Comment by mcguire | Hacker News Reader

Comment by mcguire | Hacker News Reader

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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 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.

Reminds 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

Negative numbers work just fine for directions. -3 feet west makes plenty of sense, even if -3 oranges doesn't.

Negative numbers work fine for bookkeeping too. You owe someone 3 oranges.

On the other hand double entry was invented so bookkeepers didn't need to deal with negative numbers.

it 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

> 10 (base 10)

Even this notation assumes base ten for the number written inside the parenthesis. Otherwise it works for any base.

So 10 (base ten).

That'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).

The 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.

"practically useful"

The Mathematician blinks several times, looking at you as if you'd just grown a third elbow. :-O :-)

The problem is that there's nothing arbitrary about Cantor's work. Now, you could (along with Aristotle, IIRC) just declare "there are no actual infinities, only potential infinities" and thus that the whole investigation is verboten. But that's not very mathy.

Another alternative would be to redefine the fundamentals to outlaw actual infinite sets, but if you do that then you have redo everything, because some of the things that you might consider useful and related to other things go out the window. (Offhand, I think the result would be constructive mathematics. Putting on my computer scientist hat, I'm perfectly happy with constructive math. But most mathematicians aren't really.)

But if you allow infinite sets, then the properties of them, including infinite sets of different sizes, are derived directly from basic set theory. They are as "true" as anything else in math. (I'm a formalist---we're just playing a game that has a specific set of rules. :-))

"Sedenion neural networks provide a means of efficient and compact expression in machine learning applications and were used in solving multiple time series forecasting problems." "Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm" (https://ieeexplore.ieee.org/document/9160921/).

> 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.

The distinction between countable and uncountable is very practically useful. You can always encode the elements of a countable set such that every element is represented in the computer by a finite-length bit string, but you can never design an encoding that does this for every member of an uncountable set. No exceptions.

In addition, this notion of distinguishing countability from uncountability can be applied to show that the set of Turing machines is countable (up to isomorphism). This immediately informs one of the limits of computation, and why, e.g. trying to design a computer or program that performs exact arithmetic operations on real numbers in general, or say finding the limit of a series in real numbers in general, is as achievable as squaring the circle.

And make no mistake, computational modeling that today already permeates every aspect of society would look very different if exact operations on real numbers were computable and efficient. (and for the general programmer, no need to worry about the imprecision of floats and especially with financial stuff)

> 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.

> 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

You'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.

This is not true. The usual calculus that people learn (and usually learn with rigor in a real analysis course) does not touch infinitesimals.

The approach via infinitesimals is even called "nonstandard analysis".

https://en.wikipedia.org/wiki/Nonstandard_analysis

You're absolutely right. I was going for a different concept. See explanation above. I should stop writing after 02:00.

The problem here is that you're basing your conception of what is "good," "interesting," and "useful" based off of what you might discuss in an undergraduate class in Real or Complex Analysis.

The notion of cardinality and how we can have different "sizes of infinity" is an enormously important and interesting idea that comes up all the time in mathematics.

How 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

The 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.

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