Tuesday, September 29, 2020

A Brief Discussion of a YouTube Debate (Which is Really Mostly About Some Finitist Arguments)

  I would like to discuss a case study in the contemporary internet apologetics and counter-apologetics ecosystem, namely an ongoing YouTube debate between Cameron Bertuzzi (CB) of Capturing Christianity, and Steven Woodford (SW) of Rationality Rules. Just the names of these channels should give you a fair idea of the sort of discourse in store. Unsurprisingly, this debate is on no more precise a topic than the existence of God. CB has given the opening argument, to which SW has replied and CB has recently given his response in turn.

The videos can be found here. I wish mainly to examine this debate as a window into what apologetics can (and often does) look like today online, and to some extent, in certain academic circles. 

Cameron's Main Argument

CB states that he will be defending a version of the Kalam Cosmological argument (KCA). This is a favorite argument of the well-known apologist William Lane Craig (WLC), who almost invariably states it as follows:
  1. Everything that begins to exist has a cause.
  2. The universe began to exist.
  3. Therefore the universe has a cause.
  4. If the universe has a cause, that cause is God.
  5. Therefore, God is the cause of the universe (and thus exists)
The premises are usually confined to 1-3, with the others being less formally explored once (3) is established. Once arriving at (3), WLC will proceed into what the cause of the universe must be like, arguing that it must be spaceless, timeless, powerful, etc. and thus worthy of being called God. This is not too dissimilar from an argument of St. Thomas Aquinas, who argues to divine properties in a similar sort of way, thus the ground there is fairly well-trod. 

CB, however, does not offer the argument in this exact way. After going over some of what he sees as historical inaccuracies on SW's part (and alleging that SW consulted only the Wikipedia article), he instead offers the following:
  1. There is a First Cause.
  2. If there is First Cause, then God exists.
  3. Therefore, God exists.
Cameron makes a point of the fact that this is a valid syllogism (stating, for anyone having trouble keeping up, that it follows the form of modus ponens). In his response, SW points out that a valid syllogism is really only what is expected of anyone who understands the basics of logic and thus is not something to be so stressed. CB retorts that validity is crucial for an argument. All I'll say is that validity is akin to showing up to a duel with a pistol rather than a broom, and is really hardy something that should be brandished as worthy of much attention or emphasis. 

Let us make some remarks on the argument as given. It relies on the notion of a First Cause (FC), which will presumably function in some metaphysically important way. Moreover, it is arguing through causes. Now, the concept of a cause is ubiquitous in daily locution, and in all manner of diverse, and mutually incompatible ways. Language that sees much use in many forms often acquires a breadth of meaning, and as such is notoriously difficult to work within a philosophical argument, which is ideally as precise and unambiguous as possible. That is, it is prone to equivocation, ambiguity, concept creep, etc. It's not even clear how much validity such a concept has, at least in its most basic form. Perhaps we can one day come to an analysis of Edward Feser's Thomistic arguments and the issues that arise there, but those likewise rest on this idea of causation, which is somehow built from everyday usage and then ends up resembling it very tenuously, or not at all (I will argue). Thus, we may already have a very good idea of what the sort of argument CB will offer and likely what issues it will encounter.

CB explains what he means by a First Cause: "As we look back further and further into the past, we'll eventually arrive at something that has no prior causes. In other words, there is an uncaused starting point." Thus the First Cause can be read as an Uncaused Cause. By way of analogy, he says that there must be a "first domino" in the cascade of dominos that is the past series of causes. Seemingly as a way to calm the antagonistic viewer, he assures us that this First Cause could still, at this point in the argument, be considered potentially to be something that isn't God, perhaps some physical object or state of affairs.

Cameron's First Premise

CB states that he would like to get at an argument for his first premise through an argument for causal finitism, which is just the position that every event has a finite causal history. That is, for all events, there is some finite number N of causes that lead to that event. This is defended by the philosopher Alexander Pruss, known for offering and defending such arguments. CB claims that causal finitism would lead to his premise (1) being justified. We will return to this claim later.

For a sort of warm-up, CB offers a version of Thompson's Lamp. As CB describes it, the lamp is programmed to switch positions (if off, turn on; if on, turn off) at all of {11:00 PM - \(2^{-n}\) hours}, for n={0,1,2,3,...}, that is, at 10:00PM, at 10:30PM, at 10:45PM, etc. Assuming the lamp was off at 9:59PM, what is the state of the lamp at 11:01 PM? It seems that there can't be any good reason to say it is off or that it is on, as that is equivalent to saying infinity is even or odd, whereas that seems absurd. There has, of course, been much argumentation on this question, but CB proceeds from the difficulty in answering this question as evidence that there can't be an infinite number of switches. This is not a common conclusion drawn by philosophers examining this problem, and this serves as an insight into the issues to come. For what it's worth, I think the solution is simple: there is not sufficient information to determine the answer, or, stated another way, there is no contradiction either way. We can be assured the lamp is either off or on, but each is compatible with the problem as given. It is both true that \(\frac{\infty}{2}=\infty\) and that \(\frac{\infty-1}{2}=\infty\), and thus infinite quantities can function as both even and odd. 

The Grim Reaper Argument

CB offers a rendition of one of Pruss' arguments, namely the Grim Reaper Argument (GRA). Summarizing, the argument goes like this:

A Grim Reaper (GR) is an entity that has certain god-like powers, only to the extent that it can instantaneously kill someone at a given specific time, if they are alive at that time (it is assumed that death and life are perfectly well defined with no intermediate states, etc.). We suppose that, for all the numbers \({0,1,2,3,4,...}\), there is a corresponding GR (GR(n)) which is programmed (fated?) to kill a certain Fred at \(2^{-n}\) hours after 12:00PM. That is, GR(0) will kill at 1:00PM, GR(1) will kill at 12:30PM, GR(2) at 12:15PM, etc. Fred is alive at 11:59 AM.

The question is, then, is Fred alive at, say, 1:01PM? The argument goes that, clearly, he can't survive past 1:00PM, as if he's alive by then, GR(0) would kill him. But which GR kills him, then? If you say GR(n), kills him, that would mean that GR(n+1) should have killed him instead, as that GR's allotted time came before GR(n)'s, and if Fred was alive for GR(n) to kill, Fred would have been alive for GR(n+1) to kill him. Thus, we cannot point to which GR killed Fred. This inability to point to the guilty GR is offered as a true paradox (as CB puts it, Fred is dead and also not-dead) that shows the impossibility of a past-infinite causal chain. CB claims that the fatal flaw in the GRA setup is the infinite number of GRs.

CB then proceeds as follows:
  1. The GR scenario of the GRA is impossible
  2. If the GR scenario of the GRA is impossible, then causal finitism is true.
  3. Thus, causal finitism is true.
  4. If causal finitism is true, then there is a First Cause
  5. Thus, there is a First Cause.

A (Lengthy) Response to the Grim Reaper Argument

So let's examine this argument, beginning with (4). The impossibility of the scenario is clearly derived from the difficulty in naming the specific GR that killed Fred. We can see this as being the same difficulty in answering "what is the lowest element in the set {1/1,1/2, 1/3,1/4,...}?" The answer is that the set does not contain a minimum (though it has an infimum of 0). This is equivalent to the question "What is the largest member of the set {1,2,3,...}?" to which the answer is that there is no largest member: we can find an element of the set exceeding any given value. 

