I was musing on the fact that I have never heard a psychologically plausible account of the appeal of pure mathematics. (I say “pure” mathematics because I suspect pure and applied mathematics have different sources of appeal). By “psychologically plausible”, I mean one grounded on the psychology of individual mathematicians. Lots of mathematicians have written explanations of the appeal, but most of these are either of the form “Because mathematics is awesome.”, or “Because I’m awesome” While mathematics is awesome, and while I’m willing to grant the premise that I’m awesome pretty much any time it comes up, these explanations lack the kind of specificity I have in mind. One common explanation, for example, is that math is like music, which relies on the presupposition that music is intrinsically valuable, and that math has value by analogy. But why do we like music? What in the psychology of mathematicians makes math seem like music to them? These are harder questions than the original one. Another explanation is that math is challenging, which is a subspecies of the “I’m awesome”. But in what way is mathematics challenging to mathematicians? Mathematicians, as a group, do not strive to be Nietzschean superman endlessly trying to overcome their limitations, so why this particular challenge, rather than the Nathan’s hot dog eating contest, or climbing Everest?
There are psychological explanations floating around as stereotypes, most of which are immensely unflattering, but are least examples of the kind of explanation I have in mind. One example is that mathematicians are like the Rain Man in that they just like repetitive tasks like counting or adding. Another example is that mathematicians can’t handle the real world, and so retreat to the safety of the world of numbers. These are both wrong and insulting, but they are at least grounded in the psychology of individual mathematicians. If anyone has a non-wrong explanation, I’d be curious to hear it.
Are you open to a theistic explanation?
Some mathematicians do, of course, climb mountains. I think you’re pretty close to the right explanations in the first paragraph. The answer is because they find it interesting, challenging, and they’re good at it. If they’re good at eating hotdogs or climbing mountains and find those interesting and challenging then they will probably pursue those things as well as pure mathematics. People will pursue things that they aren’t good at just because they find them interesting and challenging only for so long before getting frustrated. I pursued pure mathematics for a while alongside my interest in philosophy. I found out that I was better at philosophy and that’s why I’m finishing a PhD in that rather than math. I’m pretty good at math though, and still find it interesting and challenging, so I read math blogs now and pop open my old Hungerford text from time to time.
But no mathematician has won the Nathan’s hot dog eating contest, right? There’s still that frontier to cross.
Fredrich Nietzsche once wrote, “Out of chaos comes order.” I think that from an evolutionary or Judeo-Christian view we can argue for the value of order over chaos. I think mathematics is the dance between order and chaos and the most beautiful of those dances is when the most chaotic is simplified to small steps of rhythm and order.
Mathematics is playful wondering, not knowing an answer, but rather trying, failing, adapting, and solving… then discovering a simpler path, or a fantastic diversion that brings the same result based upon other truths.
For example:
A 6 × 8 metal plate is resting inside a hemispherical bowl, whose radius is 13. The
plate is parallel to the rim of the bowl, which is parallel to the tabletop on which the bowl
is sitting. How far is it from the plate to the bottom of the bowl? [This problem is taken from the Exeter math curriculum... an amazing resource for excellent problems!]
I love this problem because when I gave it to my PreCalculus students and tried to solve it myself, we derived equations for circles, took tangents of angles, and searched for a geometric solution within Sketchpad… only to discover that the problem can be solved by some nice sketches and the Pythagorean theorem (twice).
Again, my point is that yes, math is awesome, but the awesomeness is a by-product, not the ends in itself. Down deep inside of us, we are curious and the world is full of questions. We want to know, to order, to conquer. It is pleasing to see meaning in the world and while mathematics does not speak to purpose, it does it gives a framework for the pursuit of meaning.
I think the appeal is that a mathematical result lives forever and math satisfies a strong desire to create something that will last.
I study pure mathematics because I find it interesting. I like the mind games.
How about the feeling of mental power that one has, when with a single shift in point of view one swallows several theories in one gulp? Thurston has spoken of this, and speaks on the remarkable mental compression that can take place as a result. I’m certain many mathematicians feel that (and so this is nothing like a case of “I’m awesome” — in some sense this is just an ordinary experience).
Or the sheer pleasure in objective form, as well as pride in personal craftsmanship, when one polishes an argument so that it becomes very elegant and memorable. A pleasure which is again quite commonplace, and not unlike the pleasure a sculptor must feel when she smooths her stone to her sensual delight and satisfaction (and so this type of pleasure is not specific to mathematics, particularly).
There are all sorts of replies one could give. As far as analogies to music is concerned: one could compare the vertiginous feeling one has of the architecture of mathematics, from the most sweeping conception down to the finest nuance, with the grand architecture found in some of the compositions of J.S. Bach, for instance. Or to certain architecture in its common sense, for that matter.
