Waxing Platonic…

The Platonic divide in math....

Ramanujan

The book I've highlighted recently, "The New York Times Book of Mathematics," ends with a chapter of readings on various notable mathematicians… Erdos, Ramanujan, Conway, Gödel, Wiles, etc. I suspect most (if not all) of the brilliant figures profiled were/are Platonists (mathematics is discovered, not merely created). Yet many other recent math figures (Reuben Hersh, William Byers, Keith Devlin, Jim Holt, and more) have forcefully argued that mathematics is indeed a mental creation that might even differ considerably in a different Universe than ours -- indeed some almost seem to find the notion of mathematical Platonism so wrong-headed as to be silly (while Martin Gardner found the non-Platonist view almost silly). And occasionally such writers cause me to sway toward their non-Platonist stance though I always seem to float back toward Platonism.

One thing that so many of the greatest, most productive mathematicians seem to share is an uncanny, almost inexplicable ability to tap into a realm of intuition or mental landscape not readily accessible to most of us. Ramanujan is certainly the unparalleled, most inexplicable, example of this; producing amazing mathematical results that are still today being explored and proven. Reading James Gleick's portrait of Ramanujan in the Times volume it really hit me… was Ramanujan, who routinely produced such results/theorems without ever showing the steps that led to the outcome, in direct access of the "Platonic realm?" He himself claimed his insights came in dreams and trances directly from the Indian Goddess Namagiri... Who are we to argue (and where did she reside)?!!

In many ways, Ramanujan's extraordinary talents are reminiscent of the incredible abilities of various mathematical savants and prodigies who usually can't explain how they do what they do. Their brains seem clearly to operate, or even be wired, differently from those of 'ordinary' people.

My point in all this is simply that such rare, yet nonetheless real, individuals DO give an appearance of tapping into a realm… call it perhaps the Platonist realm… that the rest of us lack ready access to, where numbers and math really DO exist apart from our day-to-day world.  Naysaying non-Platonists will simply argue that however Ramanujan and the rest gain their special knowledge, it ultimately still arises via the firing of neurons within a physical human brain situated between two ears… i.e. it is still a human creation. I can't prove that reductive view wrong, but the notion that there are worlds out there that only some of us can easily tap into, and only some of the time, through means we don't even comprehend… is so much more appealing! As Shakespeare put it long long ago, “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.”

I think Martin Gardner might well relate to this idea too… For all his empirical skepticism, Gardner also described himself as a "Mysterian," a philosophical view which holds that ultimately consciousness cannot be explained by any human brain. In the famous words of computer scientist Emerson Pugh, "If the human brain were so simple that we could understand it, than we would be so simple that we couldn't."  Is it possible that humans are able to draw upon a Platonic world, and can recognize 'consciousness,' yet perhaps never, with our limited minds, fully grasp either? Does the 'Platonic world' exist, but like the Continuum Hypothesis, fall into a nether land of things that simply can't be proved true or false by human logic?
Speaking of certain mathematical proofs, Paul Erdos would famously say, "This one is from The Book!" I'm not so sure he was speaking in metaphor... perhaps The Book, in some (Platonic) manifestation, exists. Is the alluring beauty of math only in our heads, or is it an integral part of all creation? MIT physicist Max Tegmark has argued for some time now that the entire physical universe, as we perceive it, is nothing more than mathematics, or a mathematical structure (called the MUH, or "mathematical universe hypothesis").

Anyway, read Gleick's beautiful 1987 portrait of Ramanujan and just imagine the Indian mystic-mathematician dreaming and dipping into a realm where numbers are as 'real' as rocks and chairs are to most of us:

http://www.nytimes.com/1987/07/14/science/an-isolated-genius-is-given-his-due.html?

a couple of brief lines from therein:

" 'When he [Ramanujan] pulled extraordinary objects out of the air, they weren't just curiosities but they were the right things,' said Jonathan M. Borwein of Dalhousie University in Halifax, Nova Scotia...
" 'He seems to have functioned in a way unlike anybody else we know of,' Dr. Borwein said. 'He had such a feel for things that they just flowed out of his brain. Perhaps he didn't see them in any way that's translatable.' "

