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\centerline{\bf Confessions of an Apostate Mathematician}\medskip
\centerline{by Edward Nelson}
\centerline{Department of Mathematics}
\centerline{Princeton University}
\centerline{\eightrm http:$\scriptstyle//$www.math.princeton.edu$\scriptstyle/\scriptscriptstyle\sim$nelson$\scriptstyle/$papers.html}
\bigskip
Midway between Moses and Jesus there appeared a figure of like
char\-isma: golden-thighed Pythagoras. Son of a Greek mother and
a Phoenician father, he spent years in Egypt, endured a period of
captivity in Babylon, and founded the Pythagorean Brotherhood in
southern Italy. The Brotherhood was
something rare for its time, and alas!~for ours: a true
Brotherhood and Sisterhood. The legends surrounding Pythagoras
differ wildly, but all attest to an extraordinary radiance of face
and person.
To develop my thesis, I must speak of the mathematics of
antiquity. My knowledge in this field is second-hand and
superficial. There is no following ``but'': I am simply admitting
at the outset a weakness of the presentation.
There was no mathematics before Pythagoras, just as there was
no Christianity before Jesus. The story that the Egyptians
discovered geometry because the annual flooding of the Nile
washed away boundary marks is in a class with the story of
Newton and the apple. The Babylonians were wizards at computation,
they had at least one of the two ways of generating what we now
call Pythagorean triples (as Otto Neugebauer brilliantly
demonstrated from the Plimpton cuneiform tablets), they could
approximate the square root of two to as many sexagesimal
places as desired, but the question of the rationality or
irrationality of the square root of two
simply would have made no sense to them, because
they had no numbers in their conceptual framework. This is not
surprising, because numbers did not yet exist in those days.
Numbers were invented (or revealed, as believers would maintain)
by Pythagoras. Numbers are divine, the only true divinity, the source
of all that is in the world, holy, to be worshiped and glorified.
Such is the Pythagorean religion, and such is the origin
of mathematics. This is the religion from which I am apostate.
What an extraordinary religion it is! Had it not been for Pythagoras,
would it ever have entered anyone's head that it is a good and
normal thing to have communities of people think deeply about
numbers and magnitudes, to develop an unimaginably complex web
of formal deductions -- a web of reasoning covering
the globe and spanning millennia?\bigskip\goodbreak
{\noindent\bf Eudoxus}\medskip
Plato traveled all the way to Sicily to obtain a
Pythagorean manu\-script. Plato was a dreadful fellow, the source
of a persistent evil from which the world has not yet
been liberated, but he was the teacher of Eudoxus. When I was
in the Athenian Agora, thoughts of Socrates, Plato, and
Aristotle, of St. Paul, of the dramatists, soon faded in the
realization that I was walking where Eudoxus once walked!
Eudoxus was the greatest mathematician that ever lived. It was
he who invented the method of exhaustion (definite integrals)
that Archimedes developed so successfully; it was he who enunciated
the subtle principle we call the Archimedean axiom (but which
Archimedes himself attributes to Eudoxus); he did
extremely intricate work on the apparent motion of the planets
and he founded mathematical physics with a work on dynamics.
Not one of his manuscripts survives.
We have not yet spoken of the deepest invention of Eudoxus. To their
consternation, the Pythagoreans discovered that the diagonal of
a square has no ratio, of numbers, to the side: the square root of two
is irrational, with all the emotional and intellectual connotations
of that word. This was a true crisis in religion. If numbers are
the only divinity, the source of all, how can this be? Must we
admit magnitudes into the pantheon as well? Seemingly so, for the
late Pythagorean description of the quadrivium was this:
\bigskip
\halign{\qquad\qquad#\hfil&\quad#\hfil\cr
arithmetic:& numbers at rest\cr
music:& numbers in motion\cr
geometry:& magnitudes at rest\cr
astronomy:& magnitudes in motion\cr
}\bigskip
But Eudoxus eliminated the dualism of number and magnitude. His
idea was this: rather than say what the ratio of two magnitudes
{\it is}, it suffices to define a notion of {\it two\/}
such (possibly nonexistent) ratios {\it being equal}, and this he did
by a subtle quantification over all the Pythagorean numbers.
