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\centerline{\bigrm Mathematics and Faith}\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
In this reflection about mathematics I shall confine myself to
arithmetic, the study of the numbers 0, 1, 2, 3,~4, \dots.
Everyone has at least the
feeling of familiarity with arithmetic, and the issues that concern
the human search for truth in mathematics are already present
in arithmetic.
Here is an illustration of research in arithmetic.
About 2500 years ago, the Pythagoreans
defined a number to be {\it perfect\/} in case it is the sum
of all its divisors other than itself. Thus
$6=1+2+3$ and \hbox{$28=1+2+4+7+14$} are perfect. The Pythagoreans,
or perhaps Euclid himself, proved that if $2^n-1$ is a prime,
then $(2^n-1)\cdot2^{n-1}$ is perfect.$^1$ More than 2000
years later, Euler proved that every even perfect number is of
this form. This left open the question whether there exists an
odd perfect number. The search for an odd perfect number or,
alternatively, for a proof that no odd perfect number exists,
continues today, several centuries after Euler and in the
fourth millennium from Pythagoras.
No other field of human endeavor so transcends the barriers
of time and culture. What accounts for the astounding ability of
Pythagoras, Euler, and mathematicians of the 21st century to
engage in a common pursuit?
Of the three schools of thought on the foundations of
mathematics---Platonic realism, intuitionism, and formalism---the
Platonists offer what seems to be the simplest explanation.
The sequence of numbers is the
most primitive mathematical structure. Mathematicians
postulate as axioms certain
self-evident truths about the numbers and then deduce by logical
reasoning other truths about numbers, theorems such as those of
Euclid and Euler. This is the traditional story told about
mathematics. Can there be any philosophical---let alone
theological---problem with it? Let us look more
closely.
\bigskip
{\bf Reasoning in mathematics.}
The logic of Aristotle---the greatest logician before G\"odel---is
inadequate for mathematics. It was already inadequate for the
mathematics of his day. Only relatively recently was the
logic of mathematical reasoning clarified. Boole$^2$ brilliantly
began the clarification and Hilbert$^3$ perfected it.
They succeeded where Frege and Russell
went partly astray, because they sharply distinguished syntax
from semantics.
How is the syntax of arithmetic formulated?
One begins with a few marks whose only relevant property is that
they can be distinguished from each other. An {\it expression\/}
is any combination of these marks written one after the other.
Certain expressions, built up according to simple definite rules,
are {\it formulas}, generically denoted by~$F$.
Certain formulas are chosen as {\it axioms}.
With a suitable choice of axioms, we obtain what is called
{\it Peano Arithmetic\/} (PA).
A {\it proof\/} in PA is a succession of formulas such that
each of its formulas~$F$ is either an axiom or is preceded
by some formula $F'$ and by \hbox{$F'\imp F$}. (One of the marks
is~$\to$.)$^4$ Each formula in the proof is a {\it theorem}.
And this is what mathematics is. Mathematicians are people who
prove theorems; we construct proofs.
The salient feature of syntax is that it is concrete.
The question whether a putative proof
is indeed a proof is a matter simply of checking. Disputes about
the correctness of a proof are quickly settled
and the mathematical community reaches permanent consensus.
The status, age, and reputation of the parties to the
dispute play no role. In this we are singularly blessed.
(Of course, being human, we do squabble---not about the correctness
of a proof but about priority and value. The two can conflict.
Once at a party a friend spoke eloquently and at length. He was irate at
the credit accorded to another's work. When he concluded, I said
with some trepidation, ``You seem to be saying two things: the
work is utterly trivial, and you did it first.'' I was relieved
when my friend replied, ``That is {\it precisely\/} what I am
saying.'')
Admittedly, I have somewhat exaggerated the mechanical nature of
proof checking as it occurs in the year 2000, but we can expect
it to be fully mechanized, by computer programs, in the near future.
To check a proof is mechanical, to construct a proof is not.
Living mathematics is one of the glories of the human spirit, and
the sources of mathematical creativity are deep and mysterious.
