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## Bahnsen’s TAG I

[Map: Intro, I, II, III, IV, V, VI]

Last time, I analyzed the atheist demand to demonstrate God, as an unreasonable standard of proof. This time, I flip to the opposite side of the logical coin, and consider the (mis)conception that you can’t prove a negative.

Gordon Stein, in “The Great Debate” with Greg Bahnsen, redefines “Atheist” as “Agnostic”:

Atheists do not say that they can prove that there is no God. An atheist is not someone who denies that there is a God. Rather an atheist says that he has examined the roofs that are offered by the theist and he finds them inadequate.

Stein goes even further, purporting to disallow even the possibility of positively proving the nonexistence of God (or nonexistence of anything, for that matter):

If I wanted to prove that unicorns do not exist, I can examine this room and we can find out that there are definitely no unicorns in this room (that small area), but to prove the general non-existence of something like unicorns, we would have to search the entire universe simultaneously. Then we could only say that no unicorns existed at the moment we searched the universe. But you know, maybe they were there five minutes before, or if we only searched the whole earth, they were on another planet at the time; I mean there are all kinds of other possibilities; so you cannot prove that something does not exist.

Stein shows that he suffers from the same fallacy as Sansone — a belief that existence proofs must be constructive. If that were the case, then certainly it would be impossible to prove nonexistence, because it would be impossible to distinguish a “nonexistence proof” from “not looking hard enough”.

But as Stein (a physical scientist) should well know, every mathematician worth his weight in salt proves that things don’t exist every day. For instance, here are a dozen proofs that no fraction (ratio) exists which is the square root of 2 (i.e. $\sqrt{2}$ is irrational). Following Stein’s reasoning, it would not be possible to know for sure whether $\sqrt{2}$ might actually be rational, because we could never search the infinite space of all pairs of integers — maybe there are a p and q out there such that p/q is the square root of two, but we just haven’t looked long enough, or hard enough, or cleverly enough?

But all of the irrationality proofs linked above preemptively debunk any potential p and q. The arguments all begin with “Suppose that some amazingly smart or amazingly lucky guy finds the missing two numbers that divide to create $\sqrt{2}$“. However many thousands or googols of digits long these hypothetical numbers are, each proof proceeds to deduce in its own various ways, until a contradiction is reached (for instance q is simultaneously even and odd). Since such a contradiction cannot exist, the starting assumption (that such p and q exist) is invalidated. This logic applies to any and all pairs of numbers p and q, so we can be certain there is no such p and q out there, and we shouldn’t waste our time searching.

To be fair, Stein may be correct in his particular statement, since he spoke of proving “the general non-existence of something like unicorns“. It may be that the only acceptable standard of proof for the existence of unicorns is to see a unicorn. (Or perhaps the definition of unicorns extends to certain properties of unicorn poop, by which one could non-constructively prove the recent existence of a unicorn).

But God is not something like a unicorn. God is a being that by definition possesses certain characteristics (omnipotence, omniscience, omnibenevolence, …), and the existence of a being with those characteristics can be proven to be either logically inconsistent (as the atheist must attempt to prove) or logically necessary (as the Christian apologist must attempt to prove).

So Stein cannot cannot shift the burden of proof away from the atheist by means of a categorical assertion that “you cannot prove that something does not exist.” He’s still got some work to do.

### 17 Responses

1. he has examined the roofs that are offered by the theist and he finds them inadequate.

So that’s why the rain falls on the just and on the unjust alike!

2. Ha ha. My bad copy & paste from the PDF. It’s amusing enough to let stand though…

3. Well, Ruberad, you are either a pastor with a gift of reading or you are single or some kind of professor.

You do good work here. I’ll have to come back to read more.

4. Thx Howard, I hope you do keep coming back and participating! And close guess: the “some kind” of professor that I am is “former”…

5. Nutty professor – Jerry Lewis
Absent minded professor – Fred McMurray
Little professor – Casey Stengel

Not to get off the thread.

6. Are we just as blessed if we believe in Him, love Him and follow Him, and simply believe what the Bible says, or do you think we are more blessed if we wrestle with man’s doubts about Him?

