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June 28, 2010

Inevitability in Mathematics

Posted by David Corfield

I’m just back from a conference in Nancy – From Practice to Results in Logic and Mathematics. As you see from the abstract, I was speaking about the inevitability of certain things appearing in mathematics: entities, facts, theories, ideas. We had a discussion of the related robustness a while ago.

Someone posed me the question of the inevitability of Lie groups. Forgetting worries about anachronism, what is our very best account of why the Lie group construct was going to be reached by a mathematics as sophisticated as ours? Can we do better than

Groups are interesting as models of symmetry, manifolds are interesting as models of smooth space, therefore group objects in the category of manifolds will be interesting as models of smoothly varying symmetry?

The other side of the coin is to think of constructs which are arbitrary and contingent. If someone had told me about hypergroups and hyperrings, where instead of a binary operation sending a pair of elements to a single element, it takes them to a nonempty set of elements, I would have imagined that they would have been good candidates for arbitrariness. When, however, you find that Alain Connes is using them in search of an absolute arithmetic, you naturally take notice:

We show that the trace formula interpretation of the explicit formulas expresses the counting function N(q)N(q) of the hypothetical curve CC associated to the Riemann zeta function, as an intersection number involving the scaling action on the adele class space. Then, we discuss the algebraic structure of the adele class space both as a monoid and as a hyperring. We construct an extension R convexR^{convex} of the hyperfield SS of signs, which is the hyperfield analogue of the semifield R + maxR_+^{max} of tropical geometry, admitting a one parameter group of automorphisms fixing SS. Finally, we develop function theory over Spec(S)Spec(S) and we show how to recover the field of real numbers from a purely algebraic construction, as the function theory over Spec(S)Spec(S).

I wonder whether Durov’s generalized rings are related.

I see the Connes and Consani paper followed on from The hyperring of adèle classes:

We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space 𝕂=𝔸 𝕂/𝕂 ×\mathbb{H}_{\mathbb{K}} = \mathbb{A}_{\mathbb{K}} / \mathbb{K}^{\times} of a global field 𝕂\mathbb{K}. After promoting 𝔽 1\mathbb{F}_1 to a hyperfield K, we prove that a hyperring of the form R/GR/G (where RR is a ring and GR ×G \subset R^{\times} is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G{0}G \union \{0\} is a subfield of RR. This result applies to the adèle class space which thus inherits the structure of a hyperring extension 𝕂\mathbb{H}_{\mathbb{K}} of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field 𝕂\mathbb{K} of positive characteristic, the groupoid of the prime elements of the hyperring 𝕂\mathbb{H}_{\mathbb{K}} is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field 𝕂\mathbb{K}.

Hyperstructures are at least 40 years old:

D. Stratigopoulos, Hyperanneaux non commutatifs: Hyperanneaux, hypercorps, hypermodules, hyperespaces vectoriels et leurs propriétés élémentaires. (French) C. R. Acad. Sci. Paris Sér. A-B 269 (1969) A489–A492.

Posted at June 28, 2010 11:43 AM UTC

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Re: Inevitability in Mathematics

I think it is inevitable that things we now consider inevitable will be seen as mere distractions by future generations.

Posted by: Eric Forgy on June 28, 2010 2:27 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

PS: if you think my previous comment was a distraction, then you’ve proven my point :)

Sorry. I shouldn’t post anything when exhausted and delirious.

Posted by: Eric Forgy on June 28, 2010 2:32 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

I think it is inevitable that things we now consider inevitable will be seen as mere distractions by future generations.

It’s not easy to come up with an example. How about explicit construction of curves (1-dim submanifolds of 2\mathbb{R}^2)? The “Deutsches Museum” in Munich has an impressive collection of mechanical devices that let you draw a plethora of specific curves that I did not even recognize the names of (and I spent a considerable amount of time learning differential geometry).

Posted by: Tim van Beek on June 28, 2010 3:08 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

Hi Tim,

I did have some thoughts in mind, but am in no shape to enunciate them right now. But you did get close to what I meant. If we could go back in time, things that seemed obvious and inevitable at some point in the past, would probably seem naive or simply wrong today.

My initial gut reaction was to the example of Lie groups. I’m somewhat continuum averse especially when it comes to models of fundamental physics. Lie groupoids require infinitesimally small morphisms and I’m not convinced such things have a role to play in anything other the phenomenological models.

I suppose there is not much to complain about (as far as I know) when thought of as pure mathematical objects though. But only future generations can tell.

Posted by: Eric Forgy on June 28, 2010 3:24 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

I am somewhat continuum averse especially when it comes to models of fundamental physics.

Hear, hear!

Posted by: Kea on June 28, 2010 11:17 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

Can we do better than “Groups are interesting as models of symmetry, manifolds are interesting as models of smooth space, therefore group objects in the category of manifolds will be interesting as models of smoothly varying symmetry?”

