Log24

Monday, September 17, 2012

Pattern Conception

Filed under: General,Geometry — Tags: , , , — m759 @ 10:00 AM

( Continued from yesterday's post FLT )

Context Part I —

"In 1957, George Miller initiated a research programme at Harvard University to investigate rule-learning, in situations where participants are exposed to stimuli generated by rules, but are not told about those rules. The research program was designed to understand how, given exposure to some finite subset of stimuli, a participant could 'induce' a set of rules that would allow them to recognize novel members of the broader set. The stimuli in question could be meaningless strings of letters, spoken syllables or other sounds, or structured images. Conceived broadly, the project was a seminal first attempt to understand how observers, exposed to a set of stimuli, could come up with a set of principles, patterns, rules or hypotheses that generalized over their observations. Such abstract principles, patterns, rules or hypotheses then allow the observer to recognize not just the previously seen stimuli, but a wide range of other stimuli consistent with them. Miller termed this approach 'pattern conception ' (as opposed to 'pattern perception'), because the abstract patterns in question were too abstract to be 'truly perceptual.'….

…. the 'grammatical rules' in such a system are drawn from the discipline of formal language theory  (FLT)…."

— W. Tecumseh Fitch, Angela D. Friederici, and Peter Hagoort, "Pattern Perception and Computational Complexity: Introduction to the Special Issue," Phil. Trans. R. Soc. B  (2012) 367, 1925-1932 

Context Part II —

IMAGE- Wikipedia article 'Formal language'

Context Part III —

A four-color theorem describes the mathematics of
general  structures, not just symbol-strings, formed from
four kinds of things— for instance, from the four elements
of the finite Galois field GF(4), or the four bases of DNA.

Context Part IV —

A quotation from William P. Thurston, a mathematician
who died on Aug. 21, 2012—

"It may sound almost circular to say that
what mathematicians are accomplishing
is to advance human understanding of mathematics.
I will not try to resolve this
by discussing what mathematics is,
because it would take us far afield.
Mathematicians generally feel that they know
what mathematics is, but find it difficult
to give a good direct definition.
It is interesting to try. For me,
'the theory of formal patterns'
has come the closest, but to discuss this
would be a whole essay in itself."

Related material from a literate source—

"So we moved, and they, in a formal pattern"

Formal Patterns—

Not formal language theory  but rather
finite projective geometry  provides a graphic grammar
of abstract design

IMAGE- Harvard Crimson ad, Easter Sunday, 2008: 'Finite projective geometry as a graphic grammar of abstract design'

See also, elsewhere in this journal,
Crimson Easter Egg and Formal Pattern.

Sunday, September 16, 2012

FLT

Filed under: General,Geometry — Tags: , , — m759 @ 8:28 PM

The "FLT" of the above title is not Fermat's Last Theorem,
but Formal Language Theory (see image below).

In memory of George A. Miller, Harvard cognitive psychologist, who
reportedly died at 92 on July 22, 2012, the first page of a tribute
published  shortly before his death

IMAGE- Design and Formal Language Theory

The complete introduction is available online. It ends by saying—

"In conclusion, the research discussed in this issue
breathes new life into a set of issues that were raised,
but never resolved, by Miller 60 years ago…."

Related material: Symmetry and Hierarchy (a post of 9/11), and
Notes on Groups and Geometry, 1978-1986 .

Happy Rosh Hashanah.

Tuesday, September 11, 2012

Symmetry and Hierarchy

Filed under: General,Geometry — Tags: , , — m759 @ 1:00 PM

A followup to Intelligence Test (April 2, 2012).

Philosophical Transactions of the Royal Society
B  (2012) 367, 2007–2022
(theme issue of July 19, 2012

 
Gesche Westphal-Fitch [1], Ludwig Huber [2],
Juan Carlos Gómez [3], and W. Tecumseh Fitch [1]
 
[1]  Department of Cognitive Biology, University of Vienna,
      Althanstrasse 14, 1090 Vienna, Austria
 
[2]  Messerli Research Institute, University of Veterinary Medicine Vienna,
      Medical University of Vienna and University of Vienna,
      Veterinärplatz 1, 1210 Vienna, Austria
 
[3]  School of Psychology, St Mary’s College, University of St Andrews,
      South Street, St Andrews, KY16 9JP, UK
 
Excerpt (added in an update on Dec. 8, 2012) —
 
 
Conclusion —
 
"…  We believe that applying the theoretical
framework of formal language theory to two-dimensional
patterns offers a rich new perspective on the
human capacity for producing regular, hierarchically
organized structures. Such visual patterns may actually
prove more flexible than music or language for probing
the full extent of human pattern processing abilities.
      With the results presented here, we have taken the
first steps in decoding the uniquely human fascination
with visual patterns, what Gombrich termed our
‘sense of order’.
      Although the patterns we studied are most similar
to tilings or mosaics, they are examples of a much
broader type of abstract plane pattern, a type found
in virtually all of the world’s cultures [4]. Given that
such abstract visual patterns seem to represent
human universals, they have received astonishingly
little attention from psychologists. This neglect is particularly
unfortunate given their democratic nature,
their popular appeal and the ease with which they
can be generated and analysed in the laboratory.
With the current research, we hope to spark renewed
scientific interest in these ‘unregarded arts’, which
we believe have much to teach us about the nature of
the human mind."
 
[4]  Washburn, D. K. & Crowe, D. W.,1988
      Symmetries of Culture
      Theory and Practice of Plane Pattern Analysis
.
      Seattle, WA: University of Washington Press.
 
Commentary —
 
For hierarchy , see my assessment of Gombrich.
For culture , see T. S. Eliot and Russell Kirk on Eliot.

Thursday, February 18, 2010

Theories: An Outline

Filed under: General,Geometry — Tags: , — m759 @ 10:31 AM

Truth, Geometry, Algebra

The following notes are related to A Simple Reflection Group of Order 168.

1. According to H.S.M. Coxeter and Richard J. Trudeau

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”

— Coxeter, 1987, introduction to Trudeau’s The Non-Euclidean Revolution

1.1 Trudeau’s Diamond Theory of Truth

1.2 Trudeau’s Story Theory of Truth

2. According to Alexandre Borovik and Steven H. Cullinane

2.1 Coxeter Theory according to Borovik

2.1.1 The Geometry–

Mirror Systems in Coxeter Theory

2.1.2 The Algebra–

Coxeter Languages in Coxeter Theory

2.2 Diamond Theory according to Cullinane

2.2.1 The Geometry–

Examples: Eightfold Cube and Solomon’s Cube

2.2.2 The Algebra–

Examples: Cullinane and (rather indirectly related) Gerhard Grams

Summary of the story thus far:

Diamond theory and Coxeter theory are to some extent analogous– both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side.  Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+”characteristic two” (or “2“) and for generators+relations+”GF(2)”. (This last search is the source of the Grams reference in 2.2.2 above.)

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