I find that Marc's suggestion ties in interestingly with a prior subject I've dealt with in this blog: Subjective Reality.
I think it is probably not the best approach to think about the universe as a formal system. I find it more useful to consider formal systems as approximate and partial models of the universe.
So, in my view, the universe is neither consistent nor inconsistent, any more than a brick is either consistent or inconsistent. There may be mutually consistent or mutually inconsistent models of the universe, or of a brick.
The question Marc has raised, in this perspective, is whether the "best" (in some useful sense) way of understanding the universe involves constructing multiple mutually logically inconsistent models of the universe.
An alternative philosophical perspective is that, though the universe is not in itself a formal system, the "best" way of understanding it involves constructing more and more comprehensive and sophisticated consistent formal systems, each one capturing more aspects of the universe than the previous. This is fairly close to being a rephrasing of Charles S. Peirce's philosophy of science.
It seems nice to refer to these two perspectives as Inconsistent versus Consistentist views of the universe. (Being clear however that the inconsistency and consistency refer to models of the universe rather than the universe itself.)
Potentially the Inconsistentist perspective ties in with a previous thread in this blog regarding the notion of Subjective Reality. It could be that, properly formalized, the two models
A) The universe is fundamentally subjective, and the apparently objective world is constructed out of a mind's experience
B) The universe is fundamentally objective and physical, and the apparently subjective world is constructed out of physical structures and dynamics
could be viewed as two
- individually logically consistent
- mutually logically inconsistent
- separately useful
Inconsistentism also seems to tie in with G. Spencer Brown's notion of modeling the universe using "imaginary logic", in which contradiction is treated as an extra truth value similar in status to true and false. Francisco Varela and Louis Kauffmann extended Brown's approach to include two different imaginary truth values I and J, basically corresponding to the series
I = True, False, True, False,...
J = False, True, False, True,...
which are two "solutions" to the paradox
X = Not(X)
obtained by introducing the notion of time and rewriting the paradox as
X[t+1] = Not (X[t])
In Brownian philosophy, the universe may be viewed in two ways
- timeless and inconsistent
- time-ful and consistent
creates(subjective reality, objective reality)
creates(objective reality, subjective reality)
creates(X,Y) --> ~ creates(Y,X)
and then a resolution such as
I = subjective, objective, subjective, objective,...
J = objective, subjective, objective, subjective,...
embodying the iteration
creates(subjective reality[t], objective reality[t+1])
creates(objective reality[t+1], subjective reality[t+2)
If this describes the universe then it would follow that the subjective/objective distinction only introduces contradiction if one ignores the existence of time.
Arguing in favor of this kind of iteration, however, is a very deep matter that I don't have time to undertake at the moment!
I have said above that it's better to think of formal systems as modeling the universe rather than as being the universe. On the other hand, taking the "patternist philosophy" I've proposed in my various cognitive science books, we may view the universe as a kind of formal system comprised of a set of propositions about patterns.
A formal system consists of a set of axioms.... OTOH, in my "pattern theory" a process F is a pattern in G if
- F produces G
- F is simpler than G
In this sense, any set of patterns may be considered as a formal system.
I would argue that, for any consistent simplicity-evaluation-measure, the universal pattern set is a consistent formal system; but of course inconsistent simplicity-evaluation-measures will lead to inconsistent formal systems.
Whether it is useful to think about the whole universe as a formal system in this sense, I have no idea...