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Music: [Here's my key. Philosophy. A freak like me, just needs infinity.]
This is Gary Geck of the GaryGeck.com website. I’m just a normal guy, but I wanted to make
this my very first YouTube video talking about the secret side of history.
Music: [Relax. Take your time. So take your time and trust in me and you will find infinity.
Infinity. Infinity.]
Today I’ll cover the Absolute Infinity, its contradictions and mathematics as explored
by Georg Cantor who lived 1845 to 1918.
Well, a lot of this stuff isn’t so secret, but why should just a handful of scholars
and grad students have all of the fun. So I’m going to be bringing some of this stuff
to normal guys and gals everywhere.
So today I want to talk, specifically, about the metaphysics of Georg Cantor. He is one
of the most important mathematicians who ever lived. I'm going to get into some of the mind-blowing
mathematics, but that stuff is talked about quite often. What I'm going to get into is
some of the deeper stuff that isn't talked about very often.
Cantor was the father of our mathematical understanding of Infinity today. He is also
the father of Set Theory. which he used to understand Infinities. Set Theory is foundational
today in not only mathematics, but also is important in philosophy, logic and even in
Computer Science as well. Although they usually leave out some of the metaphysics and the
mystical philosophy that Cantor clearly held and which partly led him to his discoveries.
I won’t be leaving those out today. in fact I will be focusing on them as a historical
study. This is not an attempt to endorse these views, but they deserve the light of day.
So this is my attempt to bring as honest and truthful of a study as I can of them in this
video.
To quote David Hilbert, “No one shall expel us from the paradise that Cantor has created
for us”.
But like so many other Geniuses, in his life, Cantor struggled to get a good job and his
work was often bitterly attacked by his critics.
Despite the fact that Cantor could rigorously demonstrate his most important ideas, they
were often far too radical for his time and so, at first, he had to bury his results in
unassuming ways to get them published among less radical topics. And even later on when
he had gained some fame later in life, he would have to strip out philosophical or metaphysical
statements especially in French and English translations. The preface of the English translation
of “Contributions to the founding of transfinite Numbers” (which will be called Contributions
here on out) refers to itself as “logically purified” Basically this means they forced
him to strip away much of the philosophy and metaphysics in his German writings.
Yet this very same preface also admits on page vi says quote:
“The philosophical revolution brought about by Cantor’s work was even greater, perhaps,
than the mathematical one. With few exceptions, mathematicians joyfully accepted the foundations
of Cantor’s undying theory. but very many philosophers combated it. This seems to be
because very few understood it.” unquote.
Many factors, including the caustic attacks he endured from his philosophical and mathematical
enemies, and the sudden tragic death, of his son contributed to the roughly 4 years Professor
Cantor collectively spent in mental asylums throughout his life. plus, many today believe
he suffered from manic depression. It is clear that Cantor felt alone in his interests and
was ignored and isolated from the larger mathematical community. By all accounts he was a loving
father and a husband but still, Cantor’s biography is generally not a very happy one.
Two excellent books that I used extensively to prepare for this youtube video are Georg
Cantor his mathematics and philosophy of the Infinite by Joseph Warren Dauben published
in 1979 by Princeton University press and an essay by Rüdiger Thiele on Cantor which
appeared in the book called Mathematics and the Divine published in 2005 by Elsevier B.V.
Both books are available at Gary Geck.com if you click on the ‘Books’ button at
the top of my site menu.
I have also studied Contributions in detail and it is now in the public domain and a link
to a free copy is also in the books section of GaryGeck.com. I have also sought out as
many primary works as possible and i urge everyone to use primary sources. Wikipedia
is great but it is only going to get you so far.
[Image: GaryGeck.com books section]
Now there are other YouTube videos about Cantor and infinity and I highly recommend them,
I especially like the videos titled “The Infinities between 1 and 2 by the Cellar Academic”
so i would like to give him a shout out. But my video is going to focus on a lesser known
story that is not yet on YouTube. That is: Cantor’s metaphysics and how Cantor viewed
his work as revealing the secret knowledge of God. So this will be some light-hearted
stuff.
[laughter].
To Cantor, God existed. Mathematics was God’s language and an extension of God’s possibilities
and power. Cantor accepted the actual infinite. This is in real dramatic contrast to the potential
infinite.
One of the underlying thesis’s of GaryGeck.com is that there are two main branches in Western
Philosophy but we are only really familiar with one of them. We are familiar with the
one that is very much opposed to metaphysics and respects physical science above philosophy.
The tradition of Plato, Leibniz, Cantor, Kurt Gödel saw things very differently and put
much more emphasis on the philosophy and metaphysics of mathematics as these3 individuals and members
of this tradition made their groundbreaking discoveries.
