Mathematics as a love of wisdom:
Saunders Mac Lane as philosopher
Colin McLarty
Department of Philosophy, Case Western Reserve University
Keywords
naturalism, philosophy, mathematics, Aristotle.
Go, read, and disagree for yourself.
(Mac Lane, 1946, p.390)
T
his note describes Saunders Mac Lane as a philosopher, and
indeed as a paragon naturalist philosopher. Obviously he approaches questions in philosophy the way a mathematician would. He
is one. But, more deeply, he learned philosophy by attending David
Hilbert’s public lectures on it, and by discussing it with Hermann
Weyl, as much as he did by studying for a qualifying exam on it
with the mathematically informed Göttingen Philosophy professor
No 69 (2020), pp. 17–32 • CC-BY-NC-ND 4.0
This note describes Saunders Mac Lane as a philosopher, and indeed
as a paragon naturalist philosopher. He approaches philosophy as
a mathematician. But, more than that, he learned philosophy from
David Hilbert’s lectures on it, and by discussing it with Hermann Weyl,
as much as he did by studying it with the mathematically informed
Göttingen Philosophy professor Moritz Geiger.
Philosophical Problems in Science (Zagadnienia Filozoficzne w Nauce)
Abstract
18
Colin McLarty
Moritz Geiger (McLarty, 2007; 2020).1 Before comparing Mac Lane
to Penelope Maddy’s created naturalist, the Second Philosopher, we
relate him as a philosopher to Aristotle.
This is not to disagree with Skowron’s view of Mac Lane as a
platonist in ontology (Skowron, manuscript). This is about Mac Lane
on scientific and philosophic method. He understood philosophy very
much as Aristotle did: Philosophy is the love of wisdom, and every
science pursues wisdom and not mere facts. I do not claim Mac Lane
got the ideas from Aristotle. So far as I know, Mac Lane himself had
no interest in or opinion of Aristotle, though his teachers Weyl and
Geiger certainly talked of Aristotle.2
1. Aristotle on wisdom and science
In general the sign of knowing or not knowing is the ability to
teach, so we hold that art rather than experience is scientific
knowledge; some can teach, some cannot. Further, the senses
are not taken to be wisdom. They are indeed the authority for
acquaintance with all individual things, but they do not tell the
why of anything, for example why fire is hot, but only that it
is hot.
It is generally assumed that what is called wisdom is concerned
with the primary causes and principles, so that, as has been
already stated, those who have experience are held to be wiser
1
Mac Lane’s close contact with Paul Bernays in Göttingen deserves more attention.
But my research so far has not identified strong philosophic influence from Bernays.
See Section 3.
2 Browder and Mac Lane (1978) give a beautiful survey of mathematics which mentions Plato and Aristotle on ontology. However, the discussion of Plato and Aristotle
summarizes the longer discussion by Browder (1976).
Mathematics as a love of wisdom: Saunders Mac Lane as philosopher
19
than those who merely have any kind of sensation, the artisan
than those of experience, the craft master than the artisan.
(Aristotle, Metaphysics 981f.)3
Here Aristotle says scientific knowledge is better than acquaintance, or even experience, in two related ways:
1. scientific knowledge can be taught, and
2. scientific knowledge gives the why of things.
He says wisdom deals with primary causes and principles. And those
are his touchstone for scientific knowledge: “We do not know or have
scientific knowledge of objects of any methodical inquiry, in a subject
that has principles, causes, or elements, until we are acquainted with
those and reach the simplest elements” (Physics 184a).
Throughout his career Mac Lane tied pedagogy to research and
research to craft. Among many examples see his early notes on presenting mathematical logic to university students (Mac Lane, 1939),
and his impassioned argument that theoretical education prepared men
and women well to do the applied mathematics which he supervised
in World War II (Mac Lane, 1989; 1997).
Mac Lane also insists mathematical understanding includes knowing the reasons for a given theorem. Some proofs of a theorem may
reveal the reason, while other technically sufficient proofs will not
reveal the reason. In his book for philosophers, Mac Lane sketches
proofs for major theorems from many subjects, like linear algebra, or
complex analysis. He often describes several alternative proofs for a
single theorem and then eventually singles out one as giving the real
reason. That book is (Mac Lane, 1986) and some examples are on
pages 145, 189, 427, 455.
3
Translations of Aristotle here use “know” for oida, “scientific knowledge” for episteme, and “acquaintance” for gnosis. Of course “wisdom” is sophia.
