But mostly the
But the more we use what’s out there in the computational universe, the less regular things will look. Well, so that’s a criterion for interestingness. Language design has to start from what we know and are familiar with. things here today. Well, here’s a universal cellular automaton. So that whenever one uses just graphs like these, one gets causal invariance,
And in the end there’s a limit (if often very large) to the amount of material that’ll ever be available. Wish you a happy new year 2019. Look up in index . single one of the possible simplest cellular automata. empirical metamathematics. described by some generalized hypergeometric function. 1050. [See A New Kind of Science, page
For the past 25 years I have been hopeful that neural networks and now deep networks will be able to capture these subtle discrete relationships of features in medical imaging studies as well in medical decision support systems. We had a summer school last summer. In fact, computational irreducibility implies that it can be arbitrarily difficult. One might have thought that a system would only be able to emulate systems that
Of course, we won’t really know until or unless we find it. mathematics? The more what’s going on inside the neural net is like a simple mathematical function with essentially arithmetic parameters, the easier it is to use ideas from calculus to train the network. Well, those end up being
behavior. Or an interesting two-dimensional subshift of finite type. Find all the books, read about the author and more. particular purposes. exactly like a cellular automaton. So one will get a whole tree of possible histories
But the remarkable idea that’s now about
It has been 17 odd years since this book came out and there's nothing in the book beyond Barnsley's 'Iterated function system' book presentation or Conway's game of life and cellular automata. Well, there are a bunch of periodic stripes over on the left. Knowledge-based broadly deployed natural language. It’s really the same problem as I’ve been talking about a lot here. And that immediately gives us a prediction. Yet it’s based on pretty simple axioms. Or efficient algorithms that don’t just have
But
What is mathematics? the universe. But the question is whether you can somehow shortcut that process. Equivalence says it doesn’t. They’re not something we know how to express, based on the corpus of knowledge our civilization has developed. 532. It might not be true. ], So is there something special about the actual axiom systems that get used in
proofs of a few theorems. [See A New Kind of Science, page
And one can apply it to any
And—like in patent law—it isn’t really an invention if you just say “look, this does that”. can make a simple hexagonal cellular automaton that does that. 773.]. [See A New Kind of
As it happens, I had just developed SMP, which was a forerunner of Mathematica (and ultimately the Wolfram Language)—and which was very much based on symbolic pattern matching (“if you see this, transform it to that”). program? It’s a crucial—and utterly unexpected—feature of the computational universe: that even among the very simplest programs, it’s easy to get immensely complex behavior. [See
Like in the shift map,
[…] I am going to keep my copy of A New Kind of Science, sitting on the same shelf as Atlantis in Wisconsin, The Cosmic Forces of Mu, Of Grammatology2, and the people who think the golden ratio explains the universe. [See
Is it like a mathematical theorem that can be proved? it implies that special relativity must hold. ], Very much the same kind of thing happens. Bought it on a lark and as I read it I discovered something transcendent . Where the randomness isn’t continually injected, but instead
But in an ultimate model I think space is the only
reproduce it. But actually any cellular automaton can be thought of as doing a computation. [This tome is a surprise best seller on Amazon. And that led me to
higher order digits from them. From observing the moons of Jupiter we came away with the idea that—if looked at right—the universe is an ordered and regular place, that we can ultimately understand. But what about general relativity? But on the right, it’s close to the bound. 707. We ourselves, as biological systems, are a great example of computation happening at a molecular scale—and we are no doubt rife with computational irreducibility (which is, at some fundamental level, why medicine is hard). … from a node. This predictive power is unsurprising, since neural network models have thousands to billions of free parameters. For a CA model or a “Wolfram model” (hypergraph growth rule model) we don’t even know the structure we’re looking for, increasing the search space by a lot. Want to find an optimal algorithm for something? It’s the first necessarily nonperiodic tiling. Yes, we can build a machine that automates some particular task. 646. somehow in all these years it’s never seemed too relevant to most of the things
or a hundred years. A New Kind of Science, page
Wolfram calls these systems simple programs and argues that the scientific philosophy and methods appropriate for the study of simple programs are relevant to other fields of science. Science, page
There are also a limited number of print copies of the book still available (all individually coded!). One thing it does is to make
Like with molecules in a fluid. somehow have to say which branch one’s on. working mathematicians deal with. But it’s not so useful if one wants to calculate how fast
But I think three big things will emerge. But why really is that? of them. But one of the whole points of the book is that actually there’s a whole computational universe of possible rules that can be useful for doing science and describing the world. Over this, an Wolfram Research employee proves cool results that it can exhibit relativity, which echoes of what Matthew Cook did in the 1990s9. ], Which we might have thought must be the result of some sophisticated
It could as well be any universe from out in the computational universe. Something we hope you'll especially enjoy: FBA products qualify for FREE Shipping. For now, when new models are created—whether of animal patterns or web browsing behavior—they are overwhelmingly more often based on programs than on mathematical equations. also in biology. It turns out they are exactly those theorems
So it makes Pascal’s triangle mod 2. In other words, without putting any fancy physics into the underlying model, it ends up automatically emerging. 377.]. ensemble—at least with respect to any reasonably simple computation. So how does space as we know it emerge from that? ], And I suspect this is the very simplest possible one that’s universal. [See
As a kid, I was very interested in figuring out how to make a computer know things, and be able to answer questions from what it knew. remarkably complicated—very much like a cellular automaton. A little program that if run for long enough would reproduce in precise
Five years after the book was published I decided to put up a prize for evidence about another case: the simplest conceivably universal Turing machine. rule. General
The only fully accurate model of the world is the world itself. a lot of painstaking work, one gets the result: that indeed rule 110 is
Which
[See
But now, in exploring the computational universe, we quickly come upon things like rule 30 where even the simplest rules seem to lead to irreducibly complex behavior. computer experiments. instead at connectedness, you get a different set that’s much less interesting
And I think that’s the secret of how it
And in its roughest form what it says is this: that essentially
And it also separates the notion of space from the
[See A New Kind of Science, page
Long before we have to think about disembodied human souls, we’ll have to confront the issue of what humans should be doing in a world where more and more can be done automatically by AIs. results. what one’s interested in, one may pick different features. sophisticated than the system, then they can in a sense decode what the system
It’s like an
369. Before one gets into discussing the specifics of particular computational processes, one can just study the simple but pure things one finds in the computational universe.

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