Tag Archives: mathematics

Mathematics, things, objects and brains

This is about my understanding of the post Mathematics and the Real by Louis Kauffman.

I start from this quote:

One might hypothesize that any mathematical system will find natural realizations. This is not the same as saying that the mathematics itself is realized. The point of an abstraction is that it is not, as an abstraction, realized. The set { { }, { { } } } has 2 elements, but it is not the number 2. The number 2 is nowhere “in the world”.

Recall that there are things and objects. Objects are real, things are discussions. Mathematics is made of things. In Kauffman’s example the number 2 is a thing and the set { { }, { { } } } is an object of that thing.

Because an object is a reification of a thing. It is therefore real, but less interesting than the thing, because it is obtained by forgetting (much of) the discussion about it.

Reification is not a forgetful functor, though. There are interactions in both directions, from things to objects and from objects to things.

Indeed, in the rhino thing story, a living rhinoceros is brought in Europe. The  sight of it was new. There were remnants of ancient discussions about this creature.

At the beginning that rhinoceros was not an object, not a thing. For us it is a thing though, and what I am writing about it is part of that thing.

From the discussion about that rhinoceros, a new thing emerged. A rhinoceros is an armoured beast which has a horn on its back which is used for killing elephants.

The rhino thing induced a wave of reifications:  nearby the place where that rhino was seen for the first time in Portugal, the Manueline Belém Tower  was under construction at that moment. “The tower was later decorated with gargoyles shaped as rhinoceros heads under its corbels.[11]” [wiki dixit]

Durer’s rhino, another reification of that discussion. And a vector of propagation of the discussion-thing. Yet another real effect, another  object which was created by the rhino thing is “Rinoceronte vestido con puntillas (1956) by Salvador Dalí in Puerto Banús, Marbella, Spain” [wiki dixit].

Let’s take another example. A discussion about the reglementations of the sizes of cucumbers and carrots to be sold in EU is a thing. This will produce a lot of reifications, in particular lots of correct size cucumbers and carrots and also algorithms for selecting them. And thrash, and algorithms for dispensing of that trash. And another discussions-things, like is it moral to dump the unfit carrots to the trash instead of using them to feed somebody who’s in need? or like the algorithm which states that when you go to the market, if you want to find the least poisoned vegetables then you have to pick them among those which are not the right size.

The same with the number 2, is a thing. One of it’s reifications is the set { { }, { { } } }. Once you start to discuss about sets, though, you are back in the world of things.

And so on.

I argue that one should understand from the outset that mathematics is distinct from the physical. Then it is possible to get on with the remarkable task of finding how mathematics fits with the physical, from the fact that we can represent numbers by rows of marks |  , ||, |||, ||||, |||||, ||||||, … (and note that whenever you do something concrete like this it only works for a while and then gets away from the abstraction living on as clear as ever, while the marks get hard to organize and count) to the intricate relationships of the representations of the symmetric groups with particle physics (bringing us back ’round to Littlewood and the Littlewood Richardson rule that appears to be the right abstraction behind elementary particle interactions).

However, note that   “the marks get hard to organize and count” shows only a limitation of the mark algorithm as an object, and there are two aspects of this:

  • to stir a discussion about this algorithm, thus to create a new thing
  • to recognize that such limitations are in fact limitations of our brains in isolation.

Because, I argue, brains (and their working) are real.  Thoughts are objects, in the sense used in this post! When we think about the number 2, there is a reification of out thinking about the number 2 in the brain.

Because brains, and thoughts, are made of an immensely big number of chemical reactions and electromagnetic  interactions, there is no ghost in these machines.

Most of our brain working is “low level”, that is we find hard to account even for the existence of it, we have problems to find the meaning of it, we are very limited into contemplating it in whole, like a self-reflecting mirror. We have to discuss about it, to make it into a thing and to contemplate instead derivative objects from this discussion.

However, following the path of this discussion, it may very well be that brains working thing can be understood as structure processing, with no need for external, high level, semantic, information based meaning.

