Yes, this stuff is fishy, and yes we can blame ZFC which is a bad formalization in comparison to what we've developed since. But the real scandal is why does our definition of geometry "leak" the underlying set theory it's built atop so much? Surely it's bad to have such a leaky abstraction in pure math!
The series goes on to show that by abandoning "points" — which pull all the funny set theory stuff into geometry/topology/whatever is the topic at hand, one can still have a classical foundation — e.g. with the axiom of choice and law of excluded middle — that makes mathematicians feel at ease, but also purge this Banach–Tarski gobbledygook.
I think things like the Banach-Tarski theorem are the other side of that coin: they're showing some of the places where the formalisation we're starting with isn't a great fit for some things we might hope to use it for.
I don't think I'd go as far as to say that makes the formalisation outright bad, but looking at alternate systems which don't admit Banach-Tarski-like results is surely a worthwhile way of spending time.
See https://golem.ph.utexas.edu/category/2021/06/large_sets_1.ht... for tackling the "large cardinal pissing contest" that is much of modern set theory.
Your very statement is a good retreat from platonism with blinders, acknowledging the inherit "moral relativism" that there are many possible foundations, and it is up to usflawed humans to decide what we like to work with best.
The earlier intuitionists like Brouwer were polemicists, perhaps because they felt very alone. Now there is a good network of CS-mathematician hybrids to keep everyone feeling more sane.
Here we see the dual track that you can question your foundational choices and your higher level abstractions (point-set topology vs locales which are distilled to being purely order-theoretic) concurrently. It's nice to take the same skepticism and interest in finding definitions the work with not alienste the working mathematician at multiple levels.
Because, for all the trepidation about abandoning ZFC, the mainstream formalizations have clearly failed in that mathematicians that aren't logicians or set theorists would rather engage with them as little as possible.
> I think things like the Banach-Tarski theorem are the other side of that coin: they're showing some of the places where the formalisation we're starting with isn't a great fit for some things we might hope to use it for.
I don't follow. You can view the Intermediate Value Theorem as something that motivates the definition of "continuous function", so that once you have the definition it had better conform to the theorem, sure.
But the Banach-Tarski theorem isn't like that. It's just a cool result of some other things that work well. It's not motivating anything or being motivated by anything.
Yeah... Those crazy HoTT people, trying to actualize the goal of putting mathematics on an actually firm foundation and removing the rest of the gobblygook handwaved into the religion of math as opposed to the pure logic it represents...
Also you can use HoTT WITH AC / law of excluded middle... It's just not there by default and there are some really nice things you get without it, so it's pretty much only the lazy crutch of mathematics since forever. If you see proof via excluded middle, consider it a code smell (and recall by the Curry-Howard correspondence the proof is essentially code)
[0]: https://en.wikipedia.org/wiki/Homotopy_type_theory#Special_Y...
[1]: http://math.andrej.com/2016/10/10/five-stages-of-accepting-c...
Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite.
Happy to hear disagreements tho.
This is true of all mathematical objects. The number 7 doesn't exist in the universe either. It's not a physical object.
Honestly I think that's a continuous claim, and comes down to differences in understanding. I can certainly have 7 of some object, does the 7-ness exist in the collection? Not really, but what about another phenomenon: colour? An object appears blue, and we say it is blue, and the blueness is due to physics, but it's a subjective delineation. A table is a delineation too, the leg is part of the table and the White House is not. In some sense, the table-ness category is just as real as the 7-ness category.
Of course you could just say that all that actually exists is some collection of particles/fields, but then you've abused all the words we're using until they stop being useful.
So, while I can't "point" to an infinite number of things like I can point to 9 things or 3.62 things, I still think it exists.
I'm not sure how well this generalizes to all infinite cardinals, ordinals, or to transfinite induction/construction. It is certainly strange that Cantor's theorem (the cardinality of a set is strictly smaller than that of its power set) implies there are different sizes of "all" implicit in my usage of the word.
What this means is that a universe that contains infinities is, even in theory, entirely indistinguishable (in finite time) from an universe that contains really large/small but finite quantities.
Also wouldn’t your argument also apply to zero? You can never know if a quantity is zero as opposed to some enormously small epsilon that you haven’t detected yet. Is zero “unscientific?”
That depends on the model of computation you pick, doesn't it?
Even whole numbers are an abstraction that makes sense only when you can clearly define what is the thing you're counting.
Mathematics has nothing to do with laws of physics. Even if the laws of physics[0] were different, these mathematical[1] truths would remain the same.
[0] Laws of physics don't actually exist. They're shorthand generalizations about features of particulars. The notion of some kind of abstract disembodied "laws" that somehow "govern" everything is absurd.
[1] For clarify: mathematics is a field that studies such things.
What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large.
