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Date: Thu, 3 Nov 1994 15:44:26 +0200 (EET)
From: Karlis Podnieks <podnieks@mii.lu.lv>
To: QED discussions <qed@mcs.anl.gov>
Subject: Semantics
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K.Podnieks, Dr.Math.
podnieks@mii.lu.lv

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