Now, it's time for the final chapter in my "visits with old friends" series, which brings us
back to the Good Math/Bad Math all-time reader favorite crackpot: Mr. George Shollenberger.
Last time I mentioned George, a number of readers commented on the fact that it's cruel to pick on poor George, because the guy is clearly not all there: he's suffered from a number of medical problems which can cause impaired reasoning, etc. I don't like to be pointlessly cruel, and in general, I think it's inappropriate to be harsh with someone who is suffering from medical problems - particularly medical problems that affect the functioning of the mind.
But I don't cut George any slack. None at all. Because much of what spews from his mouth isn't the
result of an impaired mind: it's the product of an arrogant, vile, awful person. Since our last contact
with George, aside from the humorous idiocy, he's also taken it upon himself to explain how we'll never
have a peaceful society in America until we get rid of all of those damned foreigners, who have
"unamerican mindsets". That post was where I really started to despise George. He's not just a senile
old fool - he's a disgusting, horrible person, just another of the evil ghouls who used a horrible
event, committed by a severely ill individual, as a cudgel to promote a deeply racist agenda.
There's been some great humor coming from George, particularly in the areas of math and logic.
George claims to be a big fan of Cantor, and likes to babble about the transfinite numbers. Unfortunately, he doesn't really have a clue about them - and in fact, in one of his comment threads, he
admitted to never actually reading any of Cantor's work, but just having read a popularized biography of Cantor that briefly mentioned the idea of the transfinites. He continually comes up with new and ever-stranger ways of characterizing Cantor's work. For a recent example, last month, in this post, he said this:
The discovery of new truths about a monotheistic God by modern philosophers and scientists are correcting the teachings of today's religions. One discovery is the new truth about God's infinity. Specifically, in the 15th century, Nicholas of Cusa spoke of the new infinity as 'the infinite qua infinite' beyond all finite things. In the 19th century, Friedrich Hegel called it 'the 'Absolute' beyond all relative things and Georg Cantor called it 'the genuine infinity' beyond all bad infinities. In the 21st century, I call it 'the determinate infinite' beyond all indeterminate infinities.
Did Cantor talk about bad infinities? No. What Cantor did was define the basis of what became modern set theory, and then use it to construct two fundamentally different kinds of numbers using sets: Cardinal numbers (which measure size), and Ordinal numbers (which measure position). Then, looking at sets - like the set of natural numbers, he realized that there's a cardinal (a size) that describes
the size of the set of natural numbers. But that number can't possibly be finite. So there is at least
one "infinite" cardinal number. Then moving on from there, he was able to show that there must be
more than one infinite cardinal; and in fact, that there can be different sizes of infinities. But there's no value judgement in Cantor's studies: just the discovery of this seemingly
paradoxical (but fascinating) idea of infinite cascades of ever-larger infinities.
George is incapable of understanding that. But he continually insists that he understands
Cantor, and that we evil atheistic scientists and mathematicians (even the theistic Jewish ones like me),
are refusing to acknowledge the work of Cantor because of its supposed theological implications. For
example, here, he explains why we refuse to solve the worlds problems by ignoring the transfinite numbers:
Since mathematicians cannot extend the number domain beyond the transfinite numbers and into some higher world of numbers, it seems time for them to accept the infinite qua infinite and start developing Cantor's transfinite numbers. By developing the transfinite numbers, I expect science to uncover many hidden secrets such as new energy sources.
Alas for George, modern math is virtually all done in terms of constructions built using the ideas of
Cantor, and no serious math student can get through school without seeing at least a little bit of
Cantor's transfinites - at least enough to see the famous diagonalization argument. But aside from
the laugh value of that, it's a demonstration of how profoundly he doesn't understand Cantor - and how much he doesn't understand why he doesn't understand Cantor.
Cantor's conception of the transfinite numbers wasn't a religious thing. And it also was never
intended to be a practical description of a real thing in the world. It's an idea: a concept based
on understanding the meanings of numbers, and of understanding the different kinds of
numbers. It's a profound step to realize that even though the set of natural numbers is infinite,
it's infinitely small compared to the set of numbers that can't be written as fractions. That's an astonishing fact - even a useful fact for understanding the world. But that doesn't mean
that we're going to find a real-live infinite set with a transfinite cardinality - or that we're
going to find an object that occupies a transfinite ordinal position in an infinite set.
