Incompleteness in the natural world


Gödel Incompleteness Theorem
Gödel Incompleteness Theorem (Photo credit: janoma.cl)

A post inspired by Godel, Escher, Bach, Complexity: A Guided Tour, an article in this week’s New Scientist about the clash between general relativity and quantum mechanics and personal humiliation.

The everyday incompleteness: This is the personal humiliation bit. For the first time ever I went on a “Parkrun” today – the 5km Finsbury Park run, but I dropped out after 2.5km 2km – at the top of a hill and about 250 metres from my front door – I simply thought this is meant to be a leisure activity and I am not enjoying it one little bit. I can offer some excuses – it was really the first time ever I had run outdoors and so it was a bit silly to try a semi-competitive environment for that, I had not warmed up properly and so the first 500 metres were about simply getting breathing and limbs in co-ordination – mais qui s’excuse, s’accuse.

But the sense of incompleteness I want to write about here is not that everyday incompleteness, but a more fundamental one – our inability to fully describe the universe, or rather, a necessary fuzziness in our description.

Let’s begin with three great mathematical or scientific discoveries:

The diagonalisation method and the “incompleteness” of the real numbers: In 1891 Georg Cantor published one of the most beautiful, important and accessible arguments in number theory – through his diagonalisation argument, that proved that the infinity of the real numbers was qualitatively different from and greater than the infinity of the counting numbers.

The infinity of the counting numbers is just what it sounds like – start at one and keep going and you go on infinitely. This is the smallest infinity – called aleph null (\aleph_0 ).

Real numbers include the irrationals – those which cannot be expressed as fractions of counting numbers (Pythagoras shocked himself by discovering that \sqrt 2 was such a number). So the reals are all the numbers along a counting line – every single infinitesimal point along that line.

Few would disagree that there are, say, an infinite number of points between 0 and 1 on such a line. But Cantor showed that the number was uncountably infinite – i.e., we cannot just start counting from the first point and keep going. Here’s a brief proof…

Imagine we start to list all the points between 0 and 1 (in binary) – and we number each point, so…

1 is 0.00000000…..
2 is 0.100000000…..
3 is 0.010000000……
4 is 0.0010000000….
n is 0.{n – 2 0s}1{000……}

You can see this can go on for an infinitely countable number of times….

and so on. Now we decide to ‘flip’ the o or 1 at the index number, so we get:

1 is 0.1000000….
2 is 0.1100000….
3 is 0.0110000….
4 is 0.00110000….

And so on. But although we have already used up all the counting numbers we are now generating new numbers which we have not been able to count – this means we have more than \aleph_0 numbers in the reals, surely? But you argue, let’s just interleave these new numbers into our list like so….

1 is 0.0000000….
2 is 0.1000000…..
3 is 0.0100000….
4 is 0.1100000….
5 is 0.0010000….
6 is 0.0110000….

And so on. This is just another countably infinite set you argue. But, Cantor responds, do the ‘diagonalisation’ trick again and you get…

1 is 0.100000…..
2 is 0.110000….
3 is 0.0110000….
4 is 0.1101000…
5 is 0.00101000…
6 is 0.0110010….

And again we have new numbers, busting the countability of the set. And the point is this: no matter how many times you add the new numbers produced by diagonalisation into your counting list, diagonalisation will produce numbers you have not yet accounted for. From set theory you can show that while the counting numbers are of order (analogous to size) \aleph_0 , the reals are of order 2^{\aleph_0} , a far far bigger number – literally an uncountably bigger number.

Gödel’s Incompleteness Theorems: These are not amenable to a blog post length demonstration, but amount to this – we can state mathematical statements we know to be true but we cannot design a complete proof system that incorporates them – or we can state mathematical truths but we cannot build a self-contained system that proves they are true. The analogy with diagonalisation is that we know how to write out any real number between 0 and 1, but we cannot design a system (such as a computer program) that will write them all out – we have to keep ‘breaking’ the system by diagonalising it to find the missing numbers our rules will not generate for us. Gödel’s demonstration of this in 1931 was profoundly shocking to mathematicians as it appeared to many of them to completely undermine the very purpose of maths.

Turing’s Halting Problem: Very closely related to both Gödel’s incompleteness theorems and Cantor’s diagonalisation proof is Alan Turing’s formulation of the ‘halting problem’. Turing proposed a basic model of a computer – what we now refer to as a Turing machine – as an infinite paper tape and a reader (of the tape) and writer (to the tape). The tape’s contents can be interpreted as instructions to move, to write to the tape or to change the machine’s internal state (and that state can determine how the instructions are interpreted).

Now such a machine can easily be made of go into an infinite loop e.g.,:

  • The machine begins in the ‘start’ state and reads the tape.  If it reads a 0 or 1 it moves to the right and changes its state to ‘even’.
  • If the machine is in the state ‘even’ it reads the tape. If it reads a 0 or 1 it moves to the left and changes its state to ‘start’

You can see that if the tape is marked with two 0s or two 1s or any combination of 0 or 1 in the first two places the machine will loop for ever.

