Bookman

:Bookman wades in, aware that he is in over his head...:

I haven't read the thread that spawned this (and perhaps I should before offering my meager two cents worth), but here goes. The purpose of the discussion that has been advanced, as I understand it, is to shed some light on the interplay between applied and (for lack of a better term) theoretical mathematics and the tendency for mathematicians to "[tend] towards axiomatization -- to push everything up to the syntactic level, so that the process of derivation becomes a mechanical manipulation of symbols with no regard to their `underlying' meaning."

It seems to be to be beyond a tendency -- isn't that what mathematics is?

We can't have this discussion without raising the issue of Goedel and the incompleteness theorem. In short, any formal system must be either incomplete or inconsistent. In an inconsistent system, both logical proposition A and logical proposition ~A could be true; there is not a broad usefulness for such systems. So, useful formal systems have the characteristic that they are incomplete: that there are statements expressible within the syntax of the formal system which can not be proven.

What this means is that we never run out of axioms. Now, it is true that some simple systems can be modeled with a simple set of axioms which desribe their behaviour with apparent completeness. (Technically, Goedel's theorem applies to systems of sufficient complexity to represent arithmetic.) But, I take it that the discussion in the other thread was about consciousness? I'll take it as an unproven assertion that the system which underlies consciousness is sufficiently complex to model arithmetic.

Which leads to this:
It isn't possible. One never runs out of axioms. The tricky bit is not in using the axioms to derive the formal system; rather the hard bit is determining what the axioms are, and no matter how complete our mathematical model there will still be questions about real-world consciousness which we our mathematical model can not answer (but which observation likely can).

Mathematics can neither provide a short-cut to knowledge, nor even take us the whole journey.

Kim o' the Concrete Jungle

A perfect system, I think, would be a system that could solve every problem you could ask. Mathematics is the most purely abstract syntactical system there is, and yet, there are certain mathematical problems for which you cannot derive an answer. Moreover, it's not just that we don't currently know how to solve these problems, we can prove that they are fundamentally insoluble to mathematics.

If a system of pure, abstract logic like mathematics is imperfect, then what hope is there for a system that deals more directly with the messiness of the real world?

I have felt for some time now that a unified theory of everything -- a theory that could distil the essence of the universe down into a few fundamental axioms -- is a bit of a pipe dream. Instead, our human quest for knowledge will leave us with a raft of different models, each one of which works only within certain limits (like the plus or minus 100 years in radio-carbon dating).

I don't think that's necessarily a bad thing. It's just something that we, as a species, are going to have to work around. It could even have some positive influence, if it goes some way towards diminishing the arrogance of certainty. (Dubya could do with a bit of this before he winds up alienating the US from the entire rest of the world.)

jpbrooks

Again, sorry for the brevity of my comments.

I know. I recently even made the mistake of answering a comment that was addressed to you in this forum!

We might even be facing an infinite number of mysteries ahead of us waiting to be uncovered! But I think that's a good thing.

Yes. Our brains seem to be able to do things with symbols that current computing machines can't do.

I tend to agree with people like Searle who hold that Semantics cannot be reduced to Syntactics. Syntactics and Semantics appear to be related but independent aspects of symbol systems.

True. Even if we assume that consciousness resides in some realm outside of the material realm, neurological research would still be important because the brain, in such a view, would be (part of) an "interface" between the material realm and the realm of consciousness.


This seems right because the denial of this would appear to be "Rationalism" (e.g., Cartesian Foundationalism) which cannot ultimately establish its own most fundamental assumptions on purely rational grounds. You are right to point out that we can't reason our way to reality.

jpbrooks

Possibly. But surely there must be some part or aspect of the "mechanism" of the brain that is directly related to (if not the cause of) our ability to overcome the limitations imposed on current computational machinery.

I have to leave soon. I"ll be back later.