These questions and their replies are instructive. Surely the metaphysical properties of the GRs are related to the mathematical way they are constructed and placed. The GR "paradox" arises from the fact that there is no smallest number greater than 0: for any given x>0,  we can form x/2, and as x>x/2>0, x isn't the smallest. This was well-known to the ancient Greeks, even: each time I move my hand a distance D, I must move it through al the distances D/2, D/3, D/4, D/5, etc. Thus, there is no first point for my hand to move to, as it must move to the preceding point first. There is no way to begin moving my hand, then, as it's impossible to specify the first point to move to. Does this mean movement is impossible? Most philosophers (and people generally) don't think so. So how can it be resolved?

There are several solutions that can be proposed, and we will explore a few.

One is that Fred was killed, but not by any nameable GR, or perhaps by a GR not strictly bearing a natural number. This may be referred to as "the infinity-th GR" or "the omega-th GR" (omega being the smallest infinite ordinal). This need not be taken to refer to a GR with the name or index of "infinity" or "\(\omega\)", but rather as a way of saying "The GR beyond any GR one names". Though not rigidly specific, this does always have a referent: for any given GR, there is a GR that came before it. If we lined up all the GRs in ascending index, and asked each if the guilty GR was before (of smaller index) or after (of larger index) him, all would answer "after". The inability to single out a specific slayer may seem problematic, but that is merely the byproduct of dealing with the infinite. Given that "the largest natural number" is meaningless on its face, we shouldn't be surprised when an attempt to find it in a more (meta)physical context likewise fails to be meaningful. 

Another potential solution is to introduce a temporal metaphysical principle that precludes the difficulty. As a motivating example, suppose a lamp is off at 12:00PM and it turned on at exactly 1:00PM. What was the last time that it was off? Either the question can't be answered (for reasons like those discussed above), or else we must bring up some metaphysical notions of time. Surely we would like to answer that the last time it was off was at 1:00PM, even though, explicitly, it was on at that time. Perhaps we say that transitions may be counted in both, or in neither, in a meaningful sense.

In the GR case, we may say that temporal sequences must contain their infima/suprema, or that they de facto do. That is, in terms of temporal metaphysics, there either must be or de facto is a GR at 12:00PM, and it is that GR (which may be labeled by its time, or perhaps by "infinity" in the context of index) that killed Fred. This sort of reasoning may apply to any sort of "completed infinite" set. 

Suppose a Grim Apparition is, by definition, an opaque planar phantasm of the size of a normal human (say 6 feet tall and 3 feet wide) positioned and facing in a given location and direction. Now suppose that there is an infinite set of these, each indexed by natural number n. All are facing me, as I look due North, but the nth one is \(5+2^{-n}\) feet away. As the apparitions are opaque, I can only see one, the rest being hidden behind the ones preceding it. Which one do I see? It seems that I can't name the one that I see, supposing I do in fact see one. Does this prove that all infinite sets are impossible? Say this is not impossible, and I merely see some apparition but don't know its index, as all are indistinguishable apart from their positions. What I do is command each to form a number on itself (they may do this) that I may distinguish and count them. The first one forms a 1, then moves 5 feet East. The second, infinitesimally behind it (is it not even in the same position?) forms a 2, and steps to the East, behind the first. This continues for some time, perhaps some googolplex apparitions get a number and move aside. But it would seem I have not made any progress: the apparitions are precisely as far from me as they ever were (as \(2^n\) times an infinitesimal is still an infinitesimal, for any finite n). So I can remove any (finite) number and not make any perceptible progress. Effectively, there are arbitrarily many apparitions just stacked at (or just behind) 5 feet North of me.

This suggests another possibility: that "completed" infinite sequences cannot have all their members named. The natural numbers can be used to index arbitrarily deep, but cannot provide an index to all of them. They thus demand some sort of extension in order to be applied to "completed" cases, or cases where the order is reversed.  The following may serve as a useful example: it is easily provable (e.g. by induction) that
As the series converges absolutely, we may permute the order of addition arbitrarily without worrying about convergence or changing the sum. Thus, let's write:
This clearly follows from the first. Likewise, we can prove that
That is, the sum of the reciprocals of 2 to the power of each natural number is 1. But what happens if we try to reverse this expression, as we did with the first? It's not clear that we meaningfully can. We could write something like:
 But this seems like an abuse of notation, to some extent, though it does capture the sense of the Grim Apparitions scenario. We could merely write it as
But then this seems not to get at the "reversal" we had in mind, at least not fully (also an ellipsis at the beginning of an expression seems at the very least quite difficult to interpret). Each expression expresses something truthful, and yet also not to fully capture it. Is there merely a deficiency in language or description? Ought some new mathematical way of description be formulated to fill this gap, if indeed there is a gap? That doesn't seem altogether wrongheaded. As \(\omega\) often represents the smallest infinite ordinal, perhaps we can signify the largest finite ordinal by \(\psi\), for obvious reasons. Then \(\psi+1=\omega\) in a certain meaningful sense, though this results in passing from finite to infinite.  We can then write:
in a perfectly meaningful way, the notation making clear that the sum is across all natural numbers. This then would give a clear way to label the guilty GR. Granted this is mathematically speculative and dubious. But then so are such extensions to the real numbers as the hyper-reals and surreal numbers, both of which would label the guilty reaper as "\(\omega\)". 

Yet another potential avenue could be to examine the set of GRs as a whole, or perhaps just the concept of "the limit of GR(n) as n increases". Recall that we defined a GR as an entity that has certain god-like powers, only to the extent that it can instantaneously kill someone at a given specific time, if they are alive at that time. Given such a vague notion (that is, unless this is further specified), it is altogether arguable that the set or limit of GRs is itself a sort of GR, and one that kills Fred at 12:00PM (or, what is metaphysically the same thing, at a time infinitesimally removed from 12:00PM). Thus, the guilty GR could be meaningfully said to be an entity of this sort.

The seemingly counterintuitive or absurd is merely what one may run into when dealing with the infinite. All it means is our intuitions often break down when approaching the infinite. Is this a surprise? We don't have much contact with the infinite, in all its various forms, in everyday life, after all. At the end of the day, arguments against the infinite seem always to make that clear: the main reason they give for outlawing the infinite is that the infinite is difficult and gives rise to difficulties. Rathe than working top-down from intuitions, an alternate (if not strictly superior) approach is to work bottom-up: that is, we would use intuitions to build the fundamental framework and derive consequences of that framework. As long as the consequences are compatible with the underlying fundamental intuitions, we would simply go with the derived consequences, any resulting top-down counter-intuitiveness notwithstanding. 

As an example, the notion that the part cannot ever be equinumerous to the whole certainly seems intuitive and holds true for all finite wholes and parts, but it fails in the case of the infinite set of whole and even numbers. That doesn't mean we discard the idea of an infinite number of whole and even numbers (some may disagree), but rather that we discard or qualify the intuition. Surely not all intuitions are valid, after all, and it is only through examination and testing that we can discern the valid ones from the invalid ones. 

