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Transcending Proof: A Reply to Richard Carrier (2014)

Don McIntosh


Many years ago I remember hearing that while becoming the first man to orbit the Earth, Russian cosmonaut Yuri Gagarin had looked out into the black void of space and remarked: "I don't see any God up here." Gagarin's 1961 journey into space was thus supposed to have signified not merely the Soviet Union's temporary lead over the United States in the "space race," but a triumph for atheism. As it turns out, Gagarin was not much of an atheist. According to a friend, Colonel Valentin Petrov, Gagarin had been baptized into the Orthodox Church and openly sought to restore the historic role of religion in Russian society. The remark about God, often attributed to Gagarin but nowhere to be found in the transcripts of his communications from space, actually came from a propaganda speech by Nikita Khrushchev outlining the position of the state on religion.[1] Regardless of the source, it made for good Cold War rhetoric.

Of course, the failure to detect God in outer space hardly constitutes a meaningful refutation of theism. But many believers would maintain further that, in principle, atheists can never "prove the negative" that God does not exist. Most of the atheists that I know would consider this objection to atheism to be nothing more than a quibble. The burden of proof, they say, rests solely upon the believer who claims knowledge of a supernatural being, not the skeptic who merely doubts or denies that claim. On this account, atheism makes no positive assertions, but only withholds assent to theism.[2] This view is sometimes called "weak atheism" or "negative atheism." For "strong atheists," or "positive atheists," however, weak atheism is too modest. In his Secular Web article "Proving a Negative," Richard Carrier argues, contrary to received wisdom, that proving a negative is actually relatively easy. In particular, Carrier seems to think that he can prove the nonexistence of God in the space of a few paragraphs with passing appeals to the role of evidence in epistemology and the presumed incoherence of Christian theology.[3]

For the most part, believers and nonbelievers alike would acknowledge that there is rather more to theology and spiritual experience than a handful of ridiculous notions subject to simple refutations, such as the belief that if God existed he would visibly permeate the vast expanse of space. To this observer Carrier's depiction of Christian theism is similarly misleading, and his attempts to prove the nonexistence of God are consequently misguided. Moreover, as a strong atheist who claims that God's nonexistence is provable (and proven), Carrier has assumed a substantial burden of proof. In this article I present some reasons to think that his argument fails to meet that burden.

The Weakness of Induction

Now I do agree with Carrier that at least certain negative propositions can be proven. Inference by modus tollens, for instance, essentially runs as follows: Given that p entails q, and given that q is false, it follows that p is false. If werewolves can only attack us during a full moon, and right now the moon is at a waxing crescent, it follows that right now werewolves cannot attack us. This kind of reasoning also applies to more concrete empirical observations. Carrier offers the example of crows in a box—or rather, the absence of crows in a box. To falsify through simple observation that there are some crows in the box is also to confirm that there are no crows in the box. Thus in certain well-defined situations we may have access to positive knowledge of an existential negation, in this case by simply peering into the box. As Carrier notes, "since crows being in the box (p) entails that we would see crows when we look in the box (q), if we find q false, we know that p is false."

But Carrier recognizes that this scientific (or inductive) argument for the negative faces certain restrictions: "Of course, we could be mistaken about what we saw, or about what a crow is, or things could have changed after we looked, but within the limits of our knowing anything at all, and given a full understanding of what a proposition means and thus entails, we can easily prove a negative in such a case." So whereas negative proofs are available, they only work "within the limits of our knowing anything at all"—which in the context of his example seems to mean knowing strictly by direct observation. That seems to mean, in turn, that where the purpose of proofs is to ascertain knowledge that cannot be directly verified, very few such proofs are available.

