Monthly Archives: August, 2011

The -logy and -nomy Suffixes

I had mentioned before about how philosophy and mathematics used to be equivalent before math diverged by making solid, technical assumptions, and proving what followed.  Similarly is the case with regards to astrology and astronomy.  Both began by looking for correlation between the stars and X, where X was pretty much any Earthly phenomenon.  Then astronomy used counterexamples to the anthropic principle to establish more consistent theories, and astrology was properly buried.  Then for some reason in the 20th century horoscopes reincarnated astrology.1  While technically the position of the stars infinitesimally changes the net gravitational force on a human, the fact that cognitive activity is based on electric potentials that globally dominate any infinitesimal net gravitational change on the body renders their study on personality irrelevant.

This being the case, the suffix -logy on astrology seems insulting to the scientific community.  The suffix -log means “to speak”.2    Coupled with -y, it takes the meaning of “speaker” which appears to conventionally translate to “generic speaker”, or, “department/field”.  Xlogy thus becomes “study/department of X”.  In this case, X being astro–deriving from “stars”.  At the same time, -nomy derives from “law”.  How should we distinguish the two suffixes?  We could say Xlogy is the general inquiry into X, which builds upon the object Xnomy.  Xnomics could then be reserved for when Xlogy leads to multiple instances of an Xnomy.  Hence we should have the following conventional-to-literal semantic assignments:

Astrology \mapsto\varnothing

Astronomy (inquiry component) \mapsto Astrology

Astronomy (law component) \mapsto Astronomy.

What would academic departments call themselves then?  We could pick the Xlogy form.  In this case “economics” would be called “ecology”?  The prefix eco- derives from the study of environments of living organisms.  In this sense, conventional “economics” is far more abstract than it semantically pretends to be.  I would consider something along the lines of

Economics \mapsto Elogy

where E derives from something like “token”.  Although trade itself may prove to be more fundamental and inherently connected to physics (think conservation of energy).  We could then have

Physics \mapsto Cosmology,

which one could plausibly argue.  I’m not sure to what we could assign mathematics.  It may not even matter since math, physics, and economics may share a fundamental equivalence (all could be construed as structures of symbols=0 volume energy regions=tokens).

[1]  Campion, Nicholas, 2009. A History of Western Astrology, Vol. 2, The Medieval and Modern Worlds. London: Continuum. (via “Astrology” on Wikipedia).

[2]  http://www.etymonline.com/index.php?search=-logy

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Laplacian and Energy in Graphs

If E is a digraph (say, irreflexive binary relation on a set V), we then defined the incidence matrix [E] as the |V|\times |V| matrix whose ij entry was 1 iff (i,j)\in E for vertices i,j.  NOTE:  This definition is not the traditional one;  I accidentally wrote it incorrectly in a previous post and have made the correction to that definition.  The correct definition of [E] is the |V|\times |E| matrix where the i,(j,k) entry is 1 iff 1=i or 1=j.  That is, the entry i,(j,k) is 1 iff i is a vertex on the edge (j,k).  We can then define the Laplacian of E as

L(E)=[E][E]^\top.

We will simply write L as the Laplacian when it is clear what digraph we are discussing.

Now consider a map f:V\to\mathbb{R}^m and the corresponding |V|\times m matrix whose ith row is f(i).  We will denote this matrix f(V).  The authors ([1]) call the map f a representation of the graph (or digraph in our case) in \mathbb{R}^m.  As the m-vectors f(i) aren’t operators on \mathbb{R}^m, it seems counterintuitive to call the map a representation, but I will discuss an m\times m matrix they mention in a bit.

Definition 1.  Let \|\cdot\|=\|\cdot\|_2.  We define the energy with respect to f of the graph E by

\displaystyle\mathcal{E}_f(E)=\sum_{(i,j)\in E}\|f(i)-f(j)\|^2.

Furthermore we can define each edge to have a weight w((i,j))=w_{ij}\in\mathbb{R}^+ and define the weighted energy of E with respect to f and w by

\displaystyle\mathcal{E}_{f,w}(E)=\sum_{(i,j)\in E}w_{ij}\|f(i)-f(j)\|^2.

Now let W be the diagonal |E|\times |E| matrix whose diagonal (i,j),(i,j) entry is w_{ij} for a chosen weight function w.

Proposition 2.  Let E be an antisymmetric ((i,j)\in E\Rightarrow (j,i)\notin E) digraph on V, f:V\to\mathbb{R}^m, and w be a weight function on E.  Then

\displaystyle\mathcal{E}_{f,w}(E)=\mbox{tr~}f(V)^\top[E]W[E]^\top f(V).

One may call the matrix [E]W[E]^\top the weighted Laplacian of E.  Now the matrix f(V)^\top [E]W[E]^\top f(V) is an m\times m matrix which depends upon a triple (E,f,w) with E an antisymmetric digraph on V, f:V\to\mathbb{R}^m, and w:E\to\mathbb{R}^+.  So we could define a map \rho_E:F\times W\to L(\mathbb{R}^m) where

\displaystyle\rho_E((f,w))=f(V)^\top [E]W[E]^\top f(V).

Unfortunately this map isn’t a homomorphism, and hence not technically a representation;  I couldn’t find a simple structure on F\times W compatible with addition of m\times m matrices.

[1]  Godsil, Chris and Gordon Royle.  Algebraic Graph Theory.  Graduate Texts in Mathematics.  Vol. 207.  Springer Science and Business Media.  2004.

