When Absence of Evidence is Evidence of Absence

There is a popular and catchy saying which I myself have been caught repeating in the past, but which, for all its intuitive appeal, is false; namely, that the absence of evidence isn’t evidence of absence. Many a new-atheist has repeated the mantra that there is no evidence for God’s existence, insinuating thereby that this absence of evidence is good evidence for atheism. William Lane Craig, a noted philosopher, theologian and tireless Christian apologist has responded as follows:

[Atheists] insist that it is precisely the absence of evidence for theism that justifies their claim that God does not exist. The problem with such a position is captured neatly by the aphorism, beloved of forensic scientists, that “absence of evidence is not evidence of absence.” The absence of evidence is evidence of absence only in cases in which, were the postulated entity to exist, we should expect to have more evidence of its existence than we do.1

He has reiterated as much more informally (but more elaborately) on his podcast, ReasonableFaith, where he says:

The absence of evidence will count as evidence of absence when if the thing existed, then having surveyed the grounds, so to speak, we would expect to see evidence of their existence, and we don’t see it. And so, for example, in the case of fairies, if they existed then we ought to be able to find traces of their existence – their dead bodies when they die, their carcasses, other sorts of remains, little clothing factories where they build their clothes, and we ought to detect them flying about just as we detect dragon flies and bumblebees – but we don’t. So this would be a case where I think the absence of evidence would count as evidence of absence.”2

On this view, the absence of evidence only counts as evidence of absence when we have some reason to expect to see the evidence ex hypothesi. This has enormous intuitive appeal; consider the hypothesis that there is at least one tiger in India. Can the fact that I, sitting in Canada, currently see no tiger really count as evidence that there is not at least one tiger in India? Surely not; presumably because that evidence isn’t expected on the assumption of the relevant hypothesis’ truth. Elliott Sober, reflecting on absence of evidence, notes that in the case of arguments from absence “it is easy to see how each can be turned into a valid argument by adding a premise. The arguments have the form:

I do not have any evidence that p is true.
p is false.

Just add the premise

(P1) If p were true, then I’d have evidence that p is true.”3

This further highlights the fact that it is natural for us to think that absence of evidence is evidence of absence only when we expect the evidence ex hypothesi.

For years I found this response intellectually satisfying, but in recent years I have come to think that it is woefully mistaken. It is true that my failure to observe a tiger in Canada provides no evidence against there being at least one tiger in India, but it is not because I wouldn’t have anticipated seeing a tiger in Canada given that there is at least one tiger in India. All my affection and respect for Craig notwithstanding, if Craig means that absence of evidence E for hypothesis H is only evidence of absence (i.e., not-H) when the probability of E on H is greater than 0.5, then he is, I think, incorrect. In what follows I will try to explain why, as well as explore what to me seem interesting corollaries of Bayesianism.4

John Hawthorne, speaking about probability theory and the fine-tuning argument at a conference back in 2015, warned:

“Human beings, even intelligent human beings, are terrible at reasoning about probabilities. There’s enormous empirical evidence that human beings are terrible at reasoning about probabilities, and so we have to proceed with care.”5

Playfully picking on (presumably) a student in the audience, Hawthorne says: “Justin gave us the kind of awesome sounding principle… [that] if you don’t see something then that can be evidence of its absence only if you expect that you would get evidence were the thing there.”6 Not the cleanest off the cuff articulation, but clearly Hawthorne had in mind the principle for which W.L. Craig advocates. He continues; “that’s wrong… and I can prove to you that it’s wrong.”7 He proceeds to give an illustration using a hypothetical creature he calls a Dynx, where he stipulates that 75% of Dynx are invisible to the naked eye, and the probability that there is a Dynx in a box placed before us is 50%. We open the box, and we see no Dynx. The probability that there is no Dynx given our background knowledge and this new piece of information (namely that we do not see any Dynx) is approximately 57%. You can satisfy this for yourself by simply dividing up the space of possibilities (i.e., ‘seeing a Dynx in the box,’ ‘not seeing the Dynx in the box,’ and ‘there being no Dynx in the box’), eliminating the possibility of ‘seeing a Dynx in the box,’ and then expressing your updated probability assessment accordingly. So, even though we ought not to expect to see a Dynx in the box if there is one in the box, our failure to observe one is still evidence for their being no Dynx. This simple illustration (and others like it) seems to be entirely compelling. What, then, is the genuinely Bayesian determination of evidence?

On the Bayesian theory of confirmation,8 some evidence E will count as evidence for some hypothesis H (given background knowledge B) just in case E (conjoined with B) raises the (prior) conditional probability of H. To put it more formally, E will count as evidence for H just in case: P(H|E&B)>P(H|B). However, [P(H|E&B)>P(H|B)]⊃[P(~H|~E&B)>P(~H|B)]. In other words, if E provides any evidence for H, then ~E provides some evidence against H. It needn’t, of course, be the case that E provides as much evidence for H as ~E does for ~H, but it strictly follows from Bayesianism itself that ~E would be evidence against H just in case E would be evidence for H.

To illustrate with an example, let us take a hypothesis H1: “that aliens exist,” and evidence E1: “I am being abducted by aliens.” Obviously P(H1|E1&B)>>P(H1|B). What is not so obvious is that P(H1|~E1&B)<P(H1|B). The reason it isn’t so obvious is that ~E1 provides negligible evidence for ~H1 (even though E1 would provide compelling evidence of H1). If aliens abduct me, that’s really good evidence that they exist. If aliens do not abduct me that’s really poor evidence that they don’t exist. It may be some evidence, but it isn’t very much evidence.

