In our ongoing effort to provide our readers the finest in easily digestible yet ultimatley informative and entertaining reading, I felt it was time to delve into one of the deepest pockets of punk rock minutiae, one of the richest veins of golden age lore, one of the most profligate wells in music geek fandom, the Angry Samoans Files. Collected over years and years of investigative work into a band that could rightfully take the throne of Greatest Punk Outfit of All Time (and even if they didn't take the throne, they'd be there to throw tomatos at and write songs about whoever did), we shall begin puking forth these files as a running feature. Each installment will tackle a different Samoans-related topic, continuing until we run out of material (which will happen sometime in 2012). If you have any ideas you'd like to see discussed in future installments, by all means let us know.
In this episode, we drop in on Professor Gregg Turner for a little Q&A...
TB: First of all, what do you do in terms of where you work, what you teach, and what kinds of research you do? Feel free to be as specific as you want. i.e.: x'(t) = Ax(t) + Bu(t) and u(t) = Kx[t-r]
Gregg: I'm an assistant professor of mathematics at New Mexico Highlands University, a small liberal arts college in Las Vegas, NM (that's about 60 miles from Santa Fe). I teach the full spectrum of upper and lower division mathematics courses to our majors and as service courses to other departments. My teaching load is 12 credits per semester (roughly 4 classes) which more than sucks - that does not support release time for extensive research. When I completed my PhD in 1991, the job market in mathematics had tanked (suddenly). The net consequence was that you'd have routinely like 300-500 applications per advertised tenure track position. Friends of mine at UCLA, to brighten an otherwise grim prognosis, would wallpaper their offices with the myriad of rejection notices they'd receive! I finished my degree at Claremont Graduate University and my dissertation mentor was a rather well known biomathematician at Pomona College (Dr. Kenneth Cooke - his mentor at Stanford was a really famous guy, Richard Bellman! So I guess despite winding up in the sticks with a less than prestigious Ivy League employer, I can at least enjoy this genealogical pedigree!!). Despite this, there were only post doc opportunities -- so I completed a 2-year post doc at UCLA (where I went undergraduate). Now it was 1993 and the job market wasn't substantially better than ‘91 (despite all the assurances that all the old guys employed circa Sputnik in the 60's would have to be retiring soon -- problem was, their positions were not being readily replaced). So the only tenure track position I was offered was at this small private liberal arts art-college in Santa Fe, NM, the College of Santa Fe. It wasn't Princeton (ha ha) not even say uhm University of Texas at El Paso. But it was an offer and there were still a lot of talented fresh PhD’s with no offer at all. So I grabbed it - and that was my ticket out of Los Angeles and into New Mexico. Anyway -- I apologize for rambling here, just thought I'd set the context for my entry into the professional big leagues. By the way, this move essentially ceased any continuance with the Angry Samoans, but that for all intents and purposes had dead-ended in ‘91. I was loaded down with teaching, sometimes 15 credits per semester, otherwise 12. The pay wasn't red hot either, so it necessitated teaching summer school. This played itself out over a 5-year period. The major downside, due to the heavy teaching, advising responsibilities and committee work, etc, was virtually a dead end as far as time to produce quality research papers. I didn't realize at the time that if tenure wasn't a sure thing at the College of SF, then shopping the next position would become doubly difficult (then don't provide these insights into academic politics in grad school). But after 5 years, the College decided it could not recruit sufficient numbers of math majors to justify tenuring my position (convenient for them to fall back on this). Eventually I moved down the road and wound up at NM Highlands U. My research area of interest is in applied linear algebra, matrix theory, finite dimensional control theory, and delay differential equations. My 3 (count em!) published results have involved stabilization of hybrid control systems in the presence of feedback delays. This involves producing algebraic criteria to stabilize an essentially continuous system with a discrete control vector involving a delay.
where [.] denotes the greatest integer function. In the above, A, B, K are time-invariant and my main result characterized the spectrum of K in terms of A to assure stabilizability.
