Boxing Pythagoras

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Archive for the category “Mathematics”

On Teaching Calculus

Almost universally, when Calculus is taught to modern students, we preface the entire subject by introducing those students to a concept known as a “limit.” The reason for this, historically, was to ensure that mathematics was taught in a rigorous and well-defined manner. When Leibniz (and, independently, Newton) first developed methods for performing calculus, the concept of a limit was nowhere to be found. However, the tool which these men did utilize in their work was something which they had not rigorously defined, at the time. Newton called it a “fluxion” and Leibniz called it a “differential,” but the concept was the same: a number which was not zero, but which was so small that adding it to any Real number did not yield a different Real number.

Many other mathematicians and philosophers of the time rightfully balked at the notion. It seemed entirely ludicrous. Bishop George Berkeley famously scoffed at Newton, asking if his fluxions were “the ghosts of departed quantities.” However, it was quite plain that the mathematics which Leibniz and Newton presented worked. When the results which could be found from the methods of Calculus were able to be confirmed using other methods, they were found to be accurate and true. Indeed, the Calculus was such a powerful tool that even most mathematicians and philosophers who recognized its flaws continued to utilize it in their work. Many began searching for some way to make the Calculus just as rigorous as the rest of mathematics. These efforts culminated in the work of Karl Weierstrass, who found a way to base Calculus upon a different tool. Instead of the Newtonian “fluxion” or the Leibnizian “differential,” Weierstrass gave mathematics a well-defined notion of the limit.

It is Weierstrass’ method of limits which is still taught, even to this day, in nearly every Calculus textbook in the world; but perhaps it is time to abandon this notion and return to the concept which Newton and Leibniz pioneered.

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A Variation on the Grim Reaper Paradox

In one of my earlier posts, I addressed the Grim Reaper paradox and offered my input on a possible resolution of the thought experiment’s curious implications. However, some of my readers may have been dissatisfied with my answer, thinking that it sidestepped around the issue rather than addressing the conundrum directly. A few people asked me why I thought that obscure philosophy on the nature of Time might have any relevance to the question, in the first place. To that end, I have decided to offer a bit more clarification and to attempt to illustrate why I think the Grim Reaper paradox is inherently flawed.

Consider this slightly modified version of the thought experiment…

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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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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 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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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.

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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}.

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