Erdős Number of 4

For many years, I was happy enough that I had an Erdős number of 5. It seemed appropriate given my tangential connection to mathematics, using and sometimes developing, around the edges anyway, but not working on any deep mathematical truths, like how to cut a birthday cake in an ultra-fair way given that your friends are remarkably suspicious and back-biting. But recently, as I’ve gone back to doing a little more math in my day-to-day life, I found myself clicking about in the Erdős graph and, oh joy of joys, discovered a new lower bound, a fellow integer with the previous on that smallest integral right triangle. 

Now, some of my friends might say, Erling, if 5 was good enough, if you were happy in that life, why search for more? Why risk the coconuts that might fall from the shaken tree? Ah, I would say, there is nothing to fear, as when one searches for a shorter path, that search is defined by the shortest path found so far, and a bound that has been discovered can be made better but never worse. 3 and even 2 are now possible, if unlikely, but 5 will never be again. 

As it happens, this new shorter path wends through the same author and colleague as the last, the late Oscar Rothaus, mostly of Cornell University, best known for work on the very useful Hidden Markov Models - you know, that every speech recognition system uses. The paper was Fast Fourier transform processors using Gaussian residue arithmetic by Alvin M. Despain, Allen M. Peterson, Oscar S. Rothaus, and Erling H. Wold [Ed: note the order of names was suggested by Al, one might think in a spirit of bonhomie and all-for-one, but an order most often suggested by those who spend their life in the glow of the beginning of the alphabet, knee to knee with the camp counselor and first across the street holding the teacher’s hand]. Although it was already well known that some computational problems would lend themselves to attack using a residue number system, our technique was a clever combination of Al’s beloved CORDIC rotations along with residue arithmetic using not the regular old primes, but some small complex - aka Gaussian - primes, breaking the problem into very small pieces that could then be computed with tables. 

The paper was supported, like so much work on computation then and now, by the Department of Defense. Oscar and Allen and Al were all up in it, all part of the JASON Defense Advisory Group, a group of people who were tasked with searching through every bit of technology and science to see if there was something in it that speed up the process of killing or of being killed, and who have been favorites of the conspiracy theorists, controlling the weather and magnetizing the children. At the time, the reason that was given for faster and faster transforms was that, in a dogfight, one needs to decode and block the other fellow’s frequency-hopping radar while at the same time hopping your own and detecting the other fellow's attempts to block you and hopefully blowing him out of the sky during that moment of arbitrage when you’ve hopped and he yet hasn’t.  But, unlike the other paper I wrote with Al, Pipeline and Parallel-Pipeline FFT Processors for VLSI Implementations, I don’t believe our clever Gaussian residue approach was ever used in an instrument of death. I don’t know for sure, as I didn’t have the clearances to know, but it never came up again, while I was questioned in some detail about the parallel-pipeline stuff by engineers from Westinghouse, that same Westinghouse who built the radar that detected the Japanese attack on Pearl Harbor, but which wasn’t believed by anyone, a seemingly unbelievable suggestion on a beautifully crisp Hawaiian winter day. 

But, getting back to my happier graph traversal, from there, the next node on the new path is to Further results on p-automorphic p-groups by James R. Boen, Oscar S. Rothaus and John G. Thompson, a paper which tightened some constraints on counterexamples to a conjecture by Boen as to whether certain p-automorphic p-groups are Abelian. Boen, whose conjecture was later proved by Shult, is interesting in a number of ways: very active in mathematics and science, but also an activist quadriplegic for the last fifty or so years of his life. But even more interesting to us here is that, in 2012, Hugh L. Montgomery and the same John G. Thompson above published an article in Acta Arithmetica on Geometric properties of the zeta function. In it, they summarize the state of topographical knowledge of Herr Riemann's delight, and by so doing, laid the extra edge that allowed my number to drop, as Hugh Montgomery had, many years before, published Sums of Numbers with Many Divisors, which looks at representing large integers as sum of highly divisible … oh let’s just quote their abstract in all its poetry: 

Let k be a fixed integer, k⩾2, and suppose that ε>0. We show that every sufficiently large integer n can be expressed in the form n=m₁+m₂+…+m_k where d(m_i)>n^{(log 2−ε)(1−1/k)/log log n} for all i. This is best possible, since there are infinitely many exceptional n if the factor log 2−ε is replaced by log 2+ε.

squee!