After all, chemistry is structure processing.

A proof of principle argument for this is Distributed GLC.

The best part, in my opinion, of Kauffman’s post is, as it should, the end of it:

The key is in the seeing of the pattern, not in the mechanical work of the computation. The work of the computation occurs in physicality. The seeing of the pattern, the understanding of its generality occurs in the conceptual domain.

… which says, to my mind at least, that computation (in the usual input-output-with-bits-in-between sense) is just one of the derivative objects of the discussion about how brains (and anything) work.

Closer to the brain working thing, including the understanding of those thoughts about mathematics, is the discussion about “computation” as structure processing.

UPDATE: A discussion started in this G+ post.


Uniform spaces, coarse spaces, dilation spaces (II)

(1) Uniform spaces, coarse spaces, dilation spaces

(2) What’s the difference between uniform and coarse spaces (I)

I shall use the idea explained in (1), in the groupoid frame of (2), classical situation of the trivial groupoid over a space X. In this case the uniform and coarse structures are just the classical notions.

This idea says that all we need to have is a field of dilations. With such an object we may construct an uniformity, or a coarse structure, then ask that the field of dilations has some properties with respect to the said uniformity (or coarse structure). If the field of dilations has those properties then it transforms into an emergent algebra  (in the case 0 below; in the other case  there is a new type of emergent algebra which appears in relation to coarse structures).

Remark. Measure can be constructed from a field of dilations, that’s for later.

Fields of dilations. We have a set X, call it “space”. We have the commutative group (0,\infty) with multiplication of reals operation, (other groups work well, let’s concentrate on this one).

A field of dilations is a function which associates to any x \in X and any \varepsilon \in (0,\infty)
an invertible transformation  \delta^{x}_{\varepsilon} : U(x) \subset X \rightarrow U(x,\varepsilon) \subset X which we call “the dilation based at x, of coefficient \varepsilon“.

We ask:

1. x \in U(x) for any point x \in X

2. for any fixed x \in X the function \varepsilon \in (0,\infty) \rightarrow \delta^{x}_{\varepsilon} is a representation of the commutative group (0,\infty).

3. fields of dilations come into 2 flavors (are there more?), depending on the choice between (0,1] and [1,\infty), two important sub-semigroups of (0,\infty).

(case 0) – If you choose (0,1] then we ask that for any \varepsilon \in (0,1] we have U(x,\varepsilon) \subset U(x), for any x,

This case is good for generating uniformities and  for the infinitesimal point of view.

(case \infty) – If your choice is [1,\infty) then we ask that for any \varepsilon \in [1,\infty) we have U(x) \subset U(x,\varepsilon) , for any x,

This case is good for generating coarse structures and for  the asymptotic point of view.

Starting from here, I’m afraid that my latex capabilities on wordpress are below what I need to continue.

Follow this working paper to see more. Thanks for comments!

PS. At some point, at least for the case of uniformities, I shall use “uniform refinements” and what I call “topological derivative” from arXiv:0911.4619, which can be applied for giving alternate proofs for rigidity results, without using Pansu’s Rademacher theorem in Carnot groups.

Heisenberg group, convex analysis, alike or not?

I posted on mathoverflow a question, with the purpose of clarifying the feelings I have concerning the formal resemblance between Heisenberg group and cyclically monotone operators from convex analysis. The question got an answer which made me realize something trivial (but sometimes we need somebody else to point to the obvious). However, I still think there is something worthy of further consideration here, that’s why I post it.

Setting: Let S be a semigroup (i.e. has an associative operation with neutral element e) and let (A,+) be a commutative group (with neutral element 0).

Let’s say that a semigroup extension of S with A is any operation on S \times A, of the form

(s,a)(s',a') = (s s' , a+a'+ \lambda(s,s'))

where \lambda: S \times S \rightarrow A is a function such that S \times A with the mentioned operation is a semigroup with neutral element (e,0). Obviously, the operation is encoded by the function \lambda, which has to satisfy:

\lambda(s s', s") + \lambda(s,s') = \lambda(s, s's") + \lambda(s',s")  (from associativity)

\lambda(e,s) = \lambda(s,e) = 0  (from the neutral element condition).

