Frustrating, that list contains almost no usable links. Its as if he doesn’t want you to actually read or understand what he is saying.
2,464 karma · joined July 25, 2010
https://tali.link
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I'm quite happy to see more work on discrete generative models -- probabilistic programming languages are still wrestling (or simply ignoring!) the problem of "disintegration", where conditioning changes the base measure because it collapses the dimensionality of the probability manifold (similar to the issue Arjovsky identified with GANs). See e.g. http://homes.sice.indiana.edu/ccshan/rational/disint2arg.pdf and https://probprog.cc/assets/posters/thu/78.pdf. To me (although this is probably too radical a move to be palatable to most people) we should largely abandon continuous distributions and start building out discrete probability spaces and methods with the same vigor that continuous probability got in the form of measure theory. These probability trees some like a natural data structure to begin this. I'd also like to see representations for working with probabilities on discrete manifolds that approximate continuous space in computationally efficient ways -- there could be work in this direction already but I'm not aware of it.
Also, a fun implication of these probability trees I'd like to see explored is structural sharing: you need not copy an entire tree to represent the result of a conditioning or intervention. In general you need to copy only a number of nodes given by the size lying above the cut set, in a similar way to how immutable data structures like HAMTs can represent modified hash maps etc with persistent space efficiency by reusing unchanged nodes. If one expects to condition often with certain variables, it would then be useful to hoist such cut sets as high as possible -- does such a 'transpose' operation exist? I admit I did not read the paper thoroughly enough to know if this was mentioned.
That's not to say it is definitely not useful! It could be that, for some bizarre reason, reading the mathematical tea leaves of our current incomplete (and incoherent) theories might well yield insight into post-quantum physics. I do doubt it though.
A great example of a mathematics popularizer along these lines is Norman Wildberger, who has an astonishing range of videos on youtube about an amazing range of mathematical topics. Here’s an interview with him: https://youtu.be/E_dGqavx5AU .
My personal feeling is that a lot of beautiful and interesting mathematics will arise as a result of an intuitionist, finitist, computationalist rethinking of the foundations of mathematics.
How this can be connected to gravity I do not exactly know. I suppose the change in the refractive index of the water is akin to g_{00} component of the Riemann metric, since it produces a compensating change in the wave vector at that point. Whether you could imagine GR consistently in terms of changes in the local speed of light a.k.a. changes in the permittivity of free space, again, I do not know.
I think the value of Stephen's big book is that served as a kind of manifesto to "take computation seriously". By that I mean: to think about computations, in the abstract, as a kind of new mathematics about which we know almost nothing, and about which our naive intuitions from other domains is almost totally inapplicable. It is an injunction to explore the "computational universe", in other words, the "universe of simple computational systems", where CAs live as one of several kinds of maximally simple form of computation (which Wolfram made some attempt to categorize and explore).
By analogy, think about the development of algebra. It didn't come naturally! Yet Algebra is one of the most natural ideas in abstract mathematics, incredibly simple and incredibly powerful, lying as a rosetta stone connecting so many other topics in mathematics. But people didn't immediately accept or describe it; it took many hundreds of years to crystalize and mature as a topic. It's a very old piece of the operating system of mathematics that underwent a lot of hacking and refactoring.
Wolfram proposes we think about computational systems in the same way, as a nascent field that needs exploration, mapping, the irrigation of young minds and new ideas. It may be some time before it yields a harvest. We shouldn't expect it to immediately revolutionize everything.
Even after Turing, computational systems existed as a kind of diaspora in mathematics, having sat unrecognized in all kinds of places, never having had a sort of independent state in which they are not considered as an aspect of something else, as somehow alien and unworthy of respect because of their confusing aspects. Largely, there were two reasons they were treated so shoddily:
1. they required computers (and good computer tools) to actually explore, since they produced complex and irreducible computations 2. they resisted any kind of analysis by prior mathematics
In other words, we were not ready to really probe them until a few decades ago, and even now our software is not well suited to explore them (Mathematica remains the best tool for the job, though I anticipate Julia will surpass it rapidly). Furthermore, traditional mathematical fields have not adapted to the presence of computation very gracefully.
I guess I would summarize: NKS is an imperfect book and Stephen an imperfect herald of the ideas within. But he was the first person to really articulate those ideas crisply and push them hard into the imagination, and I'm very glad he did. The hand he overplayed was the application to the natural world -- I think that will take longer to pay off than he predicted.
