# Tagged: philosophy of mathematics

# On Bayes’ Theorem and Its Proper Application

By R.N. Carmona

There are times in where one’s assumptions fall into question. This could happen in a number of ways, but one of the more common ways is when one’s assumptions are confronted by facts of the matter. I had assumed that Bayes’ Theorem was well understood by my fellow atheists. Unfortunately, that assumption was wrong. It isn’t well understood. Like some of our academic tools (e.g. Bayes’ Theorem; statistics in general) and even entire disciplines (e.g. philosophy), our perception of things can be tarnished by misuse–specifically, the misuse or more accurately, misapplication of religious apologists and theologians. I see that that’s the case with Bayes’ Theorem.

Bayes’ Theorem is just another way to properly confront one’s assumptions. Nat Napoletano stated that “Bayes’ Theorem is a plan for changing our beliefs in the face of evidence.”1 Before we cross that bridge, however, it will be useful to present the theorem in its entirety. I will then explain its variables. Bayes’ Theorem appears as follows:

P( A | B) = [P (B | A) * P (A)] / P(B)]

Prior to proceeding, some definitions are required. As Mario Triola stated:

One key to understanding the essence of Bayes’ theorem is to recognize that we are dealing with sequential events, whereby new additional information is obtained for a subsequent event, and that new information is used to revise the probability of the initial event. In this context, the terms prior probability and posterior probability are commonly used.2

Given this, we learn that the prior probability “is an initial probability value originally obtained before any additional information is obtained” and the posterior probability “is a probability value that has been revised by using additional information that is later obtained.”3 Another way of putting this is that the theorem shows the relation between a conditional probability with its inverse. In other words, this is the probability of a hypothesis given some observation as related to the probability of an observation given some hypothesis. Let’s work with an example. I’ll first state it as a word problem and then I’ll flesh out the variables.

Let’s first assume that there’s an equal chance that Jack or Zack get the job. Then, let us find the probability that either Jack or Zack get promoted some time after being hired. However, we find that since Zack is less suited for the position, the probability of him being promoted is lower. We therefore have the following probabilities.

P(J): the probability that Jack gets the job. (.5)

P(Z): the probability that Zack gets the job. (.5)

P(pr): the probability of promotion. (.55)

P(t): the probability of termination. (.45)

P(pr|J): the probability that Jack is promoted after getting hired. (.8)

P(pr|Z): the probability that Zack is promoted after getting hired. (.25)

P(t|J): the probability that Jack is terminated. (.2)

P(t|Z): the probability that Zack is terminated. (.75)

As you can see, I already assigned probabilities to each case. I want the numbers to be easy to work with, so that this demonstration is clear. So, let us calculate the probability that one of them is promoted regardless of who’s hired.

P(pr) = P(pr|J) * P(J) + P(pr|Z) * P(Z)

P(pr) = (.8 * .5) + (.25 * .5)

P(pr) = .4 + .125 = .525 or 52.5%

The probability that either of them is terminated can be shown in the same way. It can, however, be simplified as follows:

P(t) = 1 – (.525) = .475 or 47.5%

So Bayes’ Theorem comes in when you want to find out either of the following: the probability that termination occurred given that Zack was hired or the probability that termination occurred given that Jack was hired. Alternatively, you can find out the probability of promotion given that either Zack or Jack was hired. Note that this is *after* the hiring. The equation for the probability that termination occurred given that Zack was hired looks as follows:

P(Z|t) = [P(pr|Z) * P(Z)] / P(pr)

So, P(Z|t) = (.25 * .5)/(.45). P(Z|t) is .278 or 27.8%. Of course, Bayes’ Theorem isn’t used for these kind of cases. This example is simply for ease. As stated earlier, the misapplication of Bayes’ Theorem leads some of us to doubt its utility. Napoletano touched on this exactly. He stated that it has been used by theologians (e.g. Richard Swinburne) to prove the existence of god or to demonstrate that the resurrection actually happened. He does, however, state that “it is difficult to separate out the variables and make them independent”4 in such cases. He also stated the theorem can expose when someone is being irrational. Aside from that, it can expose when someone is being biased–when someone is favoring one hypothesis over the other. In such cases, the theorem is often misapplied.

In medicine, for instance, a correct application will keep one from undergoing a series of medical tests. Like in our example above, doctors will be able to know the probability that you have either x disease or y disease given the symptoms you have. This goes back to showing a relation between the conditional probability and its inverse. This is, in other words, showing the relationship between what we currently know and what we want to know. This is a correct application.

