Archive for Kolmogorov

Cosmology Talks – To Infinity and Beyond (Probably)

Posted in mathematics, The Universe and Stuff with tags , , , , , , , , , , , , , on March 20, 2024 by telescoper

Here’s an interestingly different talk in the series of Cosmology Talks curated by Shaun Hotchkiss. The speaker, Sylvia Wenmackers, is a philosopher of science. According to the blurb on Youtube:

Her focus is probability and she has worked on a few theories that aim to extend and modify the standard axioms of probability in order to tackle paradoxes related to infinite spaces. In particular there is a paradox of the “infinite fair lottery” where within standard probability it seems impossible to write down a “fair” probability function on the integers. If you give the integers any non-zero probability, the total probability of all integers is unbounded, so the function is not normalisable. If you give the integers zero probability, the total probability of all integers is also zero. No other option seems viable for a fair distribution. This paradox arises in a number of places within cosmology, especially in the context of eternal inflation and a possible multiverse of big bangs bubbling off. If every bubble is to be treated fairly, and there will ultimately be an unbounded number of them, how do we assign probability? The proposed solutions involve hyper-real numbers, such as infinitesimals and infinities with different relative sizes, (reflecting how quickly things converge or diverge respectively). The multiverse has other problems, and other areas of cosmology where this issue arises also have their own problems (e.g. the initial conditions of inflation); however this could very well be part of the way towards fixing the cosmological multiverse.

The paper referred to in the presentation can be found here. There is a lot to digest in this thought-provoking talk, from the starting point on Kolmogorov’s axioms to the application to the multiverse, but this video gives me an excuse to repeat my thoughts on infinities in cosmology.

Most of us – whether scientists or not – have an uncomfortable time coping with the concept of infinity. Physicists have had a particularly difficult relationship with the notion of boundlessness, as various kinds of pesky infinities keep cropping up in calculations. In most cases this this symptomatic of deficiencies in the theoretical foundations of the subject. Think of the ‘ultraviolet catastrophe‘ of classical statistical mechanics, in which the electromagnetic radiation produced by a black body at a finite temperature is calculated to be infinitely intense at infinitely short wavelengths; this signalled the failure of classical statistical mechanics and ushered in the era of quantum mechanics about a hundred years ago. Quantum field theories have other forms of pathological behaviour, with mathematical components of the theory tending to run out of control to infinity unless they are healed using the technique of renormalization. The general theory of relativity predicts that singularities in which physical properties become infinite occur in the centre of black holes and in the Big Bang that kicked our Universe into existence. But even these are regarded as indications that we are missing a piece of the puzzle, rather than implying that somehow infinity is a part of nature itself.

The exception to this rule is the field of cosmology. Somehow it seems natural at least to consider the possibility that our cosmos might be infinite, either in extent or duration, or both, or perhaps even be a multiverse comprising an infinite collection of sub-universes. If the Universe is defined as everything that exists, why should it necessarily be finite? Why should there be some underlying principle that restricts it to a size our human brains can cope with?

On the other hand, there are cosmologists who won’t allow infinity into their view of the Universe. A prominent example is George Ellis, a strong critic of the multiverse idea in particular, who frequently quotes David Hilbert

The final result then is: nowhere is the infinite realized; it is neither present in nature nor admissible as a foundation in our rational thinking—a remarkable harmony between being and thought

But to every Hilbert there’s an equal and opposite Leibniz

I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author.

You see that it’s an argument with quite a long pedigree!

Many years ago I attended a lecture by Alex Vilenkin, entitled The Principle of Mediocrity. This was a talk based on some ideas from his book Many Worlds in One: The Search for Other Universes, in which he discusses some of the consequences of the so-called eternal inflation scenario, which leads to a variation of the multiverse idea in which the universe comprises an infinite collection of causally-disconnected “bubbles” with different laws of low-energy physics applying in each. Indeed, in Vilenkin’s vision, all possible configurations of all possible things are realised somewhere in this ensemble of mini-universes.

One of the features of this scenario is that it brings the anthropic principle into play as a potential “explanation” for the apparent fine-tuning of our Universe that enables life to be sustained within it. We can only live in a domain wherein the laws of physics are compatible with life so it should be no surprise that’s what we find. There is an infinity of dead universes, but we don’t live there.

I’m not going to go on about the anthropic principle here, although it’s a subject that’s quite fun to write or, better still, give a talk about, especially if you enjoy winding people up! What I did want to say mention, though, is that Vilenkin correctly pointed out that three ingredients are needed to make this work:

  1. An infinite ensemble of realizations
  2. A discretizer
  3. A randomizer

Item 2 involves some sort of principle that ensures that the number of possible states of the system we’re talking about  is not infinite. A very simple example from  quantum physics might be the two spin states of an electron, up (↑) or down(↓). No “in-between” states are allowed, according to our tried-and-tested theories of quantum physics, so the state space is discrete.  In the more general context required for cosmology, the states are the allowed “laws of physics” ( i.e. possible  false vacuum configurations). The space of possible states is very much larger here, of course, and the theory that makes it discrete much less secure. In string theory, the number of false vacua is estimated at 10500. That’s certainly a very big number, but it’s not infinite so will do the job needed.

