The Beginning of Infinity: Explanations That Transform the World

by David Deutsch

reach always has an explanation. Just as in science a simple formula may summarize a mass of facts, so a simple, easily remembered rule can bring many additional words into a writing system, but only if it reflects an underlying regularity.

As the rules of a writing system were improved, a significant threshold could be crossed: the system could become universal for that language – capable of representing every word in it.

a writing system based on an alphabet can cover not only every word but every possible word in its language,

Some historians believe that the idea of an alphabet-based writing system was conceived only once in human history – by some unknown predecessors of the Phoenicians, who then spread it throughout the Mediterranean

It is sometimes suggested that scribes deliberately limited the use of alphabets for fear that their livelihoods would be threatened by a system that was too easy to learn.

indeed it seems to be a recurring theme in the early history of many fields that universality, when it was achieved, was not the primary objective, if it was an objective at all. A small change in a system to meet a parochial purpose just happened to make the system universal as well. This is the jump to universality.

Something new has happened here, which is more than just a matter of shorthand: an abstract truth has been discovered, and proved, about seven, eight and fifteen without anyone having counted or tallied anything. Numbers have been manipulated in their own right, via their numerals.

knowledge is information which, when it is physically embodied in a suitable environment, tends to cause itself to remain so.

People consist of abstract information, including the distinctive ideas, theories, intentions, feelings and other states of mind that characterize an ‘I’.

The only way to emancipate arithmetic from tallying is with rules of universal reach. As with alphabets, a small set of basic rules and symbols is sufficient. The universal system in general use today has ten symbols, the digits 0 to 9, and its universality is due to a rule that the value of a digit depends on its position in the number.

This curious lack of enthusiasm for universality was repeated in medieval Europe: a few scholars adopted Indian numerals from the Arabs in the tenth century (resulting in the misnomer ‘Arabic numerals’), but again these numerals did not come into everyday use for centuries.

It is as though everyone in the ancient world was avoiding universality on purpose.

ancient Greek culture in general, may not have had the concept of an abstract number at all, so that, for them, numerals could refer only to objects – if only objects of the imagination.

The jump to computational universality should have happened in the 1820s, when the mathematician Charles Babbage designed a device that he called the Difference Engine – a mechanical calculator which represented decimal digits by cogs, each of which could click into one of ten positions.

Today, your washing machine is almost certainly controlled by a computer that could be programmed to do astrophysics or word processing instead, if it were given suitable input–

Without error-correction all information processing, and hence all knowledge-creation, is necessarily bounded. Error-correction is the beginning of infinity.

Because of the necessity for error-correction, all jumps to universality occur in digital systems. It is why spoken languages build words out of a finite set of elementary sounds:

DNA. How that intricate mechanism evolved is not essential here, but for definiteness let me sketch one possibility.

Genes are replicators that can be interpreted as instructions in a genetic code. Genomes are groups of genes that are dependent on each other for replication. The process of copying a genome is called a living organism. Thus the genetic code is also a language for specifying organisms. At some point, the system switched to replicators made of DNA, which is more stable than RNA and therefore more suitable for storing large amounts of information.

considered as a language for specifying organisms, the genetic code has displayed phenomenal reach. It evolved only to specify organisms with no nervous systems, no ability to move or exert forces, no internal organs and no sense organs, whose lifestyle consisted of little more than synthesizing their own structural constituents and then dividing in two. And yet the same language today specifies the hardware and software for countless multicellular behaviours that had no close analogue in those organisms, such as running and flying and breathing and mating and recognizing predators and prey. It also specifies engineering structures such as wings and teeth, and nanotechnology such as immune systems, and even a brain that is capable of explaining quasars, designing other organisms from scratch, and wondering why it exists.

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

SUMMARY

in a paper entitled ‘Computing Machinery and Intelligence’, he famously addressed the question: can a machine think?

nineteen years after Eliza, not one of the Eliza-like programs of the day resembled a person even slightly more than the original had.

Programs written today – a further twenty-six years later – are still no better at the task of seeming to think than Eliza was.

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

language (of subroutines) would have to evolve along with the adaptations that it was expressing. This is what was happening in the biosphere before that jump to universality that finally settled on the DNA genetic code.

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

although it seems obvious that an AI will have qualia and consciousness, we cannot explain those things. So long as we cannot explain them, how can we expect to simulate them in a computer program?

Quale (plural qualia) The subjective aspect of a sensation.

SUMMARY 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. Once that has been solved, programming it will not be difficult. Even artificial evolution may not have been achieved yet, despite appearances. There the problem is that we do not understand the nature of the universality of the DNA replication system.

The best explanation of anything eventually involves universality, and therefore infinity. The reach of explanations cannot be limited by fiat.

The defining property of an infinite set is that some part of it has as many elements as the whole thing.

intuitions about infinity are often illogical.

Every room is at the beginning of infinity. That is one of the attributes of the unbounded growth of knowledge too: we are only just scratching the surface, and shall never be doing anything else. So there is no such thing as a typical room number at Infinity Hotel. Every room number is untypically close to the beginning. The intuitive idea that there must be ‘typical’ or ‘average’ members of any set of values is false for infinite sets.

the intuitive notion of a proportion of the members of a set does not necessarily apply to infinite sets either.

the attributes probable or improbable, rare or common, typical or untypical have literally no meaning in regard to comparing infinite sets of natural numbers.

Scientific explanations cannot possibly depend on how we choose to label the entities referred to in the theory. So anthropic reasoning, by itself, cannot make predictions.

a meter might move by one centimetre, which is a finite distance, but it consists of an uncountable infinity of points.

Only the laws of physics determine what is finite in nature. Failure to realize this has often caused confusion.

Thus the laws of physics determine the distinction not only between rare and common, probable and improbable, fine-tuned or not, but even between finite and infinite.

the same sequence of events can be finite or infinite depending on what the laws of physics are.

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

almost all mathematical truths have no proofs. They are unprovable truths. It also follows that almost all mathematical statements are undecidable: there is no proof that they are true, and no proof that they are false. Each of them is either true or false, but there is no way of using physical objects such as brains or computers to discover which is which. The laws of physics provide us with only a narrow window through which we can look out on the world of abstractions.

So, there is nothing mathematically special about the undecidable questions, the non-computable functions, the unprovable propositions. They are distinguished by physics only. Different physical laws would make different things infinite, different things computable, different truths – both mathematical and scientific – knowable. 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.

Three closely related ways in which the laws of physics seem fine-tuned are: they are all expressible in terms of a single, finite set of elementary operations; they share a single uniform distinction between finite and infinite operations; and their predictions can all be computed by a single physical object, a universal classical computer

It 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.

there is something special – infinitely special, it seems – about the laws of physics as we actually find them, something exceptionally computation-friendly, prediction-friendly and explanation-friendly.

the lack of a proof does not necessarily prevent a proposition from being understood. On the contrary, the usual order of events is for the mathematician first to understand something about the abstraction in question and then to use that understanding to conjecture how true propositions about the abstraction might be proved, and then to prove them.

At present we do not know why the laws of physics seem fine-tuned; we do not know why various forms of universality exist (though we do know of many connections between them); 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.