In science, the theorem can be used to arrive at the probability of a hypothesis given observations. In other words, given MOND (Modified Newtonian Dynamics), what is the probability that some phenomena would be the case? Chris Wiggins explains:

One primary scientific value of Bayes’s theorem today is in comparing models to data and selecting the best model given those data. For example, imagine two mathematical models, A and B, from which one can calculate the likelihood of any data given the model (p(D|A) and p(D|B)). For example, model A might be one in which spacetime is 11-dimensional, and model B one in which spacetime is 26-dimensional. Once I have performed a quantitative measurement and obtained some data D, one needs to calculate the relative probability of the two models: p(A|D)/p(B|D). Note that just as in relating p(+|s) to p(s|+), I can equate this relative probability to p(D|A)p(A)/p(D|B)p(B). To some, this relationship is the source of deep joy; to others, maddening frustration.5

Religious apologists have misused the theorem. This is most apparent in their absurd calculations. “Richard Swinburne, for example…estimated the probability of God’s existence to be more than 50 percent in 1979 and, in 2003, calculated the probability of the resurrection [presumably of both Jesus and his followers] to be “something like 97 percent.”6 These numbers are, as Dawkins stated, “not measured quantities but personal judgments, turned into numbers for the sake of the exercise.”7 Put simply, it’s blatant bias.

This abuse of the theorem is likely behind people’s suspicion of the theorem. Whether or not a theist or an atheist can use Bayes’ Theorem to arrive at a probability of god’s (non-)existence or of the resurrection is perhaps an open question. There are, however, accurate historical applications. It must first be pointed out that Bayes’ Theorem cannot be used in isolation. “The only way to argue for a claim is to look at the evidence and compare that claim to alternative claims.”8 More importantly, however, assumptions reduce prior probability. Rather than reduce their prior probability, religious apologists and theologians inflate it. This is perhaps at the core of people’s suspicions. Your suspicion, however, isn’t in Bayes’ Theorem itself but rather in how it is being applied. It must be noted that a reduction in the prior implies a reduction in the posterior. Richard Carrier explains:

[I]f the evidence doesn’t match your claim so well (as theism always doesn’t, e.g. the Problem of Evil, or to continue the last analogy, the fact that when miracle claims are properly investigated they never turn out to be genuine), you can invent “excuses” to make your claim fit that evidence better. This means, adding assumptions to bolster your claim. But what Bayes’ Theorem teaches us is that every such assumption you add reduces the probability that your claim is true, rather than increasing it (as that excuse was supposed to do). Because adding assumptions to any claim C reduces the prior probability of C, and any reduction in the prior probability of C entails a reduction in the posterior probability of C, which is simply the probability of C. This is because the prior probability of several assumptions is always necessarily lower than the prior probability of any one of them alone, a basic fact of cumulative probability, which when overlooked produces the Conjunction Fallacy.9

When it comes to the existence of god or the occurrence of a miracle or a purported past occurrence like the resurrection, a neutral probability of .5 can serve as our prior. An atheist will no doubt assume that the prior probability of Jesus’ resurrection is much lower. A theist will assume it’s much higher. If, however, we set our assumptions aside, our posterior probability also goes unaffected. This will allow the numbers to do the work, so to speak. Though it remains difficult to separate the variables and make them independent, our results aren’t laden with bias. Ultimately, we can do closer analysis of Richard Swinburne’s misapplication or Richard Carrier’s application of Bayes’ Theorem. Such analyses are, however, cumbersome and for our purposes, unnecessary.

Ultimately, if you have any suspicions, I must reiterate, they do not lie with the theorem itself but rather with the way it is being applied. We must always mind the source. Not in the sense that we’re dismissive in an *ad hominem* or perceived guilt by association sort of manner, but in the sense that we’re acquainted with their biases. If, for instance, s/he is a believer in god, if it turns out that the calculations are suspicious or even dubious, we must single out the bias. There is, for instance, no way that the probability of the resurrection of Jesus is “something like 97 percent.” This is *prima facie* ludicrous and tenuous. It, of course, must be questioned. Common sense dictates that if the probability of his resurrection is that high, the probability of all miracles, regardless of which religious adherent makes the claim, is comparably high. This certainly isn’t the case, especially when we account for the dubious epistemic support for such claims (e.g. eyewitness testimony). I repeat, this irresponsible and to be blunt, non-academic misapplication must be questioned. We, however, must learn to do so responsibly and that isn’t possible when we’re directing our suspicions at the theorem rather than the theorist.

**Works Cited**

1 Napoletano, Nat. “Bayes’ Theorem for Everyone 01 – Introduction”. *YouTube*, *YouTube, LLC*. 29 Jan 2012. Web. 1 Mar 2015.

2 Triola, Mario F. “Bayes’ Theorem”. *University of Washington*. ND. Web. 1 Mar 2015.

3 Ibid. [2]

4 Ibid. [1]

5 Wiggins, Chris. “What is Bayes’s theorem, and how can it be used to assign probabilities to questions such as the existence of God? What scientific value does it have?”. *Scientific American*. 4 Dec 2006. Web. 1 Mar 2015.

6 Ibid. [5]

7 Ibid. [5]

8 Carrier, Richard C. “If You Learn Nothing Else about Bayes’ Theorem, Let It Be This”. *Freethought Blogs*. 26 Apr 2014. Web. 1 Mar 2015.

