Boxing Pythagoras

Philosophy from the mind of a fighter

More on 0.999…=1

In my last post, I discussed a particular video which I found to be more than a bit misleading. The discussion centered around a simple, but extremely counterintuitive notion of mathematics: the fact that the number 0.999…, or zero-point-nine-repeating, is equal to 1.

Well, as I mentioned, the very counterintuitive nature of the result led at least one of my readers to question its validity. As such, I thought I would lay out one proof of this concept, in order to make it easier for those who do not accept the result to pinpoint exactly where they disagree. I’ll break my proof down into numbered steps, to ease in that venture.

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Yet another failed attempt at showing 0.999…≠1

I’ve discussed before how mathematics can sometimes lead to very counterintuitive results. One of the most common, and famous, of these counterintuitive properties of math is that the number 0.999… (that is, zero point nine, nine, nine, repeating) is equal to 1. This one is so well known that it is fairly often taught even to Elementary and High School students. If you are unfamiliar with this discussion, I highly recommend that you watch this video from Vi Hart, in which she discusses 10 different reasons to accept this concept. Additionally, you may have fun watching this video, in which she lampoons the common objections to the concept.

Despite the fact that it is fairly simple to prove that 0.999…=1, the concept is so counterintuitive that I find people try to struggle against it– even when they know and accept the reasoning behind the equality. One such attempt comes from Presh Talwalkar. In the following video, Mr. Talwalkar attempts to demonstrate that on the Surreal number system, 0.999…≠1.

Unfortunately for Mr. Talwalkar, he is wrong. Even on the Surreals, it is still true that 0.999…=1.

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Commentary on my Catholic Answers call

On Monday night, I called into the Catholic Answers radio program to give the reason why I am an atheist. My stated reason was that I have not been offered any convincing reasons to believe that deity exists, and the discussion quickly turned to the subject of the Cosmological family of arguments. Unfortunately, a live call-in program does not offer the best forum for back-and-forth discussion, so I wanted to take some time to respond to a number of the things which Trent Horn said, in our dialogue.

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My brief call to the Catholic Answers radio show

This past Monday night, the Catholic Answers Live radio show on EWTN hosted an episode in which they asked, “Why are you an Atheist?” They restricted callers to atheists or agnostics, and asked those respondents to tell the hosts, Patrick Coffin and Trent Horn, why they are either atheist or agnostic. Anyone who made it on the air would be sent a free copy of Trent Horn’s book, Answering Atheism. Curious to see how the apologists would respond to my position– and also, to be sure, looking to get a free book– I called in to the show to offer my position. The audio recording of Monday night’s discussion is available here, but for those of you who do not want to take the time to listen to the whole show, I’ve transcribed my discussion in this article.

I would like to say that it was an absolute pleasure to talk to Trent about this subject, and I found him to be utterly sincere, entirely respectful, and genuinely interested in having a dialogue. I can honestly say that, brief as it was, this was one of the most enjoyable conversations I’ve ever had with an apologist. I am very much looking forward to receiving Trent’s book, now, and I promise that I will review it here on Boxing Pythagoras.

Also, I would be remiss if I didn’t give a huge thank you to Elaine, the call screener for the show. She seemed an absolutely lovely woman, extremely kind and respectful. After my call dropped due entirely to problems on my end, she very graciously moved me back to the head of the line when I called back in.

The transcript follows, edited only slightly at the very beginning, due to my phone troubles.

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WLC doesn’t understand infinity, Part 2

In my previous article, we began to take aim at William Lane Craig’s misconceptions regarding the nature of infinity. We continue on that theme, today, by taking a look at the further arguments which Dr. Craig makes in Part 10 of his Excursus on Natural Theology. While most of the objections which Dr. Craig espouses in this episode fall prey to the same mistakes which he was making last time, I still thought it might be instructive to respond to each one, in turn. Suffice to say, the arguments which Dr. Craig levies this time around are absolutely no better than the ones which he raised previously.

In fact, I’d argue that– for the most part– they are far worse.

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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.

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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.

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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.

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The Grim Reaper Paradox

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.

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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.

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