The Easy Life

I'm sitting in our apartment in Firenze in the former Palazzo Baccio Valori looking out the window at the Duomo. The lights on the monuments have just come on as the sun has set, the last of the bells are dying out, and the mosquitos are now let loose in the city.  In the days of the Medici, the mosquitos were fierce and malarial, but in the modern age they move slowly, engorged on drunken tourists.  We sometimes find the opportunity to dissect them, not with scalpel but with shoe or paperback or oven mitt, and this is ok, the way of things, and with some touch-up paint we can remove their little lives from history, much as we ourselves will disappear one day, when our great-great or great-great-great grandchildren, cleaning the urns on the mantel, no longer remember which urn is which and whether this one is great granduncle Teddy and maybe they need to make some room for more hyper-photos of people they do remember, at which point one hopes that enough life force has endured in those remains to give them just the slightest jolt as they toss you out into the mulched hyper-roses causing them to trip and fall into the pond.

Like so many of those who wish not to be tossed into the hyper-roses, I have vainly attempted to leave a mark on the world. My CV goes on a bit, and although we are constantly reminded of those who have accomplished so very much more, I do take it out of the old box of clippings and read it from time to time, along with all the fading notices of my time on the stage, the limelight and the girls on shoulders of boys and the applause. I was doing this just the other day in fact, and came across Joshua Kosman's article on me from a few years back and found this bit:

At 50, he has the amiable demeanor of a practiced collaborator and the buzzing nervous energy of someone with a long history of getting twice as much done in a day as the rest of us.

Is that true?  I've always thought of myself as fundamentally lazy and fundamentally not so bright. When I was an underage boy at Caltech, freshly deflowered and still wet behind the ears, I met people whose brains seemed to be running at a speed I could not imagine, and I spent most of my time drinking and sleeping all day and skipping class, but I remember that my friend Billy B. was envious about how much I got done even with all the drinking and sleeping through class, so maybe he felt the same. And in my later years - in graduate school and in the working world - I've been pretty sure that I am in fact a fraud, just clever enough to hide my inadequacies and hoodwink everyone around me, the snake oil salesman in their midst, somehow able to knock something together that works, and as long as no one looks too closely for the shoddiness (and don't even talk to me about my doctoral thesis) one might miss it, but luckily I've been able to find jobs where such slipshod work is enough to get by, and fortunately I'm not building airplanes or pacemakers or nuclear bombs.

Of course I know of the inverse Dunning-Kruger effect, but just knowing that that effect doesn't help you - you yourself can't tell if you are an actually competent person who downplays their abilities or if you are simply mediocre.

But there is the other side, the looming shadow-side of the self-thought-to-be-incompetent Wold boy, the one who secretly thinks that he is smarter than everyone else, who wants everyone to know, who carries intellectual books about with covers displayed so that others can see, who does look through the aforementioned box and remembers that he did fight long and hard for those Scholar of the Year awards, printed on fake parchment, remembering too that when he was called up to the podium to receive the first one, his name was proceeded by "and now for the Freshman girls" and although he wasn't quite yet the cross dresser he would later become, his hair was well down his back and that did earn him some cheek-reddening catcalls from construction workers on his way to school. And then there are those moments when, expecting that well of course everyone knows this or understands that, he discovers with mind-splitting incredulity that people he assumed were as competent as he is actually don't understand some bit of mathematics so totally obvious and straightforward, or find Ulysses "too hard," or haven't read the vi or emacs manual and learned all the arcane details, or whatever else he can hold over everyone's head, even though some of accomplishments actually were kind of hard back when he faced them himself but seem so easy now in retrospect.  And all of this happens in both aspects of his life, the science-y math-y engineering-y bits and the art-y music-y literature-y bits.

