By James A. Lindsay, Onus Books, November 14, 2013, 0956694896

Dot Dot Dot is a fascinating book. I have read some about infinity, but it is a difficult concept to grasp. Does infinity exist? No, it is an abstraction. Lindsay’s argument in a nutshell is that if infinity doesn’t actually exist than neither does a omni-grade God. I love the term omni-grade.

Modern math is simply beyond me. I’ve tried to read a few books that explain it, and never finished. Lindsay does a great job at explaining the basic ideas to me. Particularly being very clear that infinity is an abstraction, and that we had to go beyond the Peano Axioms to reason about infinity (Zermelo–Fraenkel set theory).

But the math isn’t really important. We aren’t going to prove or disprove the existence of an omni-grade God. Instead, Lindsay says there is no God, almost surely. The probability Almost Surely was new to me. It makes Lindsay’s argument philosophically defensible. For those who say he must prove God doesn’t exist, he replies that he’s not saying that, just that the probability is so low as to be inconsequential.

The argument is complex, really, and requires you read the book, which I highly recommend. The reasoning is solid, even if I can’t understand the math.

Lindsay doesn’t mind an abstraction called God. In fact, he thinks it is very helpful to lots of people. Abstractions do not exist, and can not be omnipotent, omniscient, or eternal. Abstractions cannot create, destroy, or judge.

Math is an abstraction. It’s a map. The terrain is reality, and math can guide us but math is not reality. Mistaking the model for the reality, or as Lindsay says, confusing the map for the terrain, is another important argument of Lindsay’s, which I really like.

I’m trying to explain the book. Needless to say, I enjoyed the book, and I learned a lot. Lindsay is an excellent writer.

[k376] Quantum fields are such abstractions. No one has ever observed a quantum field. Furthermore, there is simply no way we can know what is ultimately real. All we know about the world is what we observe.

[k391] For some time, the mathematical concept of the infinite–that which is limitless and impossible to count–has been tied up with the mythological concept called “God.” I don’t claim to know why this is the case. In the past, I have suggested that perhaps it is a result of an arms race between memes.

[k399] I say so because a wide majority of arguments, both for and against belief in God, that call upon infinity are deeply problematic. It is probably not controversial in the least to point out that this fact may rest heavily upon a simple reality: math is hard. Most people just don’t know enough mathematics to talk competently about infinity.

[k420] Platonism holds that abstractions have meaningful reality in their own realm known as the “realm of ideals,” independent of minds to think them, and Platonism is relevant because it leaves us confused between the abstract and the real.

I do not accept the Platonist position.

[k423] In short, then, I think mathematical and other Platonists are reading too much into their own ideas.

[k478] Once the axioms at the foundations are agreed upon, a mathematical proof is in a sense a timeless thing–and there’s a reason for this near the heart of my purposes.

[k489] Instead, here I want to set the overarching tones of the collection, which is to say first that infinity and God are incompatible and second that I lean away from Platonism as a philosophical foundation of mathematics and, by extension, mature philosophy of all kinds.

[k540] This idea seems very natural and self-evident: it essentially states that however many things we have, we can conceive of the idea of adding another, even if we were to know that in the physical universe there isn’t another thing to add. Of course, considering the previous sentence, we could digress into endless and mostly fruitless postmodern discussions about what “self-evident” really means if in reality we couldn’t add another thing even if we wanted, but we needn’t do that.

[k577] The immediate consequence here, as with the successorship property of numbers, is that if we have one size of infinity, then we have infinitely many sizes of infinity. It was later shown that the infinity that tells us the size of the counting numbers is the smallest infinity and, slightly ironically and yet meaningfully, this size is known as countable infinity.

[k587] Cantor’s work in the 1870s motivated Goedel’s, which itself wasn’t even completed in this regard until Paul Cohen put it together in 1963, so for many decades these ideas were highly contentious because, not only did we not know the answers to some apparently fundamental questions, we didn’t know that we couldn’t know.

[k625] Of particular note, the axiom of choice that allows us to select from infinitely many nonempty bins is generally accepted, only lately, as a foundational part of set theoretic mathematics, and the resulting framework is abbreviated ZFC.

[k634] A recurring theme that I encounter from religious apologists is their strong tendency to try to argue God into existence via any number of claims to necessity. What sticks out to me, though, is that they always resort to philosophical-style arguments instead of being able to present verifiable evidence for the existence of their God (among other theological claims). This is truly bizarre.

