*The Ideal
Mathematician*
=We will construct= a portrait of the
`ideal mathematician.' By this we do not
mean the perfect mathematician, the mathe
matician without defect or limitation. Rather,
we mean to describe the most mathematicianlike mathe
matician, as one might describe the ideal thoroughbred
greyhound, or the ideal thirteenthcentury monk. We will
try to construct an impossibly pure specimen, in order to
exhibit the paradoxical and problematical aspects of the
mathematician's role. In particular, we want to display
clearly the discrepancy between the actual work and activ
ity of the mathematician and his own perception of his
work and activity.
The ideal mathematician's work is intelligible only to a
small group of specialists, numbering a few dozen or at
most a few hundred. This group has existed only for a few
decades, and there is every possibility that it may become
extinct in another few decades. However, the mathemati
cian regards his work as part of the very structure of the
world, containing truths which are valid forever, from the
beginning of time, even in the most remote corner of the
universe.
He rests his faith on rigorous proof; he believes that the
difference between a correct proof and an incorrect one is
an unmistakable and decisive difference. He can think of
no condemnation more damning than to say of a student,
`He doesn't even know what a proof is.' Yet he is able to
give no coherent explanation of what is meant by rigor, or
what is required to make a proof rigorous. In his own
work, the line between complete and incomplete proof is
always somewhat fuzzy, and often controversial.
To talk about the ideal mathematician at all, we must
have a name for his `field,' his subject. Let's call it, for in
stance, `nonRiemannian hypersquares.'
He is labeled by his field, by how much he publishes, and
especially by whose work he uses, and by whose taste he
follows in his choice of problems.
He studies objects whose existence is unsuspected by all
except a handful of his fellows. Indeed, if one who is not
an initiate asks him what he studies, he is incapable of
showing or telling what it is. It is necessary to go through
an arduous apprenticeship of several years to understand
the theory to which he is devoted. Only then would one's
mind be prepared to receive his explanation of what he is
studying. Short of that, one could be given a `definition,'
which would be so recondite as to defeat all attempts at
comprehension.
The objects which our mathematician studies were un
known before the twentieth century; most likely, they were
unknown even thirty years ago. Today they are the chief
interest in life for a few dozen (at most, a few hundred) of
his comrades. He and his comrades do not doubt, however,
that nonRiemannian hypersquares have a real existence as
definite and objective as that of the Rock of Gibraltar or
Halley's comet. In fact, the proof of the existence of non
Riemannian hypersquares is one of their main achieve
ments, whereas the existence of the Rock of Gibraltar is
very probable, but not rigorously proved.
It has never occurred to him to question what the word
`exist' means here. One could try to discover its meaning
by watching him at work and observing what the word
`exist' signifies operationally.
In any case, for him the nonRiemannian hypersquare
exists, and he pursues it with passionate devotion. He
spends all his days in contemplating it. His life is successful
to the extent that he can discover new facts about it.
He finds it difficult to establish meaningful conversation
with that large portion of humanity that has never heard of
a nonRiemannian hypersquare. This creates grave diffi
culties for him; there are two colleagues in his department
who know something about nonRiemannian hyper
squares, but one of them is on sabbatical, and the other is
much more interested in nonEulerian semirings. He goes
to conferences, and on summer visits to colleagues, to meet
people who talk his language, who can appreciate his work
and whose recognition, approval, and admiration are the
only meaningful rewards he can ever hope for.
At the conferences, the principal topic is usually `the de
cision problem', (or perhaps `the construction problem' or
`the classification problem') for nonRiemannian hyper
squares. This problem was first stated by Professor Name
less, the founder of the theory of nonRiemannian hyper
squares. It is important because Professor Nameless stated
it and gave a partial solution which, unfortunately, no one
but Professor Nameless was ever able to understand. Since
Professor Nameless'~ day, all the best nonRiemannian hy
persquarers have worked on the problem, obtaining many
partial results. Thus the problem has acquired great pres
tige.
Our hero often dreams he has solved it. He has twice
convinced himself during waking hours that he had solved
it but, both times, a gap in his reasoning was discovered by
other nonRiemannian devotees, and the problem remains
open. In the meantime, he continues to discover new and
interesting facts about the nonRiemannian hypersquares.
To his fellow experts, he communicates these results in a
casual shorthand. `If you apply a tangential mollifier to the
left quasimartingale, you can get an estimate better than
quadratic, so the convergence in the Bergstein theorem
turns out to be of the same order as the degree of approxi
mation in the Steinberg theorem.'