But even writing this now brings the fear on for the Wold boy, the fear that he is one of those people to whom Dunning-Kruger really does apply, who worries above everything, even that maybe he is worse than stupid, maybe he is actually delusional, as in delusions-of-grandeur delusional. So we'll set that aside for a time when the room isn't so dark and the candles aren't guttering.

What I really wanted to talk about here - and the reason for the photo of the totally gorgeous telescopes in the Galileo Museum above - is the epiphany I had the other day while walking through this beautiful city, which is that, even though I've come out in some ways, although in the article above there is this bit:

Wold is a little reticent about his sexual autobiography, despite the fact that his Web site identifies him as a composer, producer of operas, and "libertine." He volunteers only that after his divorce, he moved across the bay in part out of an attraction to San Francisco's gay scene - despite the fact that he is, by his own description, "queer but not gay."

and finding this bit the other day made me wonder if I'd ever actually read the whole interview, as I didn't remember it at all, but as I was saying, even though I've come out in some ways, I tend to avoid coming out as an engineer/math guy to the art world and as an art/lit guy to the engineering world, thinking that somehow there is a stigma of un-seriousness about being one to the other, but my epiphany the other day was that there is of course no stigma, not the slightest at all. The Galileo Museum is filled with objects that are both gorgeously scientific and gorgeously artistic, and being here in goddamned Firenze makes one remember that we intellectual types used to gather to make decisions about all aspects of the world and that article one on the agendum list that afternoon might be do mathematical objects exist in a Platonic reality of their own and article two might be be shouldn't we create a whole new art form? Leonardo was given a commission one day to paint the adoration of the magi or whatever and then the next day hired to figure out better methods to slaughter the good citizens of Pisa and it didn't seem to matter to either commissioner about the other. So where did that change? Or is this perceived stigma just my own problem? Hey, that reminds me that I did in fact work on a weapons system once, a parallel-pipeline FFT processor inside the F-16 or maybe it was the -15 radar, being built by Westinghouse Electric Corp, and I had published a paper on fast parallel-pipeline FFT construction, so was the go to guy, and I needed some money because I was a poor graduate student and like everyone else whose hands are bloodied for some bit of money, I just had to wash them with a little bit stiffer brush when I got home in the evenings, and I wonder now if they still would have hired me if I had told them about the other stuff, the settings of the Antonin Artaud poems and the readings of the Kathy Acker books and suchlike.

Oh wait, that reminds me of the other bit that gnaws at me - the piece of paper that says whether you are capable of doing a job - a notion that is so obviously crazy that one wonders how it even got started. I remember years ago reading some nonsense by Charles Wuorinen (note I may be misremembering this and maybe it was someone else but whatever, it's my memory so there) about how Charles Ives couldn't be that good a composer because he didn't really have that much schooling.  Hey, I have a PhD but the only reason I have it is because 1) one of my first girlfriends basically dared me to do it and 2) I didn't really like having a real job. Smart people can do smart things regardless of the particulars, and having the piece of paper doesn't even say that much anyway. I'll tell you the simple way to figure out whether someone is capable of doing something - ready? - have them actually do it. The best engineering/math work I did in graduate school wasn't even for my thesis, and almost none of what I do now was what I studied then - in art or the other.

And finally there is the last part of the engineering/art equation or should I say minuet, and the basis of the title of this essay: money. It's the weird and strange specter of the art world, a world in which no one has enough unless they inherited it from someone or they happened to be one of the lucky few that connected with the mass market, or they have a day job. Engineering gives me the easy life, and pays for this trip to Firenze, and gives me the support to do my art, and asks only that I work all the fucking time, day and night, never to see the beauty of summer, never knowing the joy of a day wasted without care.

