[HN Gopher] Banach-Tarski and the Paradox of Infinite Cloning
___________________________________________________________________
Banach-Tarski and the Paradox of Infinite Cloning
Author : daviddisco
Score : 48 points
Date : 2021-08-29 12:05 UTC (10 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| smiley1437 wrote:
| Is this basically the same premise as Zeno's arrow, but with more
| steps?
| derbOac wrote:
| I had the exact same thought -- it seems like some
| multidimensional version of a Zeno's paradox, with all the
| attendant issues.
| Viliam1234 wrote:
| An interesting difference is that the Banach-Tarski trick works
| in 3D, but not in 2D or 1D.
| Sinidir wrote:
| Can someone correct me if im wrong?
|
| 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.
| ouid wrote:
| "no matter how many you add together", is where this argument
| breaks down in ZFC. The sphere is indeed the union of all of
| the singletons consisting of its points, all of which are
| measure zero. Banach-Tarski is mainly considered "weird"
| because it describes a partition into so few pieces, and they
| are rearranged via rigid motions only. It is trivial to come up
| with bijections between compact finite dimensional manifolds,
| (https://en.wikipedia.org/wiki/Space-filling_curve). For
| another example of the axiom of choice wreaking havoc on the
| notion of measure, see https://en.wikipedia.org/wiki/Vitali_set
| .
| thaumasiotes wrote:
| > "no matter how many you add together", is where this
| argument breaks down
|
| 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.
| thaumasiotes wrote:
| > 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.
|
| 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.
| pfortuny wrote:
| "Cutting" it is certainly not: none of those sets is given by
| the zeroes of a continuous function (they would be
| measurable, and they cannot be).
|
| 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.
| thaumasiotes wrote:
| Come to think of it, the fact that two spheres contain the
| same number of points as one sphere does would seem to be
| closely related to why it's possible to produce two spheres
| from one sphere just by rearranging the points.
|
| 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.
| throwaway81523 wrote:
| It is deeper than that. There is no way to do a similar
| duplication of a 2-dimensional disc. Why is it different in
| 3 dimensions? That is a property of the transformation
| group (rotation and translation) rather than 3-space
| itself.
| karmakaze wrote:
| This comment and the parent sheds some light on what
| makes this interesting. It didn't occur to me that we
| were allowing rotations and translations and that scaling
| is excluded. The paradox isn't about getting more from
| less as they're all comparably infinite, but rather being
| able to arrange them to be so.
| thaumasiotes wrote:
| > Why is it different in 3 dimensions? That is a property
| of the transformation group (rotation and translation)
| rather than 3-space itself.
|
| Can you be more specific? Rotations and translations also
| exist in 2-space. It seems difficult to argue that this
| difference between 2-space and 3-space is "not a property
| of 3-space".
| throwaway81523 wrote:
| What I mean is if you pick different transformation
| groups you can get different spaces where the "paradox"
| occurs. Likewise you can get rid of the paradox in
| 3-space by picking a different group.
| [deleted]
| joiguru wrote:
| "cut the ball into 5 pieces" is not the best description. A
| better one is: 2a. Split the ball into infinite pieces 2b.
| Divide the infinite pieces into 5 groups
| thaumasiotes wrote:
| > 2a. Split the ball into infinite pieces 2b. Divide the
| infinite pieces into 5 groups
|
| Huh? The ball is already composed of infinite points. So in
| 2a you recognize that the ball exists, and then in 2b you
| cut it into pieces. But it seems superfluous to mention 2a
| separately.
| joiguru wrote:
| In regular natural language, cutting the ball into 5
| pieces implies cutting 5 contiguous pieces.
|
| If I say a cake is cut into 5 pieces, no person will
| consider that each piece contains parts from all parts of
| the cake.
| ajuc wrote:
| > A point has no volume so no matter how many you add together
| you don't get something with a volume
|
| 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.
| stan_rogers wrote:
| Points, though, as traditonally defined, are _not_
| infinitessimal. They are literally zero in extent, having
| only a defined location.
| pacifist wrote:
| A sphere in the mathematical sense is the 2D shell on the
| surface of the sphere you're thinking of. So, no height, no
| volume.
| pacifist wrote:
| My bad. Should have RTFA. We're talking about 3D here.
| impendia wrote:
| In your example, I would say that it's _addition_ that 's
| failing.
|
| 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.