This breezy style is not to be found in his published
writings. There he piles up formalism on top of formalism.
Three pages of definitions are followed by seven lemmas
and, finally, a theorem whose hypotheses take half a page
to state, while its proof reduces essentially to `Apply
Lemmas 17 to definitions AH.'
His writing follows an unbreakable convention: to con
ceal any sign that the author or the intended reader is a
human being. It gives the impression that, from the stated
definitions, the desired results follow infallibly by a purely
mechanical procedure. In fact, no computing machine has
ever been built that could accept his definitions as inputs.
To read his proofs, one must be privy to a whole subcul
ture of motivations, standard arguments and examples,
habits of thought and agreedupon modes of reasoning.
The intended readers (all twelve of them) can decode the
formal presentation, detect the new idea hidden in lemma
4, ignore the routine and uninteresting calculations of
lemmas 1, 2, 3, 5, 6, 7, and see what the author is doing and
why he does it. But for the noninitiate, this is a cipher that
will never yield its secret. If (heaven forbid) the fraternity
of nonRiemannian hypersquarers should ever die out,
our hero's writings would become less translatable than
those of the Maya.
The difficulties of communication emerged vividly when
the ideal mathematician received a visit from a public in
formation officer of the University.
_P.I.O.:_ I appreciate your taking time to talk to me. Math
ematics was always my worst subject.
_I.M.:_ That's O.K. You've got your job to do.
_P.I.O.:_ I was given the assignment of writing a press re
lease about the renewal of your grant. The
usual thing would be a onesentence item, `Pro
fessor X received a grant of _Y_ dollars to con
tinue his research on the decision problem for
nonRiemannian hypersquares.' But I thought
it would be a good challenge for me to try and
give people a better idea about what your work
really involves. First of all, what is a hyper
square?
_I.M.:_ I hate to say this, but the truth is, if I told you what
it is, you would think I was trying to put you
down and make you feel stupid. The definition
is really somewhat technical, and it just wouldn't
mean anything at all to most people.
_P.I.O.:_ Would it be something engineers or physicists
would know about?
_I.M.:_ No. Well, maybe a few theoretical physicists. Very
few.
_P.I.O.:_ Even if you can't give me the real definition, can't
you give me some idea of the general nature
and purpose of your work?
_I.M.:_ All right, I'll try. Consider a smooth function _f_ on
a measure space "O" taking its value in a sheaf of
germs equipped with a convergence structure
of saturated type. In the simplest case...
_P.I.O.:_ Perhaps I'm asking the wrong questions. Can you
tell me something about the applications of
your research?
_I.M.:_ Applications?
_P.I.O.:_ Yes, applications.
_I.M.:_ I've been told that some attempts have been made
to use nonRiemannian hypersquares as models
for elementary particles in nuclear physics. I
don't know if any progress was made.
_P.I.O.:_ Have there been any major breakthroughs re
cently in your area? Any exciting new results
that people are talking about?
_I.M.:_ Sure, there's the SteinbergBergstein paper.
That's the biggest advance in at least five years.
_P.I.O.:_ What did they do?
_I.M.:_ I can't tell you.
_P.I.O.:_ I see. Do you feel there is adequate support in re
search in your field?
_I.M.:_ Adequate? It's hardly lip service. Some of the best
young people in the field are being denied re
search support. I have no doubt that with extra
support we could be making much more rapid
progress on the decision problem.
_P.I.O.:_ Do you see any way that the work in your area
could lead to anything that would be under
standable to the ordinary citizen of this
country?
_I.M.:_ No.
_P.I.O.:_ How about engineers or scientists?
_I.M.:_ I doubt it very much.
_P.I.O.:_ Among pure mathematicians, would the majority
be interested in or acquainted with your work?
_I.M.:_ No, it would be a small minority.
_P.I.O.:_ Is there anything at all that you would like to say
about your work?
_I.M.:_ Just the usual one sentence will be fine.
_P.I.O.:_ Don't you want the public to sympathize with your
work and support it?
_I.M.:_ Sure, but not if it means debasing myself.
_P.I.O.:_ Debasing yourself?
_I.M.:_ Getting involved in public relations gimmicks,
that sort of thing.
_P.I.O.:_ I see. Well, thanks again for your time.
_I.M.:_ That's O.K. You've got a job to do.
Well, a public relations officer. What can one expect?
