# Boxing Pythagoras

## WLC doesn’t understand Infinity, Part 1

One of the topics which William Lane Craig often discusses is a question which has been argued in the Philosophy of Mathematics for at least 2300 years. Can an infinite number of things actually exist? Dr. Craig asserts that such actual infinites cannot exist. This is actually a topic which I have discussed before, on this blog, but Dr. Craig attempts to tackle the question quite differently than does Dr. Wildberger. Interestingly, Dr. Wildberger is a mathematician, and most of my objections to his argument pointed out his unfamiliarity with philosophy; while Dr. Craig, on the other hand, is a philosopher, and most of my objections to his argument will point out his unfamiliarity with mathematics.

Dr. Craig has discussed the topic of actual infinities in a number of different places, but I will be referring to his Excursus on Natural Theology, Part 9, for our discussion today. These are the same arguments which I have generally seen Dr. Craig present in his other work, but this happens to be the most recent exploration of the topic from WLC which is available to us.

Unfortunately, just as he has done many times before (see here and here, for example), William Lane Craig demonstrates that he has a rather poor grasp of the mathematics he’s attempting to discuss.

## WLC on the Speed of Light

I’ve been listening to a series from William Lane Craig’s Defenders podcast entitled “An Excursus on Natural Theology,” over at the Reasonable Faith website, of late. Needless to say, I have a lot I would like to say about almost the entirety of the series. However, today, I’m going to focus on a minor point which Dr. Craig makes in Part 6 of the series. Now, to be completely fair, this point is only tangential Dr. Craig’s overarching claims. By no means am I attempting to imply that the problems with this one issue somehow refute his whole Excursus– I’ll be dedicating a whole new series of posts to that, in the future. However, I chose to focus on this very minor point made by Dr. Craig for another reason entirely.

Once again, William Lane Craig has demonstrated himself to be rather ignorant in regards to the science which he attempts to discuss.

## The Universe Has Always Existed

As will be patently obvious to anyone who has read much of my blog, I am incredibly fascinated by the question of Time and the description of the universe’s history. The topic is incredibly complex and wonderfully intricate. Unfortunately, these peculiarities can very often lead to very common misconceptions. One of the misconceptions which I encounter most often is the idea that there was once a state in which the universe did not exist.

This misconception has arisen because, over the past century, it has become increasingly plausible that the universe may not extend infinitely into the past. Thanks to Big Bang Cosmology, the previously prevailing view of Aristotle that the universe is static and eternal has been almost entirely abandoned. It is entirely possible– and perhaps even likely, given certain assumptions– that the universe has a finite history. That is to say, there was a first Moment of Time. Given this, people naturally wonder, “Well, what happened before that?” Unfortunately, these people don’t realize that the question which they are asking is entirely nonsensical.

Whether the universe is past-finite or past-infinite, it has always existed.

There is a long tradition, in philosophy, of employing paradoxical thought experiments in order to show that our understanding of some subject is either incomplete or incorrect. Quite famously, the paradoxes of Zeno of Elea puzzled philosophers and mathematicians for millennia. These enigmas can be, at once, immensely entertaining and thoroughly maddening to contemplate.

About a year ago, I was introduced to one such thought experiment which I had not previously encountered. It is known as the Grim Reaper Paradox, and the version with which I will interact today is presented by philosopher Alexander Pruss. The thought experiment proceeds as follows:

Fred is sitting in a room at 8:00 am. There exists an infinite number of Grim Reapers along with Fred, each of which is currently dormant. When any individual Grim Reaper becomes activated, if Fred is still alive, then that Reaper will instantaneously kill Fred; however, if Fred is not alive, the Reaper will return to a dormant state and continue to do nothing. Each of the Grim Reapers is timed to activate at a specific time after 8:00 am. The last Reaper will activate at 9:00 am. The second to last activates at 8:30 am. The third from last at 8:15 am. In general, the nth from last Grim Reaper will activate after $\frac {1}{2^{n+1}}$ hours have passed.

