Uploaded by GaryGeckDotCom on 02.01.2011

Transcript:

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]

[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]