Here are two examples. Both are written in the setting of bipotentials, namely   X and Y are topological, locally convex, real vector spaces of dual variables x \in X and y \in Y, with the duality product \langle \cdot , \cdot \rangle : X \times Y \rightarrow \mathbb{R}.

The spaces X, Y have topologies compatible with the duality product, in the sense that for any continuous linear functional on X there is an y \in Y which puts the functional into the form x \mapsto \langle x,y\rangle (respectively any continuous linear functional on Y has the form y \mapsto \langle x,y\rangle, for an x \in X).

Example 1: (Heisenberg group) Let S = X \times Y with the operation of addition of pairs of vectors and let A = \mathbb{R} with addition. We may define a Heisenberg group over the pair (X,Y) as H(X,Y) = S \times A with the operation

(x,y,a)(x',y',b) = (x+x', y+y', a+a'+ \langle x, y'\rangle - \langle x', y \rangle)
Fact: There is no injective morphism from (X\times Y, +) to H(X,Y).

Example 2: (convex analysis) Let this time S = (X \times Y)^{*}, the free semigroup generated by X \times Y, i.e. the collection of all finite words with letters from X \times Y, together with the empty word e, with the operation of concatenation of words.

Let A be the set of bi-affine real functions on X \times Y, i.e. the collection of all functions a: X \times Y \rightarrow \mathbb{R} which are affine and continuous in each argument. A is a commutative group with the addition of real valued functions operation.

We define the function \lambda: S \times S \rightarrow A by:

\lambda(e, c)(x,y) = \lambda(c,e)(x,y)=0 for any c \in S and any (x,y) \in X \times Y.
– if c, h \in S are words c = (x_{1},y_{1})...(x_{n}, y_{n}) and h = (u_{1},v_{1})...(u_{m}, v_{m}), with m,n \geq 1, then

\lambda(c,h)(x,y) = \langle u_{1} - x , y_{n} - y \rangle for any (x,y) \in X \times Y.

This \lambda induces a semigroup extension operation on S \times A.

Fact: there is an injective morphism F: S \rightarrow S \times A, with the expression F(c) = (c, E(c)).

Here, for any c = (x_{1},y_{1})...(x_{n}, y_{n}) the expression E(c)(x,y) is the well known circular sum associated to the “dissipation during the discrete cycle” (x_{1},y_{1})...(x_{n}, y_{n})(x,y), namely:

E(c)(x,y) = \langle x_{1},y\rangle + \langle x,y_{n}\rangle - \langle x,y \rangle + \sum_{k=1}^{n-1}\langle x_{k+1}, y_{k}\rangle - \sum_{k=1}^{n} \langle x_{k},y_{k} \rangle

which appears in convex analysis, related to cyclically monotone operators.

The main difference between those two examples is that the example 2. is a direct product of a semigroup with a group. There are many resemblances though.

Uniform spaces, coarse spaces, dilation spaces

This post is related to the previous What’s the difference between uniform and coarse structures? (I).  However, it’s not part (II).

Psychological motivation.  Helge Gloecker made a clear mathscinet review of my paper Braided spaces with dilations and sub-riemannian symmetric spaces.  This part of the review motivated me to start writing some more about dilatation structures, some of it explained in the next section of this post. Here is the motivating part of the review:

“Aiming to put sub-Riemannian geometry on a purely metric footing (without recourse to differentiable structures), the author developed the concept of a dilatation structure and related notions in earlier works [see J. Gen. Lie Theory Appl. 1 (2007), no. 2, 65–95; MR2300069 (2008d:20074); Houston J. Math. 36 (2010), no. 1, 91–136; MR2610783 (2011h:53032); Commun. Math. Anal. 11 (2011), no. 2, 70–111; MR2780883 (2012e:53051)].