To reduce any of this to merely saying “mathematics is awesome” or “I’m awesome” would seem to me to be pretty wooden and insensitive. But many people have expressed these sorts of things much more eloquently, so I guess I still don’t understand what you’re after. Surely you’ve experienced these sorts of things yourself?
No, I have never experienced those sorts of things myself. I am an emotionless automaton. Is that a serious question?
Beauty. I chose a beautiful cat, and then a seriously goodlooking husband. I enjoy beautiful prose, and I’m willing to believe that music has esthetic value to some people as well (I’m basically tone deaf). But mathematics is where I have the feeling of seeing beauty naked. Beauty in the platonic, world of ideas sense. There’s nothing comparable in the real world we all live in.
As a physicist, I often find myself wondering why people enjoy pure maths.
Ok, ok, just kidding! 
I do find myself spending a lot of time reading pure maths journal papers and texts, and I can see the appeal. This may not sound so flattering, yet it seems to me there is that certain something that is deliciously sexy about learning the `dark arts’; of a far off, forgotten language known only to a few eclectic others?
Is there appeal of being part of a special club?
An obvious root psychological motivation is a deep anxiety about, for some, poorly ordered or understood ideas (e.g. “it’s awesome”) , for others, it’s for their place in the hierarchy (e.g. “I’m awesome”). The successful mathematician manages these (and other) anxieties to build a career based on recognized work.
Walt, don’t get mad; yes it was obviously a rhetorical question. I thought it was clear what I was really asking: what about those descriptions of these various appeals of mathematics is unsatisfactory from your point of view, and why? Clearly I don’t understand what it is you’re asking for.
Perhaps I could ask it this way: name me any field of human endeavor (let’s say music for sake of discussion) for which from your point of view, the appeal could be based on something specific in the psychology of the practitioners, which is not reducible to the statement “I’m awesome” or “music is awesome”. Then maybe I’d more clearly understand the question.
The appeal of mathematics lies in the pleasure of insight. Solving a problem can be frustrating, and there is a flood of pleasure and relief when insight is finally attained. Some get hooked on this pleasure and become “insight junkies” i.e. mathematicians. People who are “thrill junkies” put themselves in situations that are dangerous or feel dangerous, and then get out safely … danger followed by safety leads to relief and pleasure. Likewise mathematicians put themselves in frustrating situations, that are “dangerous” in the sense that there’s no guarantee that they’ll ever solve the problem and relieve their frustration … frustration followed by insight leads to relief and pleasure, just as danger followed by safety does. This is addictive for some. I wouldn’t be surprised if the two are related … could explain why so many “insight junkies” spend their free time doing rock climbing etc.
IIRC correctly Hardy gave a fairly cynical explanation of how one becomes a pure mathematician in his Apology. While you’re a student you do it because you have some talent for it and after that you discover you aren’t qualified to anything else. The last part is probably a lot less true now than then but the fact that what starts us on the road may be something as shallow as filling a curricula requirement holds.
I think it’s also a mistake to look for a single account of the psychological appeal of pure maths. I suspect we find different joys in different parts of the process.
Although you dismiss it, I actually think that in some cases an attempt to retreat into a world with some certainty may be part of the appeal. I myself, remember being perturbed at loosing that sense of certainty when I first thought through computer proofs, that they could be knocked off course by cosmic rays and such (although it is unlikely), and then that so could our brains when we did proofs. Being left with only a high probability that even logical implications held bothered me, which suggests that I was craving certainty from mathematics. (I got over it).
Todd: Many practitioners of medicine claim altruistic motives. They heal the sick, which (arguably) makes the world a better place.
Matt: don’t be so sure things are different. As I look on the possible death of my career I have no idea what else I know how to do.
John: agreed. I really wanted to see how Walt was going to respond (i.e., what area of human activity would evade the kind of reduction which I took him to be claiming for the appeal of mathematics). Walt, I’m sorry if I misunderstand you.
But your reply reminds me of another source of appeal of mathematics: most mathematicians love to explain things to each other, to share their insights, to teach [i.e., teach those who are prepared to learn]. That would have to consist in more than just serving the glory of one’s self: there’s something wonderful about the fact that all this mathematics we carry in our minds can be communicated so precisely. There’s no other mental discipline quite like that of which I’m aware, and the resultant shared sense of community must be one of the most powerful sources of mathematics’ appeal.
John: be of good cheer; I’m told that banks and the civil service are often keen to hire maths PhDs (even from very pure fields and without programming backgrounds) to hack around with their numerical models. A colleague from when I was doing my PhD (a C*-algebraist and before that always a secondary teacher) took a job like this in finance and describes it as easy, well paid and kind of dull.
