B’OR HA’TORAH 9 (1995)
Zvi Victor Saks completed an MS in computer science at the State University
of N ew York at Buffalo and a PhD in mathematics in 1972 at Wesleyan Univer sity. Having taught on a professorial level at various universities, including the
University of Costa Rica and SUNY, Saks now conducts research and development in applications of artificial intelligence to solve real world problems at
Carnegie Group in Pittsburgh, Pennsylvania. He has just completed a program of
automated logistics distribution planning of commodities through a distribution
network for the US Army, and is now working on a program that automatically
schedules medical evacuation of wounded people from a military situation or a
natural disaster.
An internationally reputed topologist , Saks's speciality in abstract infinite
space prepared him well for learning and teaching Kabbala and hasidic philosophy. He learned two years at
the Hadar Ha'Torah Yeshiva and one year at Yeshiva Derekh Hayyim, both in Brooklyn Frequently lecturing at Habad activities - including the Ivy League Torah Study Program - Saks is currently teaching two
community classes, "Reintroduction to Judaism: An Adult Approach to Judaism for Beginners" and
"l:!asidism."
On the Nature of Truth in Mathematics
by Professor Zvi Victor Saks
This paper was originally delivered at the Second International B'Or Ha'Torah Conference. It appeared in
Hebrew in B'Or Ha'Torah 8H and has been revised by the author for publication here .
Introduction
Given that the focus of this conference is "Truth and Bias in Scientific Research," I am sure
many of you are wondering how these issues can relate to mathematics. After all, in mathematics, the results of one mathematician can be verified by anyone else who is competent in that
field. What room is there for bias and controversy in mathematics? As I will discuss in this
paper, there has in fact been a great deal of controversy in mathematics. In this century, some of
the greatest mathematicians have disagreed about what is valid mathematics. In addition, over
the centuries, mathematicians have dramatically changed their view of what mathematics is and
how it relates to "reality."
In the modem view, a distinction is made between pure mathematics and applied mathematics. 1 In pure mathematics, the axioms of a deductive system are arbitrary assumptions,
subject to the sole restriction that they must be consistent with each other, that is, they must not
lead to a contradiction. Theorems are logical deductions from the axioms, and all have an
implicit "if...then" form; if the axioms are true, then the theorems are true. There is no concern
about whether or not the axioms or the theorems are actually true.
1
Irving Adler. A New Look at Geometry (New York : John Day, 1966) page 78.
96
On the Nature of Truth in Mathematics
This is the prevailing view, although not
all mathematicians are comfortable with it.
For example, Richard Courant and Herbert
Robbins say that:
A serious threat to the very life of
science is implied by. the assertion that
mathematics is nothing but a system of
conclusions drawn from definitions and
postulates that must be consistent but
otherwise may be created by the free
will of the mathematician. If this description were accurate , mathematics
2
could not attract any intelligent person.
Actually, this description is accurate as far
as many pure mathematicians are concerned .
I would say that for the mathematics that I
specialize in, and surely for other branches of
mathematics, no relationship with reality is
required, or even relevant.
In applied mathematics, a pure mathematical system is used as a model for some "real
world" system of objects, in which the axioms are true. Then the theorems are true, and
the pure mathematical system can be used to
make inferences and calculations about the
real world system.
In this paper I shall go back to Greek
mathematics, specifically geometry, and discuss some of the highlights in the evolution
of geometry to the present with respect to the
issues of truth and validity. Then I shall visit
the domain of mathematical infinity, and discuss its evolution and the divergent views of
some of the greatest mathematicians of the
twentieth century regarding truth and validity.
Finally, I shall discuss the hasidic view and
offer some personal insights into these issues .
B'Or Ha'Torah
metry according to the system of Euclid . In
fact, Euclid has been called the most successful textbook writer in history, since Euclidean
geometry is still studied in high schools, with
rather minor variations from his original text.
In Euclid's view, axioms are "self-evident
truths," that is, a description of "reality."
There was no real distinction as far as truth is
concerned between pure mathematics and applied mathematics. Euclid had one controversial axiom , the so-called "parallel postulate ."
