Cosmological Arguments

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Cosmological Arguments

The Cosmological Argument is founded on a basic human intuition that there is a reason for everything [1], and this is why it is very ancient. It tries to show that searching for the ultimate reason of things leads to God as this reason [2]. The earliest surviving formulation is by Plato (4th century BCE). In his book ‘’Laws’’ (chapter 10), Plato argues for the existence of god on the basis of movement - while tracking down the origin of movement, he finds a proof for the existence of the gods [3]. His student Aristotle provided a more detailed argument in his book ‘’Metaphysics’’, which greatly influenced the development of the argument in later generations. Christian and Muslim philosophers developed the argument in two new directions. First, they developed arguments that did not depend on a causal relation in time - that weren’t founded on looking for a cause that preceded the event in time (e.g. deciding to move the hand causing the hand to move), but rather one that preceded it in some other sense. Secondly, they developed variants that did not depend on rejecting an infinite chain of causes. These developments culminated with the formulations presented in the ‘’Summa Theologica’’ of St. Aquinas in the 13th century, and in the arguments made by the Muslim school of the Kalam, like those made by Al-Ghazali (in the 11th century). The next significant development occurred only at the 17th century, when Leibniz formulated the Principle of Sufficient Reason, which provided a clearer and more general formulation for the arguments that did not rely on a temporal causal relation.

In the 18th century the argument suffered two critical blows, from which it had never recovered. First, the Scottish philosopher David Hume attacked the causality principle that the argument stands on. In addition, Immanuel Kant attacked the possibility of doing metaphysics is general, and of talking about a Necessary Being in particular. In more modern times, the work of Georg Cantor (late 19th century) showed how to deal with infinities mathematically, undermining the arguments for rejecting an infinite chain of causes. Modern science also shed much light on the structure of causality and the nature of time. Scientific progress has, however, also led to the discovery of the Big Bang, which contributed a scientific argument supporting the belief that the past is finite - which is a key assumption of the Kalam cosmological argument.


Philosophical Overview

Nearly all Cosmological Arguments are of the following structure [4]:

(1) The Chaining Rule: For everything of type (A) there is a “reason”, in the sense (B) which includes the existence of another thing of type (A).
(2) The Existence Claim: There is a thing of type (A).
(3) An Argument Against Infinity: The chain of reasons of (A) must end with a first cause, because of argument (C).
(4) The Identity Assumption: This first cause is God.

Different arguments are set apart by what they fill in for (A), (B), and (C). The argument from movement, from example, would fill in “movement” for (A).

Accordingly, there are five possible objections to the Cosmological Argument [5]:

(I) The Glendower Problem: Things of type (A) don’t have a reason, or at least don’t always have a reason in the sense (B).
(II) The Existence Problem: There isn’t anything of type (A) in existence; type (A) things just don’t exist.
(III) The Regress Problem: There is no good reason to reject an infinite chain of (A).
(IIIb) The Taxicab Problem: There is a contradiction between proposition (1), which implies the existence of an infinite chain of (A)s, and premise (3), which denies the existence of such a chain.
(IV) The Gap Problem: Even if a first cause exists, it isn’t God.

Depending on the content of (A), (B) and (C), different objections would be more or less relevant. Generally the gap problem (IV) is the most severe, and relevant even to the most careful formulations of the argument. It is also, in a sense, the only relevant one - atheism doesn't necessarily imply that there is no First Cause, only that this cause isn't god. Despite this, it is often the case that the other premises are also incorrect. The argument is usually put forward as addressing things that are commonly believed to exist, so the existence problem (II) is rarely important - but it can be relevant in some cases, for example in arguments that discuss creation out of nothingness. All other premises are usually suspect at best.

We will divide the detailed discussion to five parts, according to the problems and historical development. First we will present the most ancient argument, the argument from motion. We will use it as a background to discuss relevant issues, so its analysis will be somewhat more extensive than what is strictly necessary. We will then turn to discuss other formulations of the argument (the argument from causality, contingency, sufficient reason, and a finite past), but for each we will discuss only the particular aspects peculiar to that argument and refer the reader to the discussion of the argument from motion on more general points.