|
| https://mathworld.wolfram.com/CountableAdditivity.html
| BeetleB wrote:
| > A point has no volume so no matter how many you add together
| you don't get something with a volume.
|
| 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.
|
| [1] https://en.wikipedia.org/wiki/Non-measurable_set
| dcminter wrote:
| If I remember rightly there'a a Feynman anecdote where he points
| out that as the real universe is quantised this is a purely
| mathematical notion.
|
| I used to riff with a friend that we were "the two members of the
| Banach-Tarski quartet." :)
| tsimionescu wrote:
| Current physical models don't have the universe itself (space-
| time) quantized - only matter is quantized. Even the planck
| time and planck length only represent minimal _measurable_
| distances /durations - the maths still assume that two things
| can be separated by fractional multiples of these.
|
| 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.
| rob_c wrote:
| Physics doesn't require infinities, but it's scary how well
| QED/QFT approximates the g-factor
| (https://en.m.wikipedia.org/wiki/G-factor_(physics) ) for
| electrons given the amount of renormalization (cancelling of
| infinities) needed to estimate the true value in nature.
| dcminter wrote:
| I didn't remember quite correctly - here it is, from the
| section "A Different Box of Tools" in "Surely you're joking Mr
| Feynman". He doesn't state it explicitly, but I think it's
| clear they must have been talking about Banach-Tarski:
|
| ---
|
| ...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_. "
| mugwumprk wrote:
| I do enjoy articles like this. They are such good ways to make
| math compelling for laymen such as myself.
| paulpauper wrote:
| I don't understand the paradox. Obviously if you dissaemble or
| scamble something u can reararange it?
| bobthechef wrote:
| When's the last time you came across something that you could
| disassemble and could then reassemble into two things identical
| to the first thing you disassembled?
|
| It is natural to suspect that foundational axioms are somewhere
| flawed.
| x3n0ph3n3 wrote:
| One thing worth pointing out is that our universe operates on
| the integers rather than the real numbers, and the Banach-
| Tarski requires operating on the reals.
| Smaug123 wrote:
| Is that known? It's an appealing idea, but bearing in mind
| that general relativity is very resistant to quantisation,
| I'm not sure I'd be comfortable to declare it as fact.
| roywiggins wrote:
| It's like taking a bed apart and rearranging it into two beds,
| each identical to the original bed, without adding any more
| material.
| paulpauper wrote:
| Oh that is pretty cool. I wonder if it could be proved with
| differential geometry for the sphere, which has a simple
| paramertization
| Smaug123 wrote:
| The ability to do it is equivalent to the axiom of choice,
| so my guess is going to be "not without considerable
| effort".
| KirillPanov wrote:
| Spoiler: the cut line between the two apple-halves is fractal in
| shape, with infinite surface area and taking forever to cut at
| any finite cutting speed.
|
| 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.
| desperate wrote:
| To me this is proof that infinity is something only present in
| our math and not in the universe.
|
| 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.
| robot_no_419 wrote:
| Mathematics doesn't really "exist" in the first place. It's
| more of a language that's rich enough and with enough logic to
| describe/approximate the laws of physics that actually do
| exist.
| bobthechef wrote:
| So quantity, structure, formal necessity don't exist?
|
| 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.
| throwawaygal7 wrote:
| I have never understood the opposition to the law of
| physics you seem to hold.
|
| What is the alternative? That we merely observe rigid
| patterns that are baked into physical reality? Isn't
| whatever is 'baked in' more or less a 'law of physics'?
|
| If these are just 'brute facts' are they not then 'laws'?
| Maybe governance is too strong an word for the
| correspondence but what is the alternative?
| snet0 wrote:
| Laws in the physics sense don't govern, though, they
| describe. It's akin to saying moral laws don't _decide_
| what 's immoral, they simply describe it.
|
| Given that, "laws of physics" are certainly describable. We
| simply write the formulae that tell us what the next state
| of the dynamical system is. They are ways of delimiting
| what is physically possible, given the current state of the
| art.
| dlkmp wrote:
| What's your concept of time and its extension then?
| tsimionescu wrote:
| Well, infinity is inherently unscientific, as there is no way
| to scientifically differentiate between an infinite quantity
| and a really huge quantity (same for infinitesimals), in finite
| time.
|
| 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.
| ithkuil wrote:
| What about negative numbers? Or complex numbers? They are
| only tools, which can be quite useful to build models of the
| world with predictive powers but shouldn't be confused for
| the underlying reality.
