French mathematician Jacques Hadamard once wrote that a sense of beauty is almost the only useful drive for discovery in mathematics.
Now, Rolf Reber, Morten Brun, and Karoline Mitterndorfer of the
University of Bergen, in Norway, claim to have evidence that beauty
does indeed lead to truth in such contexts.

Their findings appear in the paper “The Use of Heuristics in Intuitive
Mathematical Judgment,” published in the December Psychonomic Bulletin
& Review.

The team was inspired by much anecdotal evidence that points to the
use of beauty as an indication of the validity of a solution. In two
experiments, students without training in mathematics were presented
with pairs of arrays of dots and asked to quickly determine whether
the given totals were correct. Half of the additions were based on
symmetric dot patterns, and the other half on asymmetric patterns.

The researchers found that participants were more likely to judge
symmetric additions than asymmetric additions to be correct, even when
the additions were, in fact, incorrect.

“Speeded decisions about the correctness of these equations led to
higher endorsements for both correct and incorrect equations when the
addend and sum dot patterns were symmetrical,” the researchers
reported.

They related their findings to a theory known as “processing fluency,”
which focuses on the experienced ease with which mental content is
processed. The researchers argue that such fluency may come from
familiarity with problems or the attributes of a task and leads to an
increase in intuitively judged truth.

Source: University of Bergen, Nov. 24, 2008.

“Beauty is truth, truth beauty,” - that is all
Ye know on earth, and all ye need to know.

“Ode on a Grecian Urn”, lines 49-50, John Keats [1795–1821].

Written in 1819, ‘Ode on a Grecian Urn’ was the third of the five
‘great odes’ of 1819. These are the most discussed two lines in all of
Keats’s poetry. The exact meaning of those lines is disputed by
everyone; no less a critic than T.S. Eliot considered them a blight
upon an otherwise beautiful poem. Scholars have been unable even to
agree to whom the last thirteen lines of the poem are addressed. There
is further confusion due to the change in quotation marks between the
original manuscript copy of the ode and the 1820 published edition.
Thomas Stearns Eliot, [26 September 1888 – 4 January 1965], was a an
important poet, dramatist, and literary critic, to me and to the
English-speaking world. He received the Nobel Prize in Literature in
1948 and, though I deplore his slight antisemistism, and unkindness to
his mad first wife, I discussed him with his beloved second wife, and
have published works about him, including the verse play “John Lennon
Meets T.,S. Eliot, published in the anthology 13 Rock Fantasies, in
Germany.

In 1997, Dennis R. Dean published an article in the Philological
Quarterly titled ‘Some Quotations in Keats’s Poetry’. In it, he
discussed the problem of the final quotation, linking it with the work
of Sir Joshua Reynolds. Reynolds was an English Rococo Era Painter,
[16 July 1723 – 23 February 1792] and probably the most important and
influential of 18th century English painters, specializing in
portraits and promoting the “Grand Style” in painting which depended
on idealization of the imperfect. He was one of the founders and first
President of the Royal Academy. George III appreciated his merits and
knighted him in 1769. But Reynolds was attacked by many of the
Pre-Raphaelites, and William Blake, the latter having published his
vitriolic “Annotations to Sir Joshua Reynolds’ Discourses” in 1808.
This is important to me, as an admirer of Blake, and (more subtly)
that I have discussed with George Wald (former Chairman of Biology at
Harvard) and Allen Ginsberg [3 June 1926 – 5 April 1997] in a long
conversation of which I alone survive to tell of it, backstage at a
concert by Pete Seeger. George Wald, 1906-1997, was the Higgins
Professor of Biology at Harvard University from 1968 to 1977, a
Nobel-Prize winning biologist, and a promoter of progressive political
and social causes.

The crux of my contribution to what was mostly a dialogue between Wald
as Scientist and Ginsberg as Artist on the relationship between their
professions.

Shakespeare, other poets and other literary figures were grappling in
their own ways with the Big Questions. Science has developed into an
alternative approach. William Blake made his own etchings, by his own
invented technology, to illustrate his own quirky take on cosmology
and other weighty issues. Blake considered himself radically opposed
to Isaac Newton [4 January 1643 – 31 March 1727], even though both
Blake and Newton were influenced by a common metaphysical thinker,
Jakob Boehm [1575 — 21 Nov 1624] mystic and theosophist who founded
modern theosophy; and influenced others such as George Fox
[1575-1624].