One sound alternative negative proof would be a double negation: Given that p is true, the negation of p is false. If it is true that Barack Obama is President of the United States, then Barack Obama cannot not be President of the United States. The negation of the proposition that Obama is not the President can be positively proven by the simple observation that Obama, well, is the President. That basic logical principle works well as far as it goes... which is not very far. How would one go about proving the proposition (inspired by Douglas Adams' character in the Hitchhiker's Guide to the Galaxy) that Zaphod Beeblebrox, purported President of the Imperial Galactic Government, simply does not exist? To confirm the nonexistence of President Beeblebrox would require us to explore every corner of the galaxy, and all at once: Otherwise he could slip off to Volgon while we're looking for him in Damogran, or to planet X while we're seeking him on planet Y, and so on, for as many planets as you please.

At the heart of the conviction that negative existential statements generally cannot be proven lies what Carrier calls "the eternal problem of induction." Since this age-old philosophical stumbling block has persisted to the present, we have good reason to expect most proofs of existential negatives to fail. Nonetheless, Carrier is right to argue that a negative inductive inference—what I would call "existential falsification"—may be rationally justified within certain confined situations and given the generous concession that all positive reports to the contrary are false. Although it cannot be shown that there are no nine foot tall phosphorescent green Martians anywhere in the universe, simple observation suffices to demonstrate that there are no such critters currently bathing in my tub. Perhaps if enough places are explored beyond my bathtub—including Mars—where we can detect no Martians, and if all reports of Martians can somehow be safely presumed false, then there may be an inductive justification for the proposition that Martians do not in fact exist. As Carrier concludes, in the meantime we have (at least in certain carefully selected and circumscribed cases) genuinely proven a negative, if not a universe-sized "sweeping proposition."

In light of these caveats, Carrier describes his proof of the negative in more humble terms as the "method of the best bet." This is a decision-making process based on degrees of evidence, analogous to the operations of a computer program. Most of today's better software programs are written to bypass the execution of logically recursive routines that trigger infinite feedback loops—the sort that cause systems to shut down. They accomplish this by implementing subroutines that seek suboptimal or "best guess" solutions. Such programs set up a barrier preventing infinite loops while avoiding simply "giving up" and ceasing to function. According to Carrier, human reasoners do something very similar. Instead of depending solely on absolute proofs, we continuously suboptimize by rationally choosing among alternatives. In the process we recognize that many propositions may be true, but that we also lack sufficient reason to believe them. These propositions, he adds, are "nearly all of them false, so it is a safe bet to assume they are all false until proven otherwise." The implication for the proposition that God exists is that, at bottom, theism is false because it lacks supporting evidence. Supposing for the sake of argument that theism does in fact lack supporting evidence, however, one could just as easily argue that strong atheism (in affirming the proposition "God does not exist" or "no gods exist") amounts to another proposition that lacks supporting evidence. By parity of Carrier's reasoning, then, assuming that strong atheism is false would also be a safe bet. That the method of the best bet obligates us to assume the falsity of each of two jointly exhaustive propositions suggests that something is wrong with that method.

The problem is that Carrier's best bet method equates insufficient evidence (or the perception of it) with outright falsification. The best bet method is therefore self-defeating in much the same way as the criterion of verification proposed by the logical positivists of the previous century: once we assert that unverifiable claims are false or meaningless, the assertion itself must be deemed false or meaningless. Proving a negative must involve more than simply alleging insufficient evidence for a positive assertion, for in general lack of evidence cannot prove much of anything. It is one thing to assert that no evidence justifies belief that Bigfoot roams the woods of the Pacific Northwest; it is quite another to assert that the proposition "Bigfoot exists" has thereby been proven false. To infer the latter from the former would be to commit the argument from ignorance fallacy. The very act of discovery seems to imply, as the cosmologist Martin Rees put it, that "absence of evidence is not evidence of absence."[4] If absence of evidence were truly evidence of absence, then scientific theories presently considered strongly confirmed should have been considered falsified by the lack of evidence for them prior to their confirmation. It would hardly make sense, however, to argue that the big-bang hypothesis was actually false prior to the discovery of cosmic background radiation and red-shifted wavelengths of light from distant galaxies, or that atoms only began to exist with observations of Avogadro's number and Brownian motion.