The Rest of the Argument for the First Premise

Getting at long last back to the argument, let's see how the GR argument would even fit into the place where CB uses it. Having cast serious doubt on (4), let's now examine (5), which is "If the GR scenario of the GRA is impossible, then causal finitism is true." As he doesn't offer any independent argument for (5), CB seems to take it as obvious, but is it? First of all, it is worth noting that in the GRA, there is no infinite chain of causes. The cause of any GR killing Fred is only that Fred is alive when that GR's appointed killing time arrives. In a weak sense, Fred's being alive is "caused" by previous GRs not killing him, but that seems incorrect in a major way: absences are now causes of lack of change? Is a window remaining unbroken in any meaningful sense caused by the absence of rocks thrown at it? That seems both silly and metaphysically incorrect. The only way that the GRA could support causal finitism is by supporting a more general notion of finitism which would then be applied to the case of causes. Thus the argument is more accurately framed as follows:
  1. If the GR scenario of the GRA is impossible, then finitism over class X is true.
  2. If finitism over class X is true, then causal finitism is true.
  3. Therefore causal finitism is true.
Let's assume class X merely includes causal chains as a subset, rendering (10) unarguably true. But what of (9)? The GRA is merely one instance of a type of infinite set or sequence. How could one case imply that a much broader set of cases shares the same property? As a rule, this is not the case: P being true of an element rarely serves to show that P is true of the whole set (not even the composition fallacy partakes of this!). So there must be yet another line of reasoning implicit in the argument, namely:
  1. If a case of non-finitism of type T is impossible, then finitism over class X is true.
  2. The GR scenario of the GRA is a case of non-finitism of type T.
  3. Therefore, if the GR scenario of the GRA is impossible, then finitism over class X is true.
But what could the type T be that would render (12) true? The only possibility is that it is in some way equivalent to "belonging to the class X, of which causal chains are a subset". However, this seems hopeless for the finitist: the class X/type T must be such as to include the GR scenario and causal chains generally, but this connection seems simply nonexistent, as the GR scenario doesn't include infinite causal chains at all. We may conclude from the GR scenario that, perhaps, scenarios that rely on identifying a "largest natural number" are impossible (as this is the singular difficulty of the GR scenario, evidenced by its being resolved if such a concept is given a referent). But beyond this, there is nothing standing in the way, as we go into below.

Something worth noting, as a general metaphysical principle, is that impossibilities are, well, impossible. That is, an impossible sort of thing is not merely sometimes impossible, but rather always so i.e. necessarily. Married bachelors aren't impossible only in certain circumstances, but rather in all circumstances. This seems to follow directly from the axiom of S5 modal logic that "necessarily X" implies "necessarily, necessarily X". Given that "necessarily not-X" is equivalent to "not possibly X," it follows that "not possibly X" implies "necessarily, not possibly X." Let us then make the following tiny variation to the GR scenario, with the addition of a single GR (let's call it GR(-1)) which is to kill Fred at exactly 12:00PM. Now we can easily answer which GR killed Fred: it's GR(-1). The difficulty disappears. But we still have an infinite number of GRs! Wasn't it allegedly the infinity of GRs that was at issue? How come we still have infinitely many GRs but now with no paradox? The answer is clear on even a cursory consideration: now there is a well-defined "earliest GR", and it was the absence of this that was the cause of the difficulty in the original GR scenario. This serves as a conclusive proof that it is only a circumstance that can arise with infinity that was at issue, rather than the infinity itself. We can distinguish between infinite scenarios that are and aren't problematic, and thus it cannot possibly be the case that all infinities are impermissible. 

Here's a simple demonstration that you can even do at home that shows that it's not the infinite that's the problem, but rather the scenario in which it arises. It is easy to show that the fraction \(1/99\) has the decimal expansion 0.01010101... My challenge is simply this: write the digits 0 and 1 on a piece of paper. Now underline the digit that is the last decimal digit of \(1/99\) (if you don't think it's either of these, feel free to list out all 10 decimal digits and then underline the correct one). Can you do it? I'm quite certain you can't, and simply for the reason that the number has no final decimal digit. Suppose I define a p-rock as any solid rock larger than 1 inch in each dimension that has the last digit (excluding any trailing 0s) of the number p carved on its surface. Some p-rocks are entirely possible: a \(p=1/2\)-rock is possible. However, a \(p=1/99\)-rock is not possible. Thus p-rocks generally are neither all possible nor all impossible: it depends on the p of the p-rock. Suppose I say "Imagine a world with a \(p=1/99\)-rock...". At that point, it can be assured that such a world is impossible, as it includes an impossible object. It is not different than supposing a world with a married bachelor, and concluding that such a world is impossible. This serves merely to highlight an important point: among a class of things related to the infinite, some may be possible, while some may not be: the impossibility of some of them doesn't prove the impossibility of all of them. 

Another quick illustrative example is to consider the infinite sequences {1,1,0,0,0,0,...,0,0,0,...} and {1,0,1,0,1,0,...,1,0,1,0,...}. The infinite is equally present in both cases. And yet, if we ask the question "what is the index of the last '1'?" we can answer simply "2" in the case of the first, while there is no such answer in the second. Clearly, then, it is not that the sequence is infinite that causes the difficulty, even if its being infinite allows for the difficulty to arise: the infinite makes possible that certain questions won't have answers, but it doesn't guarantee it. If it was important that we answer such questions of infinite sequences, we could merely restrict ourselves to those of the first sort and outlaw those of the second, or perhaps allow for such answers as "infinity".

From Causal Finitism to a First Cause?

However, there is still the matter of CB's premise (7): does causal finitism show there is a First Cause? Here is a scenario in which causal finitism holds but there is no universal First Cause: supposing finitism generally doesn't hold, let's imagine an infinite array of dominos separated by one unit (an inch, say), and we will specify the locations of the dominos as though they were in a cartesian plane. There are dominos at all of (m,n) where m is any integer {...,-2,-1,0,1,2,...} and n is any non-negative integer {0,1,2,3,...}. The dominos are oriented so that the domino at (m,n) can only be knocked over by the one at (m,n-1) and will then proceed to knock over the domino at (m,n+1). It takes one second between a domino getting hit and striking the next one. This leaves all the dominos at (m,0) free of any domino to knock them over. However, there is also an infinite set of demons D(m) corresponding to each column of dominos. The Demon D(m) knocks over the domino at (m,0) at precisely the time m, measured relative to the time at which the domino at (0,0) was knocked over. Thus, the domino at (m,n) gets knocked over at exactly the time m+n, and its causal chain was started at time m. As all of these are finite, it holds that causal finitism holds. However, there is no First Cause for al the dominos: each column has a separate "First Cause", while there is no single First Cause for all domino falls. demonstrates, then, that CB is making a quantifier shift fallacy: "all causal chains have a first cause" doesn't imply that "there is a (singular) first cause of all causal chains" (confer "all people have a mother" vs. "there is a (singular) mother of all people").

Thus (4), (5), and (7) have been called into serious doubt, which leaves (1) quite unsupported. This is obviously a grave issue for CB's argument.

Cameron's Second Premise

Going on, CB proceeds to argue in support of his second main premise, namely "If there is First Cause, then God exists." In support of this, he begins by inviting us to consider what distinguishes the caused and the uncaused. Despite correctly observing that we have no experience of any such uncaused things (thus rendering almost all his argument moot), he proceeds along the standard apologetics story of some hikers in the woods who find a strange object in the woods. The usual apologetics tack (following WLC) is to say that no matter how strange the object (a glowing orb, say) is, we nevertheless would be incorrect to conclude that it's uncaused, and that would be independent of its particular size or sort, such that even if it were the size of the universe, we would still be warranted in concluding it has a cause. 

CB proceeds along these lines at some (tiresome) length, then stressing that "being very different" is not in itself a relevant difference for concluding that something is uncaused. He (again tiresomely) considers then eliminates such features as shape, size, color, and power (oddly referring to battery wattage or FLOPS rather than the more abstract potency), before arriving at his point, which is that limits are how we can distinguish the caused vs. the uncaused. He argues that we understand that rocks, people, cars, etc. have causes because they have limits, and "limits have causes". As the First Cause has itself no cause, he argues, the First Cause must be unlimited, and unlimited in the usual maximal-god ways (e.g. omnipotent, omniscient, etc.).