John Playfair gives the form of this axiom as
follows:
Parallel Postulate : Through a point
not on a given line, there is one, and
only one, straight line that does not intersect the given line .3 (See Figure 1.)
Figure 1
point
parallel line
given line
For around 2000 years , mathematicians tried
to prove that this axiom was a logical consequence of Euclid's other axioms and that Euclid should not have assumed it as an axiom .
The outstanding fact about all of these efforts
is that they failed. The major breakthrough
came from Karl Friedrich Gauss in about
1800. Gauss, who was probably the greatest
mathematician of all time, realized that no
one had successfully proved the parallel pos2
Geometry
Virtually all of us have studied plane geo-
3
Richard Courant and Herbert Robbins . What Is
Mathematics ? (Oxford , England : Oxford University Press, 1969) page XVII .
Adler, page 198.
B'Or Ha'Torah
tulate as a consequence of the other axioms
because it was really independent of the other
axioms. Gauss assumed a negation of the
parallel postulate, that there exist infinitely
many parallel lines through a given point that
is not on a given line, which are parallel to
the given . line. This was the first example of
non-euclidean geometry. The results may appear strange, but it has been shown that his
new kind of geometry is as valid as Euclid 's.
Thus Euclid was both vindicated and dethroned.4 Euclid was correct in assuming the
parallel postulate, since it was not a consequence of his other axioms. However, his
view that axioms are "self-evident truths" was
incorrect. Other alternative geometries are
possible and equally valid to his.
An interesting aspect of Gauss's discovery
is that he kept it secret, informing only a few
close friends by letter of his new geometry.
Irving Adler explains that Gauss kept his discovery secret because the then dominant philosophy of Immanuel Kant stated that "properties of space supposedly expressed in Euclidean geometry were necessary properties imposed on reality by the process of perception. "
Gauss was afraid to buck the tide and expose
himself to ridicule by people who were his
intellectual inferiors .5
Thirty or forty years later, other mathematicians, most notably John Bolyai, Nikolai
Lobatchevski, and Georg Riemann, developed different non-euclidean geometries and
publicized them. Riemann's geometry is probably the most important of these non-euclidean geometries , since it was used by Albert
Einstein as his model of the universe in his
theory of relativity. His geometry can probably most easily be described by the following example.
In Riemannian geometry, there are no parallel lines whatsoever. The simplest model of
On the Nature of Truth in Mathematics
97
this geometry is the surface of a sphere, for
which lines are defined as arcs of great circles. In other words, given two points on the
surface of a sphere, there is exactly one arc
going around the sphere, which - if filled in
to make a circular plane - would go through
those two points and the center of the sphere.
That arc is defined to be the line going
through those two points . As can be seen in
Figure 2, any two such lines intersect in two
points , so there are no parallel lines. Also
notice that any two of the lines in the picture
intersect in a right angle .6
Figure 2
p oint
point
The plane is th e ar c "fill ed in " whi ch goes through
the center .
Thus each of the angles in the triangle is a
right angle, and the sum of the angles in a
triangle is 270 degrees. That may
seem
strange, but it is a valid geometry, and the
correct geometry for the surface of a sphere.
4
5
6
Adler, page 196.
Adler, page 212.
Morris Kline. Mathematics in the Modern World:
Readings from Scientific American, with introductions by Morris Kline (San Francisco : W. H . Freeman, 1968), page 129.
98
On the Nature of Truth in Mathematics
As I mentioned before, Einstein used Riemannian geometry to formulate his theory of
relativity, and this is certainly one of the
greatest examples of applying pure mathematics to solve physical problems. There are a
variety of interesting issues concerning the
applicabili'ty of pure mathematics, which I
will discuss in the context of this example.
First of all, as Kline asks,
Was Riemannian geometry just right
for the theory of relativity? Almost certainly not. There is ample reason to believe that Einstein merely did the best
he could with the mathematics he found
available.7
In other words, for pure mathematics to be
applicable to a given situation, it must only
provide a good enough model. It need not be
perfectly applicable.
Kline goes on to argue that the reason
why Riemann's pure mathematics turned out
to be useful several generations later was that
Riemann was rooted in physical problems for
his research. 8 Since Riemann was trying to
understand physical space, it is natural that
Einstein found his work useful. But I don't
agree with Kline's opinion that pure mathematics must be grounded in physical reality
in order to possibly be useful in the future.