Fuzzy Logic

Recall the definition of an n-ary logic system as a homomorphism \varphi:F[X]\to V with X a theory, F a structure, and V a structure of cardinality n such that F and V have the same signature.  Fuzzy logic is hereby defined as the study of c-ary logic systems where c is the cardinality of the continuum.  In this sense propositions can be thought of as having valence values in some interval, like [0,1].  So classical binary boolean logic is a type of fuzzy logic, where \mbox{ran}\,\varphi=\{0,1\}.

Example 1.  Recall in our construction of a utilitarian set we had a set X together with a utility function u:X\to [-1,1).  Now suppose T is a set of terms which is also a utilitarian set.  Suppose we define a structure F on T/\sim_u by

\displaystyle [x]\vee [y]=u'^{-1}\left(\min\{u'[x],u'[y]\}\right)=u'^{-1}\left(\min\{u(x),u(y)\}\right)

\displaystyle [x]\wedge [y]=u'^{-1}\left(\max\{u'[x],u'[y]\}\right)=u'^{-1}\left(\max\{u(x),u(y)\}\right).

This structure has signature (0,0,2,0,...).  Then T/\sim_u is also a set of terms, with certain terms in T identified, and if X is a theory/subset in T/\sim_u, then u':F[X]\to [-1,1) is a c-ary logic system where [-1,1) has the min and max operations.

In the case of Example 1, if we think of the set T of terms as a set of behaviors, which could be construed as terms of persons (acting as words), then the structure F[X] can be interpreted as equivalence classes of compound behaviors that yield the same utility where logical valence of compound behaviors is simply based on their utility.

Since the valence set in a fuzzy logic system is an interval, let us look at some common structures on [0,1] to discuss some intuitive structures on terms.  We already mentioned min and max functions and a corresponding structure on class utilitarian sets.  [0,1] is closed under binary multiplication.  The corresponding binary connective would thus send two propositions to a proposition whose valence is a product of the original two.  Treating binary boolean logic as a fuzzy logic, propositional conjunction satisfies this condition.

Example 2.  Let (X,\Sigma,P) be a probability space.  Let [0,1] be closed under addition where any sum exceeding 1 is defined as 1 and define a unary operation by r^{-1}=1-r. Now note that \Sigma is closed under union and complementation; denote this structure on \Sigma by F.  If measurable sets are construed as formulas, select a theory X of disjoint sets.  Then P:F[X]\to [0,1] is a fuzzy logic system.

Update of Language Definition

Note that I have removed the factorization requirement from the definition of a language in Fundamental Knowledge Part 1;  so we will just have \mathcal{L}_{F,T,W}=F[T[W]].  This will remove some triviality in examples of fuzzy logic systems in the upcoming post.  The original motivation behind the factorization was that traditionally compound terms are considered formulas, but terms themselves are not considered formulas.  I don’t really see why we can’t let terms be formulas;  let us assume “substitutions” have already been made.

I have also removed the requirement that \varphi(\phi)=\varphi(\psi) for all \phi,\psi in a theory X where \varphi is a logic system.  Instead I have defined a logic system that satisfies this condition as a normal logic system.

Matrices and Graphs

Recall in the post “Canonical Representation of Finitely Presented Coxeter Groups” we defined the Coxeter matrix of a Coxeter group G by a_{ij}=m_{ij} where (r_ir_j)^{m_{ij}}=1 with r_i,r_j as the generators of the group.  We can also define the Coxeter graph of a Coxeter group G whose vertices are the generators of the group.  Then since m_{ij}\geq 2 for i\neq j, we will define edges between two generators iff m_{i,j}\geq 3, and label the edge with m_{ij} when m_{ij}\geq 4.

In a generic graph (recall our presentation with V a set and E a symmetric relation on V^2), we can define a |V|\times |E| matrix, called the incidence matrix of E, denoted [E], where

a_{i,(j,k)}=\left\{\begin{array}{lcl}1&\mbox{if}&i=j \mbox{~or~} i=k\\ 0&\mbox{otherwise}&\end{array}\right .

So a_{i,(j,k)} is 1 iff the vertex i is a vertex on the edge (j,k).  This assumes some ordering of V.  So if we assume irreflexivity of E, then every column, indexed by (i,j), will have precisely two entries that are 1, specifically in the ith and jth row.

In the case of the incidence matrix of a Coxeter graph, the nonzero rows thus coincide with generators r_i such that m_{ij}\geq 3 for some j.

We can remove the symmetric requirement of E; so E is just a subset of V^2.  We can then use “binary relation” synonymously with “digraph“.  From here on we will assume a digraph on V is an irreflexive binary relation on V, and that a graph on V is an irreflexive symmetric binary relation on V.

Definition 1.  Let E_1 and E_2 be digraphs on V_1 and V_2 respectively.  A map \varphi:V_1\to V_2 is an (E_1,E_2)-digraph homomorphism if (x,y)\in E_1\Rightarrow(\varphi(x),\varphi(y))\in E_2.  An (E_1,E_2)isomorphism of digraphs is a bijective (E_1,E_2)-homomorphism of digraphs.

Definition 2.  Let E be a digraph on V.  We define the adjacency matrix of E, denoted (E), as the |V|\times |V| matrix where a_{x,y} is the number of paths from x to y.

So an entry a_{i,(j,k)}\in [E] being 1 means one of the entries a_{i,j} or a_{j,i} is 1 in (E) since being a vertex on an edge means there is a path to some other vertex (namely the other vertex on that edge).

Proposition 3.  Let E_1 and E_2 be digraphs on V.  There is an (E_1,E_2)-automorphism of V iff there is a permutation matrix P such that

\displaystyle P^\top(E_1)P=(E_2).

[1]  Godsil, Chris and Gordon Royle.  Algebraic Graph Theory.  Graduate Texts in Mathematics.  Vol. 207.  Springer Science and Business Media.  2004.