An Unmoved Mover

In his book Summa Theologica (“all that is known on god”) the Dominican friar Thomas Aquinas [6] (1225-1274) presents “five ways” to prove the existence of god. The first three are three cosmological arguments - from motion, causality, and contingency. We will deal with all of them in the order they appear, and start with the argument from movement, or motion:

- Thomas Aquinas, Summa Theologica (First Part, Second Question, Article 3)

The argument from “movement” can be understood in two ways. It can be understood as an argument from a metaphysical understanding of movement, emphasizing the Aristotelian philosophical understanding of causality that underlies Aquinas' reasoning. This kind of argument will be considered below, under the First Cause section. However, Aquinas' argument is arguably simply an argument from physical change, understood in light of physics. Aquinas’ formulation is based on Aristotelian physics, because that was the "science" of his day. Now we know better, and we can attempt to reformulate the "Argument from Phyiscs" in modern terms. This is the argument we will deal with in this section.

If we ignore errors caused by time [7], we could perhaps rephrase the “first way” approximately like this [8]:

(1a) Every state of affairs (A) is the result (B) of a previous state of affairs that is the cause of the present one (i.e. the existence of the previous state led to the existence of the present one and not a different state or no state).
(2a) There exist states of affairs, that is there are different states of affairs at different times.
(3a) The chain of changing states of affairs must begin with some initial state that is not caused by a prior state, for (C) without such an existing initial state there would be nothing that will cause any state to exist.
(4a) This initial state is God.

The phrasing of (1a) may look cumbersome, and apologists usually prefer to use simpler formulations like “everything happens for a reason”. But this phrasing is at least somewhat scientifically based, so it makes for a better starting point. Less careful phrasings (like “everything happens for a reason”) would be dealt with within the discussion of the argument from causality.

Traditionally the conclusion of the argument is known as an Unmoved Mover, since the argument seems to talk about movement. But Aquinas actually discussed any type of change (not necessarily movement), and similarly we arrive at an “initial state” instead of an “unmoved mover”. Our formulation shows the absurdity of the argument by identifying this initial state with god, but this is the conclusion Aquinas’ argument (and many others’, once carefully formulated) actually leads to. To deduce something that lies beyond the chain of states (or moving objects etc.) and that forms a reason for the existence of the first state requires a wider chaining rule (1), and we will discuss such eventualities in the treatment of the argument from sufficient reason, below.

Even atheists don’t doubt the existence of physical states, so the existential objection (II) doesn’t apply to this case. All other objections, however, are important.

The Glendower Problem (I)

In his play “Henri IV”, Shakespeare gave the following immortal lines to the players:

Glendower: I can call Spirits from the vastie Deepe
Hotspur: Why so can I, or so can any man: But will they come, when you doe call for them?

The chaining rule - assumption (1) in any cosmological argument - suffers from the problem of poor Glendower. The argument from motion assumes a causality principle, which conjures up past reasons for the existence of the present state. But are there reasons for its existence at all, and if so - are those the ones that the principle raises? Why should we believe the suggested causality principle? This is the Glendower problem. In general one can offer two kinds of justifications for the causal principle: philosophical and scientific justifications.

The Philosophy of Causality

Presumably people have always raised causal arguments, but the philosophical analysis of causality essentially begins with Aristotle (384-322 BCE). Aristotle’s causal theory is too extensive and complicated to be worth describing here, but it is enough to mention that it is founded on exploring the different ways one can answer the question “why?” (as in “Why was this brick building built?”). Many philosophers continue to our day to proceed in similar analysis of the concept of causality and characteristics of satisfying answers, but this is not what we are looking for - we are looking for an argument for why we should hold a particular causal principle, like the chaining rule (1a). Most of the philosophical treatment of causality is therefore irrelevant, and we will not cover it.