Not only can the absence of evidence be negligible evidence of absence while the presence of that evidence would be altogether compelling, but the absence of evidence can even be inscrutable evidence of absence while the presence of evidence is scrutable and enormously supportive of the hypothesis in question. Take the example of a miracle, and for simplicity let us use the miracle of the bodily resurrection of Jesus of Nazareth. The bodily resurrection of Jesus, if it did occur, would be relatively good evidence for God’s existence; P(G|R&B)>>P(G|B). However, if Jesus had not been raised from the dead, would that provide any evidence against God’s existence? According to Bayesianism it would, but it seems like it would be not only negligible evidence, but even inscrutable evidence. There is no way one could put a figure (with any justification) on how much more confident it should make us in atheism that some miracle, like Jesus’ resurrection, did not occur. If we could give any estimate of what the probability is that God would perform a miracle when called upon to do so, for instance, then we could make some predictions about how many hospitalized people with terminal diseases (according to medical diagnosis) under observation get better when prayed for. We can’t make these predictions not because there is no actual probability of God doing a miracle, but because we aren’t at an epistemic vantage point from which we can assess that probability with any level of confidence at all.

Further, the evidence may not be merely negligible, but can in special instances be literally infinitesimal (an infinitesimal is a non-zero infinitely small quantity). Consider Hempel’s paradox9 for a moment; any observation of a pink shoe provides some evidence for the hypothesis that all ravens are black. The hypothesis that all ravens are black is logically equivalent to the statement that all non-black things are non-ravens. It follows, therefore, that any observation of a black raven is evidence that all non-black things are non-ravens, and any observation of a non-black non-raven is evidence that all ravens are black. An observation can’t be evidence for one without being evidence for the other precisely because they are logically equivalent statements, at least interpreted at face value; this is just what Hempel called “the equivalence condition.”10 However, it seems as though there are potentially infinitely many things which are non-black non-ravens which, at any moment, we will fail to observe. If this is so, then each of these instances of absence of evidence will count as instances of infinitesimal evidence of absence (or, at least, infinitely many of these instances will count as instances of infinitesimal evidence of absence). One thinks of the infinitely many miracles God could have performed at any given moment (e.g., growing a lost limb, bringing a dead child back to life, parting the Atlantic ocean); is it really the case that every instance of a miracle not happening provides some evidence against God’s existence? If so, and if there are infinitely many opportunities for God to perform a miracle of some kind (in infinitely many of which God decides to perform no miracle), does that not entail that the probability of theism is literally infinitesimal, or else that each instance (or, at least, infinitely many instances) of a non-miracle provides at most infinitesimal evidence against theism? This gets a little tricky, of course, because Bayesian theory isn’t really equipped to deal with cases of what we might call ‘transfinite probabilities,’11 but if we take its implications seriously even in such cases we will plausibly think that at least some things provide literally infinitesimal evidence for a conclusion or hypothesis.

An interesting objection to this suggests that there is not, even potentially, an infinite number of unobserved observables. Given the limited bandwidth of the human body as a kind of measuring apparatus,12 there may be infinitely many different but observationally indistinguishable events. Imagine, for instance, two pairs of pink shoes whose colours or sizes differ by so little as to make it impossible for any human being to tell the difference between them. For any of the attributes assessed by the five senses, there will be limited empirical bandwidth given the human body as a tool of observation. What this seems to entail is that there is not a potentially infinite number of different possible observations, in which case we needn’t concede the absurdity of infinitesimal probabilities. This objection is appreciably practical, but I’m not entirely confident that it settles the matter. After all, I can imagine a human being with “electron-microscope eyes”13 or with any number of other physical alterations which would allow them to observe an apparently potentially infinite number of different events. For any such alteration, I can imagine God miraculously bringing it about that observer S has precisely the alterations necessary to observe some miracle M1 which would have previously been indistinguishable from miracle M2, but is not now indistinguishable from M2 for S. Moreover, I’m not convinced that observational indistinguishability is terribly relevant; there are infinitely many possible pink shoes which I could now be observing, but am not, and even if infinitely many of them would be indistinguishable to me, failing to observe any one provides some evidence against the hypothesis that all ravens are black. So it seems to me that we’re stuck with conceding that at least some things provide literally infinitesimal evidence.

In summary, I think we have seen why the absence of evidence is evidence of absence in all cases except those in which the presence of so-called evidence would do nothing to raise the conditional probability of the hypothesis in question. Thus, my failing to observe a tiger in Canada provides no evidence against the hypothesis that there is at least one tiger in India not because I wouldn’t expect that evidence if there were at least one tiger in India, but because even if I were observing a tiger in Canada it would provide no evidence that there is at least one tiger in India.14 We have also seen that even when absence of evidence is negligible evidence of absence, or inscrutable evidence of absence, or infinitesimal evidence of absence (or any combination of those three), it will still provide some evidence of absence; if E would have been evidence for H, then the absence of E provides evidence against H.

Post Scriptum: I want to thank Tim Blais, Cale Nearing and Sean Boivin who provided me, in discussions subsequent to the original article, with food for thought without which I would never have made the improvements I have lately introduced above.

