You don't have to have slogged through Part 1 or Part 2 to get here, but it may help to have skimmed the backstory. A dog, not yet sixteen, is starting college.
1965 - 1967: The Prime Years
Gauss called mathematics the 'queen of sciences' and number theory the 'queen of mathematics'. Then Primes must be the Queen of Numbers.
The Infinitude of Primes. There exist an infinity of prime numbers. One of the deepest of theorems and the most beautiful of proofs[2] from, who else, Euclid himself.
The Fundamental Theorem of Arithmetic. All positive integers (other than 1) are either prime or the product of two primes. Not an easy proof.
The Goldbach conjecture. All even numbers above 2 can be expressed as the sum of two primes. Not proven.
So many paths to explore. See in the series above how so many of the primes P have P+2 as "twins" (3,5; 5,7; 17,19.... 41,43....)? Can we theorize about that? Proofs had to be pretty. Theorems had to be deep. Hardy's Theory of Numbers was the Bible. Ramanujan a god. Partitions. Diophantines. On to analysis. Topology.
"Now Euler! now, Fermat! now, Leibniz and Newton!
On, Cauchy! On, Cantor! on, on Hilbert and Riemann!"[3]
It was like an incandescent Christmas every day. Who needed LSD?
Of course, you could add something more practical like the Philosophy of Mathematics (or Foundations of Mathematics, depending on which department was doing the teaching). Argue all night about the nature of mathematical objects. Debate the status of mathematical truth. Discover answers far afield in the drawings of Escher and in the stories of Borges[4] or at his very feet.
Sex (not enough), Drugs (not a lot) and Rock and Roll (Yes!). Till we come to the numbers for the non-prime year of:
1968
4.4.1968 Dr. Martin Luther King Jr. He was 39.
6.6.1968 RFK. He was 42. For those who didn't watch his final
journey, here's a link.
8.28.1968 Democratic Convention police riot. The Chicago 7.
11.22.1968 5th Anniversary of JFK assassination at age 46.
11.23.1968 "Harvard Beats Yale 29-29" in The Game. The high point
of the year for many.
Turmoil on campus. Ban Dow Chemical. Bash McNamara. Burn draft cards. Abolish ROTC. Long Live SDS (Students for a Democratic Society). Hell no, we won't go. Ho ho ho, Ho Chi Minh. Gimme an F... Gimme a U...
Turmoil in my head. They won't leave me alone to do math. The worst system devised, next only to the Cambridge tripos described by Hardy. ABD (all but dissertation) TA's and preceptors. Rote and regurgitate exams. Memorize stupid human tricks.[5]
There was hardly any consolation in philosophy. The "foundational crisis"[6] in mathematics was in full flood, of little interest to anyone but its promoters. On the other hand, a "joint concentrator" had to be careful not to tread on any philosophical corns if he wanted his thesis to be more than a doorstop -- cause for significant angst. Is any of this real? Does any of it matter?
Turmoil in my heart. Was I good enough? Was I good enough for original work? Sure, I could be a high school teacher or maybe even a college professor doing "A note on Hilbert's ..... " ad infinitum. But would I ever discover anything? Would I live to discover anything? Would I live? In short, I was afraid.
I have seen the moment of my greatness flicker,
And I have seen the eternal Footman hold my coat, and snicker,
And in short, I was afraid.
And would it have been worth it, after all ...
-- T. S. Eliot The Love Song of J. Alfred Prufrock (1917)
1969
It was the perfect early June day for a commencement that is still talked about. Red armbands. Rants by all and sundry. Begowned Grouchos passing out cigar-sized joints. Fist-raised walkouts during the president's speech and David Rockefeller's honoris causa doctorate. And of the few that remained (and understood), some still retain the memory of the peroration in stoned dog Latin : Praeses et Socii Colegii, Bruti, Cumani, sedens in ripam Carolus, suscipite amorem, non bellum... and ended with a Roman salute to the assembled personages: Ave caesar! Morituri te salutamus. Ave atque vale. [7]
Summa cum laude with a joint concentration in Mathematics and Philosophy. But Dawg never pursued his beloved Pure Mathematics further.
Coda:
I wandered for a few years in a purple haze. Was rescued by the words: "You can go on killing yourself. But I won't be around to see you doing it."
In and out of the academy. A couple more degrees. A couple or three disciplines. Worked for the Man. Became the Man. Back again to being a dog.