|
| Even whole numbers are an abstraction that makes sense only
| when you can clearly define what is the thing you're
| counting.
| tsimionescu wrote:
| Whole numbers can be defined and proven to be necessary to
| describe the world pretty easily. From there, rational
| numbers are trivial to define. Negative numbers are
| somewhat more abstract, but they have very intuitive
| definitions in many domains, such as accounting. It may be
| possible to avoid them in a theory of physics, though.
|
| The complex numbers (well, at least those with a rational
| imaginary part and a rational real part) have been recently
| proven to be necessary to describe the universe[0]
| (assuming quantum theory is correct).
|
| The irrational numbers are then are the only numbers that
| are harder to pin down, and I'm not sure that there is a
| way to prove that any physical quantity has an irrational
| value, vs a rational value that is arbitrarily close to
| that irrational value.
|
| [0] https://arxiv.org/abs/2101.10873
| ithkuil wrote:
| Infinity may be also "necessary to describe the world".
| But like every tool, you need to know its limits.
| tsimionescu wrote:
| I'm not sure that it could be, actually. You can't use a
| finite amount of evidence to verify that something is
| infinite, so any infinity can always be replaced with a
| huge (or minuscule) number and the theory would make the
| same measurable predictions.
| ithkuil wrote:
| I'm not talking about proving that there exist infinite
| things.
|
| I'm talking about using the abstract concepts of infinity
| as a useful mathematical tool to produce predictions.
| Notable example: calculus
| tsimionescu wrote:
| Sure, but calculus makes infinity sufficient, but not
| necessary for describing the physical world. Integers,
| rationals, and apparently complex numbers (presumably
| those with rational components) are actually necessary
| for describing the physical world, given our current
| understanding. Irrational numbers and infinities are
| extremely useful, but not strictly necessary.
| canjobear wrote:
| I don't see how this makes infinity "unscientific." Infinity
| is part of the language of mathematics. It's no more
| scientific or unscientific than the definition of matrix
| multiplication.
|
| 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?"
| tsimionescu wrote:
| Well, I can say that there are precisely 0 african
| elephants in the room with me right now, so no, 0 and other
| integers don't have this problem. Similarly, the rationals
| are clearly realizable with perfect precision.
|
| The reals however are a different problem, and it's not
| scientifically possible to prove that the ratio between the
| length and radius of any object is exactly pi (that it is a
| perfect circle). However, it's also impossible to prove
| scientifically that it is 3 or 3.14 or any other number.
|
| Now my use of "unscientific" is more of a hyperbole or
| click-bait. I thought I explained my actual claim pretty
| well - that you can't measurably/scientifically distinguish
| between a universe that contains actual infinities and one
| that only contains some arbitrarily large numbers.
| snet0 wrote:
| That's just because you use the word "exact", though.
| Exactitude doesn't exist in the universe as we understand
| it.
|
| There's a difference between something not being
| instantiated in this universe and being unscientific,
| though.
|
| If we produce a model of the universe that doesn't make a
| single incorrect prediction given all data available, and
| it predicts infinities to exist in some strange but quite
| real cases, is it unscientific?
| tsimionescu wrote:
| > Exactitude doesn't exist in the universe as we
| understand it.
|
| Of course exactitude exists. For example, two electrons
| have exactly the same charge. A photon has exactly 0
| charge.
|
| > There's a difference between something not being
| instantiated in this universe and being unscientific,
| though.
|
| Well, science is a particular way of studying what
| exists. Studying something that doesn't exist is
| unscientific (of course, you can use science to try to
| determine IF something exists).
|
| But there are also things that are outside the reach of
| the methods of science, so they are unscientific in this
| sense. Questions such as "did some god create the
| universe" are unscientific because it is simply
| impossible to apply the methods of science to arrive at
| an answer to this question.
|
| Similarly, asking "is the universe infinite in size" is
| unscientific, because it is impossible to apply the
| methods of science and arrive at a definite answer to
| this question.
|
| > If we produce a model of the universe that doesn't make
| a single incorrect prediction given all data available,
| and it predicts infinities to exist in some strange but
| quite real cases, is it unscientific?
|
| If it predicts actual infinities exist in certain
| conditions, than it is not going to be a testable theory
| in those conditions. It may still be a perfectly workable
| model, just as GR is perfectly workable despite
| predicting singularities at the center of black holes.
