> But the Cantor vision of the real numbers is just wrong and completely unphysical.
They're unphysical, and yet the very physical human mind can work with them just fine. They're a perfectly logical construction from perfectly reasonable axioms. There are lots of objects in math which aren't physically realizable. Plato would have said that those sorts of objects are more real than anything which actually exists in "reality".
There are two things being talked about here, and worth teasing them out.
On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically; two quantities are interestingly "unequal" only at the precision where an underlying process can distinguish them. Turing tells us that any underlying process must represent a computation, and that the power of computation is a law of the underlying reality of the universe (this is my view of the Universal Church-Turing Thesis, not necessarily the generally accepted variant).
The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. It leads to rabbit holes that are just uninteresting; trying to distinguish inifinities (continuum hypothesis) and leading us to counterintuitive and useless results. Fun to play with, like writing programs that can invoke a HaltingFunction oracle, but does not tell us anything that we can map back to reality. For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.
> On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically
I didn't mean to suggest that the reals are the floor of reality, rather that they are more floorlike than the integers.
> The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no.
Tools are created by transforming nature into something useful to humans. Is Cantor's conception of infinity more natural? I can't really say, but the uselessness and confusion seems more like nature than technology.
> the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful
it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment
That a bijection exists is incredibly useful. But the idea of "measuring" infinite sets in the cardinality sense is not very interesting or useful.
Saying that two sets have the same cardinality is equivalent to saying there is a bijection between them. I don't understand how the latter can be useful but not the former?
It isn't very interesting or useful... to you.
> For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.
I am not sure what you are arguing here. We’ve been teaching this to all undergraduate mathematicians for the last century; are you trying to make the point that this part of the curriculum is unnecessary, or that mathematics has not contributed to the wellbeing of society in the last hundred years? Both of these seem like rather difficult positions to defend.
Yeah, we teach it. It ends up showing up again in measure theory, assuming that anyone still bothers to teach the mostly useless Lebesgue integral instead of the gauge integral. Measure theory shows up again in probability theory if you're not using Kolmogorov for some sadistic reason and you have to deal with countability.
Otherwise it's pretty much a dead end unless you're in the weeds. You just mutter "almost everywhere" as a caveat once in a while and move on with your life. Nobody really cares about the immensely large group of numbers that by definition we cannot calculate or define or name except to kowtow to what is in retrospect a pretty bad theoretical underpinning for formal analysis.
> For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.
Well, there are the same number. So, uh, sorry?
> They're unphysical, and yet the very physical human mind can work with them just fine.
Can it? We can only work with things we can name and the real numbers we can name are an infinitesimal fraction of the real numbers. (The nameable reals and sets of reals have the same cardinality as integers while the rest are a higher cardinality.)
We can work with unnameable things very easily. Take, for instance, every known theorem that quantifies over all real numbers. If you try to argue that proving theorems about these real numbers does not constitute “working with” them, it seems you have chosen a rather deficient definition of “working with” that does not match with how that phrase is used in the real world.
I would argue that all of those theorems work with nameable sets of real numbers but not with any unnamable real numbers themselves.
The human mind can't work with a real number any more than it can infinity. We box them into concepts and then work with those. An actual raw real number is unfathomable.
I don’t know about you, I can work with it just fine. I know its properties. I can manipulate it. I can prove theorems about it. What more is there?
In fact, if you are to argue that we cannot know a “raw” real number, I would point out that we can’t know a natural number either! Take 2: you can picture two apples, you can imagine second place, you can visualize its decimal representation in Arabic numerals, you can tell me all its arithmetical properties, you can write down its construction as a set in ZFC set theory… but can you really know the number – not a representation of the number, not its properties, but the number itself? Of course not: mathematical objects are their properties and nothing more. It doesn’t even make sense to consider the idea of a “raw” object.
You can hold a two in your head, but you can't hold a number with infinitely many decimal places. Any manipulations you do with the real 2 are done conceptually whereas with the natural 2, its done concretely.
The decimal places are just a way of representing it.
The infinite number of decimal places is the definitional feature of a real number. No matter how's it represented they are still there and cannot be contained in our brains. We can say pi and hold the concept of pi in our heads, but not the actual number.
No, it really isn’t. The real numbers can be constructed in a number of ways, and it is more common to define them as either Dedekind cuts, or equivalence classes of Cauchy sequences of rational numbers.
Personally, I’d go with the sideline cut definition.
Dang autocorrect. “sideline” should be Dedekind
Or maybe we can know them equally well? The function f(x) = x(0^(sin(πx)^2)) for example "requires" infinities, but only returns integer values.
I felt also something like this before. Also integers seem pretty close to the reality around us. One of their functions is to symbolically represent the similarity of objects (there might be a better way to put it). Like, if you see 5 sheep in one group and 6 in another, after that point they’re no longer just distinct sheep with unique properties - the numbers act as symbols for the groups. Real numbers still can work in the brain, but they're most distant from the world around us, at least when it comes to going from visual to conceptual understanding.
> They're unphysical, and yet the very physical human mind can work with them just fine
Nah, you're likely thinking of the rationals, which are basically just two integers in a halloween costume. Ooh a third, big deal. The overwhelming majority of the reals are completely batshit and you're not working with them "just fine" except in some very hand wavy sense.
the rationals are 3 naturals with in a "2,1" structure.
the first 2 naturals form an integer.
that integer and a 3rd natural constitute a real (but this 3rd natural best be bigger than zero, else we're in trouble)
what I choose to focus after observing the "unphysical" nature of numbers. is the sense of natural opposition (bordering on alternation) between "mathematical true" and "physical true". both are claiming to be really real Reality.
in the mathematical realm, finite things are "impossible", they become "zero", negible in the presence of infinities. it's impossible for the primes to be finite (by contradiction). it's impossible for things (numbers or functions of mathematical objects) to be finite.
but in the physical reality, it's the "infinite things" which become impossible.
the "decimal point" (i.e. scientific notation i.e. positional numeral systems) is truly THE wonder of the world. for some reason I want something better than such a system... so I'm still learning about categories
Huh?