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October 27, 2007

Steve Fever

Posted by John Baez

Yesterday when I read my MIT alumni magazine, I was pleased to see a short story by Greg Egan. This magazine is available for free online if you submit to a mildly annoying registration process, so I’ll advertise the story here:

It’s about an artificial intelligence so stupid it believes what it reads on the internet.

Egan also has a new story out called “Dark Integers”. It’s a sequel to “Luminous”, which I wrote about back in week123:

It all started out as a joke. Argument for argument’s sake. Alison and her infuriating heresies.

“A mathematical theorem,” she’d proclaimed, “only becomes true when a physical system tests it out: when the system’s behaviour depends in some way on the theorem being true or false.”

It was June 1994. We were sitting in a small paved courtyard, having just emerged from the final lecture in a one-semester course on the philosophy of mathematics — a bit of light relief from the hard grind of the real stuff. We had fifteen minutes to to kill before meeting some friends for lunch. It was a social conversation — verging on mild flirtation — nothing more. Maybe there were demented academics, lurking in dark crypts somewhere, who held views on the nature of mathematical truth which they were willing to die for. But were were twenty years old, and we knew it was all angels on the head of a pin.

I said, "Physical systems don’t create mathematics. Nothing creates mathematics - it’s timeless. All of number theory would still be exactly the same, even if the universe contained nothing but a single electron."

Alison snorted. "Yes, because even one electron, plus a space-time to put it in, needs all of quantum mechanics and all of general relativity - and all the mathematical infrastructure they entail. One particle floating in a quantum vacuum needs half the major results of group theory, functional analysis, differential geometry - "

"OK, OK! I get the point. But if that’s the case… the events in the first picosecond after the Big Bang would have `constructed’ every last mathematical truth required by any physical system, all the way to the Big Cruch. Once you’ve got the mathematics which underpins the Theory of Everything… that’s it, that’s all you ever need. End of story."

"But it’s not. To apply the Theory of Everything to a particular system, you still need all the mathematics for dealing with that system - which could include results far beyond the mathematics the TOE itself requires. I mean, fifteen billion years after the Big Bang, someone can still come along and prove, say… Fermat’s Last Theorem." Andrew Wiles at Princeton had recently announced a proof of the famous conjecture, although his work was still being scrutinised by his colleagues, and the final verdict wasn’t yet in. "Physics never needed that before."

I protested, "What do you mean, ‘before’? Fermat’s Last Theorem never has - and never will - have anything to do with any branch of physics."

Alison smiled sneakily. "No branch, no. But only because the class of physical systems whose behaviour depend on it is so ludicrously specific: the brains of mathematicians who are trying to validate the Wiles proof."

"Think about it. Once you start trying to prove a theorem, then even if the mathematics is so ‘pure’ that it has no relevance to any other object in the universe… you’ve just made it relevant to yourself. You have to choose some physical process to test the theorem - whether you use a computer, or a pen and paper… or just close your eyes and shuffle neurotransmitters. There’s no such thing as a proof which doesn’t rely on physical events, and whether they’re inside or outside your skull doesn’t make them any less real."

And this is just the beginning… the beginning of a wild tale of an inconsistency in the axioms of arithmetic — a "topological defect" left over in the fabric of mathematics, much like the cosmic strings or monopoles hypothesized by certain physicists thinking about the early universe — and the mathematicians who discover it and struggle to prevent a large corporation from exploiting it for their own nefarious purposes.

I think you can download the story “Luminous” for $1.59 here, though I’m too old-fashioned to have tried this sort of thing. The sequel, “Dark Integers”, seems right now to be available only through the magazine Asimov’s Science Fiction — the October/November 2007 issue. (Strangely, this futuristic magazine seems to be available only in the form of hand-delivered pulverized tree fragments.)

Posted at October 27, 2007 3:23 AM UTC

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6 Comments & 0 Trackbacks

Re: Steve Fever

one could get “Dark Integers” here.

Posted by: ericv on October 27, 2007 9:27 AM | Permalink | Reply to this

Re: Steve Fever

Not bad — that’s 6 bucks for a bunch of stories including “Dark Integers”.

Posted by: John Baez on October 28, 2007 12:06 AM | Permalink | Reply to this

Re: Steve Fever

In that issue of the Tech Review, I also liked the article by Dan Stroock and the cover article both about the role of quants on Wall Street.

Posted by: Jason Starr on October 28, 2007 3:15 AM | Permalink | Reply to this

Re: Steve Fever

Jason, can you please tell me if either article about quants discusses any failures in mathematical finance? I am asking because it is my job to study such failures and then try to find new solutions. I don’t want to go on a long and off-topic rant about these failures but I will mention the two most famous failures of 2007:

1) James Simons is the mathematician who coinvented Chern-Simons (CS) theory, and last year he made $1.7 billion just for himself. However, this year his fund, Renaissance Technologies, has significantly underperformed the market and, e.g., lost a lot of money in August.