Some scholars think that Dennis R. Dean’s attribution to the epigram
from Reynolds in the Keats couplet reasonably settles the ‘quotation
issue’:

Is there in truth no beauty?
[George Herbert, "Jordan"]

“John Keats often used the rhetorical device of quotation in his
poetry. He did so in a variety of ways and sometimes with unclear
directions to his reader. His aberrant use of quotation marks has, in
particular, created editorial problems. In this exploratory essay, I
review the several mechanisms of quotation used by Keats and then
discuss two particularly well known examples—’pure serene’ in the
Chapman sonnet and the Beauty-Truth conclusion to ‘Ode on a Grecian
Urn,’ suggesting sources and appropriate punctuation for both. If I am
right about the latter, especially, then a long-standing textual crux
may at last be resolved.”

“In his “Ode on a Grecian Urn” Keats will say exactly the same thing,
more elegantly but more cryptically also: “Beauty is truth, truth
beauty”—which some English professors have called “surely the most
famous equation in English literature and precisely correct in
suggesting the Newtonian origin of the unstated ‘proof.’”

“The urn, in other words, begins by quoting Sir Joshua (for Keats and
his readers, the world’s greatest authority on art of all kinds),
implicitly affirms the sufficiency of human intellect, explicitly
affirms the equation of beauty and truth, and pronounces this
knowledge entirely sufficient to create the elegant geometry of such
superb art as the urn. Because of the uniformity of human minds and
passions, moreover, the figures inscribed on the urn (which puzzle the
observer at first glance) become intelligible as we relate them to our
own experience. The first stanza of the poem is filled with questions;
the last, with none. Being art, the urn retains its ability to ’speak’
to all who observe it, reminding us of our paradoxical dilemma as
mortals who exist in finite time.”
['Some Quotations in Keats's Poetry' by Dennis R. Dean. From the
Philological Quarterly. Volume: 76. Issue: 1, 1997.]

Oh yes, we do indeed exist in finite time, yet my life as
Mathematician and artist is deeply connected to Infinity.

So here’s what these lines mean to me as a professional Mathematician,
Scientist, Poet, and Teacher.

Everything that I tell my students is the Truth. I tell them so. I
want them to respond as complete human beings, aware of beauty and
ugliness in the world and within themselves. That gives me a chance,
(again, regardless of the content area) to discuss with them what
“Beauty” and “Truth” are, to me, and to them.

How do we reach the highest level in Bloom’s Taxonomy (of pedagogical theory), synthesizing and judging the major players in what C. P. Snow famously (and I think incorrectly) identified as “The Two Cultures?” How can I teach both Newton and Keats, and make them part of the same tapestry, to my students?

My mentor’s mentor’s mentor Albert Einstein, when pressed on the
subject, would say that he believed in the God of Spinoza, that is,
that all matter, energy, time, space existed “in the mind of God.” The
phrase “the mind of God” was used by Hawking and others since, to
indicate what some Physicists think that they are trying, by
mathematico-scientific means, to read from what Galileo called “the
Book of Nature.” But in the secular world (including my classroom),
how can we find a Humanist framework in which to address and
appreciate both art and science?

I have come to believe, from many sources that there are are least 5
kinds of “truth” that each have their own notions of “proof”, of
deduction, of evidence, of social protocol.

(1) Axiomatic Truth, the beating heart of pure Mathematics, from
Euclid on. Given a set of axioms, and rules of deduction, and two
people can sit down together or apart and prove the same truths or
disprove the same falsehoods, up to the limits described by Godel,
Turing, Church, Post [Emil Post, famous Logician, not a relative], et
al. — but that is not the Physical world.

(2) Empirical Truth, from the Scientific methods, and, more recently,
from Experimental Mathematics a la Borwein et al. That is, an evolved
articulation of trial and error, with open publication and peer
review, with a standard of independent verifiability in diverse
laboratories.

(3) Legal-political Truth. If a jury declares O.J. Simpson “not
guilty” of murder, then he is, by law, not guilty of that criminal
charge. Another jury may find him guilty of a civil charge of wrongful
death, as did happen. Or of kidnapping and grand larceny of sports memorabilia. If a politician is elected by a plurality, he or she may claim a mandate from the people, and that is a political truth, regardless of circumstance.

