*International Conference on Philosophy*in Athens. I was in Europe for other matters and thought it might be worthwhile to include this conference.

It was an interesting experience. There were about 60 people present, presumably (almost) all philosophers. I did not meet any other theists. There was a philosopher from Notre Dame University, whom I thought would probably be Roman Catholic. However, when I chatted with him, it turned out that he was in fact a lapsed Roman Catholic; it was the problem of evil that drove him to atheism. It was intriguing to hear him tell of his interactions with his colleagues Alvin Plantinga and Nicholas Wolterstorff, two well-known Christian philosophers at Notre Dame.

I presented a paper on the role of models and presuppositions in science. My thesis was that one’s worldview plays a decisive role both in justifying the presuppositions needed for doing science and in determining the actual content of science. I argued that naturalism fails to provide an adequate basis for science whereas Christianity does so very well. Allowance for Christianity, I noted, has significant implications for ontology (e.g., the existence of a spiritual reality), epistemology (e.g., divine revelation), and causality (e.g., supernatural events). These, in turn might well have great impact on the content of science, particularly with regards to questions of eschatology and origins..

My purpose for attending this conference (aside from the opportunity to visit Greece) was to test these ideas against a hostile audience. The response I received was polite disagreement regarding my theist presuppositions but a general acknowledgment that naturalism did have its problems.

Indeed, throughout the conference I noticed a widespread sense of frustration. One philosopher complained that Nietzsche left him feeling empty. The prime difficulty was the lack of absolutes. It was generally agreed, for example, that absolutes were essential for genuine morality. However, in the absence of an Absolute, there was no way to establish these. One philosopher proclaimed, “give me rationality and I can give you morality.” His idea was that morality could be established on a utilitarian basis once one had a properly grounded rationality. But he had no idea how the latter could be rigorously established. Another philosopher defended Descartes in arguing for a platonic realm for absolutes. Yet he had no way of connecting the inert platonic ideals with the actual, concrete material world. On this point he was unwilling to invoke theism, as Descartes had done.

In one of the last papers, the presenter defended “constructionist” mathematics. He objected to classical mathematics since this was based on the theist notion of an ideal Mathematician who knew everything and could manipulate infinite sets. Instead, he argued for a type of mathematics that allowed only for finite, human constructions. Such a restriction places severe limits on the content of mathematics. When I pointed out that constructionist mathematics was insufficient to prove advanced theorems needed for modern physics (particularly quantum mechanics and general relativity), he acknowledged this. He also agreed with my further claim that, if mathematics is just a convenient convention, with no claims to absolute truth, then the same must apply to physics, which is strongly dependent on math. Where, I asked, does this leave Big Bang cosmology, or the materialist basis of your philosophy? He just smiled and shrugged his shoulders. Ultimately, it seems, particularly given its nihilistic implications, naturalism can be accepted only on the basis of a rather a tough-minded faith that is firmly resolved to evade God.

If this is the best that naturalism can do, Christians have little to fear from its challenges. I left this conference feeling thankful for the joy, meaning, and hope that we have in Christ. It is clear that denying God leads only to intellectual frustration, not fulfillment.

That must have been an interesting experience!

ReplyDeleteAbout the "constructionist" mathematics; I have read a little bit about this before; doesn't their finiteness also disallow them from using the law of the excluded middle (i.e. not(not(A)) doesn't necessarily imply A)?

If this is the case, then it is not just "advanced theorems" that are lost, but even very simple ones. I've been wondering for years whether it is possible to prove that sqrt(2) is irrational -- WITHOUT a proof by contradiction (i.e. law of excluded middle). The only proofs I can find or think of are all of the form:

Suppose there exist integers p and q such that (p/q)*(p/q)=2

...

Contradiction! Therefore sqrt(2) is irrational.

I mean, if there are only "finite, human constructions", I don't even see how we could make a statement like "sqrt(2) is irrational", but only "none of the integers I've ever heard of have a ratio of sqrt(2)"

Hi Ruberad

ReplyDeleteWelcome to my blog. You raise an interesting question. There are constructionist proofs for most classical theorems. These proofs can be much longer than classical ones, but once established those theorems can be utilized.

Here is one possible constructive proof:

Define x < y iff there is positive integer n such that y-x > 1/n.

Define x neq y iff x 0).

Define x irrational iff for all rational a/b, x neq a/b. (Note that

this is stronger than not(x=y).)

Consider rational a/b. We can prove (by usual means) that a^2 - 2b^2

neq 0. (This is integer neq, of course.)

