Boxing Pythagoras

Philosophy from the mind of a fighter

Wildberger says that Banach-Tarski is Nonsense

Dr. N.J. Wildberger has added a new video to his “Sociology and Mathematics” series in which he discusses the Banach-Tarski Paradox. If you are unfamiliar with this particular concept, it suffices to say that Banach-Tarski illustrates some very peculiar and counterintuitive properties of infinite sets. Fairly unsurprisingly for anyone familiar with Dr. Wildberger’s work, he considers the entire discussion undertaken by Banach-Tarski to be nothing but nonsense. In the video, Dr. Wildberger explicitly notes that he rejects the Axiom of Choice (one of the major axioms upon which Banach-Tarski relies) and I have discussed previously that he also rejects the Axiom of Infinity (which is similarly necessary for Banach-Tarski). Thus, Dr. Wildberger’s video (and his original blog post which inspired the video) seemed fairly curious to me.

Yes, of course the Banach-Tarski Paradox is nonsense if you reject the axioms upon which it depends. Any and every mathematical theorem in existence would be nonsensical to a person who rejected the axioms underlying that theorem.

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Review of Craig v. Malpass, Part 1

On March 24th of this year, Cameron Bertuzzi’s channel on YouTube, Capturing Christianity, streamed a discussion between William Lane Craig and Alex Malpass. Nominally, the topic of debate was “Did the universe begin to exist?” However, their actual discussion was quite a bit more focused onto two very particular subjects. In part one of this review, we’ll look at the discussion of whether actual infinites are metaphysically possible. In the forthcoming part two, we’ll discuss the manner in which actual infinites are constructed.

As I am keenly interested in these particular questions, I was very excited for this discussion. I’ve discussed my contention with Dr. Craig’s treatment of the mathematics of infinity on a few occasions (most directly, here and here) but this particular debate brings forth some issues with which I have not previously engaged.

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On Time, Aristotle, and Relativity


As I have noted many times over the years, I believe Aristotle’s metaphysics to be every bit as antiquated and outmoded as are his physics. I have expressed wonderment at the fact that everyone seems to have rejected his notions that the Earth is the center of the universe, that heavy things fall faster than light things, that the sky is composed of aetherial spheres, and a great many other things; and yet there are philosophers who ardently and doggedly remain attached to the ideas of hylomorphism, finitism, and– particularly– act and potency.

This latest notion has been a topic of discussion on Boxing Pythagoras very nearly since the start. One of my earliest articles was on William Lane Craig’s Theory of Time which is not explicitly Aristotelian but which is nonetheless predicated upon similar notions to act and potency. This has factored into my discussions on a range of other topics, including the Kalam Cosmological Argument, which explores the implications of the temporal finitude of the universe; the Grim Reaper Paradox, which purports to give good logical reasons to doubt the existence of actual infinities; Free Will and Determinism, regarding how to reconcile the notion of free-will with wholly extant Time; Infinity and Eternity, wherein I discuss how even a universe which does not extend infinitely into the past can be eternal; and most germane to our discussion today, Thomas Aquinas’ Five Ways, in which the eminent 13th-Century philosopher attempted to demonstrate the necessity of God’s existence explicitly through the Aristotelian notion of act and potency.

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Answering 36 Questions for Atheists

Every now and again, I stumble across a list of questions posed by a person on one side of an argument towards those on the other side. These questions are intended to be somewhat Socratic, leading a person towards an intended conclusion solely through his or her own answers; however, they very rarely actually have that effect.

I came upon just this sort of list from a blog called Adherent Apologetics. I’ve interacted with that blog’s author a few times before and found him very respectful and sincere; as such, I decided to take a few minutes to set my answers to his questions to page in the hope that we might begin to get a better understanding of one another.

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A review of the Dillahunty v. Jones debate, Part 2

In my first post on this debate, I focused on the issues which I had with Matt Dillahunty’s performance in this debate, particularly in the ways in which he responded to Michael Jones’ claims. Matt generally did not deal directly with Jones’ positions, but rather attempted to undermine them on foundational reasons and, unfortunately, I think he missed the mark with his responses.

In this second article on the debate, I intend to discuss the arguments which Michael Jones raises, despite the fact that he does not really get much time to discuss their nuances directly in the particular debate. His arguments are interesting and different than the usual rehashing of the Kalam or Fine-Tuning or Anselm’s Ontological arguments. That, alone, is often more than enough to grab my attention. After having been interested in apologetics for 20 years prior to being an outspoken atheist for another 10 years, it’s a rare thing, indeed, to be presented with an entirely unfamiliar argument for the existence of God.

