A related passage —

An image that led off the year-end review yesterday in
the weblog of British combinatorialist Peter J. Cameron:

See also this  weblog's post final post of 2014,
with a rectangular array illustrating the six faces
of a die, and Cameron's reference yesterday to
a die-related post…

"The things on my blog that seem to be
of continuing value are the expository
series like the one on the symmetric group
(the third post in this series was reblogged
by Gil Kalai last month, which gave it a new
lease of life)…."

A tale from an author of Prague:

See also a passage quoted in this  weblog on the original
date of Cameron's Prague image, July 26, 2014 —

"The philosopher Graham Harman is invested in
re-thinking the autonomy of objects and is part 
of a movement called Object-Oriented-Philosophy
(OOP)." — From “The Action of Things,” a 2011
M.A. thesis at the Center for Curatorial Studies,
Bard College, by Manuela Moscoso 

— in the context of a search here for the phrase
     "structure of the object." An image from that search:

From an obituary for a Kennedy advisor
who reportedly died at 94 on February 23, 2014*—

“He favored withdrawing rural civilians
into what he called ‘strategic hamlets’
and spraying defoliants to cut off
the enemy’s food supply.”

Other rhetoric:  Hamlet and Infinite Space  in this journal,
as well as King of Infinite Space , Part I and Part II.

These “King” links, to remarks on Coxeter  and  Saniga ,
are about two human beings to whom Hamlet’s
phrase “king of infinite space” has been applied.

The phrase would, of course, be more accurately
applied to God.

* The date of the ‘God’s Architecture’ sermon
at Princeton discussed in this afternoon’s post.

Edward Frenkel on Eichler's reciprocity law
(Love and Math , Kindle edition of 2013-10-01,
page 88, location 1812)—

"It seems nearly unbelievable that there
would be a rule generating these numbers.
And yet, German mathematician Martin
Eichler discovered one in 1954.¹¹ "

"11.   I follow the presentation of this result
given in Richard Taylor, Modular arithmetic:
driven by inherent beauty and human
curiosity , The Letter of the Institute for
Advanced Study [IAS], Summer 2012,
pp. 6– 8. I thank Ken Ribet for useful
comments. According to André Weil’s book 
Dirichlet Series and Automorphic Forms ,
Springer-Verlag, 1971 [pp. 143-144], the
cubic equation we are discussing in this
chapter was introduced by John Tate,
following Robert Fricke."

Richard Taylor, op. cit. —

With this account and its context, Taylor has
perhaps atoned for his ridiculous remarks
quoted at Log24 in The Proof and the Lie.

The Cube Model and Peano Arithmetic

The eightfold cube  model of the Fano plane may or may not have influenced a new paper (with the date Feb. 10, 2011, in its URL) on an attempted consistency proof of Peano arithmetic—

The Consistency of Arithmetic, by Storrs McCall

"Is Peano arithmetic (PA) consistent?  This paper contains a proof that it is. …

Axiomatic proofs we may categorize as 'syntactic', meaning that they concern only symbols and the derivation of one string of symbols from another, according to set rules.  'Semantic' proofs, on the other hand, differ from syntactic proofs in being based not only on symbols but on a non-symbolic, non-linguistic component, a domain of objects.    If the sole paradigm of 'proof ' in mathematics is 'axiomatic proof ', in which to prove a formula means to deduce it from axioms using specified rules of inference, then Gödel indeed appears to have had the last word on the question of PA-consistency.  But in addition to axiomatic proofs there is another kind of proof.   In this paper I give a proof of PA's consistency based on a formal semantics for PA.   To my knowledge, no semantic consistency proof of Peano arithmetic has yet been constructed.

The difference between 'semantic' and 'syntactic' theories is described by van Fraassen in his book The Scientific Image :

"The syntactic picture of a theory identifies it with a body of theorems, stated in one particular language chosen for the expression of that theory.  This should be contrasted with the alternative of presenting a theory in the first instance by identifying a class of structures as its models.  In this second, semantic, approach the language used to express the theory is neither basic nor unique; the same class of structures could well be described in radically different ways, each with its own limitations.  The models occupy centre stage." (1980, p. 44)

Van Fraassen gives the example on p. 42 of a consistency proof in formal geometry that is based on a non-linguistic model.  Suppose we wish to prove the consistency of the following geometric axioms:

A1.  For any two lines, there is at most one point that lies on both.
A2.  For any two points, there is exactly one line that lies on both.
A3.  On every line there lie at least two points.