9 Ibid. [8]

# Arguments For Atheism: The Argument From Cosmology

By R.N. Carmona

Though the strongest argument of the arguments for atheism, this argument is the most philosophically involved. That is to say that, due to the implications of the argument, the argument ventures into philosophical territory–much of which is the subject of continued disagreement and lack of consensus. On the surface, the argument is strong. However, the argument’s tacit assumption has to be qualified so that the strength of the argument is augmented. In doing so, however, some problems will arise. Though these problems aren’t damning to the argument and though these problems do not lend credence to antithetical arguments, the problems must be addressed. In other words, at the very least, solutions must be suggested.

It is time now to turn to the argument. Groundwork will then be laid out to qualify its assumption. Problems will then be presented and addressed. Then it will be suggested that the uncertainty of the solutions to these problems lends no support to antithetical arguments because such arguments carry their own onus. The argument is as follows:

P1 If there is a naturalistic explanation for the origin of the universe, a creative agent does not exist. (P -> Q)

P2 There is a naturalistic explanation for the origin of the universe. (P)

C Therefore, a creative agent does not exist. (∴ Q)

Prior to qualifying the assumption of the argument, the conclusion will be qualified. The implication here is that if there is a naturalistic explanation for the origin of the universe, a creative agent is not necessary. However, as is argued below, if a creative agent is not necessary, then it follows that a creative agent doesn’t exist.

To see how the conclusion follows, it is perhaps best to shrink the scope. This is to say that rather than focus on the universe as a whole, the focus should turn to a smaller aspect. Suppose that the argument instead argues that if there is a naturalistic explanation for the formation of planets, a creative agent doesn’t exist with respect to planets. Antithetical arguments would no doubt focus on the formation of Earth and thus, would engage in special pleading or question begging. In other words, such arguments would have no choice but to accept that there is a naturalistic explanation for planet formation, but that this explanation is somehow inapplicable to the Earth.

However, if a creative agent isn’t necessary for the formation of the Earth, then there’s no respect in which it can be said to exist in relation to the Earth. Now, in broadening the scope once again, the same argument applies to the universe. If a creative agent isn’t necessary for the origin of the universe, then there’s no respect in which it can be said to exist in relation to the universe. If it isn’t necessary for the creation of the universe, then it doesn’t exist within the universe or transcendentally in respect to the universe. The latter is to say that if it played no role in the origin of the universe, even the assumption that it exists outside of the universe doesn’t lend support to its existence. With respect to this universe, it simply doesn’t exist.

The assumption of the argument is where much of this discussion will focus. The discussion will center around the question of what constitutes a *naturalistic* explanation. Thus, it will center around the question of what is meant by *natural*. Keith Augustine offers three definitions. The one he seems to accept is problematic given that supervenience lends credence to non-natural or even supernatural explanations of the mind. Though non-reductive physicalism states that mental states are contingent on physical states, mental states and physical states aren’t congruent. This implies that mental states are non-physical.1

Along these lines, a religionist who is, for example, a Cartesian dualist can argue that the soul is supervenient on the brain. She can argue that mental states are non-physical precisely because mental states are a property of the soul. For such reasons, non-reductive physicalism is to be rejected by one looking to provide a case for strict naturalism. A case for strict naturalism would entail reductive physicalism or reductionism–which agrees with the thesis of physicalism but adds that complex phenomena can be reduced to physical processes.2 This is to say, for example, that morality reduces to the mind of the moral agent. Reductionism could therefore be seen as the view that a given phenomenon is reducible to another phenomenon; alternatively, in the philosophy of science, reductionism is the view that one theory is reducible to another (e.g. Modified Newtonian Mechanics (MOND) is reducible to dark matter). This thesis runs into, at least, two problems.

The first of these issues is qualia–“the felt or phenomenal qualities associated with experiences.”3 This is often referred to as the *what it’s like*-ness of an experience. For example, a sharp pain in the foot, the smell of wet dog fur, and the taste of chocolate have a subjective quality that vary from one person to the next. These can only be accessed via introspection and are thus a marquee example of the so called privacy of consciousness. Qualia, however, aren’t as pervasive as the second problem. The second problem will therefore receive much warranted attention.

The second problem for reductionism is the purported existence of *abstract**a. **Abstracta* are abstract objects like propositions, letters, and numbers. Of these, arguably the most seriously debated are numbers. The debate between mathematical realists and non-realists should occupy one who is attempting a clear case for strict naturalism. If numbers exist, then at least one non-natural object exists; furthermore, this non-natural object isn’t reducible to anything physical. The existence of numbers would therefore refute reductionism.

