Science is destroyed.

jeeprs wrote:

What Craig is arguing is that many scientific explanations have also no further explanation. Science doesn't go 'all the way down' to put it very crudely

I don't know what this means. Can you give me an example?

I'm curious, how many explanations would you need, for something to be fully explained to you? If you asked what the chemical composition of water is, and I answered "One hydrogen molecule and two oxygen molecules", would that not suffice for you?

More importantly, do you think you may be misusing the word "explanation"? Many of you mystical folk tend to confuse the word with "absolute X".

Quote:

We all know that mathematical physics is extraodinarily powerful, but at the same time, to ask 'why does mathematics work?' is to seek an explanation for why mathematical physics is as effective as it is. Why can't really explain the explicatory value of mathematics, we just know that it works very well.

I don't know what sort of question "Why does mathematics work?" is. What does that mean? If I told you 1+1=2, and you asked me "Why does that work", I wouldn't know how to respond. I would tell you it works because those are the values of the numbers, and those are the functions of those signs.

Consistency and criticisms

According to Carnap's "Logicist Foundations of Mathematics", Russell wanted a theory that could plausibly be said to derive all of mathematics from purely logical axioms. However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility. Since the first two were existential axioms, Russell phrased mathematical statements depending on them as conditionals. But reducibility was required to be sure that the formal statements even properly express statements of real analysis, so that statements depending on it could not be reformulated as conditionals. Frank Ramsey tried to argue that Russell's ramification of the theory of types was unnecessary, so that reducibility could be removed, but these arguments seemed inconclusive.

Beyond the status of the axioms as logical truths, the questions remained:

* whether a contradiction could be derived from the Principia's axioms (the question of inconsistency), and
* whether there exists a mathematical statement which could neither be proven nor disproven in the system (the question of completeness).

Propositional logic itself was known to be consistent, but the same had not been established for Principia's axioms of set theory. (See Hilbert's second problem.)

In 1930, Gödel's completeness theorem showed that propositional logic itself was complete in a much weaker sense - that is, any sentence that is unprovable from a given set of axioms must actually be false in some model of the axioms. However, this is not the stronger sense of completeness desired for Principia Mathematica, since a given system of axioms (such as those of Principia Mathematica) may have many models, in some of which a given statement is true and in others of which that statement is false, so that the statement is left undecided by the axioms.

Gödel's incompleteness theorems cast unexpected light on these two related questions.

Gödel's first incompleteness theorem showed that Principia could not be both consistent and complete. According to the theorem, for every sufficiently powerful logical system (such as Principia), there exists a statement G that essentially reads, "The statement G cannot be proved." Such a statement is a sort of Catch-22: if G is provable, then it is false, and the system is therefore inconsistent; and if G is not provable, then it is true, and the system is therefore incomplete.

Gödel's second incompleteness theorem shows that no formal system extending basic arithmetic can be used to prove its own consistency. Thus, the statement "there are no contradictions in the Principia system" cannot be proven in the Principia system unless there are contradictions in the system (in which case it can be proven both true and false).

Wittgenstein (e.g. in his Lectures on the Foundations of Mathematics, Cambridge 1939) criticised Principia on various grounds, such as:

* It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental; for if a persistent discrepancy arose between counting and Principia, this would be treated as evidence of an error in Principia (e.g. that Principia did not characterize numbers or addition correctly), not as evidence of an error in everyday counting.
* The calculating methods in Principia can only be used in practice with very small numbers. To calculate using large numbers (e.g. billions), the formulae would become too long, and some short-cut method would have to be used, which would no doubt rely on everyday techniques such as counting (or else on non-fundamental - and hence questionable - methods such as induction). So again Principia depends on everyday techniques, not vice versa.

However Wittgenstein did concede that Principia may nonetheless make some aspects of everyday arithmetic clearer.