For example, let's look at what Philip David
and Reuben Hersch say about groups .
A group is an abstract mathematical
structure , one of the simplest and most
pervasive in the whole of mathematics.
The notion finds applications, for example, to the theory of equations ...to crystallography, to atomic and particle physics. The latter applications are particularly interesting in view of the fact that
in 1910 a board of experts including
B'Or Ha'Torah
Oswald Veblen and Sir James Jeans,
upon reviewing the mathematics curriculum at Princeton, concluded that
group theory ought to be thrown out as
useless. So much for the crystal ball of
experts. 9
Although group theory is quite abstract
and not grounded in physical reality, within a
generation it turned out to be tremendously
useful. I agree with the attitude just quoted
that it is impossible to predict which pure
mathematics some genius has produced will
be applicable in the future. History teaches us
that there is pure mathematics being produced
today which appears totally useless but some
day will become useful.
Mathematical Infinit y
For many mathematicians, including myself, modern mathematics begins with the revolutionary work of Georg Cantor, who formulated the theory of mathematical infinity in
the 1870s. Before Cantor, mathematical infinity was "infinity as a potential." For example ,
the sequence of positive integers 1, 2, 3,... is
infinite because there is no last number. But it
was only potential infinity because there was
no concept of reaching the end of the sequence. Cantor's basic contribution was that
mathematically infinite objects can be considered to be actual well-defined objects that can
be manipulated in many of the same ways as
finite objects. For example , the set of positive
integers {1, 2, 3,...} is an actual object. This
can be formulated as an axiom :
7
8
9
Kline, page 233.
Kline, page 234 .
Philip J. Davi s and Reuben Hersh . The Mathematical Experienc e (Boston : Birkhauser, 1981) page
204-205 .
B'Or Ha'Torah
Axiom of Infinity: There exists an infinite set, or more precisely, {1, 2, 3,...}
is a set, that is, an actual object.
Cantor proved many remarkable results,
including the notion of different levels of
mathematical infinity, so that some mathematical infinities are greater than others.
But paradoxes and contradictions were
found in Cantor's theory and there was a
"crisis in foundations" in mathematics. Some
mathematicians wanted to throw the whole
theory out the window. An extreme example
is L .E.J. Brouwer's intuitionism, which in the
words of Kurt Godel, "is utterly destructive in
its results." 10
Brouwer's school holds that "difficulties
arise only where the totality of integers is
involved in some way," rejecting the notion
of actual infinity. 11 Others felt that the basis
of the theory was sound and with proper
axioms; the core would be preserved and the
contradictions removed .
The great mathematician David Hilbert
said:
Where there is any hope of salvage,
we will carefully investigate fruitful definitions and deductive methods. We
will nurse them, strengthen them , and
make them useful. No one shall drive us
out of the paradise which Cantor has
created for us. 12
I would now like to discuss several specific aspects of the development of theory of
mathematical infinity.
Is Mathematics Complete ?
One of the important features of Hilbert's
program for mathematics was that mathematics is complete, that is, every mathemati-
On the Nature of Truth in Mathematics
99
cal problem is solvable. 13 This was shown
false in 1930 by Godel, who produced perhaps the most remarkable theorems ever proven . Godel proved that:
No axiomatic system containing arithmetic can demonstrate its own consistency, so we can never know for sure
whether our system is consistent.
Any such system must have true
statements which are unprovable within
the system.
These results were extremely disappointing to many mathematicians. They are a tremendous expression of the limitation of
mathematics. Thus mathematics is never
going to contain objective truth. Moreover,
for any finite system of axioms, there will
always be undecidable statements within the
system. Thus a mathematical system will always be describing only a limited portion of
its domain of investigation. On the other
hand, Godel's results are a wonderful expression of the vitality of mathematics to have
that level of self-knowledge and to be able to
pinpoint its limitations so precisely.
Now let's discuss the attempts made to put
Cantor's theory on solid foundation.