In December 1996 it was announced that a computer had solved
the Robbins problem. This is a problem that the eminent mathematician
Alfred Tarski, and others, had failed to solve. I was interested in
this, and having reached what Irving Kaplansky calls the age of
ossification when the only way to learn something new is to teach it,
I gave a graduate course on this work. What impressed me was not
the speed and power of the computer, but the great ingenuity with which
William McCune$^5$ had developed a program
for this general kind of problem, and his artful strategy that steered
the search between the Scylla of exponentially expanding
generation of useless formulas and the Charybdis of premature
termination of the search. Computers will play an increasing
role in the construction of proofs, but I predict that 2500 years
from now they will not have replaced mathematicians.
Hilbert's logic expresses something very deep within human
nature. The work of the best and clearest mathematicians of
antiquity---Archimedes is the prime example---in no way sounds
alien to our ears. Archimedes reasoned in mathematics just as
we reason today.
In describing mathematical syntax, I have implicitly advanced an answer
to the question raised in the beginning: we can continue the work
of Euclid and Euler because we reason in the same way. One might
object that this answer is inadequate because it makes no mention of
truth.
\bigskip
{\bf Truth in mathematics.}
Now let us turn to the semantics of mathematics, and of arithmetic
in particular. I shall describe Platonic semantics.
Among the marks are 0 and S. A {\it numeral\/} is an expression
of the form 0, S0, SS0, and so forth, generically denoted by~$n$.
The mark~0 denotes the number zero, and if the numeral~$n$ denotes
a certain
number then $\S n$ denotes its successor, the next number after it.
Thus the numerals serve as names for the numbers.
Certain expressions are {\it variables}, generically denoted by~$v$.
Occurrences of variables in formulas are syntactically divided into
{\it free\/} and {\it bound\/} occurrences, and a formula with no
free occurrences of variables is {\it closed.} The result of
substituting the numeral~$n$ for each free occurrence of the
variable~$v$ in the formula~$F$ is denoted by $F_v[n]$.
One assigns to each closed formula a {\it truth value}, true or false.
This proceeds iteratively, starting with the simplest formulas
and working up to more complex formulas. One of the marks is~$\forall$,
and one step in the iterative assignment of truth values to formulas
is the clause:
$$\hbox{$\forall vF$ is true
in case for all numerals~$n$, the formula $F_v[n]$ is true.}$$
Platonists maintain that by virtue of the iterative assignment
of truth values, a closed formula acquires meaning
as a statement about numbers and is either true or false.
The clause displayed above
is the battlefield where clash the armies of Platonists,
intuitionists, and formalists. It differs from syntactical definitions
because it invokes the notion of an infinite search.
We have left the realm of the concrete for the speculative.
The intuitionist, epistemologically
inclined, says, ``What you said makes no sense. How can I possibly
examine for every numeral~$n$ the truth of $F_v[n]$?'' The Platonist,
ontologically inclined, replies, ``Nevertheless, either $F_v[n]$ is
true for every numeral~$n$ or there exists a numeral~$n$ for
which it is false.'' The formalist is content to observe that the
definition of truth value can be formalized in set theory (a branch
of mathematics more complex than arithmetic), thus sweeping the
semantic dirt under the carpet of syntax.
Intuitionism was the creation of Brouwer, who proposed it as a
restricted, and truthful, replacement for classical mathematics.
He developed a new syntax and a new semantics, which for arithmetic
were clarified and made precise by Heyting and Kleene, respectively.
There is a specific closed formula of arithmetic that is false
for Platonists and true for intuitionists.$^6$
When speaking of truth in mathematics, one must
always specify truth according to whom.
The notion of truth in mathematics is irrelevant to what
mathematicians do, it is
vague unless abstractly formalized, and it varies according to
philosophical opinion. In short, it is formal abstraction
masquerading as reality.
At this point I pause to say something with all the emphasis I can
muster. In these days when postmodernism is still in
vogue and some people seriously proclaim that scientific truth
is a social construct, I do not want to be misunderstood.