The following diagram shows the axioms to be consistent:

The consistency proof is not a 'syntactic' one, in which the consistency of A1-A3 is derived as a theorem of a deductive system, but is based on a non-linguistic structure.  It is a semantic as opposed to a syntactic proof.  The proof constructed in this paper, like van Fraassen's, is based on a non-linguistic component, not a diagram in this case but a physical domain of three-dimensional cube-shaped blocks. ….

… The semantics presented in this paper I call 'block semantics', for reasons that will become clear….  Block semantics is based on domains consisting of cube-shaped objects of the same size, e.g. children's wooden building blocks.  These can be arranged either in a linear array or in a rectangular array, i.e. either in a row with no space between the blocks, or in a rectangle composed of rows and columns.  A linear array can consist of a single block, and the order of individual blocks in a linear or rectangular array is irrelevant. Given three blocks A, B and C, the linear arrays ABC and BCA are indistinguishable.  Two linear arrays can be joined together or concatenated into a single linear array, and a rectangle can be re-arranged or transformed into a linear array by successive concatenation of its rows.  The result is called the 'linear transformation' of the rectangle.  An essential characteristic of block semantics is that every domain of every block model is finite.  In this respect it differs from Tarski’s semantics for first-order logic, which permits infinite domains.  But although every block model is finite, there is no upper limit to the number of such models, nor to the size of their domains.

It should be emphasized that block models are physical models, the elements of which can be physically manipulated.  Their manipulation differs in obvious and fundamental ways from the manipulation of symbols in formal axiomatic systems and in mathematics.  For example the transformations described above, in which two linear arrays are joined together to form one array, or a rectangle of blocks is re-assembled into a linear array, are physical transformations not symbolic transformations. …" 

— Storrs McCall, Department of Philosophy, McGill University

See also…

• Finite Geometry and Physical Space,
• McCall in the book search of this morning's Salvation for Gopnik, and
• the post Declarations of Jan. 29, 2012.

Peter J. Cameron yesterday on Galois—

"He was killed in a duel at the age of 20…. His work languished for another 14 years until Liouville published it in his Journal; soon it was recognised as the foundation stone of modern algebra, a position it has never lost."

Here Cameron is discussing Galois theory, a part of algebra. Galois is known also as the founder* of group theory, a more general subject.

Group theory is an essential part of modern geometry as well as of modern algebra—

"In der Galois'schen Theorie, wie hier, concentrirt sich das Interesse auf Gruppen von Änderungen. Die Objecte, auf welche sich die Änderungen beziehen, sind allerdings verschieden; man hat es dort mit einer endlichen Zahl discreter Elemente, hier mit der unendlichen Zahl von Elementen einer stetigen Mannigfaltigkeit zu thun."

— Felix Christian Klein, Erlanger Programm , 1872

("In the Galois theory, as in ours, the interest centres on groups of transformations. The objects to which the transformations are applied are indeed different; there we have to do with a finite number of discrete elements, here with the infinite number of elements in a continuous manifoldness." (Translated by M.W. Haskell, published in Bull. New York Math. Soc. 2, (1892-1893), 215-249))

Related material from Hermann Weyl, Symmetry , Princeton University Press, 1952 (paperback reprint of 1982, pp. 143-144)—

"A field is perhaps the simplest algebraic structure we can invent. Its elements are numbers…. Space is another example of an entity endowed with a structure. Here the elements are points…. What we learn from our whole discussion and what has indeed become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity  Σ try to determine is group of automorphisms , the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of Σ in this way."

For a simple example of a group acting on a field (of 8 elements) that is also a space (of 8 points), see Generating the Octad Generator and Knight Moves.

* Joseph J. Rotman, An Introduction to the Theory of Groups , 4th ed., Springer, 1994, page 2

Conclusion of "The Place of Pure Mathematics" —

"Dogmas and philosophies, it would seem, rise and fall. But gradually accumulating throughout the ages, from the earliest dawn of history, there is a body of doctrine, a reasoned insight into the relations of exact ideas, painfully won and often tested. And this remains the main heritage of man; his little beacon of light amidst the solitudes and darknesses of infinite space; or, if you prefer, like the shout of children at play together in the cultivated valleys, which continues from generation to generation.