Of the four criteria Otávio Bueno offers, two have been at the center of the debate: indispensability and explanation versus description. The indispensability criterion states that mathematics must be more than a useful part of an explanation; it must be indispensable to that explanation.4Mathematical realists don’t doubt that mathematics meets this criterion. The explanation versus description criterion states that mathematics, aside from describing a given phenomenon, must explain the phenomenon.5 On these two grounds, the nominalist has the most to say.

Mathematics, for example, doesn’t explain Kirkwood gaps. It merely provides a description for the relevant interactions between the gravitational tugs of Jupiter, the Sun, and asteroids in the asteroid belt. Briefly, Kirkwood gaps are regions within the asteroid belt that contain few asteroids; the distribution of asteroids in the belt are therefore non-uniform. There have been attempts to explain this non-uniformity mathematically (see Vrbik 2014).6 However, as argued by Bueno, proper interpretation is required before a mathematical description is relevant to the explanation of the phenomena.

This failure to explain Kirkwood gaps doesn’t harm the realist case, however. Realists have offered other phenomena that mathematics might explain: the hexagonal cells of beehives; the lifespan of cicadas, which is either 13 or 17 years–both of which are prime numbers; the bridges of Königsberg; the plateau soap film. Each phenomenon is explained semantically. Despite this, Mark Steiner suggested that when one “remove[s] the physics, we remain with a mathematical explanation—of a mathematical truth!”7

However, Baker [forthcoming] has argued that the proposal is false. And it is false for a very simple reason: there are mathematical explanations of empirical facts where the mathematics involved has no (known) mathematical explanation. Baker has argued convincingly, I think, that the example of the bees is a case in point—i.e., that the proof of the Honeycomb Theorem given by Hales [2001] does not explain the theorem. Therefore, whatever is doing the explanatory work, it isn’t the proof of the theorem.8

Steiner’s proposal has been replaced with program explanations. Conversely, program explanations make use of dispositions. John Heil provides an example:

Consider the dispositional property of being fragile. This is a property an object—this delicate vase, for instance—might have in virtue of having a particular molecular structure. Having this structure is held to be lower-level non-dispotional property that grounds the high-level dispositional property of being fragile; the vase is fragile, the vase possesses the dispositional property of being fragile,

in viture ofpossessing some non-dispositional structural property.9

A disposition is “not causally efficacious” and “ensures the instantiation of a causally efficacious property or entity that is an actual cause of the explanandum.”10 This is precisely what program explanations make use of.

A fragile glass is struck and breaks. Why did it break? First answer: because of its Fragility. Second answer: because of the particular molecular structure of the glass. The property of fragility was efficacious in producing the breaking only if the molecular structural property was efficacious: hence 3(i) [there is a distinct property G such that F is efficacious in the production of e only if G is efficacious in its production]. But the fragility did not help to produce the molecular structure in the way in which the structure, if it was efficacious, helped to produce the breaking. There was no time-lag between the exercise of the efficacy, if it was efficacious, by the disposition and the exercise of the efficacy, if it was efficacious, by the structure. Hence 3(ii) [the F-instance does not help to produce the G-instance in the sense in which the G-instance, if G is efficacious, helps to produce e; they are not sequential causal factors]. Nor did the fragility combine with the structure, in the manner of a coordinate factor, to help in the same sense to produce e. Full information about the structure, the trigger and the relevant laws would enable one to predict e; fragility would not need to be taken into account as a coordinate factor. Hence 3(iii) [the F-instance does not combine with the G-instance, directly or via further effects, to help in the same sense to produce e (nor of course, vice versa): they are not coordinate causal factors].11

Though this is a remarkable case, it doesn’t accomplish the type of explanation the realist is looking for. There are, however, other program explanations such as the explanation of the radiation emitted by a piece of uranium over a period of time (see Jackson and Pettit, Ibid.). This kind of program explanation is both indispensable and cannot be superseded by a process explanation–i.e. “a detailed account of the actual causes that led to the event to be explained.”12 So given program explanations, it looks as though the mathematical realist has won. Not so fast.

One must ask whether program explanations bear any relation to mathematics. In other words, one must question whether program explanations are mathematical and whether they meet the indispensability criterion. There are two options for the nominalist: Kim’s exclusion principle: a principle which “hinders the acceptance of two causal explanations for a single effect unless an acceptable relation exists between the two purported causes” or “view mathematics as playing a broadly representational role in scientific explanations.”[13][14] On the exclusion principle, the fragility example can be revisited:

The problem with this example is that there seems to be some avenue to a conceptual reduction of the two properties, “anyone who had access to the [molecular] account would have all the significant information at his disposal which is offered by the fragility explanation.” This, I think, again gives support to the heterogeneous nature of Jackson and Pettit’s examples; if a conceptual reduction of some kind is possible on a particular occasion then explanatory exclusion will effectively remove the program explanation in the same way that metaphysical reduction will remove the higher-level cause.15