| That doesn't mean that the singularities exist, it means
| that GR breaks down at certain points.
|
| But even if you had a physical theory that relied on
| something like a Banach-Tarski construction, you could
| never distinguish between an actual infinity of points,
| leading to two perfectly solid, perfectly identical
| spheres; and an arbitrarily large number of points,
| leading either to two perfectly solid but slightly
| different-sized spheres; or two identically-sized spheres
| with small holes.
|
| Of course, without some need to specify the number of
| points, you would be well positioned to use the infinite
| variant. But if someone asked you if this means that the
| sphere really has an infinite number of points, the
| answer would have to be that you can't be sure.
| canjobear wrote:
| > Of course exactitude exists. For example, two electrons
| have exactly the same charge. A photon has exactly 0
| charge.
|
| Aren't claims like this unscientific according to your
| standard? You will never be able to measure that two
| electrons have the same charge to infinite decimal
| precision. You might have a theory that says they should
| have the same charge, but you won't be able to test that
| theory to infinite precision either.
| tsimionescu wrote:
| Hmm, actually you may be right. I'm not entirely sure how
| powerful the 'requirement' in QM for these quantities to
| be quantized is though, but most likely you are right -
| the theory wouldn't be able to distinguish between
| identical charges and veeeeery slightly different
| charges.
| rytill wrote:
| As the other comment implied, infinity and exactitude are
| two sides of the same coin. Exactitude is infinite
| precision. No finite amount of empirical evidence can
| afford infinite precision, so you're back in math-land.
| tsimionescu wrote:
| That's true, though I have a hard time squaring that with
| observations of classical objects. Would it make sense to
| say that I have approximately 2 legs, or can I actually
| be confident in saying I have exactly two legs? Even with
| things like MWI, in any particular world I would still
| have an exact integer amount of legs, as far as I
| understand.
|
| Perhaps the problem here is one of mixing intuition (the
| idea of 'an object') with rigorous physics and
| mathematics, perhaps this is where I am going a bit
| wrong.
| snet0 wrote:
| We have a very precise (although one might struggle to
| describe it) idea of what qualifies as a human leg. The
| number 2 is basically defined (at least in common usage)
| as the number of things you have when you have one thing
| and then another thing. I'd point out that, as you
| mentioned, this is a different level of abstraction to
| physics and mathematics.
| tsimionescu wrote:
| I would also need to know more QM. I don't know if the
| theory would actually allow a small fraction of an
| electron or an electron and a bit to actually exist -
| common descriptions suggest that it wouldn't. If it
| doesn't, then electrons could be counted just as much as
| mathematical objects and legs.
| snet0 wrote:
| You were talking about exactitude in space, rather than
| charge.
|
| >Studying something that doesn't exist is unscientific
|
| What about things that _could_ exist, _might_ exist, or
| even _aren 't expressly forbidden_ from existing? These
| have all been used as perfectly valid reasons for
| scientific inquiry, historically.
|
| Asking "did some god create the universe" is unscientific
| by your reasoning so long as it is known that there is no
| in-universe trace or evidence that it was indeed created
| by a god. Proving that is proving a negative. I think it
| is not impossible for us to prove that the universe was
| created by a god, if we found some hidden message in
| subatomic particles or cosmic dust or something. It does
| certainly feel impossible that we will prove that the
| universe _wasn 't_ created by a god, though. The inquiry
| is deemed unscientific because we have no reason to go
| down that pathway, not because the question is
| fundamentally intractable.
|
| Multiverse theory, on the other hand, would qualify as
| unscientific by your reasoning. If it were true, the
| different universes would be fundamentally inaccessible,
| according to our understanding. The model does not
| suggest that evidence could even possibly exist, as far
| as I understand.
|
| A result being untestable doesn't, in my opinion, lead to
| it being unscientific. We cannot test whether black holes
| exist, except by looking for them. We cannot test whether
| wormholes exist, except by looking for them. These are
| predictions that we cannot "test" except by looking at
| the universe and seeing what we find, and even then we
| are not guaranteed a positive result, just because maybe
| it is the case that our model is correct but there was
| never the appropriate state of the universe to prove our
| prediction.