Yes, and continues for ever! A universe which has the potentiality of becoming thus conscious of itself is not without something of which that which we call memory is but an image. Somewhere, somehow, in ways we dream not of, when you and I have merged again into the illimitable whole, when all that is material has ceased, the faculty in which we now have some share, shall surely endure; the conceptions we now dimly struggle to grasp, the joy we have in the effort, these are but part of a greater whole. Some may fear, and some may hope, that they and theirs shall not endure for ever. But he must have studied Nature in vain who does not see that our spiritual activities are inherent in the mighty process of which we are part; who can doubt of their persistence.

And, on the intellectual side, of all that is best ascertained, and surest, and most definite, of these; of all that is oldest and most universal; of all that is most fundamental and far-reaching, of these activities, Pure Mathematics is the symbol and the sum."

— From a 1913 address by geometry saint Henry Frederick Baker, who died on this date in 1956

The feast of another saint, Patrick, also falls on 3/17.  The date itself is related, if only by chance, to the following remark—

“317 is a prime, not because we think so,
or because our minds are shaped in one way
rather than another, but because it is so,
because mathematical reality is built that way.”

— From a 1940 book by the somewhat less saintly number theorist G. H. Hardy

Indiana Jones and the Magical Oracle

Mathematician Ken Ono in the December 2010 American Mathematical Society Notices—

The "dying genius" here is Ramanujan, not Galois. The story now continues at the AMS website—

      (Excerpt from Jan. 27 screenshot;
      the partitions story has been the top
      news item at the site all week.)

From a Jan. 20, 2011, Emory University press release —
"Finite formula found for partition numbers" —

"We found a function, that we call P, that is like a magical oracle," Ono says. "I can take any number, plug it into P, and instantly calculate the partitions of that number. P does not return gruesome numbers with infinitely many decimal places. It's the finite, algebraic formula that we have all been looking for."

For an introduction to the magical oracle, see a preprint, "Bruinier-Ono," at the American Institute of Mathematics website.

Ono also discussed the oracle in a video (see minute 25) recorded Jan. 21 and placed online today.

See as well "Exact formulas for the partition function?" at mathoverflow.net.

A Nov. 29, 2010, remark by Thomas Bloom on that page leads to a 2006 preprint by Ono and Kathrin Bringmann, "An Arithmetic Formula for the Partition Function*," that seems not unrelated to Ono's new "magical oracle" formula—

Click to enlarge

The Bruinier-Ono paper does not mention the earlier Bringmann-Ono work.

(Both the 2011 Bruinier-Ono paper and the 2006 Bringmann-Ono paper mention their debt to a 2002 work by Zagier—  Don Zagier, "Traces of singular moduli," in Motives, Polylogarithms and Hodge theory, Part II  (Irvine, CA, 1998), International Press Lecture Series 3 (International Press, Somerville, MA, 2002),   pages 211-244.)

Some background for those who prefer mathematics to narrative—
The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series ,
by Ken Ono, American Mathematical Society CBMS Series, 2004.

* Proc. Amer. Math. Soc. 135 (2007), 3507-3514.

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.)

See also the rest  of Little Gidding.

Barnes & Noble has an informative new review today of the recent Galois book Duel at Dawn.

It begins…

"In 1820, the Hungarian noble Farkas Bolyai wrote an impassioned cautionary letter to his son Janos:

'I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life… It can deprive you of your leisure, your health, your peace of mind, and your entire happiness… I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind. Learn from my example…'

Bolyai wasn't warning his son off gambling, or poetry, or a poorly chosen love affair. He was trying to keep him away from non-Euclidean geometry."

For a less dark view (obtained by simply redefining "non-Euclidean" in a more logical way*) see Non-Euclidean Blocks and Finite Geometry and Physical Space.

* Finite  geometry is not  Euclidean geometry— and is, therefore, non-Euclidean
  in the strictest sense (though not according to popular usage), simply because
  Euclidean  geometry has infinitely many points, and a finite  geometry does not.
  (This more logical definition of "non-Euclidean" seems to be shared by
  at least one other person.)

  And some  finite geometries are non-Euclidean in the popular-usage sense,
  related to Euclid's parallel postulate.

  The seven-point Fano plane has, for instance, been called
  "a non-Euclidean geometry" not because it is finite
  (though that reason would suffice), but because it has no parallel lines.