If conceptual reduction of the efficacious and programming non-efficacious properties is possible, there is no longer room to assume that the non-efficacious property plays any role in the explanation. More pointedly, Juha Saatsi states the following: “For it seems that mathematical properties cannot ensure the instantiation of causally efficacious properties in any realist view of mathematics without some unduly *ad hoc* metaphysical connection being postulated between the concrete world and mathematical abstracta.”16 Given these epistemic and metaphysical difficulties, challenges are presented to one looking to defend mathematical realism. At best, the question of whether or not numbers exist in a Platonic sense remains to be seen. The fact that the floor is still open to this question shouldn’t hinder naturalism. Such an uncertain proposition–namely the existence of numbers–shouldn’t pose problems for naturalism. Though qualia were given less attention, the same conclusion applies. Therefore, reductionism is still a plausible thesis for one attempting a case for strict naturalism. It follows that *natural* is that which is physical or reducible to that which is physical.

Even if applied mathematics presented an issue for naturalism, pure mathematics says nothing about the world. It is concerned with objects, relations, and structures. These, however, are abstract. They’re not spatio-temporaral and aren’t causally active.17 Assuming that the existence of numbers caused a problem for naturalism, this wouldn’t lend any support to antithetical arguments. One, for example, wouldn’t be able to argue that the existence of numbers implies the existence of a god. For the same reasons one cannot assert that there’s a moral arbiter for morality, one cannot assert that there’s a programmer for program explanations or more simply, that there’s a being who designed a mathematical universe. This is to say, a being who designed the universe so that it adheres to laws fully explainable in terms of mathematics. Such a proposition would require justification.

There’s also the fact that in assuming that numbers exist, this existence could be entirely mind-dependent. This is to say that wherever there’s a being that is sufficiently intelligent, mathematical representations will not only emerge but might be required. Given, for example, the inferior fitness of *h.neanderthalensis*, one is justified in regarding them as less intelligent than *h.sapien*. Yet there’s evidence to suggest that neanderthals crafted and used tools; there’s also evidence to suggest that they produced art.[18][19] In the case of cave art, rudimentary mathematical thinking is required. For instance, a cave artist didn’t depict herself and aspects of nature (e.g. buffalos; trees) in *actual *size. She, instead, scaled down actual sizes, but still represented herself and her environment in an accurate manner. That is to say that, though she didn’t depict buffalos at their actual sizes, she still depicted them as larger than herself. This scaling down is possible evidence for rudimentary mathematical thinking. Therefore, if an intellectually inferior being is capable of thinking in this manner, it is reasonable to expect that an intellectually superior being is also capable of such thinking. When presented with given circumstances (e.g. predators that hunt in packs), the capacity to identify *multiple* threats becomes advantageous to survival. This, in turn, are the first fruits of mathematical thought. Mathematics, thence, could have a real *contingent *existence rather than, as the Platonists/Mathematical Realists argue, a real necessary existence. Thus, despite the strength of nominalism, a watered down realism could also dispel with the problem the existence of numbers would pose for reductionism and therefore, naturalism.

With a definition of *natural* now established and given the care taken in addressing possible problems, it is time now to turn to examples of naturalistic explanations for the origin of the universe. Prior to doing this however, it is necessary to point out that naturalism is the prevailing view in science. Sean Carroll states:

[I]f a so-called supernatural phenomenon has strictly no effect on anything we can observe about the world, then indeed it is not subject to scientific investigation. It’s also completely irrelevant, of course, so who cares? If it does have an effect, than of course science can investigate it, within the above scheme. Why not? Science does not presume the world is natural; most scientists have concluded that the world is natural because that’s the best explanation for what we observe. If you are ever confused about what “science” has to say about something, just ask yourself what actual scientists would do. If real scientists were faced with a purportedly supernatural phenomenon, they wouldn’t just shrug their shoulders because it wasn’t part of their definition of science. They would investigate it and try to come up with the best possible explanation.20

If a supernatural explanation presented itself, however, one should remember “that most naturalists would agree that naturalism at least entails that nature is a closed system containing only natural causes and their effects.”21 This is precisely what cosmological models present: a causally closed universe and explanations showing that the universe is self-contained. Even if one wrongfully assumes, like William Lane Craig, that the Borde-Guth-Vilenkin theorem yields evidence for an absolute beginning of the universe, when one considers that the theorem says that the ability to explain the universe classically gives out, such a theorem isn’t useful to argue for a beginning.22Even if we assumed, however, that this theorem implies an absolute beginning to the universe, this wouldn’t imply that a supernatural explanation is the only resort.

With this in mind, naturalistic explanations can be presented. The consensual theory, i.e. the Big Bang, will be discussed. The multiverse, which might be the mark of a paradigm shift, will also be discussed. Then an exotic possibility will be explored, namely that the universe is the product of a four-dimensional black hole.