John Page

Yes, one could view it that way. How, though, does the human mind quantize and what internal modus operandis permits it to manipulate and detect the relations between quantities (more, less etc.) and types of quantities (negatives, primes etc.).

Beyond that, isn't axiomatization an approach that we use to (try and) explain what numbers are and "fundamental relations" between them. In this way I'm agreeing with you but is there an underlying phenomenology of math and/or psychology of math that explain why we think of numbers the way we do. Why do we 'need' to axiomatize, for example?
Interesting. But if we can explain why it appears that we never run out of axioms.....

Cheers, John

Bookman

Well, an axiom is simply a proposition that is assumed to be true.

By Goedel, for any formal system (with the proviso given above) there exists a statement A, such that the truth value of A can not be determined using the formal system.

In other words, for the real-world system which we have modeled with mathematics there exist conjectures (A or ~A) which can not be proven using the mathematical model. Therefore, if we need the formal system to reflect the real-world truth of the system which it models, we must take A or ~A as a new axiom. Euclid's parallel postulate is a good example here.

And on and on and on...no matter how many axioms we add to the system, there will still exist statements (A') which we can not deduce the truth value of. We never run out of conjecture.

Is that on the track you were inquiring about?

Bookman (who knows a little bit of math, but not a thimbleful of philosophy)

Kim o' the Concrete Jungle

I agree. And since this mechanism of the mind has had several million years to evolve, it is undoubtably much more sophisticated than man-made computational machinery. I wouldn't say, though, that this pattern matching/intuition mechanism of the mind is necessarily superior (as opposed to older, and more developed). It is weak in some areas that computers are strong in, such as data storage and the sort of large scale mathematical manipulation of data, like you would see in, for example, climate modelling.

Brains and computers are just different, that's all. You can use them for doing different kinds of things. In the same way, mathematics is different from the scientific method, inasmuch as they are two different systems that are good for solving different kinds of problems. In order to obtain the broadest understanding of our universe, we should be using all of the tools available to us.
Some of the information we gain from different models will, no doubt, be contradictory. For example, the fact that a photon can't decide if its a particle or a wave seems to bother some people of a logical bent. Classical logic insists that if you have two opposing and mutually exclusive ideas, then one of them has to be wrong. It doesn't allow that both ideas can be right in different contexts.

The way that classical logic insists on a dichotomy between true and false is merely a peculiarity of that system. For a long time, people optimistically assumed that this peculiarity of classical logic was also some sort of fundamental truth about the universe itself, but they were wrong. The wave/particle question proved them wrong.

It is the wave/particle question in particular, that convinces me we need the concept of "context" in a modern philosophical system. We acknowledge that there are many different possible models, and many different possible syntax systems. But we must also admit that any "truths" we discover in the context of a particular model may not necessarily be true in other contexts.

I have spoken about the context sensitivity of "truth" before, and it has proved a rather controversial idea in this forum, but I don't see any other sensible way of resolving the deep philosophical problem caused by the wave/partical question, without going stark raving bonkers. In one model a photon is a wave. In another model, a photon is a particle. Neither model represents an "absolute truth" about the universe, and it's fairly useless to speculate what the "absolute truth" about light might be. We can't know what the "absolute truth" is, all we can do is build more models.

So while I say here that we should not allow ourselves to get too caught up in syntactical arguments at the expense of semantic meaning, I acknowledge that we can never entirely dispense with syntax and abstraction. We need to seek a good balance, that's all. We need to avoid the situation (common amongst Christian apologists) where the words we use become meaningless, because we refuse to pin them down to any specific meaning.

John Page

Not really. I understand that an axiom is an assumption (of an undoubtable truth w.r.t the system under consideration). Try this: "What is the mental representation of an axiom and how do we select one axiom over another?"

Furthermore, let me argue that an axiom is little more than a brain state that is used as the "template" by which to judge other conjectures (represented within other parts of the brain). Very simply, when we report "This is a man" we are merely reporting that the subject under consideration fits our axiomatic concept of a man.

