# Boxing Pythagoras

## 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.

## 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.

## 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.

## 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, repeat 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.

## Theology and Indeterminate Infinity

Apologists often claim that actual infinites are logically impossible. One of the arguments which they utilize to support this claim deals with subtracting quantities from infinite quantities. One example of this comes from Blake Giunta’s Belief Map:

Infinity minus an infinity yields logically impossible scenarios. Notably, one can take away identical quantities from identical quantities and arrive at contradictory remainders.

On the face of it, this claim appeals to our intuitive understanding of subtraction. If I were to claim that there exists some Integer, $x$, such that $x-4=7$ and $x-4=19$, then we stumble upon the contradiction that $11=23$. Subtracting identical quantities from identical quantities should yield identical results.

## On Wronski’s Definition of π

Joseph Nebus has recently written a couple of posts (here and here) in which he discusses an interesting attempt by Józef Maria Hoëne-Wronski to create a purely numerical definition of the mathematical constant π which is independent of the classical, geometric definition of “the ratio of the circumference of a circle to its diameter.” This has been a goal of many mathematicians, since the idea of π seems like it is more fundamental to mathematics than a definition based on circles would make it seem– as evidenced by the fact that it shows up in areas of mathematics which are seemingly unrelated to circles. Wronski’s idea, to this end, was the following formula:

$\pi = \frac{4\infty}{\sqrt{-1}}\left\{ \left(1 + \sqrt{-1}\right)^{\frac{1}{\infty}} - \left(1 - \sqrt{-1}\right)^{\frac{1}{\infty}} \right\}$

At first glance, the formula seems inherently nonsensical. After all, $\infty$ is not a number, and therefore cannot be utilized in numerical operations in this way. However, one can get a sense of what Wronski may have intended by this equation. It appears that Wronski wanted to utilize $\infty$ to represent an infinite number, and modern mathematics actually gives us several tools for handling this sort of idea. One which might be of particular use, here, is Non-Standard Analysis with its infinite and infinitesimal Hyperreal numbers. In NSA, we have the ability to perform calculations with and upon infinite numbers perfectly consistently and reasonably.

## Theology and the Actually Infinite

One of the common claims which is utilized in arguments for the existence of God is that actual infinities cannot exist, implying that there cannot be an infinite regress of causal events in the history of the universe. If there cannot be such an infinite regress, then there must be some First Cause. Theologians then put forth other arguments attempting to show that this First Cause must be God. Blake Giunta, a Christian apologist, has constructed a very interesting and quite useful website cataloging common lines of argumentation from both sides of the debate (color coded Green for Christian arguments and Red for opposing arguments), along with citations and documentation for those claims, called BeliefMap.org. It does not take very long for a fairly cursory perusal of Belief Map to bring one to this exact claim regarding the actually infinite.

While I disagree with Mr. Giunta on many of his views, I have a great deal of respect for him and I think that his work with Belief Map is absolutely fantastic. He truly does attempt to give an irenic and charitable view to the positions of his opposition, and he does sincerely want to discuss the actual arguments being made, instead of being content to knock down Straw Men. To that end, I would like to help Mr. Giunta add to his encyclopedia of apologetics by addressing the manner in which one might answer the claims about actual infinities.

## 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?

## 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…

## 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.