Tuesday, May 31, 2011

Stewart Delivers... Again

"Instead of isolated clusters of scientists, obsessed with their own narrow specialty, today's scientific frontiers increasingly require teams of people with diverse, complementary interests. Science is changing from a collection of villages to a worldwide community. And if the story of mathematical biology shows anything, it is that interconnected communities can achieve things that are impossible for their individual members.
Welcome to the global ecosystem of tomorrow's science."
-- the end of Ian Stewart's latest book, "Mathematics of Life"

I received a review copy of Ian Stewart's latest book, "Mathematics of Life," causing me to ask just one simple question: does Dr. Stewart ever sleep!? Stewart is one of the most prolific popular math writers out there, and virtually always worth reading. I won't do a full review since I assume most readers likely already know and enjoy Ian Stewart's works... or, if you don't, than there's nothing I can do or say to cure your malady...

Certainly some of the material in this book has appeared elsewhere in previous Stewart books, but still this book is different (and unlike most popular math writing), in being so focused on the intersection of math and biology; a relationship that is rapidly expanding, but not often concentrated on.

From the title (and Stewart's past writing) one might assume the volume would be mostly math with biology sprinkled in, but actually I'd say the reverse is true: it is largely biological discussion with the math sprinkled in. The book is a great introduction to basic ideas for interested laypersons, or high schoolers and middle-schoolers interested in these fields, but also includes material for more advanced readers. The first 100 pages or so are largely historical and I thought a bit of a yawner, but following that are 200 very engaging and interesting pages surveying subjects from evolution, genetics and DNA, virus and protein structure, knots, symmetry, extraterrestrial life... a smorgasbord of topics. I especially liked Stewart's discussions of the biological concept of "species" and population dynamics, but there is much else of interest here, and the book progresses nicely as it moves along. Stewart consistently offers enough information for the reader to chew on and prod one's interest, but without being so technical or jargon-laden as to intimidate readers.
... so in short, don't be put off if the first 100 pages don't grab you; the next 200 pages are a great ride, that will have something appealing for every math-lover.

I recommend it for your shelf... right alongside a couple dozen other Stewart volumes... and, I also recommend Dr. Stewart go get some sleep now!

Monday, May 30, 2011

Kickin' Back With Cantor

...Whenever I have a day off (like today), and just want to relax a bit with some pizza and imported beer, I enjoy kicking back and... like yourself no doubt... watching some rollicking video on Georg Cantor ;-) :

Friday, May 27, 2011

Friday Puzzler

Here's one of Richard Wiseman's Friday puzzles (re-phrased) from a couple weeks ago:

A column of a particular building is exactly 200 feet high and 16 feet 8 inches in circumference. If a garland evenly wraps around the column in a spiral from top to bottom exactly five times than what is the length of the garland?

Wiseman's presentation and answer here:


Thursday, May 26, 2011

Wednesday, May 25, 2011

"Big Questions" from Tony Crilly

I've previously mentioned liking a "bite-size" math book from Brit Tony Crilly called "50 Mathematical Ideas You Really Need to Know," and also briefly mentioned that he had a new book out, "The Big Questions," that I hadn't had a chance to see yet.

It's finally available in the States and I'm enjoying it even more than the first volume. It is again a short (200 very accessible pages), delightful read for the person with a non-professional interest in math (...why are there so many excellent British math writers!?) He hits upon a wide range of key mathematical topics in a clear, succinct, enjoyable, instructive way: prime numbers, set theory, infinity, calculus, statistics, probability, chaos theory, topology, Godel, Cantor, philosophy, etc. I highly recommend the volume, and can't write any better review than the one already up at plusmaths.org:


...and on a side-note (speaking of Brits), hat-tip to Sol at "WildAboutMath" for pointing me to this collection of BBC math radio shows on the Web:


Tuesday, May 24, 2011

'When will I ever use this stuff?'

This non-profit site is trying to answer that question for students questioning what is math good for:


Check out their site, including their blog:


Monday, May 23, 2011

"All roads lead to mathematics"

Sort of a bizarre and interesting outcome utilizing Wikipedia:


...I've tried it a couple of times, and indeed arrived at 'mathematics' in some number of steps.

Sunday, May 22, 2011

In Memoriam

Can't let the day pass without acknowledging that Martin Gardner died one year ago today.

My single favorite book of his (and it's hard to choose) is still his older essay collection, "The Night Is Large," but for sheer fun, the compendium of his recreational math writings, "The Colossal Book of Mathematics" probably can't be beat.
All his books at Amazon here:


His memories spread across the mathscape.....

Saturday, May 21, 2011

Go Play...

For math students... of all ages, the latest 'Math Teachers At Play' blog carnival is up here:


Thursday, May 19, 2011

Tuesday, May 17, 2011

The Blind Spot

An overview of emeritus math professor William Byers' newest work, "The Blind Spot," a book more for the general science reader, but having philosophical application to mathematics as well, and not an easy book to give justice to in a review!:

Many years ago I didn't have much patience for philosophy of science, but since reading the ideas of Godel and Cantor, and more generally about uncertainty, the underlying tenets of mathematics and science have fascinated me. I've previously mentioned that William Byers' book "How Mathematicians Think" is my favorite book about mathematics. With the usual admonition that we all have different tastes, I can now add that I love this new book wherein he adapts the ideas from that earlier volume to the broader realm of science.