The language that i see in this article and elsewhere however is suggesting that we actually duplicated the sphere (doubled the volume).
This seems incorrect.
> This seems incorrect.
It isn't incorrect. You're right that the number of points in the sphere does not equate to the volume of the sphere. But the Banach-Tarski theorem does in fact let you double the volume. It is considered to be of interest because it does the following:
1. You have a ball.
2. You cut the ball into 5 pieces in a very clever way.
3. You move the pieces around.
4. Now you have two balls, each the same size as the first.
The key, interesting part of this is in step 3, where we only use translations and rotations. Those preserve volume. (By contrast, it's easy to scale a ball of radius 2 to become a ball of radius 3, but that's not a volume-preserving transformation.) The part of the process that doesn't preserve volume is actually step 2, where we cut the ball into pieces. People find it unintuitive that this step doesn't preserve volume.
You can also cut your ball into several pieces and move the pieces around such that you end up with a much larger ball.
So the paradox breaks down when you start to realize that you are not CUTTING but “choosing some points” and rearranging them. The fact that this rearrangement can be done with Euclidean moves is the surprise.
You can obviously produce a large sphere from a small sphere by rearranging the points, as long as you're willing to handle one point at a time -- that's what scaling is. But that requires an uncountably infinite number of translations. The Banach-Tarski theorem says we can do the same thing in only a finite number of translations.
Addition is defined as an operation with two inputs. You can't add more than two things, unless there is some particular rule that lets you.
If you have finitely many things, then this rule is the associative law: add them pairwise in whatever order, and you are guaranteed to get the same result.
To add infinitely many numbers, you need to talk about limits. Formally, when you say something like
1 - 1/2 + 1/4 - 1/8 + 1/16 - ...
you mean: look at the sum of the first two; then, look at the sum of the first three; then, look at the sum of the first four; and so on -- this sum converges to a limit, which is 2/3.
This sum is "absolutely convergent", which means you get the same result no matter how you order the summands, but some infinite sums change if you reorder things!
With points on the sphere the situation gets even worse, as there is no way to "list them in order". These sets are "uncountable", which means don't even try to sum any function defined on them.
To say approximately the same thing using technical jargon, one has countable additivity for Lebesgue measure on the reals, but uncountable additivity does not hold.
Not true. If you add uncountably many infinitesimal objects they can add up to noninfinitesimal object, that's how integration works in math, it's pretty confusing cause there's many kinds of infinity and they allow some unintuitive things to happen, but if they didn't worked we couldn't move (see Zeno paradox).
Banach-Tarski is formally correct, you add a finite number of sets with uncountably many points in each so you can get something with volume (depending on how they are positioned).
And yes - a line in math is just a set of points, same with a sphere (but it has 0 volume cause a sphere is just the "skin" without the insides) and a ball (which is what Banach-Tarski talks about). In fact every geometric object is just a set of points.
You're on the right track.
The Banach-Tarski paradox requires accepting that non-measurable sets[1] exist. A non-measurable set is a set with a an inspecifiable volume. Note: That's non-measurable - not 0. It means you have a quantity of something, whose volume is not 0, but it's also not any other number.
Once I realized that the paradox requires it, all the WTF aspect went away. Of course - if you can accept quantities for which you cannot specify a volume, you can probably accept about anything.
Interpret it as "adding more points will not necessarily increase the volume, no matter how many points you add". There are plenty of measure-0 sets containing as many points as the continuum does.
I'll be using spatial dimensions as a conceptual framework to tackle this exact issue in future videos.
That part is easy - for each point on a unitary sphere move it to a point at position 2x ( ie. to a corresponding location on the sphere of the 2 units radius) - you've just doubled the volume, i.e. you've just built a 2 units radius sphere out of the points belonging to 1 unit radius sphere. Banach-Tarski of course more fun and illustrates much more than just volume.
In the paragraph on nonstandard analysis in the Wikipedia page on infinity, it says:
"The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers"
https://en.wikipedia.org/wiki/Infinity
I can't say anything precise or mathematical, but after I read the above, I have an "obvious in hindsight" feeling. If H=inf is different from H + 1, how much different is it? 1/inf or an infinitesimal amount! And an infinitesimal is not nothing.
The quanta article says "You can add or subtract any finite number to infinity and the result is still the same infinity you started with" but this seems like just a dogma for non mathematicians?
They really aren’t connected. The first statement (the positive integers can be partitioned into two sets, each of which has the same size as the original set) follows from the usual axioms of set theory (ZF), while the Banach–Tarski paradox cannot be proven to work without the Axiom of Choice or a similar axiom.