Without surveying the history of the Big Bang, an outline of its properties can be presented: singularity, inflation, baryogenesis, cooling, structure formation, accelerated cosmic expansion. Stephen Hawking describes the singularity as follows:

At this time, the Big Bang, all the matter in the universe, would have been on top of itself. The density would have been infinite. It would have been what is called, a singularity. At a singularity, all the laws of physics would have broken down. This means that the state of the universe, after the Big Bang, will not depend on anything that may have happened before, because the deterministic laws that govern the universe will break down in the Big Bang. The universe will evolve from the Big Bang, completely independently of what it was like before. Even the amount of matter in the universe, can be different to what it was before the Big Bang, as the Law of Conservation of Matter, will break down at the Big Bang.23

Hawking later adds that “the Big Bang is a beginning that is required by the dynamical laws that govern the universe. It is therefore intrinsic to the universe, and is not imposed on it from outside.”24 Baryogenesis is the period in the early universe that resulted in the prevalence of matter over antimatter. It is useful to note here that this doesn’t make a difference given the assumption that the universe was created.

Because antiparticles otherwise have the same properties as particles, a world made of antimatter would behave the same way as a world of matter, with antilovers sitting in anticars making love under an anti-Moon. It is merely an accident of our circumstances, due, we think, to rather more profound factors…that we live in a universe that is made up of matter and not antimatter or one with equal amounts of both.25

Given this, the prevalence of matter over antimatter is arbitrary when assuming that the universe was created. This, however, serves as evidence against the notion since it serves as an example of chance in the universe. After the universe began to cool, stars, galaxies, and planets began to form. Expansion, as discovered by Edwin Hubble, is accelerating. What was accelerating the expansion of the universe was, at the time, unknown. Today, it is held that dark energy is responsible for the expansion of the universe and this, Sean Carroll states, is because dark energy is persistent and doesn’t dilute as the universe expands. It is, he explains, a feature of space itself and therefore constant throughout space and time.26

There have been recent suggestions that the universe is a dynamical fluid. Or, at the very least, the universe is a medium that appears to have more than three phases; in fact, as many as 10^500 and maybe even an infinite amount. So aside from curving and expanding, space could be doing something like freezing or evaporating.27

Inflation was intentionally set aside because it has received a lot of recent attention. On March 17th of this year, John Kovak, with the Harvard-Smithsonian Center for Astrophysics, announced the detection of gravitational waves. Initially, the BICEP2 collaboration had ruled out the possibility that cosmic dust in the Milky Way accounted for the polarization pattern in the Cosmic Microwave Background (CMB).

The BICEP2 collaboration, on June 19th, published a paper acknowledging that cosmic dust in the Milky Way could account for more of the b-mode polarization signal than previously thought. If the signals originate from primordial gravitational waves, this would be a smoking gun for inflation. Inflation theory proposes a short burst of exponential expansion in the early universe. This rapid expansion would produce gravitational waves.

It wasn’t long before the results came under fire and were eventually proven wrong. Charles Choi writes:

The controversy hinges on their handling of the dust emission, which relied on a preliminary map based on about 15 months of data from the European Space Agency’s Planck spacecraft. Falkowski noted the BICEP2 group might have misinterpreted the Planck data, thinking that it only contained emissions from the Milky Way when it also included unpolarized emissions from other galaxies. If the BICEP2 team did not account for this fact, they might have underestimated how polarized the foreground from the Milky Way actually was. This could mean the inflation signal the group thought it saw might only be a spurious result from Milky Way emissions.28

Gravitational waves would be a smoking gun for inflation. Though the Bicep2 results were eventually shown to be wrong, a recent suggestion could prove promising. A team, consisting of Nora Elisa Chisari, a fifth year graduate student with the Department of Astrophysical Sciences at Princeton, Cora Dvorkin with the Institute of Advanced Study at the School of Natural Sciences, and Fabian Schmidt with the Max Planck Institute for Astrophysics, is now proposing that weak lensing surveys may be able to detect the cross-correlation between b-mode polarization in the CMB and cosmic shear. The team suggests the possibility that cosmic shear is the best way to confirm primordial gravitational waves resulting from b-mode polarization in the CMB.

If inflation is empirically established, it would serve as indirect evidence for an inflationary multiverse. An inflationary universe, according to Tegmark, results in a Level I and Level II multiverse. Inflation would eventually end in parts of a rapidly expanding region, forming u-shaped regions. Each of these regions constitute a Level I multiverse while the amalgam of the universes constitute a Level II universe.29 Another way to imagine this type of multiverse is by imagining an enormous block of swiss cheese. As Brian Greene explains:

[T]he cheesy parts [are] regions where the inflaton field’s value is high and the holes [are] regions where it’s low. That is, the holes are regions, like ours, that have transitioned out of the superfast expansion and, in the process, converted the inflaton field’s energy into a bath of particles, which over time may coalesce into galaxies, stars, and planets. In this language, we’ve found that the cosmic cheese requires more and more holes because quantum processes knock the inflaton’s value downward at a random assortment of locations. At the same time, the cheesy parts stretch ever larger because they’re subject to inflationary expansion driven by the high inflaton field value they harbor. Taken together, the two processes yield an ever-expanding block of cosmic cheese riddled with an ever-growing number of holes. In the more standard language of cosmology, each hole is called a

bubble universe(or apocket universe). Each is an opening tucked within the superfast stretching cosmic expanse.30

