Incompleteness

Is all knowledge necessarily incomplete? We might intuit as much (in my first post you’ll see I’m vaguely of that persuasion). But more profoundly, is all knowledge demonstrably incomplete?

Reading Logicomix I was reminded of Gödel’s ‘incompleteness theorems’, which seem to claim that ‘there will always be unanswerable questions.’ Let’s see how this was depicted in Logicomix:

Logicomix Godel 1

Logicomix Godel 2

Bertrand Russell wasn’t actually at that lecture – the authors of Logicomix explain at the end that this was a narrative device. John von Neumann, however, really was, and really did say afterwards, ‘It’s all over.’

So it’s tempting for some (continental philosophers, say, and other anti-positivist types) to see this, jump up and shout Hah! In your face logic and positivism! But reading more about the incompleteness theorems you quickly realise that they’re (unsurprisingly) hugely complex and, more importantly perhaps, quite specific in what they claim. They concern the interrelations of axioms within a particularly defined system of logic – not all logic necessarily. For instance, it only applies to second order logic, while Gödel himself actually proved an earlier completeness theorem for first order logic.

As the Stanford Encyclopedia of Philosophy explains, ‘These results have had a great impact on the philosophy of mathematics and logic. There have been attempts to apply the results also in other areas of philosophy such as the philosophy of mind, but these attempted applications are more controversial.’ And indeed my own sporadic reading quickly threw up the idea that it is widely seen as simplistic and ill-conceived to draw broad, or even any philosophical implications from the theorems.

But that didn’t stop Gödel. One of the main analytic-philosophical debates around the theorems seems to be that of human mechanism – is the human brain a finite, deterministic machine, or does it have some irreducible quality, what we might call free will? According to Stanford, people have claimed that Gödel’s incompleteness theorem ‘proves that Mechanism is false, that is, that minds cannot be explained as machines (J.R. Lucas 1961); but ‘These Gödelian anti-mechanist arguments are, however, problematic, and there is wide consensus that they fail.’ For his part, Gödel believed the theorems did prove that ‘either … the human mind (even within the realm of pure mathematics) infinitely surpasses the power of any finite machine, or else there exist absolutely unsolvable diophantine problems.’ And furthermore, that the latter possibility above ‘seems to disprove the view that mathematics is only our own creation … that mathematical objects and facts … exist objectively and independently of our mental acts and decisions,’ i.e. would prove mathematical Platonism!

With all this uncertainty I was content to leave incompleteness be. But then I stumbled across this episode of the Radiolab podcast: ‘Loops‘.

https://www.wnyc.org/widgets/ondemand_player/radiolab/#file=%2Faudio%2Fxspf%2F161744%2F

From 38mins in they speak with mathematician Steve Strogatz about incompleteness – its history, its claims and its implications. And it’s a great story, so I’ll give it some illustration below.

We start with Gottlob Frege’s quest to find the foundations of mathematics (Bertrand Russell shared this quest throughout his early career). All the way back in the late 19th century, Frege dreamed of a rational machine that could, if programmed with the foundations of mathematics, begin autonomously discovering all the truths of the universe.[1] But what are the foundations of mathematics? He thought that it isn’t number, but that number rather expresses the common foundational quality of ‘sets’. And this is wonderfully explained by Strogatz using this sesame street sketch:


Humphrey takes a food order from some penguin guests: fish, fish, fish, fish, fish, fish. Ingrid tries to remember the order, but makes a mistake: ‘fish, fish, fish, fish, fish?’ ‘No, fish, fish, fish, fish, fish, fish.’ Ernie comes to the rescue to explain that it’s much easier to count the fish and remember that there are 6 of them. ‘Does it work on other stuff??’ asks Humphrey, blown away by this profound breakthrough; making one wonder what it was like before hominids had words for numbers, and how people reacted to their invention. What’s this got to do with anything? Well Strogatz explains that this shows how sets exist deeper than numbers themselves, as the quality of six-ness that’s shared by sets of 6 things. This can be represented by the number 6, but the number 6 is not necessarily the same thing as the foundational quality of six-ness per se.

