The End of Infinity (Infinity of Fun Part III)

by David Siegel Bernstein, PhD

To conclude my series on infinity, I will now attempt to answer the question: does size does matter? Answer: maybe (at least for infinity . . . which is what I am talking about).

According to the mathematician Georg Cantor there is an infinite number of infinities including "little bits" of infinity. Meaning, not all infinities are created equal. No fair! Or is it? Who knows? Remember infinity is an idea not a number.

Consider the -- artfully designed -- Venn diagram below. It represents infinite sized sets nestled inside other infinite sets. The inner most set represents the universe of whole (counting) numbers, but as shown this is only a subset of integers (whole + negative numbers) which is a subset of rational numbers (whole + negative + fractions/decimals), which is a subset of real numbers (all the above + irrational numbers).

Cantor demonstrated that these sets cannot be completely matched up; what he called a lack of 1-1 correspondence. Simply put, if you were to keep using a whole number to cancel out an integer number, once you've exhausted the your stash of whole numbers you would still have integers left over after -- even though you had an infinite number of whole numbers!

By the way, there are sets that lay beyond real numbers called transfinite numbers. Good luck thinking about those! If anyone is interested in learning more about them, let me knew and I'll cover it in a future post.

See you next month in time and space with a new topic.

Infinity of Fun, Part II

By David Siegel Bernstein, PhD

Welcome back to Part II of our discussion on infinity.

Drum Roll. Now as promised last month, I'm going to take a bullet in the name of math. My arch nemesis, who oddly enough volunteered for this job, aims her rifle at me and fires. According to math, do I get hit? (And yes, she is a good shot.)

If we believe Zeno of Elea (living during the 400s BC), I'm safe. Of course he didn't know about rifles, but he did like to devise devious paradoxes similar to this situation. The bullet flying toward me passes half way across the room, then half the remaining distance, then half again, and so on. I've divided its journey into infinite number of shorter and shorter segments. So in this case, Zeno would have argued that motion is an illusion.

The idea is (sort of) consistent with math because any line segment can be divided into infinite number of points. In the sad story of my shooting, the bullet must pass through an infinite number of points making me infinitely far away from the bullet. It's always getting closer and closer to me but never hitting me.

Math has saved me . . . well, except for when it doesn't. You see according to physics, and my foe's true aim, I'm going to get hit. Why?

I'll give you Aristotle's (384 BC – 322 BC) answer. Like Zeno, he didn't know about rifles, but he was aware of the halving paradox. He would have considered how long it took the bullet to reach me. Let's assume it's a very slow bullet, taking a full second to reach me and I'm slow to react. He'd argue that each interval of the series (all the halves) have a corresponding time. Assuming a constant bullet speed, as the distance gets shorter so does the time until impact. So I'm running out of time. All the time intervals added together can not exceed a second. Therefore it can not take an infinite amount of time to reach me.

So there is a limit (Do you recall limits from Part I? No? You could give it a re-read. I won't tell . . . I swear.) and it isn't infinity. If it took a half of a second to reach the halfway mark, then it can not be the case that the total time it takes to hit me to be greater than a second. You can't pass an INFINITE number of points in FINITE time.

So I've been shot (damn you physics). But don't worry it was only a glancing wound, I'll survive to write more.

INFINITY OF FUN

This month I thought we'd have some fun with math. And what's more fun than infinity? Nothing! Get it? You know infinity—being nothing (I can hear you gentle readers groaning.) . . . oh never mind. Anyway, this is Part I of II where we discuss the fun (and headaches) mathematicians, scientists, and philosophers have with infinity.

If you leave this article remembering only one thing let it be that infinity is NOT a number—it's an idea—and as such it can't be used in algebraic operations. To demonstrate, I will prove that 2 = 1.

1. A = B (There is some A that equals some B. This is a given)
2. A + A = B + A (Adding A to both sides)
3. 2A = B + A (Grouping)
4. 2A - 2B = B + A - 2B (Subtracting 2B from both sides)
5. 2(A - B) = B +A - 2B (Factoring left hand side)
6. 2(A - B) = 1(A - B) (Simplifying right hand side)
7. 2 = 1 (Cancel (A-B) from both sides)

The above are all traditional algebraic operations—however, I tortured them into confessing to an absurd conclusion. One of the steps above is forbidden. Do you know which? I went from step (6) to (7) by dividing both sides by (A - B).

You might say: So?

Well since you asked. Recall line (1) states A = B, meaning that (A - B) = 0. And, as your middle school teacher taught you, dividing by zero is a no no. The reason being (if you recall my lame opening joke) there is an infinite number of NOTHING(s) in EVERYTHING. Therefore dividing anything by 0 is infinity. And infinity is not a number.

Since infinity doesn't make any mathematical sense, scientists have a cheat called "taking the limit" to solve these division by zero issues. In other words, taking a limit of an equation means the variable in question may approach infinity, but must always fall short. Consider it a stop sign. Here is an example where I show you how to use a "limit" to avoid the "division by zero" problem AND I'll do this while using clichés to confirm: The less you know the more money you make.

Cliché 1: Knowledge is Power.
Cliché 2: Time is Money.

1. Power = Work / Time (Physics formula)
   
2. Knowledge = Power (Restating Clichés)
   Time = Money
   
3. Knowledge = Work / Money (Combining (1) & (2) using substitution)
4. Money = Work / Knowledge (Solving (3) for Money)

So as you can see from (4)—the moral of this proof—as the Limit of Knowledge approaches zero, Money approaches infinity, regardless of the amount of Work done.

In other words by taking the downward "limit," knowledge (the denominator) never quite reaches zero and the amount of Money is capped, never reaching infinity.

Before leaving, here are some parting questions to consider. I might even give the answer in a future post.

