PLATONISM, INTUITION
AND
THE NATURE OF MATHEMATICS
Continued from #1.
Studying mathematics Plato came to his surprising philosophy of
two worlds:
the "world of ideas" (strong and perfect as the "world" of geometry) and
the world of
things. According to Plato, each thing is only an imprecise, imperfect
implementation
of its "idea" (which does exist independently of the thing itself in the
world of ideas).
Surprising and completely fantastic is Plato's notion of the nature of
mathematical
investigation: before a man is born, his soul lives in the world of ideas
and afterwards,
doing mathematics he simply remembers what his soul has learned in the
world of
ideas. Of course, this is an upside-down notion of the nature of
mathematical method.
The end-product of the evolution of mathematical concepts - a fixed
system of
idealised objects, is treated by Plato as an independent beginning point
of the
evolution of the "world of things".
Nevertheless, being an outstanding philosopher, Plato tried to
explain (in his
own manner) those aspects of the human knowledge which remained
inaccessible to
other philosophers of his time. To explain the real nature of idealised
mathematical
objects, Greeks had insufficient knowledge in physics, biology, human
physiology
and psychology, etc.
Today, any philosophical position in which ideal objects of human
thought are
treated as a specific "world" should be called platonism. Particularly,
the philosophy
of working mathematicians is a platonist one. Platonist attitude to
objects of
investigation is inevitable for a mathematician: during his everyday work
he is used
to treat numbers, points, lines etc. as the "last reality", as a specific
"world". This sort
of platonism is an essential aspect of mathematical method, the source of
the
surprising efficiency of mathematics in the natural sciences and
technology. It
explains also the inevitability of platonism in the philosophical
position of
mathematicians (having, as a rule, very little experience in philosophy).
Habits,
obtained in the everyday work, have an immense power. Therefore, when a
mathematician, not very strong in philosophy, tries to explain "the
nature" of his
mathematical results, he unintentionally brings platonism into his
reasoning. The
reasoning of mathematicians about the "objective nature" of their results
is, as a rule,
rather an "objective idealism" (platonism) than the materialism.
A platonist is, of course, in some sense "better" than the
philosophers who
consider mathematical objects merely as "arbitrary creatures of human
mind".
Nevertheless, we must distinguish between people who simply talk about
the
"objective nature" of their constructions, and people who try to
understand the origin
of mathematical concepts and ways of their evolution.
Whether your own philosophy of mathematics is platonism or not,
can be
easily determined using the following test. Let us consider the twin
prime numbers
sequence:
(3, 5), (5, 7), (11, 13), (17 ,19), (29, 31), (41, 43),
...
(two prime numbers are called twins, if their difference is 2). In 1742
Chr.Goldbach
conjectured that there are infinitely many twin pairs. The problem
remains unsolved
up to day. Suppose that it will be proved undecidable from the axioms of
set theory.
Do you believe that, still, Goldbach's conjecture possesses an "objective
truth value"?
Imagine you are moving along the natural number system:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
...
and you meet twin pairs in it from time to time: (3, 5), (5, 7), (11,
13), (17 ,19), (29,
31), (41, 43), ... It seems there are only two possibilities:
a) we achieve the last pair and after that moving forward we do
not meet any
twin pairs (i.e. Goldbach's conjecture is false),
b) twin pairs appear over and again (i.e. Goldbach's conjecture
is true).
It seems impossible to imagine a third possibility ...
If you think so, you are, in fact, a platonist. You are used to
treat the natural
number system as a specific "world", very like the world of your everyday
practice.
You are used to think that any sufficiently definite assertion about
things in this world
must be either true or false. And, if you regard the natural number
system as a specific
"world", you cannot imagine the third possibility that, maybe, Goldbach's
conjecture
is neither true nor false. But such a possibility will not surprise us if
we remember
(following Rashevsky [1973]) that natural number system contains not only
some
information about the real things of the human practice, but it also
contains
many elements of fantasy. Why do you think that a fantastic "world" (some
kind
of Disneyland) will be completely perfect?
As another striking example of platonist approach to nature of
mathematics let
us consider an expression of N.Luzin from 1927 about the
continuum-problem
(quoted after Keldish [1974]):
"The cardinality of continuum, if it is thought to be a set of
points, is some
unique reality, and it must be located on the aleph scale there, where it
is. It's not
essential, whether the determination of the exact place is hard or even
impossible (as
might have been added by Hadamard) for us, men".
The continuum-problem was formulated by Georg Cantor in 1878:
does there
exist a set of points with cardinality greater than the cardinality of
natural numbers
(the so called countable cardinality) and less than the cardinality of
the continuum
(i.e. of the set of all points of a line)? In the set theory (using the
axiom of choice) one
can prove that the cardinality of every infinite set can be measured by
means of the so
called aleph scale:
A0 A1 A2 ... An An+1 ... Aw ...
|___|___|_ ..._|___|__ ... __|__ ...
Here A0 (aleph-0) is the countable cardinality, A1 - the least
uncountable cardinality
etc., and Aw is greater than An for every natural number n .
Cantor established that A0<c (c denotes the cardinality of
continuum), and
then he conjectured that c=A1. This conjecture is called
continuum-hypothesis. Long-
drawn efforts of Cantor itself and of many other outstanding people did
not lead to
any solution of the problem. In 1905 D.Koenig proved that c is not equal
to Aw, and
that was all ... .
Now we know that the continuum-problem is undecidable if one uses
commonly acknowledged axioms of set theory. Kurt Goedel in 1939 and Paul
Cohen
in 1963 proved that one can assume without contradiction any of the
following
"axioms":
c=A1, c=A2, c=A3, ...,
and even (as joked N.Luzin): c=A17. Thus, the axioms of set theory do not
allow to
determine the exact place of c on the aleph scale, although we can prove
that
(Ex) c=Ax,
i.e. that c "is located" on this scale.