Now, we are guaranteed that Fred will not survive past 9:00 am. After all, if he is alive at 9:00 am, then the last Grim Reaper will activate and kill him. However, he can’t have lasted that long, either, since the previous Grim Reaper would have killed him if he had survived until it activated. In fact, we can generalize this: the nth from last Grim Reaper cannot have killed Fred, because if he had survived until $\frac {1}{2^{n+1}}$ hours after 8:00 am, then the (n+1)st from last Grim Reaper would have killed him.

Therefore, we see that Fred cannot survive until 9:00 am, and yet we have also shown (by mathematical induction) that none of the Grim Reapers can have been the one which killed Fred. Thus, we have come to a paradox.

## Never step onto the road without your weapons

Let a man never stir on his road a step
without his weapons of war;
for unsure is the knowing when need shall arise
of a spear on the way without.

-Hávamál 38

Though I do agree with the literal interpretation of this passage, there is far more wisdom in it than the bare surface reading would suggest. This is a proverb of preparedness. While the terms “weapons of war” and “spear” certainly carry their obvious meaning, they also stand as metaphors for any tool which might aid one in a struggle. This applies just as much to knowledge and wisdom as to weaponry; it refers to planning one’s finances as much as planning for a fight; it’s as applicable to a good pair of boots just as much as to a spear.

Now, obviously, one cannot be prepared for all situations at all times, but the more one considers his own preparedness, the better he will be when necessities arise. Personally, I find this passage to be extremely poignant with respect to one’s understanding of his own philosophy. As someone who engages in conversation and debate rather frequently over the subject, I have found myself often calling on some esoteric body of trivial knowledge which many people would wonder that I had bothered to learn in the first place: the incredible examples of Corvid intelligence, the strange implications of a particular grammatical construction in ancient Greek, the intricate symmetries of the openings in a game of Go, and many other beautiful– but peculiar– bits of knowledge. These are all things which a person could well live without knowing, and yet each of them has been incredibly useful to me in philosophical discussions on subjects which are seemingly unrelated to those bodies of knowledge.

I have never in my adult life been in a fight outside of the parameters of my martial arts. And yet, my Jiu-Jitsu training has been immensely useful in many other aspects of my life. Though the passage explicitly mentions “weapons of war,” the need of a spear does not necessarily refer to a martial need. Yes, a spear can be used for war. But it can also provide food. It can steady a weary tread. It can lever a wheel out of the mud. It can perform any number of tasks beyond its primary intention. The same is true for all weaponry, whether made of wood and steel or knowledge and wisdom. Never step onto the road without your weapons.

## WLC on Time, Part 6: Did the Universe Begin?

William Lane Craig has dedicated a good portion of his career to the concept of Time. Unfortunately, he has not invested the time necessary into learning the mathematics and physics which are necessary to discuss the concept cogently. Dr. Craig is a philosopher of religion, not a philosopher of science. He is a theologian, not a scientist. So, when William Lane Craig posts a podcast to his Reasonable Faith website in which he upbraids someone who is an accomplished and well-respected scientist for that person’s understanding of science, I have to say that I am more than a bit skeptical.

In the podcast, Dr. Craig is responding to an interview of Dr. Sean Carroll, a prominent cosmologist, by Robert Kuhn for the program, Closer to Truth. If you would like to see the relevant portions of this interview, you can find them here, along with several other clips. Dr. Craig’s podcast makes specific reference to the clips entitled What would an Infinite Universe Mean? and Did the Universe Begin?, but I recommend the other clips, as well– particularly, Is Time Real?, as it is closely related to our topic at hand.

William Lane Craig has a very poor understanding of the science which he attempts to discuss, and as a result, he once again leaps to false conclusions.

## Math is Really Weird: On Strange Sums and Counterintuitive Results

Whenever you add a finite integer to another finite integer, you always get a sum which is, itself, a finite integer. This, by itself, is not very shocking. When you add 1 to 1, you get 2. When you add 5 and -9, you get -4. When you add 0 and 299,792,458, you get 299,792,458. This is all rather unsurprising.