[k645] Generally speaking, I find the conflation of logic and reality to be an enormously common mistake, including in Platonism, which is something of a refined form of committing this error.

[k648] These arguments essentially try to logic “God” into existence by showing that it is a logical necessity.

[k680] To emphasize: “This must necessarily exist” only ensures that some abstraction “exists” in some abstract sense, and it “exists” then in a particular axiomatic framework. Particularly, “this must necessarily exist” confers no responsibility on reality whatsoever! Platonism should have a hard time weathering this storm.

[k696] Quantum mechanics very successfully explains evidence from reality with the tools we have, but it appears not to be able to be properly understood.

[k718] Once we choose the axioms and know what we are calling logic, the truth values of every statement within that system are already determined, though we don’t know those values until we prove them.

[k724] Generally, we accept the fact that axioms are baldly asserted, and since this is true of all axiomatic systems, it is not a strike against any of them to point it out.

[k727] Axioms have to be judged against how “self-evident” they really are, how useful they are, how little they assume, and in other such ways. This, then, is why the theistic worldview axioms seemed more reasonable in the past than now; we now see that the purported existence of God is not self-evident, has limited utility with little or no explanatory power, and yet assumes an awful lot.

[k742] Also often lost in the shuffle, the axioms and logic are abstract things that do not “exist” in reality.

[k765] This creates a powerful illusion that our logic must be the way that it is because it is very, very difficult to conceive of a different way to do logic.

[k772] Thus, the Peano Axioms predict the concept of infinity, but they do not give us a way to get there. Adding one always gives us another number, not infinity.

[k839] For my purposes, this is very powerful. If those who believe in God want to claim it has infinite properties, which they must to maintain statuses like “Most High” and “Almighty,” which they do by official dogma, then we have to wonder what they are implying about their deity. It seems to follow that God, thusly defined, condemned to abstract unreality, at the very least unless physical infinities can be proven to exist. Critically, though, as I keep saying, abstractions don’t do anything except serve as the basis for potentially useful ideas. Particularly, abstractions don’t have agency and thus can’t create, destroy, judge, forgive, or answer prayers.

[k846] They become very vulnerable in this position for the same reason the Peano Axioms are unresolvable with set theory: here, infinite Gods cannot exist but neither can there be a limit on finite ones if they are truly to be omni-grade Gods.

[k928] The Kalam argument, in fact, only tries to bridge the gap between “began” and “was caused,” which is far less than Craig needs, and, as has been pointed out in many ways, it apparently fails to do so.

[k931] The Kalam asserts that “everything that began has a cause,” but can the Universe be considered something within that “everything”? It is utterly unclear that it can be.

[k938] Their conclusion that the Universe must have had a beginning because it would be impossible for an infinite number of moments to have passed already, though, commits the error of jumping over the ellipsis straight to the strong limit cardinal of (negative) infinity.

[k949] In other words, the Craigian argument, even on the A-Theory of time, has no purchase without jumping outside of a framework that has no outside.

[k951] The Craigian argument against actual infinities presents another glaring problem: the omni-grade properties of the Christian God for which they apologize depend upon infinity. Omni-grade properties are often taken to mean “infinite in scope,” particularly omnipotence and omniscience. The Craigian solution to this problem is to argue for a qualitative, not quantitative, understanding of infinity, which would make him something of an finitist–people who reject the notion of infinity as a quantity, even in the abstract.

Very few mathematicians are finitists, though, and there are good reasons. One reason is that infinity, even as a qualitative thing, has roots in the quantitative.

[k960] Theologians using infinity, again, seem to want it both ways. They want to say there is infinity–in God–but not deal with the consequences of accepting the axiom that allows it.

[k968] Particularly, the questions of whether or not “God” could do one more thing, move one ounce more, or know one more detail are left hanging and said to be meaningless questions.

[k997] The difference between Craigians and me is that they want to argue that their favorite abstraction, “God,” is actual while infinities are not.

[k1114] The theological relevance of this statement hits hard upon the ontological definition of “God” as “Most High” given by Anselm in the eleventh century and explains much of William Lane Craig’s philosophical chicanery on the topic of infinity. Anselm’s “God” is that than which nothing higher can be conceived (see Chapter 16 for more on this important point). Therein lies his problem: we immediately realize that the moment we conceive of the highness of some conception of “God,” not only can we conceive of something higher, as “highness” implies a metric at least in principle, but the “God” we have conceived of is lower than almost every conceivable conception.