It's not clear where even to begin with this. CB hasn't given much specificity to the notion of what a cause even is. The causes of the first premise seem to be efficient causes, whereas the causes of the second are pretty clearly formal causes. Aristotelians are often happy to lump these together, but it's worth keeping a solid logical distinction between them. Suffice it to say that the logic of the defense of the first premise can't be used to substantiate the second without transgressing into equivocation. Implicitly, CB argues that something having a limit implies that it has some cause in the sense of not being the first part of a causal chain. Is this true? It's easy to imagine a violation of this, perhaps an eternal infinite void, or a stone forever floating alone in such a void, or perhaps an inferior deity that is not maximal but still functions as a first cause. Is this not a refutation of CB's principle? It seems perfectly coherent that the first element of the universe's causal chain (supposing there is one) has limits: try to imagine it and you'll likely find that you can with almost no difficulty. (You'll notice that many world religions think of a highly limited and imperfect being causing the universe. If nothing else, this is some amount of evidence for the conceivability of it.) 

In fact, CB is arguing from "if caused then limited" (using as evidence all manner of everyday examples) to then illicitly conclude "if limited then caused". This doesn't follow logically, of course, but CB seems to want to argue it inductively. Induction has notorious issues (just because every swan we've seen thus far is white doesn't mean all swans are white, after all), but then applying induction here seems to open quite the can of worms. From experience, all limited effects have limited causes (in fact, there is often a certain proportionality: larger causes generally have larger effects), thus unlimited causes would have unlimited effects. Can something perfect produce something imperfect? From our experience, all imperfect effects have imperfect causes. Very arguably producing an imperfect effect would seem to render the cause itself imperfect. Perhaps a better case could be made if we all lived in Eden, but we clearly don't (arguments from the high imperfection of the world can go here). 

CB seems to be quietly invoking some sort of "Principle of Sufficient Reason". Perhaps such a principle is strong enough to justify his desired conclusion that limits have causes (read: reasons), but then not only should he state so plainly, but then also defend such a principle.

Regardless, is the God that CB argues the First Cause has to be really limitless? Does God have infinite density and spatial curvature like a black hole? How many lies can God tell? How many miracles would god do to make his nonexistence totally untenable to even the staunchest atheist? The answer to all these are zero, and so zero seems to be another "good" sort of limit. Christians would also answer that other limits on God have the values "three" or "one" so these seem allowable as well. If Occam's razor is acceptable, is not the simplest cause of the universe some entity with exactly one ability, namely, whatever it takes to cause that something that we call the universe? We have no justification to infer that the cause of the universe is perfectly good when "amoral" seems a much more justifiable lack of limitation. We have no reason to suppose that it is omniscient when "lacking any knowledge of any kind" is a much more natural limit. 

Thus, CB's premise (2) likewise runs into serious, perhaps insurmountable difficulties. As (1) and (2) are both now gravely undermined, the argument itself needs considerable work and certainly can't be considered to have succeeded.

Cameron's Response

Rather than discuss SW's rebuttal, we will skip straight to CB's response to that rebuttal. This is mainly because CB's arguments are my main object of interest, and, frankly, SW's rebuttal was not nearly as strong as it could have been. My hope is to offer what I consider the best rebuttal available, perhaps, even, in hopes that SW will employ arguments more to this effect.

The Grim Messenger Scenario

As a way to reformulate the GR scenario to cover an infinite past, CB offers a variant of the original GR scenario as follows. We imagine an infinite sequence of Grim Messengers (GMs), each assigned to a natural number index n. GM(n) is assigned to January 1st of the year 2021-n. Each GM is tasked with receiving and then transmitting a message, according to the following rules: GM(n) receives a message from GM(n-1). If the message doesn't contain a natural number, GM(n) is to pass to GM(n-1) the message of "n" (i.e. GM(n)'s index). If the message does contain a natural number, GM(n) passes the message, unaltered, to GM(n-1). CB describes this as "writing down a number", but since, for instance, (100!)! has well over a googol digits, writing it down on any sort of actual piece of paper seems infeasible, thus merely transmitting the message by whatever means seems better. The alleged paradox is thus that the message can't contain any number, as each number has a predecessor (if the message contains N, it should really contain N+1, N+2, etc.).

The response to this is quite simple: it is an impossible scenario, but not because of an infinite past. The only number the message could contain is "the largest natural number" which doesn't exist. Perhaps the message could contain \(\omega\) which each and every messenger passes to his successor, but I suspect CB would say this is not a natural number. The scenario stipulates rules which cannot be followed, and they cannot be followed simply because the rules are unfollowable, just as unfollowable as a form telling you to fill in a space with the last digit of pi. At least the GR scenario could be described in a reasonably coherent way. The GM scenario, in contrast, can at no time be described meaningfully and coherently. Either some GMs (always an infinite subset) fail to faithfully follow their rules or all of them do (by, e.g. passing an \(\omega\)). The scenario is as meaningfully possible as a \(p=1/99\)-rock. 

As discussed above, this has no implication that an infinite past is impossible. Here is a simple proof: instead of being required to follow the rules as described, we simply modify the rules, changing nothing else. Now the messengers follow this rule: GM(n) always passes "n" to his successor. Now there is still an infinite number of GMs and yet any paradox has vanished. Thus, it is not that there are infinitely many GMs that caused the problem but merely the rule that they were stipulated to follow. It is not surprising that some rules can't be followed, as there are some rules that can't be followed even in finite time (e.g. write down the last digit of pi). 

CB's Responses to SW's Objections to the GRA

SW objects that the GRA can't be applied to an infinite past. CB responds that, while the GRA doesn't itself imply a First Cause, his proposed resolution of the difficulty (i.e. causal finitism) does imply a First Cause. As we've discussed above, this is highly questionable if not fallacious. CB argues that the Grim Messenger scenario does imply the impossibility of an infinite past. Again, this is not at all clear or established and, as I've argued, fails quite resoundingly. 

SW objects that one sort of infinity being impossible doesn't imply that all infinities are then impossible. CB responds that he didn't argue that all infinities are impossible. Rather than strict or general finitism, he argues, he has only concluded with causal finitism. Now, this seems either disingenuous or altogether misguided. As discussed above, the GRA has no clear connection to causal finitism except through finitism generally. This is the thrust of SW's objection and CB has tried to avoid it in a rather obtuse way. 

CB's Responses to SW's Objections to the 2nd Premise

CB begins by latching on to a comment SW makes and spending a good chunk of time rebutting it. Namely, SW states "As an empiricist, I would say that, to concretely know something, we must verify it through the scientific method". CB labels this as "scientism" and proceeds to offer some standard apologetics rebuttals to it. No real attempt seems to have been made to interpret it in as favorable light as is reasonable. Moreover, all the efforts of the well-known empiricists (e.g. Hume) are not touched on. One simple way to interpret SW's claim more favorably is as stating that intuitions, especially when they conflict with other intuitions, often need to be tested in order to be validated and used confidently. In fact, CB is making what is fundamentally an empiricist argument, appealing to our experiences of things in order to infer certain conclusions. Further, it seems CB missed that SW follows up his statement with "I agree", which would render CB's knee-jerk reaction a case of responding to the part, not the whole.

