I’m long fascinated by the varying lives and minds of mathematicians. Thusly, one of my favorite sections of Thomas Lin’s (editor) fabulous volume, “The Prime Number Conspiracy,” is Part 4 “How Do the Best Mathematical Minds Work?,” offering fascinating profiles of several individual prominent mathematicians. One of them, Freeman Dyson held a well-known disdain for PhD. degrees, and I’ll just pass along these lines (that just maybe will inspire some):
“…I’m very proud of not having a Ph.D. I think the Ph.D. system is an abomination… It’s good for a very small number of people who are going to spend their lives being professors. But it has become now a kind of union card that you have to have in order to have a job, whether it’s being a professor or other things, and it’s quite inappropriate for that… The Ph.D takes far too long and discourages women from becoming scientists, which I consider a great tragedy. So I have opposed it all my life without any success at all…
…So I’m very proud that I don’t have a Ph.D. and I raised six children and none of them has a Ph.D., so that’s my contribution.”
[Dyson is still going strong at 95, contemplating unsolved math problems, and 5 of his 6 children, by the way, are women.]
...and then, on a separate note, a bit ago I came across this wonderful Paul Lockhart quote (from "A Mathematician's Lament") in a piece by Sunil Singh:
“Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion—not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.”
"What about four-dimensional space? Is there such a thing? If we're asking whether four-dimensional space is real we might as well ask about three-dimensional space: Is there such a thing? I suppose it appears that there is. We're walking around (apparently), and things certainly look and feel as though they are part of a three-dimensional universe, but when you come right down to it, three-dimensional space is really an abstract mathematical object -- inspired by our perception of reality, to be sure, but imaginary nonetheless. So I don't think we should put four-dimensional space in any special mystical category. Spaces come in all sorts of dimensions, and none are any more real than any other. There are no one-dimensional or two-dimensional spaces in real life, and the only thing that gives the number 3 any special status is that our senses appear to offer us that particular illusion."
"Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively. The truth is, I don't know of any human activity as demanding of one's imagination, intuition, and ingenuity. But I do it anyway. I do it because I love it and because I can't help it. Once you've been to the jungle, you can never really leave. It haunts your waking dreams….
"The solution to a math problem is not a number; it's an argument, a proof. We're trying to create these little poems of pure reason. Of course, like any other form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying."
-- Paul Lockhart from "Measurement"
[…If
you have a favorite math-related passage that might make a nice Sunday
morning reflection here let me know (SheckyR@gmail.com). If I use one
submitted by a reader, I'll cite the contributor.]
"Truth" arising from an "infinite sequence of lies"….
One of my favorite 2012 math books, "Measurement" by Paul Lockhart, is now out in paperback, so in honor of that, some brief passages to ponder therefrom on this Sunday morning:
"The solution to a math problem is not a number; it's an argument, a proof. We're trying to create these little poems of pure reason. Of course, like any form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying." "Geometry, then, is not so much about shapes themselves as it is about the verbal patterns that define them. The central problem of geometry is to take these patterns and produce measurements -- numbers which themselves must necessarily be given by verbal patterns."
After explaining calculating the area of a circle from an infinite-sided polygon inscribed within, Lockhart writes (in one of my favorite bits from the entire book):
"Something really serious has just happened here. We have somehow obtained an exact description of the area of a circle using nothing but approximations. The point is that we didn't just make a few good approximations, we made infinitely many. We constructed an infinite sequence of increasingly better approximations, and there was enough of a pattern in those approximations that we could tell where they were heading. In other words, an infinite sequence of lies with a pattern can tell us the truth. It is arguable that this is the single greatest idea the human race has ever had."
And finally, this: "Maybe the bottom line is that I don't have that much to say about the real world. Maybe part of it is that I'm not altogether entirely here a lot of the time. Maybe the point of this book is to give you a glimpse of what it is like to live a mathematical life -- to have the better part of one's mentality off in an imaginary world. At any rate, I know that I am by nature permanently isolated from reality -- my brain is alone, receiving only the (possibly illusory) sensory input that it does -- but mathematical reality is me."
"My idea with this book is that we will design patterns. We'll make patterns of shape and motion, and then we will try to understand our patterns and measure them. And we will see beautiful things! But I won't lie to you: this is going to be very hard work. Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively. The truth is, I don't know of any human activity as demanding of one's imagination, intuition, and ingenuity. But I do it anyway. I do it because I love it and I can't help it. Once you've been to the jungle, you can never really leave. It haunts your waking dreams."
