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\centerline{\bigrm Mathematics and the Mind}
\medskip
\centerline{by Edward Nelson}
\smallskip
\centerline{Department of Mathematics}
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\centerline{Princeton University}
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\centerline{\eighttt http://www.math.princeton.edu./$\scriptstyle\sim$nelson/papers.html}
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The claim of J. R. Lucas [7, 8], expounded by Roger Penrose in the
section entitled ``The non-algorithmic nature of mathematical insight''
in {\sl The Emperor's New Mind}
[9,~Chap.~10], can be phrased as follows:
\medskip
{\openup.5\jot\narrower\narrower\narrower\noindent \Und{G\"odel's theorem} shows that there
is no \Und{algorithm} producing all \Und{statements of Arithmetic}
that mathematicians can \Und{see} to be \Und{true}. Therefore
consciousness acts non-algorithmically.
}
\medskip
\noindent I shall analyze this claim and then present some contrasting
thoughts.
\medskip
What are \Und{statements of Arithmetic}? Here there is no dispute.
They are expressions formed from symbols
$$\forall\quad \exists\quad \neg\quad \&\quad \vee\quad \to\quad =\quad 0\quad \ro S\quad +\quad \.\quad (\quad )\quad x\quad y\quad z\quad \ldots$$
according to certain rules. For example,
$$
\forall x\exists y(\exists a
\,x+a=y\ \&\ \forall m\forall n
(m\.n=y\ \vee\ m\.n=y+\ro S\ro S0\ \to\ m=\ro S0\ \vee\
n=\ro S0))$$
is a statement of Arithmetic.
\medskip
What does it mean for a statement of Arithmetic to be \Und{true}?
It depends on whom you ask.
Students of the foundations of mathematics fall into three schools:
{\it Platonists}, such as G\"odel, {\it intuitionists}, such as
Brouwer, and {\it formalists}, such as Hilbert. I shall first explain
the Platonic notion of truth in our example.
A lexicon helps:
\medskip
\halign{\qquad\qquad\hfil$#$\hfil&\qquad#\hfil\cr
\forall&for all\cr
\exists&there exists\cr
\neg¬\cr
\&&and\cr
\vee&or\cr
\to&implies\cr
=&equals\cr
0&zero\cr
\ro S&successor\cr
+&plus\cr
\.×\cr
}\medskip
\noindent Think of the variables $x\ y\ z\ \ldots\,$\ as ranging over
the natural numbers
$$0,\quad\ro S0,\quad\ro S\ro S0,\quad \ro S\ro S\ro S0,\quad\ldots$$
(i.e., 0, 1, 2, 3, \dots). Then Platonists say that our formula is
true in case for all numbers~$x$, there exists
a larger number~$y$ (larger since $x+a=y$) such that both~$y$ and
$y+2$ are primes (if either one can be factored as $m\.n$, then one
of the factors $m$ or $n$ is~1).
This is a famous unsolved problem of Arithmetic,
the twin primes problem. So far, no mathematician can prove it
or disprove it.
Intuitionists
demand an effective method for constructing the twin primes greater
than any given~$x$ before assenting to the truth of the formula.
Our example is a very simple statement for which it is easy to
acquire an intuition, but a truth definition for statements of
Arithmetic must apply also to statements containing thousands of
symbols and not corresponding
to any notions for which we have built up an intuition.
The Platonic truth
definition for a statement
$\exists x_1\forall y_1\exists x_2\forall y_2\ldots$ reads: there
exists a number~$x_1$ such that for all numbers~$y_1$ there exists a
number~$x_2$ such that for all numbers~$y_2$ \dots~This notion of
interleaved infinite searches and verifications makes sense only to
Platonists; to intuitionists and formalists the idea lacks coherence.
Kleene [5,~\S82] has made a deep analysis of the intuitionistic
truth notion in terms of recursive realizability. He shows that
intuitionistic truth is
not simply a restriction of the Platonic truth notion
but actually conflicts with it. That is, he exhibits a statement
of Arithmetic that is Platonically true and intuitionistically
false, and another that is Platonically false and intuitionistically
true.