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A review of the Dillahunty v. Jones debate, Part 1

As anyone who has read much of Boxing Pythagoras will likely already know, I am rather fond of debate. I quite enjoy listening to, reading, and watching debates between two people on a great variety of subjects– all the moreso when the subject happens to be one with which I am heartily interested. Recently, two YouTubers got together for just such a discussion. Matt Dillahunty is the president of the Atheist Community of Austin and one of the hosts of a broadcast called The Atheist Experience. Michael Jones is a Christian apologist who created a YouTube channel called “Inspiring Philosophy.” Both men are very intelligent and I have greatly enjoyed listening to both, in the past. When I heard that they were going to debate the topic, “Are there good reasons to believe in God?” I was quite excited to give it a listen.

Unfortunately, I was very let down by the debate. I spent a great deal of the two-hour long discussion cringing and yelling at my computer. While it might be easy for one of my readers to think my discomfort was caused by the Christian conversant, the truth is that it was the atheist who had me so upset. This was honestly one of the worst debate performances on the part of Matt Dillahunty which I have ever seen.

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Classical Limits vs. Non-Standard Limits

One of the most important and fundamental concepts taught in modern Calculus classes is that of the Limit. I have discussed this idea once before, but I thought I would revisit it, here. In that first article, I noted that the classical definition for a Limit is fairly complex and that we can utilize a more intuitive notion of infinitesimals to accomplish the same task, insofar as derivatives are concerned. However, there are other uses and purposes for limits, in mathematics, so we would not want to simply omit them entirely, even using a non-standard approach to Calculus lessons.

Thankfully, even the very difficult and complex definition of “limit” can be simplified and made easier to understand by use of non-standard infinitesimals.

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Intuitionism and the Excluded Middle

Introductory lessons on Logic often make note of three basic, but powerful, principles which are so universally recognized that they are commonly referred to as the Laws of Logic. The first is the Law of Identity which states something like, “A thing is equal to itself.” The second is the Law of Non-Contradiction, sometimes phrased as, “A proposition cannot be both true and false at the same time.” The third is known as the Law of the Excluded Middle which declares, “Either a given proposition is true or else its negation is true.”

A classic example of the Law of Identity might be, “Socrates is Socrates.” An illustration of Non-Contradiction could be, “Socrates cannot both be mortal and not be mortal at the same time.” For the Excluded Middle, we would say, “Either Socrates is mortal or else Socrates is not mortal.” This all seems perfectly obvious and simple, even to complete beginners in the study of Logic.

However, one might be surprised to learn that the Law of Excluded Middle is actually a source of some controversy in philosophy– particularly in the Philosophy of Mathematics, where there exists a small but strong community which rejects this principle vehemently.

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On Causal Simultaneity

Several times over the life of this blog, I have discussed the Kalam Cosmological Argument– as well as other, similar cosmological arguments for God. They are exceedingly popular topics within theistic apologetics and are therefore levied quite often. As a reminder, the Kalam is often formulated as follows:

  1. Anything which begins to exist has a cause for its existence.
  2. The universe began to exist.
  3. Therefore, the universe has a cause for its existence.

The theistic apologist will usually then argue that the only possible cause for the universe’s existence must be God. Detractors and critics of these cosmological arguments, like myself, often point out that this doesn’t seem to make much sense when applied to the universe, as a whole. After all, “the universe” includes time, and if the universe began to exist, then that implies that there must have been a first moment of time. However, if there is a first moment of time, the universe exists in that moment; and since there are no moments prior to the first, there is literally no time prior to the universe’s existence during which it could have been caused.

There cannot have been anything which existed before the universe because there is literally no such thing as “before the universe.”

The prolific philosopher and theologian, Dr. William Lane Craig, addresses this issue in a manner which I find to be rather curious. Dr. Craig acknowledges that it is nonsensical to assert that there must have been something before the universe which caused the universe to exist. Instead, he invokes the notion of causal simultaneity— that is, the idea that a cause can be simultaneous with its effect. With such a notion in place, Dr. Craig argues that the implications of the Kalam are salvaged and that God must still be the cause of the universe.

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The Axiom of Infinity

In my previous post introducing the concept of Set Theory, we discussed one method for constructing the Natural numbers– a method often referred to as a Von Neumann construction. Using that method, we start with the Empty Set (\emptyset) and then systematically build the Natural numbers by following a rule. As described in that post, this was a step-wise process: look at a number, find its successor, look at the new number, find its successor, repeatĀ ad infinitum. Now, obviously, given a finite amount of time there would be no way to perform this process enough times to generate every Natural number, since every new number we create would still have yet another number succeeding it.

But what if we want to discuss the whole set of Natural numbers?

As we just noted, we cannot construct the Natural numbers in a step-wise manner in order to get all of them. However, mathematicians like Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem devised a very clever way to take the very same ideas from our step-wise construction in order to discuss a whole, completed set. We refer to this notion as the Axiom of Infinity, and it is one of the premises which underlies the vast majority of modern mathematics.

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