7. Never thought of BELIEF as needing proof. You simple do, or you don’t. Simple as that.

8. I, meant to say “simpley”.

9. You’ve used a completely described mathematical system with a complete set of operators and found that you can prove that something doesn’t happen. I could also prove that there are is not a 6th node in a k5. This doesn’t relate to disprovability in the general case. Try proving that George Bush doesn’t have any dark, disturbing secrets relating to smooth peanut butter.

10. [Albino Hayford:] Are we just as blessed if we believe in Him, love Him and follow Him, and simply believe what the Bible says, or do you think we are more blessed if we wrestle with man’s doubts about Him?

Complex truths require more than just simple belief; would you rather I didn’t wrestle with man’s doubts, but instead let them run rampant? If you are asking whether I am more blessed than you, I can’t answer that. But certainly I am more blessed than I would be if I didn’t wrestle. Not wrestling, not understanding enough of the fullness of God’s truth, is the reason many fall away — how blessed is that?

[Tisziggy:]Never thought of BELIEF as needing proof. You simply do, or you don’t. Simple as that.

Certainly only God imparts faith to overcome the stumbling block to the Jews, foolishness to the Greeks. But some people don’t believe, either because they misunderstand the truth, or because they suppress the truth. Proof is helpful (as an earthly means God uses towards his heavenly ends) in either case.

This doesn’t relate to disprovability in the general case.

Actually, the general case is exactly what it relates to, as the existence of one nonexistence proof invalidates the general statement “you cannot prove that something does not exist”. What you mean to say is that proof of the irrationality if sqrt(2) doesn’t relate to disprovability in the particular case of God (or GWB & smooth peanut butter).

However, your point helps me make my point. I make no assertion/assumption that proofs about God (or peanut butter) would look anything like proofs about sqrt(2). Stein is the one who commits the error of making an assertion about proofs about God, based on assumed similarity to proofs about unicorns.

Bottom line, it is disingenuous for Stein to claim that Atheists are merely Agnostics, that Atheists merely poke holes in Theistic proofs, and don’t attempt any positive nonexistence proofs of their own.

Finally note that the statement “There exists no proof of the nonexistence of God” is actually true — it’s a trivial corollary to “God exists”.

11. ‘Actually, the general case is exactly what it relates to, as the existence of one nonexistence proof invalidates the general statement “you cannot prove that something does not exist”.’

That’s true, but I thought we were talking about an inability to disprove the existence of God? I don’t think I can disprove it. Disprovability in Mathematics is irrelevant to this.

‘Finally note that the statement “There exists no proof of the nonexistence of God” is actually true — it’s a trivial corollary to “God exists”.’

If you presume that God exists, then there’s no proof of his non-existence. If you can’t disprove the existence of God, this does not mean that he exists, it means that he might exist. In what way does this help anybody?

12. I thought we were talking about an inability to disprove the existence of God? I don’t think I can disprove it. Disprovability in Mathematics is irrelevant to this.

Stein categorically denied the possibility of proving the nonexistence of anything (either that, or he was sloppy). I am saying that he cannot logically justify the statement “The nonexistence of God cannot be proven”, based on a discussion of unicorns. If his position (like yours) is really just “I don’t think I can disprove it”, then he should yield the podium to a stronger Atheist, because there are plenty out there who think they can prove that God doesn’t exist.

But he can only remove the burden of proof from himself if he can show that it is not logically possible for a valid nonexistence proof to exist (and the unicorn argument doesn’t cut it). That puts us in the realm of meta-non-existence proof. This is getting complex, so I’m going to use variable NG to represent a possible proof for the Non-existence of God; Stein uses the unicorn argument to claim the nonexistence of NG, I have shown that his argument is insufficient, thus he is still on the hook for producing NG. Now the only way that I can think of to prove that NG doesn’t exist — is to prove that God actually exists! And I don’t think that is an acceptable strategy for Stein.