I don’t think so. Besides the historical fact that this was Sophus Lie’s motivation for inventing his groups, I think that this really is the reason why they are inevitable, but I would phrase it a little bit differently: A finite dimensional object will have a finite dimensional symmetry group, so that a finite number of parameters suffice to parametrize the group (locally). And since we believe that the effect of a transformation should approach zero as the transformation approaches the identity, the dependence of the group elements on the parameters should at least be continuous.

Posted by: Tim vB on June 28, 2010 2:58 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

Can we do better than “Groups are interesting as models of symmetry, manifolds are interesting as models of smooth space, therefore group objects in the category of manifolds will be interesting as models of smoothly varying symmetry?”

Why would you want to do better than that? That sounds to me like asking whether Lie groups could be more inevitable than ordinary groups or manifolds are, which doesn’t seem possible. Unless you extend “Lie group” to refer to arbitrary Dynkin-diagram-like things.

Posted by: Mike Shulman on June 29, 2010 5:48 AM | Permalink | Reply to this

Re: Inevitability in Mathematics

I’m not remotely an expert of Lie history, but wikipedia confirms what I vaguely remembered:

Lie’s idee fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Evariste Galois had done for algebraic equations: namely, to classify them in terms of group theory.

which I hadn’t seen mentioned here as part of the motivation. As such it’s important to note that mathematicians needing to work with concrete differential equations was an important thing in the historical development. This naturally leads one to wonder whether the same impetus for mathematicians developing analytical attacks would have been there had there been some mechanical device capable of numerically solving systems of differential equations at the time to engineers and physicists satisfaction. If not, then they would need to arise from the purely spatial motivation described in David Corfield’s original post. I don’t doubt that this would arise, but I wonder what it would have been.

Posted by: bane on June 30, 2010 5:20 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

Yes, a thorough investigation would take in all the historical work done, such as Thomas Hawkins’ Emergence of the theory of Lie groups and Armand Borel’s Essays in the History of Lie Groups and Algebraic Groups:

Lie groups and algebraic groups are important in many major areas of mathematics and mathematical physics. We find them in diverse roles, notably as groups of automorphisms of geometric structures, as symmetries of differential systems, or as basic tools in the theory of automorphic forms.

Multiple roles hints at inevitability.

Posted by: David Corfield on July 1, 2010 11:16 AM | Permalink | Reply to this

Re: Inevitability in Mathematics

I may be in the minority when I say that I don’t think Lie groups are inevitable because I don’t think the real numbers are inevitable, but on the other hand I do think that Dynkin diagrams are inevitable. Close enough, right?

Posted by: Qiaochu Yuan on June 28, 2010 4:36 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

And there above I quote Connes and Consani talking about

how to recover the field of real numbers from a purely algebraic construction, as the function theory over Spec(S)Spec(S),

where SS is the ‘hyperfield of signs’. Natural?

But yes, Dynkin diagrams are everywhere, as Hazewinkel explained.

Posted by: David Corfield on June 28, 2010 5:11 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

Not all Lie algebras have Dynkin diagrams, not even all physically relevant ones. Example: Virasoro algebra.

Posted by: Thomas Larsson on July 1, 2010 2:46 AM | Permalink | Reply to this

Re: Inevitability in Mathematics

Another recent paper with multivalued arithmetic:

Title: Multifields for Tropical Geometry I. Multifields and dequantization
Authors: Oleg Viro

Abstract: New multifields, that is fields in which addition is multivalued, are introduced and studied. In a separate paper these multifields are shown to provide a base for Tropical Geometry. The main multifields considered here are classical number sets, such as the set of complex numbers, the set of real numbers, and the set of real non-negative numbers, with the usual multiplications, but new, multivalued additions. The new multifields are related with the classical fields and each other by dequantisations. For example, the new complex tropical field is a dequantization of the field of complex numbers.

Posted by: Allen Knutson on June 28, 2010 10:08 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

Thanks.

This paper is devoted to multifields. The notion of multifield is an immediate generalization of the notion of field. A multifield is just a field, in which the addition is multivalued. Multifields are very natural and useful algebraic objects. However, to the best of my knowledge, they appeared in literature as late as in 2006 in a paper [6] by Murray Marshall, and yet has to find their way to the mainstream mathematics. Probably, the main obstacle is that a multivalued operation does not fit to the tradition of set-theoretic terminology, which avoids multivalued maps.

I believe the taboo on multivalued maps has no real ground, and eventually will be removed. Multifields, as well as multigroups and multirings, are legitimate algebraic objects related in many ways to the classical core of mathematics. They provide elegant terminological and conceptual opportunities. In this paper I try to present new evidences for this.

