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"The idea of "differently sized infinities" has always felt like a mathematical game to me." How do you feel about negative numbers? There is absolutely no wa
by mcguire 5y ago
"The idea of "differently sized infinities" has always felt like a mathematical game to me."
How do you feel about negative numbers?
There is absolutely no way you can have -3 oranges. There are metaphors, and we're all taught what you can do with a negative number, and there are all those lovely proofs of the properties of negative numbers, but you will never see -3 oranges sitting on someone's dining table.
(And I'm not even going to go into the reals---they're just ridiculous.)
The commonsense notion of "size" is one of those metaphors, but it's a good one: for non-infinite sets, the idea of a bijection and the idea that two sets have the same size match exactly. Consider three blocks, one labelled "1", another "2", and a third "3", and three oranges. Block "1" goes with that big, fat orange. Block "2" goes with the smaller orange. And Block "3" goes with the orange that's kind of pear-shaped.
Now, the extension of "size" to infinite sets is a little weird, mostly because infinite sets are very weird. As is the phrase "almost all"; it's not immediately obvious that, because there are more Rs than Ns, that there are a lot more Rs than Ns.
But it's true (in the sense that it's how math works), and you just have to get used to it the same way you got used to -3.
- GuB-42 5y agoReminds me of a joke: A biologist, an engineer and a mathematician see two people enter a seemingly empty building, and a moment later, three people leave. Biologist: They must have reproduced Engineer: It must have been a measurement error Mathematician: If one person enters the building, it will become empty
- crazygringo 5y agoThe difference is that negative numbers are 1) practically useful, and 2) extremely interrelated with other useful mathematical concepts. We build up an entire system of real (and even complex) algebra. While with differently-sized infinite sets, the point is that they aren't (to the best of my knowledge) either 1) directly useful or 2) particularly interrelated with anything else. E.g. they're in addition to the set theory used for real and complex analysis -- not a foundation for it. And so it seems more apt to put the notion of measuring different "sizes" for infinite sets in the same category of, say, quaternions and sedenions. I don't think too many people would say sedenions are "true", just that they're a construction with certain properties. They're not derived from anything, in the way that negative numbers are derived from inverting addition. Yet math textbooks and courses indeed attempt to present the "sizes" of infinite sets as "true" -- as real as negative numbers -- when I still fail to see how they're anything but a relatively arbitrary curiosity like sedenions.
- porb121 5y agoHow can you say that infinities of different sizes are not useful? All of analysis is built on the fact that R is the unique complete ordered field, and therefore uncountable. And if you want every infinite set to have size continuum, then N (and so Z and Q) break horribly
- btilly 5y agoThe field of constructive analysis would beg to differ with you. An amazing amount of real analysis can be done with absolutely no reference to the excluded middle, completeness axiom, or accepting the existence of uncountable sets.
- jyounker 5y ago> While with infinite sets, the point is that they aren't (to the best of my knowledge) either 1) directly useful or 2) particularly interrelated with anything else. Formal treatments of calculus are based around infinitesimals. Infinitesimals are the same set of ideas about infinite sets, but applied to the spaces between arbitrarily close points. Specifically formal treatments of calculus work because between any two points there are an infinite number of points and the size of that infinity is the same size as the entire set of real numbers. At least this is my undergrad level of comprehension. I'm certain there are more exotic treatments, but this is the one that you have to do all the proofs for.
- porb121 5y ago> Formal treatments of calculus are based around infinitesimals formal treatments of calculus don't use infinitesimals, and analysis using infinitesimals is explicitly separate from traditional analysis
- jyounker 5y agoYou're absolutely right. I don't know what I was thinking when I wrote this last night. I meant to say that formal treatments are based around being able to find arbitrarily small neighborhoods around a point, and that this only works when you have something the size of the reals, for which you can find a real-sized space between any two points. It's this infinite self-similarity at any arbitrarily small scale that is at the heart of calculus.
- 6gvONxR4sf7o 5y agoNegative numbers work just fine for directions. -3 feet west makes plenty of sense, even if -3 oranges doesn't.
- mc32 5y agoNegative numbers work fine for bookkeeping too. You owe someone 3 oranges.
- agumonkey 5y agoit seems there's a little issue in math, in that the concept and notation often assume enough intuition and imagination from the reader -3 looks odd if you interpret it as material existence but as you say, read in bookkeeping context, it's totally natural (npi). it's a bit like numbers 10 is actually 10 (base 10) but we elude it
- tonyedgecombe 5y agoOn the other hand double entry was invented so bookkeepers didn't need to deal with negative numbers.
- mcguire 5y agoThat's one of those metaphors plus what you have been taught about how negative numbers work. It's a product of Cartesian geometry (which had to be invented).