Item 3 requires a process that realizes every possible configuration across the ensemble in a “random” fashion. The word “random” is a bit problematic for me because I don’t really know what it’s supposed to mean. It’s a word that far too many scientists are content to hide behind, in my opinion. In this context, however, “random” really means that the assigning of states to elements in the ensemble must be ergodic, meaning that it must visit the entire state space with some probability. This is the kind of process that’s needed if an infinite collection of monkeys is indeed to type the (large but finite) complete works of shakespeare. It’s not enough that there be an infinite number and that the works of shakespeare be finite. The process of typing must also be ergodic.

Now it’s by no means obvious that monkeys would type ergodically. If, for example, they always hit two adjoining keys at the same time then the process would not be ergodic. Likewise it is by no means clear to me that the process of realizing the ensemble is ergodic. In fact I’m not even sure that there’s any process at all that “realizes” the string landscape. There’s a long and dangerous road from the (hypothetical) ensembles that exist even in standard quantum field theory to an actually existing “random” collection of observed things…

More generally, the mere fact that a mathematical solution of an equation can be derived does not mean that that equation describes anything that actually exists in nature. In this respect I agree with Alfred North Whitehead:

There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain.

It’s a quote I think some string theorists might benefit from reading!

Items 1, 2 and 3 are all needed to ensure that each particular configuration of the system is actually realized in nature. If we had an infinite number of realizations but with either infinite number of possible configurations or a non-ergodic selection mechanism then there’s no guarantee each possibility would actually happen. The success of this explanation consequently rests on quite stringent assumptions.

I’m a sceptic about this whole scheme for many reasons. First, I’m uncomfortable with infinity – that’s what you get for working with George Ellis, I guess. Second, and more importantly, I don’t understand string theory and am in any case unsure of the ontological status of the string landscape. Finally, although a large number of prominent cosmologists have waved their hands with commendable vigour, I have never seen anything even approaching a rigorous proof that eternal inflation does lead to realized infinity of  false vacua. If such a thing exists, I’d really like to hear about it!

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The Monkey Complex

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , on November 15, 2009 by telescoper

There’s an old story that if you leave a set of monkeys hammering on typewriters for a sufficiently long time then they will eventually reproduce the entire text of Shakespeare’s play Hamlet. It comes up in a variety of contexts, but the particular generalisation of this parable in cosmology is to argue that if we live in an enormously big universe (or “multiverse“), in which the laws of nature (as specified by the relevant fundamental constants) vary “sort of randomly” from place to place, then there will be a domain in which they have the right properties for life to evolve. This is one way of explaining away the apparent fine-tuning of the laws of physics: they’re not finely tuned, but we just live in a place where they allowed us to evolve. Although it may seem an easy step from monkeys to the multiverse, it always seemed to me a very shaky one.

For a start, let’s go back to the monkeys. The supposition that given an infinite time the monkeys must produce everything that’s possible in a finite sequence, is not necessarily true even if one does allow an infinite time. It depends on how they type. If the monkeys were always to hit two adjoining keys at the same time then they would never produce a script for Hamlet, no matter how long they typed for, as the combinations QW or ZX do not appear anywhere in that play. To guarantee what we need the kind their typing has to be ergodic, a very specific requirement not possessed by all “random” sequences.

A more fundamental problem is what is meant by randomness in the first place. I’ve actually commented on this before, in a post that still seems to be collecting readers so I thought I’d develop one or two of the ideas a little.

 It is surprisingly easy to generate perfectly deterministic mathematical sequences that behave in the way we usually take to characterize indeterministic processes. As a very simple example, consider the following “iteration” scheme:

 X_{j+1}= 2 X_{j} \mod(1)

If you are not familiar with the notation, the term mod(1) just means “drop the integer part”.  To illustrate how this works, let us start with a (positive) number, say 0.37. To calculate the next value I double it (getting 0.74) and drop the integer part. Well, 0.74 does not have an integer part so that’s fine. This value (0.74) becomes my first iterate. The next one is obtained by putting 0.74 in the formula, i.e. doubling it (1.48) and dropping  the integer part: result 0.48. Next one is 0.96, and so on. You can carry on this process as long as you like, using each output number as the input state for the following step of the iteration.

Now to simplify things a little bit, notice that, because we drop the integer part each time, all iterates must lie in the range between 0 and 1. Suppose I divide this range into two bins, labelled “heads” for X less than ½ and “tails” for X greater than or equal to ½. In my example above the first value of X is 0.37 which is “heads”. Next is 0.74 (tails); then 0.48 (heads), 0.96(heads), and so on.