The justification of causality can be divided to before and after David Hume (1711-1776). Before Hume it was taken for granted that causal laws, like (1a), existed, and we could derive them from reason and experience. David Hume pointed out that there is no way to derive a causality principle from pure reason, as there is no self-contradiction in it failing to hold. On the other hand, in fact we never see the causal connection itself, only the regularity of events. We see that we push an object and that it moves, but we don’t actually see that the push is the cause of the movement. We can conjecture that it is the cause, but we will always remain with the theoretical possibility that we were wrong, that one day we will push and the object will not move [9]. There is, therefore, no philosophical justification for any causality principle at all - only a scientific one. This conclusion of Hume still holds, despite the attempts of substantive opponents to undermine it.

The most significant attempt to provide a philosophical justification for a causal principle was given by Immanuel Kant (1724-1804). Kant effectively claimed that the structure of human consciousness compels us to see the world through “glasses” of cause and effect, so we impose a description in terms of cause and effect onto the world. But this explanation does not suffice. It is true, as Kant maintained, that Reason is only possible if one can apply the categories of cause and effect (for otherwise one cannot think, make plans, learn from experience, and so on) - but this shows only that reason is not prior to causality, that it is possible only given sufficient causality (causality applying to its own workings and its relations to the world). This does not imply that causality applies to all physical states of affairs (as we will see below, it doesn’t).

Philosophy is hence unable to justify causality at all or principle (1a) in particular. We must now turn to science.

Causality in Light of Science

The chaining rule as we formulated it in (1a) does not fit what we know scientifically. However, it can be easily changed to a weaker principle, that does fit. We will soon see, however, that this weakened principle is insufficient to carry the rest of the argument.

There are two big problems with the phrasing of (1a). The first is that it talks about causality as a relation of certainty - given a certain state of affairs at time t0, it would with certainty lead to some other state of affairs at a later time t1. Such a certain relation is known as Determinism, and is rejected by modern physics - in particular, by Quantum Theory [10]. Quantum theory teaches us that in general events will occur at a probability, and the effect of a past event (or state of affairs) can increase or lower this probability but would not usually change it to 0 or 1. The putated event at time t1 will hence occur only if there would be enough luck, for there is a random element in the laws of nature. This is known as Indeterminism. It is therefore widely accepted that some event should be seen as the “cause” of another event if it raises its probability.

In daily situations we don’t notice the probabilistic nature of causality, because daily phenomena occur in such high probabilities that they have no exceptions in practice. When you throw a rock on the wall, it will always collide with it. But the importance of probabilities increases as we go back in time, for the probabilities multiply [11] and occasionally an event with low probability will be involved. For example, the emergence of a particular evolutionary lineage depends on the occurrence of a particular mutation, whose probability can be very low - so that the appearance of a lineage that is very significant in the history of life on our planet can be very unlikely. As you go back enough in time you will find such random events in the history of every event or object, so everything that matters (the existence of our sun, for example, or of the human race) is in fact very improbable and the result of luck more than of causes.

In other words, every state of affairs is the consequence of a preceding state of affairs and the laws of nature, and in the final analysis the state we find ourselves in is due to luck and without any real cause.

The second glaring problem with the phrasing of (1a) is that it talks about a causal relation in time [12]. Understanding time is not easy, but in this context it is important to notice that the physical and philosophical understanding of time does not fit our intuitive notion that the future does not yet exist, and that time “flows” from the past to the future. Philosophically, we can see the problem as follows: when we think of the flow of time we are thinking of something like the change of a certain state (the “present”) as time flows. But change is already a phenomena in time! As an analogy, we could think of a cop shining a flashlight on the poles of a picket fence [13], passing one pole after another as time passes. The movement of the flashlight to shine on the next pole leaves the past poles in the dark, and similarly the flow of time leaves the moment that was once present in the past. But when we think of the movement of the flashlight (of the flow of time) we are already thinking in terms of time, so clearly we have not explained how time flows but merely moved to a kind of other “time”, a “higher” time - time in which the flashlight moves (or the moment that is the present changes), instead of time as poles on the fence (or moments in time). This is no solution. In short, thinking in terms of the flow of time is incoherent - time cannot flow.