I have expressed doubts about the coherence of the notion of
truth in mathematics, but I am speaking only about {\it
mathematics}.
However much amplification the following description
of truth may require,
truth is a correspondence between a linguistic formulation and
reality. My claim is that there is no Platonic reality underlying
mathematics; mathematicians prove theorems, but the theorems are
not {\it about\/} anything. This is how mathematics differs
profoundly from science.
Mathematicians no more {\it discover\/} theorems
than the sculptor discovers the sculpture inside the stone.
(Surely you are joking, Mr.~Buonarroti\hskip.3pt!)
But unlike sculpting,
our work is tightly constrained, both by the strict requirements
of syntax and by the collegial nature of the enterprise.
This is how mathematics differs profoundly from art.
To deny the cogency of the Platonic notion of truth in mathematics
in no way deprives mathematics of meaning. In mathematics, meaning
is found not in a cold, abstract, static world of Platonic ideas
but in the human, historical, collegial world of mathematicians
and their work. Since Boole, mathematics is understood as the
creation and study of abstract patterns and structures; this replaces
the old understanding of mathematics as the study of number and
magnitude. Meaning is found in the beauty and depth of these
patterns, in unsuspected relationships between structures that
previously seemed unrelated, in the fierce struggle to bring
order to seemingly insurmountable complexity, in the
joy of providing new tools for the better understanding of the
physical world, in the pleasures of collegial cooperation and
competition, in the visual beauty of geometry and dynamics in
three dimensions and their inner visual beauty in higher dimensions,
and above all in the awe of confronting the potential infinity
that is the world of mathematics.
It is sometimes claimed that mathematicians have direct insight
that goes beyond proof. A young mathematician would be ill advised
to attempt to make a career by advancing such claims; it would
violate collegiality. Share with us how you come by your insights
so that we can examine your methods, test them, formalize them,
see how they fit into the rest of mathematics.
If your conjecture is deep and beautiful, if it resists proof or
disproof for a long time, you will have made an important
contribution to the collegial enterprise---but the final criterion
is proof. Proof is the essence of mathematics.
Many, many writers have stated that mathematics is the most
certain human acquisition of knowledge. This misperception leads
to such embarrassments as the pseudo-Euclidean form that Spinoza
gave to his {\it Ethics}. These writers are too pedestrian in
their view of mathematics and yet they give us too much credit.
All we do is construct proofs, and at present we do so from
axiom systems for which no one can give a convincing demonstration
of consistency.
\bigskip
{\bf Consistency in mathematics.}
Unlike truth,
consistency is a syntactical matter. One of the marks is~$\neg$,
and the system is consistent if there do not exist proofs of both
a formula~$F$ and its {\it negation\/}~$\neg F$.
Hilbert, stung by the attacks of Brouwer, initiated his
program: to establish the consistency of classical mathematics
by finitary methods, methods that would be acceptable to the
intuitionists. Hilbert called himself a formalist but I suspect
that he was a crypto-Platonist. Referring to Cantor's set theory,
he declared,
``No one shall expel us from the paradise that Cantor has created
for us.'' (It is curious that mathematicians confronting the
depths of our subject often fall into the language of faith.
Kronecker wrote, ``God made the numbers, all else is the work
of Man.'' Once when I expressed my skepticism about the consistency
of mathematics to my thesis adviser, the late Irving Segal,
he asked, ``But you {\it believe in\/} the natural numbers,
don't you?'')
Hilbert's hopes were dashed by G\"odel's great paper$^7$ of 1931.
We have seen that mathematical syntax is a matter of combining marks
in simple ways; G\"odel's first step was to express these
combinations within arithmetic. It is frequently said that G\"odel's
theorem is based on the liar paradox of antiquity (``this
statement is not true''), but this misses the point. G\"odel
replaced the superficial semantic paradox by deep syntax, constructing
a formula~$F$ of~PA that expresses ``$F$~is unprovable in~PA''.