Todd, my question is pretty vague. Let me take your original comment as an example. I would say that your first paragraph constitutes a psychological explanation, while the rest shades into an aesthetic explanation (a subspecies of “math is awesome”). Most explanations I’ve heard have been either that math is beautiful, or math is challenging. Neither explanation is very specific — do we have a special section of the brain that is triggered by Chartres, the Brandenberg Concertos, and the Stone-Weierstrauss Theorem? How does it work, exactly? What in human psychology makes mathematics aesthetically compelling or challenging?
I agree with coward. The way I’d phrase it is that people love figuring things out (for obvious evolutionary reasons). Pure math takes this to the extreme where the objects of study fade out into abstraction and all that’s left is the essence of figuring out.
I add my voice to those who have mentioned a need to understand and explain. This is a desire that, to my mind, cuts across nearly all intellectual endeavour, both artistic and scientific. The quest to find truth to give it a slightly melodramatic title. The quest is certainly a concept with deep psycological resonance. One only has to see how often it turns up in stories.
In different areas the ability to have a complete understanding differs. However in mathematics we can simply define away complications until we have something we understand. Mathematicians, especially those working in pure, are perhaps those people not satisfied with uncertainty in understanding, they want the sense of completeness.
Of course it should be mentioned that completeness comes at a price. Something that can be understood completely is probably less relevant than something that still has unknowns. There is therefore a desire to add to what is understood. This is the mathematical quest, to conquer the dragons of a theorem, and be the first to understand something, perhaps in the history of humanity.
A final thought is the additional nature of competition. A result can gain reputation beyond its utility of the potential illumination it might provide. The quest then changes slightly. The goal is no longer understanding alone but also the idea of besting the great mathematicians of the past who failed to solve this problem. For example as you look down the list of Clay Millenium Maths problems ask yourself how much of their importance comes from the knowledeg and understanding that their solution will bring to humanity and how much comes from the fact that they are long-standing problems that have beaten all of the greats of the twentieth century.
Walt — thanks. I think I have at least a slightly better idea of what you’re after now, and also I think I owe you an apology: I see that your “math is awesome”, “I’m awesome” were mere shorthands for umbrellas of possible explanations, and I don’t think you meant them in a reductionistic spirit. If anyone was being insensitive then, it was probably me.
Anyway, I don’t think anyone has quite mentioned another source of appeal which might conform to what you are after: that mathematics partially satisfies longings which could properly be called “religious”. Religion here should be broadly construed; perhaps “longings for transcendence” better captures what is meant.
This can be seen on all sorts of levels. There is for example a craving for truth or certainty of a permanent nature, of a nature which transcends earthly contingencies. Related is a Platonic belief in objective mathematical reality, that is reality which always was and ever shall be, independent of human existence. I think for example Platonic belief in actual infinities is very close to religious in character; at any rate it is “natural” to feel that mathematics is an Other somehow larger than and beyond ourselves.
Historically, there have been close ties between religion and mathematics. It would be reasonable to say that some of the ancient fascinations with numbers or numerology was driven by belief in their numinosity, and geometry itself was largely driven by interest in the motions of heavenly bodies and their religious significance. (Recall also the ancient quaternity arithmetic: music :: geometry: astronomy.)
I personally think the same kind of thing can be seen in contemplating the architecture of mathematics. Looking at ordinary architecture: the great cathedrals of Europe were designed to promote contemplation of the glory of God and His Creation, and certainly there is something transcendent about the many scales at which one can observe these great creations, both in the grand conception and in the fine details. So too there may be something in the contemplation of the many scales at which one can view mathematics, from the most sweeping views to the subtlest nuance, that satisfies in some degree such transcendent longings. (The thrill of being able to survey huge sweeps of mathematics at a glance is not unlike the thrill of being at the top of a mountain overlooking a vast landscape, and the appeal of both has a religious character for many people.)
The idea that mathematics is an escape from reality has been remarked upon. There’s some truth to that, but of course this is a negative way of formulating the thought. More positively, it is contemplation of a world which many feel transcends or is beyond mere earthly bonds. Say what you will about religion, but it would be silly to deny that such longings exist!
Well, I do math, set theory in particular, because I am fascinated with the way you can build immensely complicated structures starting with just a simple empty set and using a few operations. Probably the same reason I liked playing with lego blocks or programming the computer.
There’s been a lot of work on the psychology and neurophysiology of music - I don’t know much about it, but Oliver Sacks’ book Musicophilia sounds like an entertaining place to start. I have a hunch that understanding why almost everyone likes music will someday help us understand why a few people like math.
