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Beginning Mathematica and Wolfram for Data Science. Fourier Transforms Using Mathematica. Introduction to Probability: Models and Applications. Introduction to Statistics with the Wolfram Language. Exercise Modern quantum mechanics, second edition Physics series. Can't find what you're looking for or don't see your title listed? Contact us �. All rights reserved. Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how �. Advanced Search. Balakrishnan, Markos V.

Could my little rule 30 now be the seed for another such intellectual revolution, and a new way of thinking about everything? And certainly for years I have just quietly used such ideas to develop technology and my own thinking. 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 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.

What the Principle of Computational Equivalence says is that above an extremely low threshold, all processes correspond to computations of equivalent sophistication. It might not be true. It might be that something like rule 30 corresponds to a fundamentally simpler computation than the fluid dynamics of a hurricane, or the processes in my brain as I write this. But what the Principle of Computational Equivalence says is that in fact all these things are computationally equivalent. For one thing, it implies what I call computational irreducibility.

The mathematical tradition in exact science has emphasized the idea of predicting the behavior of systems by doing things like solving mathematical equations. One of the things I did in A New Kind of Science was to show how simple programs can serve as models for the essential features of all sorts of physical, biological and other systems.

Back when the book appeared, some people were skeptical about this. And indeed at that time there was a year unbroken tradition that serious models in science should be based on mathematical equations. But in the past 15 years something remarkable has happened.

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. Three centuries ago pure philosophical reasoning was supplanted by mathematical equations. Now in these few short years, equations have been largely supplanted by programs.

Traditional mathematics-based ways of thinking have made concepts like force and momentum ubiquitous in the way we talk about the world. But now as we think in fundamentally computational terms we have to start talking in terms of concepts like undecidability and computational irreducibility. Will some type of tumor always stop growing in some particular model? It might be undecidable.

Is there a way to work out how a weather system will develop? It might be computationally irreducible. These concepts are pretty important when it comes to understanding not only what can and cannot be modeled, but also what can and cannot be controlled in the world.

Computational irreducibility in economics is going to limit what can be globally controlled. Computational irreducibility in biology is going to limit how generally effective therapies can be�and make highly personalized medicine a fundamental necessity. And through ideas like the Principle of Computational Equivalence we can start to discuss just what it is that allows nature�seemingly so effortlessly�to generate so much that seems so complex to us. Want to automatically make an interesting custom piece of art?

Just start looking at simple programs and automatically pick out one you like �as in our WolframTones music site from a decade ago. Want to find an optimal algorithm for something? Well, then just enumerate cellular automata as I did in , and very quickly you come upon rule 30�which turns out to be one of the very best known generators of apparent randomness look down the center column of cell values, for examples. In other situations you might have to search , cases as I did in finding the simplest axiom system for logic , or the simplest universal Turing machine , or you might have to search millions or even trillions of cases.

One finds some tiny program out in the computational universe. One can tell it does what one wants. We may notice that some particular substance is a useful drug or a great chemical catalyst, but we may have no idea why. But in doing engineering and in most of our modern efforts to build technology, the great emphasis has instead been on constructing things whose design and operation we can readily understand. In the past we might have thought that was enough. What will the world look like when more of what we have is mined from the computational universe?

Today the environment we build for ourselves is dominated by things like simple shapes and repetitive processes. But sometimes they may look quite random, until perhaps suddenly and incomprehensibly they achieve something we recognize.

For several millennia we as a civilization have been on a path to understand more about what happens in our world�whether by using science to decode nature, or by creating our own environment through technology. But to use more of the richness of the computational universe we must at least to some extent forsake this path. 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.

I was fortunate enough that my own very first field�particle physics�was in its period of hypergrowth right when I was involved in the late s. But today, the obvious field in hypergrowth is machine learning , or, more specifically, neural nets. I actually worked on neural nets back in , before I started on cellular automata, and several years before I found rule But I never managed to get neural nets to do anything very interesting�and actually I found them too messy and complicated for the fundamental questions I was concerned with.

I was also inspired by things like the Ising model in statistical physics, etc. At the outset , I thought I might have simplified too far, and that my little cellular automata would never do anything interesting. But then I found things like rule But about 5 years ago I suddenly started hearing amazing things: that somehow the idea of training neural nets to do sophisticated things was actually working. But then we started building neural net capabilities in the Wolfram Language, and finally two years ago we released our ImageIdentify.

There are lots of tasks that had traditionally been viewed as the unique domain of humans, but which now we can routinely do by computer. A neural net is really a sequence of functions that operate on arrays of numbers, with each function typically taking quite a few inputs from around the array.

And instead of taking inputs from all over the place, in a cellular automaton each step takes inputs only from a very well-defined local region. Because it shows that out in the computational universe, away from the constraints of explicitly building systems whose detailed behavior one can foresee, there are immediately all sorts of rich and useful things to be found. Is there a way to bring the full power of the computational universe�and the ideas of A New Kind of Science �to the kinds of things one does with neural nets?

I suspect so. And perhaps even it will be possible to invent some major generalization of things like calculus that will operate in the full computational universe. I have some suspicions, based on thinking about generalizing basic notions of geometry to cover things like cellular automaton rule spaces.