Science

Hearing the premiere of Paul Dresher's latest piece, a concerto of his Quadrachord long string instrument with the Berkeley Symphony Orchestra, and especially his quick lecture beforehand on the details of the Just Intonation aspects of his piece (natural harmonics on his instrument, natural harmonics in the horns, and maybe in the trumpets, a subgroup of 15 strings tuned 40 cents lower to bring them into correspondence with some of the higher partials), I was reminded once again of the lazy connection between music theories and scientific theory. Even in my deepest and darkest days spent connected to the JI world, nothing I read - and I read a lot - ever explained, except in the most mystical terms, why Just Intonation was better than any other pitch-choosing mechanism. The purity of the intervals was mentioned, the transcendental perfection of small integers, the lack of beats, the sacredness of the harmonic series, and of course the music of the spheres. But does any of that say anything about badness or goodness or are such considerations irrelevant in the face of such revealed truth? The realities are mundane in comparison, e.g. that people like beats, that they tune pianos and 12 string guitars and mandolins and Balinese Gamelans to emphasize them, and why are small integers better than irrational number, since as Cantor found, there are more of the latter than the former? And I wouldn't be the first to point out that the JI-ness of the vibrating world exists only in some Platonic ideal, as real strings and real columns of air have harmonic series which diverge from that ideal, sometimes by a lot, and brake drums and bells and lots of other wonderful musical machines have harmonics that are decidedly inharmonic, a lovely and self-contradictory description of the partials whose frequency relationships one might otherwise think defined the word.

In music, I love such nonsense. I think it is important somehow, like reading a kōan, putting the mind in a place where mere truth is irrelevant, but I also do have a deep lasting long term relationship with science - and maybe even a love for it - which goes back to my youth, discovering one day a large cache of old Scientific Americans and reading through them all from cover to cover and cardboard box to cardboard box. It leads me to still spend days reading through scientific and mathematical articles, scribbling down my own calculations and pondering the deep search for truth. Although I sometimes dissemble when discussing it, my doctorate is not in music at all, but in Electrical Engineering, and I recall a story from those days. My research advisor was Al Despain, a wild-haired crazy man who was willing to skim off some money from his various defense grants to support me, a poor graduate student interested in the intersection of music and technology in those heady times, when one had to write one's own file system to get samples off a disk fast enough to achieve audio rates, when one had to build one's own D/A converter to listen to the audio in real time.  But Al's true love was all things military, and one day he told me to be at his house the next morning early, where we were met by a limo and, quickly chewing through several columns of Fig Newtons, headed to the airport and a quick flight to San Diego. Again, a limo, and bustled into a room, I found myself giving a talk on my thesis to a room full of JASONs, the notorious and/or acclaimed MITRE-related Defense Advisory Group, including the esteemed Freeman Dyson. I bumbled through, in awe, and wondered at the attendees most celebrated, not able to say what I really wanted to say: in fact some kind of gushing fanboy babble.

Last week, driving back from visiting my mother and my in-laws, I was reminded of this experience when listening to a Relatively Prime podcast on Paul Erdős. The subject is dear to my heart, and I glow with a very small respectability due to a paper I published with the mathematician Oscar S. Rothaus on Gaussian Residue Arithmetic, giving me an Erdős number of 5 to his 4. The podcast featured three mathematicians with Erdős numbers of 1, and one of them told how he was invited by Erdős to give a talk at a symposium where no one showed up except Erdős, the organizer of the symposium and Stanislaw Ulam, and how proud he was to give his talk to such a small but illustrious audience. In life and work, we love our icons and we hope that someday they may love us.

Mathematics, as it is

Whenever I am asked whether I make a living as a composer and I have to reveal that no, like so many, I have a day job, and then I'm asked what pray tell might that be, and I say I'm a mathematician or whatever category into which I'm dropping my job as Chief Scientist that day, I brace myself for the inevitable insight that well, Music is Mathematics, isn't it now.  Being a Very Nice Fellow, I smile wanly and nod and then try patiently to explain that no, it is nothing like Mathematics, any more than Cooking is Mathematics, Writing is Mathematics, Painting is Mathematics.

But then, they might say, even those Things are in fact Mathematics. Remember the words to the children's song by Tom Lehrer:

Counting sheep

When you're trying to sleep,

Being fair

When there's something to share,

Being neat

When you're folding a sheet,

That's mathematics!

