31–40 of 359 posts
Earlier quoted context omitted.
I don’t like this framing because it implies any piece of pure math becoming useful is just a matter of time, and I see no reason to suspect this is the case.
> I see no reason to suspect this is the case Isn’t history enough reason?
> "Not all infinities are equal" In other words, there are different categories of infinite, and it might be inappropriate to represent infinity with just one symbol! This article is about how many types of infinity might exist. I was taught there is countably and uncountably infinite. Integers are countably infinite because the number of integers between any two numbers if finite. Real numbers are uncountably infini…
This is unsound: a set is countable if there's a mapping that assigns an integer to each element of that set. There are an infinite number of rational numbers between any pair of rational numbers, but you can pretty easily construct a mapping between integers and rationals. Imagine representing a rational x/y as a point on the plane, and draw a square spiral around the origin. Every rational number lays on that spiral, and there's a unique "shortest arc-length of the spiral from the origin" for each rational.
Earlier quoted context omitted.
We know that the cardinality of the natural numbers is less than the cardinality of the real numbers. The Continuum Hypothesis, which is a long unsolved problem, states that there are no sets with cardinality between the two. The posted article states that this new result strengthens the case against the hypothesis, that is that’s it’s probably false. All of this is nuanced but is important to mathematics and philoso…
Why does “the set containing the natural numbers and a sandwich” not have cardinality between the two?
Earlier quoted context omitted.
We know that the cardinality of the natural numbers is less than the cardinality of the real numbers. The Continuum Hypothesis, which is a long unsolved problem, states that there are no sets with cardinality between the two. The posted article states that this new result strengthens the case against the hypothesis, that is that’s it’s probably false. All of this is nuanced but is important to mathematics and philoso…
Why does “the set containing the natural numbers and a sandwich” not have cardinality between the two?
0 sandwich
1 0
...
n+1 n
By definition, if you can biject two sets, they have the same cardinality.
I'm not trying to be flippant, although it may come off that way: why does any of this matter?
It took many years to understand that! It’s not a flippant question at all. The answer is, it doesn’t matter. And that’s the joy of it. It wasn’t until I got into ML that I learned the value of doing unimportant work. When you’re free to think about inconsequential matters very seriously, you end up discovering so many useful things. It was how I independently rediscovered what is apparently called the Discrete Hartl…
Do I 'go home' - leave my home office - and think about it? No. Do I muse on how to solve a work-related issue when I'm showering? Also no. Will I forget almost everything about this job as or when I move onto the next? Definitely yes.
But academically, I'm interested, and actively publish albeit modest, low-impact research, in relational databases. I love the set-based approach to all matters SQL and after many years in the field still fool around trying to embellish, attack, improve and invent the core ideas.
Why bother? Even if I come up with some magical new improvement to RDBMSs, it doesn't matter. If I fork MySQL and try out my ideas, no one cares. I'll never get the traction or the FOSS community support, I'll spend more time playing politics and managing collaborators than I'd like (zero) and I enjoy the independent thought that my interests bring.
So my academic work is pointless. But it's meaningful to me. And I think that's what matters. I don't understand one-tenth of the ideas in this article, but that doesn't matter. What matters is that it's interesting to somebody.
It depends on what you define as a number. For example reals and complex have different properties from integers. Is there a mathematical reason why both are considered numbers? You can count (with) integers but not with reals. Thus one appears to be a number while the other appears to be a measure.
Generally speaking, a good chunk of the number systems mathematicians worked with involve taking some existing number system (tracing back to the natural numbers), finding some operation that takes you outside that system, and then building a new system to cover that hole. A system that has no such holes given a particular operation is said to "close over" that operation. So, integers close over subtraction; rationals close over division, and complex numbers close over exponentiation. Since all of these systems ultimately derived from operations over the natural numbers, it makes sense to also call them numbers, even though they might lack certain symmetries or properties of simpler number systems.
Also, the word "measure" is already taken by a different concept.
Earlier quoted context omitted.
We know that the cardinality of the natural numbers is less than the cardinality of the real numbers. The Continuum Hypothesis, which is a long unsolved problem, states that there are no sets with cardinality between the two. The posted article states that this new result strengthens the case against the hypothesis, that is that’s it’s probably false. All of this is nuanced but is important to mathematics and philoso…
Why does “the set containing the natural numbers and a sandwich” not have cardinality between the two?
If we're ok with extending that to sets of infinite things (we can still pair elements of each set, we'd just never be able to finish listing all the pairs), then we can say that the natural numbers and "the set containing the natural numbers and a sandwich" are of equal size because we could pair 1 from the first set with the sandwich from the second set, pair 2 from the first set with 1 from the second set, 3 from the first set with 2 from the second set, etc etc.
There's no element of either set without a match in the other set, so they have the same cardinality as the natural numbers, with or without the sandwich.
Great article. Since it looks like a lot of folks are interested in this article, some extra background. First, what is forcing? The article actually has a great description of ultrapowers (a key part of the construction) but it goes by a little fast, so you might like Tim Chow's "A beginner's guide to forcing" [1] which does a good job not only laying out the mathematical details at a high level, but also really cle…
Why does forcing work? To me it seems flawed (which obviously means I don't understand it fully). For diagonalization argument: 1) Assume every real can be assigned a natural number. 2) Do a bunch of steps that essentially find a new real that differs from any real you have listed from step 1. 3) Conclude that either your steps are flawed, or your initial assumption is wrong 4). Because your steps aren't flawed then…
Cantor's diagonalization proof proves that it's impossible to list all the real numbers with a list of size aleph-0, which is the cardinality of the set of natural numbers.
The forcing proof is an attempt to prove you also can't do it with the a list of the size aleph-1, which is the size of the power set of aleph-0. It purports to prove that you need a list of size aleph-2, which is the size of the powerset of aleph-1.
You can kind of think of this not so much as whether "Can Crysis run?" ("can you have a set of all real numbers?") but "Can Crysis run on this machine?" Do the real numbers need aleph-1 "resources", or aleph-2 "resources" to list?
I'm not trying to be flippant, although it may come off that way: why does any of this matter?
Others have addressed why it matters (or doesn't matter) when viewed from outside mathematics. But within mathematics, unsolved problems usually matter for two reasons: (1) When lots of really smart people spend lots of time trying to solve something, and fail to do so, it becomes even more interesting for other smart people. A well-known example is the Collatz Conjecture[0] which, by most accounts is a meaningless p…
> If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found.
There are lots of histograms and empirical data supporting the conjecture.
My question is, why is this empirical data not "good enough" for mathematicians? As something closer to a physicist, I do wonder what the fascination is with trying to prove conclusively that 3n+1 xor n/2 will eventually encounter 1. It seems a bit like trying to prove conclusively that the gravitational constant is so-and-so, when it seems the best you can do is to measure it as precisely as possible:
> Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics."
As someone who loved mathematics but was never much good at it, what's the fascination here? (I'm only trying to understand the motivations of people smarter than I am.)
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