The n-Category Café

It gives me a very odd feeling to be looking up references on a topic and stumble across a comment that I wrote and had completely forgotten about. But perhaps I can make that comment more useful, historically. The original reference is

• J. H. Conway, S. P. Norton and L. H. Soicher, “The Bimonster, the Group $Y_{555}$, and the Projective Plane of Order 3,” in Computers in Algebra, M. C. Tangara, ed. (Marcel Dekker, 1988), pp. 27–50.

This paper was tricky to get my hands on. A more recent item that was more accessible, both in terms of library access and of prose, is the following:

• C. S. Simons, “Deflating infinite Coxeter groups to finite groups,” arXiv:math/0105197 (2001).

A paper I have made repeated attempts to understand is the following:

• M. Miyamoto, “21 involutions acting on the Moonshine module,” Journal of Algebra 175 (1995), 941–65.

… which I found via the following:

• T. Gannon, “Monstrous Moonshine: The first twenty-five years,” arXiv:math/0402345 (2004).

All of this bears upon my favorite mathematical coincidence that probably means nothing at all — that the finite affine plane on 9 points, and its structure of point-line incidences, appears both in the quantum information theory of a three-level system and in the study of the Monster group.

Yes, that sentence seems absurd. Conway and Sloane published the first edition of Sphere Packings, Lattices and Groups in 1988. Since Sloane worked at AT&T in New Jersey, I’d guess they wrote that book after Conway moved to Princeton. Having read this book, I find it hard to believe that it was the work of someone “desperate never to look at a Group again”.

Capitalizing the word “Group” also seems peculiar… sort of archaic, no?

David wrote:

I took a course on group representations with Conway and sat in on a couple of classes given by Norton. The latter certainly struck an odd figure, but I’m surprised he is said to have suffered a “catastrophic intellectual collapse”.

Interesting! But apparently this “collapse”, or whatever it was, occurred after Conway went to Princeton:

With Conway gone, group theory work on the Atlas finished, and with no one to goad and tease him into attacking fresh problems, Simon was lost. The popular image of a brilliant mathematician is a man who looks like Simon, and spends 23 hours a day alone in his mother’s attic solving the most difficult problem in existence. But Simon is a different and much more common type of mathematician. For his genius to flourish, he needs liveliness and company. Simon had no champions and few mathematical friends. There was no one to work with, so he did not work. The mathematics department refused to renew his contract. Never, said mathematicians, had they seen such a spectacular and thorough demotion. The career of one of the great mathematical prodigies of the 20th century was over.

Simon has two explanations for why his genius collapsed. The first is that everyone is mistaken – he never was a great brain, just a very quick one. At five, he could do the mathematics of a 12-year-old; by 20, the equal of a professor. Then his brain stopped developing. Others began to catch up. They mistook equality for his decline, and declared he’d suffered a catastrophic intellectual failure. What else, Simon argues, explains the fact that, despite his infamous “collapse”, he is doing maths today that is as good as, if not better than, he’s ever done? Witness his discovery of “the appearance of Conway Group in the projective plane presentation of the Monster” (Simon: “I don’t think I can make it any more comprehensible than that”), done two years after Conway had left for America, long after Simon’s supposed “first mistake”.

Simon’s second explanation of his loss of mathematical direction is heartbreaking. There is no one in the mathematical world who will work with him. They say he is too peculiar, too shabby, too old. His talent, suited to an extraordinary moment in algebraic history (the symmetry work at Cambridge in the 70s and 80s), is out of fashion.

Tom wrote:

The only thing I remember from that interaction was that he wrote the cyclic group of order $n$ as simply “$n$”

Perhaps a bit like Jim Dolan calling a bicategory ‘$x$’. :)

I guess “power of mind” had been felt as something “magical” since very early times, e.g. this very old hunting method (more).

“I suspect that this ties in with the Promethean myth of stolen fire from the gods, treating fire as symbolic of consciousness. Which I think has interesting parallels with the Biblical story of Eve stealing an apple. In both cases both were punished severely: Intelligence/Knowledge not being unmixed blessings.”

A better explanation is that there is a long-standing ancient belief that there are things humans are allowed to do, and things gods are allowed to do, and if humans do a god-like thing, then we’re breaking the rules, and we’ll be punished for it. We’ll be punished for hubris. We’ll be struck down for our arrogance. Some bad thing will happen as a result. This is a reoccurring theme throughout the history of Western mythology, religion, philosophy, fiction, and history. Here are just a few examples.

1. Prometheus stole fire from the gods, and was chained to a cliff.

2. Phatheon drove his father Helios’ chariot and set fire to the heavens.

3. Adams and Eve ate the apple, and were cast out of the garden.

4. God sent the biblical flood of Noah to punish humanity.

5. Xerxes built a pontoon bridge across the Hellespont, linking two continents which the gods intended to be separate, which Herodotus implies is part of the reason why the superior Persian force was defeated by a smaller inferior Greek force.

6. King Arthur was killed by his son Mordred, who was conceived in an unholy union with Arthur’s half-sister Morgana.

7. In Mary Shelley’s “Frankenstein”, Dr. Frankenstein creates a monster out of pieces of human corpses, and therefore playing god or violating the natural order, and so the monster is a murderer who kills several people including the doctor’s bride on their wedding night.

8. When the vaccine to smallpox was invented, some people had an irrational fear of it.

9. When Niel Armstrong walked on the Moon, a woman wrote a letter to the editor expressing surprise that God allowed it to take place.

10. When in vitro fertlization was invented, in 1972, which was recently awarded the Nobel Prize, the Catholic church, as well, as other people were vehemently opposed to it.

11. Today, people have an irrational opposition to human cloning, claiming that it is “playing god”.

12. People thought the Y2K bug would cause the end of the world. In other words, they believed our arrogant over-reliance on computers would lead to our own destruction.

13. Many people claimed that attempting to recreate conditions similar to the Big Bang at particle accelerators would lead to the End of the World. Some people claimed that the Relativistic Heavy Ion collider would create strange matter that would destroy the world. Other people claimed that the Large Hardonic Collider would create black holes that would destroy the world.

14. Enviromentalists claim that if humans modify the planet in any way, such as by causing acid rain, ozone depletion, or global warming, that would cause the end of the world. To modify the planet sounds god-like. If humans do a god-like thing, such as modify the planet, the real gods will punish is by causing the end of the world.

This last example is the main point of my post. The belief that global warming is bad is just one more example in this long tradition of assuming that if humans do a supposedly god-like thing, we must be punished for it. That is a religious belief since it presumes some sort of god/gods/goddess that sets the rules and will punish us if we break the rules.

Of course, humans are increasing the average temperature of the Earth. It’s not harmful to people. Nobody is hurt by it. There’s nothing bad abouyt it. There’s no reason to want to stop it. The belief that it’s bad is a religious belief.

Modern environmentalism originated with the counter culture of the late 1960’s. They were rebelling against mainstream society in many ways, including rebelling against mainstream religion, which is why they inventing New Age religion. Part of that New Age religion is worshipping the Mother Earth Goddess, which includes believing in a mythical balance of nature, and that humans are not allowed to modify the Earth in any way.

Modern environmentalists originated with the New Age pagan wiccan hippies that worshipped the Mother Earth Goddess, and according to their New Age religion, humans somehow don’t have the moral right to increase the average temperature of the Earth, by any amount, regardless of how small, because according to their New Age religion, this would supposedly offend the Mother Earth Goddess.

Obviously, that belief is not scientific, and so there is nothing scientific about the people who are against global warming.

Jeffery Winkler