The Beginning of Infinity: Explanations That Transform the World (33 page)

How do all those drastic limitations on what can be known and what can be achieved by mathematics and by computation, including the existence of undecidable questions in mathematics, square with the maxim that
problems are soluble
?

Problems are conflicts between ideas. Most mathematical questions that exist abstractly never appear as the subject of such a conflict: they are never the subject of curiosity, never the focus of conflicting misconceptions about some attribute of the world of abstractions. In short, most of them are uninteresting.

Moreover, recall that finding proofs is not the purpose of mathematics: it is merely one of the methods of mathematics. 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. One does not understand a mathematical proposition merely by proving it true. This is why there are such things as mathematics lectures rather than just lists of proofs. And, conversely, 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.

A mathematical theorem can be proved, yet remain for ever uninteresting. And an unproved mathematical conjecture can be fruitful in providing explanations even if it remains unproved for centuries, or even if it is unprovable. One example is the conjecture known in the jargon of computer science as ‘P ≠ NP’. It is, roughly speaking, that there exist classes of mathematical questions whose answers can be
verified
efficiently once one has them but cannot be
computed
efficiently in the first place by a universal (classical) computer. (‘Efficient’ computation has a technical definition that roughly approximates what we mean by the phrase in practice.) Almost all researchers in computing theory are sure that the conjecture is true (which is further refutation of the idea that mathematical knowledge consists only of proofs). That is
because, although no proof is known, there are fairly good explanations of why we should expect it to be true, and none to the contrary. (And so the same is thought to hold for quantum computers.)

Moreover, a vast amount of mathematical knowledge that is both useful and interesting has been built on the conjecture. It includes theorems of the form ‘
if
the conjecture is true then this interesting consequence follows.’ And there are fewer, but still interesting, theorems about what would follow if it were false.

A mathematician studying an undecidable question may
prove
that it is undecidable (and explain why). From the mathematician’s point of view, that is a success. Though it does not answer the
mathematical question
, it solves the
mathematician’s problem.
Even working on a mathematical problem without any of those kinds of success is still not the same as failing to create knowledge. Whenever one tries and fails to solve a mathematical problem one has discovered a theorem – and usually also an explanation – about why that approach to solving it does not work.

Hence, undecidability no more contradicts the maxim that problems are soluble than does the fact that there are truths about the
physical
world that we shall never know. I expect that one day we shall have the technology to measure the number of grains of sand on Earth exactly, but I doubt that we shall ever know what the exact number was in Archimedes’ time. Indeed, I have already mentioned more drastic limitations on what can be known and achieved. There are the direct limitations imposed by the universal laws of physics – we cannot exceed the speed of light, and so on. Then there are the limitations of epistemology: we cannot create knowledge other than by the fallible method of conjecture and criticism; errors are inevitable, and only errorcorrecting processes can succeed or continue for long. None of this contradicts the maxim, because none of those limitations need ever cause an unresolvable conflict of explanations.

Hence I conjecture that, in mathematics as well as in science and philosophy,
if the question is interesting, then the problem is soluble.
Fallibilism tells us that we can be mistaken about what is interesting. And so, three corollaries follow from this conjecture. The first is that inherently insoluble problems are inherently uninteresting. 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. And the third corollary is that the interesting problem of
why
every problem that is interesting is also soluble is itself soluble. 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.

The most important of all limitations on knowledge-creation is that we cannot prophesy: we cannot predict the content of ideas yet to be created, or their effects. This limitation is not only consistent with the unlimited growth of knowledge, it is entailed by it, as I shall explain in the next chapter.

That problems are soluble does not mean that we already know their solutions, or can generate them to order. That would be akin to creationism. The biologist Peter Medawar described science as ‘the art of the soluble’, but the same applies to all forms of knowledge. All kinds of creative thought involve judgements about what approaches might or might not work. Gaining or losing interest in particular problems or sub-problems is part of the creative process and itself constitutes problem-solving. So whether ‘problems are soluble’ does not depend on whether any given question can be answered, or answered by a particular thinker on a particular day. But if
progress
ever depended on violating a law of physics, then ‘problems are soluble’ would be false.

One-to-one correspondence
   Tallying each member of one set with each member of another.

Infinite (mathematical)
   A set is infinite if it can be placed in one-to-one correspondence with part of itself.

Infinite (physical)
   A rather vague concept meaning something like ‘larger than anything that could in principle be encompassed by experience’.

Countably infinite
   Infinite, but small enough to be placed in one-to-one correspondence with the natural numbers.

Measure
    A method by which a theory gives meaning to proportions and averages of infinite sets of things, such as universes.

Singularity
   A situation in which something physical becomes unboundedly large, while remaining everywhere finite.

Multiverse
   A unified physical entity that contains more than one universe.

