## August 30, 2011

### The Genius In My Basement

#### Posted by John Baez

Simon J. Norton was a math prodigy as a child. He went to Cambridge for grad school. Together with his advisor John Conway, he did some amazing work on group theory. In 1985, based on an idea from John McKay, they conjectured an astounding relation between the Monster group and the modular $j$-function. Conway dubbed this “Monstrous Moonshine”. The proof turned out to involve ideas from string theory, but the full implications are yet to be understood.

But in 1985 — when some mathematicians claim he suffered a “catastrophic intellectual collapse” — Simon took to collecting thousands of bus and train timetables. What happened to him? What is he doing now?

You can learn a bit of the answer here:

This is an excerpt from Master’s new book The Genius In My Basement. Here’s a review:

From the excerpt, Norton is apparently still doing good math. He says he discovered “the appearance of Conway Group in the projective plane presentation of the Monster”. I would like to know more about that!

I find it fascinating how much people enjoy tales of eccentric mathematicians. I do too, but I don’t think that counts: I’m a mathematician, and I also enjoy tales of non-eccentric mathematicians, packed with details of their actual work.

Why would a non-mathematician want to read about an expert on finite simple groups… but only if they switch to filling their apartment with bus schedules? Here’s one possibility. Most non-mathematicians eventually ‘hit the wall’ in their studies of math: it becomes too hard, and they quit. So, they’re left wondering what they’re missing, and what sort of mind it would take to learn that stuff. It’s fascinating, and somewhat reassuring, to hear that it takes someone crazy.

But that doesn’t seem to be all there is to it. Perhaps in our times the mathematician has replaced the poet as the mythical figure who — some like to think — must approach the brink of madness to see beyond what everyone else has seen. From the Wikipedia article on artistic inspiration:

In Greek thought, inspiration meant that the poet or artist would go into ecstasy or furor poeticus, the divine frenzy or poetic madness. He or she would be transported beyond his own mind and given the gods’ or goddesses own thoughts to embody.

Inspiration is prior to consciousness and outside of skill (ingenium in Latin). Technique and performance are independent of inspiration, and therefore it is possible for the non-poet to be inspired and for a poet or painter’s skill to be insufficient to the inspiration.

Posted at August 30, 2011 4:11 AM UTC

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### Re: The Genius In My Basement

I see your Wikipedia citation and raise you a TV Tropes entry, Mad Mathematician:

Clearly, many writers (and viewers) fear and dread mathematics. On rare occasion, a character is found who possesses a defining trait of having delved deeply into the study of profound mathematical knowledge. These characters are, as a rule, insane. It is not necessarily clear whether advanced number theory is itself destructive to sanity (as with some forms of Formulaic Magic), or whether the insane are drawn to maths; nonetheless, the correlation seems to exist.

A spot of Google-Scholaring turned up the following:

What does it mean to be able to compute in a given group? If the group is defined as a set of elements of given type (e.g. matrices), we need to be able to multiply or invert them, and an optional extra is to distinguish elements of the group from non-elements. However, even if there is no explicit algorithm for the latter, it can usually be solved in practice by multiplying by random group elements and determining the order of the product. […] A practical method of computing in the Monster would be valuable for several reasons: “because it’s there”, i.e. to tidy up the last case; to settle certain problems about groups attacked by means of the classification of finite simple groups; to enumerate the maximal subgroups of the Monster; and to study some of its special properties.

The projective-plane stuff seems to go back to “Constructing the Monster” (1990), in Groups, Combinatorics and Geometry, Liebeck and Saxi, eds.

Posted by: Blake Stacey on August 30, 2011 6:19 AM | Permalink | Reply to this

### the projective plane presentation of the Monster

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:

… 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.

Posted by: Blake Stacey on November 25, 2018 3:05 AM | Permalink | Reply to this

### Re: The Genius In My Basement

Masters has past form in this genre, it would seem…

On a quick skim, I’m rather taken with the prose style he’s adopted for the Norton piece, and rather like its general tone, although I wince at passages like

As soon as possible after the manuscript was published, Conway emigrated to a professorship in America, desperate never to look at a Group again.

Posted by: Yemon Choi on August 30, 2011 6:48 AM | Permalink | Reply to this

### Re: The Genius In My Basement

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?

