Latest posts of: hwhw

After wiking the theorem and still not understanding it to any functional degree, I'm going to have to go off of only what you said. In order to only answer the very narrow question of the effect of “unprovable truths” on epistemology, I must first define epistemology narrowly.
Epistemology is a field of philosophy concerned with the following questions about knowledge (also from the wiki page, but fairly accurate so I'll use it):
What is knowledge?
How is knowledge acquired?
How do we know what we know?

One relevant question then becomes: are unprovable truths knowledge or not? If not, then apparently it does not have anything to do with epistemology. If a concept or theory cannot be proven, then it seems the only other way for it to be considered true in the human mind is for it to be assumed. Theoretically, assuming a theory to be true does absolutely nothing to show its validity. Functionally, however, assuming a theory to be true allows us to use it as if it had been proven. In applying the theory, we can ignore the fact that it hasn’t been proven by assuming it.

Here, we discover an important answer to the first question. That is, intuitively, one property of knowledge must be that it is true (“bunnies are made out of plutonium” is not knowledge only because it’s not true). This leads us to distinguish between theoretical knowledge, or knowledge that is proven (and thus actually “known”), and functional knowledge, or knowledge that has been assumed. “unprovable truths”, then, are only functional knowledge.

But if the assertion that “in any consistent formulation of number theory, undecidable propositions must exist” is correct, then there can be no purely theoretical knowledge. This is because any formulation must include within it some functional knowledge (unprovable truths). If the conclusions of a theorem are derived from certain assumptions of logic (functional knowledge), then its power of proof is completely dependent on the correctness of the functional knowledge it uses. It is only true if the assumptions it uses are true. This is just a different form of functional knowledge.

By this deconstruction of theoretical knowledge into functional knowledge, seems that functional knowledge is the dominant form of knowledge existent, and maybe the only form of existent knowledge possible. This seems to answer the third question of epistemology: How do we know what we know? In large, we just assume we do and leave it at that.

It appears the second question of epistemology has already been touched upon. We use proofs that still rely on assumptions. Along with those, we use observation and experimentation, assuming that such methods yield reliable results.

Thus, even though the existence of “unprovable truths” may seem like it presents an interesting problem to epistemology, it seems quite boring that the entire problem may simply be ignored by the concept that knowledge is a series of assumptions which don’t appear to need to be purely “proven” to be considered knowledge. In a nutshell, even though these unprovable truths are not truly “known”, no one really seems to care, when the decision is made to apply them anyway. Of course, this raises issues about whether anything can be truly “known”, but such a question matters little except as an exercise of theoretical rumination. In other words, this question doesn’t serve any functional purpose.

Since I understand the theorem you present very little, I may have been ranting on about something related to, but not identical to, your question. It may be more helpful if I talk about the implications of the statement “P: P is not provable”. But I can only quote what someone else said regarding this statement since I know little about it, but it seems to make sense:

“What Goedel shows is that no singe system can account for all possible expressions. Which is to say any set of axioms what wishes to account for all possible statements will either be incomplete or contradictory.
However, if you have various axiomatic systems which complement eachother, you can have a situation where P is True form one system and false form another. If you take expreince of reality as the co-existince and gestalt of multiple axiomatic systems or "paradigms", such as is the case with Postmodern thought and especially the works of Robert Anton Wilson, there is no contradiction, and all possible statments can be accounted for
This is evident in Quantum mechanics, where from the X-axis set of axioms, the Y-spin must occur as a super postion of 2 the posisble X spins. Which is to say that P may infact be True and False at the same time, until measured by an apropriet axiomatic system. The choice of systems, determins the collaps of P as either True or not True. When measured from some particlar systems, P must always remain a maybe, until an external paradigm of thought or a pragmatic necessity is introduced, that can determine or force a decision about P.”