  (See the finite geometry page at the Centre for the Mathematics
   of Symmetry and Computation at the University of Western Australia.)

Where Entertainment is God, continued—

The following paragraphs are from a review by Piotr Siemion of Infinite Jest, a novel by David Foster Wallace. Illustrations have been added.

"Wallace was somehow able to twist together three yarns…. …there's a J.D Salinger for those who like J.D. Salinger. There's William Burroughs for those hardy souls who like some kick in their prose. And there's a dash of Kurt Vonnegut too. All three voices, though, are amplified in Infinite Jest beyond mere distortion and then projected onto Wallace's peculiar own three-ring circus….

… there's entertainment. Make it a capital E.

Illustration by Clint Eastwood
from Log24 post "E is for Everlast"

Infinite Jest revolves, among its many gyrations, around the story of the Entertainment, a film-like creation going by the title of 'Infinite Jest' and created shortly before his suicidal death by the young tennis star's father. The Entertainment's copies are now being disseminated clandestinely all over Wallace's funny America. Problem is, of course, that the film is too good. Anybody who gets to watch it becomes hooked instantly and craves only to watch it again, and again, and again, until the audience drops dead of exhaustion and hunger. Why eat when you're entertained by such a good movie? Wallace's premise brings you back to that apocryphal lab experiment in which rats were treated to a similar choice. When the rat pushed one button, marked FOOD, it would get a food pellet. The other button, marked FUN, would fire up an electrode rigged right into the orgasm center somewhere in the rat's cortex. Needless to add, one rat after another would drop dead from hunger, still twitching luridly and trying to finesse one last push of the button. Same thing in Wallace's story, especially that even those characters who have not seen the Entertainment yet, keep on entertaining themselves by different means."

The title of the Entertainment, "Infinite Jest," might also be applied to a BBC program featuring mathematician Peter J. Cameron. The program's actual title was "To Infinity and Beyond." It was broadcast the night of Feb. 10 (the date of this journal's previous post).

Few, however, are likely to find the Infinity program addictive. For closer approaches to Wallace's ideal Entertainment, see instead Dante (in the context of this journal's Feb. 4 posts on Cameron and the afterlife) and the BBC News.

The Galois Quaternion

Related material:
The Galois Quaternion

Click for context.
(See also Nativity and the end
of this morning's post.)

Wikipedia:

“In the Roman Catholic tradition, the term ‘Body of Christ’ refers not only to the body of Christ in the spiritual realm, but also to two distinct though related things: the Church and the reality of the transubstantiated bread of the Eucharist….

According to the Catechism of the Catholic Church, ‘the comparison of the Church with the body casts light on the intimate bond between Christ and his Church. Not only is she gathered around him; she is united in him, in his body….’

….To distinguish the Body of Christ in this sense from his physical body, the term ‘Mystical Body of Christ’ is often used. This term was used as the first words, and so as the title, of the encyclical Mystici Corporis Christi of Pope Pius XII.”

Pope Pius XII:

“83. The Sacrament of the Eucharist is itself a striking and wonderful figure of the unity of the Church, if we consider how in the bread to be consecrated many grains go to form one whole, and that in it the very Author of supernatural grace is given to us, so that through Him we may receive the spirit of charity in which we are bidden to live now no longer our own life but the life of Christ, and to love the Redeemer Himself in all the members of His social Body.”

Seeing the Finite Structure

The following supplies some context for remarks of Halmos on combinatorics.

From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991–

Interviews with Halmos, “Paul Halmos by Parts,” by Donald J. Albers–

“Part II: In Touch with God*“– on pp. 27-28:

The Root of All Deep Mathematics

“Albers. In the conclusion of ‘Fifty Years of Linear Algebra,’ you wrote: ‘I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight… I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need– the pendulum needs– to swing back, and will swing back in the combinatorial direction.’ I always thought of you as an analyst.

Halmos: People call me an analyst, but I think I’m a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.

Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the ‘somehow’ would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions– that’s silly. The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.

Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it’s sometimes deep, but it is nowhere near as fundamental as seeing the finite structure.”