Now I'm generalizing here because as you get into the higher reaches of scholarship this
topic do come up. But to the average person the normal guys and gals this stuff is not
very well known.
My video series will expose many of the secrets of mathematics, many not even known but professors
and grad students. Here is one great example in the difference between potential and actual
infinity. Now it was believed that Cauchy he had finally put the calculus on a firm
grounding without recourse to metaphysics that had been smelled prior to Cauchy's formalization.
In fact we use Cauchy’s famous equation today when we learn Calculus 99% of the time.
We all know this equaiton.
But as Abraham Robinson pointed out in his 1969 article on the Metaphysics of The Calculus
(in a book also available on GaryGeck.com), what Cauchy did here was simply just put the
problem off further and no one seems to have noticed this. To quote Abraham Robinson when
talking about the weaknesses of perceived Leibniz’s method:
Quote “The weaknesses had been associated throughout with the introduction of entities
which were commonly regarded as denizens of the world of actual infinity. It now appeared
that Cauchy was able to remove them from that domain and to base Analysis on the potential
infinite (compare to Cantor and Carrucio). He did this by choosing as basic the notion
of a variable which, intuitively, suggests potentiality rather than actuality. And so
it happened that a grateful public was willing to overlook the fact that, from a strictly
logical point of view, the new method shared some of the weaknesses of its predecessors
and, indeed, introduced new weaknesses of its own.” unquote. that's from page 161
from the philosophy of mathematics that i will link to on my site.
In other words, the sober-minded mathematical public had been forced to stare at actual
infinity for over 100 years and absolutely despised it philosophically in the form of
an infinitessimal. But the works of Robinson and other in the 1960s showed that Leibniz’s
method was actually perfectly sound and rigorous. and that the attempts of Cauchy and others
to sweep infinity under the carpet and replace it with potential infinite variables was more
smoke and mirrors than anything rigorous. but yet we still use Cauchy’s methods today
in calculus textbooks, not Leibniz’s so I guess Robinson’s words fell on deaf ears.
But I digress.
I would like to make a quick footnote about Abraham Robinson. He really is such an interesting
guy that he deserves his own video, and perhaps you'll see one on GaryGeck.com soon enough.
I recommend a biography also by Joseph Warren Dauben called Abraham Robinson: The Creation
of Nonstandard Analysis A Personal and Mathematical Odyssey. You can get his book at GaryGeck.com.
This book makes an interesting parallel between Robinson and Cantor. On pages 354 to 355,
it makes the astute point that Robinson helped define and legitimize the infinitely small
just as Cantor did the same with the infinitely large in the form of the transfinite. That
Cantor rejected the infinitely small is especially ironic because, Robinson seems to have had
serious doubts about the actual infinite. So you have Robinson basically proving the
existence of the infinitely small while disbelieving the infinitely large and you have Cantor proving
the existence of the infinitely large while rejecting the infinitely small. Robinson relied
highly on Cantor’s work and he considered himself as operating in the “Set Theoretic”
tradition.
I would like to build on this by noting that both Cantor and Robinson were basically Platonists,
certainly knew their way around the works of Plato and even the Neo-Platonists. In fact
Robinson had a dual doctorate in mathematics and in Philosophy, so like Cantor he was highly
educated in philosophical matters. Both men can be described as generally well rounded
with interests in music, theatre, etc.. Both had happy marriages and loving wives. On page
98 of Dauben’s biography on Robinson, he writes, when talking about the discussion
Robinson had with Chimen and his wife, Miriam Abramsky while the three where staying together
through the horrible and deadly London Blitz:
quote. “Abramsky was impressed by Robinson’s wide range of interests which included, not
only philosophy but a very broad spectrum of German culture. Abby’s substantial knowledge
of Goethe was, he thought, remarkable. Robinson was just as happy discussing Maimonides which
Abramsky admitted, ‘again surprised me because Maimonides, I thought, was quite remote from
modern aspects of mathematics’. Bombs were falling outside, but inside there was much
song, hilarity. ” unquote.
Robinson was the head of mathematical departments in universities such as UCLA, Yale, the University
of Jerusalem, and several other world renown institutions.
Robinson was an advocate of Non-standard Analysis being the new way that mathematics and mathematical
physics would be done (following in a more Leibnizian tradition). It was far more intuitive,
he argued. to this day I know of many engineers who agree and use it in their projects simply
to save time. Many mathematical discoveries have been made using it, but afterwards those
discoveries have been translated back into traditional standard mathematics.
I’m sure Robinson is rolling in his grave.