This sequence now mimics quite accurately the tossing of a fair coin. It produces a pattern of heads and tails with roughly 50% frequency in a long run. It is also difficult to predict the next term in the series given only the classification as “heads” or “tails”.

However, given the seed number which starts off the process, and of course the algorithm, one could reproduce the entire sequence. It is not random, but in some respects  looks like it is.

One can think of “heads” or “tails” in more general terms, as indicating the “0” or “1” states in the binary representation of a number. This method can therefore be used to generate the any sequence of digits. In fact algorithms like this one are used in computers for generating what are called pseudorandom numbers. They are not precisely random because computers can only do arithmetic to a finite number of decimal places. This means that only a finite number of possible sequences can be computed, so some repetition is inevitable, but these limitations are not always important in practice.

The ability to generate  random numbers accurately and rapidly in a computer has led to an entirely new way of doing science. Instead of doing real experiments with measuring equipment and the inevitable errors, one can now do numerical experiments with pseudorandom numbers in order to investigate how an experiment might work if we could do it. If we think we know what the result would be, and what kind of noise might arise, we can do a random simulation to discover the likelihood of success with a particular measurement strategy. This is called the “Monte Carlo” approach, and it is extraordinarily powerful. Observational astronomers and particle physicists use it a great deal in order to plan complex observing programmes and convince the powers that be that their proposal is sufficiently feasible to be allocated time on expensive facilities. In the end there is no substitute for real experiments, but in the meantime the Monte Carlo method can help avoid wasting time on flawed projects:

…in real life mistakes are likely to be irrevocable. Computer simulation, however, makes it economically practical to make mistakes on purpose.

(John McLeod and John Osborne, in Natural Automata and Useful Simulations).

So is there a way to tell whether a set of numbers is really random? Consider the following sequence:

1415926535897932384626433832795028841971

Is this a random string of numbers? There doesn’t seem to be a discernible pattern, and each possible digit seems to occur with roughly the same frequency. It doesn’t look like anyone’s phone number or bank account. Is that enough to make you think it is random?

Actually this is not at all random. If I had started it with a three and a decimal place you might have cottoned on straight away. “3.1415926..” is the first few digits in the decimal representation of p. The full representation goes on forever without repeating. This is a sequence that satisfies most naïve definitions of randomness. It does, however, provide something of a hint as to how we might construct an operational definition, i.e. one that we can apply in practice to a finite set of numbers.

The key idea originates from the Russian mathematician Andrei Kolmogorov, who wrote the first truly rigorous mathematical work on probability theory in 1933. Kolmogorov’s approach was considerably ahead of its time, because it used many concepts that belong to the era of computers. In essence, what he did was to provide a definition of the complexity of an N-digit sequence in terms of the smallest amount of computer memory it would take to store a program capable of generating the sequence. Obviously one can always store the sequence itself, which means that there is always a program that occupies about as many bytes of memory as the sequence itself, but some numbers can be generated by codes much shorter than the numbers themselves. For example the sequence

111111111111111111111111111111111111

can be generated by the instruction to “print 1 35 times”, which can be stored in much less memory than the original string of digits. Such a sequence is therefore said to be algorithmically compressible.

There are many ways of calculating the digits of π numerically also, so although it may look superficially like a random string it is most definitely not random. It is algorithmically compressible.

I’m not sure how compressible Hamlet is, but it’s certainly not entirely random. When I studied it at school I certainly wished it were a little shorter…

The complexity of a sequence can be defined to be the length of the shortest program capable of generating it. If no algorithm can be found that compresses the sequence into a program shorter than itself then it is maximally complex and can suitably be defined as random. This is a very elegant description, and has good intuitive appeal.  

I’m not sure how compressible Hamlet is, but it’s certainly not entirely random. At any rate, when I studied it at school, I certainly wished it were a little shorter…

However, this still does not provide us with a way of testing rigorously whether a given finite sequence has been produced “randomly” or not.

If an algorithmic compression can be found then that means we declare the given sequence not to be  random. However we can never be sure if the next term in the sequence would fit with what our algorithm would predict. We have to argue, inferentially, that if we have fit a long sequence with a simple algorithm then it is improbable that the sequence was generated randomly.

On the other hand, if we fail to find a suitable compression that doesn’t mean it is random either. It may just mean we didn’t look hard enough or weren’t clever enough.

Human brains are good at finding patterns. When we can’t see one we usually take the easy way out and declare that none exists. We often model a complicated system as a random process because it is  too difficult to predict its behaviour accurately even if we know the relevant laws and have  powerful computers at our disposal. That’s a very reasonable thing to do when there is no practical alternative. 

It’s quite another matter, however,  to embrace randomness as a first principle to avoid looking for an explanation in the first place. For one thing, it’s lazy, taking the easy way out like that. And for another it’s a bit arrogant. Just because we can’t find an explanation within the framework of our current theories doesn’t mean more intelligent creatures than us won’t do so. We’re only monkeys, after all.

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