For me it was when he said that the cardinality of integers is the same as real numbers. Then I saw his twitter and all the politics and crazy stuff about QM.
Either he's: Trying to be edgy for edgy's sake or he bragging about how he's smarter than the experts in the field while demonstrating a lack of understanding (thinking he doesn't have to prove it to others, they should just trust him). Neither give me that great of an opinion of him. If you don't understand something don't tell everyone that studies the thing that they're wrong. If the experts are wrong, show them, embarrass them, get a fields metal, another million dollars, and a shit ton of fame. Essentially, pony up or shut up.
This got me thinking, is there a scenario where the number of new guests is uncountable? Seems to me that every kind of ferries/buses/guests story is just going to be countable, since a finite number of countables is still countable.
Maybe something that pretends to be the real numbers, like a matrioshka doll of infinite containers inside containers.
The easiest analogy I can come up with is an infinite pipe that's completely full with water. When a new amount of water arrives, say 1 Liter, all the water just flows along the pipe a bit further to make space for the new water.
In the hotel fashion, might be difficult. Think of it this way, if we make the Hilbert Hotel infinitely tall and each floor has infinitely many rooms such that they correspond to each rational number, we can fill any number of people from any source on the first floor alone.
I think we could only do this with an even more absurd scenario like if each room was filled with pregnant women who are giving birth, and the children rapidly age, and also give birth at an infinite rate? That would create an infinite nesting doll like situation for each guest
Some light, coffee reading "Cardinality of the continuum" [1]: in short, the cardinality of real numbers (ℝ) is often called the cardinality of the continuum, and denoted by 𝔠 or 2^ℵ_0 or ℶ_1 (beth-one [2); whereas, interestingly [3], the cardinality of the integers (ℤ) is the same as the cardinality of the natural numbers (ℕ) and is ℵ_0 (aleph-null) [perhaps what was meant initially?].
Related: the Schröder–Bernstein theorem [4], "if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B.".
Not related, but great: Max Cooper (sound) and Martin Krzywinski (visuals) did a splendid job visualising "ℵ_2" [5].
adding upon this comment to why the two cardinalities are not equal, on one hand we have the set of integers {..., -2, -1, 0, 1, 2, ...} and they can be put into a bijection with the set of natural numbers {1, 2, 3, 4, ...}, this is done by rearranging the set of integers like {0, -1, 1, -2, 2, -3, 3, ...}. so this is a countably infinite set (one that has a cardinality of ℵ_0)
As for the set of real numbers, we have the subset of irrational numbers which are uncountably infinite (see cantors diagonalization argument) thus making the whole set of real numbers, a set whose cardinality is ℵ_1.
The annotated turing book goes into this pretty well in the first couple pages.
One way to think about it would be to replace or with and: the continuum hypothesis can be true and false: it is a 'polycomputational object' [1].
[1] Using the concept of polycomputing from There’s Plenty of Room Right Here: Biological Systems as Evolved, Overloaded, Multi-Scale Machines: "Form and function are tightly entwined in nature, and in some cases, in robotics as well. Thus, efforts to re-shape living systems for biomedical or bioengineering purposes require prediction and control of their function at multiple scales. This is challenging for many reasons, one of which is that living systems perform multiple functions in the same place at the same time. We refer to this as 'polycomputing'—the ability of the same substrate to simultaneously compute different things, and make those computational results available to different observers.", https://www.mdpi.com/2313-7673/8/1/110
Here is Michael Levin, one of the paper's author, speaking at length about the polycomputing concept and more: "Agency, Attractors, & Observer-Dependent Computation in Biology & Beyond" [1].
(You can skip to the part that discusses Cantor's arguments, but I suspect that if you haven't heard about related concepts you probably want to understand what it is first.)
I love that the "hater" thread turns into a discussion of uncountable infinities :)
The "cardinality" section of that Wikipedia page describes my objection well. I don't doubt the real numbers are not recursively enumerable, but that doesn't mean they have a larger cardinality than the integers.
