In the ‘Tractatus’ Wittgenstein attempted to reduce the problems of language and philosophy down to logic – it was this little book that provided much of the backbone to the movement of logical positivism in the 30s/40s. But, as we have seen ( http://wp.me/p1IfS5-40 for the previous discussion on this topic) in doing so Wittgenstein uncovered the paradoxical nature of the structure of language and logic – and it was this that proved to be the death of the positivism movement.
The central problem of positivism (and its more recent relation, scientism) is this:
‘Any proposition whereby one attempts to restrict the scope of human knowledge to some narrow domain, necessarily involves a temporary step “outside” that domain in order to get the birds-eye-view needed to see the broader epistemic landscape so as to formulate the restrictive proposition in the first place. In short, one is sneaking a quick glance from beyond the limits of the proposed knowledge boundary in order to assert knowledge of the boundary itself: a knowledge claim derived from a venue which the knowledge claim itself declares as “off-limits” to human knowledge.’
In short, it is a self-refuting idea – it cannot justify itself within its own system. This idea was key for mathmetician Kurt Godel in his – see http://www.miskatonic.org/godel.html for an excellent introduction and discussion for Godel’s thought and its consequences in philosophy/logic/mathematics.
‘He proved it impossible to establish the internal logical consistency of a very large class of deductive systems – elementary arithmetic, for example – unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves … Second main conclusion is … Gödel showed that Principia, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of arithmetical axioms, there are true mathematical statements that cannot be derived from the set… Even if the axioms of arithmetic are augmented by an indefinite number of other true ones, there will always be further mathematical truths that are not formally derivable from the augmented set.‘ (http://www.miskatonic.org/godel.html)
That, briefly, then are some sketches of the limits of logic and language within formalized systems, which is precisely what Wittgenstein was trying to achieve in the ‘Tractatus – Bertrand Russel and A.N. Whitehead in their ‘Principia Mathmatica,’ but, as has been shown, was ultimately a futile effort.
What this shows are the limits of these systems – but, as noted previously, there seems to be a rather paradoxical trait of limits, in that one has to be on both sides (or at least have been on both sides) for a limit to really make any sense. It seems that even limits themselves are subject to the first quotation above – any limit supposes that one is able to step outside that limit in order to affirm that said limit is indeed a limit.
This then raises the question of certainty – of what can we be certain?