The halting problem is this – can we design a Turing machine that will tell us if a given machine and its instructions will fall into an infinite loop? Turing proved  we cannot without having to discuss any particular methodology … here’s my attempt to recreate his proof:

We can model any other Turing machine though a set of instructions on the tape, so if we have machine T we can have have it model machine M with instructions I : i.e., T(M, I)

Let us say T can tell whether M will halt or loop forever with instructions I – we don’t need to understand how it does it, just suppose that it does. So if (M, I) will halt T writes ‘yes’, otherwise it writes ‘no’.

Now let us design another machine T^\prime that takes T(M,I) its input but here T^\prime loops forever if T writes ‘yes’ and halts if T writes ‘no’.

Then we have:

M(I) halts or loops – T(M, I) halts – T^\prime loops forever.

But what if we feed T^\prime the input of T^\prime(T(M, I)?

M(I) halts or loops – T(M, I) halts – T^\prime(T(M,I)) loops forever – T^\prime(T^\prime(T(M,I))) – ??

Because if the second T^\prime(T^\prime(T(M,I))) halted then that would imply that the first had halted – but it is meant to loop forever, and so on…

As with Gödel we have reached a contradiction and so we cannot go further and must conclude that we cannot build a Turing machine (computer) that can solve the halting problem.

Quantum mechanics: The classic, Copenhagen, formulation of quantum mechanics states that the uncertainty of the theory collapses when we observe the world, but the “quantum worlds” theory suggests that actually the various outcomes do take place and we are just experiencing one of them at any given time. The experimental backup for the many worlds theory comes from quantum ‘double-slit’ experiments which suggest particles leave traces of their multiple states in every ‘world’.

What intrigues me: What if our limiting theories – the halting problem, Gödel’s incompleteness theorem, the uncountable infinite, were actually the equivalents of the Copenhagen formulation and, in fact, maths was also a “many world” domain where the incompleteness of the theories was actually the deeper reality – in other words the Turing machine can both loop forever and halt? This is probably, almost certainly, a very naïve analogy between the different theories but, lying in the bath and contemplating my humiliation via incompleteness this morning, it struck me as worth exploring at least.

How I discovered the fundamental theorem of arithmetic by chance


Cover of "Gödel, Escher, Bach: An Eternal...
Cover via Amazon

Actually, of course, I rediscovered it.

I have been attempting to read, for the third time Douglas Hofstadter‘s celebrated Godel, Escher, Bach: I bought a copy in Washington DC in 2009 and loved it (though didn’t get very far before I put it down for some reason) but I have always struggled to get deeply into it: it’s plainly a great book – though having read more on Turing these days I am not sure I’d subscribe to what appears to be the core thesis – but it’s not an easy read.

(Though Hofstadter’s preface did inspire me three years ago to read the brilliantly compact Godel’s Proof– buy, steal or borrow a copy if you are at all interested in maths.)

This time round I am feeling more confident I’ll make progress, it just seems a bit easier to grasp – possibly because I am feeling more confident about maths these days (not that there is any higher maths in it – but grasping some of the more complex stuff does seem to help with all of it.)

So I got the part known as the “MU puzzle” – can you go from the axiom MI to the theorem MU following the rules of the formal system he sets out. I thought – oh, yes, easy, just find a power of two that is also a multiple of three – and in my head I start going 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096 – a list I have been familiar with for thirty three years or so. But none of them are multiples of three.

Now, presumably anyone who knows anything about prime numbers has by now said “of course they are not, haven’t you heard of the fundamental theorem of arithmetic?” – well it seems the answer to that is “err, no”.

Or, if I had, I never really grasped its meaning. Each number cn be factorised into a unique set of primes (if a prime then just itself, of course). That precludes any power of two having as a factor any other prime number (of which three is of course one). It’s really epically important in terms of understanding numbers and I am slightly shocked and puzzled I have never really understood it before now.

Gödel, Turing and decidability


Portrait of Kurt Gödel, one of the most signif...
Image via Wikipedia

I am still wandering around in the world opened to me by The Annotated Turing
– perhaps a little lost, but I have been reading some guide books.

The most recent of these has been Gödel’s Proof which gives a gentle(ish) introduction to Gödel’s incompleteness theorem.

This is a short, but fascinating book that I repeatedly made the mistake of trying to read when tired or not able to give it my full attention (eg on a flight from London to Budapest last week when three days of hard work had really taken it out of me). But I finally managed to finish it last night.

I think the really startling point it makes – which admittedly Charles Petzold’s book also makes but I didn’t fully grasp at the time – is about the nature of the human mind.

Gödel’s theorem states that no consistent system of axioms whose theorems can be listed by an “effective procedure” (such as a computer program) is capable of proving all facts about the natural numbers.

Before I read the book I had not really thought about what this meant: after all Turing and Church had shown that most numbers/mathematical problems were not computable and so this seemed of a part with that conclusion.

But the key thing here is that metamathematics can show the correctness of theorems that axiomatic proofs cannot. In other words – I think – that computer programs are actually a poor model of our ability to solve mathematical problems.