The new volume is a relatively short, 180+ pages, for such a broad topic, but very rich, even if at times wordy or redundant. And I'd call it a heavier, denser, sloggier read, than the earlier math book covering similar material. I've sometimes remarked that I think our educational system would benefit if we threw Shakespeare in the trash bin, but required all high-schoolers to read Godel and Cantor (well, their interpreters)... or perhaps now, just substitute Byers! (I may be biased though, as I've never encountered a volume that more precisely matched my own take on science than this offering -- it's uncanny how often some of the phrases and examples he uses mimic those I've employed myself at times.)

Byers' theme is laid out in the Preface... the rest of the book fleshes out, from different angles and sometimes in subtle, nuanced ways, the same message and ideas over and over.  I understand Byers' need to keep repeating things in different ways (almost to the point of redundancy) to make us consider basic points that are easily overlooked or taken for granted. We are entrenched in a viewpoint and bias that he is trying to penetrate and overcome.

The Preface opens this way:
"This book is about science, what it is as opposed to what people say it is; what scientists do as opposed to what most people believe they do...
"The popular belief in scientific certainty has two aspects: first, that a state of objective certainty exists and second, that scientific kinds of activities are the methods through which this state can be accessed. Yet I will make the case that absolute certainty is illusory and that the human need for certainty has often been abused with noxious consequences."
But actually the best summary of the book, perhaps, comes from this longer passage toward the end:
"It is certainly conceivable that the clarity we perceive in the world is something we bring to the world, not something that is there independent of us. The clarity of the natural world is a metaphysical belief that we unconsciously impose on the situation. We consider it to be obvious that the natural world is something exterior of us and independent of our thoughts and sense impressions; we believe in a mind-independent reality. Paradoxically, we do not recognize that the belief in a mind-independent reality is itself mind-dependent. Logically, we cannot work our way free of the bubble we live in, which consists of all of our sense impression and thoughts. The pristine world of clarity, the natural world independent of the observer, is merely a hypothesis that cannot, in principle, ever be verified.
"To say that the natural world is ambiguous is to highlight this assumption. It is to emphasize that the feeling that there is a natural world 'out there' that is the same for all people at all times, is an assumption that is not self-evident. This is not to embrace a kind of solipsism and to deny the reality of the world. It is to emphasize that the natural world is intimately intertwined with the world of the mind. In consequence, the natural world is a flow just like the inner world. By stabilizing the inner world through language, logic, mathematics, and science, we simultaneously stabilize the outer world. The result of all this is the recognition that the clarity we assume to be a basic feature of the natural world merely masks a deeper ambiguity.
"One of the functions of mathematics and science is precisely to deny this ambiguity. This is really the motivation behind the science of certainty." Yet, as he writes at another point, "...in the deepest and most profound sense, the things that make up the world cannot be defined, nor can they be understood or pinned down in any definitive way."
Byers argues through these pages that there is both a "science of certainty" (that most people recognize or assume) and a "science of wonder," which is closer to what science ought be -- in fact he calls the former a "simplistic misinterpretation of the latter." What differentiates the two is whether or not what he calls "the Blind Spot is acknowledged."  It is because subjectivity and objectivity CANNOT be consistently or definitively separated, that the science of certainty is illusory. Ambiguity, not certainty, is the true central element of science and is recognized by the 'science of wonder.' In turn, the reason for this is because we cannot escape the bubble of self-reference that all our thought and reason is trapped in, never permitting true certainty to prevail. This is our blind spot (metaphorically similar to the blind spot in our visual system, that we're usually unaware of, but is there nonetheless). There are in his words "limits to reason, to deductive systems, to certainty, and to objectivity." He is often echoing the work/thought of Gregory Chaitin.

Further, "Human beings have a basic need for certainty. Yet since things are ultimately uncertain, we satisfy this need by creating artificial islands of certainty. We create models of reality and then insist that the models are reality."
This is easily seen in the areas of religion and finance (and not always to positive outcomes) as Byers alludes to, but is endemic to science as well.

In the world of science there are 'participants' (or actors) and 'observers' (the scientists) who are considered to be more detached and objective, and thus have a more accurate view of things. But the observers themselves are participants in the larger scale (participating in the act of observing/recording -- the observers could be observed by other observers, who could be observed by....). The dichotomy is itself ambiguous or self-referential, or as Byers writes: "...it is impossible to definitively separate the objective from the subjective; they are joined in a unity whose complexity arises from the inevitability of self-reference."

