Dialogue between Barry Mazur & Peter Pesic

On mathematics, imagination
& the beauty of numbers

peter pesic: Many intelligent people
only see in mathematics a wasteland of
dreary formalism, a mind-numbing ex-
panse of theorems and proofs expressed
in very abstract language. Doubtless this
is partly due to the way it is taught, mais
such teaching is widespread, the product
of good intentions and much effort. Le
disconnection between the inner, lived
world of mathematicians and the main-
stream of intelligent people is very deep,
despite the sensual character of mathe-
matics that you describe so well in your
recent book, Imagining Numbers: (particu-

Barry Mazur, a Fellow of the American Academy
depuis 1978, is the Gerhard Gade University Profes-
sor at Harvard University, where he has taught for
over four decades. He has done research in many
aspects of pure mathematics and is the author of
“Imagining Numbers: (particularly the square
root of minus ½fteen)» (2003).

Peter Pesic is a tutor and musician-in-residence at
St. John’s College in Santa Fe, New Mexico. He is
the author of “Labyrinth: A Search for the Hid-
den Meaning of Science” (2000), “Seeing Dou-
ble: Shared Identities in Physics, Philosophy, et
Literature” (2001), “Abel’s Proof: An Essay on
the Sources and Meaning of Mathematical Un-
solvability” (2003), and “Sky in a Bottle”
(2005).

larly the square root of minus ½fteen). Ce
raises a hard question: How–if at all–
can this living world of mathematics
become accessible?

barry mazur: I can’t answer that
question, but I can offer some com-
ments. A person’s ½rst steps in his or
her mathematical development are ex-
ceedingly important. Early education
deserves our efforts and ingenuity. Mais
also here is a message to any older per-
son who has never given a thought to
mathematics or science during their
school days or afterwards: You may be
ready to start. Starting can be intellectu-
ally thrilling, and there are quite a few
old classics written in just the right style
to accompany you as you begin to take
your ½rst steps in mathematics. I’m
thinking, Par exemple, of the old T. C.
Mits series, or Tobias Dantizg’s wonder-
ful Number: The Language of Science, ou
Lancelot Hogben’s Mathematics for the
Millions. De plus, one should not be
dismayed that there are many steps–
there is no need to take them all. Just
enjoy each one you do take.

Bill Thurston, a great geometer, uses
the word ‘tall’ to describe mathematics:
math is a tall subject in the sense that
skyscrapers are tall. C'est, one piece of
mathematics lies on top of a prior piece
of mathematics and lies under the next
piece of mathematics, etc.. To get to the

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½ftieth story you must traverse all the
prior forty-nine, and in the right order. je
like this image, but would want to insist
that it may be more of a Gaudi-esque
structure, with a wide choice of alternate
staircases joining and crossing so if you
are ever uncomfortable with one route
–if the risers are too high, or not high
enough–there are other, more accom-
modating stairwells. And besides, même
the view from the ½rst story is a marvel.

pp: What is your earliest memory of
mathematics?

bm: The very earliest was when I was
seven or eight years old. My father, OMS
was always fascinated with numbers,
would shower me with arithmetical
queries like, What is the number that
when you double it and add one gives
you eleven? I don’t think I was particu-
larly adept at ½nding the answers to
these problems, but I did love them. My
method was, bien sûr, trial and error.
Alors, after an especially long barrage of
such queries, my father smiled at me and
said, “I’ll tell you a secret. Here is how
you can do these problems more quick-
ly.” The “secret” he imparted to me was
to invoke the magical X of algebra, concernant-
state the problem in the language of al-
gebra, and then to simplify the algebraic
sentence, where by simplify he meant
solve for X. That X became, after sim-
pli½cation, an actual number, which as-
tonished me. My father also insisted on
the ritual palindrome of analysis and
synthesis, in the sense that once the val-
ue of X was found, I was to redo the steps
of the derivation in reverse order to
check that the number I came up with
for X really worked.

I suppose that I had an especially liter-
al mind, for I actually did think that this
information was some sort of secret. UN
family secret, peut-être, as there might

be family secret recipes for particular
dishes. I remember being stunned a few
years later in a math class, for somehow
the teacher had gotten wind of this se-
cret and seemed to be in the process of
explaining it to the entire class.

pp: As you confronted this secret, comment
did it act on you, and especially on your
imagination?

bm: I think it acted more on my sense of
wonder than on any concrete imagin-
ings.