Still with the one who saved my life. More to the point, she's still with me. And a daughter who is smarter by a factor of n, who is charting her course through the academy without detours, diversions or derailments, unlike dear old dad.
What a ride it has been. Sometimes I wonder if I "fulfilled my potential". What if I had stayed with my first love - Pure Mathematics? It's a cliché , true, but you never forget your first love, you know.
WOOF
P. S. : My apologies for a long and what may be considered a somewhat self-indulgent piece. But it was hard to stop the flow once it started. And believe me, it's been mercilessly pruned, though with some of the excisions showing up as "Notes" :-).
___________________________________________________
Notes:
[1] M. C. Escher: Though Escher (1898 - 1972) titled the print Relativity, it seems to me he could as easily have called it Mathematics. He studied the works of and worked with some of the best mathematicians of his day (such as Roger Penrose) and incorporated these concepts into his prints.
Bill Thurston, a great geometer, calls mathematics a "tall" subject, like a skyscraper with many staircases that must be ascended in an orderly fashion. Barry Mazur, another of the greats, calls the structure Gaudi-esque, with a wide choice of alternate staircases. This is the representation I see in the Escher drawing.
Gödel, Escher, Bach by Douglas Hofstadter (Basic Books, 1979) is, of course, the now-classic work which weaves many of these themes together.
Illustration credit: Wikipedia Commons.
[2] Euclid: A short proof of the infinity of primes theorem, after Hardy.
Posit P is the highest prime number possible (i.e. there are not infinite primes)
Let Q = (2*3*5*7*11.....*P) +1
Then:
Either Q is prime (dividing by all known primes gives a remainder of 1) and obviously greater than P
Or It is not prime, having another combination of prime divisors, one of which must be greater than P
QED via contradiction.
[3] All the names are of mathematicians, of course.
G.H. Hardy's A Mathematician's Apology (1940) is a classic, providing a window into the creative aspects of the mathematician's craft.
Cambridge University Press, 1967. Available in paperback Canto edition.
Robert Kanigel's The Man Who Knew Infinity: A Life of the Genius Ramanujan is an excellent book, describing the life of this largely self-taught prodigy from India, without short-changing the math. Ramanujan's genius was recognized by G.H. Hardy who brought him to Cambridge (England) and his work to the notice of the professional world. He died in 1920 at age 33 but his work in number theory is still providing grist for the paper mill.
[4] Jorge Luis Borges (1899 - 1986) incorporated elusive and profound mathematical themes in many of his stories. Two of my favorites are A Survey of the works of Herbert Quain and The Library of Babel.
Jorge Luis Borges Collected Fictions (Trans. Andrew Hurley) Penguin Books, 1999
[5] Upon rereading, the paragraph seems overly whiny and snide, but it captures my feelings at the time.The remarks about TA's and preceptors is off the mark. They were a great group, for the most part. The American TA's (no surprise, they were all male, and the influx of "foreign" TA's was not yet in full spate) were grappling with their own issues vis a vis the draft. And the math professors who taught undergraduates were truly of All-Star future Hall of Fame caliber: Barry Mazur, Dave Mumford, the late George Mackey.....
But the criticism of "the system" and exams still stands. Two examples selected at random from recent exams:
(i) Learn by rote and regurgitate:
State and prove Ramsey’s Theorem, Hall’s Theorem, Erdös-Ko-Rado Theorem and the Erdös lower bound on Ramsey.
(ii) Stupid human tricks:
For z ∈ C Z, set
(a) Show that this limit exists, and that the function f defined in this way
is meromorphic.
(b) Show that f(z) = π cot πz
You can work forever and a day without success unless you know "the trick": separate the summation into three parts -N to -1; 0; 1 to N and then it's "easy". Meromorphic? Don't ask.
[6] The foundational crisis
The Philosophy of Mathematics may be said to encompass ontology/metaphysics (do numbers exist?), epistemology (how do we know the truth, say, of proofs? ) and more.
The battle-lines are roughly drawn as follows:
- Plato: Mathematical concepts have an objective existence, just like the cup on the table. Platonism
- Aristotle: Not so, they are merely abstractions of empirical experience. Empiricism
- And so across the centuries to Kant, of course, and then
- Frege/Russell: Forget ontology, mathematics is logic and logical truth is its truth. Logicism
- Hilbert: Bosh to the above. Mathematics is just a bunch of signs, a system for replacing one bunch of symbols with another. Create the syntax, you get the semantics (truth). Formalism
- Gödel: Utter nonsense, the previous two. Incompleteness theorem says no mathematical system can be both consistent and complete in itself, blowing Hilbert's project to smithereens.