It's probably also worth noting that the very first entry in the entire NY Times anthology is a 1998 George Johnson piece also addressing the subject of Platonism:

"Useful Invention or Absolute Truth: What Is Math?" by George Johnson

At the end of the piece, Johnson cites a 1995 book, "Conversations on Mind, Matter and Mathematics" that covered a debate between French mathematician Alain Connes and French neurobiologist Jean-Pierre Changeux over the subject of math Platonism. An interesting and rich review of that book here (even makes brief reference to Ramanujan):
 
http://www.timeshighereducation.co.uk/161513.article 

Connes and Changeux didn't resolve the debate... and we won't here... but still, nourishing food-for-thought.

Math via The New York Times

Non-technical math anthologies are rare critters… when one comes along my instinct is to pounce on it. "The New York Times Book of Mathematics," edited by Gina Kolata, was worth the pounce!

This volume covers a wide and interesting array of topics. Here are the 7 chapter headings (though they don't fully hint at the range of material touched on):

1) What Is Mathematics?

2) Statistics, Coincidences and Surprising Facts
 
3) Famous Problems, Solved and As Yet Unsolved
 
4) Chaos, Catastrophe and Randomness
 
5) Cryptography and the Emergence of Truly Unbreakable Codes
 
6) Computers Enter the World of Mathematics
 
7) Mathematicians and Their World

That should give you a sense of the breadth of topics on display here. The pieces are vibrant, terse treatments (no doubt only intended to fit within a certain column length). The writing is so good that the pithiness leaves one reaching the end of most pieces wanting more... just one more page pl-e-e-ease.

I think Gina Kolata sets the tone and 'feel' of this engaging volume very aptly when she writes in her Introduction:

"A mathematician once dismissed the very idea that people outside his circle could ever understand the true essence of the field. Mathematics is an art form, like music or painting. Translating math into the English language, he said, is harder than translating Chinese poetry. The beauty is lost, the elegance, and a proof that is a thing of ineffable iridescence becomes reduced to a baffling or mundane-sounding bottom line....

"But even if the rest of us cannot appreciate mathematics as an art form, are we really shut out? Articles in the New York Times may not give the details of proofs, but they reveal a rich world that can be exciting, surprising, and can even tug at the heartstrings."

Yet several reviews I've seen of the volume are rather ho-hum about it, but these are usually from professional mathematicians -- for the working mathematician there may not be that much here to excite -- although I think any math lover will find at least a few pieces that strike a chord. But for lay folks with an interest in math (my core readership!!) this may be the BEST anthology I've ever come across! There is no technical material or equations to weigh down your enjoyment, nor slow your consumption. It is all about math and mathematicians... without doing math.

One downside is that because this is limited to NY Times' writers, many excellent popular math writers are absent.  Indeed, I'd normally be skeptical of an "anthology" that was restricted to the number of writers this one is -- it is very heavy on pieces from Gina Kolata and James Gleick -- but these writers are SO good at their craft that skepticism quickly fades away. While Kolata and Gleick's pieces are perhaps the best, there are numerous fine contributions as well from George Johnson, John Markoff, Dennis Overbye, and others. Oddly, there are no entries from Steven Strogatz here (author of some of the most popular math pieces the Times has carried in recent years), but perhaps his offerings simply didn't make the 2010 cutoff for the volume. The one other thing that may be missing from the collection, it seems to me, are more articles which relate math to the other sciences, particularly physics and biology (I believe the Times has run several such pieces).

Most of the entries come from the last 3 decades or so, but some go back as far as the late 1800s. I wasn't particularly enamored of several of the older entries that were probably included more for the sense of history or progression they illustrate than for the math covered. Still, overall the mix is appealing.
 
Chapter 5, focusing on cryptography, would have been interesting in its own right, but became even more-so, in light of current events, as almost every article makes mention of the NSA and its relationship with mathematicians (by most accounts, by the way, NSA is the largest employer of mathematicians in the world). But there isn't a bad chapter in the volume.

