No seminar today (Sorry!)

Due to an illness among the teaching staff, today’s statistical-mechanics session will be postponed until later in the week. In the meantime, check out science writer Carl Zimmer and his experience being quote-mined by global warming denialists. To cheer up after that study in human folly, try Mark Chu-Carroll’s ongoing series on surreal numbers. Based on his prior habits, I’m curious to see if he takes a crack at explaining the relation between surreals and category theory (definitely one of the branches of mathematics you need to study if you want to be like the guy in Pi).

One point should be raised about the surreals which has not yet appeared in Mark Chu-Carroll’s exposition. You can get the reals — that ordinary, familiar number line — from the surreals by imposing an extra condition on the construction. It’s exercise 17 in the back of Knuth’s Surreal Numbers (a problem originally suggested to Knuth by John Conway). A number x is defined to be real if -n < x < n for some integer n, and if x falls in the same equivalence class as the surreal number

({x – 1, x – 1/2, x – 1/4, …}, {x + 1, x + 1/2, x + 1/4, …)}.

This topic is also discussed in chapter 2 of Conway’s On Numbers and Games. Theorem 13 proves that dyadic rationals are real numbers, and Conway then deduces that each real number not a dyadic rational is born on day ω (“Aleph Day” in Knuth’s book).

The practical upshot of all this is that surreals may provide a better pathway to understanding the real numbers than the standard way of teaching real analysis! Dealing with Dedekind cuts, for example, leads you to an explosion of special cases and general irritation. Conway says:

This discussion should convince the reader that the construction of the real numbers by any of the standard methods is really quite complicated. Of course the main advantage of an approach like that of the present work is that there is just one kind of number, so that one does not spend large amounts of time proving the associative law in several different guises. I think that this makes it the simplest so far, from a purely logical point of view.

Nevertheless there are certain disadvantages. One that can be dealt with quickly is that it is quite difficult to make the process stop after constructing the reals! We can cure this by adding to the construction the proviso that if L is non-empty but with no greatest member, then R is non-empty with no least member, and vice versa. This happily restricts us exactly to the reals.

The remaining disadvantages are that the dyadic rationals receive a curiously special treatment, and that the inductive definitions are of an unusual character. From a purely logical point of view these are unimportant quibbles (we discuss the induction problems later in more detail), but they would predispose me against teaching this to undergraduates as “the” theory of real numbers.

There is another way out. If we adopt a classical approach as far as the rationals Q, and then define the reals as sections of Q with the definitions of addition and multiplication given in this book, then all the formal laws have 1-line proofs and there is no case-splitting. The definition of multiplication seems complicated, but is fairly easy to motivate. Altogether, this seems the easiest possible approach.

Irony: Dead? Transparency: In Embryo?

Via Bad Astronomy, we hear that “there are 28 CCTV cameras within 200 yards of George Orwell’s house.” This bit of information is brought to you by, whose rather panicky article concludes in the following manner:

This week, the Royal Academy of Engineering (RAE) produced a report highlighting the astonishing numbers of CCTV cameras in the country and warned how such ‘Big Brother tactics’ could eventually put lives at risk.

The RAE report warned any security system was ‘vulnerable to abuse, including bribery of staff and computer hackers gaining access to it’. One of the report’s authors, Professor Nigel Gilbert, claimed the numbers of CCTV cameras now being used is so vast that further installations should be stopped until the need for them is proven.

One fear is a nationwide standard for CCTV cameras which would make it possible for all information gathered by individual cameras to be shared — and accessed by anyone with the means to do so.

The RAE report follows a warning by the Government’s Information Commissioner Richard Thomas that excessive use of CCTV and other information-gathering was ‘creating a climate of suspicion’.

Now, I’m not so sure having the feeds from such cameras widely available would be a bad thing. Given that all technologies have upshots and every silver lining has its own cloud, etc., it might actually be pretty cool.