Briefly, an inflaton field is a field corresponding to a given inflationary model. Like the Higgs field corresponds to the Higgs boson and an electromagnetic field corresponds to electromagnetism, an inflaton field corresponds to inflation. Some have written off multiverses as too hypothetical. However, Einstein’s general relativity suggested both an expanding universe and black holes long before there was evidence for either. Likewise, quantum mechanics (e.g. Everett’s Many Worlds Interpretation), string theory, and other equations suggest a multiverse. Some consider the multiverse the best explanation for the so called Fine-Tuning problem in physics. When the mathematics of such equations suggests as observable aspect of a given theory or a yet to be observed phenomena, it isn’t long before these phenomena are found to be actual.

The Everettian interpretation isn’t the only interpretation of quantum mechanics that results in a multiverse. Howard Wiseman, a theoretical quantum physicist at Griffith University, along with his team of colleagues, suggested the “many interacting worlds” approach. On this interpretation, each world is governed by Newtonian physics. However, given the interaction of these worlds, phenomena that are associated with quantum mechanics will arise. To test this approach, Wiseman suggested that the collision of two worlds could lead to the acceleration of one and the recoil of another; this would result in quantum tunneling. Wiseman and his team go through other examples, including the interaction of 41 classical worlds resulting in the type of phenomena observed in the double-slit experiment.31

It was suggested that the multiverse could mark a paradigm shift. This could be the case because the multiverse is able to explanatorily absorb Big Bang cosmology. In other words, inflation, for example, though a property of the Big Bang theory, isn’t well understood without introducing the inflationary multiverse. For Big Bang cosmology to remain the paradigm, inflation would have to be explained without recourse to the inflationary multiverse. The multiverse, aside from explaining inflation, via string theory, it can explain the behavior of dark energy. As aforementioned, as many as 10^500 phase changes of space have been proposed by string theorists.

As surveyed above, the Big Bang and the multiverse are self-contained, naturalistic explanations for the origin of the universe. There are, however, exotic explanations. One of the more recent suggestions offered by Niayesh Afshordi and his team is that the Big Bang was the result of a star that collapsed in a higher dimension. This four-dimensional star collapsed into a black hole. Interestingly enough, one of the properties of black holes is the singularity. The Big Bang, coincidentally, began as a singularity. “When Afshordi’s team modelled the death of a 4D star, they found that the ejected material would form a 3D brane surrounding that 3D event horizon, and slowly expand. The authors postulate that the 3D Universe we live in might be just such a brane—and that we detect the brane’s growth as cosmic expansion.”32 This, Afshordi argues, led astronomers to extrapolate back to the early universe and reason that it must have begun in a Big Bang. According to Afshordi, the Big Bang is a mirage.

This brief survey offers the consensual explanation: the Big Bang; a good candidate to shift the current paradigm: the multiverse; and an exotic explanation: the universe resulted from the collapse of a four-dimensional star into a black hole. The survey is by no means exhaustive. There are, for example, many inflationary theories (e.g. hybrid inflation; inflation as is related to loop quantum gravity). There are also a number of theorems (e.g. quantum eternity theorem). The feature they all share, however, is that they represent a self-contained universe–i.e. a causally closed universe.

A religionist may object and say that his god chose to work via naturalistic processes. William Provine offers a perfect reply to this notion:

A widespread theological view now exists saying that God started off the world, props it up and works through laws of nature, very subtly, so subtly that its action is undetectable. But that kind of God is effectively no different to my mind than atheism.33

This sort of suggestion is also in violation of Ockham’s razor or the principle of parsimony, which states that one shouldn’t multiply entities beyond necessity. Augustine quotes J.J.C. Smart:

“Ockham’s Razor does not imply that we should accept simpler theories at all costs. The Razor is a method for deciding between two theories that equally account for the agreed-upon facts” (Smart 1984, p. 124).

Augustine continues:

Given this basic heuristic, if we have no evidence for likely candidates for a supernatural event, we should adopt the simplest explanation for this fact–that only natural causes are operative within the natural world. If every caused event we have encountered can be explained in terms of natural causes, there is no reason to invoke supernatural causes that do no explanatory work for any particular events.34

Given this, such a god would be an added, unnecessary appendage. Attaching a god to naturalistic explanations doesn’t change the nature of the explanations. It doesn’t support the case for the supernatural. Therefore, given the naturalistic explanations surveyed above, creative agents are unnecessary. Attaching them to such explanations doesn’t make them necessary since they don’t lend support to the explanatory work. As argued earlier, if such gods are unnecessary, then they are also nonexistent with respect to the object purportedly created.