TB: In your opinion, what has been the most important advancement in mathematics since differential and integral calculus was discovered independently by Isaac Newton and Gottfried Leibniz in the 17th century?
Gregg: This is probably not shared by most mathematicians, but personally I feel the work of Georg Cantor in the late 1800's - his creation of hierarchical infinities, or more precisely, the notion of equivalently sized infinite sets, was a watershed. This allowed a rigor into analysis that previously did not exist.
TB: The concept of zero (a number that symbolizes the absence of a number) did not come about until its discovery in India in the 7th century, and was not introduced into Western mathematics until the 9th century. What took so long? Was it the lack of need for such a number in primitive mathematics, or do you think it was more of an intuitive leap?
Gregg: There are many ways of looking at this, I think. The structure of the way we understand the real numbers, for example, depends on an additive identity. So, from the modern constructive viewpoint, 0 functions as a necessary logical piece of the puzzle for additive groups, rings and so forth. One could make the case that 1 is equally significant (multiplicative identity for mult. groups, rings etc). And then there's the whole numerological schtick - but let's resist the temptation to indulge that (otherwise we'd be knocking on 666's door next!). Phi, the empty set, can be analogized to 0 as a quantitative measure. Why the delay with baptizing 0 into the fold of other numeric symbols? Probably (I'm guessing) because there was greater utility for symbols describing tangible quantities. I mean, when did -1 come into vogue? e, pi or i? These were all context derived per need. E.g., how to solve x2 + 1 = 0? So you need to invent an i and then the Pandora’s box that opens. Pythagorean cultists did not believe in irrational numbers - i.e. all quantities were to be expressed as the ratio of integers. But that didn't fly with the length of the hypotenuse of an isosceles right triangle with sides = 1. There's a very elementary proof demonstrating that the square root of 2 cannot be rational.
Suppose √ 2 is rational. Then it can be expressed as a ratio of 2 integers a and b. I.e. √ 2 = a / b and without any loss of generality we may assume that a / b is an irreducible fraction (cos if not, then reduce it until it is). Square each side, get 2 = a2 / b2 or equivalently that 2b2 = a2. This implies, then, that a2 must be an even integer. But this would imply that a must be an even integer. So we can then write a = 2k where k is some integer. Hence we now have 2b2 = (2k)2 = 4k2 or that b2 = 2k2. But this means that b2 must even and consequently that b must be even. Finally we arrive at a contradiction, a logical incongruity. Since from the initial premise that √ 2 can be expressed as a / b with a / b irreducible, it follows that a and b are both even! Therefore √ 2 cannot be expressed as the ratio of two integers, and it is therefore irrational. Well that was big deal for the Pythagoreans. Supposedly, the story goes, that the fellow who demonstrated this reality to the higher up Pythies was forced to walk the plank (if you were a Pythagorean, one of the major taboos was to [accidentally or intentionally] walk over someone else’s used toenail clippings [I'm not making this up]).
TB: Although calculus can be a powerful tool, it does have its limitations; for example, it could never predict a sharp drop in a population curve. Could there be something out there that we haven't discovered? Something that could serve as a better tool for predicting our fate?
Gregg: Any scientific model is a randomization of the real world it attempts to describe. Equations, formulas, symbols are just window dressing of short hand for symbolic description of these randomized estimates. Something different? Mandelbrot in describing fractal geometries as models of natural physical processes? If nothing else, this spawned the tidal wave (to some degree - it was there before Julia sets became fashionable wallpaper) of chaos and complexity to model discontinuous and nonlinear systems. Some of these new models remain suspect in the eyes of peers - until the theory delivers on its promise to explain phenomena that otherwise had remained elusive or impenetrable. There was this guy, Rene Thom, who was the figurehead of what was called Catastrophe Theory. This was very hot a few decades ago. He constructed this simple machine -- two nails on a short board of wood, connected by a twisted rubber band. If you move the rubber band in a continuous fashion, at one point it snaps back - voila! The discontinuity (catastrophe!).....Soon everyone who had any sense of scientific fashion had one of these contraptions on their desk in their ivory tower academe offices. But my understanding is that the hype never delivered the goods (results), tho some of the mathematics created was quite elegant. Now no one has any Zeeman catastrophe machines in plain view.
