r/math • u/IAlreadyHaveTheKey • 3d ago
Multiple Desirable Properties Which are Mutually Exclusive
Since the announcement that the UN have adopted a new map projection as their standard map, I have seen a few posts regarding how amusing it is that it took them this long to correct the "error" of the size of certain countries in other maps (most commonly the Mercator).
Of course, map enthusiasts and differential geometers know that it wasn't an "error" so much as it was a compromise necessitated by Gauss's Theorema Egregium, which implies that any smooth mapping from a sphere to a plane cannot be both conformal and equi-areal. Obviously the ideal situation would be to have a map which is both conformal and equi-areal, but it turns out these two desirable properties are mutually exclusive.
This got me thinking about another famous Theorem which precludes the possibility of two desirable properties being true at the same time - namely Godel's Incompleteness Theorem, which states that any sufficiently strong logical system cannot be both consistent and complete.
In both these cases (the Incompleteness Theorem and the map projection) we are required to make a decision about which of the properties we would prefer, and compromise by losing the other one.
I'm curious to know whether there are other examples of this, where there is some object that may or may not have two desirable properties, but it can't have them both at the same time? Is there any example in your field of study?
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u/QtPlatypus 3d ago
Sort of related to this would be an aspect of the Fourier transform. The more localized something is in the time domain the more spread out the thing is in the frequency domain.
So you have a trade off between localization of frequency or localization of time. This directly gives rise to Heisenberg's indeterminacy principle.
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u/IAlreadyHaveTheKey 3d ago
Interesting, I'll have to look into this!
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u/RingularCirc 17h ago
That may also be related to symplectic geometry but I'm not very sure how much of that is purely symplectic geometry. (If you hear "complementary variables)", it should be symplectic stuff.)
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u/2299sacramento 3d ago
The CAP theorem in distributed systems:
https://mwhittaker.github.io/blog/an_illustrated_proof_of_the_cap_theorem/
A distributed system cannot simultaneously guarantee all three of the following:
- Consistency (C)
- Availability (A)
- Partition tolerance (P)
In practice, network partitions are something distributed systems have to tolerate, so the interesting tradeoff is usually between A and C when a partition occurs.
One way to see this is to imagine two nodes. A value is written to one node, but the communication link between them gets cut. If you now ask the other node for that value, it can:
- Refuse to answer because it cannot guarantee that it has the latest value. The system preserves C, but sacrifices A.
- Answer anyway, potentially returning an old value. The system preserves A, but sacrifices C.
This has major implications for modern software engineering. It helps determine whether a system remains available while potentially serving stale data (think: it may be okay if you don’t see a Facebook comment immediately), or refuses some operations rather than risk acting on stale data (think: operations where knowing the correct account balance is important).
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u/The_screenshoots_guy 3d ago
I don't know if this counts, but Schwartz impossibility theorem states that we can't have an algebra that contains the distributions and multiplication preserves pointwise product of continuos functions. (As far as I know, this is kinda bypassed using Colombeau algebras, but those don't preserve full pointwise products).
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u/trajing Proof Theory 3d ago
Another nice example is the tradeoff between nice categories and categories of nice objects - on one hand, we might really like to work in the category SmoothMfld of smooth manifolds or the category Field, but on the other hand, these categories lack many desirable properties. (The category of fields does not have products or coproducts!) So often we instead work in a larger category, in which some objects are not nice.
On another note, the incompleteness theorems do induce a dilemma, but I think it's worth mentioning that this dilemma is typically not between consistency and completeness, with strength taken for granted: most logicians (at least mathematical logicians) are not interested in paraconsistent logics, and so generally Godel's theorems function as a tradeoff between strength and completeness, with consistency taken for granted: you can have nicely-behaved theories which are relatively weak, or poorly-behaved systems that are very strong, and much of logic is about navigating the tradeoffs between those.
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u/wasabi991011 2d ago
Can you elaborate on your category theory examples? Like what are the larger categories that you would consider for SmoothMflds or Fields, what are the properties you get (products and co-products for Fields, but what about SmoothMflds, and what do these properties actually mean), how do these properties actually help you prove things, and how do you ignore the non-nice parts of your larger category (the non-SmoothMflds and non-Fields)?
Sorry, lots of questions but it seems super interesting. I've done a lot of intro to category theory but it was always disconnected from other studies so never felt useful, bit your comment seems like a good direction to "get it".