This led to his two incompleteness theorems. They are often
paraphrased like this:
\medskip
{\narrower\noindent(1) there is a true but unprovable formula of PA;
\vskip3pt
\noindent(2) if PA is consistent, the formula expressing
``PA is consistent'' is unprovable in PA.
}
\medskip
\noindent It is instructive to compare these two statements.
The first is a statement about the Platonic world, for as we have
seen, the notion of truth for a formula of arithmetic is
an extra-mathematical point of contention among the various schools
of foundations (unless it is formalized in set theory). The
second is a statement about the world we live in, and it demolished
Hilbert's program. The first is abstract, the second concrete:
G\"odel explicitly shows how to derive a contradiction in~PA from
a proof in~PA of the formula expressing ``PA~is consistent''.
The consistency of PA cannot be concretely demonstrated. What are
the options? First, Platonism---the option followed by G\"odel
himself. Second, intuitionism---but this is no solution, since G\"odel
in a very short paper$^8$ in 1933 gave an interpretation of classical
arithmetic within intuitionistic arithmetic, showing that one is
consistent if and only if the other is. Third, pragmatism---to stop
worrying about foundations and get on with mathematics; this has
been the general response. Fourth, radical formalism---to explore
whether PA might not after all be inconsistent.$^9$
Abstract ideas have concrete consequences---this is their power,
and in human affairs this is their terror, as in the abstract idea
of the Aryan race. The Platonist argues
that the axioms of PA are true,$^{10}$
the rules of inference preserve truth, so all theorems are true,
but a formula and its negation cannot both be true; hence PA is
consistent. An attempt to construct a contradiction
in~PA is a concrete act which the Platonist dismisses as futile.
For the Platonist the consistency of PA is a matter of faith.
\bigskip
{\bf Faith in mathematics.} The title of this section should
give us pause.
Our culture instills respect for other religions and
our faith requires respect for people who practice other
religions; nevertheless, I was dismayed when I saw
an idol---an actual metal effigy---being worshiped. We who are
children of Abraham by adoption may have something to learn from
our Jewish and Muslim siblings concerning the vehemence of God's
abhorrence of idolatry. God demands our faithful faith.
Christianity in its early years confronted in Hellenistic culture
not only pagan idolatry but a refined strain of religious
thought in philosophy. In {\it Fides et Ratio\/} we read, concerning
Greek philosophy,
``Superstitions were recognized for what they were and religion was,
at least in part, purified by rational analysis.''$^{11}$ Let us pay
close attention to the phrase ``at least in part''.
Mathematics as we know it originated with Pythagoras, and Pytha\-goras
founded a religion in which numbers played a central role.
For the Pythagoreans, the numbers were an infinite, real, uncreated
world of beings, and most mathematicians retain that faith today.
With no training in history,
philosophy, theology, Greek, or Latin, I can only
listen as others speak of the early encounter of Christianity with
Pythagoreanism. Elaine Pagels writes movingly of Justin's
conversion to Christianity after a previous unsatisfactory
experience with a Pythagorean teacher who required him first to
learn mathematics before he could be enlightened.$^{12}$ Perhaps the
Holy Spirit had implanted in Justin what today is called
``math anxiety''.
I must relate how I lost my faith in Pythagorean numbers.
One morning at the 1976 Summer Meeting of the American Mathematical
Society in Toronto, I woke early. As I lay
meditating about numbers, I felt the momentary overwhelming
presence of one who convicted me of arrogance for my belief in
the real existence of an infinite world of
numbers, leaving me like an infant in a
crib reduced to counting on my fingers.
Now I live in a world in
which there are no numbers save those that human beings on occasion
construct.
The Pythagorean religion is no longer practiced. But Pythagoras
strongly influenced Plato, and through Plato us. The numbers
of Pythagoras are the type of Platonic ideas.
What are Platonic ideas? Are they created---contingent and
subject to change?
