Conjectures and Meditations

With the recent holidays behind and proper resuscitation, a little bit of expendable time and the boat seasonally stuck in the garage, I seem to have started reading again.  A special gift this Christmas was Visions of Infinity: The Great Mathematical Problems by Ian Stewart.

I am no mathematician!  (This will soon be obvious.)  But I was always able to do the work assigned, and I occasionally found mathematics to be inspiring.  (As I write this, I have to question myself; Was it the pursuit?  Or the solution? ...that provided the drive?)  For a month or so I even considered mathematics as a college major (perhaps to counterbalance a second major in The Classics), and I took a bit more math than was specifically required for my eventual biological studies.  But in the end, I have to repeat, I am no mathematician.

So here’s another book, on a pretty deep topic and written for a broad, yet sharp audience by an eloquent expert in the field; and, a gift from a loved one!  Once again, my kind of book!  I'm a Big Fan of Prologues and Epilogues, and I quickly extracted some value from the Prologue (and introductory Chapter 1) on Christmas morning:

• Early on, mathematics is described as “Notions, not Notations.”   Now, if Karl Gauss believes this about mathematics, then I do, too!  The beauty of this notion or type of book is that one can be freed from many of the technical details and logistics, and be allowed to actually think of what’s behind the subject matter.  Why is it important?  Why is it interesting?  Why is it worth our time and effort?  Surely, math and science (or any other discipline) should be taught (or at least offered) in this manner.  But more importantly, this type of idea affirms process over outcome; and as I’ve noted before, one can often only try to influence/improve the process.  You can’t necessarily guarantee any specific result.
• Alongside this philosophy, the author characterizes mathematicians’ view-point of these Great Problems as (in my words) “Not really caring about the specific solution, but rather caring with extreme depth that they’ve not yet provided one. “  What a wellspring of motivation!  There will be no resting on laurels or coasting through issues, because there are always new, interesting problems to be addressed.  These might arise naturally, in their due time, or be revealed through what we learn via our efforts.
• And so the actual quest is to identify and develop useful and productive methods.   In the end, it is the mastery of Methods – not simply Answers – that matters the most; at least with respect to productively engaging the next situation we find ourselves in.
• Finally, a compelling aspect of this book (and of math, science and academic pursuit in general) is that this work is being tackled by clearly engaged humans, with emotional investment and interactions among disciplines.  It takes personal investment and the help of others to succeed at most things.  (Dare I go on?  Is there really anything else of value that I could add?)

I know that I didn’t capture everything the book offers.  Occasionally I was on the fringe of fully understanding the matter at hand, and quite often I was less so.  Neither The Four Color Theorem for Coloring Maps (nor its proof) exactly excites me.  

I mostly get the ideas behind a Moebius Strip, but beyond that, Topology largely alludes me.  Unfortunately, it seems to be pretty important to Modern Mathematicians.

The mathematics got further and further from my experience and comfort level, and the last problem posed was difficult for the author to even phrase.  And yet I could feel the author becoming less mathematical and more philosophical in his writing.  So while I'd gotten off to a great start with this book, and had progressively slogged through the middle, he really brought things together for me in the last couple of chapters.

So let me declare my appreciation for the following “mathy” concepts from this book that rang true for me:

• Let us Accept the Computer, but Appreciate the Human.   Computers are capable of doing the work, but they need to be directed.  Use of computerized iteration only makes sense and can improve accuracy while reducing the drudgery of addressing many problems, but (for the most part) humans will provide the vision and direction for these efforts.  We’ll benefit most from some sort of Balanced Partnership between Computer Scientists and Mathematicians; who knows where their collaboration will lead us?  But we do know that each expertly wields a set of powerful tools; and there is a synergistic, complementary relationship between these sets.  Again, who knows where their collaboration will lead us?
• Where’s the Value?  Does it reside in The solution or in The Process?  It depends on perspective and immediate needs, of course; the recipient of successful emergency surgery is most grateful for the results; society in the main should be most grateful for the processes, atmosphere and institutions that enabled the development and timely delivery of the employed surgical techniques.  So the specific resolution to Fermat’s Last Theorem or any of these Great Problems doesn’t matter to most as much as the quest for this resolution.  That quest spun off many tools and understandings that will ultimately affect society as a whole; and while many of the participants in this quest probably foresaw no specific practical applications from this effort, it is inevitable that these will arise.
• Mordell’s Conjecture that (for certain, known types of equations), “Yes, there are a finite number of solutions from which all (infinite) solutions can be obtained” lets me appreciate the juxtaposition of finite with infinite; but ultimately, isn’t the search for, and the identification and appreciation of, those Finite Solutions/ Core Essences/Numena  what this is all about?  My role as father or spouse, my endeavors, my blogging, my motivations and happiness ultimately define my identity, purpose, and influence.
• Pi and e are Transcendental.   They are powerful and irrational.  They simply arise from our versions of mathematics, from multiple directions, and yet they can only be (albeit very closely) approximated.  They are the glue to many, basic concepts, and I’ve often wondered about them.  It’s a wonder we (collectively) found them.  It’s a further wonder that we can use them so productively, repeatedly.  They have lives of their own, and they seem to represent (in a Quest for Knowledge sense) Truth and Beauty about as much as anything. 

So this is a math book, but let’s face it, it’s not necessarily about the math.  It’s about recognizing and respecting the monumental effort behind this human endeavor and the broad perspective that this can provide.  It’s also about passionately tackling a portion of it (especially for the fun of it!) with the tools at hand.   It’s about trying to link our interests and assets, with the prospect of some unspecified Added Value in our lives.

Now IMHO, Fishing is more fun than math!  But many of the same lines of thought hold true, and while I’ve stated that I am no mathematician; I think I can say that I am a fisherman!  Fishing’s not necessarily about the catching; but it is about having caught; and about having understood the experience to the extent that we go fishing with the expectation of catching.   It’s about using this understanding to develop the skills and methods we use to present our baits; with the best tools we’ve acquired along the way to do so.  It’s about Fishing’s Great Problems, which are represented by the fish we know or suspect to be present but haven’t yet caught; the places we’ve not yet visited; and the ecosystems we need to protect.  Outcomes will certainly matter, but they will ultimately pale in comparison to the methods utilized and efforts expended.  What we experience and learn along the way will certainly inspire future efforts, and those efforts will be aided with sharpened and refined tools.

Fishing is my Big Effort of choice, and it certainly shapes my Perspective (on the natural world, what to do with time and money, how to figure things out, how to engage and learn and progress and share), and it might be the Living Well portion of my efforts.  In the end, I simply appreciate all of the fishing-related instructions, advice, camaraderie, assistance and experiences that I’ve shared (and will share) with family, friends and customers.

I was fortunate to share this bruiser Maine surf striper with my good friend, F.