20
Colin McLarty
For me, though, the deepest connection between Aristotle’s and
Mac Lane’s loves of wisdom is how they say we gain this scientific
knowledge. Both believe in foundations, or “first principles” if you
prefer, but neither believes we start with those. Aristotle’s theoretical
demand of philosophy was a theoretical and practical demand in
mathematics for Mac Lane:
The natural way of [getting scientific knowledge] is to start
from the things which are more knowable and obvious to
us and proceed towards those which are clearer and more
knowable by nature; for the same things are not ’knowable
relatively to us’ and ’knowable’ without qualification. So in
the present inquiry we must follow this method and advance
from what is more obscure by nature, but clearer to us, towards
what is more clear and more knowable by nature. (Physics
184a)
Aristotle speaks of advancing from what is initially clear to us,
towards what is more knowable by nature. I am not sure if he believed
there was a final point where the absolutely first principles and simplest elements are known so that they will never change. Mac Lane
certainly did not believe it for mathematics.
From his early work on field theory (Mac Lane and Schilling,
1939; 1940) and for the rest of his career Mac Lane often worked
to find more basic concepts in some part of mathematics. He and
Eilenberg spent over a decade collaborating on ever broader uses of the
concepts in their “General theory of natural equivalences” (Eilenberg
and Mac Lane, 1945). They meant that paper to be the only one ever
needed on this technical concept for group theory and topology, but it
became the founding paper of the whole field of category theory.
Only in the 1960s, after meeting graduate student Bill Lawvere,
did Mac Lane come to believe category theory could be a foundation
Mathematics as a love of wisdom: Saunders Mac Lane as philosopher
21
for all mathematics. Even then, precisely because of all the concrete
mathematics that had gone into developing his ideas, Mac Lane insisted this, and any foundation for mathematics, must be seen as
“proposals for the organization of mathematics” (Mac Lane, 1986,
p.406). The optimal organization (i.e. the optimal foundation) will
change as mathematics develops, and will help advance those developments. He warned that excessive faith in any “fixed foundation would
preclude the novelty which might result from the discovery of new
form” (Mac Lane, 1986, p.455).
2. Mathematics as a love of wisdom
Let us come to cases with one paradigmatically philosophical question,
and one paradigmatically mathematical. For Mac Lane these questions
are inseparable:
Q1 What are mathematical objects, and how do we come to know
them?
Q2 What are solutions to a Partial Differential Equation (PDE),
and how do we come to know them?
For Mac Lane Q2 can only be a specific case of Q1 . For him, as for
Aristotle, basic questions of the special sciences are philosophy. They
cannot not be philosophy.
Let us be clear: A mathematician can learn a textbook answer to
Q2 without ever asking for a philosophy behind it. In just the same
way, a philosopher can learn the currently received answers to Q1
from philosophy books, without ever asking about live mathematics.
Admittedly the math textbook answers will be more stable over time
than the philosophically received answers. But that is not important.
22
Colin McLarty
For Mac Lane, both of those ways of learning are failures of understanding. They are failures of philosophy. For him, an answer to either
one of those questions can only be valuable to the extent that you can
see what it is good for—for Mac Lane that cannot be either a purely
technical mathematical question or a purely academic philosophic
one. Think back to his work in World War II.
Mathematicians speak of solutions to PDEs in many ways:
•
•
•
•
Smooth (or, sufficiently differentiable) function solutions.
Symbolic solutions.
Generalized function solutions (of various kinds. . . ).
Numerical solutions. . .
These different senses of solutions are sought in very different ways.
There are well understood relations between them, but the relations
are not all obvious and in particular cases they may be quite difficult,
and important, to find.
Mac Lane’s war work certainly involved relating different kinds
of solutions to PDEs. Even when an equation has a known exact solution by an easily specified smooth function, applying it also requires
numerical solutions. The worker has to choose which aspects are best
handled in theory, so as to direct and optimize the calculations, and
when best to leave theory and begin calculating. Those choices are
rarely textbook work. They are often not clear cut at all. They require
exactly what Aristotle called the wisdom of the craft master. Namely,
they require grasping the why of each kind of solution. They require
knowing not only the technical definition of each kind of solution, but
what good each one is, and especially the good of their relations to
one another.
The craft master, having wisdom, knows the whys, can teach
them, and supervise work with them.
Mathematics as a love of wisdom: Saunders Mac Lane as philosopher
23
As I write this, I imagine some practice-minded philosopher
challenging: “How are philosophies like logicism, formalism, and
intuitionism going to help anyone solve or apply a PDE?” Indeed.
This is why Mac Lane so often deprecates those philosophies. But
just to give one example, Mac Lane argued that formalism in Hilbert’s
hands was a step towards programmable computers. See Section 3.
Those unquestionably help solve PDEs.