In a recent preprint [see “Emergent algebras”, arXiv:0907.1520], the author showed that idempotent right quasigroups are a useful auxiliary notion for the introduction (and applications) of dilatation structures.


The author sketches how dilatation structures can be defined with the help of uniform irqs, suppressing the technical and delicate axioms concerning the domains of the locally defined maps. More precisely, because the maps are only locally defined, the structures which occur are not uniform irqs at face value, but only suitable analogues using locally defined maps (a technical complication which is not mentioned in the article).”

This was done before, several times, for dilatation structures, but Helge is right that it has not been done for emergent algebras.

Which brings me to the main point: I started a paper with the same (but temporary) name as this post.  The paper is based on the following ideas.

Fields of dilations which generate their own topology, uniformity, coarse structure, whatever.    In a normed real vector space (V, \| \cdot \|) we may use dilations as replacements of metric balls. Here is the simple idea.

Let us define, for any x \in V, the domain of x as U(x),  the ball with center x, of radius one, with respect to the distance d(x,y) = \|x - y \|. In general, let B(x, r)  be the  metric ball with respect to the distance induced by the norm.

Also, for any x \in V and \varepsilon \in (0,+\infty), the dilation based at x, with coefficient \varepsilon is the function \delta^{x}_{\varepsilon} y = x + \varepsilon ( -x + y) .

I use these notations to write the  ball of radius \varepsilon \in (0,1] as B(x,\varepsilon) = \delta^{x}_{\varepsilon} U(x) and give it the fancy name dilation ball of coefficient \varepsilon.

In fact, spaces with dilations, or dilatation structures, or dilation structures, are various names for spaces endowed with fields of dilations which satisfy certain axioms. Real normed vector spaces are examples of spaces with dilations, as a subclass of the larger class of conical groups with dilations, itself just a subclass of groups with dilations. Regular sub-riemannian manifolds are examples of spaces with dilations without a predefined group structure.

For any space with dilations, then, we may ask what can we get by forgetting the distance function, but keeping the dilation balls.  With the help of those we may define the uniformity associated with a field of dilations and then ask that the field of dilations is behaves uniformly, and so on.

Another structure, as interesting as the uniformity structure, is the (bounded metric) coarse structure of the space, which  again could be expressed in terms of fields of dilations. As coarse structures and uniform structures are very much alike (only that one is interesting for the small scale, other for the large scale), is there a notion of dilation structure which is appropriate for coarse structures?

I shall be back on this, with details, but the overall idea is that a field of dilations (a notion even weaker than an emergent algebra) is somehow midway between a uniformity structure and a coarse structure.

UPDATE (28.10.2012): I just found bits of the work of Protasov, who introduced the notion of “ball structure” and “ballean”, which is clearly related with the idea of keeping the balls, not the distance. Unfortunately, for the moment I am unable to find a way to read his two monographs on the subject.

Ado’s theorem for groups with dilations?

Ado’s theorem  is equivalent with the following:

Theorem. Let G be a local Lie group. Then there is a real, finite dimensional vector space V and an injective, local group morphism from (a neighbourhood of the neutral element of) G to GL(V), the linear group of $V$.

Any proof I am aware of, (see this post for one proof and relevant links),  mixes the following ingredients:

–  the Lie bracket and the BCH formula,

– either reduction to the nilpotent case or (nonexclusive) use of differential equations,

– the universal enveloping algebra.

WARNING: further I shall not mention the “local” word, in the realm of spaces with dilations everything is local.

We may pass to the following larger frame of spaces with dilations, dilatation structures or emergent algebras:

– locally compact groups with dilations instead of Lie groups

– locally compact conical groups instead of vector spaces

– linearity in the sense of dilation structures instead of usual linearity.

Conjecture:  For any locally compact group with dilations G there is a locally compact conical group N and an injective morphism \rho: G \rightarrow GL(N) such that for every x \in N the map g \in G \mapsto \rho(g)x is differentiable.