In 1905, Ernst Zermelo offered a list of
axioms for (infinite) set theory. He noticed
that the following assumption was made implicitly by Cantor and others, and proved
1
° Kurt
Godel. "What is Cantor's Continuum Problem?" in P. Benacerraf and H. Putnam , eds. Philosophy of Mathematics : Selected Readings (Englewood Cliffs : Prentice-Hall, 1964) page 261.
11
A. Heyting . Intuitionism, An Introduction (North
Holland, 1971) page 14.
12
David Hilbert . "On the Infinite" in Benacerraf and
Putnam, page 141.
13
Hilbert, page 150.
100 On the Nature of Truth in Mathematics
some remarkable
lences.
and unintuitive
equiva-
Axiom of Choice: Given any collection of nonempty sets, an element can
·be chosen from each set.
Intuitively the axiom of choice should be
true, because if a set is nonempty, then we can
simply choose an element from it. If the collection of nonempty sets is finite, then we can
simply pick a set, choose an element from it,
pick another set and choose an element from
it, and so on until we have chosen an element
from each set in the collection. On the other
hand, if the collection of sets is infinite, then
the method I have outlined here would not
work. Since we are finite beings, we can
never perform an infinite number of operations and make an infinite number of choices.
It is important to point out that the axiom of
choice is only necessary in the case when no
rule is available to choose an element from
each set. If a rule is available , then the rule
makes the choice.
Bertrand Russell explained the axiom of
choice as follows. Suppose that we have an
infinite set of people. If we want to take one
shoe from each person, then we can choose,
say, the left shoe. This is a rule which applies
to each pair of shoes and allows us to choose
one shoe from each pair. Thus in this case we
do not need the axiom of choice. However, if
we want to take one sock from each person,
then since the two socks are indistinguishable
from each other, there is no rule that applies
to each pair of socks that allows us to select
one sock from each pair. Thus in this case we
would need the axiom of choice to make the
arbitrary choices for us.
The axiom of choice is very controversial
and is interesting and important enough to
have a whole book written about it. 14 David
B'Or Ha'Torah
Hilbert wrote that the axiom of choice is "the
most attacked up to15the present in the mathematical literature ." Moore summarizes the
different degrees of opposition taken by various mathematicians who opposed the axiom
of choice. 16
***
I would like now to discuss some basic
questions:
Is the mathematics of actual infinity consistent?
From a philosophical point of view, what
allows us to make these assumptions ?
The consistency question was solved by
Godel in 1938 when he proved the following
result. If the regular axioms of set theory
(without the axiom of choice) are consistent,
then they remain consistent if the axiom of
choice is assumed. In other words, once we
assume the other axioms, from a consistency
point of view, we lose nothing by also assuming the axiom of choice. Since Godel had
already proven that an absolute proof of consistency from within the system is impossible,
this result of relative consistency is the best
result possible.
From a philosophic point of view, there is
wide divergence concerning the validity of
Cantor's theory of mathematical infinity. As I
mentioned before, Brouwer completely rejects it. At the other extreme , for Bertrand
Russell and many others consistency is the
only criterion. Since Godel showed that the
theory is as consistent as possible , from this
point of view, the system is valid . Godel
14
Gregory H . Moore . Zermelo's Axiom of Choice
(New York : Springer-Verlag , 1982).
15
Moore, Prologue.
16
Moore, pages 139-141.
B'Or Ha'Torah
defends the theory and, I believe, requires
more than just consistency. Godel writes:
However, . this (Brouwer's) negative
attitude toward Cantor's set theory, and
toward classical mathematics , of which it
is a natural generalization, is by no means
a necessary outcome of a closer examination of their foundations, but only the
result of a certain philosophical conception of the nature of mathematics, which
admits mathematical objects only to the
extent to which they are interpretable as
our own constructions or, at least, can be
completely given in mathematical intuition. For someone who considers mathematical objects to exist independently of
our constructions and of our having an
intuition of them individually, and who
requires only that the general mathematical concepts must be sufficiently clear for
us to recognize their soundness and the
truth of the axioms concerning them ,
there exists, I believe, a satisfactory foundation of Cantor's set theory in its whole
original extent and meaning , namely axiomatics of set theory. 17
On the Nature of Truth in Mathematics 101
a word, it requires the existence of G-d. This
was my feeling many years ago, when I was a
graduate student very far away from Jewish
observance. Now that I have become an observant Jew and have studied hasidic philosophy for the last 18 years, my conception of
G-d has expand d significantly, and I realize
that G-d is much greater than a being who can
make the axiom of choice work. Nonetheless,
I would like to explore with you the significant relationship between the validity of actual mathematical infinity and G-d.