The platonist, looking at the picture of the aleph scale, tries
to find the exact
place of c ... visually! He cannot imagine a situation when a point is
situated on a line,
but it is impossible to determine the exact place. This is a normal
platonism of a
working mathematician. It stimulates investigation even in the most
complicated
fields (we never know before whether some problem is solvable or not).
But, if we
pass to methodological problems, for example, to the problem of the
"meaning" of
Cohen's results, we should keep off our platonist habits. If we think
that, in spite of
the undecidability of the continuum-problem "for us, men", some
"objective", "real"
place for the cardinality of continuum on aleph scale does exist, then we
assume
something like Plato's "world of ideas" - some fantastic "world of sets",
which exists
independently of the axioms used in reasoning of mathematicians. At this
moment the
mathematical platonism has converted into the philosophical one. Such
people say
that the axioms of set theory do not reflect the "real world of sets"
adequately, that we
must search for more adequate axioms, and even - that no fixed axiom
system can
represent the "world of sets" precisely. But here they pursue a mirage,
of course, no
"world of sets" can exist independently of the principles used in its
investigation.
The real meaning of Cohen's results is very simple. We have
established that
(Ex) c=Ax, but it is impossible to determine the exact value of x. It
means that the
traditional set theory is not perfect and, therefore, we may try to
improve it. And it
appears that one can choose between several possibilities.
For example, we can postulate the axiom of constructibility (see
Jech [1971],
Devlin [1977]). Then we will be able to prove that c=A1, and to solve
some other
problems, which are undecidable in the traditional set theory.
But we can postulate also a completely different axiom - the
axiom of
determinateness (see Kleinberg [1977]). Then we will be forced to reject
the axiom
of choice (in its most general form) and as a result we will be able to
prove that every
set of points is Lebesgue-mesuarable, and that the cardinality of
continuum is
incompatible with alephs (except of A0). In this set theory
continuum-hypothesis can
be proved in the following form: every infinite set of points is either
countable or has
the cardinality of continuum.
Both directions (the axiom of constructibility and the axiom of
determinateness) have yielded already a plentiful collection of beautiful
and
interesting results. These two set theories are at least as "good" as the
traditional set
theory, but they contradict each other, therefore we cannot speak here
about the
convergence to some unique "world of sets".
Our main conclusion is the following: everyday work is
permanently
moving mathematicians to platonism (and, as a creative method, this
platonism is
extremely efficient), but passing to methodology we must reject such a
philosophy deliberately. Most essays on philosophy of mathematics
disregard this
problem.
2. Investigation of fixed models - the nature of the mathematical method
The term "model" will be used below in the sense of applied
mathematics, not
in the sense of logic (i.e. we will discuss "models intended to model"
natural
processes or technical devices, not sets of formulas).
Following the mathematical approach of solving some (physical,
technical
etc.) problem, one tries "to escape the reality" as fast as possible,
passing to
investigation of a definite (fixed) mathematical model. In the process of
formulating a
model one asks frequently: can we assume that this dependency is linear?
can we
disregard these deviations? can we assume that this partition of
probabilities is
normal? It means that one tends (as fast as possible and using a minimum
of
postulates) to formulate a mathematical problem, i.e. to model the real
situation in
some well known mathematical structure or to create a new structure.
Solving the
mathematical problem one hopes that, in spite of the simplifications made
in the
model, he will obtain some solution of the original (physical, technical
etc.) problem.
After mathematics has appeared, all scientific theories can be
divided into two
classes:
a) theories, based on a developing system of principles,
b) theories, based on a fixed system of principles.
In the process of development theories of class (a) are enriched with new
basic
principles, which do not follow from the principles acknowledged before.
Such
principles arise due to fantasy of specialists, supported by more and
more perfect
experimental data. The progress of such theories is first of all in this
enrichment
process.
On the other hand, in mathematics, physics and, at times, in
other branches of
science one can find theories, whose basic principles (postulates) do not
change in the
process of their development. Every change in the set of principles is
regarded here as
a passage to new theory. For example, the special relativity theory of
A.Einstein can
be regarded as refinement of the classical mechanics, a further
development of
I.Newton's theory. But, since both theories are defined very precisely,
the passage
"from Newton to Einstein" can be regarded also as a passage to a new
theory. The
evolution of both theories is going on today: new theorems are proved,
new methods
and algorithms are developed etc. Nevertheless, both sets of basic
principles remain
constant (such as they were during life time of their creators).
Fixed system of basic principles is the distinguishing property
of
mathematical theories. A mathematical model of some natural process or
technical
device is essentially a fixed model which can be investigated
independently of its
"original" (and, thus, the similarity of the model and the "original" is
only a limited
one). Only such models can be investigated by mathematicians. Any attempt
to refine
a model (to change its definition in order to obtain more similarity with
the "original")
leads to a new model, which must remain fixed again, to enable a
mathematical
investigation of it.
Working with fixed models mathematicians have learned to draw
maximum of conclusions from a minimum of premises. This is why
mathematical
modelling is so efficient.
It is very important to note that a mathematical model (because
it is fixed) is
not bound firmly to its "original". It may appear that some model is
constructed badly
(in the sense of the correspondence to the "original"), but its
mathematical
investigation goes on successfully. Since a mathematical model is defined
very
precisely, it "does not need" its "original". One can change some model
(obtaining a
new model) not only for the sake of the correspondence to "original", but
also for a
mere experiment. In this way we easily obtain various models (and entire
branches of
mathematics) which do not have any real "originals". The fixed character
of
mathematical models makes such deviations possible and even inevitable.
To be continued. #2