Personally, I believe it is inevitable that a vast and rich new mathematical world will spring up around simple computational systems, where they are the primary object of study and not relegated to being games, "simulations" of something else, curiosities, etc. CAs are just the "hello world" of this universe.
In that future, your attitude will be seen as a kind of parochial holdout of 20th century attitudes.
The new mathematics will:
* privilege finite, discrete systems
* be explicitly constructivist
* not shy away from computational enumeration and classification
* be much more visual, but no less rigorous than our last few hundred years of mathematics
* make deep connections with many new technological systems, like Git, distributed systems, blockchains, computer networks, trust networks, social networks, etc.
* most importantly, be way more fun than old math!
It will exploit to the full the capabilities of computer systems to make the new systems tangible, so that the idea of a 'math paper' will be itself quite old fashioned -- most mathematical work will start with forking a Git repo or the equivalent, and most communication and education will be interactive interrogation of computational systems using a kind of souped-up, computation-focused REPL. It's the kind of mathematics that will be natural when Brett Victor's ideas are actually implemented.
Wolfram's NKS book is the prototype work of this kind, but it will also be seen as quite a product of its time and of the author's prior commitments to physics, math, etc. And of course distorted by the classic Wolfram immodesty.r
What I find quite interesting to think about is how the complexity (as a function of accuracy) of a “taken-for-granted” computational process works in the context of another process. For example, the eigensystem of its adjacency matrix provides powerful understanding of a graph — but how much accuracy do you need in order to trust that understanding? If you need arbitrary precision because of the pathological features of a particular family of graph (expander graphs, say), and the scaling of the needed precision links directly to aspects of that graph pathology, this is really telling you that the eigenvector computation is not the most natural “computation basis” to approach this phenomenon — and some explicitly discrete algorithm would make this latent complexity relationship manifest. Probably individual instances of this kind of phenomena are well known to those who study the relevant structures, but I’m not sure if the general phenomena is described or modeled anywhere.
In some weird sense you can link this to a kind of “computational curvature” — ala some of Wolfram’s metamathematical musings. If one’s prior is to distrust continuous mathematics and see only discrete computational processes as “real”, it’s exactly the kind of phenomena one will be on the lookout for. Anyway, the curvature analogy is that as you attempt to maintain a particular computational basis for computing a phenomena of interest as its complexity scales, other typically-uncorrelated complexity requirements that are normally treated as constants will explode. Needs to be made more concrete though.
And by riff I mean with an actual electric guitar
I happen to be working on a new way to look at quantum mechanics, based on finite combinatorial objects called multi-way systems, and after just a few weeks I’ve made a very satisfying amount of progress.
Maybe it won’t go anywhere. But if it succeeds, it should generate the abstract algebraic structure of QM — complex Hilbert spaces — from a more elegant, intuitive, general, and natural set of axioms. If it succeeds, it will probably be how we teach quantum mechanics in 50 years. And I’ll have to write the textbook! As a side bonus, it could also reveal quantum mechanical behavior in a much wider set of systems in the natural sciences, unconnected with physics.
Sure, saying all these things out aloud sounds super grandiose and silly. But I wouldn’t be doing this if I didn’t think it could work. Obviously it’s super risky, but that’s the only reason I’m actually interested in trying it — the high stakes are what get me up every day, obsessing about it, talking about it, reading endlessly, trying different permutations of ideas, etc.
Now, I’m self funded. I would struggle to imagine someone who would fund me to do this. Or an institution or adviser that would regard it as a suitable topic for a PhD. How on earth could something so ambitious possibly work? Who the hell do I think I am? etc.
But we need hundreds of people doing this — one of them will probably succeed, and it will change the world!
Blind Watchmaker had a big impact on me as a kid. After I had read it, I programmed my own take on the biomorphs program and sent a copy of my program to Richard Dawkins. His research assistant Yan Wong replied with a very complimentary email. I wish I still had the email.
I've been playing with evolutionary algorithms ever since, such as https://taliesin.ai/projects/science/floatworld/
I lost my fob. So I took my intercom apart and wired a band pass filter circuit between the buzzer input and actuator output. If you pressed my buzzer with the right steady pattern, it would automatically let you in.
It worked great and I didn’t buy a replacement fob.
A few months later I started getting woken up by someone buzzing my buzzer. They would always scurry in before I could open my door to see who it was.
Turns out my upstairs neighbor saw my shenanigans a few times and started freeloading off my invention. Honestly amazed he/she reverse engineered what was going on!
Needless to say, I bought a new fob.