“Excluded Middle” « Search Results

Thursday, October 10, 2013

Nobel Jam

Filed under: General — Tags: Continued, Cube School — m759 @ 7:00 pm

From this date five years ago in The Guardian—

Alice Munro: An Appreciation by Margaret Atwood—

"The central Christian tenet is that
two disparate and mutually exclusive elements—
divinity and humanity— got jammed together
in Christ, neither annihilating the other.
The result was not a demi-god, or a God
in disguise: God became totally a human being
while remaining at the same time totally divine.
To believe either that Christ was only a man or
that he was simply God was declared heretical
by the early Christian church. Christianity thus
depends on a denial of either/or classifying logic
and an acceptance of both-at-once mystery.
Logic says that A cannot be both itself and non-A
at the same time; Christianity says it can. The
formulation 'A but also non-A' is indispensable to it."

Related literary material— "Excluded Middle" and "Couple of Tots."

See also "The Divided Cube" and "Mimsy Were the Borogoves."

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Tuesday, May 22, 2012

Included Middle

Filed under: General,Geometry — m759 @ 2:01 pm

Wikipedia—

"In logic, the law of excluded middle (or the principle of excluded middle) is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is.

The law is also known as the law (or principle) of the excluded third (or of the excluded middle), or, in Latin, principium tertii exclusi. Yet another Latin designation for this law is tertium non datur: 'no third (possibility) is given.'"

"Clowns to the left of me, jokers to the right"

— Songwriter who died on January 4, 2011.

Online NY Times on the date of the songwriter's death—

"A version of this review appeared in print
on January 4, 2011, on page C6 of the New York edition."

REVIEW

"The philosopher Hubert Dreyfus and his former student
Sean Dorrance Kelly have a story to tell, and it is not
a pretty tale for us moderns. Ours is an age of nihilism,
they say, meaning not so much that we have nothing
in which to believe, but that we don’t know how to choose
among the various things to which we might commit
ourselves. Looking down from their perches at Berkeley
and Harvard, they see the 'human indecision that
plagues us all.'"

For an application of the excluded-middle law, see
Non-Euclidean Blocks and Deep Play.

Violators of the law may have trouble* distinguishing
between "Euclidean" and "non-Euclidean" phenomena
because their definition of the latter is too narrow,
based only on examples that are historically well known.

See the Non-Euclidean Blocks footnote.

* Followers of the excluded-middle law will avoid such
trouble by noting that "non-Euclidean" should mean
simply "not Euclidean in some way "— not necessarily
in a way contradicting Euclid's parallel postulate.

But see Wikipedia's defense of the standard, illogical,
usage of the phrase "non-Euclidean."

Postscript—

Tertium Datur

"Here I am, stuck in the middle with you."

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Tuesday, January 17, 2012

Manning and Khora

Filed under: General — m759 @ 1:26 am

A weblog post from Saturday, Jan. 14, 2012—

"Today is the 120th anniversary of Cardinal Henry Edward Manning's death."

— A Reluctant Sinner (Thanks to Andrew Cusack for the link.)

If Manning is a saint, then Saturday was his feast day.

Some background— Manning in this journal.

See also Saturday's Derrida at Villanova. The link there to
previous posts on that topic leads to a post on Derrida's promotion
of his neologism différance as a version of Plato's khôra.

I prefer Manning's discussion of a closely related concept,
the scholastic philosophers' materia prima .

See Hugh R. King's 1956 paper sneering at the scholastics'
concept, and Heisenberg's much better-informed remarks
on the related concept of potentia —

For a related fictional account of a religious quest for "possibilities"
and "excluded middles" between "zeroes and ones," see
Ingraffia on The Crying of Lot 49 .

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Wednesday, November 17, 2010

Church Narrative

Filed under: General,Geometry — Tags: The Horcrux Narrative — m759 @ 2:22 am

Thanks to David Lavery for the following dialogue on the word "narrative" in politics—

"It's like – does this fit into narrative?
It's like, wait, wait, what about a platform? What about, like, ideas?
What about, you know, these truths we hold to be self-evident?
No, it's the narrative."

"Is narrative a fancy word for spin?"

Related material —

Church Logic (Log24, October 29) —

What sort of geometry
is the following?

"What about, you know, these truths we hold to be self-evident?"

Some background from Cambridge University Press in 1976 —

Commentary —

The Church Logic post argues that Cameron's implicit definition of "non-Euclidean" is incorrect.

The four-point, six-line geometry has as lines "all subsets of the point set" which have cardinality 2.

It clearly satisfies Euclid's parallel postulate. Is it, then, not non-Euclidean?

That would, according to the principle of the excluded middle (cf. Church), make it Euclidean.

A definition from Wikipedia that is still essentially the same as it was when written on July 14, 2003—

"Finite geometry describes any geometric system that has only a finite number of points. Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points…."

This definition would seem to imply that a finite geometry (such as the four-point geometry above) should be called non -Euclidean whether or not it violates Euclid's parallel postulate. (The definition's author, unlike many at Wikipedia, is not anonymous.)

Friday, October 29, 2010

Church Logic

Filed under: General,Geometry — m759 @ 1:23 pm

"The law of excluded middle is the logical principle in
accordance with which every proposition is either true or
false. This principle is used, in particular, whenever a proof
is made by the method of reductio ad absurdum . And it is
this principle, also, which enables us to say that the denial of
the denial of a proposition is equivalent to the assertion of
the proposition."

— Alonzo Church, "On the Law of Excluded Middle,"
Bulletin of the American Mathematical Society ,
Vol. 34, No. 1 (Jan.–Feb. 1928), pp. 75–78

It seems reasonable to define a Euclidean geometry as one describing what mathematicians now call a Euclidean space.

What sort of geometry
is the following?

   Four points and six lines,
   with parallel lines indicated
   by being colored alike.

Consider the proposition "The finite geometry with four points and six lines is non-Euclidean."
Consider its negation. Absurd? Of course.

"Non-Euclidean," therefore, does not apply only to geometries that violate Euclid's parallel postulate.

The problem here is not with geometry, but with writings about geometry.

A pop-science website —

"In the plainest terms, non-Euclidean geometry
took something that was rather simple and straightforward
(Euclidean geometry) and made it endlessly more difficult."

Had the Greeks investigated finite geometry before Euclid came along, the reverse would be true.

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