Boxing Pythagoras

Philosophy from the mind of a fighter

Infinitesimal Calculus 2: The Changes in Change

The mathematics of change are quite interesting. In a naive sense, we can often describe a change by a simple collection of data points. For example, let’s think about a little boy rolling a ball across the floor. The boy pushes the ball, and four seconds later, the ball has come to be 2 meters away from him. Given these data points, we may attempt to connect them in some meaningful analytical manner– perhaps by saying that the ball rolled at a speed of half a meter per second. But even this is a somewhat naive bit of information, as it only really tells us something about the completed journey. Mathematicians are greedy, however; they want to be able to know about every point of the ball’s travel, at any arbitrary moment in time.

We can use a function for just such a purpose. A function is a specific mathematical tool which allows us to describe an entire set of data points all at once. We encode the data by means of a mathematical formula. For example, our exemplary rolling ball might well have been encoded by the function f(x)=\frac{1}{2}x, where the x represents the time, in seconds, that the ball has been rolling, and the value of the function, f(x) tells us the distance in meters which the ball has traveled in that time. In this particular function, the coefficient of x tells us the rate at which distance changes as time passes– that is, \frac{1}{2} a meter per second. When the boy first rolls it, the ball is traveling at \frac{1}{2} a meter per second; when it finishes it had been traveling at \frac{1}{2} a meter per second; and at any single point during the journey the ball is traveling at \frac{1}{2} a meter per second.

However, this is a very simple example. It describes a situation involving a constant velocity. Things become a bit more muddied when the rate at which a change occurs is, itself, changing.

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Infinitesimal Calculus 1: The Numbers Between Numbers

If I were to ask a person to name a number which comes between 1 and 3, everyone from a three-year-old child to a white-bearded great-grandfather is likely to respond by saying, “2.” If I rephrase the question to ask about a number between 1 and 2, then the young child might be confused, but a fourth-grader might be able to respond with 1\frac{1}{2}. We have to extend our understanding of what we mean by “number” to include some concepts which are not quite so intuitive. That is to say, in between the Integers, there are other numbers which are known as Rational numbers. In fact, given any Integer, n, there are an infinite number of Rational numbers which are greater than n and yet less than any other Integer which is greater than n.

There are numbers in between the Rational numbers, too. We can define some number, r, which is not equal to any Rational number. There are Rational numbers which are greater than r, and those which are less than r, but somehow our number r squeezes itself into a gap in between the Rational numbers. In order to find such a number, we need to further extend our understanding of “number” to include the Real numbers. This should all be very familiar to the average high-school student.

Now, what happens if we extend this idea one step further? Are there more numbers which are in between the Real numbers?

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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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Theologically Loaded Language

Translating ancient documents into modern languages carries with it more difficulties than most people realize. Pretty much anyone who has ever taken a foreign language class in high school understands that it can often be quite hard to find a word which corresponds exactly between two tongues. Those who have studied outside of the modern Romance languages– classes like Arabic, Chinese, or Japanese– often realize that there are subtleties in grammatical constructions which can convey a great deal more than can be expressed in English. It is a very frequent occurrence that a phrase from one language cannot be rendered with 100% accuracy in another language. In English, this has led to the popular idiom that “something has been lost in translation.”

Ancient languages maintain these problems, but add an entirely new layer of obfuscation which is not found even in most culturally distinct modern languages. Over the past few thousands of years, human understanding of the world around us has changed quite significantly. Just one hundred years ago, no one had ever viewed the ground from five miles up in the air. Two hundred years ago, we had no idea that microscopic organisms cause disease. Three hundred years ago, humanity had no idea that oxygen exists. Four hundred years ago, the world was shocked to learn the the planet Jupiter has moons. The manner in which religion, philosophy, and science have discussed a myriad of things about reality has changed so greatly in recent millennia that very often even one word in a single language can mean something exceedingly different to people living in different periods of time.

The documents which comprise the New Testament of the Christian Bible were written 2000 years ago. In those ensuing twenty centuries, many of the words used by the original authors and many of the concepts which they espoused have engendered incredible amounts of revision, alteration, and nuance by subsequent philosophers and theologians which would have been wholly alien to those initial ancient writers. The vast majority of modern readers– including an embarassingly large number of modern scholars of the text– seem wholly ignorant of this fact when they read a passage from their Bibles.

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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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WLC dodges his own question

Recently, I have taken to addressing William Lane Craig’s Excursus on Natural Theology podcasts. These are lessons directed at the layperson with the goal of demonstrating the rationality of theism from simple arguments. As you may infer from my previous articles, I do not think that the Excursus has come even close to meeting that goal.

Today, we will be discussing Part 17 of the Excursus. If you read my article on Part 16, you might remember that I was actually quite excited for this, due to Dr. Craig’s promise to discuss the plausibility of Design as an explanation of the universe’s fine-tuning. As I mentioned, whenever I have discussed the idea of Intelligent Design with an apologist, I have brought up this very subject. Unfortunately, I’ve only ever been met with answers about the purported improbability of chance or necessity. I’ve never been proffered any answers with positive evidence for the idea of Design, nor even with a proposed mechanism by which the Fine-Tuning of the universe could be Designed.

Early on in the discussion, Dr. Craig makes a statement with which I wholeheartedly agree:

But we cannot infer immediately to design because sometimes it can be justified to believe in an improbable explanation. You would be justified in believing in some improbable explanation just in case there were no better explanation available of the phenomenon in question…

The question we are facing now with regard to the fine-tuning of the universe is: is design a better explanation than chance or physical necessity?

Yes, this most certainly is the question! So, how does Dr. Craig answer this question? Does he define what, exactly, he means by the term “design?” Does he offer some method for differentiating something which is “designed” from something which is not “designed?” Does he then apply this standard to the question of Fine-Tuning in order to show that the constants and quantities of the universe more keenly fit into the “designed” category than the “not designed” category?

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WLC doesn’t understand cosmology

Over the past few months, I have been listening to Dr. William Lane Craig’s Excursus on Natural Theology, which is a course designed to introduce an audience to reasons for accepting the positions of theism. From time to time, I find that Dr. Craig says something so egregiously wrong that I feel I should address it, here, at Boxing Pythagoras. In two previous articles, I have discussed Dr. Craig’s misconceptions in regards to the mathematical concept of infinity, from parts 9 and 10 of his Excursus. Today, I want to focus on Part 16 of the Excursus in which Dr. Craig talks about the Fine-Tuning problem of cosmology.

Unfortunately for our esteemed theologian, his understanding of cosmology seems to be just as poor as his understanding of mathematics.

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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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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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