crowe-Dauben debate.
With developments in the 19th and early 20th centuries, structuralist ideas concerning the subject matter of mathematics have become commonplace. Yet fundamental questions concerning structures and relations themselves as well as the scope of structuralist analyses remain to be answered. The distinction between axioms as defining conditions (Hilbertian conception) and axioms as assertions (traditional Fregean conception) is highlighted as is the problem of the indefinite extendability of any putatively all-embracing realm of structures. This chapter systematically compares four main versions: set-theoretic structuralism, a version taking structures as sui generis universals, structuralism based on category theory, and a quasi-nominalist modalstructuralism. While none of the approaches is problem-free, it appears that some synthesis of the category-theoretic approach with modal-structuralism can meet the challenges set out, given the notion of "logical possibility."
Weyl's inclination toward constructivism in the foundations of mathematics runs through his entire career, starting with Das Kontinuum. Why was Weyl inclined toward constructivism? I argue that Weyl's general views on foundations were shaped by a type of transcendental idealism in which it is held that mathematical knowledge must be founded on intuition. Kant and Fichte had an impact on Weyl but HusserFs transcendental idealism was even more influential. I discuss Weyl's views on vicious circularity, existence claims, meaning, the continuum and choice sequences, and the intuitive-symbolic distinction against the background of his transcendental idealism and general intuitionism.