Despite the fact that it was in the {\it Elements\/} for all
to read, this construction of the real number system was not
understood -- not even by Galilei. It was
just last century that the notion was re-invented by Richard Dedekind.
I know of no parallel to this in the history of human thought.
Eudoxus defined a notion of two things being equal in order to
construct the things themselves.
Here was a triumph of formalism, a victory of syntax over
semantics!\bigskip\goodbreak
{\noindent\bf The origin of numbers}\medskip
From the perspective of monotheistic faith, we reject the religious
idea of numbers as divine and uncreated. What then are they? Are
they created? I have written elsewhere:\medskip
{\narrower\ninerm\baselineskip=12pt
The famous saying by Kronecker that God created the numbers, all else
is the work of Man, presumably was not meant to be taken seriously.
Nowhere in the book of Genesis do we find the passage: And God said,
let there be numbers, and there were numbers; odd and even created
he them, and he said unto them, be fruitful and multiply;
and he commanded them to keep the laws of induction.
}\medskip
Everything in creation is contingent; every created thing is
dependent on the will of the Creator for its being.
If numbers are uncreated,
they are divine -- this we reject. If numbers are created, they
are contingent -- this we find absurd.
What other possibility is there? Simply that numbers do not exist --
not until human beings make them. Despite the assertion of William
Butler Yeats that ``Things out of perfection sail,'' very few would
maintain that the poem of which that is the first line existed
before Yeats made it. Why do we mathematicians, makers like
poets and musicians, describe what we do as discovery rather than
invention? This is Pythagorean religion.
What would the world be like if God had not
created numbers? Just what it is like now. There is not a shred of
evidence that the numbers have been created. If I give you an
addition problem like
$$\eqalign{
&37460225182244100253734521345623457115604427833\cr
+\ &52328763514530238412154321543225430143254061105\cr
&\overline{\phantom{52328763514530238412154321543215432143254061105}}\cr
}$$
\vskip-12pt\noindent
and you are the first to solve it, you will have created a number
that did not exist previously.
But this invention will not create a stir in the mathematical
community; it is not the kind of number that we are primarily
interested in. More interesting numbers are the $n$, $a$, $b$,
$c$, and~$d$ in Lagrange's theorem:
$$\forall n \,\exists a \,\exists b \,\exists c \,\exists d
\,[\,n=a^2+b^2+c^2+d^2\,].$$
This may be read: for all~$n$ there exist $a$, $b$, $c$, and~$d$
such that $n$~is equal to $a$ times~$a$ plus $b$ times~$b$
plus $c$ times~$c$ plus $d$ times~$d$ -- that is, every number is
the sum of four squares. What does this mean?
Various meanings have been ascribed to it. My father, whose work at
the Young Men's Christian Association
(YMCA) in Piazza Indipendenza first brought me to Rome in 1938
(when I went to first grade in a school near the Piazza Pitagora),
used while driving to look at the number on the license plate of the
car ahead of him and mentally find $a$, $b$, $c$, and~$d$.
So this is one
meaning: a pattern of computational challenges. But the formula
asserts the existence for {\it all\/}~$n$.
\bigskip\goodbreak
{\noindent\bf A first view of mathematics: realism}\medskip
Traditionally, from
the time of the Pythagorean brotherhood to the present, it meant
that for every natural number
$$n=0,\ 1,\ 2,\ 3,\ 4,\ 5,\ \ldots$$
there exist natural numbers $a$, $b$, $c$, and~$d$ such that
$n$~is equal to the sum of their squares. Until the beginning
of this century,
with the incisive intuitionistic criticism of L.~E.~J.~Brouwer,
this was accepted by almost everyone as having a
clear and definite meaning. Mathematics based on this concept
is called {\it classical}. Many proofs in classical mathematics
demonstrate that an object having a certain property
exists without offering any means to construct such an object.
One demonstrates that the assumption that all objects fail to have
the property is untenable -- it leads to a contradiction -- and
thereby concludes that there must exist an object having the property.