1 William Lane Craig, “Theistic Critiques of Atheism” The Cambridge Companion to Atheism. Edited by Michael Martin (Cambridge University Press, 2006): 70.

3 Elliott Sober, “Absence of Evidence and Evidence of Absence: Evidential Transitivity in Connection with Fossils, Fishing, Fine-Tuning, and Firing Squads,” in Philosophical Studies 143, no. 1 (2009): 64.

4 As a cautionary caveat lector; though I’m pretty confident that what I’m about to say is correct, I have not taken any class on probability theory (yet); if anyone thinks there’s some subtle mistake somewhere, they are encouraged to share it. I am more than open to updating my views.

8 Elliott Sober, “Absence of Evidence and Evidence of Absence: Evidential Transitivity in Connection with Fossils, Fishing, Fine-Tuning, and Firing Squads,” in Philosophical Studies 143, no. 1 (2009): 66.

9 James Fetzer, “Carl Hempel,” in The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta, (Spring 2017 Edition), accessed April 2, 2017. https://plato.stanford.edu/archives/spr2017/entries/hempel/

10 James Fetzer, “Carl Hempel,” in The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta, (Spring 2017 Edition), accessed April 2, 2017. https://plato.stanford.edu/archives/spr2017/entries/hempel/

11 If one dislikes this term because they think that probabilities can be no higher than 1, which makes them finite, I would suggest they think about how the conditions I just stipulated could imply that some hypothesis H is infinitely likely without having probability 1. However, if that doesn’t mollify the critic, I could agree to change the term to ‘non-finite’ probabilities.

12 I borrow here from Bas C. van Fraassen, who notes insightfully that “the human organism is, from the point of view of physics, a certain kind of measuring apparatus.” See: Bas C. van Fraassen, The Scientific Image, (Oxford: Clarendon Press, 1980), 17. 

13 Bas C. van Fraassen, The Scientific Image, (Oxford: Clarendon Press, 1980), 17.

14 If one thinks that observing a tiger somewhere raises the conditional probability that one may be observed anywhere then one will reject this conclusion, but they needn’t, in so doing, reject the principle this example is being employed to illustrate.

Math, therefore God?

“Monsier! (a+bn)/n=x, donc Dieu existe; répondez!”[1]

Thus (allegedly) spoke the mathematician Leonard Euler when, at the invitation of Russian Empress Catherine the second, he confronted Denis Diderot in a (very short) debate on the existence of God. Diderot, who was not very good at math, was dumbstruck; he had absolutely no idea how to even begin responding to such an argument. In fact, he couldn’t even understand the argument, and Euler knew it! The court laughed him literally out of town (he promptly asked the Empress for leave to return to France). The formula, of course, is entirely meaningless, and may have been sleight of hand on Euler’s part (making his argument mathemagical rather than mathematical). Additionally, the anecdote has survived only in sparse notes (of dubious historical relevance) here and there with probably varying degrees of accuracy, so it is anyone’s guess what Euler actually meant. This amusing anecdote does, however, invite us to think about what arguments there could be, in principle, from mathematics for the existence of God.[2] Without offering much commentary on how promising these arguments are, I want to distinguish three viable (or, at least, viably viable) types of arguments which could be constructed.

The Argument from Mathematical Beauty

Although the formula Euler originally spouted off didn’t signify anything of mathematical (or philosophical) consequence, the beauty of Euler’s equation, eiπ + 1 = 0, gave rise to the apocryphal anecdote that Euler argued “eiπ + 1 = 0, therefore God exists.” There is (mathematicians tell us) a sublime mathematical beauty in this equation, and there is no obvious or intuitive reason why it is true. What is so special about this equation? One savvy commentator I ran across online put it so nicely I feel compelled to quote him:

“It’s a sort of unifying identity in mathematics, containing each of the fundamental operations (additive, multiplicative, exponential) and each of the fundamental constants (e, i, pi, 1, 0) combined in a theorem that united trigonometry, analysis, and algebra and geometry. It’s really an amazing identity, and the proofs for it are diverse and fascinating…”[3]

It has, thus, been called the origin of all mathematics. Keith Devlin is purported to have said:

“like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.”[4]

Its elegance cries out for an explanation, but that explanation has proved so elusive that a desperate appeal to God begins to look almost reasonable, even to (some) mathematicians.

What should we make of this sort of argument? It seems on its face to be about as prima facie (in)admissible as any other argument from beauty. However, this argument may have more to recommend it than meets the eye. In particular, mathematical beauty has an uncanny predictive ability, at least in the sense that the more beautiful the mathematical formula, the more likely it is to describe the fundamental structure of the real world. Robin Collins has noted, for instance, that:

“To say that the beauty of the mathematical structure of nature is merely subjective, however, completely fails to account for the amazing success of the criterion of beauty in producing predictively accurate theories, such as Einstein’s general theory of relativity.”[5]

John Polkinghorne, in a lecture I recently had the pleasure of listening to (via podcast), said something similar though with less economy of words:

“It isn’t just [to satisfy] an aesthetic indulgence that theoretical physicists look for beautiful equations; it is because we have found, time and again, that they are the ones which actually do describe… a true aspect of the physical world in which we live. I suppose the greatest physicist I’ve known personally was Paul Dirac, (who held Newton’s old chair… in Cambridge for more than 30 years, who was one of the founding figures of quantum theory, [and] unquestionably the greatest British theoretical physicist of the twentieth century) and he made his great discoveries by a relentless and highly successful lifelong quest for mathematical beauty. Dirac once said ‘it is more important to have beauty in your equations than to have them fit experiment.’ Now he didn’t mean by that that it didn’t matter at the end of the day whether your equations fit the experiments (I know no physicist could possibly mean that), but what he meant was this: ok, you’ve got your new theory, and you use the solution and you find it doesn’t seem to fit what the experimentalist is telling you – now there’s no doubt that’s a setback, but it’s not absolutely necessarily fatal. Almost certainly, you will have solved the equations in some sort of approximation, and maybe you’ve just made the wrong approximation, or maybe the experiments are wrong (we have [known that] to happen even more than once in the history of physics – even in my lifetime I can think of a couple examples of that), so at least there’s some sort of residual hope that with a bit more work and a bit more luck you might have hit the jackpot after all. But, if your equations are ugly, there’s no hope. The whole 300-year history of theoretical physics is against you. Only beautiful equations really describe the fundamental structure of the world. Now that’s a very strange fact about the world… What I am saying to you is that some of the most beautiful (mathematical) patterns that our pure mathematical friends can think up in their studies just thinking abstractly… are found actually to occur, to be instantiated, in the structure of the world around us.”[6]

So mathematical beauty satisfies the empirical desideratum of predictive power in the sense that the more beautiful the mathematical expression, the more likely it is to describe reality.

Interestingly I think this kind of consideration can motivate a scientist (and perhaps even a die-hard empiricist, and/or a naturalist) to believe in the objectivity of aesthetic properties. In fact, unless they find a plausible evolutionary account for why our brains should be calibrated so as to recognize more beauty in the abstract mathematical equations which, it turns out, describe reality, than we find in other equations, there will be a residual mystery about the eerie coincidence of mathematical beauty and accurate mathematical descriptions of physics. An eerie coincidence the queerness of which can perhaps be mitigated by admitting the objectivity of aesthetic qualities.

However, the puzzling queerness of that eerie coincidence can only be (or can most plausibly be) ultimately alleviated if the universe is seen as the product of a (trans-)cosmic artist. If behind the fundamental structure of the universe there lies an intellect with aesthetic sensibilities (in some sense), then that would explain why the world showcases the mathematical-aesthetic qualities it does at the level of fundamental physics even when there is no (obvious?) reason why it should have. That, though, begins to look quite a lot like Theism.

The Argument from the Applicability of Mathematics

This segues into the next kind of argument from mathematics, which concerns the applicability of mathematics to accurate descriptions of the fundamental structure of the physical world. For the purposes of this argument beauty is entirely irrelevant. What is surprising, and in need of an explanation (according to this argument), is that the physical world would turn out to be describable in the language of mathematics (and here we are not simply referring to the basic truths of arithmetic, which are true across all logically possible worlds). William Lane Craig has become the most well-known proponent of this argument, and his articulation of it is relatively succinct.

“Philosophers and scientists have puzzled over what physicist Eugene Wigner called the uncanny effectiveness of mathematics. How is it that a mathematical theorist like Peter Higgs can sit down at his desk and by pouring over mathematical equations predict the existence of a fundamental particle which experimentalists thirty years later after investing millions of dollars and thousands of man-hours are finally able to detect? Mathematics is the language of nature. But, how is this to be explained? If mathematical objects are abstract entities causally isolated from the universe then the applicability of mathematics is, in the words of philosopher of mathematics Penelope Maddy, “a happy coincidence.” On the other hand, if mathematical objects are just useful fictions, how is it that nature is written in the language of these fictions? In his book, Dr. Rosenberg emphasizes that naturalism doesn’t tolerate cosmic coincidences. But the naturalist has no explanation of the uncanny applicability of mathematics to the physical world. By [contrast], the theist has a ready explanation. When God created the physical universe, he designed it on the mathematical structure he had in mind. We can summarize this argument as follows:

  1. If God did not exist, the applicability of mathematics would be a happy coincidence.
  2. The applicability of mathematics is not a happy coincidence.
  3. Therefore, God exists.”[7]

I am not sure of this argument’s philosophical quality, since it seems to me that it may be a metaphysically necessary truth that a logically possible world be amenable to mathematical description of some kind. For instance, it certainly seems true that whatever the geometry of space happens to be, there’s no necessary fact of the matter, but it also seems true that if the geometry of space isn’t Euclidean, it may be hyperbolic, or elliptic, (or maybe something else, je ne sais quoi) but it has got to be something, and what it happens to be may, therefore, not cry out for any more explanation than any other quaint contingent fact about the world.[8] However, maybe I’m mistaken about this; maybe the argument is, in fact, just as viable as other teleological or ‘fine-tuning’ arguments are.