Let's see how our ideal mathematician made out with a stu
dent who came to him with a strange question.
_Student:_ Sir, what is a mathematical proof?
_I.M.:_ You don't know that? What year are you in?
_Student:_ Thirdyear graduate.
_I.M.:_ Incredible! A proof is what you've been watching
me do at the board three times a week for
three years! That's what a proof is.
_Student:_ Sorry, sir, I should have explained. I'm in philos
ophy, not math. I've never taken your course.
_I.M.:_ Oh! Well, in that caseyou have taken _some_
math, haven't you? You know the proof of the
fundamental theorem of calculusor the fun
damental theorem of algebra?
_Student:_ I've seen arguments in geometry and algebra
and calculus that were called proofs. What I'm
asking you for isn't _examples_ of proof, it's a def
inition of proof. Otherwise, how can I tell what
examples are correct?
_I.M.:_ Well, this whole thing was cleared up by the logi
cian Tarski, I guess, and some others, maybe
Russell or Peano. Anyhow, what you do is, you
write down the axioms of your theory in a for
mal language with a given list of symbols or al
phabet. Then you write down the hypothesis
of your theorem in the same symbolism. Then
you show that you can transform the hypoth
esis step by step, using the rules of logic, till
you get the conclusion. That's a proof.
_Student:_ Really? That's amazing! I've taken elementary
and advanced calculus, basic algebra, and to
pology, and I've never seen that done.
_I.M.:_ Oh, of course no one ever really _does_ it. It would
take forever! You just show that you could do
it, that's sufficient.
_Student:_ But even that doesn't sound like what was done
in my courses and textbooks. So mathemati
cians don't really do proofs, after all.
_I.M.:_ Of course we do! If a theorem isn't proved, it's
nothing.
_Student:_ Then what is a proof? If it's this thing with a for
mal language and transforming formulas, no
body ever proves anything. Do you have to
know all about formal languages and formal
logic before you can do a mathematical proof?
_I.M.:_ Of course not! The less you know, the better.
That stuff is all abstract nonsense anyway.
_Student:_ Then really what _is_ a proof?
_I.M.:_ Well, it's an argument that convinces someone
who knows the subject.
_Student:_ Someone who knows the subject? Then the defi
nition of proof is subjective; it depends on par
ticular persons. Before I can decide if some
thing is a proof, I have to decide who the
experts are. What does that have to do with
proving things?
_I.M.:_ No, no. There's nothing subjective about it!
Everybody knows what a proof is. Just read
some books, take courses from a competent
mathematician, and you'll catch on.
_Student:_ Are you sure?
_I.M.:_ Wellit is possible that you won't, if you don't
have any aptitude for it. That can happen, too.
_Student:_ Then _you_ decide what a proof is, and if I don't
learn to decide in the same way, you decide I
don't have any aptitude.
_I.M.:_ If not me, then who?
Then the ideal mathematician met a positivist philoso
pher.
_P.P.:_ This Platonism of yours is rather incredible. The
silliest undergraduate knows enough not to mul
tiply entities, and here you've got not just a hand
ful, you've got them in uncountable infinities!
And nobody knows about them but you and your
pals! Who do you think you're kidding?
_I.M.:_ I'm not interested in philosophy, I'm a mathema
tician.
_P.P.:_ You're as bad as that character in Molire who
didn't know he was talking prose! You've been
committing philosophical nonsense with your
`rigorous proofs of existence.' Don't you know
that what exists has to be observed, or at least ob
servable?
_I.M.:_ Look, I don't have time to get into philosphical con
troversies. Frankly, I doubt that you people know
what you're talking about; otherwise you could
state it in a precise form so that I could under
stand it and check your argument. As far as my
being a Platonist, that's just a handy figure of
speech. I never thought hypersquares existed.
When I say they do, all I mean is that the axioms
for a hypersquare possess a model. In other
words, no formal contradiction can be deduced
from them, and so, in the normal mathematical
fashion, we are free to postulate their existence.
The whole thing doesn't really mean anything,
it's just a game, like chess, that we play with
axioms and rules of inference.
_P.P.:_ Well, I didn't mean to be too hard on you. I'm sure
it helps you in your research to imagine you're
talking about something real.
_I.M.:_ I'm not a philosopher, philosophy bores me. You
argue, argue and never get anywhere. My job is
to prove theorems, not to worry about what they
mean.