Prior to its discovery in Myanmar in 2010, there was little evidence—other than the collective testimony of local tribesmen—for the "snub-nosed monkey," a primate with an upturned nose and a consequent tendency to sneeze when it rains. The creature was discovered by following the advice of local tribesmen to wait for a spell of rain and then listen for the sneezes.[5] On Carrier's line of reasoning, it was irrational to believe such reports before the creature was actually discovered, and again irrational to disbelieve them after it was found—yet the discovery would not have been made without believing the reports. The case of the snub-nosed monkey makes it clear that eyewitness testimony can make considerable contributions to the store of human knowledge. On the other hand, reports by local tribesmen and strange sounds in the jungle do not directly entail the existence of the snub-nosed monkey. Analogously, even given a lack of any direct human contact with God at any point in history—a premise that Christians would certainly dispute—such lack of contact would have little bearing on the strength of evidence for God's existence one way or another. Here it is important to distinguish evidence for a thing from the thing itself. One need not directly examine dark matter or physically dissect a black hole to rationally infer the existence of such objects. Though dark matter and black holes remain strictly unobservable, sufficient evidence is available to strongly suggest their existence. Theists would say the same of evidence for God.[6]

The Rational Necessity of Unprovables

As for negative propositions in general, Carrier advises: "When it comes time to decide what to believe, if we did not assume such 'unprovables' were false, we would either have to choose which unprovables to believe by some totally arbitrary means ... or else we would have to assume that all such statements are true." But this is simply not how beliefs arise. As intellectually commendable as such a detached approach may seem, human beings do not typically sit down and "decide what to believe," measuring the assets and liabilities of a given worldview against one another like accountants creating a balance sheet. Indeed, it seems that we would have to draw upon an implicit set of preexisting beliefs just to begin the process.

As Carrier himself notes, the "proof" sought in these matters is not that of the logical-deductive variety, but "proof in the scientific sense and in the sense used in law courts and in everyday life." Science presumes that its hypotheses are null until provisionally confirmed through repeated testing, courts presume that defendants are innocent of a crime until proven guilty, and people generally don't accept the truth of a controversial proposition without some grounding in evidence. According to the school of epistemology known as foundationalism, to which Carrier apparently belongs[7], this is all just as it should be: Any rationally justified belief should be either properly basic—self-evidently true—or else directly supported by some form of evidence. Alvin Plantinga famously raised the question of whether belief in God meets the foundationalist's standard because it is already properly basic (at least for believers).[8] But the more immediately relevant question here is whether any proposition can be proven without some measure of prior belief in unprovables.

According to foundationalism, candidates for properly basic propositions are those that are logically self-evident (or else experientially incorrigible). Plantinga argues that this winnowing process clearly leaves out a host of seemingly foundational propositions, such as my belief in the reliability of memory to inform me about the past. Proof in Carrier's "scientific sense" depends on some not entirely self-evident operating assumptions, such as that of causality (that one thing is usually caused by another thing) and of the uniformity of nature (that future events will be like past events). However, these assumptions are not themselves scientifically provable. Nor (by Carrier's own tacit admission) are they logically provable. Hence they are strictly unprovable. Both atheists and theists therefore grant de facto properly basic status to beliefs that do not strictly qualify as properly basic by foundationalists' standards.

Even if Carrier demonstrated the proper basicality of the assumptions utilized by the scientific method, he would still be left with a number of unprovables, since properly basic assumptions are themselves unprovable.[9] Yet Carrier also holds that belief in unprovables is whimsical and arbitrary. Perhaps by this he means only that belief in those unprovables that are not properly basic is whimsical and arbitrary. Or perhaps he fails to mention exceptions to the rule because permitting exceptions would weaken his argument that unprovability renders belief in God unjustified. Either way, like the rest of us Carrier has no choice but to affirm the truth of certain ground-level unprovables in order to prove anything else—such as the proposition that theism is false. But since Carrier's negation of theism explicitly asserts the falsity of unprovables, his argument collapses.