|
| Of course if something was actually infinite, you
| wouldn't be able to measure it to be so, but if the model
| (that you have shown to be correct in other case)
| predicts an actual infinity and you keep counting more
| and more orders of magnitude, does it not make sense to
| assume your model is correct? Is that unscientific? Just
| like we assume that the charge on electrons is constant
| despite not actually measuring it always everywhere.
| tsimionescu wrote:
| I was talking about exactitude in general, as a
| requirement for a physical interpretation of the
| integers.
|
| > I think it is not impossible for us to prove that the
| universe was created by a god, if we found some hidden
| message in subatomic particles or cosmic dust or
| something.
|
| That's actually a good point, there could be scientific
| proof of some intelligent creator in principle. The fact
| that there is no reason a priori to believe that we will
| find such a proof is a problem, but I don't think it
| would be enough to deem the theory unscientific.
| Otherwise, many actually used theories would be
| unscientific - for example, there is no scientific reason
| to expect supersimmetry to exist, but that doesn't make
| the search for supersimmetry unscientific.
|
| > Multiverse theory, on the other hand, would qualify as
| unscientific by your reasoning.
|
| Yes, multiverse theory is unscientific by my definition.
| I don't believe speculation about a multiverse can be
| considered science in any meaningful sense. Just like
| simulation theory, it is using science-sounding
| terminology for idle speculation (though the universe
| being a simulation could similarly be proven by the same
| kind of evidence as the intelligent creator idea, to be
| fair).
|
| > These are predictions that we cannot "test" except by
| looking at the universe and seeing what we find, and even
| then we are not guaranteed a positive result
|
| But this is exactly the definition of a test. It's true
| that you can't prove that something doesn't exist in this
| way, but saying that something is untestable goes beyond
| that. An untestable hypothesis is one that by definition
| doesn't make any predictions about the universe.
| Multiverse theory is in this bucket - whether you believe
| it to be true or not, you won't expect to see anything
| different in the world.
|
| > Of course if something was actually infinite, you
| wouldn't be able to measure it to be so, but if the model
| (that you have shown to be correct in other case)
| predicts an actual infinity and you keep counting more
| and more orders of magnitude, does it not make sense to
| assume your model is correct?
|
| Of course it's OK to assume your model is correct, and
| infinity will likely be the simplest assumption in this
| case. However, any model that predicts an infinity can be
| replaced with an equivalent model that makes all the same
| measurable predictions but replaces the infinity with
| some arbitrarily large but finite number (or arbitrarily
| small but not infinitesimal). This second model may well
| be harder to work with and will contain an extra
| assumption (an explicit upper bound for the infinite
| quantity), so I wouldn't advocate for its use. But it
| would have to be accepted that it is not empirically
| distinguishable from the infinity based model.
| snet0 wrote:
| Exactitude in general is impossible, though. We assume
| that the charge on an electron is a constant, but there
| are limits to the precision.
|
| An infinite value is theoretically testable. It simply
| implies that for however long you make your ruler, the
| value is larger. That is a prediction. You may not reach
| a conclusion, as you said, in finite time, but that is
| still a prediction.
|
| The problem with replacing an infinity in a model with an
| arbitrarily large number is that, given enough time and a
| long enough ruler, you'll surpass that number, meaning
| your model is incorrect. In defence of "science", you're
| adding an arbitrary number into a model that you expect
| to be incorrect. That's not how it should work.
|
| If the model says there's a singularity, we don't then
| say "okay but well that clearly doesn't make sense, so
| put a limit on the formula that clamps the values to uh
| 10^45". _That_ is unscientific.
| tsimionescu wrote:
| > Exactitude in general is impossible, though. We assume
| that the charge on an electron is a constant, but there
| are limits to the precision.
|
| Yes, others have pointed that out and I am in fact
| conflicted right now.
|
| > The problem with replacing an infinity in a model with
| an arbitrarily large number is that, given enough time
| and a long enough ruler, you'll surpass that number,
| meaning your model is incorrect. In defence of "science",
| you're adding an arbitrary number into a model that you
| expect to be incorrect. That's not how it should work.
|
| I'm not against using infinities in scientific practice
| at all. I'm just pointing out that, when it comes down to
| it, that infinity is never necessary in the logical
| sense.
|
| > If the model says there's a singularity, we don't then
| say "okay but well that clearly doesn't make sense, so
| put a limit on the formula that clamps the values to uh
| 10^45". That is unscientific.