Argument from Mathematical Truth

Finally, the third kind of argument I can think of would go something like this: mathematical truths, like all truths, have truth-makers. These truth-makers will have to be metaphysically necessary on pain of mathematical truths being contingent – but it seems obvious that mathematical truths are necessary truths, that they hold across all logically possible worlds. Now, Nominalism about mathematical objects is incompatible with the commitments we just outlined (unless one adopts Nominalism about modal properties as well), and so seems implausible (or, at least, less plausible than it otherwise would have been in virtue of this incompatibility). Platonism also, however, seems to be problematic. Between Platonism and Nominalism, there is a wide range of views including Divine Conceptualism (according to which mathematical objects exist as necessary thoughts in the necessary mind of God), Theistic Activism, Scholastic Realism[9] and many others besides. In fact, a variety (and perhaps a majority) of the accounts of abstract objects on offer today presuppose the existence of God in different ways.

This opens the way to at least two arguments we could construct for the existence of God. First, we could argue that one of these accounts in particular is most plausibly correct (such as Greg Welty’s Theistic Conceptual Realism),[10] and work our way up from there to the implication that God exists. Second, we could take the disjunction of all the accounts of abstract objects which require the existence of God and argue that (i) if any of them is correct then God exists, but (ii) it is more plausible than not that at least one of them is correct, from which it follows (iii) it is more plausible than not that God exists.

So, there we have it, three kinds of arguments from mathematics for the existence of God; a transcendental argument (from beauty), a teleological argument (from applicability), and an ontological argument (from necessity). Could there be others? Maybe, but I suspect that they will all end up falling into one or another (or maybe at least one) of the general categories I tried to outline here. I admit that I didn’t outline them as general categories very well, but that exercise will have to wait for another day when I have more time to blog to my heart’s content.

As a quick post scriptum; if Euler had any substantive argument in mind and wasn’t merely mocking Diderot for his lack of mathematical aptitude, which of these three kinds of arguments would he most likely have had in mind? It’s hard to say, of course, but my best guess is that if he had anything in mind at all, it would fall into the third category. He may have been thinking that the fact that mathematical and purely abstract (algebraic) ‘structural’ truths exist at all requires some explanation, and this explanation must be found in God. This is just a guess, and I make no apologies for it – I am happy to think that Euler was just teasing Diderot, but I am equally happy to entertain the thought that if Diderot had not immediately asked to leave (because of his embarrassment), Euler may have been able to elucidate his point.

[1] Gillings, Richard J. “The so-called Euler-Diderot incident.” The American Mathematical Monthly 61, no. 2 (1954): 77-80. http://www.fen.bilkent.edu.tr/~franz/M300/bell2.pdf

[2] Notice that these are not to be confused with mathematical arguments per se; they are merely arguments from mathematics, in the same way as you might have arguments from physics (the argument from cosmological fine-tuning, the Kalam, etc.) for the existence of God which are not intended to be scientific proofs of God’s existence, but scientifically informed philosophical proofs/arguments for God’s existence.

[3] Russel James, Why was Euler’s Identity Supposed to be a Proof for the Existence of God, https://www.quora.com/Why-was-Eulers-identity-supposed-to-be-a-mathematical-proof-for-the-existence-of-God; Note that he finishes the quoted paragraph with the words “but It has nothing to do with god whatsoever.” I have left this out not because I think he is wrong, or to misrepresent his position, but because it has nothing to do with the formula and everything to do with the propositional attitude he adopts with respect to the question of whether the formula is any kind of reason to think there is a being like God.

[4] Paul J. Nahin, Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills, (Princeton University Press, 2011), 1. https://books.google.co.uk/books?id=GvSg5HQ7WPcC&pg=PA1&redir_esc=y#v=onepage&q&f=false

[5] Robin Collins, The Case for Cosmic Design, (2008), http://infidels.org/library/modern/robin_collins/design.html

[6] John Polkinghorne, Science in the Public Sphere, http://www.veritas.org/science-public-sphere/

[7] William Lane Craig, Is Faith in God Reasonable? William Lane Craig vs. Dr. Rosenberg, http://www.reasonablefaith.org/debate-transcript-is-faith-in-god-reasonable

[8] I am really, honestly, no more sure of this counter-argument than I am of the argument. For those interested, please do check out the debate between Craig and Daniel Came on the Unbelievable? Podcast, which you can also find here: https://www.youtube.com/watch?v=nn4ocx316dk

[9] J.T. Bridges defends this view: https://www.youtube.com/watch?v=eFU1BKxJf1k

[10] See: Greg Welty, “Theistic Conceptual Realism,” in Beyond the Control of God: Six views on the Problem of God and Abstract Objects, ed. Paul Gould, (New York: Bloomsbury Academic, 2014), 81-96.

Banach-Tarski paradox, א Infinities, Infinitesimals, and the A-theory

I will offer an analysis of what is going wrong with the Banach-Tarski paradox suggesting that points, construed as infinitesimal surface areas, are nothing more than mathematically useful fictions. I will suggest that infinitesimals raise the same kinds of modally-prohibitive paradoxes in metaphysics as positing actually infinite quantities does (and for the same or similar reasons), and then consider an argument against the A-theory (in most of its forms) which can be purchased from these insights. I will then scout out some philosophical avenues available to the A-theorist.

The Banach-Tarski paradox is a famous mathematical paradox according to which it can be proved that if you divide the surface area of a sphere into little bits, and simply rearrange the bits appropriately, you can reconstruct two spheres each with the same surface area as the original sphere. In layman’s terms, you can prove (something just a shocking as) that 1=2.[1] To explain how it works, it may be worth calling to mind the various paradoxes associated with actual infinities.