-- from the Introduction to "Measurement" by Paul Lockhart
How could anyone read the above words and not want to proceed to read the book that follows!?
I'm only 3/4 of the way through Paul Lockhart's new volume but will go ahead with an overview blurb on it, because it's obvious to me that I like this idiosyncratic offering and want to recommend it! The book is composed of just two main parts: Part 1 on "Size and Shape," and Part 2 on "Time and Space." Geometers especially will love Part 1 (a sort of geometry primer), though Part 2 (which I'm only partially into) is likely the richer, more fascinating, and slower, more arduous read (essentially an introduction to calculus).
Lockhart's writing style is conversational; refreshingly so, compared to the usual prescriptive tone of much math-writing. He stresses intuition over, or at least equal to, logic. One feels at times as if he is in the same room, ever-standing over your shoulder, talking to you, or, perhaps, like a kindly grandfather, holding your hand as he takes you on a stroll pointing out things along the way that he finds exciting. The book's tone very slightly reminds me of David Berlinski's "The Advent of the Algorithm," another writer with a unique and passionate style.
One of the things I particularly like about the book is that Lockhart is bluntly honest with his audience right from the start (as indicated in the quote above). So many popular books these days imply that they will make math fun and easy for you: "Learn Calculus In Your Sleep" or "Quantum Mechanics in 3 Easy Steps" (ok, so I made those titles up, but you get the idea). But this offering doesn't pretend to be a "Math For Dummies" book. For most of us, math (at some level) is hard, and Lockhart acknowledges that; some folks who are very bright in other areas, have real mental blockages for mathematical thinking. Once again from Lockhart's intro:
"…expect it to be slow going. I have no desire to baby you or to protect you from the truth, and I'm not going to apologize for how hard it is. Let it take hours or even days for a new idea to sink in -- it may have originally taken centuries! "I'm going to assume that you love beautiful things and are curious to learn about them. The only things you will need on this journey are common sense and simple human curiosity."
Thanks grandpa ;-)
So I'm sure there are math-challenged or -phobic individuals out there who will simply find Lockhart's effort just as dry and indecipherable as any other math volume. But for those with an inclination toward the subject matter this volume will likely be a gem.
Unfortunately though, the title doesn't convey that gem-like quality. I suspect "Measurement" was thought to be an elegantly simple title, but I fear it will sound boring and even misleading to many prospective readers, for whom the word conjures up tedious, rote procedures. This book is full of 'elucidation' or 'illumination' (through math), and even play. It is no casual or beach read of course, but nor is it a cold textbook or instruction manual either; and certainly it's a valuable book for teachers to have on hand (so many wonderful examples/ideas herein). The title is not inappropriate, but it may not be the attention-getter the book deserves.
One of my favorite quotes from the volume comes when Lockhart is explaining the calculation of the area of a circle by utilizing an infinitely-sided inscribed polygon:
"In other words, an infinite sequence of lieswith a pattern can tell you the truth. It is arguable that this is the single greatest idea the human race has ever had." [bold added]
Gotta love that enthusiasm, and it beams forth from the book repeatedly.
I do have one major quibble with the volume though:
Lockhart regularly tosses out various thought questions or problems for the reader to figure out on their own to more fully fill out the ideas being discussed. Some are easier than others, but they are good and instructive, and it is ashame (even annoying and unsatisfying) that he nowhere offers the answers to these lobbed exercises, so readers can check themselves or see the intended answer if need be. Given Lockhart's joy at elucidating logical steps for the reader I can't imagine why he drops the ball on these interspersed problems, and leaves the reader potentially hanging somewhat -- a simple appendix or addendum at the end could've covered many of them.
Toward the conclusion Lockhart notes, interestingly, that there has been precious little in the book about "reality;" rather, he is discussing mathematics in the abstract, as part of an imaginary world, or a world inside our own heads, while 'reality' is little more than the 'possibly illusory sensory input' from the world around us.
He also writes at one point, "…we love patterns. Mathematics is a meeting place for language, patterns, curiosity, and joy. And it has given me a lifetime of free entertainment."
To which I say, 'Thanks for sharing, Paul!!'
Despite the simple title, this is not a simple or altogether easy book… but it is an easy one to give a thumbs-up to for NON-math-phobes!
