Built on Facts

A Matter of Definition

Posted by Matt Springer on October 14, 2008
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Mark C. Chu-Carroll at Good Math, Bad Math has an excellent post in which he mercilessly dispatches the misbegotten idea that infinity is a number. It’s not.

How do we know? He explains it very well, but the fundamental thing to remember is that number is a word, and like all words it has a specific meaning. The words we use in mathematics and science have to be very carefully defined to make sure we’re all talking about the same thing, and the definition of infinity is simply not even remotely compatible with the definition of number.

To take an extreme example, if you define dragon as “A small carnivorous mammal (Felis catus or F. domesticus) domesticated since early times as a catcher of rats and mice and as a pet and existing in several distinctive breeds and varieties.” then my parents own two real live dragons. “But the definition is wrong!”, you exclaim. No, definitions can’t be wrong. They simply are. The only problem is when people use definitions that aren’t consistent. You run into these problems with trying to say that infinity is a number. Define “number” in such a way as to include 1, 2, 3 and all the rest, along with the usual relationships between those numbers and you’ll find your definition cannot include infinity. It just can’t happen. Might as well try to say your cat is a number.

Now if you are suitably clever you can do some finagling and come up with a way to treat infinity in a manner that doesn’t fit the field axioms but still has its uses as something similar to a number. But you don’t have anything resembling ∞ + ∞ = 2∞, so it’s still doesn’t fit the usual definition of number.

Do we ever have these definitional issues in physics? Sure, all the time.

This question was on an exam for the 201 class for which I do some TAing. Say you have a ball held a meter over a table, which is itself a meter above the floor. The potential energy is mgh, but which h should you use? One meter or two meters? It was a short answer question, which should have tipped the students off that there was more to it than just one or the other.

The answer is of course that it doesn’t matter. You can chose whatever reference point for potential energy that you want, so long as you’re consistent. Pick a definition for potential energy and stick with it and you’re golden. This is first-semester material, but it crops up all throughout physics at every level. In electrodynamics and other subjects, we often have even more elaborate choices for how we talk about potential. We have all kinds of screwball gauge choices we can use to help solve problems. You can define the gauge pretty much however you want so long as you pay attention to maintaining consistency in your definition, and so long as your gauge choice fits the definition of gauge choice in the problem at hand.

All this is an extended way of saying that a lot of misunderstandings about physics and mathematics could be resolved by clarifying definitions.

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Comments

  1. #1 Eric
    October 14, 2008

    No, definitions can’t be wrong. They simply are.

    37 Ways That Words Can Be Wrong

    See #20 for your dragon example.

    It’s a bit of a nitpick since I agree with the principle of this post, but definitions can be wrong.

  2. #2 Uncle Al
    October 14, 2008

    Many Equivalence Principle tests explore contrasted differential properties like neutron vs. proton count (isospin). Such studies cannot couple to rotation or translation because they observe internal gauge symmetries. They are default null results (barring an ultra-trace difference in trace properties arising from epsilon-sized theory violations). “The answer is of course that it doesn’t matter.” (epsilon)^3 is a terrible place to shop.

    The only discontinuous (no Noether’s theorem!) external symmetry observable coupled to translation or rotation is parity violation. Enantiomorphic mass distributions are chemistry, a lesser science. Physcis examining the obvious case would be inelegant.

    [Covariance with respect to reflection in space and time is not required by the Poincaré group of Special Relativity or the Einstein group of General Relativity. SCREAMINGLY obvious.]

  3. #3 MPL
    October 14, 2008

    Re. No. 1

    Redefining a common word arbitrarily (like dragon=cat in the post), is only in error when you are intentionally trying to create an equivocation, relying on the user to confuse your new definition and the common one.

    However, common words get redefined all the time in technical subjects. In mathematics, for instance, “field”, “group”, and “pencil” are all technical terms that have nothing to do with a place you grow hay, something that toddlers get together to play in, or something you write with. Nobody’s confused when you use them technically though, since it’s obvious from context.

  4. #4 EJ
    October 14, 2008

    I agree that amateurs trying to argue something like “infinity is a number” often try to do so without any regard for the importance of proper definitions, and therefore just make a mess.

    But some of your specific claims are don’t sit well in a blog post about being careful with definitions.

    “The definition of infinity is not even remotely compatible with the definition of number.” I have a hunch you’ve made this strong claim without a careful definition of “infinity” in mind, which would be careless of you! If I’m wrong, I regret the accusation. But there are various useful and interesting number systems with infinite numbers, even fields with infinite numbers.

    Also:
    “…number is a word, and like all words it has a specific meaning.” Hmm. In mathematics, “number” has *many* specific meanings. Is aleph_2 a number? Are the quaternions numbers (or is just the ones that happen to be complex that are numbers)? But I’d love to see the specific definition of “number” you have in mind in the claim “infinity is not a number”.

  5. #5 Matt Springer
    October 14, 2008

    You’re quite right. What I have in mind for “number” is the real numbers constructed in the standard way. I think that’s probably what the average person has in mind when they say the word. Those requirements exclude infinity, and they carry over into other definitions such as the complex numbers.

    The transfinites are another story, but then while they still don’t form a field, I’m not even sure they’d qualify as most peoples notion of “infinity” either, on account of some transfinites being larger than others!

  6. #6 CCPhysicist
    October 14, 2008

    You say that Inf is NaN, but I’ll bet you use Inf like it was a number when you do physics. Do you write a limit when you describe the gravitational potential energy “at infinity”, or do you simply evaluate it using Affine arithmetic? Do you write down a limit every time you do an “improper” integral, or do you just put infinity at the limits and crunch away? Do you close a contour “at infinity”, or go through a detailed mathematical procedure each time? Mathematicians generally cringe when they watch us do almost any calculation.

    Then there is the astoundingly amusing question as to whether your calculator or digital computer calculates with “real numbers”. It doesn’t. Not even close, but its “floating point” system does include values for Inf and NaN. In that system, only NaN is actually not a number.

  7. #7 Matt Springer
    October 15, 2008

    Yes, but I know what I’m doing!

    Famous last words, I know…

  8. #8 Nemo
    October 15, 2008

    Actually, is it possible to define “number” rigorously such that they obey all the properties you are accustomed to (specifically, they include the real numbers), and such that ∞ + ∞ = 2∞. Literally.

    They are called Surreal Numbers. I think you might enjoy Knuth’s book where they were first popularized.

  9. #9 Nemo
    October 15, 2008

    Stupid broken “preview”.

    Of course I meant ? + ? = 2?

    (Although you still cannot do 1/0. And I see your buddy at Good Math, Bad Math already has some posts about surreal numbers.)