Consider what it would be like to count upwards from -7 to infinity and stop only once you’ve arrived. Even if given an infinite amount of time you would never arrive, because no finite additions can sum up to a transfinite quantity. Subtract infinity from infinity, and what do you have? You have zero, but you also have infinity, and you also have 18.9801 (and every other real number); all of these are not just legitimate answers, they are mathematically correct answers. However, clearly 18.9801 is not equal to either zero, infinity, or anything else! Have a (Hilbert) hotel with an infinite number of rooms, all of which are occupied, and you want to check in an infinite number of new guests? No problem, just move every person from the room they are in (n) to the room with a room number equivalent to two times the original room’s room number (2n). Done; you’ve managed to move people around in such a way as to create an infinite number of vacant rooms without asking anyone to leave. Most of us (who are interested in this sort of thing) know the myriad paradoxes which arise from postulating even the possibility of an actual infinity. It seems relatively philosophically secure that there cannot be an א number of things (where א represents the first transfinite number, not to be confused with ∞ which symbolizes infinity taken as a limit rather than a quantity). If there are philosophically sophisticated caveats then so be it, but the point will remain that there are plenty of examples of things for which having an א number of them is clearly (broadly logically) impossible.

Let’s return, for a moment, to Hilbert’s Hotel, because it’s a particularly useful illustration. Suppose that the guest in room 3 checks out, while all the (infinitely many) other rooms remain occupied. The desk clerk decides that they want every room occupied, so they ask each person in room n (where n>3) to move one room over; that is, from room n to room n-1. That will fill up room 3, but the process will also leave no room empty because there is no room number n for which there is not an occupied room n+1. This works equally well for two dimensional shapes, such as circles; remove one ‘point’ from the circumference of a circle and you may have an infinitesimal gap, but simply move every other point along the circumference over (uniformly) by an infinitesimal amount and, voila, the gap is plugged and there will be no new gap. The trick in the case of the Banach-Tarski paradox is to apply the same reasoning to three-dimensional objects. For the best explanation of this paradox I’ve ever seen, (especially for readers who aren’t familiar with it, please make your life better and) check out Vsauce.

Alexander Pruss has noted on his blog that this result “is taken by some to be an argument against the Axiom of Choice.”[2] However, he argues that you can get the same paradoxical result in similar cases (and even in the same case) without the axiom of choice, so that the axiom of choice should be cleared of all suspicions. I agree (though I’m certainly no expert). Richard Feynman is purported to have said, upon being shown the proof, that “it’s fine you can do it with ‘continuous spheres’, since there’s no such thing. The important thing is you can’t do it with oranges, because oranges are made of a finite number of indivisible parts.” I think he is wrong about oranges (being actually comprised of indivisible finite parts, at least if the ‘parts’ are extended in three spatial dimensions), but his sentiment is appreciably insightful nonetheless.

The problem with the paradox, in my submission, is that it divides the surface of the sphere up into points. However, points on a sphere, like points on a line segment, are infinitesimals. This is precisely why (Aristotelians) say that line segments are not composed of points the way walls are composed of bricks, but, instead, points act as the limits between which a line segment is continuously extended. An infinitesimal is a quantity which is infinitely small. It is non-zero, but it is also smaller than any finite quantity. Sure infinitesimals are useful for doing things like infinitesimal calculus, developed by one of my all time favorite philosophers Gottfried Wilhelm Leibniz, but they remain, I believe, nothing more than useful fictions. To borrow a phrase from W.L. Craig;

“They are akin to ideal gases, frictionless planes, points at infinity, and other useful fictions employed in scientific theories.”[3]

If we are to accept the possibility of infinitesimal quantities in reality, then we will quickly run into paradoxes like the Banach-Tarski paradox (which, quite apart from being obnoxious to the rational intellect, seems to violate the law of conservation of matter and energy). Positing infinitesimals is just as paradoxical as positing sets of actually infinitely many discrete things (where ‘things’ is an ontologically loaded term). I am suggesting that infinitesimals are just as paradoxical as actual infinities, and, at bottom, for the same reason(s). In fact, I have this intuition that every argument for thinking that there cannot be any actual infinities (as opposed to potential infinities, where ‘infinity’ merely acts as a limit), admits of a parody for an argument against the existence of infinitesimals. I’m not sure I can rigorously prove it, but I think it’s very plausible.

It seems to me that there’s something conceptually parasitic about infinitesimals relative to infinities. They each conceptually supervene on each other symmetrically. To visualize this symmetry, consider plotting the function ƒ(x)=  1/x which will look like this:

[http://mathworld.wolfram.com/images/eps-gif/AsymptotesOneOverX_1000.gif]

The distance between the curved line and the x-axis (i.e., y=0) as x approaches (positive or negative) infinity is shrinking (or, at least, its absolute value is shrinking), and approaching an infinitely small non-zero measure. When X is infinite, the absolute value of the y-axis coordinate of the curved line (i.e., the distance between the curved line and it’s asymptote, here being the x axis) is infinitesimally small. This example helps to illustrate the point that the concept of an infinitesimal is bound up with the concept of infinity, so that in the absence of one the other would be inconceivable. That at least motivates the suspicion that if one turns out to be metaphysically impossible, so will the other.