3. The philosophy of mathematicians
in 1930s Göttingen
Wir dürfen nicht denen glauben, die heute mit philosophischer Miene und überlegenem Tone den Kulturuntergang
prophezeien und sich in dem Ignorabimus gefallen. Für uns
gibt es kein Ignorabimus, und meiner Meinung nach auch
für die Naturwissenschaft überhaupt nicht. Statt des törichten
Ignorabimus heiße im Gegenteil unsere Losung: Wir müssen
wissen – wir werden wissen!
We must not believe those, who today, with philosophical
bearing and deliberative tone, prophesy the fall of culture and
accept the ignorabimus. For us there is no ignorabimus, and in
my opinion none whatever in natural science. In opposition to
the foolish ignorabimus our slogan shall be: We must know –
we will know! (Hilbert, 1930, p.385)4
Hilbert’s conclusion, Wir müssen wissen – wir werden wissen!, is
engraved on his tomb in Göttingen.
4
Before we try to defend or defeat Hilbert’s slogan as an assertion in academic
epistemology, it is in fact the most important statement in philosophy of mathematics of
the past 150 years. I use the translation by Ewald (2005, p.1164). Hilbert’s address was
broadcast on the radio and a recording is available at math.sfsu.edu/smith/Documents/
HilbertRadio/HilbertRadio.mp3.
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Colin McLarty
Mac Lane arrived in Göttingen just at the time Hilbert was promoting this slogan. I do not recall Mac Lane quoting it in lectures or
conversations. He did not have to quote Hilbert. Everything Mac Lane
said illustrated this faith. As you can see in Mac Lane (1986; 2005),
or McLarty (2007), Mac Lane followed Hilbert’s mathematizing scientific optimism, rather than the specific finitist program (often called
“formalism,” though not by Hilbert) of Hilbert’s famous On the Infinite (1926). Mac Lane did see specific, productive mathematical value
in that program though, and rejected a criticism of it by Freeman
Dyson (Mac Lane, 1995).
Dyson supposed Hilbert seriously meant to reduce all mathematics to formal reasoning, and said “the great mathematician David
Hilbert, after thirty years of high creative achievement [. . . ] walked
into a blind alley of reductionism.” Specifically, Dyson claimed that
Hlbert “dreamed of” formalizing all mathematics, solving the decision
problem for this formal logic, and “thereby solving as corollaries all
the famous unsolved problems of mathematics.” Mac Lane replied:
I was a student of Mathematical logic in Göttingen in 19311933, just after the publication of the famous 1931 paper by
Gödel. Hence I venture to reply [. . . ]. Hilbert himself called
this ‘metamathematics.’ He used this for a specific limited
purpose, to show mathematics consistent. Without this reduction, no Gödel’s theorem, no definition of computability, no
Turing machine, and hence no computers [. . . ]. Dyson simply
does not understand reductionism and the deep purposes it can
serve.
Mac Lane gives a concise expert review of these issues and places
them in the context he knew at the time they arose. He insists Hilbert
did not tie the slogan “we must know, we will know” to the decision
problem. Rather, Mac Lane says, “[Hilbert] held that the problems of
Mathematics as a love of wisdom: Saunders Mac Lane as philosopher
25
mathematics can all ultimately be solved” without supposing metamathematics will do it. Full disclosure: I admit that after long consideration, drawing on Sieg (1999; 2013), I myself am unsure exactly
how Hilbert and/or Bernays intended their work on the decision problem at various times. But however that may be, Mac Lane understood
Hilbert this way. And this is Mac Lane’s own far from reductionist
faith, while recognizing reductionist methods for what they actually
have achieved in mathematics.
Mac Lane learned a lot in frequent discussions with Bernays.
But for now I have to say I see no larger trace of those discussions in Mac Lane’s philosophy than is found in the letter on Dyson.
Mac Lane’s book on philosophy of mathematics is titled Mathematics:
Form and Function. But this is clearly “form” as Mac Lane learned
about it from talking with Weyl and studying under Geiger (McLarty,
2007). It does not refer to formalism in any sense related to Bernays.
Or, at least, so it seems to me. The reader is encouraged to go, read,
and disagree if they see something else.
4. Naturalism
The decisive feature marking Penelope Maddy’s Second Philosopher
as a naturalist is that:
[She] sees fit to adjudicate the methodological questions of
mathematics—what makes for a good definition, an acceptable
axiom, a dependable proof technique?—by assessing the effectiveness of the method at issue as means towards the goals of
the particular stretch of mathematics involved. (Maddy, 2007,
p.359)
26
Colin McLarty
Lots of mathematicians, and essentially all leading mathematicians,
do the same.5
The unusual thing about Mac Lane in this regard is that he was
explicitly tasked by the US government to evaluate mathematics research and teaching methods in classified reports during World War
II and publicly after that.6 Those reports were explicitly directed to
various different specific short-term and long-term goals. All the variety he saw, and dealt with, left Mac Lane ever more deeply impressed
with the actual unity of the whole.