Another of George's humorous and lengthy series of diatribes is in the area of logic. George's
reasoning capabilities aren't exactly coherent. His "proofs" are frequently nothing but series of non-sequiturs. When some commenters pointed this out, he went into a bit of a rampage, posting numerous
articles like this one about how "sequiturs and non-sequiturs" don't matter, and how he doesn't ever misuse non-sequiturs:
Since the publishing of my book in late June 2006, logicians have been commenting on the website I use to teach complex subjects of my book. Unfortunately, they have also used my website to judge my book rather than reading it. In yesterday's blog on a critique of positivism, I was told for the nth time that I cannot use non sequiturs. This time, I decided to challenge the charge that I misuse non sequiturs. After considerable thought, I concluded that this charge is not true. Let me explain my conclusion.
In the first chapter of my book, I present the first and only proof I offer in this book. This proof of God's existence is made with one statement on page 6. The statement says --- all finite things are originated by an infinite thing. No other proofs are presented in the book. In the rest of the book, I identify and discuss the many thoughts that emerge as a result of this proof of God.
Based on my website writings alone, logicians are concluding somehow that I use non sequiturs in my book. This conclusion is false. If they read my book, they would learn quickly that only one proof is being claimed by me. All of my other writings in the book deal with the possible consequences of this single proof.
The error that logicians have been making on my book is due first to their non belief in God and second due to their ignorance of the difference between Scientific Languages and Talk Languages. The misuse of non sequiturs can occur only in a Scientific Language where a proof is being presented with fixed procedures. However, it is not possible to find a non sequitur in a Talk Language. They cannot be found in Talk languages because a Scientific Language and a Talk language are separated forever. These two symbolic languages cannot be unified in a world that has no end. The Talk Languages function as the world's engine of change. It is thus futuristic. On the other hand, the Scientific Languages is not futuristic and functions only to identify the permanence of the world.
In the ensuing comment threads, it became astonishingly obvious that George doesn't
know what a non-sequitur is, and that he's not the least bit interested in learning. For those
who actually have brains, a non-sequitur is a logical statement that does not follow from its
antecedents. For example, "Mark Chu-Carroll is Jewish, therefore Mark Chu-Carroll has green socks." is
a non-sequitur: it's structured as a logical inference statement "A implies B"; but the conclusion has no connection to the statement which precedes it - "Mark Chu-Carroll has green socks" cannot be inferred from "Mark Chu-Carroll is Jewish".
The post quoted above also demonstrates another of George's little foibles. In George's world, he's
one of the most important thinkers in the history of the world. He sees himself, to quote one of his comments, as "a philosopher, theologian, dialectician, and scientific theorist", and as one of the
greatest of all time. So when someone criticizes him, it can't just be that one lowly person
is daring to criticize the great and mighty George Shollenberger. It must be that there is a massive organized effort to attack him in order to suppress his great discoveries. So it's not "a couple of commenters" who said that he doesn't understand non-sequiturs", it's "the world's logicians"! Similarly, when I reviewed his book, it wasn't just "some geek reviewed my book and thought it was crap", but "the world's mathematicians are part of an atheistic conspiracy organized by the American Museum of Natural
History to suppress my proof of God!".
Finally, I'll conclude with another example of his vile idiocy. George believes that anyone who disagrees with him is, by definition, a criminal. Because, you see, George has discovered the perfect proof of Christianity as the only valid religion, and so if you disagree with his religious babblings, you are an atheist. What's more, according to George, because the Declaration of Independence mentions God,
that means that it's illegal to be an atheist - and that atheism should be a punishable crime. So ultimately, everyone who disagrees with George must be punished until they recant and accept his version
of Christianity. George is an inquisitor-in-waiting, dearly wishing for his opportunity to punish
all of us evil atheists.
The Declaration and the Constitution do not authorize atheism. The USA has no law of any kind that authorizes the practice of atheism. Having doubts about God is thinking not a practice of atheism. All atheistic practices are illegal and must be punished. In my experiences with atheists, atheists generally show no interest in developing knowledge of God. If they did express interests in God they would find God and would not be atheists. But, a nation under God cannot beg citizens to be believers. Every family in a civil society has personal and social responsibilities. They are also responsible for developing knowledge needed by a godly nation. With respect to prayers, my research on God/man communications shows that God and man do not talk together on a telephone but do exchange information. Thus, praying might be one way that man furnishes human information to a wise God.