What would this let one do? Likely it would let one find considerably simpler systems that could achieve particular computational goals. Because they imply that even neural nets of the kinds we have now are universal, and are capable of emulating anything any other system can do.

In fact, this universality result was essentially what launched the whole modern idea of neural nets , back in But my guess is that there are tasks where for the foreseeable future access to the full computational universe will be necessary to make them even vaguely practical. What will it take to make artificial intelligence?

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. And when I studied neural nets in , it was partly in the context of trying to understand how to build such a system. I returned to the problem every so often, and kept putting it off.

And it was this realization that got me started building Wolfram Alpha. But definition is a more difficult and central issue than we might imagine.

As powerful as anything that happens in our brains. It sounds so animistic and pre-scientific. Life , intelligence , consciousness: they are all concepts that we have a specific example of, here on Earth. But what are they in general? All life on Earth shares RNA and the structure of cell membranes. And so it is with intelligence. But human intelligence as we experience it is deeply entangled with human civilization, human culture and ultimately also human physiology�even though none of these details are presumably relevant in the abstract definition of intelligence.

We might think about extraterrestrial intelligence. We imagine that in doing the things we humans do, we operate with certain goals or purposes. After all, there are definite laws of nature that govern our brains. So anything we do is at some level just playing out those laws. And this is crucial in thinking about AI.

We know we can have computational systems whose operations are as sophisticated as anything. But can we get them to do things that are aligned with human goals and purposes?

Now what I more see myself as doing is making a bridge between our patterns of human thinking, and what the computational universe is capable of.

There are all sorts of amazing things that can in principle be done by computation. But what the language does is to provide a way for us humans to express what we want done, or want to achieve�and then to get this actually executed, as automatically as possible.

Language design has to start from what we know and are familiar with. In the Wolfram Language, we name the built-in primitives with English words, leveraging the meanings that those words have acquired. But the Wolfram Language is not like natural language. But it gives us a way to build up arbitrarily sophisticated programs that in effect express arbitrarily complex goals.

Yes, the computational universe is capable of remarkable things. But in building the Wolfram Language my goal is to do the best I can in capturing everything we humans want�and being able to express it in executable computational terms. Modern neural nets provide an interesting example. And to cater to our human purposes, what the network ultimately does is to describe what it sees in terms of concepts that we can name with words�tables, chairs, elephants, etc.

But internally what the network is doing is to identify a series of features of any object in the world. Is it green? Is it round? And so on. And what happens as the neural network is trained is that it identifies features it finds useful for distinguishing different kinds of things in the world. But the point is that almost none of these features are ones to which we happen to have assigned words in human language. Now of course new concepts are being added to the corpus of human knowledge all the time.

When I wrote A New Kind of Science I viewed it in no small part as an effort to break away from the use of mathematics�at least as a foundation for science.

But one of the things I realized is that the ideas in the book also have a lot of implications for pure mathematics itself. What is mathematics? But still, plenty has been done in mathematics: indeed, the 3 million or so published theorems of mathematics represent perhaps the largest single coherent intellectual structure that our species has built.

Why is math hard? In other words, it can be arbitrarily hard to get a result in mathematics. Because it could be that most mathematical results one cares about would be undecidable.

Well, if one considers arbitrary abstract systems it happens a lot. Even something as simple as that will often be undecidable. What about the theorems that people investigate in mathematics? But somehow mathematics picks the islands where theorems can actually be proved�often particularly priding itself on places close to the sea of undecidability where the proof can only be done with great effort. What is a proof? In the book I show that for the simple case of basic logic , the theorems that have historically been considered interesting enough to be given names happen to be precisely the ones that are in some sense minimal.

Is there just one historical path that can be taken, say from arithmetic to algebra to the higher reaches of modern mathematics? Or are there an infinite diversity of possible paths, with completely different histories for mathematics? But to me one of the most interesting things is how close�when viewed in these kinds of terms�questions about the nature and character of mathematics end up being to questions about the nature and character of intelligence and AI.

There are some areas of science�like physics and astronomy�where the traditional mathematical approach has done quite well. And there are lots of biological and social systems, for example, where models have now been constructed using simple programs.

This can be perfectly successful for making particular predictions, or for applying the models in technology. And then we look at the patterns generated in understanding some whole collection of sentences. Well, what if those patterns look like the behavior of rule 30?

Or, closer at hand, the innards of some recurrent neural network? But computational irreducibility implies that there may ultimately be no way to create such a thing. Yes, it will always be possible to find patches of computational reducibility, where some things can be said. People have gotten very worried about AI in recent years. As a practical matter, of course, AIs will be able to process larger amounts of data more quickly than actual brains. And in the end the real challenge is to find a way to describe goals.

But what exactly do those things mean? What you need is a language that a human can use to say as precisely as possible what they mean. One has to have a way for humans to be able to talk about things they care about.

Three hundred years ago people like Leibniz were interested in finding a precise symbolic way to represent the content of human thoughts and human discourse.

He was far too early. But what about the AIs?