I began a post in 2013 by recognizing something that David Deutsch said in a TED talk in 2005. I have referred back to it many times since, and here I will do it again.  But this time I would like to present it more completely. It’s a beautiful articulation of something that’s just true:

Billions of years ago and billions of light-years away, the material at the center of a galaxy collapsed towards a supermassive black hole. And then intense magnetic fields directed some of the energy of that gravitational collapse, and some of the matterback out in the form of tremendous jets, which illuminated lobes with the brilliance of — I think it’s a trillion — suns. 

Now, the physics of the human brain could hardly be more unlike the physics of such a jet. We couldn’t survive for an instant in it. Language breaks down when trying to describe what it would be like in one of those jets. It would be a bit like experiencing a supernova explosion, but at point-blank range and for millions of years at a time. 

And yet, that jet happened in precisely such a way that billions of years later, on the other side of the universe, some bit of chemical scum could accurately describe and model and predict and explain, above all what was happening there, in reality. The one physical system, the brain, contains an accurate working model of the other, the quasar. Not just a superficial image of it, though it contains that as well, but an explanatory model, embodying the same mathematical relationships and the same causal structure. 

Now, that is knowledge. And if that weren’t amazing enough, the faithfulness with which the one structure resembles the other is increasing with time.

Today I listened to a TED interview that Deutsch did, on June 26, with Chris Anderson. It was given the title The Limitless Potential of Human Knowledge. Anderson confirmed that Deutsch’s use of the term chemical scum was borrowed from Steven Hawking, because essentially the same observation appears in Deutsch’s The Beginning of Infinity. In any case, this interview, like the book, is about the big picture. It is the kind of thinking I find very satisfying. Knowledge is defined as information that causes things to happen. The information in a DNA molecule, that causes features to develop in an organism, is knowledge. What we usually call knowledge is what Deutsch calls explanatory knowledge. Explanatory knowledge, he says, is a uniquely human event. And this feature of the universe, the explanatory knowledge developed by our species, ranks alongside gravity or electromagnetism and, as Deutsch sees it, surpasses them. Take the explosion of a quasar, for example, that is fully simulated by the explanatory knowledge encoded in an astrophysicist’s brain. Which of these, Deutsch asks, is more remarkable – the explosion or its encoded model in the astrophysicist’s brain.

Deutsch argues with simple, conversational language, and tightly woven observations, that human explanatory knowledge is a substantial feature of the universe and that it provides us infinite reach. In space, he points out, there is a hierarchy of size. A black hole is hardly affected by the star it swallows, and the sun is hardly affected by the earth. But on earth, he suggests, the opposite is true. On earth, life is everywhere, and every living thing is the result of the action of just one or two molecules. On earth, “submicroscopic entities command vast resources,” even among living entities that don’t have explanatory knowledge. Humans, however, with the evolution of explanatory knowledge, have now become cosmically significant. Deutsch calls this a phase change – small things affect large things, but not with mass or energy, with just information. It is this knowledge that gives us infinite reach and, with this, our optimism should begin to grow. Deutsch often points out, that most of our knowledge concerns what is not seen. The models of reality that we create are built with patterns of ideas, and capture the aspects of our world that are not visible. Mathematics is a crucial to this.

Deutsch imagines the evolution of our ability to build explanatory knowledge as the development of memes, which he defines as “anything that is copied from one brain to another.” The development of memes was followed by a tradition of criticism and error correction. And this is where things take off. He rests his thoughts on Karl Popper’s epistemology, and an important feature of Popper’s argument is that the first thrust of scientific theory is not what is perceived but rather what is imagined. The Stanford Encyclopedia of Philosophy puts it this way:

Popper stresses, simply because there are no “pure” facts available; all observation-statements are theory-laden, and are as much a function of purely subjective factors (interests, expectations, wishes, etc.) as they are a function of what is objectively real.

We imagine solutions to the problems we face, and then test their strength. Our theories are not established empirically, they are only confirmed or falsified empirically.

But it is Deutsch’s emphasis on information, abstraction, and imagination that always captures my attention. And it is because of how those things relate to mathematics. He clarifies a distinction between the abstract and the physical in The Beginning of Infinity:

Whether a mathematical proposition is true or not is indeed independent of physics. But the proof of such a proposition is a matter of physics only. There is no such thing as abstractly proving something, just as there is no such thing as abstractly knowing something. Mathematical truth is absolutely necessary and transcendent, but all knowledge is generated by physical processes, and its scope and limitations are conditioned by the laws of nature…

Consequently, the reliability of our knowledge of mathematics remains for ever subsidiary to that of our knowledge of physical reality. Every mathematical proof depends absolutely for its validity on our being right about the rules that govern the behavior of some physical objects, like computers, or ink and paper, or brains.

This does two things: it confirms the transcendence of mathematical truth and highlights, or forces the recognition of the consistent interaction of physical and abstract things.

In another passage about numbers he writes this:

Mathematicians nowadays distinguish between numbers, which are abstract entities, and numerals, which are physical symbols that represent numbers; but numerals were discovered first. They evolved from ‘tally marks’ (I, II, III, IIII,…) or tokens such as stones, which had been used since prehistoric times to keep track of discrete entities such as animals or days…The next level above tallying is counting, which involves numerals…it (tallying) is an impractical system. For instance, even the simplest operations on numbers represented by tall marks, such as comparing them, doing arithmetic, and even just copying them, involves repeating the entire tallying process….The earliest improvement may have been to just group the tally marks…Later, such groups were themselves represented by shorthand symbols….By exploiting the universal laws of addition, those rules gave the system some important reach beyond tallying – such as the ability to perform arithmetic…Something new has happened here, which is more than just a matter of shorthand: an abstract truth has been discovered…Numbers have been manipulated in their own right, via their numerals.

I mean it literally when I say that it was the system of numerals that performed arithmetic. The human users of the system did of course physically enact those transformations. But to do that, they first had to encode the system’s rules somewhere in their brains, and then they had to execute them as a computer executes its program. (emphasis added)

This seems to be telling us something about how we might find the abstractions that facilitate knowledge. It’s likely never a direct path, but it happens.

Knowledge, Deutsch imagines, is a growing sphere against the unknown. This is a nice image. All evils, he will claim, are due to a lack of understanding. I think I have always believed something like this. Other expressions I find refreshing: knowledge is our superpower, it has infinite reach, and mistakes are a gift – the faster we make them, the faster we acquire knowledge. And all of this is not just about science, it’s about everything.

I have become fully preoccupied with finding some way to comprehend the inter-relatedness or connectedness of everything, from explosive cosmological origins to human thoughts and ideas. It seems to me that mathematics provides a lens on the ground that is common to both thoughts and objects. Deutsch’s view always pulls me in. Today, what I share without hesitation is the optimism provided by a reliance on understanding, and the truth that knowledge has infinite reach. I agree that our world contains possibilities we have not yet imagined.

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