Does infinity / infinity = 1?
Does infinity - infinity = 0?

I'll see you next month for Part II of our discussion on infinity. Here's a preview: I get shot (why would anyone want to do that?) by a gun, but the bullet never reaches me. I will be saved by math!

by David Siegel Bernstein

Splicing Science from Fiction

Welcome back. It's me again, David Siegel Bernstein, and this month (the trumpeter's fanfare begins) I want to write about science (the trumpet falls silent). Come on… science is cool. If you want a more exciting introduction to this month's theme, I suggest you listen to the They May Be Giants album Here Comes Science. Here is a track where they describe science.

Now back to your regularly scheduled post.

Within the genre of Science Fiction, science asks "What if?" and fiction speculates on what happens to people or societies when that thorny question is answered. I pretty sure anyone reading this already knows all about fiction—but what makes science so mysterious and wondrous that we connoisseurs of Science Fiction fixate on it? Broadly speaking, science is a way of knowing, getting to the truth of the physical world around us. Of course, there are other ways of knowing such as art (personal truth), religion (revealed truth), and so on.

So how does science get our fiction where we want it to go? Let's start by defining how science gets to a truth. To me science has two distinct meanings. First it is a method—a set of steps questioning the natural world by using reproducible observations and controlled experiments. Scientists use the answers gleaned from this method to explain phenomena or to make predictions. BUT WAIT that isn't all. Because (second meaning) science is also a collective term for the accumulation of what was learned from this method. For example there is "doing" Chemistry and there is Chemistry (itself). Errr, I just used self reference (SR), the curse of using a term to describe itself—a logician's nightmare.

Time out:
I'd like to thank Gödel for his (damned) incompleteness theorem that left us with the SR classic: This sentence is false. And jokes such as: 73.2% of all statistics are made up—and—anyone who sees a psychiatrist should get their head examined.

Okay, back to the coolness of science. Its language is math (don't zone out, math is awesome, and it pays my rent), but as with English it's fraught with tongue twisters that scientists sometimes take too seriously, leading them astray from science. Consider String Theory. Intuitively it is wibbly wobbly timey wimey, but mathematically it is internally consistent and it may theoretically explain phenomena that can't otherwise be explained. HOWEVER, it isn't observable and there are no direct experiments to test it. So for now it dwells more in the realm of philosophy than it does science. For writers, who cares? Because whatever else can be said of String Theory, it is a great plot generator for Science Fiction.

I'd like to posit a question to the readers of AT. What would Science Fiction be without science? Fantasy? Maybe, maybe not. I concede they are related. Science Fiction began as Fantasy of the Industrial Revolution. So SF is Fantasy's child. Cool. But the baby has grown up and moved on. Think about it and feel free to post comments on how you think it may have grown. (As food for thought, is the Singularity, a la Charles Stross, more SF or more fantasy? What about a space opera?)

Before parting, I'll leave you with Arthur C. Clarke's three laws of prediction. Popular Science Fiction from Dr. Who to Star Trek has cited his third law ad nauseum. I encourage you to embrace all of them, but for fun—try living the second.

1. When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.

2. The only way of discovering the limits of the possible is to venture a little way past them into the impossible.

3. Any sufficiently advanced technology is indistinguishable from magic.

So long for now, thanks for the fish, and above all else remember: Science tells us what is, not what we want. Science Fiction has no such restriction. See you next month.

Meet David Siegel Bernstein - Forensic Statistician

Hello. My name is David Siegel Bernstein and I have a confession to make. (I take a deep breath) okay, here we go: I love Science Fiction and Fantasy. Yes, I dream about voyages between worlds in starships fully equipped with FTL drives, battling aliens from other dimensions, traveling the multiverse while standing still in order to defeat my evil twin and rescue the princess. There I said it! And do you know what? It felt great to admit that. Thank you for listening. (I suppose I should say reading.) I'm guessing my confession isn't too shocking because if you're reading this you must have, at least, some of the same interests. Of course I could be wrong, but I doubt it.

To show my devotion to the genre, I not only read and watch the stuff, but I also write it. (See here for a bibliography.) I have to say (not really, but I want to) that the people in the Science Fiction and Fantasy writing community are an amazing and talented bunch of thinkers and dreamers. So if you aren't a writer—get to know one. Or better yet, put pen to paper, fingers to keyboard, thumbs to text(ing), and create a story. Join the club.

To support my writing habit and excessively extravagant lifestyle, I consult as a forensic statistician. Yes, that means I get paid to do math. Don't look at me that way! There's nothing wrong with that. Math is the language of science and there is science in Science Fiction. So there!

Anyway, I hope to use this space (and time… Einstein told me that they were intertwined… okay, fine, not the man himself, but I did learn that when I was a physics major) to explore topics of math and science. So in future postings I plan to discuss the use of science IN fiction and to share some math and logic puzzles. Don't worry there won't be any calculus quizzes (not that there is anything wrong with that) but may be some paradoxes!

See you then.


***  Look for David's next post soon!

Readercom

For SF/F folks unfamiliar with the fact of this famous fandom fest [don't try to say this at home], here's the info on this year's model:

"Readercon is, depending on your point of view, either an annual literary conference (except it's infinitely more fun than that) or an annual science fiction convention (except we've stripped away virtually everything except talking about and buying books). In the course of its twenty years, it has become the standard bearer and role model for conventions that focus on the literature. A typical Readercon features well over 100 writers, editors, and critics (attracting prominent figures from Canada, the U.K., and occasionally even Australia) and more than 400 of their readers. Readercon is the only small convention regularly attended by such giants of literary sf as Samuel R. Delany, John Crowley, Barry N. Malzberg, and Jonathan Lethem."

http://www.readercon.org/

It's held yearly in the Boston area.