SW objects that advancements in physics call our notions of causation deeply into question, as can be seen in the cases of Special and General Relativity, and Quantum Mechanics. CB seems to miss this point entirely and argues that such things as scale and celerity are irrelevant to metaphysics. He illustrates his point with such arguments as "if size is relevant to what boxes I can carry, it's relevant to what size boxes can spontaneously appear above my head," and "A semi-truck or race car aren't uncaused in virtue of their being larger or faster than a sedan." These arguments are technically true, but they seem hopelessly to miss the point. He's clearly working in the so-called "classical" regime. Are virtual particles caused? In a sense yes and no. Is radioactive decay caused or uncaused? In a sense yes and no. I strongly suspect cases like these are more what SW had in mind. It seems entirely reasonable that our everyday physics is really only a sort of special case or regime of a much more general and different physics, a physics where causation may take on a very different cast or be altogether inapplicable. CB would do better not to patronize either his audience or his interlocutor so grossly.

SW objects that CB has offered no examples of anything uncaused, and yet has confidently concluded that any such thing must be unlimited. He also argues that CB has inferred from all caused things having limits (i.e. since everything we observe is both caused and limited) that uncaused things don't have limits, which doesn't follow. CB responds "I don't see any reason at all to think that reasonable conclusions require concrete examples." 

Ironically, CB then offers the example that he can know that a fair 200-sided die has an equal chance of 0.5% of landing on any face. In response to this, we can simply point out that this example is extremely poor. The probability of landing on any given face follows simply from the definition of what it means for an N-sided die to be fair, namely, that there is a 1/N chance of landing on each face. That is, we can conclude the probability is 0.5% purely analytically, from the definitions of the terms. Secondly, I believe there are, in fact, 200-sided dice. But even if there weren't, there are still dice, even dice with a pretty large number of sides. That is, there are things that are quite like (if not exactly like) 200-sided dice. However, we have no experience of anything like an uncaused cause or an unlimited being. Our intuition about dice derives, in part, from our experiences with other dice, dice we've used and understand pretty well. There is no such experience from which intuitions about uncaused or unlimited beings can derive. Thus, the dice example is terrible for substantiating CB's point.

The example notwithstanding, is CB's claim true? Do reasonable conclusions require examples? This is difficult to argue in general, but examples are, at the very least, extremely useful when examining statements like "All X are Y". This sort of statement is equivalent to "There is no X that is not also a Y," and to "All non-Y are not X." Justifying such a statement, then, often goes along the lines of supposing that something is an X and not a Y and deriving some sort of contradiction. This is why examples are so important: often if we have an example of something that is both an X and a Y to examine, the relationship between X-ness and Y-ness becomes more apparent and generalization becomes more accessible. Epistemically, examples are invaluable. Argumentatively, examples are extremely useful (as an example, CB's own example-giving demonstrates this) as means of analogy. It's obvious that if CB did have an example to back up his claim he would not hesitate to offer it, so his attempt to justify not offering one seems rather dishonest. 

CB then offers that his formula "if uncaused then unlimited" is only a proposal, and invites his audience to think about it and offer alternatives. In the spirit of CB's openness, here is my own proposal: the difference between the caused and the uncaused is that the one is caused and the other is not, and nothing beyond this can be reduced.

CB states that his premise "if there is a First Cause, then it has no limits" has wide empirical support, namely, that in our experience, all limited things have causes. This runs into the problems mentioned both in my response and SW's. For example, we may instead conclude that all caused things have limits, from which the premise can't be logically derived. Or we can conclude that all things are both caused and limited, which would serve to refute the claim that there is a first cause. This is why examples would be so valuable to CB: an example of an uncaused and unlimited being would provide significant insight into how causes relate to limits. CB denies that his conclusion doesn't follow, as his argument is valid, but that isn't the allegation: SW's point is that the inference doesn't follow from the data, not that the conclusion doesn't follow from the premises.

Arguments in Favor of "Unlimited Implies Perfect" 

CB points out that "a perfect being" is simply what he means by "god" so all that he needs to do is argue that (A) If there is a First Cause, then it is unlimited, and (B) if X is unlimited, then X is perfect. Satisfied with his defense of (A), he moves on to offer some arguments in defense of (B). 

His first argument is summarized as "imperfection implies limitation". As this is logically equivalent to the claim to be proved, it's difficult for this argument to get off the ground. But beyond this, the objections I raised above would apply here. Unlimited could mean "unlimitedly small" namely zero. Thus, the First cause may have exactly zero knowledge, exactly zero love, and exactly zero hate (as these seem totally unnecessary for causing things (just ask electrons), this seems far more natural and justifiable than assuming it has infinite love and yet still zero hate). In fact, the more justifiable premise would be "limitation implies imperfection" rather than vice versa, though even this seems vexed.

His next argument "moves from some degree of value to unlimited value to god" (He never offers any sort of definition of what he means by "value"). CB asserts that "different things exemplify different degrees of value." He uses, as an example (I thought these were unnecessary?) his house and his daughter. The philosophical problems here are many. It assumes some sense of objective value which many people, particularly many atheists, would never permit. Presumably, CB has some sense of possession of his house and paid something for it, while the same can't be said of his daughter (hopefully). He seems to be flagrantly equivocating "value" among moral value, sentimental value, and monetary value. To make matters worse, he asserts that the First Cause "is the ground or source of everything else that does have value." This is extremely debatable, and at best totally unsubstantiated. Finally, CB argues that if the First Cause has some amount of value, as it is unlimited, it must have an unlimited degree of value. Given that most people would say the Big Bang singularity had no moral value and yet the sum of all humanity has quite a bit of moral value, value can grow over time, and it would not be beyond thinking that the cause of the universe has no value at all. Is not the value of the universe increased every time a child is born and does it not decrease somewhat every time someone dies? Perhaps you disagree but such a position seems not unsupportable. In sum, I find this argument totally dead on arrival. 

His third and final argument is almost identical to the second, but replacing "power" for "value". He argues that the cause must at least have the power to cause what it causes. Proceeding from his argument from limits, he concludes that the First Cause must have unlimited power (i.e. omnipotent). "We have the power to know things," he argues, and uses this formulation to conclude that the First Cause is omniscient as well. In response, we can merely refer to the above, as none of this is really new, but we can make some remarks about "power" and "power to know" in particular. 

At this point repeating, we can point out that "knowing nothing, having no ability to know anything" is a natural lack of limitation, in the same way that electrons can't know things. What even are the powers of something like an electron, or a rock, or anything inanimate? What does it even mean to have power? Abilities? Potentials? I have the potential to lie: does the First Cause have the unlimited ability to lie? I have the ability to feel sexual pleasure: does the First Cause have the ability to feel unlimited sexual pleasure? I have the ability to change, and the ability to die, and the ability to be ignorant or make mistakes. Does the First Cause have these also to an unlimited degree? I can already hear in my head all the theistic arguments in response, but then I can also envision my replies to them. We can merely conclude that this line of argumentation that CB offers is riddled with holes, to the point that it seems more hole than not.

Occam's Razor? 

CB invokes Occam's Razor as a way to justify considering the First Cause to be singular and not a plurality. He gives the example of a used cereal bowl lying in his sink and compares the explanation that his daughter placed it there rather than "a billion tiny aliens". His response to the second alternative is to make a funny face and scoot offscreen. 

Amateur theatrics notwithstanding is it, in fact, reasonable to conclude, using Occam's Razor, that the First Cause is singular rather than numerous? This seems to badly misunderstand the Razor itself. A better form of the razor is "the simplest explanation is likely correct." CB's form would have us conclude that desks are solid rather than made of billions of atoms, and yet we nevertheless think desks are, in fact, made of billions of atoms. Beyond this, the razor would have us shave off unnecessary properties of the supposed cause, like knowledge or goodness, which CB would clearly like to avoid. Going strictly by Occam's Razor, we would conclude that the First Cause has just the ability to cause what it causes and nothing more, unless any other properties can be shown to be necessary for this or entailed by this. Indeed, we would conclude that there is at least one such thing, but possibly more. But then this assumes the First Cause has already been established, which has yet to be done.