TB: Mathematics can be seen as a science; a study of the rules that govern the behavior of numbers. Its direct application to physics and other sciences seems to support this claim. However, it can also be viewed as a philosophy, an abstraction of reality. What side of the fence are you on?
Gregg: Both. My work is oriented to applications. However, what turned me on to mathematics in the first place was its reliance on its own abstract structure for justification -- of its own importance. The reasoning is that it never (ever) is clear when a seemingly abstract concept or theorem that for hundreds of years (say) has no utility other than spawning similar abstract results. Then one day, there is a bizarre superimposition of where and how this actually fits in with reality. E.g., the algebraic construct of group theory is one of the major tools for quantum mechanics, as well as elements of encryption theory and so forth. Often, physical and natural reality pose the problem, someone devises a mathematical construct. Mathematical knot theory, e.g., the topology of interconnected shapes, is motivated by the geometry of DNA strands, etc.
TB: Physicist Eugene Wigner once said, "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." It is true that mathematics is a powerful tool not only in physics, but virtually any aspect of life. But what if it wasn't? Is there any area on the cutting edge of math that has absolutely no practical use?
Gregg: All bets are off as far as "no practical use" in perpetuity! (see above). But e.g., in ’93 Andrew Wiles at Princeton proved one of the most famous open problems of all time. This was known as Fermat's Last Theorem. Simple to state (for those who haven't heard?) there are no solutions in integers x, y, z for n > 2 of the equation xn + yn = zn. The proof of this amounted to synthesizing a very sophisticated set of new (last 30 years or so) developments in the theory of elliptic curves. The proof was over 200 pages I believe. Fermat, in editing a book of algebra, would fill in unproved propositions with his own quick proofs - in the margin. This was back in the mid 1600's. However for this particular theorem, he merely noted that he had a truly wonderful proof, but unfortunately its length exceeded the marginal space provided! So it took some 350+ years to fill in the margin, but Wiles' proof was certainly not what Fermat had in mind. In any case, what I'm leading up to, is that the proving of Fermat was covered in the NY Times and papers and periodicals all over the world (to the lay public). But other than tackling a very famous and perplexing riddle it is pretty much agreed that Wile's proof and the solving of this outstanding problem, despite no one doubting it being a monumental achievement, will not probably contribute to a cure for SARS. Or offer a blueprint to the Republicrats on how to dump President Moron.
TB: Euclid took some shit in his day for proving things that were intuitively obvious, i.e. that a straight line is the shortest distance between two points. Is there anything intuitively obvious in modern mathematics that remains to be proven, or is it all just a matter of constructing new knowledge from here on out?
Gregg: My feeling is that it moves in both directions - backwards and forwards. Euclid would not have imagined that sometime in the future there would be the necessity for a NON-Euclidean geometry. There are so many outstanding unsolved problems -- the Riemann Hypothesis e.g. (the new Holy Grail of conjectures yet to be proved. The biography of John Nash [much better than the feeble parable the movie offered] alluded to Nash's schizophrenia precipitated by his inability to prove the Riemann Hypothesis) -- that it would be really reactionary to believe that what's in store, is the next generation (literally) of new ideas, new formalities, new modeling constructs.