Are they uncreated---as they were in the beginning, are now, and
ever shall be? Or are they constructs of human thought which we
come perilously close to idolizing? For us, men and women
from the world of learning,
to reify abstract ideas, even to base morality on them, is a
temptation more insidious than the worship of metal effigies, and
more corrupting.
During my first stay in Rome I used to play chess with
Ernesto Buonaiuti. In his writings and in his life,
Buonaiuti with passionate eloquence opposed
the reification of human abstractions. I close
by quoting one sentence from his {\it Pellegrino di Roma}.$^{13}$
``For [St.~Paul] abstract truth, absolute laws, do not exist,
because all of our thinking is subordinated to the construction
of this holy temple of the Spirit, whose manifestations are not
abstract ideas, but fruits of goodness, of peace, of charity
and forgiveness.''
\vfill\eject
\centerline{\bf Notes and References}\medskip
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$^1$ This is Proposition 36, Book IX of Euclid's {\it Elements},
translated with
commentary by Thomas L. Heath, 2nd. ed. revised with additions,
Vol. II, Dover, New York, 1956.
$^2$ George Boole published {\it An Investigation of the
Laws of Thought, on Which Are Founded the Mathematical Theories of
Logic and Probabilities\/} in 1854. Bertrand Russell said,
``Pure Mathematics was discovered by Boole in a work which he called
{\it The Laws of Thought},'' quoted by Carl B. Boyer in {\it A
History of Mathematics}, Wiley, New York, 1968.
$^3$ David Hilbert and Wilhelm Ackermann, {\it Gr\"undzuge der
theoretischen Logik},
Springer, Berlin, 1928.
$^4$ It is possible to choose the logical axioms so that the
only rule of inference is modus ponens, as I have assumed here for
simplicity of exposition.
For an exceedingly lucid account of mathematical logic,
see Joseph R. Shoenfield,
{\it Mathematical Logic}, Addison-Wesley, Reading, Massachusetts, 1967.
$^5$ William McCune, {\it Solution of the Robbins Problem},
Journal of Automated Reasoning, vol. 19,
263--276, 1997. {\it Robbins Algebras Are Boolean}, online at
{\tt http://www-unix.mcs.anl.gov/$\!\scriptstyle\sim$mccune/papers/robbins/}
$^6$ See Stephen Cole Kleene, {\it Introduction to Metamathematics},
North-Holland, New York, 1952,
for a detailed account of a version of Heyting's intuitionistic
arithmetic and for Kleene's intuitionistic semantics via recursive
realizability. The formula demonstrating the divergence between
classical and intuitionistic arithmetic is described in Sec.~82
and is the last corollary in the book.
Brouwer's first paper on intuitionism was published, in Dutch, in 1908
(see Kleene's bibliography).
$^7$ Kurt G\"odel, {\it \"Uber formal unentscheidbare S\"atze der
Principia Mathematica und verwandter Systeme I}, Monatshefte f\"ur
Mathematik und Physik, vol. 38, 173-198, 1931.
$^8$ Kurt G\"odel, {\it Zum intuitionistischen Arithmetik und
Zahlentheorie}, Ergebnisse eines math. Koll., Heft 4 (for 1931-2),
34-38, 1933.
$^9$ See Edward Nelson, {\it Predicative Arithmetic}, Mathematical
Notes 32, Princeton
University Press, Princeton, New Jersey, 1986,
for a detailed critique of Peano Arithmetic.
$^{10}$ More precisely, the claim is that they are valid; see
Shoenfield's book cited above.
$^{11}$ John Paul II, {\it Fides et Ratio}. The quotation is from
Article~36. Online at
{\tt http://www.vatican.va/holy\_father/john\_paul\_ii/encyclicals/}
$^{12}$ Elaine Pagels, {\it The Origin of Satan}, Random House, New
York, 1995. The story of Justin's conversion is told in Chap. 5.
$^{13}$ Ernesto Buonaiuti, {\it Pellegrino di Roma: La generazione
dell'esodo}, Laterza, Bari, 1964. The quotation is from the chapter
on the years 1915-1920.
\bye