‘Natural’, ‘useful’, ‘does not fit to the tradition’, ‘taboo’, ‘legitimate’ – it has all the makings of an interesting case study. Now where to find a grad student to undertake one?

Posted by: David Corfield on June 29, 2010 10:30 AM | Permalink | Reply to this

Re: Inevitability in Mathematics

I am not really convinced that there is a “taboo” on multivalued maps. It may just be that the majority of mathematicians nowadays is just unimpressed by the phrase “a multivalued map from XX to YY” because they know that this is the same as an ordinary map from XX to the powerset 𝒫Y\mathcal{P}Y. Moreover, in the case of e.g. a “multioperation” on XX the corresponding map from X×XX \times X to 𝒫X\mathcal{P}X can be extended to a map from 𝒫X×𝒫X\mathcal{P}X \times \mathcal{P}X to 𝒫X\mathcal{P}X, i.e. an ordinary operation on 𝒫X\mathcal{P}X.

So I guess that most people shrug their shoulders, lean back and see if introducing multifields, multigroups etc. really gives something new.

Posted by: Marc Olschok on July 14, 2010 6:25 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

Wow, I just looked at the intro to the new Connes-Consani paper. The paper is of course too much for me, but for the first time I feel that perhaps they are getting close to the R. Hypothesis. People have been saying that for years, but I didn’t get the feeling that Connes’ attitude to the Reals was correct, until now.

Posted by: Kea on June 29, 2010 12:25 AM | Permalink | Reply to this

Dynamics of Rules of Thumb; Ideocosm; Re: Inevitability in Mathematics

One precategorical way to model inevitability of mathematical ideas is by imposing a measure on trajectories in what Fritz Zwicky called “the Ideocosm” – the space of all possible ideas.

Other comments on other threads have addressed the Ideocosm. For example, what is the Topology of that space? Is there a hyperplane that separates the subspace of nonphysical theories from the subspace of physical theories?

We are biased in that we are all human, and have little understanding of what the Ideocosm might be to nonhuman intelligences, although we have useful models of the space of all possible theorems in a system, or the space of all Turing machines.

Rules of thumb about inevitability might not apply to intelligent organisms that don’t have thumbs.

Posted by: Jonathan Vos Post on June 29, 2010 9:32 AM | Permalink | Reply to this

hyperrings

There is now a beginning of nnLab: hyperring.

Posted by: Urs Schreiber on June 29, 2010 9:39 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

David wrote:

If someone had told me about hypergroups and hyperrings, where instead of a binary operation sending a pair of elements to a single element, it takes them to a nonempty set of elements, I would have imagined that they would have been good candidates for arbitrariness.

Doesn’t a hypergroup sound sort of like a group object in RelRel? Except for that ‘nonemptiness’ restriction…

Hmm, I guess it sounds even more like a group object in the category of sets and multivalued maps. This is a subcategory of RelRel. But it’s somehow less pretty than RelRel: it’s not a dagger-category, for example.

I wonder if those hyper-guys should be using groups and rings in RelRel, instead. They’re feeling proud of themselves for generalizing from 1-valued maps to 2-valued maps, 3-valued maps and so on… but not 0-valued maps! They might just be prejudiced against the number zero, which is too simple to be simple. Maybe the real hyper-gadgets should allow for 0-valued, or undefined, operations.

But then again, the category of nonempty totally ordered finite sets plays an important role in topology…

Posted by: John Baez on June 30, 2010 12:14 AM | Permalink | Reply to this

Re: Inevitability in Mathematics

I don’t know what a “group object in Rel” is, because Rel isn’t cartesian monoidal. Is there some categorical way to express whatever version of “inverses” hypergroups have?

Posted by: Mike Shulman on June 30, 2010 1:20 AM | Permalink | Reply to this

Re: Inevitability in Mathematics

Mike wrote:

I don’t know what a “group object in RelRel” is, because RelRel isn’t cartesian monoidal.

Oh, whoops — dumb of me.

So now I’m interested in what a hypergroup actually is. There’s a definition on PlanetMath. Let me try to translate it into category theory.

First, a hypersemigroup GG is a monoid object in MultMult, the category of sets and multivalued functions.

Second, a hypergroup is a hypersemigroup GG such that for all aGa \in G,

aG=Ga=Ga G = G a = G

In other words, for every gGg \in G there exists hGh \in G such that ah=ga h = g, and there exists hGh' \in G such that ha=gh' a = g. Hmm… not sure how to translate this!

A multigroup is a hypergroup with a designated identity element 11 and a designated inverse for each element. Here by an ‘inverse’ of gGg \in G I mean an element hh such that 1gh1 \in g h and 1hg1 \in h g. Hmm… not sure how to translate this, either!

Shouldn’t a category theorist, knowing some of the key examples of hypergroups and multigroups, be able to find a somewhat better definition? That is, a definition more likely to lead to a nice category of such gadgets?

