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'.

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