To work out those simple queries (par exemple.,

What is that number which when you
double it and add one you get eleven?) est
rather like seeing a concrete visual image
develop out of a blank nothing on photo-
graphic paper in a darkroom tray. You
start with something you deemed X, et
at the end of the process you discover X
to be concrete, some particular number.
There is a sense of power in this (as you
and I know, the early algebraists were
very aware of this unexpected power).
What could be more enticing than hav-
ing this power be ‘secret’ as well? Quand
I realized this was part of a much larger
common heritage, I wasn’t sad: it made
it that much larger a clubhouse. My early
fascination was that out of pure thought,
starting with nothing, something con-
crete emerges. I remember, a good deal
plus tard, being still struck by the equation
nothing + thought = something.

About a year before high school, I be-

came an avid reader of popular books
about electronics and math. When I was
building radio receivers (maladroitly, pour
the most part) I had the idea of deriving
Maxwell’s equations by pure thought.
How this was going to be accomplished
was not so clear, for it is too simple-
minded to imagine that some Saint
Anselmian strategy (making the sole
assumption, Par exemple, that the laws

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Dædalus Spring 2005

125

Dialogue
entre
Barry
Mazur &
Peter Pesic

governing radio transmission were the
“most perfect laws”) would lead to laws
none other than Maxwell’s. But when I
was in high school I had complete faith
that such a derivation was possible.

pp: In those early days, to what extent
was your access to mathematics mediat-
ed through physical devices like radios
or through visualization, as in electronic
schematics? I am thinking of Einstein’s
insistence that he was always primarily
a visual thinker, not an abstract one.

bm: Let me respond to this question go-
ing from the back to the front. I don’t
think I ever deal with things that are ab-
stract. To be more explicit, I don’t like
the word ‘abstract’ except as a compar-
ative term, even though lots of mathe-
maticians use it in a way that reminds
me of the dangling comparatives that
sometimes show up in ordinary speech.
Texts and courses have titles like Abstract
Algebra, etc.; my impulse when I see
these is to wonder, “Abstract compared
to what?” To put it another way, think
of the tactic of taking a concept that has
arisen in one context and then stripping
it from that particular context. Pour dans-
position, start with Euclidean geometry in
the full expression of its geometric intu-
ition and with all its axioms, then strip a
few axioms from the list and consider
the structure that ensues, with either no
concrete realization in mind, or at least
as an entity of thought separate from any
geometric realization. This is a situation
where I believe it is helpful to say that
one has abstracted a structure, separat-
ing it from its habitual concrete and vi-
sualizable context.

Even so, I’m hesitant to use such a Lat-
inate word as ‘abstract’ for this mode of
thought. Aristotle, Par exemple, has at
least two different ways of referring to
the activity of abstraction: He employs

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the verb aphairein, which indeed means
to ‘abstract,’ to ‘take away.’ But at times,
as in Book 13 of the Metaphysics, he em-
ploys the more explanatory phrase “to
take that which does not exist in separa-
tion and consider it separately”–a de-
scription that has, to my mind, a less
scary aspect. But once the concept has
been, as people say, abstracted, or once
it is, as Aristotle would say, taken sepa-
rately, if one is to deal effectively with it,
one must floodlight it with intuitions of
some sort or other. If one is really think-
ing about this ‘abstracted’ concept and
working with it seriously, it will become
utterly as concrete as any other concept.
Bien sûr, one may have to homegrow
the appropriate intuitions to deal seri-
ously with it.

Electronics, or at least circuitry at the
primitive level that I used as a kid, était
saturated with concretizing analogies.
As I’m sure you know, Kirchhoff’s law
and Ohm’s law are made vivid by a sim-
ple analogy to hydraulics–plumbing, si
you wish. And Maxwell, when he sought
to give vocabulary for the energy in elec-
tromagnetic ½elds, went surprisingly
further with this analogy: the somewhat
mysterious displacement current that he
denoted j (and that seemed so wonder-
ful to me when I ½rst encountered it)–a
marvelous concretization of ‘action at a
distance.’

pp: What you are saying here is consis-
tent with what you write in Imagining
Numbers, where you seek felt correlates
for an ‘abstract’ concept like i = √-1. Mais
now you are extending this view in a dar-
ing way. What happens then in ‘abstract’
thinking on the level that you and other
mathematicians pursue it, dans lequel (à
least for many intelligent people) là
seems to be no trace of any sensual, con-
crete content?