Infinite regress
   A fallacy in which an argument or explanation depends on a sub-argument of the same form which purports to address essentially the same problem as the original argument.

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 ending of the ancient aversion to the infinite (and the universal).

– Calculus, Cantor’s theory and other theories of the infinite and the infinitesimal in mathematics.

– The view along a corridor of Infinity Hotel.

– The property of infinite sequences that every element is exceptionally close to the beginning.

– The universality of reason.

– The infinite reach of some ideas.

– The internal structure of a multiverse which gives meaning to an ‘infinity of universes’.

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

We can understand infinity through the infinite reach of some explanations. It makes sense, both in mathematics and in physics. But it has counter-intuitive properties, some of which are illustrated by Hilbert’s thought experiment of Infinity Hotel. 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. Cantor proved, with his diagonal argument, that there are infinitely many levels of infinity, of which physics uses at most the first one or two: the infinity of the natural numbers and the infinity of the continuum. 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. A mere infinite sequence of universes, like the rooms in Infinity Hotel, does not have such structure, which means that anthropic reasoning by itself is insufficient to explain the apparent ‘fine-tuning’ of the constants of physics. Proof is a physical process: 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. Similarly, whether a task or pattern is simple or complex depends on what the laws of physics are.

The possibilities that lie in the future are infinite. When I say ‘It is our duty to remain optimists,’ this includes not only the openness of the future but also that which all of us contribute to it by everything we do: we are all responsible for what the future holds in store. Thus it is our duty, not to prophesy evil but, rather, to fight for a better world.

Karl Popper,
The Myth of the Framework
(1994)

Martin Rees suspects that civilization was lucky to survive the twentieth century. For throughout the Cold War there was always a possibility that another world war would break out, this time fought with hydrogen bombs, and that civilization would be destroyed. That danger seems to have receded, but in Rees’s book
Our Final Century
, published in 2003, he came to the worrying conclusion that civilization now had only a
50
per cent chance of surviving the twenty-first century.

Again this was because of the danger that newly created knowledge would have catastrophic consequences. For example, Rees thought it likely that civilization-destroying weapons, particularly biological ones, would soon become so easy to make that terrorist organizations, or even malevolent individuals, could not be prevented from acquiring them. He also feared accidental catastrophes, such as the escape of genetically modified micro-organisms from a laboratory, resulting in a pandemic of an incurable disease. Intelligent robots, and nanotechnology (engineering on the atomic scale), ‘could in the long run be even more threatening’, he wrote. And ‘it is not inconceivable that physics could be dangerous too.’ For instance, it has been suggested
that elementary-particle accelerators that briefly create conditions that are in some respects more extreme than any since the Big Bang might destabilize the very vacuum of space and destroy our entire universe.

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. He compared this with playing Russian roulette.

But there is a crucial difference between the human condition and Russian roulette: the probability of winning at Russian roulette is unaffected by anything that the player may think or do. Within its rules, it is a game of pure chance. In contrast, the future of civilization depends entirely on what we think and do. If civilization falls, that will not be something that just happens to us: it will be the outcome of choices that people make. If civilization survives, that will be because people succeed in solving the problems of survival, and that too will not have happened by chance.

Both the future of civilization and the outcome of a game of Russian roulette are unpredictable, but in different senses and for entirely unrelated reasons. Russian roulette is merely
random
. Although we cannot predict the outcome, we do know what the possible outcomes are, and the probability of each, provided that the rules of the game are obeyed. The future of civilization is
unknowable
, because the knowledge that is going to affect it has yet to be created. Hence the possible outcomes are not yet known, let alone their probabilities.

The growth of knowledge cannot change that fact. On the contrary, it contributes strongly to it: the ability of scientific theories to predict the future depends on the reach of their explanations, but no explanation has enough reach to predict the content of its own successors – or their effects, or those of other ideas that have not yet been thought of. Just as no one in 1900 could have foreseen the consequences of innovations made during the twentieth century – including whole new fields such as nuclear physics, computer science and biotechnology – so our own future will be shaped by knowledge that we do not yet have. We cannot even predict most of the problems that we shall encounter, or most of the opportunities to solve them, let alone the solutions and attempted solutions and how they will affect events. People in 1900
did not consider the internet or nuclear power
unlikely
: they did not conceive of them at all.

No good explanation can predict the outcome, or the probability of an outcome, of a phenomenon whose course is going to be significantly affected by the creation of new knowledge. This is a fundamental limitation on the reach of scientific prediction, and, when planning for the future, it is vital to come to terms with it. Following Popper, I shall use the term
prediction
for conclusions about future events that follow from good explanations, and
prophecy
for anything that purports to know what is not yet knowable. Trying to know the unknowable leads inexorably to error and self-deception. Among other things, it creates a bias towards pessimism. For example, in 1894 the physicist Albert Michelson made the following prophecy about the future of physics: