Tuesday, 15 July 2014

Gödel's Incompleteness Theorem & Leibniz’s Dream

 Albert Einstein and Kurt Godel

The mathematician and educator, Morris Kline, once made a rather grand claim about Kurt Gödel’s Incompleteness Theorems when he (in his Mathematics: The Loss of Certainty) said that it

"was a response to Leibniz’s 250-year-old dream of finding a system of logic powerful enough to calculate questions of law, politics, and ethics".

Perhaps Leibniz’s dream had nothing to do with applying logic to the content of law, politics and ethics; but only to the form of the arguments in which these things were expressed. For example, in ethics, logic can't show us “what is good”. However, it can detect good and bad arguments as to what constitutes “the Good”.
Similarly logic can show faulty reasoning in political and legal debate; regardless of the actual content of these debates.

So, in that sense, it's indeed true that logic can be applied to law, politics and ethics – indeed to anything! So just as the premises of a deductive argument needn't be true in order for the argument to be valid; so the content of political, legal and ethical statements doesn't matter to the logician - though what follows from them, logically, does matter him.

Logic can "provide the tools to resolve ethical questions by mere calculation" if it dealt only with form and not with metaphysical, epistemological and semantic content.

In any case, were Gödel’s theorems really a response to Leibniz’s dream? Perhaps it was just Gödel’s way of showing us that, well, an axiomatic system (or mathematics generally) can't be both fully consistent and complete – that’s it (without philosophical knobs on).

Much has been made of Gödel’s theorem by non-mathematicians and by many non-philosophers. Morris Kline expresses much of this here. He writes that we

"might think that Gödel’s proof implies that the rational mind is limited in its ability to understand the universe".

How a result in metamathematics could do that (even in principle), I’m not sure. In any case, the mind, again in principle, must surely be limited in some way or ways. Perhaps that means that it could never understand everything there is to know about an infinite universe. Indeed this is bound to be the case. Only an omniscient mind could know everything there is to know about the universe.

Kline makes this point. He says that

"though the mind may have its limitations, Gödel’s result doesn’t prove that these limitations exist".

What is limited isn't the mind as such; but that "axiomatic systems are limited in how well they can be used to model other types of phenomena". This has nothing to do with the mind of man taken generically! It's to do with axiomatic systems and the modelling of other types of phenomena.

Not only that: the "mind may possess far greater capacities than an axiomatic system or a Turing machine". I would say that of course the mind does actually possess far greater capacities than an axiomatic system or a Turing machine. Evidently! For a start, the mind can create great poems or pieces of music. It has memory, experience, imagination, the ability to dream, create, invent, manipulate the environment and so on. Some of these things Turing machines can do; though many of them they can’t do. And no single axiomatic system or Turning machine can do all the things a human mind can do – not even a deranged or damaged human mind!

Another common supposed result of Gödel’s theorems is to assume that his proof implies a limit to artificial intelligence. Perhaps this is a more feasible idea because it must be about the mathematical limitations of artificial intelligence – and that would be relevant to Gödel’s proof. That is, would an indefinite advance in AI be halted by the result of Gödel’s proof which showed that if a mathematical system (therefore all combined) can't be both complete and fully consistent, then a project that relies on mathematics (that is, AI) will never be both complete and fully consistent? Thus there will be a limit to what AI can do.