- Wittgenstein: Always Wittgenstein. Mathematics is all syntax, no semantics.
- And from our very own Über-philosophs W.V. O. Quine and Hilary Putnam, the latter the Faculty advisor to SDS at the time, that the reality of mathematics primarily obtains from its applicability to the natural sciences. Naturalism
[7] The Latin may be rudely translated as:
"President, overseers, animals, idiots, sitting on your asses by the banks of the Charles, make love, not war.... "
ending with the classic gladiators' tribute:
"Hail Caesar. We who are about to die, salute you. Hail and farewell."
This was preceded by liberal sprinklings from Seneca's De Providentia which describes death in gory detail:
Sive fauces nodus elisit, sive spiramentum aqua praeclusit, sive in caput lapsos subiacentis soli duritia comminuit, sive haustus ignis cursum animae remeantis interscidit....
"Whether the noose strangles the throat, whether the water suffocates the breath, whether the hard ground that breaks the fall crushes the skull, whether the fire sucked in blocks the respiration.... ".
The 40th Reunion is in a couple of months. It will be fun getting together.
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Dana Dangerous
Skip Williamson
designanator
Dennis Knight
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Comments
Glad you liked the Escher, resistance.
WOOF
I have to side with Quine on the whole reality thing, at least in my little sub-universe.
Like watching an E type Jaguar roadster whiz by at 100 mph.
As for Quine, I dunno - then I had felt mathematics being 'betrayed' to be a mere handmaiden of the physical sciences. Putnam actually was 'oil on troubled waters' sort, saying, in essence, this whole foundational crisis crap was nonsense. And now, that is probably closest to my view. But you, I'm sure, will appreciate how that sort of academic atmosphere could be scaring an undergraduate dog shitless :-).
In any case, as Marlowe said, that was in another country, besides the wench is dead.
WOOF
Btw, I put in the scl bit just for you. Now that you've seen it, I might take it out. Too embarrassing, my dear :-).
WOOF
I enjoyed the boxes with the years, all your quotes -especially the Latin ones!-, and the Harvard Strike poster. I did not think it was self-indulgent at all, rather like a self-analysis (not sure whether that's the correct term, pardon my French! -pun intended).
As a side note, I have to confess that Borges let me down at one point in his "Library of Babel":
Letizia Álvarez de Toledo has observed that this vast Library is useless: rigorously speaking, a single volume would be sufficient, a volume of ordinary format, printed in nine or ten point type, containing an infinite number of infinitely thin leaves. (In the early seventeenth century, Cavalieri said that all solid bodies are the superimposition of an infinite number of planes.) The handling of this silky vade mecum would not be convenient: each apparent page would unfold into other analogous ones; the inconceivable middle page would have no reverse.
I think that there's no reason to believe that the middle page is any different from any other page.
Your memoir reminded me very much of the movie "A Small Circle of Friends," set at Harvard in the tumultuous years you described...I grew up feeling I'd been born into the wrong era, because I missed all of this.
Rob, thank you, but I'm afraid we are both baby math now (and I bet you're just being modest). Forty years is a long time, man. Cantor's proof is indeed one of the prettiest and deepest -- so deep that I think you know that's what probably what started off the 'foundational crisis' in the first place!
I like Library of Babel, but your point is well taken. And, of course, it was Quine (boy, he does keep coming up doesn't he!) who posited, as reductio ad absurdum, that the entire library could be contained in two pages, one with a dot, one with a dash (or nowadays a zero and a one, I suppose), with the reader flipping through in some fashion and getting from the text whatever s/he brought to it!
WOOF
It's always been my opinion that the footnote's raison d'etre is for self-indulgence. A professor once commented that I used in them a tone better reserved for the long-tenured.
merwoman, I haven't seen the movie, but knowing your impeccable taste, I assume this one is better than others about that era. Generally avoid those pictures -- probably a result of a bunch of us watching "Love Story" stoned and laughing uncontrollably throughout (we were asked to leave the theater!).
Also, those years were pretty traumatizing, and this third piece is not the way I had planned the series would go (I'd pretty much blocked out the before "Purple Haze" part of my life), but it seemed to take on a life of its own. So it goes.
