# 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

Astronomy (inquiry component) Astrology

Astronomy (law component) 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 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 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).

# Laplacian and Energy in Graphs

If is a digraph (say, irreflexive binary relation on a set ), we then defined the incidence matrix as the matrix whose entry was iff for vertices 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 is the matrix where the entry is iff or . That is, the entry is iff is a vertex on the edge We can then define the **Laplacian** of as

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

Now consider a map and the corresponding matrix whose th row is We will denote this matrix The authors ([1]) call the map a *representation* of the graph (or digraph in our case) in As the -vectors aren’t operators on it seems counterintuitive to call the map a representation, but I will discuss an matrix they mention in a bit.

**Definition 1.** Let We define the **energy with respect to** of the graph by

Furthermore we can define each edge to have a **weight** and define the **weighted energy of** **with respect to** and by

Now let be the diagonal matrix whose diagonal entry is for a chosen weight function

**Proposition 2.** Let be an antisymmetric () digraph on , and be a weight function on Then

One may call the matrix the **weighted Laplacian** of Now the matrix is an matrix which depends upon a triple with an antisymmetric digraph on and So we could define a map where

Unfortunately this map isn’t a homomorphism, and hence not technically a representation; I couldn’t find a simple structure on compatible with addition of 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 -ary logic system as a homomorphism with a theory, a structure, and a structure of cardinality such that and have the same signature. **Fuzzy logic** is hereby defined as the study of -ary logic systems where is the cardinality of the continuum. In this sense propositions can be thought of as having valence values in some interval, like So classical binary boolean logic is a type of fuzzy logic, where

**Example 1.** Recall in our construction of a utilitarian set we had a set together with a utility function Now suppose is a set of terms which is also a utilitarian set. Suppose we define a structure on by

This structure has signature Then is also a set of terms, with certain terms in identified, and if is a theory/subset in then is a -ary logic system where has the min and max operations.

In the case of Example 1, if we think of the set of terms as a set of behaviors, which could be construed as terms of persons (acting as words), then the structure 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 to discuss some intuitive structures on terms. We already mentioned min and max functions and a corresponding structure on class utilitarian sets. 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 be a probability space. Let be closed under addition where any sum exceeding is defined as and define a unary operation by Now note that is closed under union and complementation; denote this structure on by If measurable sets are construed as formulas, select a theory of disjoint sets. Then is a fuzzy 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 by where with as the generators of the group. We can also define the **Coxeter graph** of a Coxeter group whose vertices are the generators of the group. Then since for we will define edges between two generators iff and label the edge with when

In a generic graph (recall our presentation with a set and a symmetric relation on ), we can define a matrix, called the **incidence matrix** of , denoted where

So is iff the vertex is a vertex on the edge This assumes some ordering of So if we assume irreflexivity of then every column, indexed by will have precisely two entries that are specifically in the th and th row.

In the case of the incidence matrix of a Coxeter graph, the nonzero rows thus coincide with generators such that for some

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

**Definition 1.** Let and be digraphs on and respectively. A map is an **-digraph homomorphism** if An –**isomorphism** of digraphs is a bijective -homomorphism of digraphs.

**Definition 2.** Let be a digraph on We define the **adjacency matrix** of , denoted as the matrix where is the number of paths from to

So an entry being means one of the entries or is in 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 and be digraphs on There is an -automorphism of iff there is a permutation matrix such that

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