Cantor’s Philosophy in Light of Pythagoras,
Plato and Aristotle. The Many and the One
[Song: Infinity...where do we go from here?
Infinity...]
Prior to Georg Cantor, the actual infinite
was often considered taboo. To quote Rüdiger
Thiele in Mathematics and the Divine on page
528:
quote “Aristotle (384–322BC), René Descartes
(1596–1650), Blaise Pascal (1623–1662),
and Carl Friedrich Gauß (1777–1855), just
to mention a few names, had rejected the actual
or complete infinite as unknowable and avoided
its application like the devil avoids holy
water.” unquote
Dauben echoes this view on page 120 of his
book. And on page 121, it is Aristotle who
Cantor blames above all else for the opposition
to the actual infinite throughout the ages.
Cantor felt that the conceptions one applies
to the finite should not be assumed to hold
for the infinite necesarily. And it was Aristotle’s
big mistake to think that the same logic applied
to both. More on this later.
Cantor belongs, not to an Aristotelian school
of thought, but rather to a Pythagorean-Platonic,
and Neo-Platonic school. And according to
Thiele, Cantor is closer to Leibniz and Bolzano.
Spinoza also had a very special place in Cantor’s
philosophy as a paper by Dr. Anne Newstead
that goes into much more detail on this.
Dr. Newstead also brings up the importance
of Cantor’s belief in some form of the principle
of plentitude. this is summed up as “what
is possible is actual” or what is consistent
with the real is also real. It’s the principle
of Parmenides when he said that “what can
be is”.
Dr. Newstead is right, I feel, as Cantor wrote,
when justifying how he was introducing the
new mathematics concepts that we will discuss
shortly, “we may regard the whole numbers
as ‘actual’ in so far as they, on the
ground of definitions, take a perfectly determined
place in our understanding, are clearly distinguished
from all other constituents of our thought,
stand in a definite relations to them, and
thus modify, in a definite way, the substance
of our mind.” (Contributions p 67).Mathematics
and abstract thought, to Cantor was free so
long as it’s well defined notions were consistent
with the whole rest of the universe of our
thoughts and free from contradictory. Hmm
i have to put an * on that and will discuss
later.
Cantor’s definition of a set as an object
of our thoughts or intuition is quite different
from what we might hear today in an English
speaking mathematical circles. And still,
an even earlier definition Cantor gives of
a set is even more philosophical in nature.
I am referring to Cantor’s Contributions
to Foundations of a General theory of Manifolds
(usually just called the Grundlagen),This
earlier definition not only associates a set
with a Platonic formal Idea, this work includes
a reference to one of Plato’s last dialogues,
written when Plato was in his 70s and represents
Plato'smature philosophy..I am speaking of
The Philebus. This is the same dialogue which
speaks of a "divine method" of dialectic by
which the human mind can “...discover the
existence of permanent, unchanging, real objects—a
method perhaps intended [Sa, p. 133] to be
the reverse counterpart of the (divine) method
of creation by which the system of Platonic
Forms is originally composed.”
[quote from The Mathematical Intelligencer
Volume 27, Number 1, 8-20, Plato and analysis,
by W. M. Priestley]
In the June 2010 editio n of The Review of
Metaphysics (Issue 252), the esteemed scholar
Professor Kai Hauser of Berlin presented a
paper which is freely available online called
Cantor’s Concept of Set in the Light of
Plato’s Philebus centered around this older
Platonic definition of Cantor’s:
Quoting Dr. Hauser:
Here, however, we shall focus on an earlier
"definition" that intimates concerns about
the ontological status of collections. It
appeared in Cantor's Grundlagen einer allgemeinen
Mannichfaltigkeitslehre (2) which summed up
the quintessence of his deepest contribution
to mathematics--the transfinite numbers--both
from a mathematical and a philosophical point
of view. In an end note Cantor emphasizes
that Mannichfaltigkeitslehre is to be understood
in a much more encompassing sense than that
of the theory of sets of numbers and sets
of points he had been developing up to that
stage.
For by a "manifold" or "set" I understand
in general every Many which may be thought
of as a One, i.e., every totality [Inbegriff]
of determinate elements that can be united
by a law into a whole. He then goes on to
explicate this further by drawing upon Plato's
Philebus.
And with this I believe to define something
related to the Platonic eidos or idea, as
well as that which Plato in his dialogue "Philebus
or the Supreme Good" calls mikton. He contraposits
thisagainst the apeiron, i.e., the unlimited,
indeterminate which I call inauthentic-infinite,
as well as against the peras, i.e., thelimit,
and he declares the [mikton] an orderly "mixture"
of the latter two. That these two notions
are of Pythagorean origin is indicated by
Plato himself;
Pythagoras, like Plato was said to have travelled
to Egypt to learn the Egyptian mystery schools
of Thoth. Pythagoras travelled the ancient
world in search of knowledge and is even said
to have travelled to India and it’s possible
he took back some ideas such as the Pythagorean
theorem from India along with introducing
the ancient West to Hindu Ideas. Of course
these events have been blurred by history.
The Grundlagen, from which Cantor’s Platonic
definition above appears, is in Dr. Hauser’s
eyes, Cantor’s highest moment. And it is
at this same time that Cantor is at his most
philosophical. I now want to now quote Godehard
Link in the Introduction to a great work he
edited called 100 years of Russell’s paradox.
