Archive for February, 2013

Godel on God

Wednesday, February 27th, 2013

My previous post refers to some aspects of Godel’s famous theorems about incompleteness in mathematics. But since the context which brought them to mind was a posting on an atheist blog it’s ironic that Godel believed he had a proof of the existence of God.

The irony is compounded by the fact that he was allegedly reluctant to publish the proof because he was worried that people might think that he actually did believe in God – but in fact he apparently did! Of course he also went essentially mad, but his wife claimed that he read the bible regularly and he once described himself as “baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza.”

And among his papers at death was a page entitled ‘My Philosophical Viewpoint’ which lists fourteen points including:
1. The world is rational
10. Materialism is false
and
14. Religions are, for the most part, bad – but religion is not.

In the light of the way he expressed the undecidability results in his 1951 Gibbs lecture (“Either . . . the human mind . . . infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems”), perhaps the result of his God proof should be expressed as “Either God exists or Modal Logic is fundamentally flawed”. As for me, not having either the will or the capacity to get into modal logic, it could go either way – or perhaps I could say that, within the powers of *my* human mind, both questions are undecidable.

On Expert non-Experts

Wednesday, February 27th, 2013

In the discussions following Landon’s Guest Post @» Butterflies and Wheels (on the value in philosophical training) there is an exchange between Landon and Harald Hanche-Olsen on the question of whether Logic is a branch of Philosophy or Mathematics. I make no claim on logic per se, but the Boolean Algebra which models its basics is a branch of mathematics and that relatively trivial bit is all that has relevance to computers – which is what Landon was on about, but that is not what I want to look at here.

Further down in that exchange the question of Godel’s proofs came up (though that definitely has nothing to do with real computers!) and Landon made some claims that I had to dispute. (In particular he denied the relevance of the well known restriction to systems which include a model for the natural numbers). This led me to revisit the little book by Nagel and Newman – and actually there is still a gap in my understanding. On page 78 N&N show how the statement that formula x is a leading part of formula y corresponds to the Godel number for x being a factor in that for y. So far so good, but on page 79 they assert that there is a similar but more complicated arithmetic characterization of the statement that number z is the Godel number for a proof of the statement with Godel number x. This seems quite plausible but I have never checked it. It could be considered an ‘exercise’ but it might be a hard one to actually carry out in detail. But that’s not the issue I want to address here either.

In order to point out the silliness of claiming that the restriction to systems which include the natural numbers is irrelevant it is of course necessary to identify a mathematical system which does not include them. Simple Boolean Algebra is one – whose consistency is in fact provable. But given Landon’s position, that might not be an effective example. Other algebraic structures naturally come to mind – except for the awkwardness that they are often (though probably not essentially so) set up in terms of a “set” of objects and so it might look at least superficially as if the axioms therefore presuppose those of ZF set theory which does include a model of the natural numbers (and on to just about everything else). Then I thought “aha the existence of Finite Geometries must show that the Euclidean axioms don’t imply the ability to construct a full model of the Natural numbers”. But fortunately I checked before posting and noted that only the incidence axioms were listed – which is fine for my purposes but raises the question of whether the standard Euclidean geometry could have been used as an example. If I had thrown out my first thought there, it would not have taken an expert to see that I was wrong (or at least hadn’t thought carefully enough about which of the often not clearly itemized axioms I was referring to). Anyone with even the slightest familiarity with high school Euclidean Geometry could have said “But Euclidean geometry allows you to construct the mid-point between any two points, and if you then construct the mid-point between that and one of the first two, and so on then don’t you get an infinite sequence of distinct points which could be identified with the ordinal numbers?” and I would have had to stop and think about what goes into that mid-point construction which has been left out of the axioms in terms of which finite models do exist.

In mathematics, and actually I suspect in other disciplines as well, it is often newcomers who make the major advances because those new to the field are less encumbered by either repeatedly reinforced preconceptions or masses of irrelevant detail. If I can be excused a bit of a play on words: they may be expert but they are not yet experts.

Does Morality Need Philosophers?

Sunday, February 24th, 2013

Ophelia Benson’s post on Patricia Churchland’s 2011 book ‘braintrust’ points out that, in contrast to the efforts of Sam Harris and Michael Shermer, Churchland makes a much more modest claim for what she is doing. Indeed Churchland’s claim “is not that science will wade in and tell us for every dilemma what is right or wrong. Rather, the point is that a deeper understanding of what it is that makes humans and other animals social, and what it is that disposes us to care about others, may lead to greater understanding of how to cope with social problems.” Harris on the other hand, and to a lesser extent Shermer, does seem to be claiming the goal of determining what is right or wrong rather than just how people in a certain context might judge it. I was happy to see this point made in a relatively high profile setting as it seems very much in line with my own earlier criticisms of Harris

But a parenthetical comment in Ophelia’s first paragraph has prompted a discussion orthogonal to that of whether science answers moral questions – namely (but putting it a bit crudely) does philosophy do so either?

And I think an exploration and continuation of that discussion may be relevant to concerns some Philosophers seem to have about public perception of their discipline – perhaps including the recent “Physics vs Philosophy” wars. (more…)

To:CRTC Re: SUN “news” Network

Friday, February 22nd, 2013

I do NOT want my cable fees used to pay for biased propaganda that undermines the caring culture that I have chosen to live in (and which, with bad luck, I may one day have to depend on).

Those who want this garbage are not prevented from buying it but please do NOT force me to join them!Sign the AVAAZ petition.

Complexity Explorer

Monday, February 11th, 2013

I heard about this Complexity Explorer course via Stephen Downes and have decided to enrol.

Our experience always involves a lot of complexity which we typically manage by isolating just a few quantities of interest which are related by compactly expressible relationships. I am curious to see whether “complexity theory” really proposes general methods for dealing with cases where this is not possible – or whether it just consists of introducing some particular new ways of extracting simplified models from more complex ones.

Sources of Success

Friday, February 8th, 2013

Alain de Botton Proposes a Kinder, Gentler Philosophy of Success.

And at the Atlantic Alexis Madrigal responds to Jack Dorsey on the role of luck (in addition to hard work and genius) in the making of a great success story.

Something similar came to mind when I read Ta-Nehisi Coates (also at the Atlantic) discussing the challenges faced by daily newspaper columnists, and frankly wondering whether he could meet them himself. But my response is that actually no-one can, and the annointing by newspaper editors of a precious few with the gift of that platform is an insult to those who pay for the papers. (Though I have to admit that if people continue to pay for it then it’s not surprising that that’s what the papers will continue to dish out.)

Whether it’s access to audience via a media pulpit, to political power, or to money, the fact is that there’s an instability in the sense that once you have a certain amount it becomes easier to get more – or as the Bible says “Unto him that hath shall be given, and from him that hath not shall be taken away – even that which he hath”.