Because of the Supreme Court ruling in favor of atheism, O!Hair built the American Atheists to protect the civil rights of nonbelievers. However, the rights of an atheist are arbitrary and are not in accordance with the Declaration. It thus seems that the American Civil Liberty Union (ACLU) might be a criminal organization whose purpose is to destroy the new political experiment and our beautiful Lockean civil society.
In a nation under God, it is not wrong to display on public land some samples of man's progress in increasing his knowledge. Such samples can teach the youth of a nation about the continual needs for theological and scientific progress. If a nation displays military statues, political leaders, technical advances, etc. in a public park, it is also right to display theological advances such as the Ten Commandments.
A more recent violation of the Declaration might have occurred when a group of atheists conspired to stop the propagation of my book. This book is propagated in order to teach new knowledge about God, theology, and science. Some of them began to stop the propagation of my book only one month after it was published in June 2006. Their effort was also aimed at assassinating my character.
That "group of atheists"? That would be me, and the commenters on this blog. And our "conspiring to stop the propagation of his book"? That would be me pointing out the giant, gaping holes in his argument, and the amorality of his behaviour with respect to that book. (If, as he claims, his book contains the secrets to world peace, cures of terrible diseases, an end to suffering, and a new age of global prosperity, then refusing to show it to anyone unless they pony up the money to pay him for the book is a thoroughly evil thing to do. If you have information that can save lives, and you withhold it in the name of personal profit, then by my moral standards, you're an
There is a recent biography of Georg Cantor which does make the claim that Cantor's investigations of the infinite did have a religious purpose and wasn't purely mathematical. It also made the claim that Cantor was not the first, and not the last, person to pursue the religious implications of the infinite and wind up insane. It mentioned Kurt GÃ¶del as a more recent example.
My personal feeling: Cantor's motivations, religious or not, have little to do with the importance in modern mathematics to his work.
Cantor was insane, however, and it is always a good idea to check and recheck any radical ideas of a madman before accepting them. Cantor's mathematical work has been examined and found to be a sound basis for modern set theory despite his mental illness. His religious work has not been so thoroughly examined, but is unimportant to modern mathematics.
So our resident old crank above doesn't have any support from me, except to suggest he didn't pull the Cantor-infinity-God connection out of his own posterior. Someone else's, maybe, but not his own.
Aha, joining in with us evil atheists in our conspiracy even though you're not really one of us, eh? In fact, didn't you write a whole post about that a while back?
Sorry that I can't give pinpoint citations to this, but I believe that Cantor, late in his life when he was accused of madness, did indeed suggest religious implications of transfinite arithmetic.
For instance, he is said to have suggested that even if there were aleph-null atoms in the world, there were at least aleph-one particles of mind or spirit in a human being, and that God was of an even greater infinity.
That does not make George Shollenberger right. The examples that you quote do strongly suggest that he has no clue as to what Cantor meant.
But we should not assume that we fully understood Cantor's intent or philosophy. I had quite an interesting conversation with Gregory Chaitin and Stephen Wolfram at the conference in Boston that I attended last week. Chaitin was quite convincing in pointing out things that Leibniz wrote that only now make sense to a modern Mathematician, and still in possible conflict with some deep musings of Feynman.
I wouldn't try to axiomatize this, but I rank the intelligence of Mark C. Chu-Carroll far above that of George Shollenberger, and Gregory Chaitin above myself and Mark C. Chu-Carroll, and Leibniz and Feynman above Gregory Chaitin. But then, from somewhere in the middle, how can I be sure? And what to do when twists are introduced by madness in various forms?
I quoted some Theomathematics I'd earlier posted on this blog when I debated the existence of God at Cal State L.A. a couple of weeks ago. Who's to say that such a field of research will not be accepted sometime in the future?
Cantor was insane, however, and it is always a good idea to check and recheck any radical ideas of a madman before accepting them.
Cantor was neither insane nor was he a madman, he suffered from clinical depression a widespread form of mental illness. Clinical depression has several causes but very commonly it is caused by a metabolic imbalance making it a physical illness of the same type as diabetes. If Cantor had lived at the end of the 20th century instead of the 19th his illness would probably been treated successfully. To call somebody who suffers from clinical depression insane or mad is both incredibly stupid and incredibly insulting.