In this frame:

– we don’t have the corresponding Lie bracket and BCH formula, see the related problem of the noncommutative BCH formula,

– what nilpotent means is no longer clear (or needed?)

– we don’t have a clear tensor product, therefore we don’t have a correspondent of the universal enveloping algebra.

Nevertheless I think the conjecture is true and actually much easier to prove than Ado’s theorem, because of the weakening of the conclusion.

Is there a positional numeral system for Carnot groups?

I advertise here a question I posed on mathoverflow, then I shall give a motivation for this question.

In the following is a reproduction of  the question (thanks for all useful comments, read them at mathoverflow) (slight modification of the title of the question here)

What is the computational complexity of multiplication in a Carnot group? 

Background:  A Carnot group is a real nilpotent Lie group N whose Lie algebra Lie(N) admits a direct sum decomposition

Lie(N) = V_{1}+ ... + V_{m}

such that if i+j \leq m then  [V_{i}, V_{j}] \subset V_{i+j},   otherwise [V_{i}, V_{j}] = 0.

Moreover, it is required that V_{1} generates the whole Lie algebra Lie(N).

Detailed question: What is the computational complexity of multiplication  in N, as a function of m, dim V_{1}, … , dim V_{m}?

More background: 

1. “Computational complexity” means giving an estimate for the running time of an algorithm for performing the multiplication operation.I am looking for nontrivial estimates, better than, for example, the one coming from matrix multiplication.Exponentiation and logarithms are not taken into account (one variant of thinking is to use BCH formula for multiplication, with a finite number of terms due to nilpotency, other variant would be to use the remark from example 2.4 below and think about multiplication as matrix multiplication).

2. Examples of Carnot groups:

2.1. When m=1, this is a real vector space and the multiplication operation is just addition of vectors (thus the complexity of the  multiplication operation is the one of the addition of vectors, much less than the one of matrix multiplication).

2.2. When m=2, the simplest example is the Heisenberg group [see posts here about it] .

2.3. The group of invertible, n \times n  upper triangular matrices, has m = n-1 and dim V_{i} = n-i. Is there any better estimate than n^{3} in this case? (I think so, but cannot find a reference to an answer.) [In fact Henry Cohn answered to this:

“The exponent for multiplication of upper triangular matrices is the same as that for full matrix multiplication. Specifically, you can reduce n \times n matrix multiplication to the 3n \times 3n upper triangular case. (Use nine n \times n  blocks, with identity matrices on the diagonal and a zero matrix in the upper right corner.) So for that case the best exponent that’s known is 2.373, the same as for the general case.

The reduction is not a morphism. Suppose A  and B are any n \times n  matrices (not necessarily invertible), and define 3n \times 3n matrices A' and B' as follows. Both are the identity matrix, except for one n \times n block. Specifically, A' has A as the middle top n \times n block [i.e. the (1,2) block is A, the diagonal blocks are n \times n identity matrices, all other blocks are n \times n  O matrices], and B' has B as the middle right n \times n  block [i.e. the (2,1) block is B, the diagonal blocks are n \times n identity matrices, all other blocks are n \times n  O matrices]. Then A'B' has AB  as its upper right block [i.e. the (3,3) block]. These larger matrices are upper triangular, so if you can efficiently multiply upper triangular matrices, then you can efficiently multiply arbitrary matrices (of one third the size).”

2.4. Any Carnot group can be seen as a subgroup of the group from example 2.3.

2.5. Carnot groups appear as metric tangent spaces for sub-riemannian, manifolds. As such, they are a kind of noncommutative vector spaces, with the “trivial” exception when they are commutative, like in the case of riemannian manifolds, when the tangent space is like in example 2.1  (other “simple” example being the metric contact manifolds, with the tangent space like in 2.2).”

What is my motivation behind this question? Actually, I wrote about this in the post “Combinatorics versus geometric …“.