Formalists find Platonic truth for statements of Arithmetic
both incoherent and irrelevant to the practice of mathematics.
A solution, positive or negative, of the twin primes problem
would be a major event in mathematics, but no solution will
depend on philosophical truth notions.
Penrose, in footnote~2 to the last chapter of~[9], says ``As to the
very dogmatic G\"odel-immune formalist who claims not even to
recognize that there {\it is\/} such a thing as mathematical truth,
I shall simply ignore him, since he apparently does not possess the
truth-divining quality that the discussion is all about!'' He
explicitly ignores formalists
and tacitly ignores intuitionists, thereby
dismissing two of the century's
greatest mathematicians and students of foundations.
Platonic truth is conceived to be an infinite, abstract structure
that assigns to each statement of Arithmetic a truth value, true or
false. Platonists believe in the existence, though not in the
physical world, of this ideal object. Intuitionists believe in a
different ideal object, intuitionistic truth, and formalists believe
in neither. These differing philosophical views are unproblematic
for mathematics---which is concerned with proof (syntax) rather
than truth (semantics)---but belief in Platonic truth is an
essential part of the Lucas argument.
It is a truism that mathematics has many applications to science.
But an application to science of the philosophy of mathematics (or,
more accurately, of one philosophy of mathematics) would indeed be a
novelty.
\medskip
What is \Und{G\"odel's theorem}? As already described, statements of
Arithmetic are simply symbols combined according to certain rules.
Arithmetic is a powerful language and the syntactical rules for
forming statements can be expressed, or encoded, within Arithmetic.
To obtain a theory of Arithmetic, call it \A, a mathematician selects
certain statements of Arithmetic as axioms and postulates certain
rules of deduction. A statement of Arithmetic is provable in~\A\
in case it follows from the axioms by the rules of deduction. This
is a syntactical notion on which there is no dispute, and it too can
be expressed in Arithmetic. But the choice
of axioms and rules of deduction varies according to one's semantic
notion of truth and one's personal aesthetics. It is essential that
\A\ be consistent, since otherwise every statement of Arithmetic could be
proved in~\A, and \A\ would be of no interest.
One much studied theory~\A\ has come to be known as Peano Arithmetic.
The liar paradox, famous from
antiquity, can be formulated as
$$\hbox{this statement is not true.}$$
The liar paradox plays no role in G\"odel's theorem.
The notion of Platonic truth is semantic, not syntactical, and cannot
be expressed within Arithmetic. (This was proved by Tarski [10].)
G\"odel does something different. He constructs a statement of
Arithmetic, call it~G, expressing
$$\hbox{this statement is not provable in \A.}$$
Using many abbreviations, the construction of~G takes a fair number
of pages. The
modern form of G\"odel's first incompleteness theorem~[3] is
$$\hbox{if \A\ is consistent, then G is not provable in \A.}$$
This is a theorem, and there is no dispute about it. Notice that
G\"odel's theorem says nothing whatever about consciousness or the
mind.
The Lucas
argument, endorsed by Penrose, can now be phrased as follows:
\medskip
{\narrower\narrower\noindent Mathematicians, by their consciousness,
see that \A\ is consistent and therefore know that G~is not
provable in \A, and therefore know that G~is true. Thus
mathematicians, by their consciousness, have insight---going beyond
proof---into mathematical truth.
Hence consciousness produces truth by non-algorithmic means.
}
\medskip
\noindent
The claim is not that mathematicians have direct insight into the
extremely complicated statement~G; rather it is that they follow the
proof of G\"odel's theorem, have direct insight into the hypothesis
that \A\ is consistent, and so conclude that G is true even
though unprovable in \A.
\medskip
But can mathematicians \Und{see}, without proof, that certain
statements of mathematics are true? A look at the history of mathematics
shows that this belief has invariably been a delusion.