One can reach the same conclusion from scientific considerations. Within Einstein’s relativity theory it is impossible to describe the state of affairs in a certain moment so that that state will be the “present” of all observers included in that state. We can imagine to ourselves that time “moves”, which means that some state of affairs that is the present for a particular observer is the one that is coming into existence and fading into the past. But this is enormously arbitrary - why is this observer the one for whom time moves “correctly”? Why not some other observer? Because of the immense size of the universe, even small differences in velocity can cause huge differences in the "present time" of different observers (such as a difference of days in the state of the Andromeda galaxy between the “present” of a pedestrian and a car driver [14]). The scientific assumption is the Copernican assumption [15], that there is no preferred observer but rather that all places and observation perspectives are fundamentally equal. Because differences in velocity will introduce different times of the same bodies (like the Andromeda galaxy) into the present for different observers, the Copernican assumption leads us to assume that these different times are equally real. All places and times exist equally, whether they are in our past, present, or future.

“...for we physicists believe the separation between past, present, and future is only an illusion, although a convincing one”
- Albert Einstein

For these reasons [16] most philosophers and scientists (that are interested in these topics) prefer to believe in the Block Universe, also known as Eternalism - the view the universe exists atemporally, an existence that contains our past, present, and future as parts of it. The flow of time is merely our subjective experience, the past and the future exist just as much as the present does. Instead of thinking of the universe as some huge three-dimensional container whose contents change with time, think of it as a four-dimensional container that the difference between different parts of it (along a curved line in the four-dimensional space) is what we call “change in time”. The line that allows us to compare the different parts is determined locally by the curvature of space-time and the observation point we choose.

This four-dimensional picture points us to an arbitrary restriction in principle (1a) as we phrased it - our formulation assumes that there is a particular direction in the four-dimensional space, the axis of time, along which we would measure the change. But why continue specifically along this direction? The laws of physics discuss continuing in all directions. Because the direction of time is set by the curvature of space [17], the laws of physics even discuss continuity where there isn’t a single time axis. It might also be possible that there might be areas in space-time without any time axis at all, but still with continuity.

The question of initial conditions is therefore converted into the question of boundary conditions - instead of talking about the end of the chain of events in time, it is better to talk about whether the universe has a rim in four-dimensional space, and how does it look like. Howver, it must be emphasized that time is still a unique direction in space time because the distribution of matter sets a particular direction in space-time as "time". It therefore still makes sense to ask specifically about the continuity in the time-direction, if one so chooses.

In short, we showed the apologist doesn’t have a good reason to believe in the chaining rule (1a) as we phrased it. However, he can still hold to a similar chaining rule, which is the continuity of space-time. To the best of our knowledge, space-time does form a continuous four-dimensional body, with statistical regularities. This description holds up to the boundary of spacetime, and the question of first cause is hence converted to the question of what does the boundary of spacetime looks like, if indeed it has one; if he chooses to, the apologist can concentrate on the boundary in the time-direction, although that is somewhat arbitrary.

The Regress Problem (III)

As stated above, the argument (3a) against the infinity of the chain of events in time was that we must assume the existence of a primal state whose existence will allow it to cause the states following it. However, the discussion of the Glendower problem revealed that existence is four dimensional. There is therefore no magical “causal power” in a physical state that causes, in an undescribed way, the emergence of the following state. All that there is to a physical state is existence, and all states (past, present, and future) exist. The laws of physics don’t describe the magical causal forces physical objects have beyond their existence, but rather the (probabilistic) order of things in space-time. There is therefore no force to argument (3a) - the First State doesn’t magically cause the existence of what follows it anyways, it and every later state just exist equally.

Modern apologists usually prefer to raise another argument against the infinity of the chain - they maintain that an infinite chain of causes is simply not possible, because an actually existing infinity (of anything) is impossible. To argue this, they bring up mathematical paradoxes related to infinity. For example, William Lane Craig writes [18]:

"...if the series of past events is actually infinite, then the number of events that have occurred up to the present is no greater than the number that have occurred [up to] any point in the past. Or again, if we number the events beginning in the present, then there have occurred as many odd=numbered events as events. If we mentally take away all the odd-numbered events, there are still an infinite number of events left; but if we take away all of the events events greater than three, there are only four events left, even though in both cases we took away the same number of events."