In short, I love this compendium, even more than I expected to. If it wasn't such a thick, heavy volume I would almost recommend it, at this time of year, as a 'beach-read'… for the mathematically-inclined. In the distressed world of print journalism, the NY Times has been cutting back on science journalism, so it is wonderful to have this hard-copy of delicious math-related essays to keep on one's shelf as a permanent source of popular math writing stretching across decades. Hats off to Ms. Kolata on a job well-done!

Flipped Classrooms, MOOCs, and Having a Blast

Timing is everything....

Well, this was great… I was planning to write a post musing a bit more about math education in regards to both "flipped classrooms" and MOOCs… but then discovered Keith Devlin has just put up a new (longish) post on his MOOC blog saying most of what I wanted to say, and with more authority than I could say it. So please read it:

http://mooctalk.org/2013/06/03/the-mooc-will-soon-die-long-live-the-moor/

Do note that I think his title may be a bit misleading so follow carefully all he has to say. I was afraid his long lapse in blogposts might mean that the 2nd rendition of his 'mathematical thinking' MOOC hadn't proceeded well (though his insanely busy schedule could also account for it), and luckily it doesn't sound like that was the case… though he does still write with caution about MOOCs, and will have more to say in the future about this last go-around.

Here are a few of the most trenchant comments he makes (I've added some emphasis):

"the vast majority of people under twenty now interact far more using social media than in person.
We could, of course, spend (I would say “waste”) our time debating whether or not this transition from physical space to cyberspace is a good thing. Personally, however, I think it is more productive to take steps to make sure it is – or at least ends up – a good thing. That means we need to take good education online, and we need to do so for the same reason that it’s important to embed good learning into video games…
"The media of any age are the ones through which we must pass on our culture and our cumulative learning."

"Something else that digital technologies and the Web make possible is rapid iteration guided by huge amounts of user feedback data – data obtained with great ease in almost real time."

The one place where I think Keith sounds a little too negative is when he writes:

"Experimentation and rapid prototyping are fine in their place, but only when we all have more experience with them and have hard evidence of their efficacy (assuming they have such), should we start to think about giving them any critical significance in an educational system which (when executed properly) has served humankind well for several hundred years. Anyone who claims otherwise is probably trying to sell you something."

Actually, I think "experimentation and rapid prototyping" may now be an integral part of our quickly evolving world and education system… more than ever before change can happen with such speed that we may try 4 failed experiments and still succeed at #5 in an acceptable/practical amount of time (even before the "hard evidence of efficacy" is fully in or agreed upon. Just the speed with which the MOOC movement has grown is a testament to that, and as Keith implies, the time is ripe for us to "make sure" they [MOOCs] work in some form.

So much for MOOCs…
What actually got me thinking again about education was a recent Twitter tweet that led me to this blog I was previously unfamiliar with:

http://flippingwithkirch.blogspot.co.uk/

Despite the uk URL appendage it's from a California high school math teacher (Crystal Kirch) focused on the "flipped classroom" concept. Just scanning over it, it looks interesting and impressive to me, but as someone not in the loop of secondary education I don't want to assume too much. What definitely caught my attention though (and those of you in secondary education likely already knew this) was the sheer number of other blogs with a similar focus on flipped instruction (as well as a network of teachers with this interest) that Mrs. Kirch links to. The "flipped classroom" has been around long enough that LOTS of teachers are trying it, tweaking it, playing/experimenting with it, blogging about it, and just generally sharing their experiences (good and bad) with their peers. What a great collaborative endeavor!!… and not brought on by some agency-directed-commissioned group-on-high, but by the spontaneous interest of those who share similar goals. Again, before the internet this sort of rapid cross-communication effort wasn't possible.

The term "flipped classroom" came about, so far as I'm aware, from early uses of Khan Academy videos (and Khan Academy still has many vocal critics), but of course there are now MANY internet resources available to choose from, and Khan itself constantly evolves. (Some have noted that the 'idea' of the flipped classroom, though not the term itself, actually long precedes Khan Academy.)

It is fascinating to me how both "flipped classrooms" and MOOCs, which in some ways share little in common, and operate on different levels of education, have simultaneously sprouted up like mushrooms in the cyber landscape, both controversial and rapidly-evolving, yet giving tremendous promise.