You can get this book on Garygeck.com in the
books section:
Quote: “Mainstream mathematicians did not
only care little about philosophy but considered
even Cantor, who was really one of them, as
too philosophical to be taken seriously. It
is a telling historical detail that Felix
Klein’s widely read lectures on the development
of mathematics in the nineteenth century [45]
mentions Cantor only in passing. [Bertrand]
Russell clearly expressed this state of affairs
in My Philosophical Development [80], where
he says:
quote: The division of universities into faculties
is, I suppose, necessary, but it has had some
very unfortunate consequences. Logic, being
considered to be a branch of philosophy and
having been treated by Aristotle, has been
considered to be a subject only to be treated
by those who are proficient in Greek. Mathematics,
as a consequence, has only been treated by
those who knew no logic. From the time of
Aristotle and Euclid to the present century,
this divorce has been disastrous.”
In addition to Dr Hauser’s paper, I also
want to bring attention to a paper written
by Dr. Chris Menzel. Dr. Menzel’s paper
is called Cantor and the Burali-Forti Paradox
which is available on his website. It also
explores Cantor’s mention of the Philebus.
Quoting Dr. Menzel’s paper:
In the Philebus Plato's use of 'peras' and
'apeiron' reflects their Pythagorean roots.
In the well known "gift of the gods" passage
he writes that "all things ... that are ever
said to be consist of a one and a many and
have in their nature a conjunction of Limit
(peras) and Unlimited (apeiron)." In a later
passage he reiterates this, saying that "God
had revealed two constituents of things, the
Unlimited and the Limit." It is in this latter
passage that 'meikton' and its cognates appear.
The passage concerns itself with a division
of "all that now exists in the universe" into
four classes. The first consists of all instances
of apeiron, i.e., the various apeira in the
world, and second all instances of peras,
the various perata. The third class consists
of meikta, the products that result from the
union or mixture of members of the first two
classes, and the last contains the causes
of the mixtures."
But to Plato, the apeira are phenomena which
can vary continuously. A phenomenon is a variable
among the infinite continuum of possible values.
This continuum, taken as a whole, is unbounded
and absolutely indeterminate.
The class of perata, on the other hand are
limited in how they vary. They are discrete
but they are determinate because they can
have equality or relationships between them
like the natural numbers. So paraphrasing
Dr. Menzel, the perata is associated with
the class of Rational Numbers.
Plato’s third class is made up of meikta
which are mixtures of apeira and peras.
The classical example is that of Pitch and
tempo being examples of apeira. The 12 notes
of the muscial scale, seen generally however,
can be represented in simple ratios between
rational numbers or as perata. These 12 notes
are very special ratios of numbers. They are
not arbitrary, however, a 12 note scale can
be tuned arbitrarily to any base frequency.
For example, in the classical world, music
was tuned to 432 Hz. In more recent times
the Nazi propaganda ministry (of all people)
favored a tuning of A = 440 Hz and the world
has followed since. I have noticed a movement
on YouTube to bring back the 432 Hz but i
digress.
So the 12 note scale tuned to a decided on
frequency is an example of meikton. It’s
a mixture of apeiron and peras. This meikton
and other meikton merge to form “the whole
art of music”.
Previously Infinity had been viewed as apeiron
and was unknowable. But to Cantor, sets were
similar to meikta. Dr. Menzel corresponds
Cantor’s principles of (mental) number generation
with the principles of Plato’s classes of
being. Much like music to the Pythagoreans
represented a sort of order taming the Infinite
making it intelligible, Cantor saw his sets
as being intelligible objects arising from
the Absolute.
Rather than an indeterminate jumble of notes,
Cantor's infinities were a well ordered sequnce
like a well ordered Beethoven SOnata.
But there was still one set which even Cantor
could not precisely extract from the world
of the Apeiron. and this is the set of all
sets or the set of everything which will remain
at least mathematically indeterminate. Knowable
only through deeper methods of thought perhaps.
Going back to the principle of Plentitude
(What is possible is actual), one can imagine
that the so-called “Ontological” proofs
of God’s existence like those ones developed
by Leibniz and Kurt Gödel would have been
valid to Cantor. To Cantor's way of thinking,
known truths are possible only because they
are consistent with God's mind not the other
way around. This is a key reversal. To Cantor,
all that can exist is what is possible in
the Mind of God. This is 180 degrees flipped
from saying God is possible because of known
truths.
-Cantor’s approach to grasp infinity was
to rethink our concept of numbers. Cantor
was only able to do this because of the deeper
philosophical and epistemological approach
to mathematics.
Cantor saw the counting numbers that we learn
as school children (1, 2, 3, 4. etc.) not
only as an ordered sequence of unique objects
naturally arising in our minds, but to also
think of them as aggregates or as they are
called in modern times, ‘sets’ the size
or cardinality of which was significant. The
German word Cantor uses for a set was “Menge”.
The central notion is that a set is a collection
or aggregate of members brought into a whole
or unity. In addition to the members, every
set has the empty set which is a pure abstraction
of this notion of unity into a whole of all
of the parts-even with its members removed.
[Grrr grrr woof]