2) The huge blow-up of subprime securitizations.

By the way, since John Baez is a top expert on quantum field theory (QFT), I will also mention that QFT has been studied for finance. See, e.g., this paper. I will now state the standard historical warning: much of the maths tried in finance will not work out over a significantly long enough time period.

Posted by: Charlie Stromeyer Jr on October 28, 2007 1:07 PM | Permalink | Reply to this

Re: Steve Fever

I bet one reason much of the math tried in finance will not work out for long is that finance is a game with competing players — so as soon as someone comes up with a good trick, it starts affecting what other people do. It’s not like physics, where the system you’re studying doesn’t know you’re watching.

Dan Stroock’s short piece in Technology Review seems designed to move blame away from the quants, towards other people. It’s so short I feel no shame in quoting the whole thing:

The role that so-called quants play in the financial world is analogous to the role batfish play in keeping coral reefs tidy. Just as batfish do not construct the reef but are essential to its health, quants do not create the structure financial markets depend on but do preserve the conditions that make markets function. So it would be misleading to suggest that quants were responsible for this summer’s meltdown in the subprime-mortgage market or for the broader troubles that followed (see “The Blow-Up”).

The functioning of financial markets relies on the general acceptance of certain assumptions. One of the most important is that the market will not sustain an opportunity for someone to have a free lunch. That is, although arbitrage opportunities will arise, market forces will eliminate them. As Fischer Black and Myron Scholes demonstrated in 1973 with their seminal model for determining the value of a stock option, the “no arbitrage” assumption provides individuals with a rational basis for putting a fair price on a variety of financial instruments. Thus, it is essential that the assumption be correct, and an important role of the quant is to make sure that it is. By scrutinizing financial data, quants spot arbitrage opportunities and alert their employers to act before others have a chance to do the same.

Another basic assumption is that risk is necessary and even beneficial. On the other hand, investors are willing to incur risk only if it’s spread out. Insurance is the classic example of a mechanism for spreading the risk of financial disaster, but recently, investment companies have introduced much more sophisticated mechanisms. They make the risk palatable by embedding it in attractive­-looking financial instruments in which it is diluted, and what remains of it is less evident. Such instruments are called derivatives, and with the help of inventive quants, the derivatives market has come to resemble a dim sum platter of enticing morsels. A further simi­larity is that overindulgence can cause indigestion.

Although I have had students who later thrived on Wall Street, I consider the role they play there closer to that of the sweepers who used to clear the ticker tape off the floor of the stock exchange than to that of a traditional investment banker. Most of the time, they have no idea what, if anything, is made by the companies with whose stocks they deal. Their mission is to blindly keep those stocks moving, not to pass judgment on their value, either to the buyer or to society. Thus, I find it completely appropriate that quants now prefer the euphemism “financial engineer.” They are certainly not “financial architects.” Nor are they responsible for the mess in which the financial world finds itself. Quants may have greased the rails, but others were supposed to man the brakes.

If you’re looking for a discussion of problems, it’s better to read the article Stroock refers to: “The Blow-Up”. (You’ll need to go through a mildly annoying registration procedure to read this article, but then it’s free, and you can delete MIT’s cookies from your web browser when you’re done.)

Posted by: John Baez on October 28, 2007 6:35 PM | Permalink | Reply to this

Re: Steve Fever

Thanks for the info, John, and you are correct about the adaptive nature of competing players. What does this have to do with category theory you might ask? Well, here I say a bit about this:

The adaptive problem you have identified is one of the reasons I am going to try out a new adaptive invention called “programming by reality”. This invention involves a variety of maths/computer science, including, e.g., a merger of Bayesian belief networks with Petri nets as you can see in my coworker’s MIT PhD thesis.

This same coworker has come up with an algebraic foundation for our approach via multi-sorted algebras. I found out that way back in 1968 Jean Benabou first defined the category of multi-sorted algebras.

Alas, Benabou had the Gaul to write his paper in French which I cannot read too well to say the least !-)
Fortunately, I have found English versions of the category of multi-sorted algebras and, furthermore, there is a generalization of many-sorted algebras called order-sorted algebras which have also been categorified (in English).

By the way, if any of you think about, e.g., the stock market then please remember these two points:

1) Over the long term, a company’s stock price follows its earnings and it is not at all easy to use mathematics to try to predict such long term earnings nor whether the same company will even be a thriving business x years hence.

2) Over shorter time periods, the trading of stocks can sometimes be overcome by either fear or greed. Say what you will about former Federal Reserve Chairman Alan Greenspan, but he has at least discussed this issue intelligently.

Posted by: Charlie Stromeyer Jr on October 28, 2007 8:27 PM | Permalink | Reply to this

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