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bm: I think that analogy is a powerful
tool, and it can extend, inde½nitely, le
range of what we are happy to call con-
crete, or sensual. Let us start with the
truism that the stock-in-trade of poets
is to concretize things by analogy. Any
snatch of poetry offers some illustration
of this. Consider, Par exemple, ces
lines of Yeats: “Like a long-legged fly
upon the stream / His mind moves upon
silence.” Here the equation is between
something that is concrete/sensual and
external (the “long-legged fly upon the
stream”) and something that might ac-
tually be even more intimately connect-
ed to us, but much harder to catch and
hold still: a curious interior state.

Mathematicians are constantly using

analogy to expand the realm of what
they hold to be concrete. The ubiquitous
activity of generalizing, which is one of
the staples of mathematical and scien-
ti½c progress, is a way of analogizing. Nous
start with a structure or concept we feel
at home with (say, multiplication of or-
dinary numbers), and we see a broader
realm for which the same or at least an
analogous structure or concept may pos-
sibly make sense (say, think of composi-
tion of transformations as a kind of mul-
tiplication operation). We make our-
selves at home with this more general
concept, initially at least, by depending
heavily on the analogy it has with the
more familiar, less general concept.

One genre of analogizing in mathe-
matics is to deal with a problem that at
½rst does not seem to be geometric by
recasting it in geometric language. Pour
example, consider Fredholm’s idea for
½nding the (unique) function that is the
solution of a certain type of equation by
translating the problem to that of ½nd-
ing a ½xed point of a certain distance-
shrinking transformation on a geometric
espace.

pp: In your view, is there, alors, any part
of mathematics that is radically divorced
from sensual intuition? What about
number theory, where there is no geo-
metric, hence visual, ½eld, at least at ½rst
glance?

bm: I don’t think there is any mathe-
matics radically divorced from some
kind of vivid intuition that illuminates
it and ties it to the sensual.

You say that number theory has no
geometric, hence visual, ½eld at ½rst
glance–but that is only at ½rst glance.
For most practitioners of number theory
these days, number theory is intensely
geometric. In the late 1950s and early
1960s I was a geometer, a topologist, et
the hook that got me fascinated with
number theory was to understand that
the set of integers

. . . -3, -2, -1, 0, +1, +2, +3 . . .

has properties closely analogous to the
three-dimensional sphere. Strange as it
may seem, the prime numbers are analo-
gous to knots (closed non-self-intersect-
ing loops) in the three-dimensional
sphère. Once you see this analogy you
begin to see deeply instructive parallels
between geometry and number theory.
Par exemple, the skew symmetry of
the linking number of one knot relative
to another is somehow formally related
to what is known in number theory as
quadratic reciprocity (a deep reciprocal
relationship, initially discovered by
Gauss, that holds between any two
prime numbers). This is hardly the only
analogy that ties number theory to ge-
ometry–there are so many that very
often it is hard to classify a theorem as
being in the one ½eld or the other.

The connections here began as far
back as in the works of Abel, in that
Galois theory itself sits–ambiguous-
ly–between geometry (the study of

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Barry
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Peter Pesic

½nite coverings of spaces) on the one
main, and algebra (the study of solutions
of polynomial equations). This relation-
ship was thoroughly understood by Kro-
necker and Weber over a hundred years
ago. The mathematical discipline of al-
gebraic geometry already expresses the
ineluctable joining of these ½elds.

Sixty years ago, André Weil dreamt up
a striking way of very tightly controlling
and counting the number of solutions of
systems of polynomial equations over
½nite ½elds (this being a quintessentially
number-theoretic problem) by surmis-
ing that there must be a tool for number
theory closely analogous to the basic to-
pological theory that ef½ciently counts
the numbers of intersections that one
geometric subspace has with another
subspace when both are contained with-
in a larger ambient space. All this appa-
ratus has now been set up and establish-
es a vivid geometric mode of under-
standing polynomials and systems of
polynomials in any algebraic context;
en effet, much of number theory is now
very comfortably viewed as a piece of a
smooth-working synthesis, usually re-
ferred to as arithmetic algebraic geom-
etry.