Notice, however, that there is no real mathematical or logical contradiction being pointed out here. The consistency of the existence of an actual infinite can be realized even by those not initiating into the mysteries of the mathematics of infinity [19] by replacing all reference to the “events” in the above paragraph with ones to the integer numbers. For the integer numbers too (including all negative numbers) the number of numbers “up to” number n is infinite, as is the number up to some smaller number n-x; there are “as many” odd-numbered negative integers as there are negative integers (since one can number the odd-numbered negative integers as “-1”, “-2”, ...), and so on. Yet there is no contradiction, we can work with negative numbers just fine.

Furthermore, our physical theories already involve actual infinities, in numerous ways. Each finite volume is made up of infinite pieces, each time segment has an infinite amount of moments, the mass of matter is made up mostly of an infinite amount of “virtual” particles [20], and more.

Why is it then that the above paradoxes feel so strange? Because we are not used to handling infinities. We are used to handling finite amounts. We are used to counting, so if we count to “infinity” then that’s a number, right? So it must be smaller than “infinity plus four”. But as the mathematicians showed, that’s not how infinities work. Counting simply doesn’t work like that when infinities are involved.

Our physical theories are the best descriptions of reality that humanity could devise thus-far. They include infinite quantities, in various forms. This includes infinite chains of moments in time, and may even include an infinitely long past. None of that harbors a contradiction, so there is no good reason to reject such descriptions. The fact that you can’t apply intuitions from finite counting to counting an infinity of moments, say, says nothing except that such intuitions are not applicable. This doesn’t imply that there isn’t, as a matter of fact, infinitely many moments. The argument against infinities is just an appeal to your unlearned intuitions and prejuduce, it is not an argument at all [21].

It appears, then, that there is no argument (C) to reject an infinite chain of causes. There is no contradiction involved, and no metaphysical need for a first cause to magically bring the rest of the chain into being. Premise (3a) is false. This may initially appear to be a serious blow to the Cosmological Argument, as it seems to imply that there is no way to deduce the existence of a first cause. That would be a premature conclusion. The Cosmological Argument doesn’t really need to reject an infinite chain, all it needs to do is assert that the chaining rule implies a first cause. How can this be done without denying an infinite chain? We’ll see in the next section.

The Taxicab Problem (IIIb)

{Needs Work}

In discussing the regress problem we rejected the idea that there is a an argument (C) that proves the First State must exist. However, let us assume for the sake of argument that such a First State does exist. That would only land the argument into a contradiction! For principle (1a) maintains that every state is followed by a prior one, and the First State would hence imply an earlier one. The chaining rule (1) in the cosmological argument seems to work like a taxicab, that once not needed is sent away. This is the taxicab problem.

There are generally two ways to deal with the taxicab problem. The first is to change the chaining rule (1) to allow for other types of causes - such as uncaused causes. The paradigmatic example is libertanian free-will - presumably, a person lifts his hand out of his own free will, not because of the causal laws of physics. In this section, however, we are discussing the argument from movement, that is from well-established physical change, described by our physics. Uncaused causes are not included in our physical theories, so we will defer dealing with this line of reasoning to the discussion of the argument from causality.

The other way is to phrase the chaining rule (1a) in such a way that it will produce prior states generally but also boundaries on space-time. Note that the whole point is to prove that there has to be boundaries - the argument from movement attempts to show that movement (physical change) implies that there has to be an unomved mover, not that there might be an unmoved mover as a matter of fact. As we will see below, this direction is more phyiscally-grounded but is philosophical folly as, simply put, there is no god in physics; to derive god, you hence have to change your assumptions into (unsupported) conjurations of godly-like causes, i.e. to adopt the first way.

The existence of a boundary in the time direction was indeed proved by the great physicist Stephen Hawking in the 1960s. The Penrose-Hawking Singualrity Theorems proved that our universe should have a singularity in its past - a point that lies in the past of any and all things and that is a boundary on space time. Space-time is built like a cone, with all events stemming from this singularity point. It appears, then, that the apologist is vindicated - time is indeed bounded in the past! There is an Unmoved Mover, which is the Singularity! Well, not so quickly.