As Kurt Gödel demonstrated in his celebrated incompleteness theorems, most any consistent, formal system is capable of generating true statements that nonetheless cannot be proven within that system. Gödel's first theorem is difficult to put into words, finding expression in systems consisting of formal language, system-specific axioms, and system-specific rules of inference. (Peano arithmetic is probably the most commonly cited example of such a system.) Formal conventions notwithstanding, various specialists have managed to "translate" Gödel's formulation into more accessible terms. Scott Aaronson, for example, has supplied a useful, nonformal distillation of incompleteness that runs basically as follows. For any Turing machine (essentially a computer) M, there is a sentence S(M) constructible in the language of M that M literally cannot compute:

S(M) = "Machine M will never output this sentence."

In Gödelian terminology, such a sentence is undecidable. As Aaronson puts it: "There are two cases: either M outputs S(M), in which case it utters a falsehood, or else M doesn't output S(M), in which case there's a mathematical truth to which it can never assent."[10] The upshot is that there is a truth coded in the language of M that is not provable in M. To put it another way, system M is incomplete with respect to the truth of S(M). Given the soundness of Gödel's paradoxical proof of unprovable truths, Carrier's proof of the nonexistence of God would be unsound, for it includes the false premise that unprovables are, if not outright false, at least intrinsically unreasonable. At a more basic level, something much like incompleteness seems to hold for science and logic themselves, for the method of scientific verification cannot be scientifically verified, and the axioms of logic cannot be proven logically. These epistemic-operational givens must be assumed true without proof—that is, believed.[11]

In certain contexts, then, "somehow truth transcends proof."[12] At some point one must believe in a truth in order to know it to be true, because truth cannot be proven in the final analysis. One may believe what is true without knowing it to be true, of course—by way of a lucky guess, for example—just as one may believe what is false without knowing it to be false. But how could one ever know what is true without also believing it? With good reason, knowledge has often been described as "justified true belief." This appeal to belief is more than sentimental religious rhetoric. To assert with no supporting evidence that only evidentially supported beliefs can be justified is, in the words of Plantinga, "self referentially incoherent."[13] By contrast, Christian theism openly acknowledges the reality of unprovables—articles of faith—as essential elements of its theology. All of this suggests that, in epistemological terms, Christian theism is prima facie more coherent than the strong atheism advocated by Carrier.

The Truth of Christian Theism

Carrier naturally disagrees, arguing that Christian theism is both falsifiable and falsified, "as easily disproven as the alien in the bathtub." To Carrier, omnipresence entails, for example, that God should be readily detectable at all times, "yet I have absolutely no sensation of any God or anything that would be entailed by a God." That argument holds so long as God is redefined as an observable entity. For most theologians, however, omnipresence is an exclusive attribute of God that is entirely consistent with his spiritual nature. God is everywhere at once in the sense that "everywhere" is immediately accessible to him as creator and sustainer of our universe—yet as a spiritual being God is himself uninstantiated within that universe. Serious theologians do not generally hold that God is empirically accessible even in principle. To this one may rightly object that theism is unscientific, but that alone would not make it any less plausible than a properly basic belief.

Next Carrier espouses an argument from evil, alleging that the existence of suffering conflicts with divine compassion: "All suffering in the world must be known and safely within the power of God to alleviate, yet it is still there, and since the Christian 'theory' entails the opposite observation, Christianity is false." The proportionately short answer to this somewhat hastily contrived sketch of the problem of evil is that Christianity does not in fact entail that all suffering must be eliminated or alleviated. For many Christian thinkers, suffering is a function of human moral freedom, itself a good. Indeed, Christianity predicts ongoing suffering as the inescapable consequence of sin in the world. Moreover, Christianity posits that Jesus Christ himself endured the heights of suffering to win eternal redemption for humankind, and therefore exhibited the heights of compassion. But for Carrier, it seems, this is all immaterial. The universe itself, he asserts, "lacks moral features," with the consequence that the laws of physics "treat the good man and the bad man equally." I don't see how these two statements can both be true. If the universe lacks moral features, then there are neither good nor bad men—indeed, there is no problem of evil to speak of—and suffering is little more than experiences that humans subjectively do not happen to enjoy or appreciate. If there are good and bad men, then their respective goodness and badness constitute moral features in the universe.