(4) Aesthetic Truth. A song is beautiful or ugly to you regardless of
what the composer, singer, or critic says. Same for a painting, a
sculpture, a building, or (to a Mathematician), an equation or a
sequence of integers. Except that one grows and changes over time,
with education and with acculturation and with maturity. What first
seems discord can become beautiful. People stormed out of Beethoven
symphony premiers, or stormed out of art museums, outraged, and we now wonder why.

(5) Revealed or religious or mystical Truth. My mention of Einstein is
about as close as I can come in the classroom to discussing this,
though I can reply to quotes from the scripture of several major
religions with my quotes from the same sources.

No two of these forms of truth are the same, and much agony comes from
the philosophical category error of confusing one with another.
Legislating the value of pi to be 3 or 22/7 (as was alleged for the
Tennessee Legislature). Outlawing an art form. China enforcing laws
about Tibetan reincarnation. Seeking beauty in a test tube, or
equations in prayer (unless you’re Ramaujan).

My students in the urban classroom, as I have seen hundreds of times
in the past year alone, within essentially all the middle schools and
high schools of Pasadena Unified School District, have a cramped,
unhappy, inconsistent, and ignorant conception of “Truth” in all its
complexity.

(1) Axiomatic Truth, and the rest of Mathematics, is something that
see as if underwater and though a cloud of squid ink. Almost all of
them hate Math. Yet I have been able to open the eyes and hearts and
minds of many students in secondary and post-secondary education, and
make Math intelligible to them, often for the first time in their
lives.

(2) Empirical Truth, from the Scientific method, is known poorly to my
students, who have had second-rate Science Teachers (who themselves do
not, by my standards, know Science). Again, I have been able to give
inspiration from glimpses of the splendor of the scientific world, and
elicted child-like delight in my students in subjects with which they
have inherent interest, such as dinosaurs, earthquakes, sunlight,
explosions, microbes, bird flight, and flower colors.

(3) Legal-political Truth is familiar, again in a degraded form, to
many of my urban students. Many have been arrested, many have been
jailed, many are on probation, many have lives blighted by gangs,
drugs, divorce, crime, violence, and death. It is important in the
classroom that these students here my mantra: “I am not a cop; I am
not a snitch; I am not a rat; I am not your boss.” It is important
that I do not act as a Judge; they tend to have a negative view of
judges and the judicial system. Nor do they, not yet of voting age,
have a record of participation in the democratic process, and know
little more than sound bytes on TV about local, state, national, or
international politics. They often do not see this arena as one
dominated by truth, but, rather, a cesspool of lies.

(4) Aesthetic Truth is real to all my students, and very inadequately
addressed by schooling. The are very aware of color and style in
clothing, tattoos, cars, make-up, hairdo, and find the black-and-white
word of the printed page and the Xeroxed handout to be as drab and
inhumane as the colors of the floor and wall of the disintegrating
classroom. Teachers castigate the music that they listen to,
confiscate their iPods and radios, and leave them in angry silence.
They are usually stunned that I can defend the lyrics of Eminem’s rap
songs, and that I have performed Rock music onstage, and wrote lyrics
that were heard on MTV. I fight to bring more beauty into their lies,
and validate their Aesthetic Truth, while leading them to a more
sophisticated context.

(5) Revealed or religious or mystical Truth. We are required by law to
respect the diversity of individual beliefs in this domain, when
teaching (as I do now, in public schools), and to respect the
“separation of church and state” in specified ways.
In conclusion, of this meandering preface, the classroom is, to me and
my students, a human microcosm in which Truth and Beauty are our
shared source of value.

Mathematical work typically speaks for itself, independent of the opinion of others including the big names of the field. No amount of “spinning” is needed and is unlikely to make a result more or less important than it is.

Doing mathematics does not require expensive experiments, which would depend on funding and ultimately the opinion of funding agencies. Anyone can work on an open math problem without the approval of anyone else. And credit is given solely to the one(s) who did the work, with no pressure to include the names of “principal investigator”, department head, colleagues, what-have-you.

Most important of all, mathematics is relevant to reality, not an escape from it. As the great mathematician Gian-Carlo Rota wisely replied when asked why there were so few women in math,

“Women are more realistic than men — they can see that it’s a flight from reality. What they don’t see is that it’s a flight from reality that works.”

John: I have a reply to your challenge up at epsilonica now.

At least one pure mathematician I’ve known has an unhealthy tendency of asceticism for the sake of abstraction.