> the mathematical universe, like our physical one, may be made up mostly of dark matter. “It seems now that most of the universe somehow consists of things that we can’t see,”
Not heaps fond of relating invisible things in the mathematical universe to dark matter! Although maybe both might turn out to be imaginary/purely-abstract? Imaginary things can absolutely influence real things in the universe, it's just that they are not usually external to the thing they are influencing. If I imagine making a cake say, and then I go ahead and make the one I imagined, the 'virtual' cake was already inside me to begin with, and wasn't 'plucked' from a virtual universe of possible cakes somewhere outside my knowledge of cake-making.
Something nags at the back of my mind around this about maths though, as if to suggest that as soon as there was one-of-anything that was kinda an 'instantiation' of the most abstract "one" object from the mathematical universe.. (irrespective of what axioms are used as long as they support something like one) But I doubt there's never been exactly-PI-of-anything in the real universe, just a whole bunch of systems that behave as if they know (or are perhaps in the process of computing) a more exact value! (spherical planets, natural sine waves etc!)
Very interesting article, I wish my math was stronger! I can just skirt the edges of what they're actually talking about and it's tantalizing! Would love to know more about these new types of cardinal numbers they've developed/discovered.
An interesting thing about the quote you highlighted is that it's already true about the set of real numbers itself. The set of real numbers that can be precisely, individually identified is a countable subset of all real numbers. That means the vast majority of real numbers, an uncountable amount of them, can not be individually defined and thought about.
That is very interesting I agree, and certainly any list of descriptions/identifiers must be countable, though I wonder if there's any validity in descriptions that describe things in aggregate?
It's certainly a brain-bender that even in the unit interval if we imagine filling in all the the rationals and then adding in the describable-irrationals like PI/4, sqrt(2)/2 and so on.. that this still does not even come close to covering the unit interval - or any interval - of Real numbers! My imagination sees a line with a heck of a lot of dots on it, but still knowing that there clearly still uncountably-more values that are not covered/described! Amazing! The continuum (Real numbers) is such a fascinating concept!
> But add a smaller cardinal to one of the new infinities, and “they kind of blow up,” Bagaria said. “This is a phenomenon that had never appeared before.”
I have to wonder just what is meant by this, because in ZFC, a sum of just two (or any finite number) of cardinals can't "blow up" like this; you need an infinite sum. I mean, presumably they're referring to such an infinite sum, but they don't really explain, and they make it sound like it's just adding two even though that can't be what is meant.
(In ZFC, if you add two cardinals, of which at least one is infinite, the sum will always be equal to the maximum of the two. Indeed, the same is true for multiplication, as long as neither of the cardinals is zero. And of course both of these extend to any finite sum. To get interesting sums or products that involve infinite cardinals, you need infinitely many summands or factors.)
I suspect they mean "add" in the sense of "add in an axiom asserting the existence of another cardinal". Things like consistency strength of the resulting theory seem to vary wildly depending on what other cardinals you throw into the mix (if I understood the article correctly, haven't read the paper, mea culpa).
If I had a semester or two of free time I'd love to hit this subject again. I once told my math prof (logician) who made a comment about transfinite cardinals: careful it's powerful but it's power from the devil. I half regret that comment in retrospect.
I've never made peace with Cantor's diagonaliztion argument because listing real numbers on the right side (natural number lhs for the mapping) is giving a real number including transedentals that pre-bakes in a kind of undefined infinite.
Maybe it's the idea of a completed infinity that's my problem; maybe it's the fact I don't understand how to define (or forgot cauchy sequences in detail) an arbitrary real.
In short, if reals are a confusing you can only tie yourself up in knots using confusing.
Sigh - wish I could do better!
There are a couple of strategies for understanding the real numbers. One is to write down a definition of real numbers, for example using rational numbers and Dedekind cuts, hoping that what you're describing is really what you mean. The other is to write down the properties of real numbers as you understand them as "axioms", and go from there. An important property of real numbers that always comes up (either as a consequence of Dedekind cuts or as an axiom itself) is the least upper bound property -- every set which has an upper bound has a least upper bound. That's what gives you the "completeness" of the real numbers, from which you can prove facts like the completeness of the real numbers (i.e., Cauchy sequences always converge), the Heine-Borel theorem (closed and bounded subsets of the reals are "compact", and vice-versa), and Cantor's intersection theorem (that the nested intersection of a sequence of non-empty compact sets is also compact).
The diagonalization argument is an intuitive tool, IMHO. It is great if it convinces you, but it's difficult to make rigorous in a way that everyone accepts due to the use of a decimal expansion for every real number. One way to avoid that is to prove a little fact: the union of a finite number of intervals can be written as the finite union of disjoint intervals, and that the total length of those intervals is at most the total length of the original intervals. (Prove it by induction.)
THEOREM: [0, 1] is uncountable. Proof: By way of contradiction, let f be the surjection that shows [0, 1] is countable. Let U_i be the interval of length 1/2*i centered on f(i). The union V_n = U_1 + U_2 + ... + U_n has combined length 1 - 1/2*n < 1, so it can't contain [0, 1]. Another way to state that is that K_n = [0, 1] - V_n is non-empty. K_n also compact, as it's closed (complement of V_n) and bounded (subset of [0, 1]). By Cantor's intersection theorem, there is some x in all K_n, which means it's in [0,1] but none of the U_i; in particular, it can't be f(i) for any i. That contradicts our assumption that f is surjective.