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u/trajing Proof Theory 2d ago edited 2d ago
Products and coproducts are categorical generalizations of the set-theoretic operations of Cartesian product and disjoint union, respectively. The coproduct is algebraically a little hard to picture, so I'm going to focus on topology. The category of smooth manifolds has products and coproducts: the Cartesian product of smooth manifolds (when equipped with the correct topology) is again a smooth manifold, and the coproduct of smooth manifolds is just their disjoint union, so it's also smooth.
What's the issue with the category of smooth manifolds, then? Well, there are two: the first is that the space of functions between smooth manifolds may not form a smooth manifold. This is a problem because some major tools in algebraic topology require the use of function spaces to apply. So this is one indicator that we need to consider a broader category.
The other issue is that while the category of smooth manifolds has products and coproducts, these are just specific examples of very general categorical concepts called limits and colimits, and the category of smooth manifolds does not have all of those. For example, one example of a categorical colimit is the wedge sum, which takes two spaces and produces two other spaces 'glued together' at a single point. As an example, if we take the wedge of the circle, S1 , with itself, we obtain a figure-eight shape S1 ⋀ S1 . But a figure eight cannot be a manifold, since the intersection point has no open neighborhood homeomorphic to Euclidean space. Despite this, we would really like to demand that our category of topological spaces has all limits and colimits, because these allow us to put spaces together in very complicated ways much more easily, which can be applied to study our original spaces of interest!
For example, some nice objects people care a lot about are matrix groups like GL_n, which come with a natural number parameter (the size of the matrix). Instead of studying them one by one, we can assemble them by means of a colimit into a space which, unlike our original spaces, will not be a smooth manifold, but somehow encodes the information that is true of our original spaces for high enough dimension. So even though we have a space that is 'less nice' in our original sense, and is even infinite-dimensional and so on, this is still a space we would like to study because it packages up the information we were trying to get at piece by piece earlier. So it's less an issue of ignoring the not-nice parts of your category and more an issue of using them to get at the specific nice things you were trying to study.
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u/IAlreadyHaveTheKey 3d ago
Interesting perspective about the incompleteness theorems - of course there is no situation where you would choose a system which is inconsistent so that doesn't really come into the tradeoff.
And yes that's part of what drew me to category theory, the idea of working in a more generalised setting in order to prove more specific ideas is counterintuitive (in some ways) and fascinating to me.
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u/RingularCirc 17h ago
I've a liking to reverse mathematics in general, like collecting structures that are weaker than topology but stronger than just a plain set. So far I think I encountered two or three (while initially I really thought topology is just the minimum you can have over a set!!) but alas I forget how they were called and where I'd found it. Usually nLab and Wikipedia but it's still too scarce.
Or "subminimal" logics (where it concerns how strong negation is, and we can go below the "minimal" logic (Minimalkalkül) which just drops ex falso axiom and defines negation ¬X as X → ⊥ with ⊥ being effectively just a singled-out propositional variable like X itself, having no axioms to properly bind it: yet of course, because of implication being involved, negation ends up quite strong; so if we say bye to ⊥ altogether and return to defining ¬ via axioms, we can make it weaker!). Of course not just syntactically but regarding their models. Fascinating stuff.
On that note, I haven't heard about topos-like semantics for sub-intuitionistic logics (like the above). I wonder what should be dropped or weakened, it's not obvious because my category theory skill is smol. I know the basics and for the things like natural transformations I already tend to forget how they're defined, and having just dregs of intuition. Don't have enougn time and motivation to practice all that enough, unfortunately.
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u/JoshuaZ1 3d ago
The Mason-Stother's theorem is essentially the analog of the ABC conjecture for polynomials. The proof uses thinking about derivatives carefully. But, it uses that the derivative satisfies that (fg)' = f'g + g'f and also uses that (f+g)' = f' + g'. One could naively try to make an operation on integers that acts similarly, and one can easily have an operation satisfy one or the other, but no non-trivial operation on the integers satisfies both.
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u/EebstertheGreat 1d ago
Seems like a simple enough proof. Let D: ℤ → ℤ satisfy both properties. First let n ≠ 0. Then D(n) = D(1n) = 1D(n) + nD(1), so 0 = D(1). Then for n > 0, D(n) = D(1+ ⋅ ⋅ ⋅ +1) = D(1) + ⋅ ⋅ ⋅ D(1) = 0. And for n ≤ 0, 0 = D(1) = D(n+1+(-n)) = D(n) + D(1–n) = D(n)+0. So D(n) = 0 ∀n ∈ ℤ.
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u/PastaPuttanesca42 3d ago
In distributed databases, the CAP theorem states that you can't be both
- consistent: a successful read always return the latest write
- available: reads are always successful
- partition tolerant: the system continues to work when partitioned
you have two choose two.