|
| Sure, picking some random big number would be
| unscientific. But saying "the model predicts a
| singularity or growing to infinity, so we're probably
| missing some piece of the picture that sets an upper bar"
| is not unscientific. It is in fact the common practice -
| just like no one believes that black holes or the early
| universe had an actual singularity at the center, we
| normally just believe the models break somewhere at those
| levels, and more powerful models (quantum gravity) will
| actually put a cap. Or how we keep saying that we know an
| upper bound for the possible mass of a photon, but don't
| actually know that it really is 0, and we keep trying to
| measure it.
| snet0 wrote:
| I think we're not disagreeing, if we were before. Adding
| limits to functions without rationale other than avoiding
| infinities is arbitrary and unscientific. Given that we
| assume that the universe doesn't _actually_ have the
| capacity for infinities (is this a scientifically
| grounded assumption, given the impossibility in measuring
| them?), finding them points us towards holes or
| limitations in the model, e.g. GR vs quantum gravity.
| turminal wrote:
| > there is no way to scientifically differentiate between an
| infinite quantity and a really huge quantity (same for
| infinitesimals), in finite time
|
| That depends on the model of computation you pick, doesn't
| it?
| tsimionescu wrote:
| Only if you want to think about models of computation that
| allow performing infinite operations in finite amounts of
| time, which I don't think are that interesting.
| auggierose wrote:
| I'd say if our universe makes such computations possible,
| then that would be very interesting.
| tsimionescu wrote:
| Absolutely! Similarly, if our universe allowed
| instantaneous travel and free energy, that would also be
| very interesting.
|
| Not holding my breath for either.
| jhgb wrote:
| > infinity is something only present in our math and not in the
| universe
|
| This is true of all mathematical objects. The number 7 doesn't
| exist in the universe either. It's not a physical object.
| snet0 wrote:
| Tell that to Plato.
|
| 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.
| danparsonson wrote:
| OP didn't argue that finite numbers are physical objects,
| they said that infinities are not _present_ in the universe.
| For example, I could in theory hand you 7 electrons but there
| are not infinity electrons for me to hand to you.
| jhgb wrote:
| That sounds like a weird interpretation of "to be present
| in the universe" to me. Also I was under the impression
| that it's unknown whether the universe contains an infinite
| number of electrons or not.
| danparsonson wrote:
| Weirder than 'numbers are physical objects'? Can you
| suggest a more suitable interpretation of the OPs intent?
| PeterisP wrote:
| It's certainly known that the _observable_ universe does
| not contain an infinite number of electrons, as it has a
| finite size and finite mass. And it 's rather moot to
| talk about the space beyond the observable universe that
| can never affect us or anything we can observe in any way
| whatsoever, so any other statements about it are
| inherently unfalsifiable, so all the science of physics
| is relevant only w.r.t. the (finite) observable universe.
| [deleted]
| layer8 wrote:
| But the electron field has different values at different
| points in spacetime, and we have no evidence that either
| the number of points (locations) or the number of different
| possible values at those points is finite. Unless we posit
| that they are finite in number, infinity is quite present
| in the universe.
| danparsonson wrote:
| Would not an infinite electron field in a finite
| (observable) universe result in an infinite energy
| density and therefore the entire universe would collapse
| into a black hole?
| layer8 wrote:
| Integrals over a finite interval can have (and often do
| have) a finite size even though the interval contains an
| infinite number of points, with an infinite number of
| different values at those point.
| danparsonson wrote:
| Right, because the integral of a function is not a
| straight sum of values of that function evaluated for
| every number in the interval; the integral of y=x dx for
| 0<=x<=1 is not 0+0.1+0.11+0.111+0.1111+...+1. Electrons
| have a fixed energy, so cramming an infinite number of
| them into a finite space necessarily requires infinite
| energy.
| hprotagonist wrote:
| it's OK. electrons don't "exist" discretely, either.
|
| At best, when you "hand me 7 electrons", you're directing
| me towards the fat part of 7 probability distributions, so
| we're back to math again...
| danparsonson wrote:
| Well that's a bold assertion about a theory with whose
| implications we are still grappling; electrons may not be
| point-like entities but they are nonetheless quantifiable
| 'packets' of energy, are they not?
| [deleted]
| cupcake-unicorn wrote:
| Great video on this: https://www.youtube.com/watch?v=s86-Z-CbaHA
| [deleted]
| inetsee wrote:
| Can someone explain the flaw in my reasoning here?
|
| 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?