If you're trolling to troll, then expect the hate because you're being annoying.
If you think you're right and all the mathematicians are wrong, pony up. Hell, you'll have a lot more lulz when you win a fields metal.
If you don't pony up I don't know why you would be surprised people assume you're arrogant. You can doubt the status quo without being arrogant. We both know that you're not going to take someone's word just because they said so, so why expect others?
No, I don't think there's very much room for controversy here. I mean, I don't know what exactly Hotz have said, since there was no quote (and, honestly, I'm not that interested either), but if somebody is simply saying "the cardinality of integers is the same as real numbers" and leaving it at that, he is just plainly wrong.
Math is all about definitions and what follows from these definitions. So, you can define "integer", "real number", "cardinality", "equals" and so on however you like, and make all sorts of correct statements — as everyone will see by following your arguments all way from the definition/axiom and until the very end of your proof. But if you don't provide any definitions of your own, then you rely on some other definitions, and everyone has no other choice than assume that these are the very much "mainstream" ones, as you are referring to them as if they are well-known.
Now, it is unquestionably true (and easily provable) that the set of all computable numbers is countable, and anyone who says otherwise is wrong. But unless you specifically define real numbers as a subset of computable numbers, as constructivists are inclined to do, your listeners won't assume that, since this is not how real numbers are generally defined, and by virtue of not providing your own definition you are implicitly referring to a "general definition". (And, honestly, you shouldn't even call any subset of computable numbers "a set of Real numbers": this name is already taken.)
These general definitions and assumptions lead to all sorts of complications, and I personally have my doubts that real numbers exist in any meaningful sense (although I'm not committed to that statement, since there are several mathematical constructs that I would like to dismiss as "clearly nonsense", except they allow us to prove some very "no-nonsense" stuff — I don't know how to deal with that, and I never heard that anybody does). But I definitely cannot say that cardinality of integers is the same as the cardinality of reals, because this is simply not true under the common definitions (which is easily provable). (And less importantly, but worth saying that the contrary is not proven by constructivist methods — as half of the actually useful math in general ends up being, unfortunately).
So, as a somebody, who doesn't quite believe in non-computable numbers, I am very sympathetic to anybody who says that Real numbers do not exist. I don't understand how could they, what does it mean for an object that we cannot define to exist. Yet, I can accept (as a game) some well-known theory which talks about these non-quite-existant "Real numbers", and prove some statements about them, and one of these easily provable statements is that cardinality of continuum does not equal the cardinality of Natural numbers.
> I personally have my doubts that real numbers exist in any meaningful sense
I think we're in agreement here in principle. (Since we're on the topic, I'd like to add that naming this suspicious set "real" numbers is a tad bit ironic)
That said, I don't like the idea of having a group of people "owning" words as if they had a monopoly over them. The statement "the cardinality of integers is the same as real numbers" can be understood to mean "real numbers should actually be computable numbers".
I didn't bother to look up what Hotz wrote on twitter that triggered this discussion, I was just providing context that the issue of cardinality isn't as settled as some might think. It's probably not fruitful to argue whether a statement from hearsay uses words accurately or not though.
It does not. Logical outcomes that use infinity as an intermediary are inherently not reliable. An example of this is the Ramanujan summation where 1+2+3+... results in -1/12, an outcome which is disputed among mathematicians due to the fact that we have not defined the concept of infinity properly.
That's not a sum in the traditional sense so don't think about it this way.
Infinities are used quite often in mathematics for rather mundane things. Calculus doesn't work without it. It is also quite important to the foundation of many other areas but this is often hidden unless you get into advanced works (in this sentence we're not considering a typical undergraduate Multivariate Calculus, PDEs, or Linear Algebra as "advanced")
Calculus works fine without infinity. Finitism is basically a philosophical position without practical consequences. Plenty of serious people have planted their flag there. I don't find it particularly surprising that someone who works with computers, especially at a low level, would be drawn to it.