Precisely that background, along with his experience as Chair of
the Chicago Mathematics Department, made Mac Lane diverge from
another feature of Maddy’s Second Philosopher:
All the Second Philosopher’s impulses are methodological,
just the thing to generate good science [. . . ]. (Maddy, 2003,
p.98)
All Mac Lane’s impulses aim at producing good science and for this
reason they are not all methodological.
Mac Lane, like Aristotle, knows methods alone generate no science. Besides evaluating methods of reaching goals, at least some
mathematicians must evaluate goals. For Aristotle, those should be
the craft masters, the wise. In his vivid words: “the wise should not
accept orders but give them; nor should they be persuaded, but the
less wise should” (Metaphysics 982a). We will see, though, Mac Lane
is less focused on command than that. He inclines more to another
5
If by axioms Maddy means specifically axioms of set theory then few mathematicians
ever learn those, let alone adjudicate them. Mac Lane is famously among those few.
6 See Mac Lane (1967; 1989) and Fitzgerald and Mac Lane (1977) and Steingart
(2011).
Mathematics as a love of wisdom: Saunders Mac Lane as philosopher
27
passage: “those who are more accurate and more able to teach about
the causes are the wiser in each branch of knowledge” (Metaphysics
982a).
Because of his broad experience, especially evaluating both methods and goals for mathematics, Mac Lane cannot agree that “the goal
of philosophy of mathematics is to account for mathematics as it
is practiced, not to recommend reform.” (Maddy, 1997, p.161) Just
sticking to the mathematician philosophers we have already named:
Hilbert, Weyl, and Mac Lane all knew reform is integral to mathematical practice. You cannot separate reform from practice if you try.
And all three made explicitly philosophic arguments for their recommended reforms along with more technically mathematical ones.7
This is important for philosophy of mathematics.
The paradigm case for anti-revisionism in philosophy of mathematics is Brouwer’s intuitionism. Brouwer is by far the favorite
illustration of a revisionist, and is the sole example that the Stanford Encyclopedia of Philosophy discusses under anti-revisionism in
the article “Naturalism in the Philosophy of Mathematics” (Paseau,
2016):
The mathematician-philosopher L.E.J. Brouwer developed
intuitionistic mathematics, which sought to overthrow and
replace standard (‘classical’) mathematics.
So it is important for philosophers to understand that the problem
with Brouwer, according to all our exemplars Hilbert, Weyl, and
7
Hilbert had sweeping success with his reforms. Among many philosophic works by
and on him see Hilbert (1923; 1930). Weyl (1918) advocated what Weyl took to be
Brouwer’s philosophy, while Weyl (1927; 1949) trace his eventual, regretful conclusion
that in fact Hilbert was right about this and Brouwer wrong.
28
Colin McLarty
Mac Lane, is not that he had philosophical motives. It is that he was
wrong. Actually, for Mac Lane, Brouwer’s philosophy was at best
wrong. At worst it was “pontifical and obscure” (Mac Lane, 1939).
Immediately upon completing his doctorate in mathematics at Göttingen, Mac Lane put a philosophy article in The Monist (Mac Lane,
1935). Fifty years later he wrote a book describing, as he told me,
what he wanted philosophers to know about math (Mac Lane, 1986).
There he asks about the large array of mathematics he surveyed: “How
does it illuminate the philosophical questions as to Mathematical truth
and beauty and does it help to make judgements about the direction of
Mathematical research?” (Mac Lane, 1986, p.409) There is a reason
he puts these questions together.
He asks about mathematical truth and beauty knowing very well
that few mathematicians want to pursue the question seriously, and
knowing philosophers who speak of it rarely know much of the wealth.
For Mac Lane both of those are failures of understanding and they
are nothing he means to promote.8 He means to promote mathematically informed philosophic pursuit of the question of mathematical
truth and beauty. And so he does of the question on the direction of
research. He seriously means to promote philosophic thought on that.
Of course he does not see philosophic thought as the sole preserve of
those with philosophy degrees. No more does he see philosophy of
mathematics as the sole preserve of those with math degrees. Mathematics for Mac Lane, when pursued with full awareness of its worth,
is philosophy.
8
À propos, I consider Edna St. Vincent Millay’s poem “Euclid alone has looked
on Beauty bare” incredibly true to its topic, despite that she apparently studied no
mathematics beyond school textbooks based on bits of Euclid’s Elements.
Mathematics as a love of wisdom: Saunders Mac Lane as philosopher
29
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