The Razor is something theists would do best to employ with some care, as it is so very often used as a way to exclude God from one's ontology. If there is nothing for God to do, nothing for him to explain or cause or ground etc., then Occam's Razor would suggest shaving him away entirely. To be candid, this is precisely my view. 


This discussion by no means offers anything like a comprehensive or representative view of all apologetics. However, this example does serve as a characteristic example of the sorts of arguments one can expect to encounter in mainstream apologetics. Many other such arguments don't differ very greatly in spirit or thrust, though obviously some are better phased or are given at more length. CB is perhaps close to the average, though he seems to lack something of originality in the arguments he presents. This, however, is not necessarily a fault of his, as his abilities are concentrated elsewhere and certainly exceed mine in a number of other areas. He is quite well-read in the realm of contemporary apologetics, though it would likely behoove him to invest some time in the fundamentals (as it would for almost all of us).

The responses, likewise, can be considered to follow a certain pattern. You may note that the refutations of the arguments require considerably more effort to expound than the stated arguments and their defenses. If there's any broadly applicable truth that I've learned from my time engaging with such arguments, it is that they are formulated and offered so that their refutations require considerably more discussion than their presentation. I consider this one of their definitive qualities. In my view, this is their greatest strength and explains their pervasiveness. It makes them well-suited for debate formats where participants are given equal time. They are strong rhetorically and quite weak philosophically. (Don't believe me? Almost all philosophers, scientists, and highly-educated people don't believe in god. Can this be said of any other position for which there are strong philosophical arguments? Note: this is not an appeal to authority or popularity, but just an invitation to consider some relevant evidence. That many people believe something isn't why one should believe a thing, but that many and especially intelligent and educated people believe something ought to give one some good reason to consider it closely.) 

This is not to say that they aren't worth examining and discussing, far from it. I think they serve as superb jumping-off points for almost all philosophical topics. A broad course or survey of philosophy could involve simply going into each of the various sorts of theistic arguments and developing an understanding of how they work, what ideas they root themselves in. Above, we have discussed such fundamental topics as the infinite, causation, and concepts of limitation, perfection, and explanation. Other arguments introduce us to such topics as metaethics, modality, ontology, design, and so on. Theists are a natural, valuable, and likely inevitable sub-species of the philosophical ecosystem. Opinions vary rather widely on their overall utility (from essential and ultimately correct, to worthless and harmful), but for my purposes, they offer a useful function. They may not like the description, but it is the same function that pseudoscience quacks offer. That is, they have a nose for the cracks in any intellectual edifice and take as firm root there as they can, thus calling to attention those areas that need to be strengthened and made less susceptible to such infestation. 

I hope this has been interesting and instructive. I hope CB and SW don't mind my honest criticism, though it may have been at their expense at times. My main motivation for writing this was my disappointment both at the weakness of CB's arguments and SW's rebuttals, and my sincere hope is that both improve so as to offer us some better, more philosophically robust arguments in the future. There's no real way to do so without some amount of patronization, so I won't bother. It's all with the best of intentions if that counts for anything, the road to hell notwithstanding.

Monday, September 14, 2020

A Fairly Rigorous Derivation of Euler's Formula


Exponential Functions

The general exponential function \(b^x\) for base \(b > 0 \) and real number \(x\) (*) is defined as the function that satisfies the conditions \[ b^x > 0 \\ b^x\cdot b^y=b^{x+y} \\ b^1=b \] It follows that: \[ \prod_{k=1}^{N}b^{x_k}=b^{\sum_{k=1}^{N}x_k} \\ b^0=1 \\ b^{-x}=1/b^x \\ (ab)^x=a^x b^x \\ b^{m/n}=\sqrt[n]{b^m}=\left ( \sqrt[n]{b} \right )^m \\ b^x=\underset{n \to \infty}{\lim}b^{\left \lfloor xn \right \rfloor/n} \]
(*) We will extend this definition to complex \(x\), for which, we will find, that \(b^x>0\) may not hold. Moreover, there is some ambiguity for non-integer \(x\), as, for example, \(4^{1/2}\) may be \(2\) or \(-2\).

Some Exponential Inequalities

Let \(b>0\). By a simple argument we find: \[ 0 \leq \left ( b^{(y-x)/2}-1 \right )^2 \\ b^{(y-x)/2} \leq \frac{b^{(y-x)}+1}{2} \\ b^xb^{(y-x)/2}\leq b^x\left (\frac{b^{y-x}+1}{2} \right ) \\ b^{(y+x)/2} \leq \tfrac{1}{2}b^y+\tfrac{1}{2}b^x \] Suppose that \(0 \leq \alpha,\beta \leq 1\) and that \[ b^\alpha\leq\alpha b + (1-\alpha) \\ b^\beta\leq\beta b + (1-\beta) \] Then \[ b^{(\alpha+\beta)/2} \leq \tfrac{1}{2}b^\alpha+\tfrac{1}{2}b^\beta \\ b^{(\alpha+\beta)/2} \leq \tfrac{1}{2}(\alpha b + (1-\alpha))+\tfrac{1}{2} (\beta b + (1-\beta)) \\ b^{(\alpha+\beta)/2} \leq \tfrac{\alpha+\beta}{2}b+(1-\tfrac{\alpha+\beta}{2}) \] As \(b^0=1\leq 0 \cdot b + (1-0)=1\), and \(b^1=b\leq 1 \cdot b + (1-1)=b\), it follows that, for all dyadic fractions of the form \(x=M/2^N\) for some whole numbers M and N with \(0 \leq M \leq 2^N\): \[ b^x \leq x b + (1-x) \] Moreover, as all real numbers \(0 \leq x \leq 1\) can be written as the limit \[ x=\underset{N \to \infty}{\lim} \frac{\left \lfloor x \cdot 2^N \right \rfloor}{2^N} \] It follows that \[ b^x \leq x b + (1-x) \] Holds for all real x in the interval \( [0,1]\) for all \(b>0\), with equality holding only at the extremes. It follows that \(2^x < 1+x\). Additionally, \((1/2)^x < 1-x/2\). We may then make the following argument: for \(0 < x < 1\) \[ x^2 > 0 \\ 1-x^2=(1+x)(1-x) < 1 \\ 1+x < \frac{1}{1-x} \\ (1/2)^x < 1-x/2 \\ 2^x > \frac{1}{1-x/2} \\ 2^x >{1+x/2} \\ 4^x >(1+x/2)^2 > 1+x \] Thus, we have \(2^x < 1+x < 4^x\).