Dreams of the Transparent Society notwithstanding, I think we all know the real reason Britain is so gear to put cameras everywhere. It’s because the newest cameras contain neural-network chips programmed to act as selective quantum observers, capable of altering wavefunction collapse and thereby mimicking the method by which natural Gorgons operate. And thus,

If we pursue this plan, by late 2006 any two adjacent public CCTV terminals — or private camcorders equipped with a digital video link — will be reprogrammable by any authenticated MAGINOT BLUE STARS superuser to permit the operator to turn them into a SCORPION STARE basilisk weapon. We remain convinced that this is the best defensive posture to adopt in order to minimize casualties when the Great Old Ones return from beyond the stars to eat our brains.

Russell Blackford on Human Enhancement

I’m not sure when the idea of “human enhancement” first bubbled up in my brain. It seems to be one of those possibilities which I just grew up with, thanks to a childhood lost in books. In Cosmos, Carl Sagan wrote,

There must be ways of putting nucleic acids together that will function far better — by any criterion we choose — than any human being who has ever lived. Fortunately, we do not yet know how to assemble alternative sequences of nucleotides to make alternative kinds of human beings. In the future we may well be able to assemble nucleotides in any desired sequence, to produce whatever characteristics we think desirable — a sobering and disquieting prospect.

The video version ends with “awesome and disquieting prospect,” by the way. Sagan’s friend Isaac Asimov was a little more cheerful; while dying of AIDS, he concluded the revision of his book The Human Brain with these words:

Man would then, by his own exertions, become more than man, and what might not be accomplished thereafter? It is quite certain, I am sure, that none of us will live to see the far-distant time when this might come to pass. And yet, the mere thought that such a day might some day come, even though it will not dawn on my own vision, is a profoundly satisfying one.

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Recycled Blake Stacey: Watchmaker Morality

I’ve spent an awful lot of time scattering thoughts into the Blagnet. It’s a valuable way of procrastinating on other things and making myself feel like an intellectual. Unfortunately, it also means that on the off-chance I do say anything worth remembering, it’s probably buried in the comment thread to some blag post and can’t be retrieved without a string of Google search terms eight words long.

So, I’ve decided to start recycling the more entertainingly pseudo-intellectual rambles of mine, editing for clarity where I can. First up is a selection from this Pharyngula thread; the rant proper can be found below the fold.

Continue reading Recycled Blake Stacey: Watchmaker Morality

Friday fun in Network Theory

Via Backreaction, I just heard about this intriguing exploration in applied network theory:

For the first time, sociologists have mapped the romantic and sexual relationships of an entire high school over 18 months, providing evidence that these adolescent networks may be structured differently than researchers previously thought.

The results showed that, unlike many adult networks, there was no core group of very sexually active people at the high school. There were not many students who had many partners and who provided links to the rest of the community.

Instead, the romantic and sexual network at the school created long chains of connections that spread out through the community, with few places where students directly shared the same partners with each other. But they were indirectly linked, partner to partner to partner. One component of the network linked 288 students — more than half of those who were romantically active at the school — in one long chain.

Out of about 1,000 students in the school, the researchers interviewed 832 and found that slightly more than half reported having had sexual intercourse. They found 63 “simple pairs”, i.e., students whose only pairings were with each other. 189 students (35% of the romantically active population) belonged to networks containing three or fewer nodes. And then, if you want some really interesting topology,

The most striking feature of the network was a single component that connected 52 percent (288) of the romantically involved students at Jefferson. This means student A had relations with student B, who had relations with student C and so on, connecting all 288 of these students.

While this component is large, it has numerous short branches and is very broad – the two most distant individuals are 37 steps apart. (Or to use a currently popular term, there were 37 degrees of separation between the two most-distant students.)

“From a student’s perspective, a large chain like this would boggle the mind,” [Ohio State professor James] Moody said. “They might know that their partner had a previous partner. But they don’t think about the fact that this partner had a previous partner, who had a partner, and so on.

“What this shows, for the first time, is that there are many of these links in a chain, going far beyond what anyone could see and hold in their head.”