Nonetheless, Carrier maintains that any proposed solution to the challenges facing Christian theism must founder upon the doctrine of omnipotence "since at every turn, God is forced to do something (to remain hidden and to wait before alleviating suffering, etc.) by some unknown feature of reality." The apologist's free will defense entails that God cannot simultaneously make men morally free and incapable of committing evil acts that result in suffering. But for Carrier omnipotence entails that God has an inscrutable power "immediately to alleviate suffering" (however one defines "suffering"). To the extent that God's power is truly inscrutable, however, it cannot be scrutinized—which probably explains why Carrier doesn't go ahead and tell us just how God could immediately alleviate all suffering (short of annihilating humanity en toto).

Setting rigorous performance requirements for omnipotence would seem to require possession of the very attribute under scrutiny. This may be the ultimate lesson of the book of Job, as Francis Andersen suggests:

There is a rebuke in it for any person who, by complaining about the particular events in his life, implies that he could propose to God better ways of running the universe than those God currently uses.... Only God can transmute evil into good. As Creator, responsible for all that happens in His world, He is able to make everything (good and bad) work together into good.[14]

The idea that God could make use of evil and suffering to create ultimate good and happiness comports with the scholarly conception of Jesus' teaching on the kingdom of God as being actualized both "already" and "not yet"—that is, powerfully inaugurated in the ministry of Christ but awaiting its final consummation in the new heaven and new earth. According to New Testament eschatology, God's work of creation has technically not been completed—the creation of this world was but the first phase of a much more expansive, eternal creative undertaking—so that the solution to the problem of evil is still underway. We could perhaps call such a view a "theodicy of transcendence," or in a nod to Gödel, a "theodicy of incompleteness." Though it may be true that the logical compatibility of an all-powerful and all-good God with suffering can be neither proven nor disproven within the system of this present world, biblical theology posits its provability within a larger system—the eternal kingdom of Heaven. In that context present suffering borne of moral freedom may well turn out to be an essential element of an everlasting and incorruptible future, and therefore, as Andersen says a bit later, "an occasion of grace."[15]

Given the inscrutability of omnipotence, our world with all of its suffering may well be the creation of a maximally compassionate God. Theists frequently argue that God has the inscrutable power to harness temporary evil and suffering toward eternally good and joyous ends. Atheists just as frequently argue that God, if he existed, would have the inscrutable power to create morally free sentient beings who are incapable of evil and suffering right now. In either case the appeal to divine inscrutability leaves us with yet another transcendent truth: omnipotence transcends human reason. And if omnipotence transcends human reason, we are left with no rational objections to omnipotence. To the degree that the repudiation of Christian theism depends on rational objections to omnipotence, then, we are left with no objection to the truth of Christian theism at all.


[1] "I am Proud to be Accused of Having Introduced Yuri Gagarin to Orthodoxy" (Interfax Religion, April 12, 2006). <>. By Petrov's account, Khrushchev took rhetorical liberties to speak for the cosmonaut when he stated: "Here is Gagarin who flew into space but saw no God there." Over time the quote became attached to Gagarin himself.

[2] For the classic statement of this position, see Antony Flew, "The Presumption of Atheism" in God, Freedom, and Immortality: A Critical Analysis (Buffalo, NY: Prometheus, 1984).

[3] Richard Carrier, "Proving a Negative," The Secular Web, October 2, 1999. <>. Except where noted otherwise, all further quotations from Carrier are from this same article.