Derivatives and Derivatives of Exponentials

The definition of a derivative of a function is: \[ \frac{\mathrm{d} }{\mathrm{d} x}f(x)=f'(x) \triangleq \underset{h \to 0}{\lim}\frac{f(x+h)-f(x)}{h} \] Thus, for an exponential, the derivative would be given by: \[ \frac{\mathrm{d} }{\mathrm{d} x}b^x\triangleq \underset{h \to 0}{\lim}\frac{b^{x+h}-b^x}{h}=b^x\underset{h \to 0}{\lim}\frac{b^{h}-1}{h}=b^x L(b) \] Where \(L(b)=\underset{h \to 0}{\lim}\frac{b^{h}-1}{h}\), provided this limit exists. This limit can be proven to exist as follows: for \(0 < q < 1 \), and \(0 < x\), for \(y = q x < x\) by the derived inequality \[ (b^x)^q = b^{qx} < (b^x -1) q +1 \\ \frac{b^{qx}-1}{qx} < \frac{(b^x -1)}{x} \\ \frac{b^{y}-1}{y} < \frac{(b^x -1)}{x} \] Thus, the limit is monotonically decreasing (from the right, increasing from the left). Moreover, the limit is bounded from below and above (for |x| < 1), as \[ 1-\tfrac{1}{b} < \frac{(b^x -1)}{x} < b-1 \] Thus, the limit exists, and so \(b^x\) is everywhere differentiable. As exponentials with \(b > 0\) are eveywhere differentiable and thus continuous, we may take the limit for \(h > 0\). From the above inequalities, we have \[ L(2)=\underset{h \to 0}{\lim}\frac{2^{h}-1}{h} < \underset{h \to 0}{\lim}\frac{1+h-1}{h}=1 \\ L(4)=\underset{h \to 0}{\lim}\frac{4^{h}-1}{h} > \underset{h \to 0}{\lim}\frac{1+h-1}{h}=1 \] As the limits are decreasing and both are bounded below ( \(L(2) > 1/2, \; L(4) > 1 \)), it follows that both limits converge. Thus \(L(2) < 1 < L(4) \). As L is clearly continous, by the intermedate value theorem, it follows that there is some real number \(2 < e < 4 \) such that \( L(e)=1 \). Let us define this number \(e\) to be that number that satisfies \[L(e)=\underset{h \to 0}{\lim}\frac{e^{h}-1}{h}=1\] This implies that \[ \frac{\mathrm{d} }{\mathrm{d} x}e^x=e^x \] This is a defining feature of the number \(e\). We may also notice that, for any real x, if \(h \to 0\) then \(xh \to 0\). Thus \[ \underset{h \to 0}{\lim}\frac{e^{xh}-1}{xh}=1 \\ \underset{h \to 0}{\lim}\frac{e^{xh}-1}{h}=x \] Thus \(L(e^x)=x\). This implies that, by definition, \(L(x)=\log_e (x)=\ln (x)\). Moreover, given the chain rule \(\frac{\mathrm{d} }{\mathrm{d} x}f(g(x))=f'(g(x))g'(x)\), we find \[ \frac{\mathrm{d} }{\mathrm{d} x} L(e^x)=L'(e^x)e^x=1 \] And thus \(L'(x)=1/x\). This is a very helpful result. For example, by rewriting and using the chain rule, we find: \[ \frac{\mathrm{d} }{\mathrm{d} x} x^a=\frac{\mathrm{d} }{\mathrm{d} x} e^{aL(x)}=e^{aL(x)} \frac{a}{x}=a x^{a-1} \] A result that is otherwise difficult to establish in the general case. We may write the limit derived above in an equivalent way as \[ \underset{n \to \infty}{\lim}n \cdot (e^{x/n}-1)=x \] Which directly implies that \[ e^x=\underset{n \to \infty}{\lim} \left ( 1+\frac{x}{n} \right )^n \] Let us expand the above expression using the binomial theorem: \[ e^x=\underset{n \to \infty}{\lim} \left ( 1+\frac{x}{n} \right )^n \\ e^x= \underset{n \to \infty}{\lim}\sum_{k=0}^{n}\binom{n}{k}\left ( \frac{x}{n} \right )^k \\ e^x= \underset{n \to \infty}{\lim}1+\sum_{k=1}^{n}\frac{x^k}{k!}\prod_{j=1}^{k}\left ( 1-\frac{j-1}{n} \right ) \] Clearly, in the limit, all the factors in the products from 1 to k go to 1. Thus, we find: \[ e^x=1+\sum_{k=1}^{\infty}\frac{x^k}{k!} \] It can be checked that this series converges for all real x by the ratio test. This is an extremely useful formula, and can be taken to be a more robust and easy-to-work-with definition for \(e^x=\exp(x)\). Note this formula directly implies that: \[ e=\sum_{k=0}^{n}\frac{1}{k!} \] Note that, as a verification, we can check that \(e^0=1=1+\sum_{k=1}^{n}\frac{0^k}{k!}\) and \[ \frac{\mathrm{d} }{\mathrm{d} x} e^x=e^x=1+\sum_{k=2}^{n}\frac{kx^{k-1}}{k!}=1+\sum_{k=2}^{n}\frac{x^{k-1}}{(k-1)!}=1+\sum_{k=1}^{n}\frac{x^{k}}{k!} \] Which verifies the differentiation formula.

Trigonometric Functions and Inequalities

The definitions of the basic trigonometric functions are given by Figure 1. The curve between points C and D is the set of points equidistant from A between the line segments \(AC\) and \(AD\), i.e. a circular arc. Let us call the length of this curve \(L\). Then the standad definition for the basic trigonometic functions is given by: \[ \theta=\frac{L}{\overline{AD}} \\ \\ \sin(\theta)\triangleq \frac{\overline{BD}}{\overline{AD}}, \; \;\; \cos(\theta)\triangleq\frac{\overline{AB}}{\overline{AD}}, \; \;\; \tan(\theta)\triangleq\frac{\overline{BD}}{\overline{AB}} \] Using these, let us look at figure 2. This figure will serve to evaluate bounds on the trigonometric functions for small angles (\(0 < \theta < 1 \)) Let us denote the length of the curve \(BE\), which is a circular arc, by \(L\). It is clear that \[ \overline{BD} < L < \overline{BF} \] (An alternative way to demonstrate this is through areas, as triangle ABD is a strict subset of sector ABE which is a strict subset of triangle ABF.) Using the definitions above, and defining \(\theta=L/\overline{AB}\), we have: \[ \frac{\overline{BD}}{\overline{AB}}=\sin(\theta) < \frac{L}{\overline{AB}}=\theta < \frac{\overline{BF}}{{\overline{AB}}}=\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)} \] And so it follows that \[ \theta\cdot\cos(\theta) < \sin(\theta) <\theta \] It follows from the Pythagorean theorem that \[ \sin(\theta)^2+\cos(\theta)^2=1 \] From which we find: \[ \cos(\theta)^2 > 1-\theta^2 > (1-\theta^2)^2 \] The last inequality following from the fact that \(0 < \theta < 1 \). We thus find \[ 1-\theta^2 < \cos(\theta) < 1 \\ \theta-\theta^3 < \sin(\theta) <\theta \] Let us now find the summation formulas for sine and cosine. These are easily found using the construction in figure 3. \[ RB=QA \;\;\;\;\;\;\;\;\;\; RQ=BA \] \[ \frac{RQ}{PQ}=\frac{QA}{OQ}=\sin(\alpha) \;\;\;\;\;\;\;\; \frac{PR}{PQ}=\frac{OA}{OQ}=\cos(\alpha) \] \[ \frac{PQ}{OP}=\sin(\beta) \;\;\;\;\;\;\;\; \frac{OQ}{OP}=\cos(\beta) \] \[ \frac{PB}{OP}=\sin(\alpha+\beta) \;\;\;\;\;\;\;\; \frac{OB}{OP}=\cos(\alpha+\beta) \] \[ PB=PR+RB=\frac{OA}{OQ}PQ+QA \] \[ \frac{PB}{OP}=\frac{OA}{OQ}\frac{PQ}{OP}+\frac{QA}{OP}=\frac{OA}{OQ}\frac{PQ}{OP}+\frac{QA}{OQ}\frac{OQ}{OP} \] \[ \sin(\alpha+\beta)=\cos(\alpha)\sin(\beta)+\sin(\alpha)\cos(\beta) \] \[ OB=OA-BA=\frac{OA}{OQ}OQ-\frac{BA}{PQ}PQ \] \[ \frac{OB}{OP}=\frac{OA}{OQ}\frac{OQ}{OP}-\frac{BA}{PQ}\frac{PQ}{OP} \] \[ \cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta) \]