Walsh Series: An Introduction
to Dyadic Harmonic Analysis,
by F. Schipp et al.,
Taylor & Francis, 1990

Halmos’s above remarks on combinatorics as a source of “deep mathematics” were in the context of operator theory. For connections between operator theory and harmonic analysis, see (for instance) H.S. Shapiro, “Operator Theory and Harmonic Analysis,” pp. 31-56 in Twentieth Century Harmonic Analysis– A Celebration, ed. by J.S. Byrnes, published by Springer, 2001.

Walsh Series states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”

The patterns on the faces of the cube on the cover of Walsh Series above illustrate both the Walsh functions of order 3 and the same structure in a different guise, subspaces of the affine 3-space over the binary field. For a note on the relationship of Walsh functions to finite geometry, see Symmetry of Walsh Functions.

Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find “the right finite question to ask,” I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.

* See the Log24 entries following Halmos’s death.

"I have another far more solid and central ground for submitting to it as a faith, instead of merely picking up hints from it as a scheme. And that is this: that the Christian Church in its practical relation to my soul is a living teacher, not a dead one. It not only certainly taught me yesterday, but will almost certainly teach me to-morrow. Once I saw suddenly the meaning of the shape of the cross; some day I may see suddenly the meaning of the shape of the mitre. One free morning I saw why windows were pointed; some fine morning I may see why priests were shaven. Plato has told you a truth; but Plato is dead. Shakespeare has startled you with an image; but Shakespeare will not startle you with any more. But imagine what it would be to live with such men still living, to know that Plato might break out with an original lecture to-morrow, or that at any moment Shakespeare might shatter everything with a single song. The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast. He is always expecting to see some truth that he has never seen before."

— G. K. Chesterton, Orthodoxy, Ch. IX

From Plato, Pegasus, and the Evening Star (11/11/99):
 

 

"O God, I could be
bounded in a nutshell
and count myself
a king of infinite space,
were it not that
I have bad dreams."
— Hamlet

From today's newspaper:

Notes:

For an illustration of
the phrase "solid and central,"
see the previous entry.

For further context, see the
five Log24 entries ending
on September 6, 2006.

For background on the word
"hollow," see the etymology of
 "hole in the wall" as well as
"The God-Shaped Hole" and
"Is Nothing Sacred?"

For further ado, see
Macbeth, V.v
("signifying nothing")
and The New Yorker,
issue dated tomorrow.

Today is the 21st birthday of my note “The Relativity Problem in Finite Geometry.”

Some relevant quotations:

“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

Describing the branch of mathematics known as Galois theory, Weyl says that it

“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”

— Weyl, Symmetry, Princeton University Press, 1952, p. 138

Weyl’s set Sigma is a finite set of complex numbers.   Some other sets with “discrete and finite character” are those of 4, 8, 16, or 64 points, arranged in squares and cubes.  For illustrations, see Finite Geometry of the Square and Cube.  What Weyl calls “the relativity problem” for these sets involves fixing “objectively” a class of equivalent coordinatizations.  For what Weyl’s “objectively” means, see the article “Symmetry and Symmetry  Breaking,” by Katherine Brading and Elena Castellani, in the Stanford Encyclopedia of Philosophy:

“The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is thus reformulated in the following group-theoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann Weyl (1952, p. 132), ‘objectivity means invariance with respect to the group of automorphisms [of space-time].^‘[22]

22. The significance of the notion of invariance and its group-theoretic treatment for the issue of objectivity is explored in Born (1953), for example. For more recent discussions see Kosso (2003) and Earman (2002, Sections 6 and 7).

References:

Born, M., 1953, “Physical Reality,” Philosophical Quarterly, 3, 139-149. Reprinted in E. Castellani (ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics, Princeton, NJ: Princeton University Press, 1998, pp. 155-167.

Earman, J., 2002, “Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity,’ PSA 2002, Proceedings of the Biennial Meeting of the Philosophy of Science Association 2002, forthcoming [Abstract/Preprint available online]

Kosso, P., 2003, “Symmetry, objectivity, and design,” in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections, Cambridge: Cambridge University Press, pp. 410-421.

Weyl, H., 1952, Symmetry, Princeton, NJ: Princeton University Press.