Ultimately, much is said about agnostic atheism. Agnostic atheism is an epistemic position, which disavows belief in gods but doesn’t claim to know whether or not they exist. Given The Argument From Cosmology, however, the fact that creative agents are unnecessary implies their nonexistence. We can know, with a high degree of certainty, that the universe is causally closed and therefore, self-contained. No outside influence can causally interact with or within the universe. It is therefore possible to know that gods do not exist. Therefore, from a much broader perspective, this argument lends strong support to gnostic atheism: an epistemic position which not only disavows belief, but claims knowledge, warrant, and justification. This implication makes for a much broader thesis that is perhaps worth exploring. Perhaps another time.

**Works Cited **

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2 “Reductionism”. 25 Nov 1999. Web. 5 Dec 2014.

3 Blackburn, Simon. The Oxford Dictionary of Philosophy. *Oxford: Oxford UP*, 1994. 301. Print.

4 Bueno, O. [2012a]: “An Easy Road to Nominalism”, *Mind 121*, pp. 967-982. Web. 5 Dec 2014. Available on Web.

5 Ibid. [4]

6 Vrbik, Jan. “Mathematical Exploration of Kirkwood Gaps”, *Mathematical Journal 14*. 2014. Web. 5 Dec 2014. Available on Web.

7 Lyon, Aidan [Sept 2012]. “Mathematical Explanations of Empirical Facts, And Mathematical Realism”. *Australasian Journal of Philosophy, Vol. 90, No. 3*, pp. 559–578. Web. 6 Dec 2014.

8 Ibid. [7]

9 Heil, John. Philosophy of Mind: A Contemporary Introduction 3rd Ed, p. 211. London: Routledge, 2013. Print.

10 Ibid. [7]

11 Jackson, Frank and Pettit, Phillip [Mar 1990]. “Program Explanation: A General Perspective”. *Analysis, Vol. 50, No. 2*, pp. 107-117. Web. 6 Dec 2014. Available on Web.

12 Ibid. [7]

13 Cooper, Wilson [Jun 2008]. “Causal Relevance and Heterogeneity of Program Explanations in the Face of Explanatory Exclusion”. *Kritike Vol. 2, No. 1*, pp. 95-109. Web. 6 Dec 2014. Available on Web.

14 Saatsi, Juha. “Mathematics and Program Explanations”. *Australasian Journal of Philosophy Vol. 90, No. 3*, pp. 579–584. Web. 6 Dec 2014.

15 Ibid. [13]

16 Ibid. [14]

17 Ibid. [4]

18 Tarlach, Gemma. “In Europe, Neanderthals Beat Homo Sapiens to Specialized Tools”. *Discovery Magazine*. 12 Aug 2013. Web. 6 Dec 2014.

19 Than, Ker. “World’s Oldest Cave Art Found—Made by Neanderthals?”. *National Geographic*. 14 Jun 2012. Web. 6 Dec 2014.

20 Carroll, Sean. “What is Science?”. *Preposterous Universe*. 3 Jul 2013. Web. 6 Dec 2014.

21 Ibid. [1]

22 William Lane Craig and Sean Carroll, “God and Cosmology (33:33)”. *YouTube*. *YouTube, LLC*. 3 Mar 2014. Web. 6 Dec 2014.

23 Hawking, Stephen. “The Beginning of Time”. ND. Web. 6 Dec 2014.

24 Ibid. [23]

25 Krauss, Lawrence. *A Universe From Nothing: Why There Is Something Rather Than Nothing*. 1st ed. New York, NY: Free Press, 2012. 61. Print.

26 Carroll, Sean. “Why Does Dark Energy Make the Universe Accelerate?”. *Preposterous Universe*. 16 Nov 2013. Web. 6 Dec 2014.

27 Tegmark, Max. Our Mathematical Universe: My Quest For the Ultimate Nature of Reality, p. 135. New York: Alfred A. Knopf, 2014. Print.

28 Choi, Charles. “Will the Bicep2 Results Hold Up?”. *The Nature of Reality*. *PBS*. 27 May 2014. Web. 6 Dec 2014.

29 Ibid. [27], p.133-134

30 Greene, B.. The Hidden Reality: Parallel Universes and The Deep Laws of the Cosmos, p.56-58. New York: Alfred A. Knopf, 2011. Print.

31 Hall, M. J. W., Deckert, D. A. & Wiseman, H. M. [2014]. Quantum Phenomena Modeled by Interactions between Many Classical Worlds”, *Phys. Rev. X 4*. Web. 6 Dec 2014.

32 Merali, Zeeya. “Did a hyper-black hole spawn the Universe?”. *Nature*. 13 Sep 2013. Web. 6 Dec 2014.

33 William Provine as quoted in Strobel, Lee. The Case For A Creator. Grand Rapids, Mich.: Zondervan, 2004. 26. Print.

34 Ibid. [1]