Mathematicians thought they could see that cubic equations could only
be solved geometrically, using conic sections---until del Ferro,
Tartaglia, and Cardano solved them algebraically; they thought they
could see that Euclid's parallel
postulate was true---until Lobachevsky and Bolyai constructed
non-Euclidean geometry.
Today we are beginning (some of us) to learn caution. The fact of the
matter is that serious students of the foundations of mathematics
are divided about the status of the hypothesis that Peano Arithmetic is
consistent. Platonists claim to see it directly; intuitionists can
be convinced by another, much less cited, theorem of G\"odel [4]
combined with their claimed insight into their own notion of truth;
some formalists regard it as a genuinely open question. And the
problem becomes more acute when, as
required by the Lucas argument, we study general theories~\A\ and
general algorithms.
\medskip
What is an \Und{algorithm}? Although important specific algorithms
date from antiquity, and the word itself comes from the name of
the ninth century algebraist al-Khwarizmi, the general
nature of an algorithm was not elucidated until the 1930s,
primarily by Turing~[11]. Today we can express the notion succinctly
by saying that an algorithm is a computer program that, for any input,
eventually halts, producing an output. This is another Platonic notion,
presupposing an infinite search over all inputs and all numbers
(the number of steps required for the computer to halt, if it
does halt). Turing proved that there is no algorithm to decide whether
any given computer program is an algorithm (undecidability of the
halting problem).
A closely related result is that there is no
algorithm that decides whether any given theory~\A\ is consistent.
Lucas and Penrose state that mathematicians have direct insight into
the consistency of some theories~\A, but they do not specify on
which~\A\ this faculty of insight operates correctly.
Now we have at hand all the ingredients of the Lucas-Penrose argument.
They postulate belief in an infinite ideal object known as Platonic
truth and they postulate a faculty of consciousness to perceive
by direct insight the consistency of certain unspecified theories~\A.
They dismiss those mathematicians who hold different views and
conclude that they have uncovered a general and fundamental feature
of consciousness.
The similarity to religious belief, with appeal to authority, is
inescapable. Here is a close analogue of the Lucas argument:
Catholics know that when the Pope speaks {\it ex cathedra\/}
on matters of
faith or morals, he speaks the truth; they then know those truths
without proof; therefore consciousness acts non-algorithmically.
(I hasten to say, sincerely, that I offer this analogy simply to
clarify my view of the Lucas argument. I respect Catholics and
Catholic beliefs, and I respect Platonists.)
I conclude that G\"odel's theorem, and the general Platonic notion
of an algorithm, are simply irrelevant to the study of the mind.
But can the study of algorithms offer anything of relevance to a science
of consciousness? I am not a student of consciousness, but it does seem
safe to say that natural selection has produced in us conscious awareness
that can perform certain computations very rapidly and efficiently.
Therefore it is plausible that a mathematical study, not of Platonic
algorithms in general but of rapid and efficient algorithms, might
have some relevance to a science of consciousness.
Such a study is a relatively new field,
more developed in computer science
departments than in mathematics departments.
The usual way of making precise
the notion of a rapid algorithm is to say that the computer program
always halts after a number of steps bounded by a given polynomial
in the length of the input. There is still a Platonic element here:
the bound is simply said to exist. But Bellantoni and Cook~[1,~2]
and Leivant~[6] have produced a purely syntactical notion and
demonstrated
that it is equivalent to the notion of a polynomial-time algorithm,
This is a surprising result. It is in sharp contrast to the Turing
result on the undecidability of the halting problem for general
algorithms. There are no arithmetic hypotheses in their characterization;
rather, their idea expresses the Aristotelian notion
of an object coming into being, as opposed to the Platonic notion
of a pre-existing ideal object.
The study of the similarities and differences between polynomial-time
algorithms and the functioning of the mind is a concrete matter,
not just philosophical speculation, and it will repay the
serious attention it is receiving.
Here is a vision that I expect to become a reality
early in the twenty-first century.