First, the existence of the singularity in practice is suspect at best. The singularity theorems are proved within the theory of General Relativity, but under the prevailing conditions one has to consider quantum effects. No one knows how to do that rigorously, but it is generally believed that quantum mechanics would not allow a singularity. This is even the opinion of Hawking himself. There are several possibilities on what the real structure of existence may be. These include the idea that there are no boundaries at all, or that there is a sort of "fuzzy" boundary which is not a singularity. While a singularity or boundary are still possible, there is no good scientific reason in the present to suppose them. This undermines the apologist's argument as this means there is no way to prove the existence of an "Unmoved Mover" or "First Event".

The second problem relates to the way this result solves the taxicab problem. Remember that in physics, just like the apologist required, every event is preceed by an earlier event. And yet, the singularity is not preceded by anything! How is this possible? How is it consistent? Because the singularity isn't an event at all. The singularity is the limit of physical events. For an analogy, consider the series of numbers 1/1, 1/2, 1/3, 1/4... Each number in this series is followed by a smaller number, just like any event has a preceding event. If we follow the chain of numbers, it is easily seen that "at infinity" we reach the number 0. There is no actual Smallest Number withint the series, however, and this is precisely why every given number in it is followed by a smaller one. The same is true for the singularity - it isn't a point in space-time, but rather a boundary of it. The singularity isn't a First State, there isn't a First State. It isn't a state at all. Rather, it is the limit of states, with each state preceded by an earlier one.

Thus, even if there is a singularity it doesn't really "exist" as such. What exists are the states of the universe, not its boundary conditions. It is therefore incorrect to state that the singularity is the First State; there is no first state.

We can conclude that there is no satisfactory way to resolve the taxicab problem. If every state has a preceding state, then there is no first state. There could still be a boundary to the length of the past, but it is currently unclear whether this is so. Assuming such a boundary exists, however, does it matter that it isn't a First Event? Is there any sense in concluding god exists from the existence of the singularity? This is the topic of the next section, the Gap Problem.

The Gap Problem (IV)

{To Be Written}

Summary on the Argument from Movement

We saw that dogmatically asserting a chaining rule (1a) is not reasonable, but that there is a reasobable scientific basis to assume the continuity of space-time up to its boundaries, if there are any.

The assumption of the existence of physical states (2a) is accepted by all, and so does not merit discussion.

The arguments (C) in (3a) against an infinite chain are not compelling, but it is possible that there are boundaries and thus the chain is not infinite - we aren’t sure yet. The presumption to show that on one hand the chain must continue (1a) and on the other that it must end (3a) is a contradiction, making the argument invalid, but this still does not preclude boundaries on the past.

But the most severe problem of the argument from movement is the gap problem. Even if boundaries exist, they won’t constitute god (4a) or even indicate his existence.

The argument from movement is hence completely falsified. A First Event impossible under the chaining rule, and even if one is content with contemplating a boundary instead the argument doesn't prove one exists. And even if one assumes the existence of a single boundary on the past (making most of the argument redundant), this does not constitute God nor indicates it in any way.

First Cause

The argument from causality.

A Necessary Being

The argument from contingency

Sufficient Reason

The argument from the principle of sufficient reason

Finite Past

The Kalam Cosmological Argument

Footnotes and References

[1] This may be a unique characteristic of man, separating us from other species. In an experiment conducted in 2001, researchers examined both Chimpanzee and human children playing with blocks. The researchers introduced blocks which appeared ordinary but were unbalanced, so that they would lead to the collapse of towers. It was found that human children examined these blocks, to ascertain why they caused the towers to collapse, but Chimpanzees didn’t (Chimpanzees did examine blocks that appeared different from the ordinary). It may be that only humans seek out hidden causes. See Daniel J Povineli and Dunphy-Leli, ‘’Do Chimpanzees seek explanations? Preliminary comparative investigation’’, Canadian Journal of Experimental Psychology, 55(2):185-93, 2001.

