definition of parsimony

definition of parsimony

[Axel Rossberg]

Dear List Members,

I am looking for a formal definition of parsimony for fundamental
scientific theories.  From the tiny bit I understood of category
theory, I had the impression it might provide the right framework for
such a definition.

The problem of motivating and defining parsimony is being discussed in
analytic philosophy.  An overview over the discussion can be found at
http://plato.stanford.edu/entries/simplicity/ , which starts off with
the sentences

  Most philosophers believe that, other things being equal, simpler
  theories are better. But what exactly does theoretical simplicity
  amount to? Syntactic simplicity, or elegance, measures the number
  and conciseness of the theories basic principles. Ontological
  simplicity, or parsimony, measures the number of kinds of entities
  postulated by the theory. One issue concerns how these two forms of
  simplicity relate to one another.

I am interested in the "ontological simplicity, or
parsimony". However, if one understands modern physics as describing
essentially only one thing, the wave-function of the universe, then
even the idea of defining parsimony in terms of numbers of kinds of
things seems to be a bit odd.

Yet, I think the idea is intuitively clear.  The minimum requirement
for a formal definition of parsimony is perhaps that it should
identify theories such as the dynamics of Newtonian point-particles or
the current "standard model" of particle physics as parsimonious, while
the same theories with some oddities added, which do not themselves
affect the "real" physics, should be identifiable as non-parsimonious.

Beyond this, such a definition should presuppose as little as possible
about the nature of the theories it applies to.

Does somebody know about applications of category theory to this
problem, or have an idea for who to do it?

Cheers,

Axel Rossberg
---
Evolution and Ecology Program
International Institute for Applied Systems Analysis
Schlossplatz 1
A-2361 Laxenburg
AUSTRIA
++++++++++++++++++++++++
reprints http://axel.rossberg.net/paper
and more http://axel.rossberg.net

Re: definition of parsimony

[Vaughan Pratt]

(Out of fairness to *the* standard model, the Higgs boson is its only
remaining unobserved particle.  The other particles being sought at
Fermilab and CERN belong to supersymmetric (SUSY) extensions of the
standard model.  Ironically such an extension while having more
particles could nevertheless claim to be more parsimonious than the
current standard model as measured by the number of its free parameters,
in particular fewer Yukawa constants.  Disclaimer: if you knew SUSY like
I know SUSY no physics lab would even think of hiring you.)

Re: definition of parsimony

[Vaughan Pratt]

Without going into its relevance to category theory, I would put the SEP
article on parsimony that Axel Rossberg pointed to alongside the
Wikipedia article on spice (the vegetative substance described at
http://en.wikipedia.org/wiki/Spice, not the rock group).

On the one hand spices, as the latter article points out, "have been
prominent in human history virtually since their inception.  Spices were
among the most valuable items of trade in the ancient and medieval world."

On the other hand what restaurant serves only spices on its menu?
Cuisine is a complex art for which spices are merely a valuable adjunct
that can make a big difference in a catalytic kind of way.

Parsimony is the catalytic converter of mathematics.  It is not the main
engine, but can be helpful in cleaning up the noxious byproducts of
inefficient thinking.

Too much however can be a bad thing: overdoing parsimony undermines its
efficacy for mathematics while adding to the cost, just as overdoing
spices does for food and platinum for catalytic converters.

Rossberg's suggestion that modern physics describes only the wave
function of the universe illustrates this nicely.  If this were really
true, physics would not be a degree major, let alone a career option,
but merely a module of a course in some other major.

In any event it is contradicted by the standard model Rossberg refers to
in the next paragraph.  Explaining the standard model by a suitably
parsimonious Theory of Everything is a nice thought, like an antigravity
belt when you're stuck in traffic, but the standard model is a complex
and evolving account of how the huge zoo of particles fits together.
"Parsimony" in any account of the standard model today is only
accomplished by leaving things out.  The Particle Physics Booklet
(formerly the Particle Properties Data Booklet) is some 200 pages of
densely packed information about uncountably many particles parametrized
by nearly a score of fundamental physical constants each determined by
careful measurement.  (The number of particles is uncountable because
many are merely conjectured to exist, although billions of dollars are
being spent today in the expectation of confirming at least some of
those conjectures.  If only the Riemann Hypothesis were so
well-endowed!)  Some idea of the parsimony achieved by the PPB can be
had from its expansion as the Review of Particle Physics, the PPB's
1100-page big brother.

Ironically the parsimony article is considerably less parsimonious than
the spice article.

As a talisman against the off-topic rule, I should relay here an
unverified report from the fourth millennium to the effect that
"categories were prominent in human mathematics virtually since their
inception, and were among the most heavily trafficked items of
metamathematical discourse during the third millennium."  They're a good
investment, I have some in my own kitchen but many on this list who take
their cooking more seriously have invested much more heavily.

Vaughan Pratt