Now,

abs( a/b - sqrt(2) ) = abs( (a/b + sqrt(2)) )^-1 b^-2 abs (a^2 -

2b^2) > ((abs(a)+2)b^2)^-1

So we get a/b neq sqrt(2) for all rational a/b, so sqrt(2) is

irrational.

http://cs.nyu.edu/pipermail/fom/2005-October/009237.html

In what way does classical mathematics depend on the existence of a creator-deity? I hear this from time to time from presuppositionalists, but I don't know of any sense in which the statement can be defended.

ReplyDeleteEven the constructivists seem to agree that invoking the existence of God won't save classical math. Bishop pointed out, for instance, that God can do all the math he wants, but it won't help us finite-minded human beings do OUR math. In other words, to the extent that we want to do math ourselves, we must rely on constructivist methods. Now, I don't agree with this view at all---I haven't found any good arguments for constructivism, or against classical math---but even assuming that the constructivists are correct, Christianity still won't help us reclaim any additional kinds of math. If there really is a problem with a classical approach, it doesn't seem that invoking God or Christianity helps us solve it.

--Ben Wallis

Here is an article from Vern Poythress that talks about that sort of stuff.

ReplyDeleteBut I agree with you; math "depends" on God in the trivial sense that if there were no God there would be no math (and no creation...). But belief in God is not necessary to do good and correct math. (Neither is it sufficient.)

Also, I don't see that classical math is in any kind of crisis that it needs to be saved by constructivism or any other such.

confessionalouthouse,

ReplyDeleteThanks for the link. It really disappoints me that a trained mathematician like Poythress could stray so far into that kind of nonsense. I guess it just goes to show that skilled mathematicians need not be good *philosophers* of mathematics.

--Ben

Yes; are these mathematical systems he describes going to take over and crumble mathematics because of failure to build from a Christian worldview? Or are they the result of a few non-mainstream sophists, who even the godless can clearly see are spinning their semantic wheels to no effect? I vote B.

ReplyDeleteMathematicians do not need to be philosophers in order to

ReplyDeletedomath. The difficulty, however, is how to account for thevalidityof their math. Recall Godel's theorems. How do we know that a particular theorem is true? what counts as a valid proof?I argue that naturalism cannot account for classical mathematics. Bishop agrees. He writes:

"Classical mathematics concerns itself with operations that can be carried out by God.I am doing my best to develop a secure philosophical foundation for current mathematical practice. The most solid foundation available at present seems to me to involve the consideration of a being with non-finite powers—call him God or whatever you will—in addition to the powers possessed by finite beings."Since Bishop doesn't believe in God, he is forced to find another basis for math. But his constructivism severely truncates math.

#$%*@$)(%@$&()%@$ Stupid blogger/google trying to make me sign up for a blogger account and lost my comment!

ReplyDeleteTrying again, if you are not immediately convinced of my self-evident brilliance, it is because my comment was better the first time...

ReplyDeleteI am not convinced by Bishop's argument (or affirmation -- I don't know his argument) that mathematical operations can only be "carried out" (

done) by an infinite being. Finite beings were doing just fine for millenia before Bishop came along, and continue doing just fine.As for Go"del, I don't see the point.

I don't think Go"del was asking epistemological questions about validity of proofs. He simply proved that there exist (must exist in systems with sufficiently complex axioms to provide elementary arithmetic) well-formed propositions which cannot be proven true or false, because they are neither true nor false, but a third category "undecidable"; self-contradictory whether you try to apply true or false to them (like "this statement is false"). Note, this is different from "We can't know whether any proof of proposition P is valid"; Go"del constructed a proposition P for which he was able to prove "P is not true and P is not false" (two implications of which are that P cannot be validly proven true, and P cannot be validly proven false).

Actually, I was thinking of Go"del's 2nd theorem: the consistency of a system S large enough to contain full arithmetic cannot be proven by methods within S.

ReplyDeleteThe problem then is to justify the soundness of full arithmetic and more advanced math, such as calculus.

These can be derived from Zermelo-Fraenkel-Choice (ZFC) set theory but the ZFC axioms posit infinite sets and two-valued logic, which are plausible from a theistic perspective but not from a naturalist view.

Interesting. Ya got me there. I don't know about ZFC and I'm not sure what two-valued logic is (an anti-Go"del assumption that all propositions are decidable?)

ReplyDeleteBut still, I don't see how infinite sets are implausible from a naturalist view. I know loads of naturalists who work with {The Set of Integers} or {The Set of Real Numbers} all the time, and they don't find them implausible.

John Byl,

ReplyDeleteThanks for the response! However, I must still disagree with your assessment.