| hodgesrm wrote:
| Your sets are finite. B-T depends on properties of infinite
| sets.
| Tomte wrote:
| > I divide the sphere into individual iron atoms.
|
| 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.
| Ericson2314 wrote:
| I highly recommend
| https://twitter.com/andrejbauer/status/1428471658088738818 and
| follow ups.
|
| 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.
| boxfire wrote:
| > But who's going to work without AC (other than crazy HoTT
| people)?
|
| 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)
| [deleted]
| creata wrote:
| You're exaggerating the significance of HoTT. Mathematics is
| already on a pretty firm foundation. And I somehow doubt
| Bauer meant any ill intent with that line...[0][1]
|
| [0]: https://en.wikipedia.org/wiki/Homotopy_type_theory#Speci
| al_Y...
|
| [1]: http://math.andrej.com/2016/10/10/five-stages-of-
| accepting-c...
| mjw1007 wrote:
| For the avoidance of doubt: Andrej Bauer is himself very much
| one of those crazy HoTT people.
| Ericson2314 wrote:
| Andreij Bauer is one of those HoTT people, so this is quite
| tongue-in-cheek.
| mjw1007 wrote:
| I remember sitting in maths lectures and wishing that when they
| did thing like prove the intermediate value theorem they'd make
| it clearer that what was going on wasn't so much "We're
| rigorously proving that this thing that seems obvious is true"
| as "We're checking that the formalisation we introduced earlier
| is fit for purpose".
|
| 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.
| thaumasiotes wrote:
| > I remember sitting in maths lectures and wishing that when
| they did thing like prove the intermediate value theorem
| they'd make it clearer that what was going on wasn't so much
| "We're rigorously proving that this thing that seems obvious
| is true" as "We're checking that the formalisation we
| introduced earlier is fit for purpose".
|
| > 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.
| Ericson2314 wrote:
| Banach-Tarski is quite arguably a red flag that all these
| non-measursble, non-open sets are barking up the wrong
| tree.
| throwaway81523 wrote:
| The Banach-Tarski theorem motivated the idea of amenable
| groups in topological group theory. Understanding exactly
| what that means is on my todo list, but I think the basic
| idea is that a given space can have additive measures
| invariant under some transformation groups but not others.
| Particularly, the Banach-Tarski paradox shows that regular
| old 3-dimensional Euclidean space doesn't have an additive
| measure invariant under rotation and translation. On the
| other hand, 2-dimensional Euclidean space does have it.
| mjw1007 wrote:
| What I mean is: if you imagine someone drawing up a
| requirements document for the team assigned to the task of
| axiomatising geometry, and somebody asked "Do we want our
| model of geometry to support cutting up a ball into five
| pieces, moving the pieces rigidly, and reassembling them
| into two copies?", I think their first idea would be to
| answer "no".
|
| So it isn't parallel to the intermediate value theorem, but
| opposite to it.
| thaumasiotes wrote:
| The whole idea of a proof system is that there are some
| things you can't have without also having other things.
| The Banach-Tarski theorem is a consequence of things we
| want. You don't get to pick and choose everything at
| once.
| ncallaway wrote:
| > The Banach-Tarski theorem is a consequence of things we
| want
|
| Is it? I think the parent comment is saying: "maybe we
| shouldn't want things that result in Banach-Tarski"
|
| Maybe it's a hint that the underlying axioms we've
| selected _aren't_ exactly what we want.
|
| You're right that we can't pick and choose the results of
| our axioms, but we do explicitly get to pick and choose
| the axioms we start with. If we choose bad axioms, we get
| nonsensical results.
|
| In general, it seems like we've picked _pretty good_
| axioms that mostly give us sensible and useful results.
| But maybe this result that seems somewhat... odd, is an
| indication that those axioms have an odd corner
| somewhere.
| thaumasiotes wrote:
| > But maybe this result that seems somewhat... odd, is an
| indication that those axioms have an odd corner
| somewhere.
|
| The only way you're going to avoid getting results like
| this is with axioms like "there is no such thing as an
| infinite number". At that point, the real line doesn't
| exist (too many points) and it becomes impossible to
| duplicate spheres by dividing them at a level of fineness
| that also doesn't exist.
|
| But that's not a productive approach to anything.
| mjw1007 wrote:
| I was taught that dropping the axiom of choice was enough
| to make Banach-Tarski go away. That seems considerably
| short of "there is no such thing as an infinite number".