Thus classical mathematics is founded on the picture of mathematical
objects eternally existing in Platonic -- or better, Pythagorean --
reality.\bigskip\goodbreak
{\noindent\bf A second view of mathematics: intuitionism}\medskip
This picture was vigorously attacked by Brouwer in the early part of
this century, as being devoid of meaning. Intuitionism is a form of
constructivism; to say that a mathematical object exists, for
Brouwer, means that one knows how to construct it.
For the four
square formula, there was no problem for Brouwer: given~$n$,
the $a$, $b$, $c$, and~$d$ must be smaller than~$n$, so there is
only a finite set to search through.
This arrogant phrase, ``only a
finite set,'' is widely used by mathematicians, and I have been
amused at mathematics parties to hear spouses of mathematicians
employ it too. At the end of this century, with the advent of digital
computers -- which are having a profound influence, whether we
like it or not (I do), on mathematics -- one may object that this
is not a feasible search: one can write down a number~$n$ on a sheet
of paper such that no computer, from now until the Big Crunch,
or whatever other end is in store for the universe, will ever be
able to search through all $a$, $b$, $c$, and~$d$ smaller than~$n$.
The question of elucidating mathematically the nature of a feasible
computation is at the forefront of computer science and mathematics,
involving deep and exciting unresolved problems, and it is a question
before which both realism and intuitionism are helpless.
Brouwer's position was vigorously attacked by Hilbert, defending
classical mathematics under the guise of a formalist. An unpleasant,
acrimonious, and unnecessary debate ensued. (May I take this opportunity
to thank the the organizers of this event for the opportunity to
debate, in a totally different spirit, with Professor De~Giorgi --
it is a great pleasure after so many years to meet him in person
to share and dispute our ideas on the foundations of mathematics.)
The Brouwer-Hilbert debate was unnecessary because both parties
shared a common misconception: that Brouwer's intuitionism
was a {\it restriction\/} of classical mathematics. But G\"odel
showed in a short paper, published two years after his epoch-making
incompleteness theorem of 1931, that it is actually an {\it extension\/}
of classical mathematics. At least, this is true for arithmetic
(or number theory), but the less said about intuitionistic analysis
the better.
Here is a slightly -- but only slightly -- simplified account of
G\"odel's argument. To an intuitionist, $\exists x\,{\rm A}(x)$
means ``I know how to construct an~$x$ with the property~A($x$).''
This certainly implies $\neg\forall x\neg\,{\rm A}(x)$ :
``it is not the case that all~$x$ fail to have the
property~A($x$)''; the converse implication certainly does not
hold. But a realist interprets $\exists x\,{\rm A}(x)$ as
``in the pre-existing abstract universe of mathematical objects,
there is at least one of them,~$x$, with the property~A($x$).''
And to the realist, this is equivalent to $\neg\,\forall x\neg\,
{\rm A}(x)$, which means the same to the realist and the intuitionist
$\big($provided they agree about the meaning of~A($x$)$\big)$.
Similarly, to an intuitionist ${\rm A}(x) \vee {\rm B}(x)$
means ``$x$~has the property~A($x$) or $x$~has the property~B($x$),
and I can say which.'' This is stronger than $\neg[\neg\,{\rm A}(x)
\;\&\; \neg\,{\rm B}(x)]$ -- ``it is not the case that
$x$~fails to have the property~A($x$) and
$x$~fails to have the property~B($x$).'' But to a realist, both
mean the same in the Pythagorean heaven. G\"odel says in effect:
replace each formula
$\exists x\,{\rm A}(x)$ in a classical proof by
$\neg\forall x\neg\,{\rm A}(x)$ and each
${\rm A}(x) \vee {\rm B(x)}$ by
$\neg[\neg\,{\rm A}(x) \;\&\;
\neg\,{\rm B}(x)]$.
The realist, regarding them as equivalent, cannot object;
the intuitionist accepts this as a reinterpretation of what the
realist is saying. And then it turns out (and this is a fact, not
depending on any view of mathematics) that the classical proof
transformed in this way gives an intuitionistic proof of the
reinterpreted theorem!