The actual infinite arises in three contexts: first when it is realised in the most complete form, in a fully independent other-worldly being, in Deo, where I call it Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number, or order type. I wish to make a sharp contrast between the Absolute and what I call Tranfinite, that is, the actual infinities of the last two sorts, which are clearly limited, subject to further increase, and thus related to the finite.
Georg Cantor, Gesammelte Abhandlungen, Ed. Abraham Fraenkel & Ernst Zermelo, Berlin 1932. p. 378. Quoted from Randy Rucker, Infinity and the Mind, Boston 1982. p. 9.
I think a fairly clear statement of Cantor's views on the infinite that displays no signs whatsoever of "insanity" or "madness" in the accepted sense of the words. If you wish to call him insane merely because he believed in god then you will also have to call our excellent host MarkCC, who is also a theist, insane.
I think if one were to grant Herr Schollenberger a modicum of exaggeration then one could reinterpret Cantor's "limited" as "bad" thus, at least from his standpoint, justifying his comment. I personally wouldn't grant the obnoxious little toad the dirt from under my toenails!
I wrote: "accused of madness" because I agree with Thony C about Cantor's depression being misunderstood. Other mathematicians accused of madness have been devastated by the accusation, as with Grassman (whose nilpotent Grassmannians are in textbooks now), and even committed suicide.
I do not make light of mental illness nor of clinical depression. Again, unable to give proper citation, I've heard that Mathematicians have the highest per capita figures for mental conditions of any science, and Logicians the highest within that.
To correct the above citation:
Rudy Rucker (1995). Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton University Press. ISBN 0-691-00172-3.
Rudy von Bitter Rucker [22 March 1946-] has a PhD in Mathematical Logic [Rutgers 1972] and is one of the leading Science Fiction authors, a self-confessed "transrealist." He, Vernor Vinge, and Greg Egan are way ahead of me in the Math + Science Fiction ranking.
To correct the above citation:
Rudy Rucker (1995). Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton University Press. ISBN 0-691-00172-3.
I own a well thumbed copy of the 1982 BirkhÃ¤user edition and as that is the one I quoted it is the one I cited. It is a book that I would unreservedly recommened to all the readers of this blog. Amongst other things it has a fairly non technical explanation of transfinite arithmetic including some really meaty stuff on transfinite cardinals. It also contains a really good section on GÃ¶del's incompleteness theorems. A real gem is his discription of his meetings with GÃ¶del.
Randy Rucker does indeed write some very good (GÃ¶delian!) Sci-Fi which I very much enjoyed when I was still reading science fiction.
There is a BBC documentary about the 'madness' (as they put it) that plagued Cantor, Boltzmann, GÃ¶del and Turing. It's a tv documentary, so not really the best source of information, but somewhat enjoyable to watch if you don't want to think to much. Anyway, it also does claim that cantor's work was largely related to his religious beliefs.
There is also the fascinating article
"Russian Mathematical Mystics and French
Rationalists: Mathematics, 1900-1930"
which discusses the connection between
the Russian school of mathematics (Kolmogorov etc.) and the esoteric mysticism of the
Jesus Prayer (yes, the one in "Franny and
It closes with a great quote from Leibniz:
"There are surely two
labyrinths for the human mind: one is concerned
with the making of continuum, the
other with the nature of freedom, and they
are born both from the same infinity."
In his plenary speech at the 7th International Conference on Complex Systems (Boston, 28 Oct-2 Nov 2007) Gregory Chaitin argued that Leibniz anticipated Complexity Theory over 320 years ago, in a hypermodern yet religious context. He expounded on these quotations, hotlinks available from his home page:
"Dieu a choisi celuy qui est... le plus simple en hypotheses et le plus riche en phenomenes"
[God has chosen that which is the most simple in hypotheses and the most rich in phenomena]
"Mais quand une regle est fort composÃ©e, ce qui luy est conforme, passe pour irrÃ©gulier"
[But when a rule is extremely complex, that which conforms to it passes for random]
--- Leibniz, Discours de mÃ©taphysique, VI, 1686
[The Discours is also available online from Gallica; see pp. 32, 33 for the above texts.]