|
| But the Twitter link at the top of this thread seems to
| have a rather more interesting way of doing so.
| thaumasiotes wrote:
| > I was taught that dropping the axiom of choice was
| enough to make Banach-Tarski go away. That seems
| considerably short of "there is no such thing as an
| infinite number".
|
| The Banach-Tarski theorem is not the only theorem out
| there that bothers some people. Anything to do with
| infinities gets a large number of outraged rejections.
| [deleted]
| Ericson2314 wrote:
| Did you read the original link? We don't want to use
| topological spaces, we want to use locales! We can get
| rid of the paradox while sacrificing very little.
| pfortuny wrote:
| The thingis exactly that in the B-T paradox you are NOT
| "cutting", you are "choosing" non-measurable subsets and
| reorganizing them. Thus, the operation of "cutting" is
| not taking place (there is no continuous function whose
| zeros gives you any of the subsets).
| Ericson2314 wrote:
| Ultimately this crowd wants to change the practice of
| mathematics in the real world, so they are very
| accomiadating.
|
| 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.
| bigbillheck wrote:
| > Ultimately this crowd wants to change the practice of
| mathematics in the real world, so they are very
| accomiadating
|
| PhD mathematician in industry here. The way I see it,
| foundations is to the rest of mathematics the way music
| theory is to music: it needs to be a describer, not a
| prescriber. (If I were less charitable I'd have said
| "ornithology is to birds").
|
| > 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
|
| On the contrary, ZFC has been a tremendous success in that
| most mathematicians don't need to worry about it at all.
| throwaway81523 wrote:
| Imho, software people should study foundations
| (particularly proof theory) much more than they do. That
| is how you ensure correct code, after all. The people
| deeply involved in software verification are basically
| logicians. Also, the test suite for the HOL Light proof
| assistant (used in software verification) uses a large
| cardinal axiom, sort of. It uses a version of itself with
| the large cardinal added, to prove the consistency of the
| normal version without the large cardinal. Neither
| version can prove itself consistent, because of Godel's
| theorem, but the one with the large cardinal can prove
| the consistency of the one without it.
|
| One can say that if either is inconsistent then they can
| prove everything, but that makes it even sharper: the
| large cardinal is used purely to give engineering
| assurance of software correctness and not real
| mathematical rigor. So it's a pure engineering use of one
| of the most "out there" mathematical objects. It doesn't
| seem worse than using IEEE floating point arithmetic to
| design airplanes....
| ncmncm wrote:
| I was initially appalled by Banach-Tarski.
|
| 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.
| perl4ever wrote:
| Until recently I never questioned the idea that, say, the
| positive integers and the odd positive integers are equivalent
| because they can be paired, but this cloning thing seems like
| something that falls out of that. And it seems like that view of
| infinity isn't actually necessary if Cantor style cardinality is
| not the last word.
|
| 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?
| mineOther wrote:
| Infinity is the axiom of paradox. Does the inclusion Infinity
| complete an otherwise incomplete set of axioms? It solves the
| halting problem for a finite Turing Machine.
|
| 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.
| Smaug123 wrote:
| From your description, I fear you don't understand the usual
| diagonalisation proof that constructs an uncounted real from
| any attempt to count the reals. Why should "the longest"
| diagonal have anything to do with it?
| ishtanbul wrote:
| Vsauce made a great video about this https://youtu.be/s86-Z-CbaHA
| karmakaze wrote:
| I recall watching the video and not being surprised by its
| paradox. The set of starting points is uncountably infinite
| (R2), and since each starting point leads to a countably
| infinite number of L/R/U/D-rotation-ending sets, each of those
| L/R/U/D sets has the same cardinality as that for starting
| points. And so on. In the end, what I took away from this was
| similar to saying the interval [0.0, 0.5] has the same
| cardinality as [0.0, 1.0] albeit in a higher number of
| dimensions. It would be surprising if an uncountably infinite
| set in a lower dimension could fill in a higher one, but
| uncountably infinities in the same number of dimensions doesn't
| seem like a paradox that needs this sphere, rotation, and
| dictionaries to demonstrate.
|
| 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.
| golemotron wrote:
| So much becomes easier to see when you see infinity not as a
| thing but as an algorithm.
| carnitine wrote:
| Algorithms are not things?
| golemotron wrote:
| You just restated the Banach-Tarski Paradox.
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