Posted by: John Baez on August 30, 2011 9:49 AM | Permalink | Reply to this

### Re: The Genius In My Basement

But in 1985 – when some mathematicians claim he suffered a “catastrophic intellectual collapse”…

85/86 was just the time I was taking Part III in Cambridge. 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”.

But I was hitting my own ‘wall’ at the time. It wasn’t so much the difficulty, as the pointlessness I felt at the time.

Posted by: David Corfield on August 30, 2011 9:19 AM | Permalink | Reply to this

### Re: The Genius In My Basement

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.

Posted by: John Baez on August 30, 2011 9:46 AM | Permalink | Reply to this

### Re: The Genius In My Basement

A to my knowledge never mentioned possibility is that inherent traits of personality are overwritten by other environmental stimulations only until an age of ca. 30 years, at least that is my observation. (That would fit e.g. to this case too.)

And: “Most non-mathematicians eventually ‘hit the wall’ in their studies of math” - non-mathematicians only??

Posted by: Thomas on August 30, 2011 10:51 AM | Permalink | Reply to this

### Re: The Genius In My Basement

I don’t know if it is relevant, but I was reminded of this: http://www.nytimes.com/2011/08/14/nyregion/children-with-autism-connecting-via-bus-and-train.html?_r=4&ref=us

Posted by: Roger Witte on August 30, 2011 2:25 PM | Permalink | Reply to this

### Re: The Genius In My Basement

I’m uncomfortable with the freak show aspect of books like this, though from the Guardian review that John links to, it seems that the author was very open with Norton, and Norton cooperated with the author.

So, I’ll add the following. For a couple of years when I was a postdoc in Cambridge, I was in the office next to Norton’s. (Presumably they were/are happy to give him visitor status.) My experience of him was that every now and again he’d come into his office, close the door, and get on quietly with whatever he was getting on with. In other words, he behaved just like my other neighbours. Once I popped in to ask him a question about finite groups, and he gave me a helpful answer. The only thing I remember from that interaction was that he wrote the cyclic group of order $n$ as simply “$n$”. That convention was actually rather suitable for the question I was investigating.

Posted by: Tom Leinster on August 31, 2011 2:59 PM | Permalink | Reply to this

### Re: The Genius In My Basement

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 ‘$x$’. :)

Posted by: David Roberts on September 7, 2011 1:04 AM | Permalink | Reply to this

### Re: The Genius In My Basement

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. And perhaps an artist rather than simply just embodying the goddesses thoughts had actually stolen them from her. And madness, being a little reminder of the punishment meted out for theft of that kind. Did someone say that good artists borrow, and great artists steal? There’s another warning in the Upanishads, about reason being pushed too far, maybe too quickly leading to madness.

I’ve read one of Alexander Masters previous books - Stuart, a life lived backwards. And found his prose both engaging and moving.

I have to say that my heart just sank when I found that this was yet another take on an eccentric mathematician. Though he should be safe in Masters hands. His equating of high intelligence and IQ also makes me feel a bit queasy. He should know better. Though I suspect he’s just building up his characters ‘mythology’.

I’m not sure I like this characterisation of mathematicians stopping doing mathematics simply when it gets too difficult, it could be that something else just got definitely more interesting or impresses itself more vividly on the senses.

Posted by: mozibur ullah on September 1, 2011 10:12 AM | Permalink | Reply to this

### Re: The Genius In My Basement

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

Posted by: Thomas on September 2, 2011 9:26 AM | Permalink | Reply to this

### Re: The Genius In My Basement

“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

Posted by: Jeffery Winkler on September 6, 2011 7:59 PM | Permalink | Reply to this

### Re: The Genius In My Basement

Jeffrey Winkler wrote:

…so there is nothing scientific about the people who are against global warming.

This isn’t relevant to this blog entry or even this blog! But instead of deleting your comment, I’ll just suggest that you read the National Academy of Sciences report Climate Stabilization Targets: Emissions, Concentrations, and Impacts over Decades to Millennia, and argue about it over at my other blog if you want.

Posted by: John Baez on September 7, 2011 4:21 AM | Permalink | Reply to this

### Re: The Genius In My Basement

And the café blog automatically link to “Some Related Entries”. Lets hope it’s not accurate :-P

Posted by: M on September 5, 2011 8:09 AM | Permalink | Reply to this

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