So Frege built set theory, a logical system that would found mathematics on grounds of certainty. This was the same quest as Bertrand Russell’s, so let’s see how he conceived of it in Logicomix:

Logicomix Russell 1

Russell studied set theory and thought it had great potential:

Logicomix Set Theory 1

And then the bugger discovered the Barber Paradox:

Logicomix Barber Paradox 1

Logicomix Barber Paradox 2

In set theoretical terms, this is the set of all sets that do not contain themselves. Like the liar’s paradox (‘this statement is false’), if it’s true it’s false, and if it’s false it’s true. It’s unprovable, and thus scuppered set theory’s claim to a certain logical foundation. Even Frege (perhaps Frege more than anyone) immediately saw the implications, and sought to stop the imminent publication of Volume 2 of his Foundations of Arithmetic. It did get published, but with an astonishing Addendum. I think every PhD student understands the terror associated with this kind of cruel ‘breakthrough’ in your field.

Logicomix Set Theory 2

Enter Gödel; and Janna Levin to the Radiolab show. She claims the incompleteness theorems stemmed from Gödel thinking about paradoxes like the barber’s paradox or the liar’s paradox, which ultimately results in the mathematisation of the statement ‘this statement is unprovable.’ After a lot of mathematical jiggery pokery, says Levin, the statement ‘this statement is unprovable’ turns out to be… true. The upshot being, says Strogatz, that Gödel felt profoundly freed and liberated, that (as discussed above) humans couldn’t be reduced to certain mechanisms, that ‘there was profound mystery forever.’ Strogatz uses Goldbach’s conjecture as an example of Gödelian undecidability. It hypothesises that ‘Every even integer greater than 2 can be expressed as the sum of two primes.’ Despite huge efforts for over three centuries, no one has yet found a way to either prove or disprove this conjecture. We can test it on every possible number, and we’re doing that, and so far every even integer up to 4 × 1018 complies to the hypothesis. But we’ll never be able test every even integer, as there are infinitely many of them. So without an abstract mathematical proof, this conjecture remains neither provable nor disprovable: undecidable. And it looks like there are plenty of other undecidable problems.

Since Gödel there has been a huge proliferation of logical systems. Axiomatic set theory, for example, apparently solves or somehow negates Russell’s barber paradox. And as discussed, trying to draw conclusions from Gödel’s theorems beyond their narrow intended application in (certain kinds of) mathematical logic is extremely fraught.

But there’s actually another segment to that Radiolab episode (from 7m7s), which is interesting to throw into the mix. They discuss transient global amnesia (TGA), and how when the sufferer’s memory resets, every 90 seconds in the early stages of onset, they tend to begin almost exactly the same conversation all over again. And again. And again. And again. (listen from 10m20s to hear this happening; it’s creepy as hell). They then speak with a doctor who’s had experience with a number of cases of this rare condition, and he confirms that the same thing happened in all of those cases. He muses about the implications of how predictable the human mind seems to become when confronted with highly similar sensory input: ‘it makes the brain seem a little bit more like a machine. You give the machine the exact same set of inputs, and see if the output ever varies, and (it doesn’t), it almost seems like the patient has no free will.’

And then consider David Eagleman’s recent book and tv show The Brain, in which he describes (among many other interesting things) the extent to which many so-called ‘decisions’ happen prior or exterior to what we would think of as our actual consciousness or free will.

So I suppose there are two main threads to this post: free will, and epistemology. Is the quest to prove all of the axioms of maths and logic a wild goose chase, and does this have anything meaningful to say about the human mind, or any kind of philosophy outside the philosophy of mathematical logic? Does this fundamentally undermine the whole project of analytic philosophy, and support the validity of interpretive epistemologies of social science? What about dialectics?

It’s probably undecidable.

But maybe once I’ve read more about all this I’ll have more to say than that. See my next post for a discussion of dialectics, in a (very) vaguely relevant manner.

 


[1]
This was of course a dream of many of the most starry eyed mathematicians and logicians. E.g. Leibniz would surely have dreamt of a combination of his Calculus Ratiocinator (universal logical calculation framework) and his step reckoner (a mechanical calculator too advanced for the manufacturing technology of time, but which became the basis of Thomas de Colmar’s Arithmometer around 150 years later, the first mass-produced calculating machine.