This long-winded answer, alors, is sim-
ply to say that to many current research-
ers, number theory is inseparable from
geometry, and much mathematical work
occurs in a realm that is–marvelously–
a synthesis of the two.

pp: But what about considerations in-
volving higher dimensions than the
three of common spatial experience?
Must we rely on analogies to that com-
mon world? To what extent would that
be possible without, peut-être, deluding
ourselves that we are really understand-
ing those more complex spaces, pas
just squashing them to ½t our limited
senses?

bm: But I think we are squashing them,
and slicing them, to ½t our limited sens-
es–or at least to ½t the limits that our
senses are constrained to at present. Et
squashing them is a prelude to under-
standing them. Without some real in-
novation, real insight, and exercise of
imagination, you don’t even know how
to begin any squashing procedure.

Squash how? Any act of squashing
takes work, and the work itself is what
expands one’s intuition–expands the
limits of our senses. Let me remind you
of some standard examples. The most
immediate source of examples does not
come from high dimensions, but is al-
ready in our three-dimensional space of
common experience. To visualize things
well in three dimensions takes some ar-
ti½ce. Think of the repertoire: the top
voir, side view, front view, etc., of archi-
tectural drawings; the Mercator and
other projections to render the globe
flat; the cat scans and mris that make
pictures of slices of three-dimensional
bodies, these slice pictures being taken
in various moving and rotating planes
and then cleverly put together to render
a more faithful understanding of the full
three-dimensionality of the examined
body. Or think, if you wish, about that
chair you are looking at, which you have
only one view of (give or take a bit of the
parallax of your two eyes and your mov-
ing head), and whose utter three-dimen-
sionality you are so at home with.

In a way, all the arti½ces, as I called
eux, which work so well for us to sub-
stantiate our common three-dimension-
al experience, are there to be employed
to bring higher dimensions into our ken
aussi. The special theory of relativity
deals with four coordinates (X,oui,z,t)
usually referred to as ‘space-time,’ and
the usual way of thinking graphically
about anything happening in this four-
dimensional geometry is as a movie of

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three-dimensional slices changing in
temps.

But there are other modes of squash-
ing the thing down to our limited senses,
thereby, in effect, extending those
senses. Par exemple, one might envision
the four-dimensional space as a planar
(c'est à dire., two-parameter) family of planes
que, taken all together, ½ll out four-
dimensional space: every point in four-
dimensional space will lie on exactly one
of these two-dimensional slices. The fun
here is that you need a two-dimensional
collection of these two-dimensional
slices to sweep out the entirety of four-
dimensional space: 2 + 2 = 4, après tout.
You then have the option of thinking of
(or visualizing, if you wish) any geomet-
ric object in four-dimensional space in
terms of how it is diced by this proce-
dure. This type of intuition is very well
developed in people who do complex
analyse.

Even this list understates the issue.
One isn’t quite ½nished if I just give you
a ½nite repertoire–a bag of tricks, so to
speak, in the art of squashing–because
at a point in one’s development of these
intuitions, one actually sees more than
the mere sum of tricks. One realizes that
there is a certain unexpected pliability of
spatial intuitions that makes spaces of
any dimension equally accessible–
equally accessible, and in certain re-
spects (and here’s a surprise) more eas-
ily accessible than lower-dimensional
les espaces. Topologists understand very well
that for certain important work, higher-
dimensional spaces are simply easier
than lower-dimensional spaces–there’s
more room to move around!