WOOF
But the out-takes did end up in the footnotes! And when the notes are equal in size to the piece itself, that's surely self-indulgence to the nth power! Your professor turns a phrase well -- I'm afraid my tone is just long-winded ;-).
WOOF
1) I followed this from pt1 and loved every minute. Your thoughts are like a manifold and I enjoyed every bizarre minutiae of significance.
2) I'm an Aristotelian with a fair smattering of Naturalism by way of the Buddha and an unbelievable amount of drugs.
3) I asked for and received Wolfram's "A New Kind of Science" for Christmas the year it was published (02). I read it. Cover to cover.
Anyway, thanks for this.
Yeah, this confessional stuff, you and me both. But once I started, the story just seemed to become more and more ME and less and less math. But maybe that's because math was me. Ah, enough confession.
As to the Wolfram book, I marvel that you finished the darn thing cover to cover. Isn't it like 1000+ pages? I've read about it and about his Turing machine challenges, but I was told he hates mathematicians :-(. Waiting avidly for next BATZ and Finn update.
WOOF
My greatest math revelation came in my high school algebra class. Mrs. Casey was a very sharp witted person and someone was discussing 'Alice InWonderland' with her when she said, "Oh, Alice In Wonderland was very mathematical." I was like, "What?? How can a story be mathematical?"
That moment remained with me and it was years before I realized the truth of what she said - for few have loved math as much as Lewis Carrol I found out - and I found what you call the poetry of mathematics. So thanks to Mrs. Casey, I can appreciate what you're saying here.
And thanks for letting us in your little Dawg world!
WOOF
Mrs. Casey was brilliant -- Alice is replete with math conceptualization, and, in fact, I believe monographs have been written on the topology of the rabbit hole and such! I should try to find some references for the cruelly Zerry truncated Cheshire cat!
WOOF
How I Lost My Innocence
A senior high school math class that introduced me to abstract number theory -- including an introduction to the binomial system that made the computer possible -- and a brief and humbling experienced in an advanced study group made of area high school wizards who sat around talking about tetrahedrons and pnp transistors, was enough to convince me my future lay in the liberal arts.
But as your story and mine attest, the journey from here to there is not easy, and there seldom ends up being where you thought it was going to be. As a result, you are left to endlessly, fruitlessly wonder about "what if?"
Still, for all the errors of my ways, they led to my son, and were I able to reverse any of my errors, I would remove him from my script -- and that I can't imagine.
SB, yes, live blogging an assemblage of 1 VPOTUS, a Nobel Prize winner, two MacArthur fellows, a Hollywood superstar and a goodly number of Bernie Madoff types should be fun! But as you know, it is the more unusual types with their out of the ordinary paths and accomplishments and the regular, "good" people that make for the best reminiscing.
Tom, that is a beautiful poem. And yes, the moving hand writes and having writ moves on. I don't look back that often -- in fact, as I told merwoman, I have consciously or sub-consciously blocked out much of that part of my life. The pain, the confusion, the fear, the anger were too real to want to revisit. And what's turned out (personally and family-wise at least) seems to have turned out for the best.
WOOF
I love the movie--I've watched it so much I've worn out the VHS --but then, I wasn't there. It may seem ridiculous to someone who experienced the times live and in person. It was made in 1980, so close enough, I think, that it was at least somewhat historically accurate.
Had a good cast, too--Jameson Parker (later of Simon & Simon), Karen Allen, Shelley Long, and one of my favorite actors ever, Brad Davis. Brad died far too young, at the age of 42, of an intentional overdose/assisted suicide (he was in the end-stage of AIDS).
He also starred in Midnight Express, another amazing slice of history (if wildly changed from the book), which I loved but can't bring myself to watch again (it's all down to John Hurt's cat).
(But lest anyone reads this comment and thinks that's the fullness of it, let me make this part quite clear: You then EDITED what you had written. Oh, lawdy, if I could convince all to do one thing it would be to edit and edit and maybe even edit again.)
Back to my real comment--I am grateful that you chanced a glimpse back at a confusing and painful and probably also hopeful and exciting time in your life and the larger world. I was genuinely moved by this, although I won't get ridiculously sappy here and scare you off the confessional mode forever ;) Do consider returning to it now and then, if only for your many friends and fans.
What can I say but I love you, dawg!!
Susan, thank you, thank you, thank you.
WOOF