Complex Numbers

Complex numbers can be defined and used in the usual way, namely, as algebraic objects with the symbol \(i\) having the property that \(i^2=-1\). Additionally, we can define the norm of a complex number as \(|a+bi|^2=a^2+b^2\). Some simple theorems we will make use of: \[ (a+bi)\cdot (c+di)=(ac-bd)+i(ad+bc) \\ |(a+bi)\cdot (c+di)|=|a+bi|\cdot|c+di| \\ \frac{1}{a+bi}=\frac{a-bi}{a^2+b^2} \] Let us define the function \[ \mathrm{cis}(x)=\cos(x)+i\sin(x) \] This function has the property that \[ \mathrm{cis}(\alpha)\cdot \mathrm{cis}(\beta)= \left (\cos(\alpha)+i\sin(\alpha) \right ) \cdot \left(\cos(\beta)+i\sin(\beta) \right ) \\ \mathrm{cis}(\alpha)\cdot \mathrm{cis}(\beta)= \left (\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta) \right ) + i\left(\sin(\alpha)\cos(\beta)+\sin(\beta)\cos(\alpha) \right ) \\ \mathrm{cis}(\alpha)\cdot \mathrm{cis}(\beta)= \cos(\alpha+\beta) + i\sin(\alpha+\beta) \] And thus \(\mathrm{cis}(\alpha)\cdot \mathrm{cis}(\beta)=\mathrm{cis}(\alpha+\beta)\). It follows by induction and the definition of the exponential that, for any natural number \(n\): \[ (\mathrm{cis}(x))^n=\mathrm{cis}(nx) \] And, thus \[ \mathrm{cis}(x)=(\mathrm{cis}(x/n))^n \] Importantly, this is true in the limit of large \(n\). We can always pick \(n\) large enough to make \(x/n\) as small as needed. Thus, we can use the inequalities derived above, namely: \[ \mathrm{cis}\left ( \frac{x}{n} \right )=1+i\frac{x}{n}-\frac{x^2}{n^2}g(x) \] Let \(g(x)=g_r(x)+i g_i(x)\) where \(g_r, g_i\) are real. Then \(0 < g_r(x) < 1\) and \(0 < g_i(x) < \tfrac{x}{n} \). Clearly, then, for \(n > |x|\), \[ |g(x)|^2 < 1+\frac{x^2}{n^2} < 2 \] And so \(|g(x)| < 2\). Also important to note is that a generic complex number can be written as \[ z=a+bi=r \cdot\mathrm{cis}(\theta) \] Where \(r=|z|\) and \(\theta\) satisfies \(r \cos(\theta)=a, \;\;\; r \sin(\theta)=b\). From the above geometric argument, assuming \(a,b > 0\) we have \[ \sin(\theta)=\frac{b}{|z|} < \theta < \tan(\theta)=\frac{b}{a} \] From the fact that \((\mathrm{cis}(x))^n=\mathrm{cis}(nx)\), we find that \[ z^n=(a+bi)^n=r^n \cdot\mathrm{cis}(n\theta) \]

A Lemma for a Family of Limits

From the above we have, for \(0 < x < 1\): \[ 2^{x} < 1+x < 4^{x} \] Let \(x=B/n^2 \), for \(B>0\) and sufficiently large \(n\). Then \[ 2^{B/n^2} < 1+\frac{B}{n^2} < 4^{B/n^2} \\ 2^{B/n} < \left (1+\frac{B}{n^2} \right )^n < 4^{B/n} \] In the limit of large \(n\), \(B/n \to 0\). As \(2^0=4^0=1\), we have \[ \underset{n \to \infty}{\lim} \left (1+\frac{B}{n^2} \right )^n=1 \] A similar argument applies to the case that \(B < 0\). In fact, suppose B is complex, then: \[ \underset{n \to \infty}{\lim} \left (1+\frac{B}{n^2} \right )^n =\underset{n \to \infty}{\lim} \left |1+\frac{B}{n^2} \right |^n \mathrm{cis}\left ( n\theta \right ) \] Where \[ \frac{1+\frac{B}{n^2}}{\left | 1+\frac{B}{n^2} \right |}=\mathrm{cis}(\theta) \] For sufficiently large \(n\), the real part is always positive. It's clear that \(-|B|/n^2 \leq b \leq |B|/n^2\), and so \(-\frac{|B|}{n^2} \leq \theta \leq \frac{|B|}{n^2}\). It clearly follows that \(-\frac{|B|}{n} \leq n\theta \leq \frac{|B|}{n}\). Thus, i nthe limit of large n, \(n\theta \to 0\), and so \(\mathrm{cis}(n\theta)\to 1\). Therefore, for all complex \(B\): \[ \underset{n \to \infty}{\lim} \left (1+\frac{B}{n^2} \right )^n=1 \] Finally, let us note that \[ 1+\frac{A}{n}+\frac{B}{n^2}=\left ( 1+\frac{A}{n} \right )\frac{1+\frac{A}{n}+\frac{B}{n^2}}{1+\frac{A}{n}}= \left ( 1+\frac{A}{n} \right )\left ( 1+\frac{1}{n^2}\frac{B}{1+\frac{A}{n}} \right ) \] For sufficiently large \(n\), we have, then \[ \left |\frac{B}{1+\frac{A}{n}} \right | < 2|B| \] It follows from the above that \[ \underset{n \to \infty}{\lim}\left (1+\frac{A}{n}+\frac{B}{n^2} \right )^n=\underset{n \to \infty}{\lim}\left ( 1+\frac{A}{n} \right )^n \] Clearly this applies to any \(B(n)\) such that, there is some M such that, for \(n>M\), \(|B(n)| < K\) for some real \(K>0\).

Euler's Formula and Identity

We recall the following from a previous section: \[ \mathrm{cis}(x)=(\mathrm{cis}(x/n))^n \] And, for sufficiently large \(n\): \[ \mathrm{cis}\left ( \frac{x}{n} \right )=1+i\frac{x}{n}-\frac{x^2}{n^2}g(x) \] Where \(|g(x)| < 2\). Combining yields: \[ \mathrm{cis}(x)=\left ( 1+i\frac{x}{n}-\frac{x^2}{n^2}g(x) \right )^n \] Equality must hold in the limit of large \(n\), and so, using the above lemma, we have: \[ \mathrm{cis}(x)=\underset{n \to \infty}{\lim}\left ( 1+i\frac{x}{n} \right )^n \] Using the limit definition of \(e^x\), this yields, at last, Euler's celebrated formula: \[ e^{ix}=\cos(x)+i\sin(x) \] This has the special case, by the definition of \(\pi\) and the trigonometric functions: \[ e^{i\pi}+1=0 \] Using the power series expansion for the exponential function, and equating realand imaginary parts yields the two power series expansions: \[ \cos(x)=1+\sum_{k=1}^{\infty}\frac{(-x^2)^k}{(2k)!} \\ \sin(x)=\sum_{k=0}^{\infty}\frac{(-1)^k x^{2k+1}}{(2k+1)!} \]