FOM: Definition of mathematics (continued)
Jan Mycielski
jmyciel at euclid.Colorado.EDU
Fri Jan 7 20:39:46 EST 2000
PHILOSOPHY OF MATHEMATICS VERSUS KNOWLEDGE
Reading my own letter of Dec. 29 1999 (DEFINITION OF MATHEMATICS)
I notice that it may be worthwile to clarify a few points, especially in
the last part of that letter. Also some correspondents have raised issues
which I would like to address. (Some of the ideas presented here were also
briefly expressed in my paper in J. S. L. 60 (1995), 191 - 198.)
1. The main objections to the ZFC-definition of mathematics, which
tells us that
Mathematics = (conjecturing and proving theorems in ZFC),
are the folloowing three:
(1) Mathematics which I know and which I do is not like that.
(2) This def. does not explain our belief (conviction) that ZFC
(and its natural extensions with large cardinal) is consistent.
(3) This definition suggests that mathematics is not only a purely
formal (symbolic) structure, but also one which is arbitrarily conceived.
I can refute (1) immediately: As a rule a description of an oject
or process is not at all like that object or process, there must be only a
certain correspondence and some explanatory power. The second will be
refuted in part 7. of this letter and the third in part 8.
2. Joe Shipman suggested that that the ZFC-def. of math. could be
simplified as follows:
Math. = (conjecturing and proving theorems in first-order logic).
I agree that this definition is correct, and that it may be more
useful and relevant for somebody who is not familiar with thedevelopment
and the interpretation of mathematical theories in ZFC.
But of course the first definition (math. = ZFC) explains more
closely what is mathathematics. Now one could object that even this
ZFC-definition is not close enough to the reality of math. But it seems to
me that most of those who think they know better what is mathematics and
that this ZFC-definition is not intresting, may have a lot of know-how but
not the real knowledge which answers that question "What is mathematics?",
and they are not able to give any other description except by providing
some examples. (Some of them complain that the ZFC definition does not
help a mathematician working, say, in mathematical physics or in algebraic
geometry. However, such complaints are not justified. They are as if
somebody complained that general relativity is not helpful for the three
body problem.)
3. In the letter of Dec. 29, 1999 I have mentioned that
quantifiers can be viewed as abbreviations of quantifier-free expressions
in Hilbert's epsilon extension of first-order logic (and in this way the
suggestion that mathematics is inherently platonic is demoted). I have
been asked to elaborate on this point.
Let fi be a quantifier-free formula. We define:
(Ex)fi(x) = fi(epsilon x fi), and
(Ax)fi(x) = fi(epsilon x (not fi)).
Then, it is easy to check that, the rules of logic concerning quantifiers
follow from Hilbert's axiom:
fi(x) implies fi(epsilon x fi).
(Hilbert 1923, see A. C. Leisenring "Mathematical logic and Hilbert's
epsilon-symbol", Gordon & Breach, New York, 1969.)
Let me repeat that, the above definitions show that quantifiers in
pure mathematics can be understood as metaphorical expressions. Those
expressions borrow our mental tools for thinking and talking about very
large actual sets (e.g. the set of atoms of my pen, or the set of stars of
our galaxy). But the metaphorical uses of those expressions do not refer
to any actual sets or universes, they express properties of certain
thought operations (tools to imagine new objects) which can be denoted by
the epsilon-terms.
4. The English version of the book of A. Sokal and J. Bricmont
"Fashionable nonsense" was published by Picador in 1998.
5. METAPHILOSOPHY. In order to criticise philosophy (of
mathematics), and eventually to build or to choose a good one, we must
construct a useful metaphilosophy which explains what is a convincing
philosophy. In fact Sokal and Bricmont (l. c.) demonstrated that it is
high time to explain what is a convincing text.
If somebody regards as acceptable any philosophy which has been
proposed by a respected philosopher, then he will think that my definition
of mathematics is one of many possible ones, and my categorical tone is
quite unreasonable. But I have reasons to believe that such a judgement is
wrong. Namely, I will state now some clear criteria which show that only
one philosophy (which I call rational) is convincing.