However, math can get weird once you start adding up an infinite collection of numbers. Take Zeno’s Dichotomy Paradox, for example. Numerically, we can represent this problem as an infinite summation: $S=\sum\limits _{n=1}^\infty \frac{1}{2^n}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...+\frac{1}{2^n}+...$ Even though we are adding up an infinite quantity of numbers, we arrive at a finite value– in this case, $S=1$. Arguably the most famous philosopher in history, Aristotle, would have vehemently objected to this formulation– and, in fact, did object rather loudly in his book Physics, when discussing this particular paradox. However, it has been over three centuries, now, since mathematicians would have found this problem to be controversial; and, in fact, similar cases of infinite summation form the entire basis of integral calculus. High schoolers are introduced to these concepts in their Pre-Calculus classes, nowadays, and you might even remember evaluating some of these limits of convergent sums from your own schoolwork.

But math can get far stranger, still. One of the most peculiar things in all mathematics occurs when you attempt to sum all of the Natural numbers. As absolutely insane as this might sound, today I’m going to demonstrate for you that $1+2+3+4+...=-\frac{1}{12}$.

## Some Unfortunate Choices in Mathematics Terminology

Words can be tricky things. The same word can often carry wholly different meanings depending upon the context in which it is used. Take, for instance, the semantic range of the word “light.” This word can carry very different meanings when used in different contexts, as the following sentences illustrate.

1. That feather is light.
2. That shade of pink is light.
3. That laserbeam is light.

Each one of these sentences is of the form “That <noun phrase> is light,” but the word “light” intends an entirely different thing, in each. In (1), “light” is a description of the weight of the feather. In (2), “light” is a description of the intensity of the shade of pink. In (3), “light” is a description of the physical nature of the laserbeam. There is a well known fallacy of logic called equivocation which involves conflating such definitions in order to arrive at a false conclusion. For example, if I said…

1. Light things weigh less than heavy things
2. This shade of pink is light
3. Therefore, this shade of pink weighs less than heavy things

…my logic would be invalid, because the definitions of “light” used in (1) and (2) are completely different.

Mathematics, unfortunately, contains some terminology which tends to lead to these same sorts of equivocation fallacies, because the common usage of a word very often differs from the mathematical usage of that word. While there are numerous examples from which I could likely choose, today I’m going to focus on a case which I believe to be particularly egregious. Today, I’m going to discuss Real and Imaginary numbers.

## A Nativity Story

The man stood, dumbfounded, attempting to comprehend the news he had just been given. The Messenger of God had just informed him that his wife– his Virgin– was now with child, and would present him with a son who would surpass all who had ever lived in beauty and wisdom. The boy to come would be a divine gift, the greatest beneficence that the human race would ever know. Though coming into this world by birth through woman, the infant boy had in fact pre-existed his human form, and was wholly divine in nature. The man rushed home to his wife to find that the Messenger of God had spoken true. She was, indeed, with child.

Thus, Pythagoras was born into this world.

What? You thought I was talking about somebody else? This account is from the Life of Pythagoras, written by the great Neoplatonist philosopher Iamblichus in the 3rd Century, CE. Now, I will completely admit that I paraphrased the story in order to obfuscate it, a bit. The “Messenger of God” that spoke to Mnesarchus (Pythagoras’ father) was the Oracle at Delphi. And the reason that I capitalized “Virgin” in my paraphrase was because that was the name of Mnesarchus’ wife, before he received this news from the Oracle– Parthenis, in Greek. As soon as he received the announcement from the Oracle, Mnesarchus changed his wife’s name to Pythais, in honor of Pythian Apollo (which, Iamblichus tells us, was also the source for Pythagoras’ name). Still, the paraphrase stands: a Messenger of God informed a man that his wife’s womb had been filled by deity, and that the child would be divine, the greatest gift the world could hope to receive.

## Proof that π=2√3

There is an inherent danger attached to blindly accepting the word of someone who sounds like they are presenting a rational, scientific claim. Too many people are willing to accept a proposition solely because they’ve heard it from someone who bears the appearance of intelligence. The line of thought seems to be, “Well, he’s smarter than me, so he must be right!” Unfortunately, this sort of fallacious reasoning goes largely unchecked, and often becomes formative in the common understanding of entire groups of people.

For almost the entirety of your mathematical education, you have been taught that the ratio of a circle’s circumference to its diameter, which we affectionately refer to as π, is something close to $3\frac{1}{7}$, or about 3.14; however, today I’m going to show you that your math teachers were wrong. In actuality, the value of π is exactly 2√3, or about 3.46.