Let's start with Cantor. Cantor was a deeply religious person who had an intimate relationship with G-d. He insisted that actual
mathematical infinity (which he called
"transfinite") exists because G-d exists . Cantor writes:
That an "infinite creation " must be
assumed to exist can be proved in many
ways ... . One proof stems from the concept of G-d . Since G-d is of the highest
perfection, one can conclude that it is
possible for Him to create a transfinitum
ordinatum. Therefore, in virtue of His
goodness and majesty we can conclude
that there is actually a created transfinitum. ...the transfinite not only expresses
the extensive domain of the possible in
G-d 's knowledge, but also presents a
rich and continually increasing field of
ideal discovery. Moreover, I am convinced that it also achieves reality and
existence in the world of the created , so
as to express more strongly than could
have been the case with a mere "finite
world" the majesty of the Creator following His Own free will. 18
Two important points in Godel's defense
of Cantor 's theory are:
l) Mathematical objects exist independently of humans.
2) We must be able to recognize the
soundness of our mathematics.
I would like to go one step further. What
would it take to make Cantor's theory of actual infinity valid in some real sense? Specifically with reference to the axiom of choice,
what would it take to make an infinite number
of individual, arbitrary choices? It requires
some type of Creative Infinite Intelligence . In
17
18
Godel, page 262.
Michael Hallett . Cantorian Set Theory and Limitation of Size (Claredon , 1984) pages 23-24 .
102 On the Nature of Truth in Mathematics
Cantor is saying that since G-d can create
infinity, then He did. Moreover, Cantor believed in the absolute truth of his set theory
because, as he once told Gosta Mittag-Leffler,
it had been directly revealed to him by G-d. 19
Joseph Dauben writes extensively about the
fundamental link between Cantor's deep religious convictions and his perception of
mathematics. Dauben also expresses surprise
that these religious convictions have received
so little attention in discussions on Cantor's
development of set theory.20
Hasidic Philosophy and Mathematical Infinity
I would now like to discuss some concepts
from hasidic philosophy that have a direct
bearing on the existence of actual mathematical infinity. With great pleasure I thank Rabbi
Simon Jacobson for helping me with the analysis and studying some of the referenced
works with me.
Several Lubavitch rabbis, including Rabbi
Jacobson, have asked me about a certain passage in the works of the third Lubavitcher
Rebbe, Menahem Mendel Schneerson,
known as the Tsemah Tsedek . The passage
seems to contradict the existence of actual
mathematical infinity:
B'Or Ha'Torah
where each troop is limited . These sources
would seem to contradict the Tsemah Tsedek 's
statement, because each world or troop is limited but there are infinitely many of them. 24
To resolve the seeming contradiction, the
present Lubavitcher Rebbe, Menahem Mendel Schneerson, writes that the fact that there
are infinitely many worlds has been revealed
to us by our sages, but there is no contradiction because G-d's power is above all limitations and contradictions. 25 There is a basic
premise that, in general, G-d creates the world
in ways that fit together with human logic. So
the Tsemah Tsedek's statement is true according to logic and will apply to any normal
situation to which logic applies. But since G-d
is completely unlimited, He can choose to
create in ways which contradict logic. In the
case of infinitely many worlds of troops, He
used His unlimited suprarational power to
create infinitely many of them, and then logic
no longer applies to these particular cases.
Let's recall that I am discussing the real
validity of actual mathematical infinity. My
claim is that since G-d has created infinitely
many worlds, then mathematics has the right
and the ability to postulate the existence of
actual infinity because it really does exist. In
19
.. .it is impossible that many finite
individuals should join together to form
21
an actual infinity.