But let me repeat that, by a philosophy (in particular a
philosophy of mathematics) I understand a chapter of knowledge or of
natural science, and many artistic descriptions of reality fall short of
this mark. [As I mentioned in an earlier letter in f.o.m. antirationalists
try to compartmentalise knowledge, or to divide the philosophy of their
opponents in artificial ways. (E.g. they invent such names as
instrumentalism or conventionalism, they oppose Einstein to Poincare,
Poincare to Hilbert, etc.) In particular they try to separate
common knowledge from scientific knowledge. Already Sokal and Bricmont
pointed out that this is a mischievious way of talking about knowledge.
Let me add that it ignores a marvelous and inspiring program, the program
to unify knowledge. It belittles such fundamental achievements as the
inseparability of biology from chemistry, of chemistry from physics, of
physics from cosmology, of thermodynamics from statistical physics, of
geometry from algebra, etc.]
By the above definition of philosophy, in order to be acceptable
a philosophy must withstand the criteria which apply to knowledge or to
science. In particular:
(A) Its intended relation to the physical reality must be clear;
it must describe or explain something which is (or can be) actually
observed. And it must agree with reality (be true) in an intended
measure.
(B) It must be the simplest or the least assuming such description
or explanation which we know; or an alternative of such simplest
descriptions or explanations.
[Notice that (A) and (B) constitute also an answer to the
important quesion: What is a convincing theory?
Philosophers who propose theories violating (a) or (b) disagree
with the previous sentence and they raise two objections:
(o) That (B) is not strongly related (or not at all related) to
the property (of a theory) of being convincing.
(oo) That the property of simplicity in (B) is too vague to be
applicable in such a definition.
I can refute (o) as follows. It is clear that there are mental
pathologies such that people affected by them reach convictions which are
not based on (A) and (B). However, I believe that sane people are
convinced by any theory which to them is interesting (significant) enough
and satisfies (A) and (B). It is true that many of them think also that
they have beliefs which do not satisfy (B). But it is not easy to
assertain if they really have those beliefs, especially when most
decisions supposedly based on them also happen to conform to their
desires. Besides one cannot believe something one does not know, so an
ignorant person can be easily induced to believe things which better
informed people cannot accept since they know or feel that they violate
(A) or (B). At this point the capacity of association of information from
distant topics is also important. Indeed, one can believe two incompatible
theories if one has not noticed their inconsistency.
I can refute (oo) as follows. First let it be clear that (in
this context) a description or a theory constitutes an algorithm for
obtaining a set of sentences. (It is not the efficiency of this algorithm
which makes it convincing it is only (A) and (B), in particular its
simplicity. Thus our most convincing theories need not be practical.) In
practice the simplicity of a description or a theory is measured by the
time it takes to explain this algorithm.
Now, it is clear that the simpler descriptions or theories are
easier to communicate and easier to remember. Hence, when evolution
developed our reason and speach it endowed us also with a preference for
simpler theories. And it gave us also a desire fo good generalisations (or
for making inductive inferences). This may stem from the same instinctive
preference of the simplest (shortest) descriptions. {It appears also
that simple general descriptions have a better chance to be true (that
is to satisfy (A)) than the more complicated ones. This is suggested by
the fact that reality is in some sense, simple or, to some extent,
learnable. "God is subtle but not mischievious". (E.g.: we have no
physical interpretations of non-Lebesgue measurable functions.)}
The need for the above definition of what is convincing is seen
from the fact there are many unconvincing philosophical theories which are
in agreement with facts, and the only property which distinguishes them
from the convincing ones is (B). E.g. solipsism which says that the world
is a product of my imagination introduces a distinction between me and the
others (I am real, they are only imagined). Thus it is more complicated
than realism which does not postulate this distinction. I do not know any
other reason for which solipsism is unconvincing or even ridiculous. We
can explain in the same way the convincing nature of the identity thesis
(that mental life, i.e., the process of thoughts and emotions, is an
electrochemical process in the brain). Whenever any equation or
equivalence between two concepts appears to be consistent with our
experience, it simplifies our view of the reality, and hence it is
convincing (and its detractors are unconvincing untill they provide a
clear counterexample).
Finally I will add to my definition the following:
OBSERVATION. We have a psychological mechanism M which
automatically (independently of will) puts into the realm of our
convictions every description or theory which is interesting to us and
appears to satisfy (A) and (B).
