The Beginning of Infinity: Explanations That Transform the World

by David Deutsch

Hence we know that, whatever that other universality of the DNA system was, the universality of computation had also been inherent in it for billions of years, without ever being used

The jump to universality The tendency of gradually improving systems to undergo a sudden large increase in functionality, becoming universal in some domain.

The fact that people are universal explainers.

All knowledge growth is by incremental improvement, but in many fields there comes a point when one of the incremental improvements in a system of knowledge or technology causes a sudden increase in reach, making it a universal system in the relevant domain.

Because error-correction is essential in processes of potentially unlimited length, the jump to universality only ever happens in digital systems.

if you can’t program it, you haven’t understood it.

Unfortunately it is very rare for practical solutions to fundamental problems to be discovered without any explanation of why they work.

That is why I said that if lack of computer power were the only thing preventing the achievement of AI, there would be no need to wait.

Becoming better at pretending to think is not the same as coming closer to being able to think.

Specifically, we do not know why the DNA code, which evolved to describe bacteria, has enough reach to describe dinosaurs and humans.

But my guess is that when we do understand them, artificially implementing evolution and intelligence and its constellation of associated attributes will then be no great effort.

The field of artificial (general) intelligence has made no progress because there is an unsolved philosophical problem at its heart: we do not understand how creativity works.

There the problem is that we do not understand the nature of the universality of the DNA replication system.

What Dawkins calls the ‘argument from personal incredulity’ is no argument: it represents nothing but a preference for parochial misconceptions over universal truths.

There is also pessimism, which (as I shall discuss in the following chapter) wants to attribute failure to the existence of a finite bound on improvement.

It is because of the requirement that they be consistent that they are counter-intuitive: intuitions about infinity are often illogical.

The intuitive idea that there must be ‘typical’ or ‘average’ members of any set of values is false for infinite sets.

If there were infinitely many, then we are left with the problem of how to count them – and the mere fact that each astrophysicist-bearing universe would give rise to several others need not meaningfully increase the proportion of such universes in the total.

Or was he experiencing only symbols? But we only ever experience symbols.

quantity is definitely neither infinite nor infinitesimal if it could, in principle, register on some measuring instrument.

Newton and Leibniz were able to use infinitesimal distances to explain physical quantities like instantaneous velocity, yet there is nothing physically infinitesimal or infinite in, say, the continuous motion of a projectile.

Only the laws of physics determine what is finite in nature.

In general terms, the mistake is to confuse an abstract attribute with a physical one of the same name.

Immanuel Kant (1724–1804), who was well aware of the distinction between the absolutely necessary truths of mathematics and the contingent truths of science, nevertheless concluded that Euclid’s theory of geometry was self-evidently true of nature.

And in this way he elevated that formerly harmless misconception into a central flaw in his philosophy, namely the doctrine that certain truths about the physical world could be ‘known a priori’ – that is to say, without doing science.

In the space near the Earth, the angles of a large triangle can add up to as much as 180.0000002 degrees, a variation from Euclid’s geometry which, for instance, satellite navigation systems nowadays have to take into account.

as the mathematician Kurt Gödel had discovered using a different approach to Hilbert’s challenge – almost all mathematical truths have no proofs. They are unprovable truths.

The laws of physics provide us with only a narrow window through which we can look out on the world of abstractions.

It is only the laws of physics that determine which abstract entities and relationships are modelled by physical objects such as mathematicians’ brains, computers and sheets of paper.

But whether the form of an explanation is long or short, and whether it requires few or many elementary operations, depends entirely on the laws of physics under which it is being stated and understood.

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.

So, a computation or a proof is a physical process in which objects such as computers or brains physically model or instantiate abstract entities like numbers or equations, and mimic their properties.

proof theory can never be made into a branch of mathematics. Proof theory is a science: specifically, it is computer science.

The object of mathematics is to understand – to explain – abstract entities.

is because the laws of physics support computational universality that human brains can predict and explain the behaviour of very un-human objects like quasars.

Computational universality is all about computers inside our physical world being related to each other under the universal laws of physics to which we (thereby) have access.

The purpose is to understand, and the overall method, as in all fields, is to make conjectures and to criticize them according to how good they are as explanations.

Then there are the limitations of epistemology: we cannot create knowledge other than by the fallible method of conjecture and criticism;

Hence I conjecture that, in mathematics as well as in science and philosophy, if the question is interesting, then the problem is soluble.

The second is that, in the long run, the distinction between what is interesting and what is boring is not a matter of subjective taste but an objective fact.

we do not know why the world is explicable. But eventually we shall. And when we do, there will be infinitely more left to explain.

But if progress ever depended on violating a law of physics, then ‘problems are soluble’ would be false.

Computation A physical process that instantiates the properties of some abstract entity.

Proof A computation which, given a theory of how the computer on which it runs works, establishes the truth of some abstract proposition.

The universality of reason. – The infinite reach of some ideas.

The unpredictability of the content of future knowledge is a necessary condition for the unlimited growth of that knowledge.

One of them is that, if unlimited progress really is going to happen, not only are we now at almost the very beginning of it, we always shall be.

Where there are infinitely many identical copies of an observer (for instance in multiple universes), probability and proportions do not make sense unless the collection as a whole has a structure subject to laws of physics that give them meaning.

whether a mathematical proposition is provable or unprovable, decidable or undecidable, depends on the laws of physics, which determine which abstract entities and relationships are modelled by physical objects.

Rees pointed out that, for his conclusion to hold, it is not necessary for any one of those catastrophes to be at all probable, because we need be unlucky only once, and we incur the risk afresh every time progress is made in a variety of fields.