TB: It has been proposed that our base-ten counting system (that is, order of magnitude increasing as multiples of the number 10) arises from the simple fact that humans have ten fingers. Trying to count in a number scheme that isn't base-ten proves to be exceptionally confusing (the first twenty numbers of a base-eight system would be 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24), and trying to come up with any sort of algorithm to convert from one counting system to another is even harder. Is there something inherently simple about our base-ten system, or is it just the fact that it's what we're accustomed to? Are there any other cultures or groups you know of that use another base number? Any advantages to such an approach?Define x(n+1) = 3x(n) if x(n) £ 0.5, and x(n+1) = -3x(n) + 3 if x(n) > 0.5
Gregg: I believe the Babylonians used base 16. Anyway, there are actually pretty simple algorithms to just as easily compute in any other base. Certainly base 2 has more utility than 10 (insofar as how binary arithmetic is used in digital computers). There's a simple formulation of what's called the Cantor Set in base 3 (although you don't need to really just use base 3). Here's the Cantor Set Game:
To start the game, you need to define an initial point x(0). If you look at the sequence of points generated by a particular x(0), various things can happen. If the sequence goes to -oo, we say the sequence is an escaping sequence, and that x(0) is an escapee. However, if you select x(0) = 0, then the entire sequence that is generated remains at 0. We call such an initial value x(0) whose sequence eventually winds up at 0, a prisoner. Note, e.g., that x(0) = 1/3 or x(0) = 1/9 are prisoners as well (since several iterates wind up at 0). The interesting question is in the interval [0,1] how to identify prisoners from escapees. How many prisoners are there in [0,1]? What is the ratio of prisoners to escapees in this interval? Given ANY input initial value x(0) an algorithm can be devised to determine whether this is an escapee or a prisoner (aside of course from iterating til you're blue in the face!). The answer involves decimal arithmetic with base 3. And beyond this being a mere mathematical curiosity, the above iterative scheme has companion results in the theory of dynamical feedback systems.
TB: I touched on the importance of mathematics in science earlier, and with theoretical physics in particular. Do you have any particular interest in the concepts that readily go hand-in-hand with the math? Have you ever collaborated with scientists?
Gregg: I've worked on a few projects involving chaos and nonlinear systems (there's a group at Los Alamos National Laboratory, a mathematical division called the Center for Nonlinear Studies), but really circumstances with my present position disallows too many adventures outside of collaboration with colleagues close at hand -- and of course teaching. But that's ok. You are aware that my interests are not constipated w/in technical stuff (despite how dry alla the above may or may not sound) -- I've written a movie treatment that we may get to work on this summer, called "Hell's Whores" (orig titled: "Necromaniacs From Hell") and am working on an anthology of short stories called "The Tapeworm Story and Other Gastrointestinal Nightmares"......and the music stuff once in a while.
TB: The visualization of extra spatial dimensions is something that has intrigued me for quite a while. Here is something that I have been pondering. Suppose there are two 2-dimensional creatures living on a sheet of paper. Arbitrarily, let's just assume that they are both shaped like the letter "F". Now suppose you reached into their two dimensional world and picked one of them up, and flipped him over, so that he appeared as a backwards "F". Of course, he would not be able to comprehend what had just happened, but his friend would be able to tell that something was strange, because no combination of two dimensional movements could cause the creature to appear "backwards". In other words, only a reversal in the 3rd dimension could produce such an effect. My question is this: If a 3-dimensional human was picked up by a 4-dimensional creature, reversed with respect to that 4th dimension, and placed back into his 3-dimensional world, what changes would his friends notice?
Gregg: Well, they haven't "found" Saddam or Osama or the weapons of mass distraction yet, so we have to assume that such technology is not yet available to the warlords in Washington. And I would posit that the result would be different if the 4D critters were Klingon as opposed to, say, polygamous Mormons. But all in all, I guess I'd speculate that the difference the friends would notice is that our 3D abducted into 4D and placed back into 3D human, would suddenly feel compelled to get tickets to John Edward's Crossing Over. Either that, or he/she would start compulsively drinking coffee out of a Klein bottle.
Pretty fucking punk, huh? If you want to read a less intellectual Gregg Turner interview go here. Thanks to whoever it was that sent this to me, and bailing me out since I didn't have anything finished for the next Files yet. Drop me a line at termibore-at-aol-dotcom if you want credit, I'm sorry I lost track of your contact info. Next issue: a more "punk" Samoans Files, which will either be Vom-related or "Angry Samoans FAQ", whichever gets finished first. Both are in the works. And Gregg Turner, drop us a line or answer our e-mails please!