For this reason people with a good mind are not sceptics since
their M is active. But in people with a critical minds the realm of their
knowledge equals the realm of their beliefs. And it is impossible to
force oneself to believe thing which do not satisfy in our minds the
conditions (A) and (B).]
Thus the ZFC-definition of mathemnatics is convincing to me
because it is the only one I know which satisfies (A) and (B).
6. Why do I believe that ZFC is consistent? Let me quote my
answer from a private letter to Moshe Mahover (from March 19, 1999).
"When we read with understanding the axioms of PA or of ZFC, we
see in our minds a small part of the Skolem hull of those theories. That
hull grows in such a regular way (we can say almost periodic way), that it
is inconceivable that it could collapse (that 0 = 1). Thus we make the
inductive inference that it will never collapse no matter who (or which
machine) will apply himself to construct it. We have a psychological
mechanism [see OBSERVATION above] which (independently of our will) the
results of such mental experiments into the realm of our convictions
(untill somebody shows us that we are wrong), just like we have such a
mechanism which puts into our convictions generalisations of other
physical observations (if the statements of those generalisations are
present in our minds).
"Now you may say, but this is just what I call intuition. Yes, but
I have I have explained a process which builds legitimate intuitions,
reduced it to certain experiments.... Now there is a difference between
legitimate intuitions based on such mental experiments (however puny) and
arbitrary claims. E.g., to be convinced that ZFC is consistent (from
reading its axioms) one has to have a normally functioning imagination and
one has to put it to work.
"I believe that no legitimate intuition which is not based on any
heuristics is possible. Of course gambling is possible, and it is fun. For
example I may propose a mathematical problem just because its statement is
pretty (say, simple and general) but If I say that such and such is true
without any supporting evidence then I am lying (even if I have no
counter-evidence).
"For example we do not feel intuitively that Frege's unrestricted
set existence principle is consistent [not because of Russell's paradox
but] because with V amember of V the Skolem hull gets a loop which
destroys its regularity (periodicity). A similar difficulty (failure of
intuition) occurs ... with Quine's NF."
7. The idea that ZFC is an arbitrary game of symbols can be
refuted as follows.
(a) Logic is a mathematical (abstract) description of natural
languages. Natural languages are real things. Thus logic is a branch of
applied mathematics.
(b) The simple theory of types is a natural extension of logic,
still very close to natural language.
(c) Set theory (ZFC) is a natural extension of the simple theory
of types.
Thus to infer from the ZFC-definition of mathematics that
mathematics is an arbitrary game of symbols is a non sequitur.
8. My critical remark about the Platonism of Godel can be
expressed more fully as follows.
(a) Godel does not seem to have considered in his philosophy of
mathematics the significance of Skolem's functions or of Hilbert's epsilon
calculus (which is a way of naming (formalizing) the Skolem functions).
In particular he does not seem to have considered the possibility that the
Skolem paradox of the existence of a countable model of set theory
(consisting of epsilon-terms) yields a mathematical description of the
ontology of pure mathematics.
[Godel used the Skolem functions in an essential way in his proof
of the consistency of GCH. But it appears that Godel the mathematician
did not speak to Godel the philosopher on this topic. (Such a case of
non-association may seem strange, but in fact such things happen quite
often. The human mind is far from perfect in its associative ability.)]
(b) Godel does not seem to have considered the possibility that
the brain (influenced by sensory experience) is the sole agent
constructing mathematics. He thought that there is another input, an
intuition about a Platonic reality (see my discussiion of intuition above
in section 7). But the first possibility is better since assumes less
and it explains more.
9. I have been told that my claim that Godel, Russell and
Wittgenstein are responsible in a large measure for the growth of a
philosphical literature about mathematics (and the mind-body problem)
which ignores the concrete achievements of the past (of Hilbert, Skolem
and Turing) and proposes inferior solutions of problems which were
convincingly solved over 70 years ago was wrong. Since I still believe
that they are responsible, let me elaborate.
First, look at the collection "The philosophy of Mathematics"
(Editor W. D. Hart, Oxford Univ. Press, 1996). As much as I was able
to read them, all papers there are inconsistent with (or ignore)
rationalism (the philosophy described above). And some are very unclear,
they violate not only (B) but also (A). Thus the bad state of the
literature on philosophy of math. which I was deploring in my letter of
Dec. 29, 1999, is real.