This caught my eye because I’ve actually dabbled in networks, studying protein structures using “motifs” — small sub-graphs with particular connection patterns whose preponderances we examine statistically. I’d be interested in getting the actual connection data, running them through MFinder and checking out their superfamily values.

Jack Cowan at MIT

Second in today’s “e-mails from Eric” department is this announcement of a talk by Jack Cowan, a mathematics professor at U. Chicago. The talk, scheduled for 16:00 on Tuesday, 3 April in room 46-3189, is abstracted as follows:

We have recently found a way to describe large-scale neural activity in terms of non-equilibrium statistical mechanics [Buice & Cowan, in preparation]. This allows us to calculate (perturbatively) the effects of fluctuations and correlations on neural activity. Major results of this formulation include a role for critical branching, and the demonstration that there exist non-equilibrium phase transitions in neocortical activity, which are in the same universality class as directed percolation. This result leads to explanations for the origin of many of the scaling laws found in LFP, EEG, fMRI, and in ISI distributions, and provides a possible explanation for the origin of alpha, beta, gamma, delta and theta waves. It also leads to ways of calculating how correlations can affect neocortical activity, and therefore provides a new tool for investigating the connections between neural dynamics, cognition and behavior.

I suspect I’ve already seen some of these results, in the video “Spontaneous pattern formation in large scale brain activity: what visual migraines and hallucinations tell us about the brain” (2006). The one review of that video is, depressingly enough, a muddled remark about “observers” in quantum mechanics and what they must mean for consciousness (and you’ll probably catch me ranting about that, anon). Fortunately, the video itself is much more informative.

For a quick primer, see “Hallucinatory neurophysics” at Sean Carroll‘s old blog, Preposterous Universe.

UPDATE (11 June 2007): Yes, I ranted more about the “quantum mind.” See here and here.

Upcoming Sessions, Week of 1 April 2007

Eric just sent along the following summary of our upcoming sessions:

This Friday at NECSI is Info Theory again: we’ll be talking specifically about the coordinate-dependence of differential (continuous) entropy and more generally, discussing the rest of Part III of Shannon’s paper. The next topic after that will be “Error-Correcting Codes in Biology“, which will probably take a few weeks at least — we’ll first cover the relevant sections of Ash (or Reza or whatever people prefer) and then talk about the biological basics.

This next Monday is Stat Mech and I will be reviewing the ensembles we have covered so far and talking at NECSI about the Gibbs-canonical and grand-canonical ensembles. Depending on time I will prove (in a physicist way) that all of the ensembles are equivalent within their own ranges of assumptions — so this may take one or two lectures. After that I will probably assign some homework so that we can get experience working with these tools.

Links added by me, because this is Xanadu 2.0, after all.

First Session on Group Theory

Yesterday evening, we had our first seminar session on the group theory track, led by Ben Allen. We covered the definition of groups, semigroups and monoids, and we developed several examples by transforming a pentagon. After a brief interlude on discrete topology and — no snickers, please — pointless topology, Ben introduced the concept of generators and posed several homework questions intended to lead us into the study of Lie groups and Lie algebras.

Notes are available in PDF format, or as a gzipped tarball for those who wish to play with the original LaTeX source. Likewise, the current notes for the entropy and information-theory seminar track (the Friday sessions) are available in both PDF and tarball flavors.

Our next session will be Friday afternoon at NECSI, where we will continue discussing Claude Shannon’s classic paper, A/The Mathematical Theory of Communication (1948). The following Monday, Eric will treat us to the grand canonical ensemble.

Welcome to Web 3.11 for Workgroups

Welcome to Science After Sunclipse, a blag devoted (more or less) to discussing mathematics and physics. Your hosts, including a soft-spoken and ever-humble Order of the Molly recipient (me), currently run a seminar series grandly entitled the “Do It Yourself University”. Typically hosted at the New England Complex Systems Institute, our sessions cover topics in statistical physics, information theory, topology and. . . well. . . whatever else strikes our collective fancy.
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