See also

Archives Henri Poincaré (research unit UMR 7117, at Université Nancy 2, of the CNRS)–

“Minkowski, Mathematicians, and the Mathematical Theory of Relativity,” by Scott Walter, in The Expanding Worlds of General Relativity (Einstein Studies, volume 7), H. Goenner, J. Renn, J. Ritter and T. Sauer, editors, Boston/Basel: Birkhäuser, 1999, pp. 45-86–

“Developing his ideas before Göttingen mathematicians in April 1909, Klein pointed out that the new theory based on the Lorentz group (which he preferred to call ‘Invariantentheorie’) could have come from pure mathematics (1910: 19). He felt that the new theory was anticipated by the ideas on geometry and groups that he had introduced in 1872, otherwise known as the Erlangen program (see Gray 1989: 229).”

References:

Gray, Jeremy J. (1989). Ideas of Space. 2d ed. Oxford: Oxford University Press.

Klein, Felix. (1910). “Über die geometrischen Grundlagen der Lorentzgruppe.” Jahresbericht der deutschen Mathematiker-Vereinigung 19: 281-300. [Reprinted: Physikalische Zeitschrift 12 (1911): 17-27].

Related material: A pathetically garbled version of the above concepts was published in 2001 by Harvard University Press.  See Invariances: The Structure of the Objective World, by Robert Nozick.

Click on picture for details.

Proof 101

From a course description:

“This module aims to introduce the student to rigorous university level mathematics….
    Syllabus: The idea of and need for mathematical statements and proofs…. proof by contradiction… proof by induction…. the infinite number of primes….”

In the December Notices of the American Mathematical Society, Brian (E. B.) Davies, a professor of mathematics at King’s College London, questions the consistency of Peano Arithmetic (PA), which has the following axioms:

From BookRags.com—

Axiom 1. 0 is a number.

Axiom 2. The successor of any number is a number.

Axiom 3. If a and b are numbers and if their successors are equal, then a and b are equal.

Axiom 4. 0 is not the successor of any number.

Axiom 5. If S is a set of numbers containing 0 and if the successor of any number in S is also in S, then S contains all the numbers.

It should be noted that the word “number” as used in the Peano axioms means “non-negative integer.”  The fifth axiom deserves special comment.  It is the first formal statement of what we now call the “induction axiom” or “the principle of mathematical induction.”

Peano’s fifth axiom particularly troubles Davies, who writes elsewhere:

I contend that our understanding of number should be placed in an historical context, and that the number system is a human invention.  Elementary arithmetic enables one to determine the number of primes less than twenty as certainly as anything we know.  On the other hand Peano arithmetic is a formal system, and its internal consistency is not provable, except within set-theoretic contexts which essentially already assume it, in which case their consistency is also not provable.  The proof that there exists an infinite number of primes does not depend upon counting, but upon the law of induction, which is an abstraction from our everyday experience…. 
… Geometry was a well developed mathematical discipline based upon explicit axioms over one and a half millennia before the law of induction was first formulated.  Even today many university students who have been taught the principle of induction prefer to avoid its use, because they do not feel that it is as natural or as certain as a purely algebraic or geometric proof, if they can find one.  The feelings of university students may not settle questions about what is truly fundamental, but they do give some insight into our native intuitions.

— E. B. Davies in
   “Counting in the real world,”
    March 2003 (word format),
    To appear in revised form in
    Brit. J. Phil. Sci. as
   “Some remarks on
    the foundations
    of quantum mechanics”

Exercise:

Discuss Davies’s claim that

The proof that there exists an infinite number of primes does not depend upon counting, but upon the law of induction.

Cite the following passage in your discussion.

It will be clear by now that, if we are to have any chance of making progress, I must produce examples of “real” mathematical theorems, theorems which every mathematician will admit to be first-rate. 

… I can hardly do better than go back to the Greeks.  I will state and prove two of the famous theorems of Greek mathematics.  They are “simple” theorems, simple both in idea and in execution, but there is no doubt at all about their being theorems of the highest class.  Each is as fresh and significant as when it was discovered– two thousand years have not written a wrinkle on either of them.  Finally, both the statements and the proofs can be mastered in an hour by any intelligent reader, however slender his mathematical equipment.

I. The first is Euclid’s proof of the existence of an infinity of prime numbers.

The prime numbers or primes are the numbers

   (A)   2, 3, 5, 7, 11, 13, 17, 19, 23, 29, … 

which cannot be resolved into smaller factors.  Thus 37 and 317 are prime.  The primes are the material out of which all numbers are built up by multiplication: thus

    666 = 2 ^. 3 ^. 3 ^. 37. 