Some may confuse Byers' viewpoint as "postmodernistic," but he is not at all a postmodernist. The postmodern view would have it that one's science outlook is molded by the culture and environment one is embedded in. For Byers (and me) it has little to do with culture or society, but the weakness or blind spot of science stems from the very intrinsic nature of our human cognition... it would be the same for all humans even if we all shared identical cultures and background. The human mind cannot get outside of itself, and science, as a product of that mind, can't achieve the objectivity it seeks or claims.  At times (and this may bother some readers), Byers is almost slightly "mystical" (my term, not his, although he might settle for the descriptor "transcendental"). This will trouble some hard-nosed science-types but I find it quite appropriate for what he is driving at, and something that science (and math), for all its self-aggrandizement, simply can't escape from.

Ambiguity and self-reference though are slippery subjects, by their very nature difficult to get a firm handle on. I've long been interested in self-reference, both in language and in science, and had hoped the creative, fertile thinker Douglas Hofstadter would have something profound to say about it in his 2007 book "I Am a Strange Loop"... but I find Byers' approach here more satisfying. Still, ambiguity is itself ambiguous, and self-reference is indeed a kind of endless loop, neither lending themselves to the empiricism of "science" -- which I think is sort of Byers' point: that empiricism is just one element in science; it ought not be seen as the equivalent (or even central factor) of science, and uncertainty is the rule, not the exception. Indeed, it becomes dangerous, when science is perceived as certainty.

How does this all apply to math... well for starters of course, math underlies science; the sense of pattern and order that is intrinsic to math is heavily impacted by the ambiguities and uncertainty Byers addresses. So too are the axioms which are the most fundamental components/assumptions of mathematics and logic. Even basic concepts of "number," "quantity," and "measurement," are far more difficult to pin down than our everyday usage would lead one to expect. The dichotomy drawn between discrete and continuous numbers is just one example of concepts in conflict that pervade mathematics.

If philosophy bores you, as it once did me, you might take a pass on this volume... but personally, I believe it should be read and contemplated by every scientist... and even applied to their own endeavors.

Anyway, another review of Byers' book here:


Monday, May 16, 2011

Close Encounters of the Math Kind

I think I can just make out Rod Serling (or, is that Paul Erdos?) in the shadows about to offer a few cogent words....

RJ Lipton reports on his own personal recent "close encounter of the proof kind" here:


Saturday, May 14, 2011

"one quadrillion calculations per second"

For those who just can't get enough of pi (or otherwise have too much free time on their hands ;-):


(....the 60-trillionth binary digit of pi-squared calculated)

Friday, May 13, 2011

Friday Arithmetic...

So as not to spook you, a fairly easy arithmetic problem for this Friday the 13th:

The last 'palindromic' year (reading the same backwards and forwards) was 2002. What is the difference (subtraction) between the next palindromic year to come and the last one prior to 2002? (answer below)
answer: 121 (2112 - 1991) interestingly, also a palindrome

Wednesday, May 11, 2011

Tuesday, May 10, 2011

Biomathematics from Ian Stewart

A little history of the intersection of mathematics and biology:


"I would be surprised if mathematics ever came to dominate biological thinking in the way it does physics, but it is rapidly becoming an essential part of the discipline: 21st-century biology makes use of mathematics in ways that no one would have dreamed of at the start of the 20th. By the time we get to the 22nd, mathematics and biology will have changed each other beyond all recognition, just as mathematics and physics did in the 19th and 20th centuries."
(from the piece by Stewart)

Friday, May 6, 2011

The Prediction

Classic Wiseman...

For a Friday jump-start, here's an older video (with a mathematical basis) from "quirkologist" Richard Wiseman:

Thursday, May 5, 2011

Play Time

The 37th "Math Teachers At Play" blog carnival, with lots of articles geared to teachers and students... of all ages, is up here:


Wednesday, May 4, 2011

Wednesday Math Miscellany

There are plenty of mathematical blog posts that are so advanced and beyond my comprehension they simply leave me in their dust. Then, for some reason, there are other posts, equally incomprehensible to me, that nonetheless catch my fancy and somehow capture my interest. Here is one such recent post from Peter Woit (centered around "octonions"):


On a different subject, here is a recent article on Millenium Prize winner/refuser Grigori Perelman:


Tuesday, May 3, 2011

The Treachery of Language

One of the things I like about mathematics is that it valiantly attempts to deal with (and mitigate) the ambiguity and vagueness that is inherent in language. Most sciences, particularly the life sciences, often seem somewhat oblivious to the imprecision of words they employ, and the degree to which concepts routinely discussed are either tautological in nature, or else simply vaguer than is implied by the discussion. Mathematicians, more than most, recognize the problem and are willing to face it head on... even when there is no resolution.

Anyway, below a post from "CTK Insights" well-illustrating the treacherous nature of language and meaning when it comes to even simple mathematical terms, in this case inquiring "is a point a part of a line?"... and the answer ain't easy!:


(On a side-note, I'm now getting around to reading William Byers' new book, "The Blind Spot," which I've referenced before, and so far it's as good or even better than I expected, delving into many of these same issues of meaning and ambiguity. A fuller review probably somewhere in the future.)

Monday, May 2, 2011

Apparently, About 40...

How many Navy Seals does it take to bring elation to America?

(CONGRATS! to all involved!!)