[4] Martin Rees, quoted in Bernard M. Oliver and John Billingham, "Project Cyclops: A Design Study of a System for Detecting Extraterrestrial Intelligent Life (NASA Technical Report CR-1144445), p. 3.

[5] Rachel Kaufman, "New Snub-Nosed Monkey Discovered, Eaten," National Geographic News, October 27, 2010. <>.

[6] Though this article does not aim to lay out the evidence for Christian theism, I do believe that plenty of such evidence is available (where "evidence" is understood as information that makes the truth of a belief more likely than it would be otherwise). I take it that viable sources of evidence supporting Christian theism include: the apparent fine-tuning of life-permitting physical constants regulating the universe; numerous instances of specified complexity in nature; general human awareness of transcendent or "objective" moral rules; the historical origin, worldwide dispersion and persecution, and subsequent physical restoration of the nation of Israel in keeping with the prophetic message of the Old Testament; the equally prophetic and miraculous ministry of Jesus Christ as attested in thousands of early manuscripts derived from originals dated to within a generation of his death and purported resurrection; the birth of the early Church in Jerusalem on the preaching of the Resurrection in the face of violent persecution; and the remarkably sudden, complete conversion of Saul of Tarsus, formerly a leader in the earliest efforts to destroy the Christian movement.

[7] Carrier writes: "To say something is 'properly basic' is to declare that it's something we get to assume without needing a reason to believe it. In my epistemology, ... only what is literally undeniable gets to be called 'properly basic'" ("Epistemological End Game," Richard Carrier Blogs, November 29, 2006. <>).

[8] Alvin Plantinga, "Is Belief in God Properly Basic?," Noûs Vol. 15, No. 1 (March 1981): 41-51.

[9] An anonymous reviewer pointed out that it would be more correct to say that properly basic propositions are simply self-evident—which implies that they are in no need of proof. Even so, they would be difficult to prove were the need to arise precisely because there are no yet more basic beliefs upon which to found them. In that sense, at least, they are unprovable.

[10] Scott Aaronson, Quantum Computing Since Democritus (New York, NY: Cambridge University Press, 2013), p. 153. It should be noted that Aaronson is an atheist whose discussion of Gödel takes place in the context of defending a materialist-functionalist conception of mental states against the more dualist-leaning ideas of Roger Penrose. I mention him because he seems to have explained incompleteness more clearly and succinctly than most.

[11] This traditional view of epistemic limitations has been challenged in recent years, most notably by the "naturalized epistemology" of W. O. Quine. According to Quine, there is no need for any ultimate "epistemic justification" of beliefs, only an empirical analysis of human psychology in the acquisition of knowledge. On this understanding epistemology is more or less another project within the larger scientific enterprise, one that describes how beliefs might arise but not whether there is any particular reason to think that they are true. But as Jaegwon Kim notes, justification of belief pretty much defines epistemology: "For epistemology to go out of the business of justification is for it to go out of business" ("What is 'Naturalized Epistemology'?" in Philosophical Perspectives, Vol. 2 (Epistemology), ed. James E. Tomberlin [Asascadero, CA: Ridgeview, 1988], p. 391).

[12] Josef Vidal-Rosset, "Does Gödel's Incompleteness Theorem Prove That Truth Transcends Proof?" in The Age of Alternative Logics: Assessing Philosophy of Logic and Mathematics Today, ed. F. A. K. van Benthem, Shahid Rahman, and John Symons [Dordrecht, Netherlands: Springer, 2006], p. 57.

[13] Plantinga, p. 44.

[14] Francis I. Andersen, Job: An Introduction and Commentary (Downers Grove, IL: InterVarsity Press, 1976), p. 287.

[15] Andersen, pp. 287-288.

Thanks to Secular Web scholarly paper editor Keith Augustine and an anonymous referee for their helpful comments on earlier versions of this paper.

Copyright ©2014 Don McIntosh. The electronic version is copyright ©2014 by Internet Infidels, Inc. with the written permission of Don McIntosh. All rights reserved.