What relevance does this have for the philosophy of time? Well, consider that on the A-theory there is such a time as the present. How long does the present last? What, precisely, is its magnitude, its duration? Let’s consider the following argument:

  1. If the A-theory is true, then the present is either infinitesimal in duration, or it is finite in duration.
  2. The present cannot be infinitesimal in duration.
  3. The present cannot be finite in duration.
  4. Therefore, the A-theory is false

Premise 3 can be established with Leibniz’ argument against the (logical) possibility of a physically indivisible element, or ‘atom’ (in the etymologically literal sense). For anything extended in three-dimensional space, however small, it will always be logically possible for me to divine it in two, even if I am physically incapable of doing so (due to some constraint, such as not having the appropriate equipment for the job, or maybe not even being able to develop any tool which could do the job). Physical impossibilities are not (all) logical impossibilities, and logically there is no constraint on how many times I could divide an object extended in space. To say that there is an object extended in space which is not logically possibly divided up into smaller constituent pieces is, according to Leibniz, incoherent. The exact same argument, mutatis mutandis, works against there being chronons (i.e., atomic chunks of time).

The denial of premise 2 is absurd given our observations that positing infinitesimals leads to modally unconscionable paradoxes like Banach-Tarski.

Ways out: I see four ways, not all of them equally viable, for an A-theorist to escape the conclusion of this argument.

First, they could challenge premise 3 on the grounds that, if there are chronons, then by definition they are entities which cannot be physically divided. The suggestion would be that the prima facie absurdity of a Chronon de dicto doesn’t entail the impossibility of a chronon de re. This dangerously dislocates rational intuition from epistemic reliability, but I can imagine extreme empiricists embracing this response.

Second, they could challenge premise 2 by arguing that positing any more than one real infinitesimal of any kind might be problematic, but that there’s no way to derive similar paradoxes from positing a maximum of one infinitesimal. In other words, perhaps paradoxes involving infinitesimals only arise when there are n infinitesimals, where n ∈ ℕ, and n>1. Multiply an infinitesimal by any natural number, or even a transfinite number, and you will still get an infinitesimal result, so it seems harder to show that from one infinitesimal you could derive some kind of contradiction.

Quick thought: Perhaps if there are rules/axioms such as (i) no infinitesimal can be larger or smaller than any other infinitesimal, but (ii) anything (other than 1) to the power of itself is larger than itself, you could derive a contradiction by taking an infinitesimal X, running it through Xx=Y, and then asking whether Y is larger than X, or the same size (it appears to be both). However, I don’t have the kind of facility in mathematics to be able to produce a rigorous proof that even a single infinitesimal would lead to some kind of contradiction or unconscionable paradox. Moreover, it isn’t entirely clear to me what relevance that kind of mathematical paradox would have for the metaphysical consideration at hand. In any case, the second challenge to premise 2 cannot be lightly dismissed.

Third, one could adopt a really wild philosophy of time, such as the Apresentism I wrote about in the last post (thus denying the first premise).

Fourth, one could deny the first premise by adopting what has been called a non-metric view of the present. This is the view preferred by William Lane Craig.[4] I have more than expended my allotted time for blogging and casual writing today, so I will leave this post here for now. I may return to the idea of non-metric present in the (near) future in another post.

[Ha, I don’t presently have time to write more. Get it?]

[1] For fun, check out and try to find the mistake in the following mathematical proof that 1=2 here: https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html

[2] http://alexanderpruss.blogspot.ca/2013/06/the-banach-tarski-paradox-and-axiom-of.html

[3] William Lane Craig, “Response to Greg Welty,” in Beyond the Control of God: Six Views on the Problem of God and Abstract Objects, ed. Paul Gould (A&C Black, 2014), 102.

[4] See: Craig, William. “The extent of the present.” International Studies in the Philosophy of Science 14, no. 2 (2000): 165-185.

Personal Theism and Contingent Persons

I sometimes hear it said that some people are ready to countenance that there exists a being which is necessary, transcends the world, and is related to the world as it’s explanation or cause, and yet for whom the suggestion that this being is ultimately personal is too much to swallow. I have previously noted that there are arguments to think that if there is such a cause of the world, then that being must be (or is much more plausibly than not) personal. Perhaps we can construct a different and additional argument to demonstrate that if Theism is true, then personal Theism is more plausibly true than impersonal Theism.

The conditional probability of their being contingent persons seems to be no greater or lesser on impersonal Theism than it would be on Naturalism. Thus Pr(CP|IT) = Pr(CP|N). Conversely we can infer that, ceteris paribus, Pr(N|CP) = Pr(IT|CP), and assuming that nothing in our background knowledge privileges either Naturalism or Impersonal Theism over one another, we can say that Pr(N|CP&BK) = Pr(IT|CP&BK). Naturalism/Impersonal-Theism gives us no more or less reason to suspect that the actual world would contain contingent persons than Impersonal-Theism/Naturalism (respectively). The rub of it is that contingent persons are surprising on a Naturalist ontology, where contingent persons are beings with intentional states, reflexive self-awareness, who inhabit first-person perspectives, have conscious experience, and introspectively apprehend themselves to be ‘free’ in some sense. On Naturalism these beings are very surprising, since if we gather up all the logically possible worlds at which Naturalism holds true, we should find that very few of them contain contingent persons (as described). Consider a thought experiment offered by John Bergsma:

“I think there is an even deeper problem with the Naturalistic, Materialist evolutionary worldview, which I will call NME,… it is that if NME is true it is unlikely and inexplicable that we would have cognitive processes at all. Restated, if NME is true we would expect a world without creatures that have mental states, however we do have a world with creatures that have mental states… Let’s engage in a mental experiment to show that this is the case. Imagine that we designed a very sophisticated mechanical robot that was able to land on other planets, locate raw materials, refine those materials, build a factory from them and proceed to build copies of itself. Suppose that we land that robot on a distant planet and let it get to work. 50 years later we return and the experiment has been successful, the planet is teeming with copies of our original robot all of them milling about in search of raw materials to make further copies of themselves. So from an evolutionary perspective these robots have been very successful; they have multiplied, exceedingly. However, would the robots have developed sentience? Would they be aware of their own existence? Would they actually think, feel, write poetry? In a science fiction movie maybe they would. But this is the real world, and we know that they would not. Although their behavior is adaptive and they have proliferated, they would not be one wit closer (pun intended) to actually having mental states… Mental states are invisible to evolution because evolution can only act on behavior; the only way mental states would become visible to evolution is if they actually effected behavior, and this is precisely the thing that most academics who hold to naturalist materialist evolution or NME vociferously deny. NME adherents generally deny that mental states have any influence on behavior because they intuitively sense that mental states are not material entities, or at the very least are difficult to analyze as material entities. Therefore, mental states fit uncomfortably into a materialist worldview and materialists want to deny the full reality of mental states.”[1]

However, arguably, contingent persons are more at home in (i.e., less surprising in) possible worlds where personal Theism is true. If God both exists and is personal then a world with contingent persons is, if not to be expected, at least not very surprising (or at least not as surprising as it would be on Naturalism). If we agree that the existence of contingent persons is relatively surprising on Naturalism (and, thus, too, on Impersonal Theism) then we can commit ourselves to: Pr(PT|CP)>>Pr(N|CP), and Pr(PT|CP)>>Pr(IT|CP).

Plausibly, if our background knowledge does nothing to privilege Naturalism over Impersonal Theism (which the Impersonal-Theist is likely to accept), then it does nothing to privilege Naturalism over Personal Theism either. Thus Pr(PT|CP&BK)>>Pr(N|CP&BK), from which it obviously also follows that Pr(PT|CP&BK)>>Pr(IT|CP&BK).

Although I can imagine a number of ways in which the indignant Naturalist might offer objections to this argument (for instance by insisting that our background knowledge really does make Naturalism more plausible than Theism), it seems to me that the Naturalist should find common cause with the (personal-) Theist in arguing that impersonal Theism does nothing to make contingent persons less surprising than they would be, or are, on Naturalism. Moreover, insofar as personal-Theism does make contingent persons less surprising than they otherwise would have been (and thus less surprising than they would have been on either Naturalism or impersonal Theism), the existence of contingent persons, combined with Theism, makes personal Theism more plausible than it otherwise would have been (and more plausible than Impersonal Theism).

Thus, we have a good argument to infer from ‘both Theism and the existence of contingent persons’ that God is personal. If somebody is willing to accept Theism, then it seems like Impersonal-Theism would be the harder pill to swallow.

Maybe this argument could have some interesting extension to pantheism. Pantheism is the view that there is no distinction between ‘God’ and ‘Everything,’ that God is nothing other than the whole of reality, and that all the parts of reality are parts of God. The term itself makes its first appearance “in the writing of the Irish freethinker John Toland (1705) and [is] constructed from the Greek roots pan (all) and theos (God).”[2] There are nuances to be attended to, such as that to which Thomas Aquinas drew his attention in distinguishing “between the doctrine that God is the form of all things (‘formal pantheism’) and the doctrine that God is the matter of all things (‘material pantheism’) (Moran 1989, 86).”[3] However, in general, Pantheism is the view that the terms ‘God’ and ‘the world’ pick out exactly the same thing. Thus Spinoza, perhaps the most famous of Western Pantheists, simply identifies ‘God’ with ‘Nature’ in his philosophical system.

I note that Pantheism is not to be confused with the increasibly popular view called Panentheism. As John Culp explains in the Stanford Encyclopedia of Philosophy:

“Panentheism seeks to avoid either isolating God from the world as traditional theism often does or identifying God with the world as pantheism does. Traditional theistic systems emphasize the difference between God and the world while panentheism stresses God’s active presence in the world. Pantheism emphasizes God’s presence in the world but panentheism maintains the identity and significance of the non-divine.”[4]

Insofar as pantheism (as opposed to Panentheism) also maintains that God is ultimately impersonal, it becomes, pace the argumentation above, less plausible than personal Theism (whether classical monotheism, or Panentheism).

[1] Alvin Plantinga and John Bergsma “Science and Faith Conference” from the Franciscan University of Steubenville http://youtu.be/mVlMK9Ejhb0?t=1h

[2] Mander, William, “Pantheism”, The Stanford Encyclopedia of Philosophy (Summer 2013 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/sum2013/entries/pantheism/&gt;.

[3] Mander, William, “Pantheism”, The Stanford Encyclopedia of Philosophy (Summer 2013 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/sum2013/entries/pantheism/&gt;.

[4] Culp, John, “Panentheism”, The Stanford Encyclopedia of Philosophy (Spring 2013 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/spr2013/entries/panentheism/&gt;.