Regarding Brouwer and Bishop: their arguments seem somewhat obscure to me, so it's difficult to tell what precisely they meant. However, as best I can make out, they objected that when math becomes too far idealized it loses its usefulness. So maybe God can make use of the idealized math if he is available to do so, but if we want to be able to use the math here on earth then we'll have to restrict ourselves to finite methods. Now, personally I don't buy into their claims. Few mathematicians do. It seems to me that non-constructivist math is enormously useful. But even if they were correct---if idealized math wasn't useful to us after all---then I don't see how the existence of God would change that. What usefulness does mathematics have with God that it hasn't got without?

Also, you expressed some concern about the "validity" of math. But validity in what sense? We seem both to agree that math is incredibly useful, so it's valid in that sense. Are you perhaps doubting the validity of the arguments of mathematicians? But then how does God help us construct valid arguments?

All in all I'm just really puzzled by how you think God helps us do/use/trust/etc. math. Hopefully you have time to say a few things about all this.

Thanks again,

--Ben

In answer to your questions please read my article

ReplyDeleteMatter, Math and God, particularly the second part.

The basic problem is: how do we know that our math is true? Math may be very useful. Yet, if it possibly contains inconsistencies,then mathematical models of, say, nuclear reactors, may turn out to be disastrously faulty.

Brouwer and Bishop try to build up math using self-evident axioms and finite methods. This yields a consistent math but at the price of leaving out much classical math.

The justification of classical math requires such things as the justification of the Axiom of Choice and the existence of transfinite numbers (higher orders of infinity).

If I were an atheist, I would concede, maybe you're right, but so what? The inductive principle may not give us any certain guarantee that the "laws" we have observed in the past will continue to hold, but it's what we have, and it's worked so far, so why shouldn't we assume they will continue? What else should we do, just give up? My plan is to continue doing math and science, to continue depending on induction. You're not arguing that I shouldn't rely on induction, and I'm not worried about it, so what's the problem? You can have fun justifying math, if that interests you, I'll keep myself busy doing math (including using the Axiom of Choice and transfinite numbers).

ReplyDeleteconfessionalouthouse,

ReplyDeleteI wouldn't even concede that much. As an atheist mathematician-in-training, here's my position:

First of all, I don't think the alleged problems have been clearly articulated. At least I still don't understand what they are supposed to be. For example, John Byl says, "The basic problem is: how do we know that our math is true?" But I would ask, what does it even MEAN for math to be "true"? Sure, individual mathematical statements have truth values, but I don't know how to make sense of assigning a truth value to something like set theory itself, or the entire field of mathematics, etc. I can say---and have said---that set theory is USEFUL. What more could we ask from it?

Equally puzzling to me is how God is supposed to help solve whatever problems we might have. In his paper for instance, John Byl remarks that God "upholds all truths" (p83). Again, though, I'm not sure what that is supposed to mean. Maybe he's just saying that God causes the world to be a certain way. But how does that help us determine what is true and what is false? Or maybe he wants to say something like, on nontheism we have no reason to trust our truth-seeking faculties, whereas theism gives us a reason. So in that case his argument would be a sort of re-statement of Plantinga's evolutionary argument against naturalism (EAAN). Of course I think the EAAN has lots of problems, many of which have been pointed out by philosophers with credentials well beyond mine. ; )

As for induction, I don't think it's appropriate to say that it has always worked in the past, and so we should continue to use it in the future. That itself is an inductive argument, and so it more or less begs the question. Instead I would simply point out that induction is part of what we mean by terms like "justification" and "warrant" and such. In other words, if you ask, "why do you believe X?" and I respond, "because I make an inductive inference to X," then that is a satisfactory response (provided my inference meets with an agreeable inductive standard). But more importantly, God doesn't help us with induction, either. Presuppositionalists sometimes want to say that God makes it the case that the future will be like the past, or something along those lines. But "makes" how? Using his causal powers, presumably. But causal powers only make sense if we already have an inductive standard in place. To say that God "causes" X to occur is to say that X is part of some regularity whereby God wills things to happen and they accordingly do. But then we still have to assume in advance that there are such things as regularities, which is just another way to express the inductive principle. Either way we have to rely on the assumption of regularities---i.e., on induction. Invoking the existence of God doesn't help us escape that.

Anyway, I hope that's clear.

--Ben

How can set theory (or other branches of math) be *useful* if they are not *true*?

ReplyDelete"

That itself is an inductive argument, and so it more or less begs the question."The question-begging was self-conscious, and part of my point. What is the "usefulness" of questioning induction? Can you somehow prepare for a hypothetical day when the laws of nature no longer apply?

I don't think your attempt to place an assumption of regularities behind/above/beyond God works. God is not bound to keep things regular, but is free to violate induction whenever he wants. It's called "miracles".

Anyways, I'm glad you want to investigate the justification of math, rather than just math itself. You can't sustain atheism if you care about questions like that. (Unless, I guess, you go constructivist like Bishop)