[2] Four our purpose of counter-apologetics, it is useful to group together arguments with a particular form or underlying rationale. We therefore consider "Cosmological Arguments" to consist of arguments with the general form given below, or at least those (like the Kalam Cosmological Argument) that are sufficiently similar. More broadly, some philosophers consider cosmological arguments to consist of all arguments that refer to features of the cosmos, or only to arguments that refer to the existence of the cosmos, or so on. All the arguments we name "cosmological arguments", however, are based around the idea that there is a reason for everything and that the ultimate reason is god. These include all the classical and inflencial cosmological arguments.

[3] To be precise, Plato concludes that movement ultimately originates in the action of at least two kinds of souls, one being the source of all evil and the other the source of all good.

[4] This scheme is based on the extensive lecture notes that are provided by Robert C. Koons free on the Internet, for his course “Western Theism”.

[5] We largely follow the scheme of Alexander R. Pruss in his article ‘’Leibnizian Cosmological Arguments’’, with slight changes.

[6] “Aquinas” actually refers to his place of birth and origin, the county of Aquin or Aquino in Italy. He is known also as Saint Thomas Aquinas, Thomas of Aquino or Aquin, and in Catholic circles also as The Angelic Doctor or the Universal or Common Doctor. He established the theological school of “Thomism”, which to this day dominates the theology of the Catholic church.

[7] The problems in Aquinas’ formulation in light of modern of physics are numerous. The main problem is that Aquinas presumes that something changes from possibility (potentiality) to actuality in regards to a certain property (such as heat) by something already in that state (already hot). We know that this isn’t so - a compound can combust on contact with air, for example, regardless of temperature. We therefore preferred to abandon entirely all talk of potentiality and actuality, and speak instead on the change of physical states, without specifying what is it about the state at a given time that causes the fact we are interested in (like heat) to emerge in situations at a later time - it is enough that something in the present state causes a later effect, it doesn’t matter what specifically that something is.

[8] This is not an entirely rigorous phrasing, as it is meant to clarify the argument rather than present it as a logical deduction. Rigor-minded readers will have to formalize it further for themselves.

[9] This problem is known as the Problem of Induction. Hume essentially showed that any causality principle is entirely unjustified - you can’t deduce it from pure reason, and you can’t deduce it from experience since you’re relying on causal laws to make such an induction from experience. Hume concluded that while you must, in practice, assume a Principle of Uniformity of Nature, it remains unjustified. The most important answer to this problem is that of modern philosophy of science, in which the Principle is held as an aesthetic of choice between different explanatory theories, while belief in particular causal laws is determined not by induction but rather by fallibility - we trust those causal principles that we can’t seem to falsify. Note that this doesn’t resolve the problem, it only means that our thinking will work in a self-correcting manner if nature is sufficiently uniform.

[10] At this point I adopt the most accepted interpretation of quantum theory, namely the Copenhagen Interpretation. Indeterminism holds in the other interpretations too, except hidden variable interpretations like Bohmian mechanics - under hidden variables ontology the universe is deterministic and the apparent indeterminism in the laws of nature reflects only our ignorance of the values of the hidden variables. The truth of the matter doesn’t matter for the philosophical consideration - the fact that we can think of a universe operating in a Copenhagen-like (indeterministic) manner is enough to doubt (1a) as we phrased it.

[11] Since the probabilities of unrelated events are not dependant on each other, the probability for both occurring is the multiplication of each of them and hence smaller. If, for example, the probability of event A is 0.9 and of event B is 0.99 then the probability of both occuring is 0.891. The calculation of the probability for an actual event conditioned on actual other events can be far more complicated, but the phenomena of the lowering of the probabilities as time and the number of events increases still generally holds.

[12] It should be noted that Aquinas affirmed also a-temporal or backwards causation, basing his chaining rules on “first philosophy”, i.e. on philosophical considerations. But a chaining rule based of physical change can now only be justified by science, so we chose to start with the simple science-inspired form (1a). We will deal with a causal order that is not temporal in the discussion of the argument from contingency.

[13] This analogy was given by C.D. Broad. It should be noted that he didn’t see it as an insurmountable difficulty, as he thought time can be considered as a stream of coming-into-being. See “Being and Becoming in Modern Physics” in the Stanford Encyclopedia of Philosophy.

