# Boxing Pythagoras

## 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.

## 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…

## 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.

## 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.

## 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.

## WLC’s Time, Part 2: Einstein the Verificationist

Originally, I had intended my first article on William Lane Craig’s Theory of Time to be a one-time affair. I stated the basics of my position, laid out my conclusions, and was ready to move on. My final thought, in the article, was that WLC’s Theory of Time is circular: he adheres to the Tensed Theory of Time due to his acceptance of Lorentzian Relativity, and he accepts Lorentzian Relativity due to his adherence to the Tensed Theory of Time. However, on his podcast released this week, Dr. Craig addresses a similarly founded accusation of circular argumentation which was given by a blogger who calls himself, “A Counter Apologist.” While the claim from A Counter Apologist deals specifically with the Kalam Cosmological Argument, he does so by addressing WLC’s Theory of Time as it conflicts with Relativity, in much the same way as my article approached the subject. In his response, Dr. Craig claims that his support of the Tensed Theory of Time is supported by more than just his preference for it, and that he has laid out his arguments for this in his published works. It occurred, to me, that perhaps I was being unfair. My first article was based on a seminar which I had seen Dr. Craig give, rather than on his books. Perhaps, in his written work, I would find that WLC provides greater support for the Tensed Theory.

I’m starting with the arguments presented in Dr. Craig’s book for the popular audience, Time and Eternity: Exploring God’s Relationship to Time (Crossway, 2001).  If I don’t find this work convincing or satisfactory, I’ll try to continue into his more scholarly works on the subject, The Tensed Theory of Time: A Critical Examination (Springer, 2000) and The Tenseless Theory of Time: A Critical Examination (Springer, 2000).

## Philosophy from the Mind of a Fighter

Pythagoras was a boxer. Plato was a wrestler. Xenophon was a soldier. Marcus Aurelius directed armies. There is a rich history of philosophers who were also fighters– or perhaps fighters who were also philosophers. And these two seemingly disparate endeavors have much more in common than most people realize. When people think of philosophy, they often conjure images of frail intellectuals discussing lofty ideals and contemplating nigh incomprehensible trivialities with like minded men. When people think of fighters, they often imagine brutish lugs thrashing at one another with neither thought nor civility. Both of these stereotypes are false. Philosophers have been some of the most brash, combative men in history; and I have personally known fighters who are absolutely brilliantly intellectual and incomparably kind. The truth is that neither philosophy nor fighting is really what most people believe them to be, and that these two concepts share a great deal more in common than most would realize.

## William Lane Craig’s Theory of Time

William Lane Craig is one of the most noted, well-known, and respected Christian apologists in the world. He is an accomplished philosopher and theologian with a very broad knowledge base and a particular acumen for public debate. He presents his positions with clarity, and he deftly anticipates most of the arguments which his opponents might present. Dr. Craig prepares his work with exceptional forethought and thoroughness, in order to present rational, cogent, and coherent arguments for his case. I have a great deal of respect for William Lane Craig and his work, despite my disagreement with it.

That said, I find myself inordinately perplexed that WLC maintains his death-grip on the idea of a Tensed Theory of Time.

## Heathen Apologetics, Part 1: Pascal’s Wager

When I was in high school, I took a class called “Myths, Dreams, and Cultures” (or MDC, for short). Now, I attended a fairly upscale private Catholic school, and all but a very few students came from Christian upbringings. As such, one of the main goals of MDC was to open the students up to the understanding that we were a product of our culture, and that other cultures often had very different pictures of the world. To that end, we would play a game called “Stump the Aborigine.” Our teacher took on the role of an idealized traditionalist Native American, living off the land, completely divorced from modern, American culture. Our job was to be the representatives of Modern America in order to convince this “primitive” of the superiority of our way of life. If anyone could succeed at convincing the aborigine to leave his culture for the one we were selling, our teacher promised to buy that student a brand-new car.

It did not take long for me to realize that the game was rigged.

## Basics of Epistemology

Pretend, for a moment, that someone has just made a claim, and you are trying to decide whether or not that claim is true. How do you go about evalutating their assertion? How do you decide if that person is right or wrong? How do you know the truth?

Epistemology is the study of how we know what we know. It is the area of philosophy which attempts to create methodologies for discerning truth from falsehood. So, for example, if I were to claim, “I am wearing a green shirt,” how could you verify or disconfirm my statement? Well, if you’re in the room with me, you could simply look at my shirt. However, what if you’re not in the room with me? You might place a phone call to someone who IS with me, and ask them– but how would you know that this new person was telling the truth, and not lying?