Indeed, a positional numeral system can be seen from two related viewpoints:

1. as the basic ingredient for the algorithm of computing the “multiplication” operation in the abelian group of integers (or reals) (that’s like in the example 2.1 above),

2. as a “positional system” in the geometric sense, by exploiting the self-similarity of the group of integers (or reals), something which has not much to do with numbers. Let me explain: I am now in Europe, Romania, Bucharest. This is a positional notation for the place where I am, because Romania is a country in Europe and Bucharest is a city in Romania. Likewise, the number 123 (in decimal basis) is that number which is “in the continent 100“, in the “country 20” and in the “city 3“.  What makes such a positional notation useful for numbers is the self-similarity (manifested by the system of orders of magnitude) of the “world of numbers”.

Let us just put all together: is there a “positional numeral system” for Carnot groups which can give fast algorithms for computing the multiplication operation (in a Carnot group), by exploiting the self-similarity of a Carnot group?

What’s the difference between uniform and coarse structures? (I)

See this, if you need: uniform structure and coarse structure.

Given a set X, endowed with an uniform structure or with a coarse structure, Instead of working with subsets of X^{2}, I prefer to think about the trivial groupoid X \times X.

Groupoids. Recall that a groupoid is a small category with all arrows invertible. We may define a groupoid in term of it’s collection of arrows, as follows: a set G endowed with a partially defined operation m: G^{(2)} \subset G \times G \rightarrow G, m(g,h) = gh, and an inverse operation inv: G \rightarrow G, inv(g) = g^{-1}, satisfying some well known properties (associativity, inverse, identity).

The set Ob(G) is the set of objects of the groupoid, \alpha , \omega: G \rightarrow Ob(G) are the source and target functions, defined as:

– the source fonction is \alpha(g) = g^{-1} g,

– the target function is \omega(g) = g g^{-1},

– the set of objects is Ob(G) consists of all elements of G with the form g g^{-1}.

Trivial groupoid. An example of a groupoid is G = X \times X, the trivial groupoid of the set X. The operations , source, target and objects are

(x,y) (y,z) = (x,z), (x,y)^{-1} = (y,x)

\alpha(x,y) = (y,y), \omega(x,y) = (x,x),

– objects (x,x) for all x \in X.

In terms of groupoids, here is the definition of an uniform structure.

Definition 1. (groupoid with uniform structure) Let G be a groupoid. An uniformity \Phi on G is a set \Phi \subset 2^{G} such that:

1. for any U \in \Phi we have Ob(G) \subset U,

2. if U \in \Phi and U \subset V \subset G then V \in \Phi,

3. if U,V \in \Phi then U \cap V \in \Phi,

4. for any U \in \Phi there is V \in \Phi such that VV \subset U,  where VV is the set of all gh with g, h \in V and moreover (g,h) \in G^{(2)},

5. if U \in \Phi then U^{-1} \in \Phi.

Here is a definition for a coarse structure on a groupoid (adapted from the usual one, seen on the trivial groupoid).

Definition 2.(groupoid with coarse structure) Let G be a groupoid. A coarse structure  \chi on G is a set \chi \subset 2^{G} such that:

1′. Ob(G) \in \chi,

2′. if U \in \chi and V \subset U then V \in \chi,

3′. if U,V \in \chi then U \cup V \in \chi,

4′. for any U \in \chi there is V \in \chi such that UU \subset V,  where UU is the set of all gh with g, h \in U and moreover (g,h) \in G^{(2)},

5′. if U \in \chi then U^{-1} \in \chi.

Look pretty much the same, right? But they are not, we shall see the difference when we take into consideration how we generate such structures.

Question. Have you seen before such structures defined like this, on groupoids? It seems trivial to do so, but I cannot find this anywhere (maybe because I am ignorant).

UPDATE (01.08.2012): Just found this interesting paper  Coarse structures on groups, by Andrew Nicas and David Rosenthal, where they already did on groups something related to what I want to explain on groupoids. I shall be back on this.