[k1150] Because infinity is a strong limit cardinal, there is no procedural way to get to infinity by repeated addition (successorship) or exponentiation, which is what repeated halving turns out to be.

[k1166] One potential solution to Zeno’s Dichotomy Paradox is simply that either of physical or actual infinities do not exist. This solution resolves the paradox by pointing out that an infinite number of tasks cannot be required because there cannot be infinitely many tasks, either conceptually (for actual infinities) or in reality (for physical infinities). Of course, as we’ve seen, this approach can be argued to create other issues. Fascinatingly, despite these and a plethora of other proposed solutions, Zeno’s Dichotomy Paradox is not considered to have been solved.

[k1179] Here, then, I wish to make a big point about infinity again: from my perspective, all of these abstractions are indicative of the map, not the terrain. We seek to understand the world using our own logical map, since it is the only way we can make sense of the terrain, but reality is not dependent upon our map–it’s quite the other way around.

[k1204] If there are no actual infinities, and thus no physical ones, though, there are some interesting and surprising consequences. One such consequence is that without actual infinities, there cannot be real-world perfect circles (or spheres, or many other geometric figures). This is because without actual infinites, we cannot have actual measurements given by irrational numbers, which have an infinite number of decimal places and cannot be represented as a fraction of integers.

[k1217] When scientists and engineers built the highly sophisticated instrument known as Gravity Probe B, used to test predictions of general relativity, they built a gyroscope using the most perfect spheres ever manufactured. These differed from being “perfect” spheres by some forty layers of atoms here or there.

[k1243] As someone who rejects Platonism, I do not accept the idea that abstract existence implies real existence in any sense, and I go on to think that conflating the abstract and the real, often via the numinous, is one of the central problems that philosophical theism cannot get beyond.

[k1811] This fact should highlight another facet of how incredibly large the infinite is: it is so big that we cannot claim that all the objects in a countably infinite set can be equally likely to be chosen at random.

[k1890] The center of my argument here, then, is that I don’t think any plausibility claims but “God exists, almost surely” and its opposite, “God does not exist, almost surely,” can be defended in this matter.

[k1941] There may or may not exist a cottage made of pancakes on the far side of the moon, but it’s ridiculous to suggest that there are even odds that one is there. Would anyone think it is reasonable to say “because either it exists or it doesn’t, there’s a 50% chance that the Force is real”? I seriously hope not! Why should agnosticism about God be treated any differently?

[k1953] Fifty percent is the cognitive hurdle. The same argument applies going downward, so we can skip quickly to 5%, just to make the point. Is there a compelling argument that the number that describes the likelihood that God exists is at least 5%?

[k1956] Would we require a good argument to believe that there’s even a 5% chance that the Force exists? I would!

[k1958] To wit, religious claims to evidence for a deity are typically rather pitiful attempts that come in a few flavors. One sort is an argument to the necessity of God to make sense of the existence of reality, but this could just as easily be done with the Force.

[k1967] The problem with the evidential approach to defending God’s existence, is simply that if there is no God, then there is no actual evidence for God, only evidence misinterpreted to that purpose. The same evidence can be misinterpreted to the purpose of the Force or to any other sufficiently broad attributional schema, and for the motivated mind, perhaps primed on Mother’s knee, each is equally believable and equally worthless as an explanation for the available evidence.

[k1979] To highlight that fact, consider how much of a religious apologists’ time has to be tied up in arguing for possible–not necessarily plausible–ways by which the God they defend might be squared with the evidence that we have. In Craig’s case, for instance, this leaves him defending the genocides of the Old Testament as objectively moral. In all cases, apologists have to provide complicated treatments to rationalize why we do not see what we expect to see on the assumptions of religious theism.

[k1985] First, one other counter might be raised that the sciences are just as guilty as religion of fitting the evidence to particular abstract constructions called “models.” I’d argue that this is precisely what scientists do, that models are abstract constructions, but that the amount of guilt here is incomparably lower.

[k1987] Were I to invent a hypothetical subatomic particle completely unknown to the Standard Model, does anyone believe that I could go to a meeting of particle physicists and simply claim that there is at least a 5% chance that it exists?

[k1990] Otherwise, anyone could make up just about anything they want with a reasonable expectation of having it believed, which at the least would be an enormous waste of everyone’s too-precious time.

