I’m Way Out of My Depth, Here

BUT SOMETIMES ASKING a really stupid question can help you learn, so, here goes.

I learned in high school that infinity was, well, infinite, and there’s no way to approach it. At the same time, I learned that there is an infinite number of increments between any pair of numbers. Or, for that matter, any pair of anythings. Including things like probable outcomes to a decision. And that carries out to the quantum level. So whether the photon jinks left or right in the double slits has an infinite number of possible outcomes, even though there’s only two — or four — options.

But this article peeled an old intellectual scab. We also learned that a number divided by itself equals one. Makes sense, right? Four fourths is a whole, right? One. So what is zero divided by itself?

Wait a minute! You can’t divide by zero!

Well, properly speaking, you can, but the answer is out of the normal bounds of our concepts of numbers. And, of course, computers lose it when you try to make them calculated it. But, really, it makes logical sense. Zero zeroths is a whole zero, right? I mean, it’s nothing, but it’s ONE nothing. A slippery concept, I’ll admit, but not as weird as n dimensions.

And this also requires admitting that dividing zero by itself to get one is a special case. And what if that means that 0/0=1 is also 0/0=∞? Talk about your special cases. And what does that imply about the question raised in the linked article as to whether infinity actually exists in the real world, or is just a mental construct? See how that blows your dress up.

Or should I stick to weaving baskets?

8 responses to “I’m Way Out of My Depth, Here

  1. What if I implied that zero really doesn’t exist except as a mental construct. Even in space there is still something. Just not a lot of something. So does that imply that even space doesn’t exist in a vacuum?

  2. Reminded me of this from Chrisstopher Langan: “…time and space (are) equivalent to cognition and information with respect to the invariant semantic relation processes, as in “time processes space” and “cognition processes information”. But when we define reality as a process, we must reformulate containment accordingly. Concisely, reality theory becomes a study of SCSPL (Self-Configuring Self-Processing Language, e.g., mathematics) autology naturally formulated in terms of mappings. This is done by adjoining to logic certain metalogical principles, formulated in terms of mappings, that enable reality to be described as an autological (self-descriptive, self-recognizing/self-processing) system.”

    Read what he has to say about set theory to really get to the bottom of your zero zeroths conundrum.

  3. I had a math/geometry professor that used to play those games. Think of this; If you establish a point on a line as zero, then everything to the right, headed toward infinity, has a value of infinity. the same is true for the left, only that would be “Negative” so to speak (From a geometric perspective)

    So the distance between zero and positive infinity is, infinity. And the distance between zero and positive infinity, is, in fact, infinity. And the distance between positive infinity and negative infinity is…. infinity. All the values in positive infinity added to all the values in negative infinity equal zero.

    I have tried, not too successfully, to drink those thoughts out of my head.

  4. “So the distance between zero and positive infinity is, infinity. And the distance between zero and NEGATIVE infinity…”
    Sorry. Not adequately caffeinated.

  5. Try reading “Infinity and the Mind” by Rudy Rucker. He is a real mathematician who is also a pretty good SF author as well. The book will bend your brain but it makes the different levels of infinity understandable. It was first published in 1984 but is still in print, so that should tell you something.

  6. Hammerbach

    Division by zero is ruled out because that operation in arithmetic, if allowed, would make it possible to prove any number equal to any other number. In other words, you could have a system, but it would be inapplicable to anything else. See Asimov’s “Science, Numbers, and I” for a pretty good essay on the subject.
    .
    I think you would enjoy it. And I’ll bet it’s in you local library.

  7. With regard to your questions about 0/0 (or dividing by 0 generally), see the intro and first 2 sections (“In elementary arithmetic,” and “In algebra”) of the following article (a total of about 2-3 pages):

    http://en.wikipedia.org/wiki/Division_by_zero

    The “In algebra” section makes a good argument why 0/0 must be indeterminate, not 1. It also shows how assuming 0/0 = 1 allows you to prove that 1 = 2.
    ————

    “I learned in high school that infinity was, well, infinite, and there’s no way to approach it.”

    Not sure what you mean by “there’s no way to approach” infinity, but calculus has, in certain respects, harnessed infinity to good effect. For example, the definite integral adds up an infinite number of infinitesimals to compute the area between a curve, y = f(x), and the x-axis (bounded by two points on the x-axis, x0 and x1). For example, a definite integral can be used to determine the area between y = x^2 + 2 and the x-axis (where “x^2” means “x squared”) between x=0 and x=2.

    Derivatives and integrals–both of which deal with infinity and infinitely small quantities–are extremely useful tools in math and science.
    ———–

    Infinity is all around us. Have you pondered Zeno’s paradoxes (see http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Dichotomy_paradox)? Another version of this might be the paradox of how you can drive 1 mile at a constant 60 mph: the first 1/2 mile takes t1 seconds; the next 1/4 mile takes t2 seconds; the next 1/8 mile takes t3 seconds; half of the remaining distance takes t4 seconds, etc. So the total time to drive 1 mile is an infinite series: Total Time = t1 + t2 + t3 + t4 + t5 + … Since there are an infinite number of time-increments, you should never complete the 1 mile!??

    The fallacy is that it is possible to add an infinite number of quantities and still get a finite result. For example, if T = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …, then T = 1. This result explains the above paradox.

  8. Actually, the first sentence of the last paragraph should read:
    The fallacy comes from not knowing that it is possible to add an infinite number of quantities and still get a finite result.

    Perhaps a decimal example might be easier. If T = 1 + 1/10 + 1/100 + 1/1000 + …, then T = 1.11111111… Now whatever T is, we know that it is less than 2. So again, it is possible to add an infinite number of quantities and still get a finite total.