Todd: What you are calling a desire for transcendence is exactly the feeling that I had hurt by considering the fallibility of mathematicians’ brains. I’m quite convinced that maths can’t offer transcendence in a metaphysical sense because in the end it is something that we “mere” physical beings do we our physical brains and hands and pens and paper. Indeed all experience is in the physical brain, so I don’t believe that any experience transcends the physical or offers transcendent certainty.
However, the desire for this sort of transcendence is clearly very real and it was scary to lose the (implicit) belief that maths offered it, just as I guess people who believe in transcendent souls sometimes fear naturalistic explanations of consciousness. As I said above, I got over it. Just as people are no less wonderful for being thinking meat, maths is no less pleasurable or beautiful for being something done by material humans.
I think it may be that mathematicians do (on average) have a more developed craving for transcendence than scholars in neighbouring areas. Anecdotally, I have come across a lot of mathematicians who also held rigid religious beliefs, which claim to offer certainty. Also you get a very bright man like Roger Penrose tying himself in knots trying to recover the idea that mathematics and the brains that make it transcend normal physical laws.
So, Matt, you’re a formalist? You seem to have a similar underlying belief that mathematics is a formal system, and a product of the activities of human minds (brains).
Not to claim a Platonic position here, but I challenge you with the same response as I’d give to a hardcore formalist: how do you explain the “unreasonable effectiveness” of mathematics in the physical sciences? Why should the output of human brains have anything to do with physical law, and how is it that truly well-formed sciences are invariably expressed in mathematics? Escapes into radical solipsism will be discarded as the jokes they are.
Neverminding this scuffle between Matt Heath and Jon Armstrong, I find myself sqarely in the middle of what I would call the Escapist Account of the Psychological Appeal of Mathematics.
As Einstein said, mathematics does not refer to real things.
It is amusing that Armstrong uses the worn term “unreasonable effectiveness.” One might be led to ask, “Why is it so effective?” Once we answer this question, we might actually not use the word “unreasonable”!!!
There are Penultimate examples to choose from, and my conclusion is that science is made for mathematics. We culturally have been predisposed to say that our physics, our chemistry, our science studies have to subordinate themselves to mathematics. If they don’t do this, they are labeled “fallacious” and sunk down the Memory Hole. So it’s rigged from the start! Thinks like Mach’s Constant Laws only work because we’ve forced them to work. We’ve explored and utilized physical truth only insofar as it corresponds to what we’ve decides are valid mathematics.
So enough of this dribble in the discussion!
NS
http://sciencedefeated.wordpress.com/
John A: I don’t want to derail this thread too far from psychological questions so I’ll answer your question at my own blog and leave a trackback.
You hit the nail right on the head, Todd. Mathematics (like good music) does satisfy certain longings. Longings for eternal truth, certainty, beauty, and justice. Yes, justice…but that’s a personal one. For me, doing mathematics is very soul-soothing and satisfying, even though it’s a lot of hard work and few people appreciate it.
“Another example is that mathematicians can’t handle the real world, and so retreat to the safety of the world of numbers.”
At least one pure mathematician I’ve known has an unhealthy tendency of asceticism for the sake of abstraction.
st: I know you say it’s personal, but could would you mind expanding on “justice”
John: I have a reply to your challenge up at epsilonica now.
Here’s a partial answer to Matt’s question:
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.”
I didn’t have time to write this more concisely, but this discussion on Math, Beauty, and Truth is irresistible to me, if you’ll grant me a minute or two.
“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.
Thanks for sharing good information. These information is very lovely and important.
Is Beauty Truth in Mathematical Intuition?
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.
Walt - The book “Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being” by Lakoff and Nunez approaches this from perspective of cognitive psychology. It is an embodied mind theory which is also an argument against Platonism. Here is a wikipedia article on Cognitive science of mathematics http://en.wikipedia.org/wiki/Cognitive_science_of_mathematics
I’m a control theorist who’s been runner-up in a cheeseburger eating competition (17 in 5 minutes) and a Russian pancake eating competition. The dream is alive Walt, the dream is alive.
I have no idea what draws the great mass of mathematicians to pure math, but I can at least tell you a little bit about what draws *me* to pure math! I’m a physics student, and in physics classes, mathematical objects are usually presented as techniques for calculating things—ways of shuffling symbols around to get the answers to certain questions. I like pure math because it gives me a way to *see* and *feel* mathematical objects—to get an intuition for what they do and how they work. I especially love defining things axiomatically, because it seems so natural and straightforward. You start with a rough idea of what you want—a way to take the derivative of a vector field, or measure the volume of a parallelepiped, or figure out whether a function is “effectively computable”—and then try to sharpen your idea until it defines a useful object. It’s like magic: if you ask for something in just the right way, your words will turn into the thing you asked for!