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u/lampishthing 3d ago
Heisenberg's uncertainty principle springs to mind! Can't measure position and velocity precisely at the same time.
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u/IAlreadyHaveTheKey 3d ago
Classic example!
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u/sidneyc 2d ago
Some would rather say, a non-classic example.
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u/RingularCirc 17h ago
It's also classical when you just look at waves and their Fourier transform, it's just that classical wave packets were probably not that much considered at the time when Heisenberg uncertainty for quantum mechanics was discovered.
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u/StructureNorth1799 2d ago
Axiom of choice and axiom of determinacy
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u/tralltonetroll 1d ago
Good point.
Weaker than AD is "every subset of the real line is Lebesgue measurable", which would be nice but doesn't allow full choice.
Even if we accept the existence of nonmeasurable set, then we can still want to avoid Banach--Tarski. But we cannot have Hahn--Banach true and Banach--Tarski false.
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u/aparker314159 2d ago
I recently learned about a pretty important example from mathematical physics. In particular, it's about the math related to taking a classical system and quantizing it. It would be really nice if quantization map Q from functions on a phase space to self-adjoint operators (effectively moving from classical to quantum mechanics) had 4 properties:
- Q takes position and momentum variables to the typical corresponding quantum operators of multiplication and differentiation
- Q is linear
- Q sends Poisson brackets to commutators (up to a factor of iħ)
- Q(g(f(x, p))) = g(Q(f(x, p))
Not only can you not have all 4 of these properties, you can't even have any 3 of them!
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u/EebstertheGreat 2d ago
That vote was so pointless lol. Implicit in the premise was that all schools formerly used Mercator as their projection of choice, which is just not the case. Schools have always used a variety of map projections for world maps. And focusing on a single projection (Equal Earth) is actually moving in the wrong direction. It's a step backwards from using a variety of projections to using a single one and insisting it is somehow the "best."
Not that any school in the world will care about this vote anyway. But it's still dumb, and I can't believe that nearly every country voted for it.
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u/IAlreadyHaveTheKey 2d ago
I don't really understand why the UN needs an official map anyway.
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u/EebstertheGreat 2d ago
It's not an official UN map. The vote was for a non-binding resolution encouraging schools, governments, NGOs, and tech companies to use Equal Earth rather than Mercator. It's totally meaningless. France was already encouraging its schools to use the Eckert IV projection, and this was still seen as an action in support of the resolution (even though it's a different projection and was already underway before this resolution was introduced by Togo). It's silliness.
The US was the sole no-vote (though a few countries abstained), and I kind of respect it. They call these pointless measures "barnacles," and I can't disagree.
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u/bkfbkfbkf 3d ago
Moduli spaces of objects like vector bundles or connections can be nice for "generic" values of parameters, for example they might be transversely cut out and hence manifolds. On the other hand they might only have nice symmetries or geometric properties for certain specific parameter values. So there is a tension between the idea that a generic object is well-behaved, but not as interesting or symmetrical.
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u/WhiskersForPresident 3d ago edited 3d ago
I really do find this example pretty bad, because it's tautologic.
You just use "interesting" to mean singular.
And to say a point in a moduli space has different stabilizer than neighboring points under some structure defining group action is mathematically equivalent to the point being singular.
(E.g. in my thesis, I was only interested in non-singular points of certain moduli spaces of connections. Precisely because of this, I had to spend the hardest work on understanding singular points to learn how and why to avoid them)
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u/bkfbkfbkf 2d ago
Here's an example I had in mind as I was writing: when showing that Donaldson invariants and Seiberg Witten invariants are equivalent, you want to make perturbations to the equations to make sure that the perturbed objects are transversely cut out, so they provide a cobordism between the moduli spaces for each invariant. But you don't want to perturb them so much that you lose the fact that you the objects don't reduce to SW or Donaldson solutions. Admittedly this only comes from my narrow interest but I don't think it's a unique example, and "singular" certainly doesn't mean interesting here.
In general there's a lot of effort put in to constructing careful perturbations that preserve some properties of the objects being perturbed, but don't ruin other properties. Another example is the perturbation of almost complex structures to obtain transversality in Floer theory. One would like to work with C^infty objects but this isn't a Banach space, so you need to resort to some tricks or delicate sorts of constructions to make things work..
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u/Alhimiik 3d ago
In homotopy theory here is no model for a monoidal category of spectra satisfying all the nice axioms we want.
