John Page

I'd like to discuss axiomatization, a trend pointed out by tk in this quote from the Dialetheism thread. I'm particularly interested in folks views on the detachment of symbol manipulation from the underlying meaning.

Cheers, John

jpbrooks

The scope of this topic is still a little beyond my understanding, but given that all of mahematics can't be "formalized", shouldn't determining when this process of axiomatization for a particular system is supposed to end be a difficulty? (Or am I just misunderstanding the whole issue?)

John Page

I think that's excatly the topic/issue. Why do you think its difficult?

Cheers, John

jpbrooks

Well again, my understanding of this subject matter is very incomplete, but if some kind of "stopping rule" for the axiomatization of each system could be put into syntactic form then wouldn't that be tantamount to showing that all of mathematics can be "formalized"? In other words, assuming the above situation to be the case, you could just put the rules into a sufficiently complex computer program (that only needs to manipulate symbols) and let it run until it has derived all of mathematics. This is not supposed to be possible. (Or perhaps I'm wrong about this.)

jpbrooks

I'll be back in a few hours (hopefully).

John Page

I believe this postulate relates to Logical Computing Machines - which use predicate logic and math. It is curious to me how we are able to compute that something is not computable.

I'm guessing that tk had this issue in mind when posting......which goes straight back to Kant and the Critique of Pure Reason.

Why does predicate logic "work"? What are its strengths and weaknesses as a representational system? Are there process approaches that can provide a better picture of the operation of the mind.

Cheers, John

jpbrooks

Sorry for the delay. I wanted to post my comments earlier, but I was pressed for time. I'm going home after I submit this reply.

Yes, this seems to show that consciousness is probably not reducible to mechanical operations that involve manipulating symbols at the syntax level. I'm not sure about this, but perhaps the pattern matching ability of the human brain is what gives us the ability to overcome the computability limitations.

I admit that my knowledge of Kant is extremely sketchy, but are you referring to Kant's concept of "pure intuition" in which he holds that "imagination" is what "synthesizes" representations in our conscious experiences?

I agree that these are indeed important and interesting questions, but Logic is supposed to be the study of inference itself apart from such epistemological (and/or ontological) issues.

The computer lab is closing, so I have to run.

John Page

Seems strange to address you as jp, since those are my initials. No problem with the delay, the mysteries of the aeons are unlikely to disappear!

I'm not sure either. Assuming that materiality is at the base of all things then all operations could be deemed mechanical, or at least physical which is kind of the same thing. I think you're right in poking at the "manipulating symbols at the syntax level" approach. There seems to be a tendency to assume that "computability" is limited to what current computer technology can do.

My perspective is that neurological research is very young and to expect it to reveal the essential secrets of consciousness (which has had many millions of years to evolve and requires many years of development in human individuals to attain the higher levels) would be naive. Unfortunately we seem to be stuck with symbols in expressing our ideas about it.

I was thinking of Kant's conclusion that reason alone cannot explain reality from first principles.
Yes, supposed to be. Now I'm going to have a crack at this in the Contradictions and Dialetheism thread.....

Cheers, John

Kim o' the Concrete Jungle

I often find myself quite bemused when a discussion devolves entirely into a syntactical argument, and completely loses sight of any semantic meaning it might otherwise have had. (It's happened to me several times in this forum alone.)

It's almost as though some people are allergic to meaning and avoid it all costs. So I find myself constantly fighting this uphill battle, trying to get people to look at the semantic content of, not just my arguments, but their own.

I suspect that some people take an entirely axiomatic approach, not because it's valid, but because it's easy. They can apply the rules and discover whether a proposition is syntactically correct. Then they can shout "true" or "false", without bothering to make the effort to actually think about the underlying idea, or trying to understand it.

It's too easy to lose sight of the fact that, just because something is syntactically correct according to some model or other, that does not necessarily mean it's actually right in reality. No logical model, however detailed, is ever going to have an exact, one to one correspondence with reality itself (because every model is an abstraction). There is always going to have to be some reference back to the reality that a model is supposed to be modelling. That's why science, with its emphasis on the objective, empirical testing of its models is the superior tool for understanding our universe.

I also find it interesting that no syntactical system (mathematics, logic, etc) has been found to be perfect. And the fact that you can mathematically prove some problems are insoluble is a particularly interesting example. What does this mean? Can we conclude from this that there will never be a unified theory of everything? That's the way I tend to think of it. An axiomatic syntactical system can take us only so far, and no further. And I assume that applies, not just to mathematics and logic, but also to the pattern matching mechanism of the human mind.

John Page

Way to go Kim! :notworthy
What would your test of perfection be?

Cheers, John