Par exemple, the Poincaré conjecture

was ½rst proved by Steve Smale in the
mid-1960s in dimensions equal to ½ve or
plus. It took well over a decade after
that for it to be proved by Michael Fried-
man in dimension four. Dimension three

is still open, although a Russian mathe-
matician, Perelman, has recently an-
nounced that he has a proof. The short
answer here is that one will always try to
reach out as far as one can with whatever
intuition one has and squash as much
as one can into it. Doing this squashing
has the effect of extending and improv-
ing our intuition.

pp: Are there no spaces that are utterly
alien to our intuition, only available
through a kind of reasoning that is not
accessible to our senses?

bm: I want to think of our intuition as
not an inert, unchanging resource, mais
rather as something that can expand
when challenged, when exercised. Et
the mechanism that forces this expan-
sion is analogy. Donc, are there spaces ut-
terly alien to our intuition? All I can say
is that I don’t think utterly.

pp: But the very struggle of human
imagination to extend itself so far past
its common limits indicates that these
spaces really may transcend our sensibil-
ville. We struggle to grasp new mathemat-
ical truths not just because it is hard to
visualize them, but also because they de-
fy our most deeply held presumptions.
Par exemple, we try to visualize the
in½nite-dimensional Hilbert space of
quantum theory using visual analogues,
but a spinning ball is utterly unlike an
electron with spin. At a certain point,
doesn’t the visual and anthropomorphic
fail just because we have gone beyond
what we can visualize? And doesn’t
symbolic mathematics then save us by
allowing us to reason securely even
when we can no longer see?

bm: What you say is unassailable. Mais
the full panoply of our mathematical
intuitions–the intuitions that mathe-

Mathematics,
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& the beauty
of numbers

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Dædalus Spring 2005

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Barry
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Peter Pesic

try–the Euclidean geometry of our high
school days–become magically avail-
able even in contexts where we would
hardly have dared to imagine that visual-
ization would have any relevance.

The example you offer of electrons
with spin modeled in terms of Hilbert
space is a great testimonial, precisely, à
the manner in which visualization as an
intuition can be ampli½ed and made
more powerful by mathematical analo-
gies. Our comprehension of Euclidean
geometry is ampli½ed, thanks to Hilbert,
to be a useful thing in understanding
even the most seemingly unvisual as-
pects of atomic particles.

pp: Here you point to new possibilities
that would surprise many people who
consider themselves mathematically
hopeless. Perhaps they think themselves
incapable of abstraction or manipulating
formalism. You are telling them that, sur
the contrary, it is the sensual side they
are missing.

bm: Living mathematics is in no way
abstract, at least to the people who live
it. Intuitions can tie mathematics to the
most concrete pictures, sensual experi-
ences, and things that are immediate to
all of us. There are always loads of alter-
nate routes. If you are blocked at one
route, no problem–try another. I believe
this is the common understanding of
just about everyone who practices math-
ematics. Mathematics is often taught
without such connections, but there is
no reason that it can’t be taught so that
a student’s intuitions are fully engaged
and exercised every step of the way.

matics helps strengthen, and re½nes–is
not limited to pure visualization or pure
symbolic combinatorial processes or, pour
that matter, pure any one thing.

I would say that the most powerful of
our intuitions are combinations: a po-
tent blending of visualization and artful
algebra, of intrinsic canniness of estima-
tion, and of all the intuitions that are the
children of sheer experience–knowing
when to approximate, when to insist on
exact calculations, when to neglect some
termes, when to pay the closest attention
to them, when to rely upon an analogy,
when to distrust it, et . . . well, Je voudrais
not want to limit this list.

But there are two things I would like to

emphasize about the ingredients of the
brew I just described. The ½rst is that
these intuitions tend to amplify, to mag-
nify, l'un l'autre. The second is that these
intuitions show up and are, in some
formulaire, perfectly available to anyone who
tries their hand at understanding any
piece of mathematics, however elemen-
tary.

Now let’s return to your example of
the in½nite-dimensional Hilbert space
that provides a model for quantum
mechanical considerations. I think the
notion of in½nite-dimensional Hilbert
space is a wonderful example of how
algebra ampli½es the range of visualiz-
ability of geometry. A Hilbert space is,
almost by formal de½nition, a space, de
never mind how many dimensions, tel
that any two-dimensional plane in it has
all the properties of the Euclidean plane.
You can think of it as being very, very
visualizable in two-dimensional slices
despite its immensity, along with a
guarantee from Hilbert that this very
feature of it–visualizability in slices–
is what is going to be most relevant.

Now once we (or initially, I suppose,

Hilbert) hit upon this idea, our basic
intuitions regarding Euclidean geome-

130

Dædalus Spring 2005

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