Through the right lens, this is precisely the idea of the diagonalization argument, with our intervals of length 2*-n (centered at points in the sequence) replacing intervals replacing intervals of length 10*-n (not centered at points in the sequence) implicit in the "diagonal" construction.
> Maybe it's the idea of a completed infinity that's my problem; maybe it's the fact I don't understand how to define (or forgot cauchy sequences in detail) an arbitrary real.
As someone who also has never fully made his peace with the diagonality argument, but just chosen to accept it as true, as a given, this kind of bumps up against an interesting implication of different cardinalities of infinity.
To precisely define an arbitrary real you'd need some kind of finite string that uniquely identifies that real number. Finite strings can be mapped, 1 to 1, to natural numbers. Therefore there can't be a finite string for any real number that uniquely identifies it. Otherwise we'd have a mapping between natural numbers and real numbers.
In fact, the set of uniquely identifiable real numbers is a countable subset of real numbers. [1]
Somehow, this realization has helped me make peace with the uncountability of real numbers.
[1] Sorry if use words like "unique", "identify", "define" in not quite the right way. I hope the meaning I'm going for comes across.
Cantor’s original proof of the uncountability of the reals didn’t use a diagonalization argument, it used order + completeness and in fact applies to any complete poset. https://en.wikipedia.org/wiki/Cantor%27s_first_set_theory_ar...
Likewise his proof that there is no surjection from a set to its power set uses a more general diagonalization argument that doesn’t make any uncomfortable assumptions: https://en.wikipedia.org/wiki/Cantor%27s_theorem
how much of modern set theory is reverse engineered from axioms rather than discovered. we're always building highways through a forest we haven't mapped, assuming every tree will fall in line. and suddenly these new large cardinals show up that don't even sit neatly in the ladder. it's maynot be failure of math,but failure of narrative. we thought the infinite was climbable, now it's folding sideways. maybe the math we're building is just a subset of what's possible, shaped by what's provable under our current tools. lot of deep shit probably hiding in the unprovable.
This isn't really how it went down, historically. They considered lots of different large cardinals, and then they turned out to be linearly orderable by consistency strength. And then it's natural to wonder if it's a general rule.
I’ve always considered math is something that is discovered, neither chaotic or orderly, it just… is. Really brilliant people make new discoveries, but they were there the whole time waiting to be found.
This article seems to kind of dance around yet agree with the discovery thing, but in an indirect way.
Math is just math. Music is just music. Even seemingly-random musical notes played in a “song” has a rational explanation relative to the instrument. It isn’t the fault of music that a song might sound chaotic, it’s just music. Bad music maybe. This analogy can break down quickly, but in my head it makes sense.
Disclaimer - the most advanced math classes I’ve taken: calc3/linear/diffeq.
What makes you consider it a "discovery" instead of a creation of us humans?
I am more on the side of seeing maths as a precision language we utilize and extend as needed, especially because it can describe physically non-existent things e.g. perfect circles.
Mathematics isn't monolithic—it depends heavily on the axioms you choose. Change the axioms, and the theorems change. ZFC, ZF¬C, intuitionistic logic, non-Euclidean geometry—each yields a different “math,” all internally consistent. So it’s not right to say math “just is” in some absolute sense. We’re not just discovering math; we’re exploring the consequences of chosen assumptions.
For instance:
Under Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), every set can be well-ordered, but we do get the Hahn–Banach paradoxes.
Under ZF without Choice, analysis as we know it no longer holds.
In constructive mathematics, which avoids the law of the excluded middle, many classical theorems lose their usual formulations or proofs.
Non-Euclidean geometries arise from altering the parallel postulate. Within their own axioms, they are as internally consistent and "natural" as Euclidean geometry. Do non-intersecting lines exist in this universe? I've no idea.
This just steps one meta level higher. Yes, you can make your object of analysis the axioms and what they lead to and proof theory etc. But now you've just stepped back one level. What are the axioms that allow you to derive that "ZFC leads to Hahn–Banach paradoxes"? Is this claim True and discovered or is it in itself also simply dependent on some axioms and assumptions?
This is part of a broader meta-ization of culture. Philosophers are also much more reluctant to make truth claims in the last century compared to centuries ago. Everything they say is just "To a Hegelian, it is {such and such}. For Descartes, {x, y, z}." If you study theology, they don't teach with conviction that "Statement A". They will teach that Presbyterians believe X while the Anglicans think Y, and the Catholics think it's an irrelevant distinction. In fact many would argue that math is not too far from theology. People who were obsessed with math limits, like Gödel, were also highly interested in theology.
I guess physics is the closest to still making actual truth claims about reality, though it's also retreating to "we're just making useful mathematical models, we aren't saying that reality is this way or that way".
> all internally consistent
Well, we hope.
Large cardinal axioms are the paradigmatic example "invented" math.
We need a word-less world of math where all meaning is derived from figures. WOrds are confusing.
"If you can't describe the meaning using only pencils and compass, you don't mean it"
And especially when we mix different categories. Like saying about any infinity as about an object is misleading, because it's rather a process
I can think of N as a process in a sense, because I can keep adding a number. But I can't think of R as a process like this, specifically because there is no surjective mapping from N to R.
How would you think of R as a process?
Infinities (transfinite cardinals) in the sense used by the article are absolutely objects. We’re not talking about infinite sums or other sequences and their limits. (And limits aren’t really “processes” either – the limit of the sequence 0.9, 0.99, 0.999, … is exactly 1, as a well-known example which nonetheless is controversial among people who don’t know what limits are.)
Sure, if you accept reifying concepts into objects as valid. But that is a gateway to misery.
Object in OOP ?
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