Every number which is not prime itself is divisible by at least one prime (usually, of course, by several).   We have to prove that there are infinitely many primes, i.e. that the series (A) never comes to an end.

Let us suppose that it does, and that

   2, 3, 5, . . . , P
 
is the complete series (so that P is the largest prime); and let us, on this hypothesis, consider the number

   Q = (2 ^. 3 ^. 5 ^{. . . . .} P) + 1.

It is plain that Q is not divisible by any of

   2, 3, 5, …, P;

for it leaves the remainder 1 when divided by any one of these numbers.  But, if not itself prime, it is divisible by some prime, and therefore there is a prime (which may be Q itself) greater than any of them.   This contradicts our hypothesis, that there is no prime greater than P; and therefore this hypothesis is false.

The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons.  It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

— G. H. Hardy,
   A Mathematician’s Apology,
   quoted in the online guide for
   Clear and Simple as the Truth:
   Writing Classic Prose, by
   Francis-Noël Thomas
   and Mark Turner,
   Princeton University Press

In discussing Davies’s claim that the above proof is by induction, you may want to refer to Davies’s statement that

Geometry was a well developed mathematical discipline based upon explicit axioms over one and a half millennia before the law of induction was first formulated

and to Hardy’s statement that the above proof is due to Euclid.

Hobgoblin?

Brian Davies is a professor of mathematics at King’s College London.  In the December Notices of the American Mathematical Society, he claims that arithmetic may, for all we know, be inconsistent:

“It is not possible to prove that Peano arithmetic is consistent”…?!

Where did Gödel say this?  Gödel proved, in fact, according to a well-known mathematician at Princeton, that (letting PA stand for Peano Arithmetic),

Remarkably, even after he has stated correctly Gödel’s result, Nelson, like Davies, concludes that

I prefer the argument that the existence of a model ensures the consistency of a theory.

For instance, the Toronto philosopher William Seager writes that

The relationship between consistency and the existence of a model is brought home by the following weblog entry that neatly summarizes a fallacious argument offered in the AMS Notices by Davies:

An important difference between A1 (the set of axioms of Peano arithmetic) and A2 (a set of axioms that describe a new, unknown, finite simple group) is that A1 is known to have a model (the nonnegative integers) and A2 is not known to have a model.

Therefore, according to Seager’s argument, A1 is consistent and A2 may or may not be consistent.

The degree to which Seager’s argument invokes Platonic realism is debatable.  Less debatable is the quasireligious faith in nominalism proclaimed by Davies and Nelson.  Nelson’s own account of a religious experience in 1976 at Toronto is instructive.

Nelson’s “Mathematics and Faith” was written for the Jubilee for Men and Women from the World of Learning held at the Vatican, 23-24 May 2000.  It concludes with an invocation of St. Paul:

Belief in the consistency of arithmetic may or may not be foolish, and therefore an Emersonian hobgoblin of little minds, but bullshit is bullshit, whether in London, in Princeton, in Toronto, or in Rome.

Holy Geometry

What was “the holy geometry book” (“das heilige Geometrie-Büchlein,” p. 10 in the Schilpp book below) that so impressed the young Albert Einstein?

“At the age of 12 I experienced a second wonder of a totally different nature: in a little book dealing with Euclidian plane geometry, which came into my hands at the beginning of a schoolyear.  Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which– though by no means evident– could nevertheless be proved with such certainty that any doubt appeared to be out of the question.  This lucidity and certainty made an indescribable impression upon me.”

(“Im Alter von 12 Jahren erlebte ich ein zweites Wunder ganz verschiedener Art: An einem Büchlein über Euklidische Geometrie der Ebene, das ich am Anfang eines Schuljahres in die Hand bekam.  Da waren Aussagen wie z.B. das Sich-Schneiden der drei Höhen eines Dreieckes in einem Punkt, die– obwohl an sich keineswegs evident– doch mit solcher Sicherheit bewiesen werden konnten, dass ein Zweifel ausgeschlossen zu sein schien.  Diese Klarheit und Sicherheit machte einen unbeschreiblichen Eindruck auf mich.”)