In "A history of Western Philosophy"(1945) Russell does not
mention Skolem, Hilbert, or Turing. And Wittgenstein ("Philosophical
Investigations" (written between 1946 and 1949) and "Culture and Value"
(written from 1914 to 1951)) does not mention them either. Therefore the
seriousness of their search for an understanding of the human mind (or of
mathematics) whenever they write about it is problematic. All the material
upon which I am drawing in this and the previous letter was available well
within the active years of those philosophers. They must have been aware
of universal Turing machines (UTM's) and of the analogies
(UTM)/(its program) = (brain)/(its knowledge),
(UTM)/(its input-output tape) = (organism)/(its environment).
It seems unlikely that they could have been unaware of such striking
insights of a person they knew. It is more likely that they consciously
avoided anything mathematical, and that they forgot that "mathematics is
the language of nature" and that one cannot explain much without it.
Wittgenstein's aims and method were definitely those of an artist. But,
philosophers took his writings as contributions to knowledge. Hence, by
ignoring solid knowledge, and recording his sensations rather than
explaining the reality, he degraded the concept of philosophy. And by
their authority and literary talents both of them gave the green lights to
a vast philosophical literature which is completely divorced from science
but is written under the illusion that it contributes to knowledge.
E.g. G. Ryle ("The concept of mind" 1949) wrote that Descartes
believed that a human being has a non-physical soul because he made a
logical error in his attempt to explain the human brain. Ryle called it "a
category mistake". Ryle did not write that in the times of Descartes
nobody knew about the complexity of the brain or about things as small as
the brain's neurons. And Ryle does not quote in this context any
scientific discoveries in anatomy, physiology or automata theory,
discoveries which give us a much bigger and more relevant picture of the
physical reality than that of Descartes.
Other "philosophers of language" followed and still follow the
path of Russell and Ryle. E.g., R. Armstrong ("A materialist theory of the
mind" 1968) does not mention any investigations of the brain, and (like
Ryle) he does not make any contribution to the knowledge of the mind. A
similar literature continues to grow till today with quite extravagant
additions made by the postmodernists (see Sokal & Bricmont, l.c.).
I believe that, in the Twentiest Century, such books were made
acceptable in the community of philosophers by a tradition deriving from
Russel and Wittgenstein.
[This is not to say that earlier philosophers were innocent of
similar sins. We know that already Plato married philosophy with
literature. But his writings survived precisely because they ware a
mixture of literary art and philosophy accesible to many readers, while
important scientific texts of other authors (e.g. Apolonius and
Diophantus) were lost. In modern times Hegel may have been the most
influential antirationalist philosopher, he had a huge success as a
lecturer and writer. And it appears that even his legacy is still living.
Why is it so? The modern academia has created a large arena for
the discussion of philosphical problems and it hired many actors for this
job. But at some point the unity of knowledge became more and more
striking and it became necessary to use mathematics in order to express
this knowledge. Philosophers who did not know mathematics found themselves
in trouble. And they tried to build dubious topics where they could still
shine. It happens that those topics do not advance human knowledge.
Still I think that this unfortunate state of affairs in the
academia could be alleviated if philosophers acted as critics. First
giving a critical appraisal of the history of philosophy (and of the
sinister influences of politics upon philosophy) and second criticising
other areas of knowledge where the kno-how is overvalued, and fundamental
problems are forgotten or insufficiently appreciated. [For example, I
think that this is exactly the case in modern physics, where the know-how
of quantum mechanics has largely suppressed the fact that it all stands on
the shoddy logic of wave-particle duality; that the fundamental question
why the absolute square of the wave function has such and such a
statistical significance is largely ignored.]
Perhaps, for the sake of justice one should add that it would be
wrong to judge art by the ammount of information which it carries, and a
piece of bad philosophy may have a redeeming artistic value. However, it
would be strange to read philosophy only for its literary style. I for one
regard it as a branch of knowledge which is in disaray. However, as I have
just mentioned, I believe that deaper philosophical criticism (and not
silly contestation of scientific knowledge) is extremly desirable and has
a great future.
10. But let me recall once more what I believe to be the main
problem which is still open in philosophy of mathematics (or in logic):
Explain how the brain finds proofs of fully stated conjectures in well
defined axiomatic theories!
Jan Mycielski
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