Godel's incompleteness theorems were proved towards the beginning of the 20th Century and to this day they are amongst the most abused mathematical theorems around. Today's post will have three parts: Firstly I shall state what the first incompleteness theorem says. Secondly I shall give you an example of how people attempt to apply it when talking of progress. Thirdly I shall explain the hidden assumptions behind their arguments and give my own viewpoint on what the implications of Godel's work are for a belief in progress.
Godel's first incompleteness theorem states (this is not the original formulation) that no computer program can, unaided, list all correct mathematical theorems about the integers.
The argument against unlimited progress then goes as follows:
'Knowledge concerning integers and mathematical objects underlies much of progress in the sciences and even to some degree the arts & humanities. So if there were a limit to our progress in mathematics then that would imply a limit to our progress elsewhere.'
Thus far I should comment the argument is fairly sound. As mathematics deals with various abstract structures and instances of those structures can appear in other areas of human understanding it seems at least plausible that a failure to progress in mathematics should lead to stagnation in other disciplines. The argument then continues:
'Therefore whatever physical laws our universe follows, since a computer could simulate them, we must have that the whole universe could be simulated by a sufficiently powerful computer.'
This in my opinion is a false step but I will finish giving you the argument before saying why:
'Humanity is part of the universe and so can in principle be simulated by a computer too. Hence Godel's incompleteness theorem applies to humanity. Hence humanity will never attain certain mathematical knowledge.'
The final part of the argument is also problematic but for different reasons. I shall give criticisms of the argument and give my opinion on them:
1) '...since a computer could simulate them...'
This betrays the exceedingly dubious assumption that the number of physical laws is finite. Without this assumption what follows in the argument does not follow logically. Physicists are trying to come up with a general unified theory (a goal which was pursued also in the 19th century but ultimately without success). However, it is by no means certain that such a theory even exists.
In fact the above argument really needs to assume that there is a General unified theory behind physics. Otherwise consistent universes can be described where the argument fails. I have more to say on the probability of this but that will have to wait for another post.
2) 'Hence Godel's incompleteness theorem applies to humanity.'
Strictly speaking we must have some way of extracting computationally the mathematical 'knowledge' that humans possess. In an absence of this statement (2) is not meaningful at all.
This second criticism is not too biting. The argument can just allow a skeptic to pick whichever computational method of extracting 'knowledge' she desires and the argument will show (assuming a GUT exists) that humanity cannot have access to all arithmetic truth according to that way of extracting 'knowledge'.
Finally what implications do Godel's incompleteness arguments have for a belief in progress?
A) If progress can continue indefinitely (and without limit) there must be infinitely many physical laws which cannot be reduced to some cleverly crafted finite set (so no GUT).
B) The structure of these laws must somehow encode the truths of arithmetic more over in a way which allows them to be discovered by appropriate means from within the universe.
(A) is not too hard to swallow but many may find (B) to be profoundly counterintuitive. I shall return to look again at (B) in another post.
This post is part of a series investigating the belief in progress. All past articles in this series are filed under the tag 'progress'.
This week I shall give two of my favorite books, with a short description of the books, a quote from them and a brief reason why I like them.
Crow - Ted Hughes
Crow is a collection of short poems with a common narrative thread. It reads like the old testament but is considerably darker. Crow (the subject of the book) is a crude, amoral force twisting the work of a naive God (Yahweh of the old testament).
Many of the poems from Crow depict humanity and nature as a nightmarish reality from which there is no escape.
Here is a passage from Crow showing an example of the way that he (gender?) twists God's work. It provides an alternative 'just so' story for the link between God and humanity.
"But Crow Crow Crow nailed them together, Nailing Heaven and earth together - So man cried but with God's voice. And God bled, but with man's blood."
I like Crow because it gives a perspective on life which, although I don't agree with it, is familiar to us all in our more depressed times. I think the depiction of decay, death and amorality is necessary for a full understanding of our world. In a way one can think of it as a literary inoculation against cruel reality. A good comparison might be made with the aesthetic behind the Hannibal lecter series.
Here [http://www.zeta.org.au/~annskea/Trickstr.htm] is a link to an essay about crow which you may find interesting.
Titus groan - Mervyn Peake
The first in the gormenghast trilogy. Set in a feudalistic earldome with rich traditions and an intensely heirarchical society, Titus Groan follows the story of Steerpike (a low born but machiovelian cook's assistant) and his thirst for power. Along the way we find out about the structures of the social order, the hidden places in Gormenghast castle and something of the life outside the castle walls.
The opening passage of Titus Groan speaks for itself:
"Gormenghast, that is, the main massing of the original stone, taken by itself would have displayed a certain pnderous architectural quality were it possilble to have ignored the circumfusion of those mean dwellings that swarmed like an epidemic around its outer walls. They sprawled over the sloping earth, each one half way over its neighbour until, held back by the castle rampartsm the innermost of these hovels laid hold on the great walls, clamping themselves thereto like limpets to a rock."
I like this book because it has many ecentric characters and their interactions alone make wonderful reading. The prose is full of long and carefully crafted descriptive passages which give the imagination much to work with. The book lends itself to being read aloud and I personally think that parts of it are best appreciated this way.