The natural numbers (and therefore Hilbert's Hotel) provide a natural way to say "whatever, just pick one" but we need to invoke the well-ordering theorem (which is equivalent to the Axiom of Choice) make the same "whatever, just pick one" statement about the reals. (and therefore Banach-Tarski)
I used to riff with a friend that we were "the two members of the Banach-Tarski quartet." :)
That's not to say that physics requires infinities, but current models also don't disallow infinity.
Of course, actual infinity is outside the purview of science - there is no way to differentiate between infinity and something too big/small to measure, even in principle. Apparent paradoxes related to infinity, such as Banach-Tarski, don't change this, as they also require infinite precision to realize, making them impossible to test as well - even if a sphere is indeed made up of an infinity of space-time points, and even if we could manipulate those, we wouldn't be able, in finite time, to extract the necessary infinite subsets of points to create the two spheres from one.
---
...It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?"
"No holes?"
"No holes."
"Impossible! There ain't no such thing."
"Ha! We got him! Everybody gather around! It's So-and-so's theorem of immeasurable measure!"
Just when they think they've got me, I remind them, "But you said an orange! You can't cut the orange peel any thinner than the atoms."
"But we have the condition of continuity: We can keep on cutting!"
"No, you said an orange, so I assumed that you meant a real orange."
It's sort of hilarious to see a physics site mention the Banach-Tarski paradox. It is, after all, the most obvious hole poked in the most basic working assumption used by physicists: that space and time are measured with real numbers.
I've seen physicists go to pretty absurd extremes to avoid thinking about the problems this creates. Fixing it properly is not easy: simply dropping the axiom of choice leaves you unable to do useful physics. Getting back to a useful state, making all sets Lebesgue, can only be done with large cardinals:
https://www.jstor.org/stable/1970696
Large cardinals are pretty exotic even by the standards of mathematicians. In many departments they are in fact the domain of logicians. In fact, the existence of certain classes of Woodin cardinals is equivalent to the Axiom of Determinacy (AD), which is the "mathematically respectable" way of investigating logics with infinitary conjunction/disjunction. In fact, AD is precisely the Law of Excluded Middle (A or not-A) for logics with infinitely-long conjunctions.
Quite odd that something so ethereal would be connected to a tangible act like cutting an apple in half.
Assume I have a sphere made of pure iron. I divide the sphere into individual iron atoms. I divide this group of atoms into two groups of atoms. I take each of those groups of atoms and form them into 2 spheres. How is it that these two new spheres are not either less dense or smaller that the original sphere?
You have highly restricted the act of choosing sets of points here. B-T doesn't say that any "division" results in that unintuitive outcome.
Note that points are infinitesimally small and infinitely many, and atoms in your iron sphere are neither.
In reading the comments for the video, I got the sense that this is different and that I was missing something but couldn't come close to guessing what that was.
Looking into it more closely, it turned out to be both trivial and not notably meaningful, like most surprising results involving uncountable infinity. Nothing that affects us involves actual infinities, so infinities are just a convenient approximation that often produces correct-enough answers. Anything infinities imply that seems crazy trivially is.
It is natural to suspect that foundational axioms are somewhere flawed.
TLDR; It is basically the same as proofs that all countable sets have the same cardinality. (TLDR of that: map set of positive integers x to the even numbers by doubling, and the odd numbers by doubling and subtracting 1. Take the union of even and odd and you end up with the set you started with, the positive integers x).
For a circle:
We can identify all the points on a circle as the points p associated with the [x,y] coordinates of the complex numbers p = e^(2.c.i.pi), where 0 <= c < 1. (And . is multiply.)
If we take each of those points p and rotate it by doubling its c, we now have the same points represented by the expression p = e^(2.c.i.pi), where x <= 0 < 2.
So the same number of points, but two passes around the circle, 0 <= c < 1 and 1 <= c < 2. We can move the second set of points in the x positive direction by 2 or more to avoid the overlap.
We have now rearranged points of one circle into two.
For the surface of a sphere:
We simply divide a sphere up into points defined by a stack of circles at real-valued vertical z positions, z <= -1 <= 1. And their real [x,y] points are the real and imaginary parts of each circle e^(2.c.i.pi).circumference(z), where 0 <= c < 1, and circumference(z) is the cos(z).
Again, rotate the points by doubling c, so that they are now located at c, where 0 <= c < 2. There are now two overlapping sphere surfaces. We can move the second in any direction by 2 to avoid the overlap.
Similar generalizations work for including the volume.
Anyone understand why this simpler proof is wrong, or why the more complex proof in the article does something better?
I don't buy the diagonalization proof as anything more than the Pythagoreom Theorom. You have infinite rows, and infinite columns. Infinity is Schrodinger's Cat. Once you check in on the state (nth row by mth column) the only thing you can say about the diagonal number is that is hasn't occurred in the rows up to that point, not beyond, nor in the columns (if n > m).
Ergo, Infinity is a paradox, and only mathematical in the absurd.
