Davies and Materialism

Paul Davies took a solid beating and responded in turn at The Edge.

Part of his response was to the effect that he didn’t mean all of science in his original discussion, but only meant instead the area of cosmology and theoretical physics which is his bailiwick. He exempts evolutionary biologists with that stroke. His opinions on cognitive science or neuropsychology would likely be similar (I bring this up since I see a few “grand problems” for science beyond the origin of the universe and some final theory of physics; the problem of mind and abiogenesis as similarly important).

But part of his discussion continued to hammer at the notion that uniformity and understandability of natural law was somehow intrinsically related to monotheism. It’s a rarefied argument that kind of bootstraps itself on the fact that Newton, Kepler and Galileo saw the hand of God in the correspondence of their mathematical abstractions to physical observations. There is an odd hint about his proclivities, though, in Davies’ mention of Lee Smolin’s evolutionary selection of universes, where other metaphorical narratives have informed the physical theory; a similar parallel exists in the use of computer metaphors in cognitive science, of course, or in ecological theories of perception. There are only a few basic algorithms available to try to explain unexplained phenomena: stochastic selection with replication (evolution), deterministic interaction (Newtonian dynamics), quantized interactive behaviors (quantum mechanics), thermodynamic uniformization, cybernetic control and feedback, computation and, yes, pure irrationality or theology.

I did some digging on some of the Davies’ arguments, passing back through the Wigner paper and the follow-on by Hamming, “The Unreasonable Effectiveness of Mathematics,” which builds-out the notion that there are essentially irrational drivers (aesthetics, play, mysticism) that push forward mathematics and that the results in turn drive scientific theory. All of this effort in some ways parallels or rediscovers the ongoing work during the same time period concerning aspects of irrationality in the philosophy of science (Kuhn, Feyeraband, etc.). The scientific method is not an acidic, scalding, and sacred pursuit devoid of irrational influences, nor are individual scientists devoid of personal faiths about their capabilities or the possibilities of their theories, but proclaiming the entire enterprise as strongly influenced by a monotheistic worldview is a strange preoccupation. Indeed, the only constant across the varied scientific pursuits is that any metaphysics is material in nature because no other explanations have provided any hint of validation or added to the task at hand (thus any metaphysics is tentative, itself), and that mathematics is a useful tool because it is simply a way of expressing relationships between objects that are not irrational but that vary according to sometimes complex but non-arbitrary ways.

Dimensional Folding and Coherence

I was reading Terence Yao's blog on mathematics earlier today, enjoying some measure of understanding since his recent posts and lectures focus on combinatorics. In addition to the subject matter, though, I was interested in the way he is using his blog to communicate complex ideas. The method is rather unique in that is less formal than a book presentation, less holographic than a journal article for professional publication, more technical than an article in a popular science magazine, and yet not as sketchy as just throwing up a series of PowerPoint slides. And of course there is interaction with readers, as well.

There is some rather interesting work in cognitive psychology and psycholinguistics on the relationship between writing styles and reader uptake. Specifically, the construction-integration model by Walter Kintsch and others tries to tease out how information learning is modulated by the learner's pre-existing cognitive model. There is a bit of parallelism with "constructivism" in educational circles that postulates that learning is a process of building, tearing down and re-engineering cognitive frameworks over time, requiring each student to be uniquely understood as bringing pre-existing knowledge systems to the classroom.

In construction-integration theory, an odd fact has been observed: if a text has lots of linking words bridging concepts from paragraph to paragraph, those with limited understanding of a field can get up to speed with greater fluidity than if the text is high-level and not written with that expectation in mind. In turn, those who are advanced in the subject matter actually learn faster when the text is more sparse and the learner bridges the gaps with their pre-existing mental model.

We can even measure this notion of textual coherence or cohesion using some pretty mathematics. If we count all the shared terms from one paragraph to the next in a text and then try to eliminate the noisy outliers, we can get a good estimate of the relative cohesion between paragraphs. These outliers arise due to lexical ambiguity or because many terms are less semantically significant than they are syntactically valuable.

A singular value decomposition (SVD) is one way to do the de-noising. In essence, the SVD is an operation that changes a large matrix of counts into a product of three matrixes, one of which contains "singular values" along the matrix diagonal. We can then order those values by magnitude and eliminate the small ones, then re-constitute a version of the original matrix. By doing this, we are in effect asking which of the original counts contribute little or nothing to the original matrix and eliminating those less influential terms.

There are some other useful applications of this same principle (broadly called "Latent Semantic Analysis" or LSA). For instance, we can automatically discover terms that are related to one another even though they may not co-occur in texts. The reduction and reconstitution approach, when applied to the contexts in which the terms occur, will tend to "fold" together contexts that are similar, exposing the contextual similarity of terms. This has applications in information retrieval, automatic essay grading and even machine translation. For the latter, if we take "parallel" texts (texts that are translations of one another by human translators), we can fold them all into the same reduced subspace and get semantically-similar terms usefully joined together.

Terence Yao's presentation is clearly aimed at grad students, advanced undergrads and other mathematics professionals, so his language tends to be fairly non-cohering (not, I note, do I think he incoherent!), and much of the background is left out or is connected via Wikipedia links. The links are a nice addition that is helpful to those of us not active in the field, and a technique that provides a little more textual cohesion without unduly bothering the expert.

Math, Science and Popular Culture

A digital video recorder (DVR) destroyed my evenings. I write that, though, with some guilty pleasure. I really was not much of a television viewer until I upgraded our technology stack, including an HD receiver with DVR pumping glorious time-shifted detail through an LCD television with a surround-sound system.

That was just over a year ago and my evening productivity has suffered. I spent probably six months just exploring features, programming a universal remote and capturing programming. Then I settled in to watch some specific programs in addition to movies and documentaries.

There are four network programs that I want to mention because of their unexpected cultural importance: the CSI franchise, House, Numb3rs and Criminal Minds. In each case, science or mathematics plays an essential role. In each case, the background material is actually well-researched, although the outcomes are almost always ridiculously neat in order to fit the format.

Numb3rs is the show closest to my background in terms of using algorithms and mathematics to solve problems. In a recent show, in fact, one mathematician used a classification and regression tree (CART) algorithm to do something. I’ve used CART before. Some of the other topics in social network analysis and covering algorithms also ring vaguely true, though they are distorted through a lens of excited elaboration that gets tiresome over time.

Northeastern University keeps track of some of the Numb3rs mathematics, here.

But more important, I think, is the cultural message that intelligent people with highly developed skill sets are heroes. Even Gregory House is a hero of sorts in his coldly analytical pursuit of truth, his anti-theism and dedication to correct diagnoses. I contrast this kind of programming with 90210, Dallas or other cultural phenomena that seemed to cater to baser ideas of wealth, power and privilege. Is the brain on the rise?

I have to admit that I am getting some kind of network television fatigue here after a year with the DVR and shows are stacking up on the hard drive. I may have to go back to a stricter media diet, but hope the science and math keeps a place on the television menu.