In the light of G\"odel's result, we can say that what Brouwer
really did was extend classical mathematics by the creation
of two new logical operators: the {\it constructive there exists\/}
and the {\it constructive or}, stronger than their classical
counterparts. Unfortunately for clarity and civility,
G\"odel's paper did not receive the proper attention or interpretation,
and the unseemly squabble dragged on.\bigskip\goodbreak
{\noindent\bf A third view of mathematics: formalism}\medskip
Formalism has a simple answer to the question of the meaning of
$$\forall n \,\exists a \,\exists b \,\exists c \,\exists d
\,[\,n=a^2+b^2+c^2+d^2\,]:$$
it means nothing. (I want to clarify a point immediately.
Perhaps it would be better to say that the formula does not
denote anything. Mathematics, like music, is meaningful, but
nowadays one no longer says of passages of music: this denotes
a little bird and that denotes a zephyr. But non-formalists insist
even today on giving a denotation to every passage of mathematics.)
The symbols in the formula are marks on paper, and the work
of the mathematician is to devise deep and beautiful concatenations
of such marks according to strict rules. In discussing mathematics,
one may speak of truth or visual images, but this has as little
relation with doing mathematics as art criticism has with doing art.
Hilbert called himself a formalist, and indeed he crowned his
mathematical career with deep and brilliant work on the foundations
of mathematics from a formalist perspective, but there is strong
reason to suspect that Hilbert never renounced his Pythagorean faith,
that formalism for him was a tactic to use against Brouwer.
Let me conclude with a brief but passionate apologia for formalism.
As a description of what mathematicians have been doing, and cherishing,
for well over two millennia, it is accurate and leaves nothing out.
What we devote our lives to is seeking for proofs; if a proof follows
the formal rules, it is correct; if it does not, it is not a proof
and is worthless unless it suggests a way to find a proof. No other
field of human endeavor has maintained such a consensus over such
a vast extent of space and time.
Formalism denies the relevance of truth to mathematics. But, one
might object, mathematics works -- the evidence is all around us.
Does this not imply
that there is truth in mathematics? Not in the slightest.
Suppose we find a primitive people, or an advanced people, but a
people with a world-view utterly alien to ours, who have an herb
that is quite effective for a certain illness. They explain
its efficacy in terms of the divine action of the
{\it shuki\/} on the body's {\it okrus\/}. We find that the
herb is equally effective in our society. How much evidence
does this provide for belief in the {\it shuki\/}? None at all.
The syntax is correct; the semantics is irrelevant.
So it is with mathematics. It works. But this is no evidence
whatsoever that the religion of mathematics has any truth in it.
In mathematics, reality lies in the symbolic expressions themselves,
not in any abstract entities they are thought to denote. The
symbol~$\exists$
is simply a backwards~E.
If we conclude that a certain entity exists just
because we have derived in a certain formal system
a formula beginning with~$\exists$,
we do so at our peril. The dwelling place of meaning is syntax;
semantics is the home of illusion.
How can I continue to be a mathematician when I have lost my
faith in the semantics of
mathematics? Why should I want to continue doing
mathematics if I no longer believe that numbers and stochastic
processes and Hilbert spaces exist? Well, why should a composer want
to compose music that is not program music? Mathematics is the
last of the arts to become nonrepresentational.
And mathematics is slowly beginning to become non-representa\-tional.
Slowly in departments of mathematics, but quickly in computer
science departments. Those who do computer science know that they
are inventing and not discovering, and they are making beautiful
and deep results concerning the nature of feasible computations.
If we who are in traditional departments don't want to miss the
boat, it behooves us to saddle a formalist horse pronto.
Abstract beliefs affect concrete actions. Despite its complete
lack of justification, the semantic view of mathematics -- the
discovery of properties of entities existing in a
Pythagorean world -- has served mathematics reasonably
well for a very
long time. But now it is time to move forward, to reject the
semantic view, and concentrate on what is real in mathematics.
And what is real in mathematics is the notation, not an imagined
denotation.
Let one brief example suffice. Abraham Robinson's creation of
nonstandard analysis was a revolutionary simplification and
extension of mathematical practice, but the mathematical
community has been very slow, or unwilling,
to adopt it because it conflicts with the Pythagorean religion.
We are too timid. If we cannot achieve the depth of Eudoxus, we can
at least emulate his willingness to break with universally held
opinion.
\bye