Now, if we reconsider the role of an axiom it becomes a kind of self-fulfilling prophesy, since the axiom effectively is the definition of a man.

Cheers, John

Kim o' the Concrete Jungle

Call me a mad bastard, but I would still prefer to try and account for mathematics along observational lines. My tentative line of reasoning for the system of integers runs as follows.

The mental pattern-recognition mechanism through which we perceive the universe has a tendency to distinguish discreet objects in our environment. We can look at, for example, a coffee cup and, while we understand it is a part of the environment, we also see it is a whole seperate entity of its own. We see it as "an object".

On the next level of complexity up, when we look at many objects together we can see that some objects are similar to other object, and can serve similar purposes. The real intuitive breakthrough here is that at some point in our history, we must have decided to create an abstract class of objects into which we could group all similar things.

The thing is that you can treat an abstract class of objects as though it was a single object in its own right. That's quite a powerful idea, because you can pick a coffee cup you have never seen before, identify it as an object in the abstract class of "coffee cups", and you will know exactly what it is good for, through its association with other objects in the same abstract class you have encountered before. Give this abstract class of objects a label and you have invented the noun form in language.

Okay. So you've got an object in the abstract class you have labelled "coffee cup". What happens when you put this with another object of the same type? You will find that you can introduce a second layer of abstraction. Not only do you have two objects of the same abstract class, you can group those two objects together and treat it as though it was a single entity. You have one group of similar objects that you will label as "two coffee cups". You can work backwards from there to the concept of having "one coffee cup". Once you have the abstract classes you have labelled "one" and "two", it is a short step from there to a system of integers by which you can identify groups of objects.

From there, you can realize that integers have a number of highly stable relationships between each other. You can observe that a group of two objects combined with another group of two objects creates a group of four objects, and that this is always the case no matter how many times you do it. From this observation you can deduce a third level of abstraction -- operations. To do arithmetic, you don't need to mess around manipulating actual groups of objects. You can do it axiomatically working directly with the system of integers itself.

I have no doubt that the high degree of stability in arithmetical (and geometrical) relationships is indicative of some fundamental quality of the universe. But that doesn't make mathematics any less dependent on the way that the human mind functions. And it certainly doesn't mean that we actually understand exactly what that fundamental quality of the universe is. All it means is that we've discovered a system of syntax with a high degree if abstraction and reliability.

jpbrooks

True. The machines that our living human brains have designed can perform some of the tasks that they were designed to do better than can the human brain. But I'm not sure why we can't use the word "superior" to describe the greater variety of things that our brains can do in comparison with computers. Until we build machines that can repair themselves, reproduce, and evolve into higher organized systems autonomously (as living systems already do), I am going to remain cautious about saying that machines are generally "superior" to human brains.

I certainly agree.

But this probably has more to do with our knowledge of the truth than with the truth itself, as you seem to point out below.

True. Formally, Logic is concerned with truth itself and not (mainly) with its contextual character.

Again, I'm not certain that this shows thar Logic is wrong in affirming a dichotomy between "true" and "not true". In fact, I'm not certain how any system that is to function as a "logical" one can avoid that dichotomy. Avoiding the dichotomy would result in a system that, unlike (Classical) Logic, would prevent us from knowing any truth with absolute certainty, including even truths about our own experiences.

Precisely! And this is why I don't believe that reality (and thus Logic) is, at basis, contradictory!

And IIRC, I was one of the few posters who generally agreed with you, once I understood your point about "context". And, If that "understanding" is correct, there is no real contradiction involved in saying that something can be true in one "context" and not true in another, etc..

Right! Wave/Particle duality just indicates that we haven't arrived at the level of "absolute truth" on this matter. So, the two statements, "a photon is a wave", and, "a photon is a particle", each may be "relative truths" that don't contradict one another.

Well, it's sometimes difficult to come to agreement on word meanings when the same word within different views may have different meanings.

I have to leave soon.