But (holding myself to the same standard), I do not categorically deny that there could be some other way that I haven’t thought of to prove that NG doesn’t exist. If Stein can convince me (using a better-than-unicorn argument) that NG doesn’t exist, then he will have successfully removed the burden of proof from himself, and I will have to allow him to retreat into the defensive position of poking holes in my (Bahnsen’s) theistic arguments.

If you can’t disprove the existence of God, this does not mean that he exists, it means that he might exist. In what way does this help anybody?

You are correct. By the discussion above, the lack of NG puts us in a P=?NP quandary, where it could be that NG actually doesn’t exist, or it could be that the atheists just haven’t looked hard enough. As for helping anybody, I said I would give the atheist a leg up, I never said I was going to do his homework for him! In any case, the Christian retains his burden of proof, to prove the existence of God.

13. Again, cool. I think Stein was being sloppy while ranting. Your NG and NP evocation seems daft, so I’ll think about it.

14. What’s NP? I am not seeing it now.

15. Sorry, kind of a private joke between Computer Scientists. I’ll summarize it as concisely as I can, and maybe limejelly won’t think my analogy so daft anymore.

There is a set of problems for which computer scientists have found “efficient” solutions. This set of problems is called P. There is a set of problems for which computer scientists have found “efficient” verifications (if somebody claims they have a correct answer to the problem, the problem can be verified as correct “efficiently”). This set is called NP. Because finding a solution “efficiently” always means that a solution can be verified “efficiently” (you could always verify by just regenerating the solution again), the set P is contained in NP.

The problem is that there are a whole lot of problems in NP, for which computer scientists have not discovered “efficient” solutions. As a matter of fact, these “NP-hard” problems are all linked, such that if somebody could find an efficient solution for just one of them, then that would automatically domino an efficient solution for all of them, thus all of NP would be included in P, so the distinction would collapse, and P would be exactly the same as NP (P=NP).

But after decades of research (admittedly, that’s not that long in the grand scheme of things), nobody has found any efficient algorithm for any NP-hard problem. I.e. nobody has proven (by demonstration) that P=NP. But neither has anybody been able to prove that, in principle, no NP-hard problem can be solved efficiently. (I.e. nobody has a nonexistence proof for efficient solutions of NP-hard problems; nobody has proven that P is less than NP).

So the community of computer scientists which focus in this area (complexity theory) have divided into two philosophical camps; Optimists hold out faith that there exist efficient solution techniques out there (and hopefully we’ll eventually find them), while Pessimists believe that NP-hard problems can never be solved efficiently (and hopefully we’ll be eventually be able to prove it). (For what it’s worth, I’m a Pessimist.) The philosophical argumentation about this question is very much like Christian/Atheist apologetics, and the Pessimists are stuck with an analogous problem as the Atheists: how do you prove that no (proof for the existence of God / efficient solution of NP-hard problems) exists? It is much easier to poke holes in any proposed (proof for the existence of God / efficient solution of NP-hard problems), but that doesn’t settle the question. There’s the possibility that the reason Pessimists/Atheists can’t come up with a proof is because they haven’t tried hard enough yet, and on the other hand, there’s the possibility that the reason they can’t come up with a proof is because they’re wrong.

I don’t know if that clarified anything for anybody who is not already acquainted with complexity theory. But if you are acquainted/interested, I HIGHLY recommend going over here, first follow the link to “Ten Reasons to Believe P!=NP”, and then follow the link to the .mp3 of the “Talmudic Exegesis”. That mp3 is the 8th in a series of ComplexityCasts, so if it baffles you to start at the end, you might want to (like me) listen to the whole series from the beginning!

People that know me might find it interesting to know what a refreshing sense of academic nostalgia filled me as I read Aaronson’s Ten Reasons, and then listened to Fortnow & Gasarch’s discussion about them. At heart, I am indeed a geek! It is also interesting to note that (at least) one of those guys (Bill Gasarch) is a christian.

16. OK. It evokes the general sense you’re looking for, but it offends my mathematical sense in the same way numb3rs does.

17. […] by RubeRad on January 16th, 2007 [Map: Intro, I, II, III, IV, V, […]