John Page

The more I think about this, the range of X and the mention of infinity are red herrings. The question "Is two times any number ever odd?" will suffice.

Computer programs typically carry out functions but do not operate on functions themselves. They generally do not perform tests for functional equivalence (although I think some pre-compilers - which are programs in their own right - do in order that object and run-time code can be optimized).

To answer the question it is necessary to understand that
a) a number is either odd or even,
b) even numbers are divisible by two,
c) the operation of dividing something by two is reversed by multiplying it by two.
d) vice versa of c) above.
e) therefore from d) any number multiplied by two can be divided by two and hence from b) any number when multiplied by 2 can never be odd.

Of course, this may not be the only way to figure the problem. Let's focus on step c) though - doesn't this seem to contradict the LOI? We take x, compute y, compute z and then we say x and z are identical. Rather, we should x and z have an equivalent numeric quality - i.e. They represent the same number.

This is not turning out as well as I'd hoped, so I'llk rush straighjt to my conclusion. Computers can perform math and logic but they don't appear to understand math and logic. The only way I know that they can hypothesize (other than being programmed to spit out strings of permutations) is in forward branching.

On the other hand, I don't consider it unfeasible to construct a computer that will learn by questioning itself.

Cheers, John

excreationist

I talked about the range because it relates to how it is implemented in the computer program - i.e. X originally equals 1 then it is tested (is 2*X odd?) and incremented indefinitely (X approaches infinity) - unless it comes across a odd number.

Computer programs typically carry out functions but do not operate on functions themselves. They generally do not perform tests for functional equivalence (although I think some pre-compilers - which are programs in their own right - do in order that object and run-time code can be optimized).

That's what Searle would say... I think he thinks "artificial brains" would be theoretically possible though - that would be able to learn to understand things.

I don't know anything about that to comment about that, but you've probably got a point.

tk

Sorry, I was caught up with other things, and sort of forgot about this thread...

Allow me to respond to a few concerns. jpbrooks said that axiomatization means that "you could just put the rules into a sufficientlycomplex computer program (that only needs to manipulate symbols) and let it run until it has derived all of mathematics". Not really true, as each term in an axiom scheme can be instantiated by an infinitely number of possible formulas, e.g. from P -> P I can derive x = x -> x = x, sin x + cos y = z^2 -> sin x + cos y = z^2, etc. etc. So unless we have a computer with infinite parallelism, even a simple theorem like 2 + 2 = 4 may take infinite time to derive! Therefore, I don't think of this as a serious objection against axiomatization, or even as an objection at all.

Kim o' the Concrete Jungle is concerned that an axiomatic approach makes people lose sight of the original semantic meaning. This I think is a valid concern: personally I advocate using the axiomatic approach mainly to debunk arguments made by charlatan philosophers (but not to "prove" arguments as valid).

Also at this point I think it's important to distinguish between the semantics of natural languages (e.g. English) and the semantics of formal languages (e.g. ZFC). Natural languages contain lots of ambiguity, and lots of weird conventional uses and exceptions accumulated over the ages (e.g. why do we say "four o'clock" but not "four o'watch"?). In contrast, the conventions for formal languages tend to be simple and well-defined.

(Then again, some technical languages such as C++ are actually specified in English prose. Thank goodness programmers tend not to interpret words with the same degree of flexibility as do e.g. lawyers...)

Finally, unfortunately the wave/particle duality isn't a good case for showing the "contradictory" nature of the universe. Because things behave as waves at certain times, and as particles at certain other times. When something behaves as a wave, it doesn't behave as a particle. When something behaves as a particle, it doesn't behave as a wave. The mistake is not in assuming the Law of Non-Contradiction, but in assuming that the terms "particle" and "wave" correctly refer to intrinsic properties of objects.

John Page

Precisely - perhaps what we need to do is specify the English language in C++, rather than the other way round.

Cheers, John

jpbrooks

Granted it's not a categorical objection against axiomatization. But (and this is admittedly an error on my part due to my lack of knowledge in rhis area) I assumed that the whole point of the trend toward axiomatizaton in mathematics, computation theory and related fields of science, was to take advantage of the calculating and data processing ability of computers. That is why I brought up the issue of computability.

Perhaps I'm misinderstanding this point also, but (assuming that you are referring to arguments that are formally invalid) if the axiomatic approach can be used to "debunk" arguments then, since any formal logical argument can be converted into a logical statement, why would that not automatically prove the denial of the statement form of the fallacious arguments that are "debunked"?

I agree with you on this point. But to be fair to the opposing view, we currently have no model that can provide an account of why light (for example) displays both wave and particle characteristics. But, what alternative to continuing the process of scientific inquiry without such a model does Science have? We thus seem to be stuck with apparently conflicting physical models of light.

tk

Fortunately, there are many other things which can be done quite easily on machines, such as verifying a known proof of a known theorem...

Agreed, it's somewhat more complicated than that. When I "debunk" an argument, it can either mean that the argument itself is bogus, or that the formal system it's based on is in fact wrong. Thus it's necessary to find out which formal system the philosopher is making use of. If there's no agreement of a formal system to use, I think only a direct empirical argument will work.

Twistor theory, perhaps?...

jpbrooks

Thanks for the link, tk. This is interesting. I am unfamiliar with this alternative to the "string" theory.