[14] This argument from relativity is known as the Rietdjik-Putnam Argument. The calculation of the state of the Andromeda galaxy specifically was done by Roger Penrose, in order to forcefully present the argument.

[15] The Copernican assumption is that we are not in a special place, such as the center of the universe. In other words, we assume that the general order we find around us - the basic laws of nature, and even the structure of the solar system and so on - are fairly typical. The assumption is not that we are not in a relatively rare place, but rather that there is nothing special in our location in a broad enough picture of reality.

The principle is named after Nicolaus Copernicus, who preached for the acceptance of the belief that the earth revolves around the sun ( rather than vice versa), and thus removed the earth from its central position. It should be noted that his heliocentric theory was just as complicated (requiring far more than a few perfect circles) and not more predictive than the old geocentric Ptolemaic system. The truly superior heliocentric model came with the work of Johannes Kepler, who abandoned circular orbits (which Copernicus favored for theological reasons) in favor of elliptical ones.

[16] There is actually another significant argument, given by J.M.E McTaggart. McTaggart noted that we speak about time in two ways: we can speak of time in terms of past, preset, or future (which he called “A Series”) or in terms of before or after (which he called “B Series”). McTaggart claimed that the A-Series is inadmissible since a change of properties (future becoming past) means that a particular event has contradicting properties. His arguments convince many philosophers, but are technical and appear to me to be related more to metaphysics and issues in communication and language so I did not present them.

[17] There are actually two relevant types of time. In General Relativity the time axis of the coordinate system that describe space-time is set by the curvature of space-time, in such a way that it is only well-defined for flat space-time. However, one can always define such an axis locally [in technical terms, there is local Lorenz covariance], so every object will see its personal “subjective” time. In curved space, i.e. space with matter, different objects will see their subjectives times as pointing at different directions in the four-dimensional space-time. The time that an object feels as passing relates to the distance along such a locally-defined time. As hinted at in the main text, in certain continuations of general relativity there may be regions where an object will feel no time at all, i.e. where it is not possible to locally define time.

[18] William Lane Craig is one of the leading current Christian apologetic, famous principally for his defense of the Kalam Cosmological Argument and his great rhetorical skill. The quote is taken from his article The Kalam Cosmological Argument in The Blackwell Companion to Natural Theology, William Lane Craig, James Poerter Moreland (Ed.) (2009)

[19] The mathematical treatment of infinity came to maturity with the work of Georg Cantor (1845-1918), who showed how to treat and count (to the degree this is possible) infinities based on set theory. The existence of infinities is the most contested area of the philosophy of mathematics, and many famous mathematicians and philosophers objected to them. Nevertheless, infinities and the quantities associated with them (such as Real numbers) form an indispensable part of modern mathematics. The work of Kurt Godel (1906-1978) showed that it is impossible to prove that axiomatic systems that include infinities (such as axiomatic set theory) are consistent, but not that they are inconsistent. The great effectiveness and beauty that mathematical structures founded on infinities allow, and the fact that no contradiction was found yet and none seems forthcoming, leads mathematicians in practice to adopt the calculus of infinities. As the great mathematician David Hilbert said, “No one shall expel us from the Paradise that Cantor has created”.

[20] Matter is made up mostly of atoms, and atoms are made up mostly of protons and neutrons. These, in turn, are made up of quarks. But most (about 95%) of the mass of a proton or neutron is not in its constituent quarks, but rather in the cloud of “virtual particles” that pop in and out of the vacuum surrounding the quarks and their interactions with them. In order to determine the actual mass of the proton one has to consider brief fluctuations of the energy into higher and higher energetic states, which include more and more such “virtual” particles. To be accurate, the calculation has to go on to infinitely high energies. Thus most of the mass of everyday things is actually due to the infinity of actual energy fluctuations. See Ab Initio Determination of Light Hadron Masses, S. Durr et al, Science 21(322) no. 5905, pp. 1224 - 1227.

[21] For a more thorough review of the vacuity of these arguments I recommend the work of Wes Morriston, in particular Craig on the actual infinite, Religious Studies 38 147-166 (2002).

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