Like most children's songs, there is a great Truth here, which I capitalize to distinguish it from actual truth. Mathematics is a study of abstract objects, and most would understand that in the sense of modern Platonism: points, lines, ideals, manifolds, rings, lattices, graphs, numbers, cardinals, propositions, sets, symbols. Those abstract objects are fun to study in their own right, but the Truths of Mathematics come into their own and touch our lives when they find a life as models of Real Objects, some of which they model well and some of which they don't. Sometimes we work to arrive at those models, but oftentimes our conceptions of real objects are greatly simplified to match a mathematical theory that we happen to have lying about.

So, let's be more precise. What one might say is that those Fields of Study above have attributes that can be modeled more or less accurately by Mathematical Objects and that one might be able to glean certain knowledge of those Fields of Study by manipulating those Mathematical Objects, assuming all of one's assumptions are more or less correct, that the initial mapping is OK, and that those mappings still remain OK even under the effect of whatever manipulations one might make in the abstract realm.

What is really really hard about applying Mathematics to the Arts, is that manipulation of anything in the Arts might yield something interesting artistically, since there is no absolute arbiter of anything, as Good and Dad and Judgement are of the past, and one can find an audience for any jumble of phonons or photons or smell-ons.  This makes the final judgement as to whether one's model is Right or Wrong well nigh impossible.

But still, limiting ourselves to the artistic area in question, one might ask: how well can available mathematical models map to something in Music that will help us compositionally or analytically?  And, like all fields of study, there are some things that work OK: in my other life, the fields of acoustics and signal processing are based on this. In that day job, I may assume that a sound is modeled by a continuous curve and that I can differentiate and integrate and take limits to infinity. This assumption is far from a Truth, but it's OK, it works OK, I can manipulate all day long and at the end of the day discover something in my Platonist Plane of Existence that I can transcribe into software and drop into an iPad app and voila!, it sells to the masses who want Groupon coupons generated by listening to the ads on the TV. And, in my musical life, I may model a musical event as a note, and further reduce that to some parameters, like a pitch, which is then further reduced to a frequency, and that to a number which, in a ratio with other numbers, can inform me as to how to tune my guitar.

We all saw many attempts in the heady days of the post war academy to model musically related parameters like crazy, to manipulate them like crazy, and to come up with maps that we hoped we might follow to some Heavenly Abode where - well, I'm not exactly sure what.  Where we might find the Perfect Music?  Or the next Perfect Musical Publication? The serialists' attempts to construct a set-theoretic world based on a small set of discrete parameters is in my opinion a model-of-a-model far removed from the world of a sound, where it's hard enough to pin down music into discrete anythings, e.g. where a musical event starts and stops, or what its please-pick-one pitch is. If notes had single pitches, Auto-tune wouldn't exist. The funny thing is that there is a lot of fabulous serialist music, but, in my opinion, I don't think the models were helpful in getting there any more than any other kick in the pants.

Of course I know the models, and I use them, in a crafty way, to solve problems that I hit here and there, just like the Painter knows the models and may compute the Golden Ratio from time to time, not knowing whether it really is Good or Bad but whatever. And, if my inquisitor by this point hasn't run for the table with the potato chips, I would then set my hand on her shoulder and explain further that the joy of Music Composition, for me, is the impossible-to-quantify or at least the uninteresting-to-quantify ineffable aspects that I don't really even want to understand: how one writes when one has stayed up all night; how one allows God and his Angels, dark or light or their familiars, to speak through us; how it is that there is that one passage of Boulez's Le soleil des eaux that gives me chills; how I can find my way to a piece of music that, when listened to later, I don't understand in the slightest. There is an aspect of the endless, of eternity here, and I might remind my partner in conversation of the end of the B section of the tune above:

If you could count for a year, would you get to infinity,

Or somewhere in that vicinity?

The answer is no, as Wittgenstein said in his Philosophical Remarks:

Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one. ... The infinite is that whose essence is to exclude nothing finite. 

A model, no matter how finely developed, is, like the runner in Zeno's paradox, no closer to reality than the model before it. Music is not Mathematics, no how and no way.