Sunday, July 3, 2022

There is no Vase

 I'm not a math nerd, so I probably shouldn't be posting on this subject. But when has a lack of expertise ever stopped me from having an opinion?

I just watched the Up and Atom video: An Infinity Paradox - How Many Balls Are In The Vase? In it, Jade describes the Ross–Littlewood paradox related to infinite pairings. I liked the video but was not satisfied with the conclusion.

I won't give the background; if you're interested in this post, go watch the video and skim the Wikipedia article. Basically, she presents the "Depends on the conditions" solution (as described in the Wikipedia article) without mentioning the "underspecified" and "ill-informed" solutions. And I guess that's an OK choice since the point of her video was to talk about infinities and pairings. But she kept returning to the question, "how many balls are there *actually*?"

Infinity math has many practical applications, especially if the infinity is related to the infinitely small. An integral is frequently described as the sum of the areas of rectangles under a curve as the width of the rectangles becomes infinitesimal - i.e., approaches zero. This gives a mathematically precise calculation of the area. Integrals are a fundamental tool for any number of scientific and engineering fields.

But remember that math is just a way of modeling reality. It is not *really* reality.

There is no such thing as an infinitesimal anything. There is a minimum distance, a minimum time, and the uncertainty principle guarantees that even as you approach the minimum in one measure, your ability to know a different measure decreases. When the numbers become small enough, the math of the infinitesimal stops being an accurate model of reality, at least not in the initially intuitive ways.

But they are still useful for real-world situations. Consider the paradox of Achilles and the tortoise, one of Zeno's paradoxes. (Again, go read it if you don't already know it.) The apparent paradox is that Achilles can never catch up to the tortoise, even though we know through common experience that he will catch up with and pass the tortoise. The power of infinity math is that we can model it and calculate the exact time he passes the tortoise. The model will match reality ... unless an eagle swoops down, grabs the tortoise, and carries it across the finish line. :-)

But models can break down, even without eagles, and a common way for infinity models to break down is if they don't converge. 1/2 plus 1/4 plus 1/8 plus 1/16 ... converges on a value (1). As you add more and more terms, it approaches a value that it will never exceed with a finite number of terms. So we say that the sum of the *infinite* series is *equal* to the limit value, 1 in this case. But what about 1/2 plus 1/3 plus 1/4 plus 1/5, etc.? This infinite series does NOT converge. It grows without bound. And therefore, we cannot claim that it "equals" anything at infinity. We could claim that the sum equals infinity, but this is not well defined since infinity is not a number.

Here's a similar train of thought. What is 1/0? If you draw a graph of 1/X, you will see the value grow larger and larger as X approaches 0. So 1/0 must be infinity. What is 0 * (1/0)? Again, if you graph 0 * (1/X), you will see a horizontal line stuck at zero as X approaches 0. So I guess that 0 * (1/0) equals 0, right? Not so fast. Let's graph X * (1/X). That is a horizontal line stuck at 1. So as X approaches 0, X * (1/X) equals 1. So 0 * 1/0 equals 1. WHICH ONE IS RIGHT???????? What *really* is 0 * (1/0)?

The answer is that the problem is ill-formed. The 1/X term does not converge. The value of 1/0 is not "equal to infinity", it is undefined. My train of thought above is similar to the fallacious "proof" that 1 equals 2. And it seems to me that the "proof" that the number of balls in the vase can be any number you want it to be is another mathematical fallacy.

The only way to model the original vase problems is to draw a graph of the number of balls in the vase over time. Even in the case where you remove the balls sequentially starting at 1, you will see the number of balls growing without bound as time proceeds. Since this function does not converge, you can't say that it "equals" anything at the end. But it tends towards infinity, so claiming that it equals some finite value *at* the end is another example of an invalid application of math to reality.

But I shouldn't complain. Jade used the "paradox" to produce an engaging video teaching about pairing elements in infinite sets. And she did a good job of that.

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