— Albert Einstein, Autobiographical Notes, pages 8 and 9 in Albert Einstein: Philosopher-Scientist, ed. by Paul A. Schilpp

From a website by Hans-Josef Küpper:

“Today it cannot be said with certainty which book is Einstein’s ‘holy geometry book.’  There are three different titles that come into question:

Theodor Spieker, 1890
Lehrbuch der ebenen Geometrie. Mit Übungsaufgaben für höhere Lehranstalten.

Heinrich Borchert Lübsen, 1870
Ausführliches Lehrbuch der ebenen und sphärischen Trigonometrie. Zum Selbstunterricht. Mit Rücksicht auf die Zwecke des praktischen Lebens.

Adolf Sickenberger, 1888
Leitfaden der elementaren Mathematik.

Young Albert Einstein owned all of these three books. The book by T. Spieker was given to him by Max Talmud (later: Talmey), a Jewish medic. The book by H. B. Lübsen was from the library of his uncle Jakob Einstein and the one of A. Sickenberger was from his parents.”

Küpper does not state clearly his source for the geometry-book information.

According to Banesh Hoffman and Helen Dukas in Albert Einstein, Creator and Rebel, the holy geometry book was Lehrbuch der Geometrie zum Gebrauch an höheren Lehranstalten, by Eduard Heis (Catholic astronomer and textbook writer) and Thomas Joseph Eschweiler.

An argument for Sickenberger from The Young Einstein: The Advent of Relativity (pdf), by Lewis Pyenson, published by Adam Hilger Ltd., 1985:

What might be the modern version of a “holy geometry book”?

Click on picture for details.

Kaleidoscope, continued

From Clifford Geertz, The Cerebral Savage:

"Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately).  And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought.  Both anecdotal and geometric, it builds coherent structures out of 'the odds and ends left over from psychological or historical process.'

These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore….  as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular.   But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort….  Levi-Strauss generalizes this permutational view of thinking to savage thought in general.  It is all a matter of shuffling discrete (and concrete) images–totem animals, sacred colors, wind directions, sun deities, or whatever–so as to produce symbolic structures capable of formulating and communicating objective (which is not to say accurate) analyses of the social and physical worlds.

…. And the point is general.  The relationship between a symbolic structure and its referent, the basis of its meaning,  is fundamentally 'logical,' a coincidence of form– not affective, not historical, not functional.  Savage thought is frozen reason and anthropology is, like music and mathematics, 'one of the few true vocations.'

Or like linguistics."

Edward Sapir on Linguistics, Mathematics, and Music:

"… linguistics has also that profoundly serene and satisfying quality which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms.  Linguistics has neither the sweep nor the instrumental power of mathematics, nor has it the universal aesthetic appeal of music.  But under its crabbed, technical, appearance there lies hidden the same classical spirit, the same freedom in restraint, which animates mathematics and music at their purest."

— Edward Sapir, "The Grammarian and his Language,"
  American Mercury 1:149-155,1924

From Robert de Marrais, Canonical Collage-oscopes:

"…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups.  You know, those things the non-professionals call . . . kaleidoscopes! *  (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism' **— then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)

* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry  (A polytope is an n-dimensional analog of a polygon or polyhedron.  Chapter V of this book is entitled 'The Kaleidoscope'….)

** … contemporary with the Johns Hopkins hatchet job that won him American marketshare, Derrida was also being subjected to a series of probing interviews in Paris by the hometown crowd.  He first gained academic notoriety in France for his book-length reading of Husserl's two-dozen-page essay on 'The Origin of Geometry.'  The interviews were collected under the rubric of Positions (Chicago: U. of Chicago Press, 1981…).  On pp. 34-5 he says the following: 'the resistance to logico-mathematical notation has always been the signature of logocentrism and phonologism in the event to which they have dominated metaphysics and the classical semiological and linguistic projects…. A grammatology that would break with this system of presuppositions, then, must in effect liberate the mathematization of language…. The effective progress of mathematical notation thus goes along with the deconstruction of metaphysics, with the profound renewal of mathematics itself, and the concept of science for which mathematics has always been the model.'  Nice campaign speech, Jacques; but as we'll see, you reneged on your promise not just with the kaleidoscope (and we'll investigate, in depth, the many layers of contradiction and cluelessness you put on display in that disingenuous 'playing to the house'); no, we'll see how, at numerous other critical junctures, you instinctively took the wrong fork in the road whenever mathematical issues arose… henceforth, monsieur, as Joe Louis once said, 'You can run, but you just can't hide.'…."