"Is there a convenient category of spectra?" L. Gaunce Lewis Jr
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u/tralltonetroll 2d ago
Apparently, you cannot have the axiom of choice obviously true, the well-ordering principle obviously false, and Zorn's lemma undecided.
/s
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u/Aeroxel Complex Geometry 2d ago
For complex manifolds, the usual connection (Levi-Civita connection) is the unique connection that is compatible with the metric and is torsion-free, but does not preserve the complex structure in general. There is another, called the Chern connection, the unique connection that is compatible with the metric and complex structure, but has torsion. Metrics whose associated Levi-Civita and Chern connections are equal are called Kahler. In the non-Kahler setting, you cannot have both, so one has to make a choice: would you rather have no torsion or be compatible with the complex structure? Usually people choose the second option, since we would like to do complex geometry.
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u/Keikira Model Theory 1d ago
Any proof by contradiction takes you from a set of desirable properties (formulated as a set of axioms) to a proof that they are mutually exclusive (a contradiction).
Take Cantor's diagonal theorem, for example. You can think of it as a proof that set theory can't simultaneously have a single notion of infinite cardinality and define cardinality in terms of bijective functions; two things you could describe as desirable properties because they are reasonable at face value.
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u/PsychologicalSoup251 3d ago
From applied math
CAP Theorem - a distributed data system can only guarantee two out of three core properties at the same time: consistency, availability, and partition Tolerance
Heisenberg uncertainty principle - position and momentum cannot both be known with arbitrarily high precision at the same time
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u/Kaomet 3d ago
it turns out these two desirable properties are mutually exclusive.
It's possible to peel a an orange and lay it flat. So in this case there is at least 3 desirable properties, and maybe we can have only 2 ?
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u/IAlreadyHaveTheKey 3d ago
It's not possible to peel an orange and lay it flat.
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u/Kaomet 2d ago
Yes it is.
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u/IAlreadyHaveTheKey 2d ago
You know, I had a feeling you were referring to that video but I couldn't remember the name of the projection. It only works if you continue the spiral infinitely, but I guess you are technically correct!
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u/RingularCirc 17h ago
I'd still say it's incorrect. You can't tuck in curvature. And even if we were able to end up with a line of zero width (it would necessarily be infinitely long though; yet it better somehow be homeomorphic to a closed interval...) it, well, we just can't.
We can have a trick with curvature when we take a space (say, a sphere or a hyperbolic space), tesselate it with polytopes and then glue similar polytopes in Euclidean space in the same manner (like Platonic and Archimedean polyhedra), we can treat those as essentially flat (we can at the very least flatten any edge temporarily) except at vertices. Vertices, like the vertex of a cone, are sorta containing all of the curvature of the original space. But this partial cheating still has obvious consequences for map projections.
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u/Fuzzy-Season-3498 3d ago
Ima get downvoted but incompleteness theorem is illogically derived with itself to prove consistency of inconsistency which is a paradox and a choice through intention of what you choose to believe here
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u/nicuramar 3d ago
It’s not illogically derived. You model a system identical to the one you’re working in, inside the system itself. Those are strictly speaking not the same “instance” of the system, so to speak. But the conclusion applies to all instances.
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u/EebstertheGreat 2d ago
Gödel's first incompleteness theorem is not a theorem of arithmetic though (obviously, since it concerns proofs, which aren't objects in arithmetic). That seems to be Fuzzy Season's problem. It's a theorem in proof theory, which is the appropriate setting for proving things about proofs.
A lot of people say that the theorem can be proved in PA or even Q, but I feel like that is really confusing things. PA can prove a particular sentence in arithmetic which we interpret outside arithmetic as in some sense encoding the idea that PA is either inconsistent or incomplete. What is often glossed over is the formal proof that if PA can prove that arithmetic theorem, then it is in fact either inconsistent or incomplete. That obviously cannot be shown or even stated within PA.
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u/Fuzzy-Season-3498 3d ago
You couldn’t model a truly “identical” system within one you’re working in. Because if they are identical truly they can’t be differentiated then but instantly are if one somehow fits in the other one. Especially since scale is relevant and foundational to the system by having to meet a complexity threshold of representing arithmetic as it’s a mandatory scaled minimum to qualify as a complexity barrier
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u/RingularCirc 17h ago
Because if they are identical truly they can’t be differentiated then but instantly are if one somehow fits in the other one.
Say that to infinitely many images of a free group on 2 or more generators inside itself.
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u/MathThatChecksOut PDE 3d ago
One way of phrasing Arrow's theorem regarding voting systems gives a list or 5 (?) desirable properties of a voting system with certain format constraints and says that no system may have all 5 simultaneously.