Mathematicians write articles with precisely structured
statements of theorems and proofs. The proofs refer to the
database of previously proved theorems, a database that lives
in a decentralized fashion on the Web. The proof is automatically
checked for correctness by a polynomial-time algorithm,
and if correct becomes a part of the
database. Such an article requires no refereeing for correctness.
The system is used interactively to help find proofs.
For mathematics that deals with concrete, syntactical
objects---numbers, graphs, anything that can be encoded by a
string of bits---and for proofs that are constructive, the
system automatically constructs from the proof a program that
produces the existentially quantified objects as functions of
the relevant universally quantified objects, together with a
worst-case bound on time and memory requirements. This will be a
polynomial-time bound if the recursions in the proof satisfy the
Bellantoni-Cook-Leivant constraint. Such a program
requires no debugging since it is a by-product of a
proof that the objects have the properties stated in the theorem
whose constructive proof has been verified.
The difficulty in searching for a mathematical proof is the
exponentially growing sea of possibilities that opens up as one
possible step among many follows another possible step.
What will be of interest to practicing mathematicians in such a system
is the interactive exploitation of the different search strategies
of conscious human minds and computers. What will be of interest to
students of Artificial Intelligence, and possibly to a science of
consciousness, is that the system will provide a laboratory for testing
whether human search abilities---which are a marvel and a mystery---can
be simulated by machine
algorithms. (My own hunch is that it will be centuries, if ever,
before computers---or, more likely, robots with a body and some
experience of the world---can begin to do this.) This is a modest
question that concerns the functioning of the mind in one area,
rather than the fundamental nature of consciousness, but I expect
it to be more fruitful than speculations about Platonic truth
and algorithms.
\medskip
\parindent=0pt
\centerline{\bf References}
\smallskip
\frenchspacing
\parskip=\smallskipamount
[1] Stephen Bellantoni and Stephen Cook, ``A new recursion-theoretic
characterization of the polytime functions'', {\sl Computational
Complexity}, {\bf2}, 97-110, 1992.
[2] Stephen Bellantoni, ``Predicative Recursion and Computational
Complexity'', {\sl Technical Report 264/92}, Department of Computer
Science, University of Toronto, 1992.
{\tt ftp:\break
//ftp.cs.toronto.edu/pub/reports/theory/cs-92-264.ps.Z}
[3] Kurt G\"odel, ``\"Uber formal unentscheidbare S\"atze der Principia
Mathematica und verwanter Systeme I'', {\sl Monatshefte f\"ur
Mathematik und Physik}, {\bf37}, 173-198, 1931.
[4] ---, ``Zur intuitionistischen Aritmetik und Zahlentheorie'',
{\sl Ergebnisse eines math. Koll.}, {\bf4}, 34-38, 1933.
[5] Stephen Cole Kleene, ``Introduction to Metamathematics'',
North-Holland, 1952.
[6]Daniel Leivant, ``Ramified recurrence and computational complexity
I: Word recurrence and poly-time'', in P. Clote and J. Remmel
(eds.), {\sl Feasible Mathematics II}, 320-343, 1994.
[7] J. R. Lucas, ``Minds, machines and G\"odel'', {\sl Philosophy},
{\bf36}, 102-124, 1961.
[8] ---, ``The Godelian Argument'', {\sl Truth}, {\bf2}, 1988.
{\tt http://www.clm.org/truth/2truth08.
html}
[9] Roger Penrose, ``The Emperor's New Mind: Concerning Computers, Minds,
and the Laws of Physics'', Oxford University Press, 1989.
[10] Alfred Tarski, ``Der Wahrheitsbegriff in den formalizierten
Sprachen'', {\sl Studia philosophica}, {\bf1}, 261-405, 1936.
Translated from the Polish original of 1933 by L. Blaustein.
[11] Alan Turing, ``On computable numbers with an application to the
Entscheidungsproblem'', {\sl Proc. London Math. Soc.}, {\bf42},
230-275, 1936. ``A correction'', {\sl ibid.}, {\bf43}, 544-546,
1937.
\bigskip
\noindent nelson@math.princeton.edu
\bye