The ideal mathematician feels prepared, if the occasion
should arise, to meet an extragalactic intelligence. His first
effort to communicate would be to write down (or other
wise transmit) the first few hundred digits in the binary ex
pansion of pi. He regards it as obvious that any intelligence
capable of intergalactic communication would be mathe
matical and that it makes sense to talk about mathematical
intelligence apart from the thoughts and actions of human
beings. Moreover, he regards it as obvious that binary rep
resentation and the real number pi are both part of the in
trinsic order of the universe.
He will admit that neither of them is a natural object, but
he will insist that they are discovered, not invented. Their
discovery, in something like the form in which we know
them, is inevitable if one rises far enough above the pri
mordial slime to communicate with other galaxies (or even
with other solar systems).
The following dialogue once took place between the
ideal mathematician and a skeptical classicist.
_S.C.:_ You believe in your numbers and curves just as
Christian missionaries believed in their crucifixes.
If a missionary had gone to the moon in 1500, he
would have been waving his crucifix to show the
moonmen that he was a Christian, and expecting
them to have their own symbol to wave back.^"*"^

 ^"*"^ Cf.~ the description of Coronado's expedition to Cibola, in 1540:
 `... there were about eighty horsemen in the vanguard besides
 twentyfive or thirty foot and a large number of Indian allies. In the
 party went all the priests, since none of them wished to remain behind
 with the army. It was their part to deal with the friendly Indians whom
 they might encounter, and they especially were bearers of the Cross, a
 symbol which ... had already come to exert an influence over the na
 tives on the way' (H.~ E.~ Bolton, Coronado, University of New Mexico
 Press, 1949).

You're even more arrogant about your expansion
of pi.
_I.M.:_ Arrogant? It's been checked and rechecked, to
100,000 places!
_S.C.:_ I've seen how little you have to say even to an Amer
ican mathematician who doesn't know your game
with hypersquares. You don't get to first base try
ing to communicate with a theoretical physicist;
you can't read his papers any more than he can
read yours. The research papers in your own
field written before 1910 are as dead to you as
Tutankhamen's will. What reason in the world is
there to think that you could communicate with
an extragalactic intelligence?
_I.M.:_ If not me, then who else?
_S.C.:_ Anybody else! Wouldn't life and death, love and
hate, joy and despair be messages more likely to
be universal than a dry pedantic formula that no
body but you and a few hundred of your type will
know from a henscratch in a farmyard?
_I.M.:_ The reason that my formulas are appropriate for
intergalactic communication is the same reason
they are not very suitable for terrestrial commu
nication. Their content is not earthbound. It is
free of the specifically human.
_S.C.:_ I don't suppose the missionary would have said
quite that about his crucifix, but probably some
thing rather close, and certainly no less absurd
and pretentious.
The foregoing sketches are not meant to be malicious;
indeed, they would apply to the present authors. But it is a
too obvious and therefore easily forgotten fact that mathe
matical work, which, no doubt as a result of long familiar
ity, the mathematician takes for granted, is a mysterious,
almost inexplicable phenomenon from the point of view of
the outsider. In this case, the outsider could be a layman, a
fellow academic, or even a scientist who uses mathematics
in his own work.
The mathematician usually assumes that his own view of
himself is the only one that need be considered. Would we
allow the same claim to any other esoteric fraternity? Or
would a dispassionate description of its activities by an ob
servant, informed outsider be more reliable than that of a
participant who may be incapable of noticing, not to say
questioning, the beliefs of his coterie?
Mathematicians know that they are studying an objective
reality. To an outsider, they seem to be engaged in an eso
teric communion with themselves and a small clique of
friends. How could we as mathematicians prove to a skepti
cal outsider that our theorems have meaning in the world
outside our own fraternity?
If such a person accepts our discipline, and goes through
two or three years of graduate study in mathematics, he ab
sorbs our way of thinking, and is no longer the critical out
sider he once was. In the same way, a critic of Scientology
who underwent several years of `study' under `recognized
authorities' in Scientology might well emerge a believer in
stead of a critic.
If the student is unable to absorb our way of thinking, we
flunk him out, of course. If he gets through our obstacle
course and then decides that our arguments are unclear or
incorrect, we dismiss him as a crank, crackpot, or misfit.
Of course, none of this proves that we are not correct in
our selfperception that we have a reliable method for dis
covering objective truths. But we must pause to realize
that, outside our coterie, much of what we do is incompre
hensible. There is no way we could convince a selfconfi
dent skeptic that the things we are talking about make
sense, let alone `exist.'