This is indeed a nice test case of ‘mathematical inevitability’ and the claims that category theory is a good guide to the tao of mathematics.

Apparently some of the important hypergroups are the canonical hypergroups. Do these have a nice definition?

Posted by: John Baez on June 30, 2010 6:40 AM | Permalink | Reply to this

Re: Inevitability in Mathematics

Maybe hyperrings occurring as quotients of rings by subgroups of their multiplicative groups could give us a category theoretical handle on them.

Posted by: David Corfield on July 1, 2010 10:39 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

I had an interesting reply to a question posed at MO on the point of hypergroups.

Posted by: David Corfield on July 3, 2010 11:15 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

I should preface this by saying that I haven’t actually looked at any examples. But anyway…

One observation that might be helpful is that Rel is equivalent to the full subcategory of Sup, the category of suplattices and sup-preserving maps, spanned by those suplattices that are power sets. And the equivalence is even monoidal, so that to give a monoid in Rel is the same as to give a quantale structure on a powerset. Since we seem to be talking a lot about multiplying subsets by each other, and identifying elements with singleton subsets, it seems that the quantale point of view might be helpful.

Note that in some ways, quantales act a lot like rings, with the role of “addition” being played by joins/unions. More generally, suplattices act like “modules,” with the powersets being the “free” ones, so quantale structures on the powerset of XX could be thought of as analogous to algebra structures on a vector space with basis XX, and the powerset-quantale arising from a group (or monoid) as analogous to the usual group algebra. Moreover, the condition on a Rel-monoid that the multiplication is a many-valued-function, rather than merely a relation, translates into saying that if AA and BB are subsets with AB=A B = \emptyset, then either A=A=\emptyset or B=B=\emptyset — i.e. the corresponding quantale “has no zero-divisors.”

The condition to be a hypergroup, in quantale language, says that the top element is absorbing. That doesn’t seem to have much analogue in ring theory, where there is no top element since we only have finite sums. It also means these quantales are very different from frame-like quantales, in which the top element is the unit.

And speaking of units, I’m confused. Do you intend a “hypersemigroup” to have a unit? The planetmath page doesn’t mention a unit until it gets down to talking about a “multigroup” – does the hypergroup condition imply some unitality as well as inverses?

Posted by: Mike Shulman on June 30, 2010 8:21 AM | Permalink | Reply to this

Re: Inevitability in Mathematics

Mike wrote:

Do you intend a “hypersemigroup” to have a unit?

No, sorry… so it’s not a monoid object in Rel, just a semigroup object!

(In my attempt to translate that page, I started out saying that a “hypergroupoid” is a magma object in Rel, and then decided that was too obscure, and noticed that associativity kicks in when we get to “hypersemigroup”, and slipped and thought “yay! a monoid object!” But of course, as the name suggests, it’s just a semigroup object.)

Posted by: John Baez on June 30, 2010 9:45 PM | Permalink | Reply to this

Re: Inevitability in Mathematics

Let’s see if Connes and Consani agree with Viro. Viro says

The notion of multigroup appeared in the literature in various contexts, sometimes under other names (such as hypergroup and polygroup).

For him, a set XX with a multivalued binary operation (a,b)ab(a, b) \mapsto a \cdot b is called a multigroup if

  • (1) the operation (a,b)ab(a, b) \mapsto a \cdot b is associative;
  • (2) XX contains an element 11 such that 1a=a=a11 \cdot a = a = a \cdot 1 for any aXa \in X;
  • (3) for each aXa \in X there exists a unique a 1Xa^{-1} \in X such that 1aa 11 \in a \cdot a^{-1} and 1a 1a1 \in a^{-1} \cdot a.
  • (4) cabc \in a \cdot b iff c 1b 1a 1c^{-1} \in b^{-1} \cdot a^{-1} for any a,b,cXa, b, c \in X.

Connes and Consani are considering canonical hypergroups. Now we have commutativity, and then expressions of conditions (1), (2) and (3). But now instead of (the commutative version of) (4)

xy+ziffxyz, x \in y + z iff -x \in - y - z,

we have

xy+zimplieszxy. x \in y + z implies z \in x - y.

Is this difference something to do with the ‘canonical’ in canonical hypergroup?

A good example to have in mind has underlying set {+1,0,1}\{+1, 0, -1 \}. Then the binary operation is given by the rule of signs

1+1=1;11=1;11={+1,0,1}. 1 + 1 = 1; - 1 - 1 = - 1; 1 - 1 = \{+1, 0, -1 \}.

Think what you know of the sign of the sum of two integers given their signs. It can be extended to a hyperring and even a hyperfield.

Posted by: David Corfield on June 30, 2010 10:17 AM | Permalink | Reply to this

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