[k2004] Now, it could be argued that there is surely more evidence for God than for the Force, given that we know the Force is a fiction, but notice the roots of this appeal. It comes down to the idea that because we know the Force is a fiction, any evidence interpreted to support the Force is misinterpreted to do so. Easily swept under the rug here are two analogous facts. God arose in our ancient superstitious past where lines between fact and fiction literally did not exist, and if God does not exist, there is no evidence for God at all either, only evidence misinterpreted to fit a God-shaped hole.

[k2009] Had the Force been thought of thousands of years ago and happened to dominate the cultures that became ours, in fact, it may very well have been the centerpiece of the discussion we’re having here instead.

[k2015] When considering the matter, it is important not to get taken in by the exchange of a possibility and a probability, or indeed a certainty. Apologists are quick to point out that there could possibly be an explanation for all of the suffering of the world and then conclude that there probably or definitely is one in God.

[k2020] If I wanted to do this in math-speak, for any small number epsilon greater than zero, assuming that the probability that God exists is epislon demands a proper and solid defense of that claim.

“But, you can’t say that!” is the main defense against this point, from both theologians and philosophers alike. In rebuttal, all there really is to say, as there are no justifications for any positive plausibility for the God hypothesis, is “but you can’t say that the probability is zero that God exists without proving it!”

[k2026] First, I haven’t. I said every positive probability needs an argument to support it. Second, actually, I might be able to say that the probability God exists is zero, so long as I qualify it with “almost surely.” Since “almost surely” admits wildly unlikely possibilities, it does not run afoul of philosophical defensibility.

[k2029] Third, would we really hesitate to say the same thing about the Force, that the probability that it is real is zero, almost surely?

[k2034] As we have discussed, though, we cannot have infinitely many equally likely hypotheses with nonzero plausibilities. Thus, logical coherence dictates at the least some mechanism by which we can assign lower plausibilities, a priori, to some kinds of hypotheses than to others. This mechanism must provide convergence of the total probability, which means that however small the plausibility of any particular hypothesis, infinitely many more must have a plausibility considerably smaller.

[k2039] So, my claim is that any attempt to argue for a nonzero plausibility for the God hypothesis requires a substantial argument to establish it.

[k2132] In fairness, we should ask if informed skeptics are also guilty of confirmation bias. Of course, we’re all susceptible to it, but in the meaningful sense in this context, I argue not. The reasons are found in the reliability of predictions made on testable hypotheses. The predictions made by science are staggeringly accurate, and this is undeniable.

[k2145] Informed skepticism doubts until it can’t, and belief believes even after it shouldn’t.

[k2391] I think it gives some insight to the overall uselessness of ontological arguments, even when they don’t invoke infinity or appeal to specific theology. Indeed, I hope this also makes the point that being an absolutely brilliant logician, as Godel unquestionably was, does not exempt someone from the possibility of conflating logic and reality–a point I consider of paramount importance in the discussion concerning the God hypothesis.

[k2487] This argument, like all of the other philosophical arguments for the existence of God, merely points to the philosopher’s God, not to any particular manifestation of it as revealed in the various scriptures, leaving the apologist to bridge the gap to the object of their belief after defending the ontological claim.

[k2492] We have St. Anselm in the eleventh century to thank for this nifty piece of imprecise thinking, which former preacher, now infidel, John Loftus generously claims gives him the sense of having just watched a “really good magician” in action.

[k2588] There is no winning here. God simply cannot be “Most High,” neither finite nor infinite, neither quantitative nor qualitative. There is no such thing.

[k2602] My conclusions are simple: infinity is easier to misuse than it is to use, mathematics is a human-made endeavor designed to help create a system by which we can better understand and communicate about reality, axiomatic systems ultimately only construct abstractions, and abstractions are not to be confused with reality.

[k2610] The applications of the notion of infinity to discussions about God force the deity further out of the realm of the real, the natural, the physical, and into the realm of the abstract, the mental, the imaginary.

[k2618] Famous Christian apologist William Lane Craig explains it thusly, “‘infinity’ is just a sort of umbrella term used to cover all of God’s superlative attributes.” He wants to justify this statement by saying,

[W]hen theologians speak of the infinity of God, they are not using the word in a mathematical sense to refer to an aggregate of an infinite number of elements. God’s infinity is, as it were, qualitative, not quantitative. It means that God is metaphysically necessary, morally perfect, omnipotent, omniscient, eternal, and so on.