In the very beginning of Chapter 2 of Stephen Wolfram’s *A New Kind of Science*, he observes:

The mathematical methods that have in the past dominated theoretical science do not help much with [the question of how simple programs behave]. But with a computer it is straightforward to start doing experiments to investigate programs, and then run them and see how they behave. [NKS, p. 23]

This might have seemed an inconsequential, introductory-type question to many readers. However, coming from a pure math background — I was just getting into toric varieties and other meaty topics in differential and algebraic geometry — I found it rather astonishing that one can model the diffusion equation with block CAs, and never know a thing about differential equations. Of course, that was just the beginning of my astonishment.

My undergraduate background is mathematical physics. I originally went into a mathematics graduate program rather than a theoretical physics program because I believed, as my electromagnetism prof Sidney Redner (a theorist himself) used to tell us — “You should learn your physics from physics books, and your math from math books.” Now, where does that leave a theorist? If one embarks on a theoretical program in quantum mechanics and string theory, one needs to know an awful lot of fairly advanced techniques in order give a meaningful answer the simplest questions. Such advanced techniques that certain mathematicians have been able to get a foothold in the field while traditionally trained physicists have struggled.

Theoretical physics has realized a decreased marginal return as they’ve increasingly built upon “standard” methodological practices (like algebraic techniques in quantum mechanics, and geometric techniques in general relativity). Yet there is the hint — not the promise, due to the computational difficulty, but the hint — that one could produce an ultimate model for the Universe [NKS, p. 465] with comparatively simple computational techniques. So simple, a high school student could, after some study, understand them with relative ease.

Is this what makes NKS techniques seem so alien and “impossible” to traditional theorists? That evolving simple programs, with mind-bogglingly simple rules, can result in modeling diffusion, or even the Universe itself? Is that why they’re willing to reject it outright, with almost a religious fervor, as if it were a piece of blasphemy that, while simply true, must not be entertained or else their whole institution, with all its traditions, could come tumbling down?

The answer to your last paragraph is YES.

I’m not a mathematician or physicist but read Wolfram’s book carefully and with relish when it first came out. After finishing it I called my brother, who is what you might call a theoretical computer scientist/mathematician (and professor) to get his take. “Utter bullshit” was his speedy reply. But said he hadn’t read it – but simply knew it and Wolfram by their reputations. It seemed clear to me that his response was mostly a knee-jerk reaction to Wolfram’s ‘ego the size of a planet’ – which I understand is somewhat justified. Plus my brother worked with (I forget his name) the person who worked up the mathematical proof at the end of NKS about universality, who wound up battling Wolfram in court over the rights to said proof…none of which is mentioned in the book. It’s all “I did this” and “I did that”…(see large ego, above)

I think I ‘got it’ because I hadn’t already been indoctrinated by math-based science…to me the Principle of Computational Equivalence is a wonderfully simple explanation of how order vs. chaos are discerned, and an amazingly transformative tool for understanding as I look at the world around me. But I’m more philosopher than scientist by a million miles. (but read math and science books for fun)

Good post – one of the few about NKS I’ve found insightful…most are simply dismissive or downright rude.

mike

I like your use of ‘decreased marginal return’ with respect to theoretical physics. It highlights the fact that, in the study of simple programs, we have exactly the opposite problem: With very little architecture, we get much more stuff than we have time to look at! It reminds me of a professor of mine, David Cope (http://arts.ucsc.edu/faculty/cope/), who, through his music-composing software, claimed once to be the world’s most prolific composer…. Of course, he has never listened to the vast majority of his compositions!

Certainly most mathematicians (and I assume most physicists as well) are still doing strictly pencil-and-paper math. So computation is just something they don’t have direct experience with all the time, and consequently I suppose they don’t see it as being fundamental to the world. For some bizarre reason, point particles in Minkowski space (or whatever) seem like better primitives to them than a grid of cells or a network.

The other thing about pencil-and-paper computation is that you need to use equations that compartmentalize a large system into a few symbols, because otherwise you just can’t carry around all the information in a system. The beauty of NKS systems is that (while they presumably take place at a vastly different scale than, say, a “formal” electron), they are easy to work out step by step. As you said, this will let a high school student — probably even an elementary school student — understand the fundamental rule of the world. I expect some day a great grandchild of mine will come home from 5th grade and say that the teacher taught them the fundamental rule of physics today.