This statement by the Tsemah Tsedek
seems to contradict the axiom of infinity, in
which the actual infinite set {I, 2, 3,...} is
composed of (infinitely many) individual
numbers. However, there are numerous references in hasidic literature which refer to G-d
22 23
having created infinitely many worlds. •
It is also stated in the Talmud that G-d
created infinitely many troops of hosts to
serve Him,
Joseph W. Dauben . Georg Cantor, His Mathematics and Ph ilosophy of the Infinite (Cambridge,
MA: Harvard University Press, 1979) page 232 .
20
Dauben, page 232.
21
Menahem Mendel Schneerson (the Tsemah Tsedek). Derekh Mitsvotekha (Brooklyn, NY: Kehot ,
1973) page 113 (in Hebrew) .
22
Yosef Yitshak Schneerson . Bosi L'Gani (Brooklyn ,
NY : Sichos in English, 1980) page 55.
23
Schneur Zalman Schneerson. Tanya (Brooklyn,
NY: Kehot, 1981) page 243.
24
Talmud, !l.agiga 13b. (Available in English from
the Traditional Press.)
25
Menahem Mendel Schneerson. Likutei Sichos , vol.
X (Br-;;oklyn, NY: Kehot , 1975) pages 178-179 (in
Hebrew) .
On the Nature of Truth in Mathematics
B'Or Ha'Torah
other words, the axiom of infinity is true
because the collection of worlds created by Gd is infinite .
What about other axioms involving mathematical infinity, for example, the axiom of
choice? Is the axiom of choice true? While Gd certainly could have performed constructions that would make the axiom of choice
true , is there any evidence that He actually
did so? In the same letter quoted above, the
Rebbe emphasizes that the world was created
by G-d to conform to logic, and that we can
make exceptions only in cases where the Torah has revealed to us that G-d made such an
exception. Thus G-d used His infinite power
and overcame human logic when he created
infinitely many worlds. But in other situations
in which the Torah did not reveal to us that
G-d made such an exception, we cannot assume that He did . Notice that this contradicts
Cantor's belief that just because G-d could
create actual infinity, then he did . Thus
although G-d certainly could have performed
constructions which would make the axiom
of choice true, I really don't know at this
point if He did or not, especially when the
axiom of choice is expressed in its full generality. Of course, these are difficult questions,
which I believe are worth more thought.
For further discussion on the relationship
between the mathematics of actual infinity
and hasidic philosophy, see my paper given at
the first B'Or Ha'Torah conference .26
In preparing this paper, I looked around
for contemporary professional literature linking G-d with the theory of actual mathematical infinity. In a fascinating book entitled The
Mathematical Experience, Philip Davis and
Reuben Hersh do just that, as follows:
Mathematics, then , asks us to believe
in an infinite set. What does it mean that
an infinite set exists? Why should one
103
believe it? In formal presentation this
request is institutionalized by axiomatization . Thus, in Introduction to Set Theory by Hrbacek and Jech, we read on
page 54: "Axiom of Infinity. An Infinite
set exists." Compare this with the axiom
of G-d as presented by Maimonides
(Mishneh Torah, Book l, Chapter 1):
The basic principles of all basic
principles and the pillar of all the
sciences is to realize that there is a
First Being who brought every existing thing into being.
Mathematical axioms have the reputation of being self-evident, but it might
seem that the axioms of infinity and that
of G-d have the same character as far as
self-evidence is concerned. Which is
mathematics and which is theology?
Does this, then, lead us to the idea that
an axiom is merely a dialectical position
on which to base further argumentation,
the opening move of a game without
which the game cannot get started? 27
While I agree with this perspective, I take
a positive stance. I believe, or rather I know,
that G-d exists and that actual infinity exists
because G-d created it. The main new idea in
this paper is that although we finite humans
could not construct actual infinity, G-d created an actual infinity of worlds, and that
therefore when mathematics asserts the existence of actual infinity, this assertion is true .
26
Zvi Victor Saks. "Applications of Mathematical Infinity on Jewish Philosophy" in A. Gotfryd, H .
Branover , and S. Lipskar , eds. Fusion : Absolute
Standards in a World of Relativity (Jerusalem/New
York: Feldheim, 1990) pages 123-142.
27
Davis and Hersh, pages 154-155.