# Boxing Pythagoras

## 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.

## What do we mean by Numbers? A simple introduction to Set Theory

The vast majority of people never think about what they mean when they use numbers to describe things. The concept is so ingrained into our early development that we simply take for granted the fact that people understand us when we apply numbers to different things. For most people, numbers are simply numbers, and questions about the meaning of those numbers are confusing and seemingly nonsensical.

Mathematicians are not most people.

For quite a long time, now, mathematicians have recognized that there are at least two very distinct ways in which we use numbers to describe things. Being the scholarly, academic types that they are, mathematicians have assigned names to these two different types of numbers which sound heady and difficult to the average person: ordinal numbers and cardinal numbers. Indeed, even mathematics students sometimes need quite a bit of work and explanation in order to really grasp the difference between these two types of number; but I’m going to do my best to explain these things in a very simple way for a casual audience.

Ordinal and cardinal numbers roughly correspond to the ideas of value and size, respectively.

## 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.

## On Wildberger’s “Inconvenient Truths”

Dr. Norman Wildberger of the University of New South Wales has a wonderful and prolific YouTube channel in which he discusses a great deal of very interesting mathematics. I have discussed Dr. Wildberger before, regarding a very similar subject, but I wanted to take a moment to discuss a video from his Math Foundations series entitled, “Inconvenient truths about sqrt(2).”

In the video, Dr. Wildberger claims that there are three different ways in which $\sqrt{2}$ is commonly discussed: the Applied, the Algebraic, and the Analytical. He does a fairly good job of discussing the manner in which the ancient Greeks discovered that there exists no ratio of two whole numbers which can be equal to $\sqrt{2}$, which is a topic I have covered here, as well. He then explains what he means by each of the above three categories.

## 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 which we symbolize as $f(x)$ (read “$f$ of $x$“). 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.

## 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?

## 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…

## 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.