As Keith writes so aptly at the end:

"Those of us in education are fortunate to be living in a time where there is so much potential for change. The last time anything happened on this scale in the world of education was the invention of the printing press in the Fifteenth Century. As you can probably tell, I am having a blast."

And some of us are just having a blast... watching those of you who are in the trenches having a blast.
To Keith, and Mrs. Kirch, and all others doing the nitty-gritty work that will shape the education of future generations... THANK YOU!

Vickie Kearn... She Reads 'em Before You Ever Hear of 'em

 Math-Frolic Interview #14

"Appreciating the power of math and what it has and can do for us is really important. It isn’t just a lot of numbers, it is about people and applications and improving everything we care about." -- Vickie Kearn


You're likely not as familiar with the name 'Vickie Kearn' as most of the other names I've interviewed here... but it's a joy for me to bring her forth from behind the professional curtain where she hangs out. As an editor for Princeton University Press (a favorite of mine) she shepherds a great many of the books and authors we come to love, to our bookstores for us to read. It's fun to gain a better sense of how that whole behind-the-scenes process works. Read on (I've emphasized a few bits with bold):

********************************

1) To start, can you tell readers a little about your background or anything else pertinent to your math interest and your editorial position with Princeton University Press?

I have always loved math and had teachers who encouraged this love. When I was 10 my parents moved to Venezuela so when I was 16 I had to come back to the States to go to boarding school since there were no English schools where we lived. I went to a very small school and had the same math teacher (Elsie Nunn) for three years. She was wonderful and we had math club every day. Now you might think that was a bit much but she was so exciting and told terrific stories about the people behind the math. She could do all sorts of things with simple tools like sting and cardboard. I remember that she was double jointed and could draw a perfect circle.

When I went to college the women took classes on one side of the lake and the men on the other. Lucky for me, the advanced math classes were all taught on the men’s side of the lake. I went to a Baptist school and men and women could only talk to one another on certain days of the week. Because of my math connection, I got to talk to them every day of the week. Perhaps not a reason to major in math, but a really neat benefit.

After college I taught school for eight years (elementary and middle school math) and then moved to New York City to begin a career in publishing. I had pretty much had it with the books I was given to use in my classrooms and thought that I could make a difference if I could find a way to get more interesting and useful books published, especially in math. I started my career at Academic Press as a developmental editor. I had to read all of the math textbooks, work all the problems (to make sure all of the needed information was there and that the problems could actually be worked) and write the solutions manuals. After three years I did not think I could work out one more calculus problem and really wanted to get on with finding those books I so sought. I moved to Marcel Dekker where I was an acquiring editor. I actually got to look for authors who could write the next great math text. I didn’t know if when I was hired as the math editor that I would end up also working on statistics, electrical engineering, quality control, and food science. Although I got a lot of experience, I was not making much progress with my math hunt. I then went to the Society for Industrial and Applied Mathematics which was heaven because it was all math. I worked on books and journals as well as conferences and membership drives. This was exactly what I was looking for.  This was a great job but I found that there were some titles that I longed to find but which were not good fits for the society. I wanted to reach out to other disciplines to show how math could be used not just in the sciences but also the humanities and the social sciences. I wanted to bring more math to general readers—the math lovers and the math haters and math phobics. It was difficult for a math society to reach all of these different audiences without publishing books in all of these areas.

Princeton University Press has been a perfect fit for me. We publish in almost every discipline you can imagine. We have sales reps who visit bookstores and publicists who visit major media outlets for print, TV, and radio. The Press and our editorial board are willing to try new things like books of puzzles and graphic novels.

2) Prior to this point, I've been interviewing individuals who are direct math communicators, bloggers and/or authors. You're sort of a layer back as a gatekeeper of the very sorts of other folks I normally interview. That makes it interesting, because many readers won't know your name and yet you are probably more personally familiar, than those readers, with the very names they are so familiar with! Can you say a little of what it's like to work with distinguished math authors as you mentor their idea or first draft from proposal to publication? And do you build personal friendships with many of these writers as you collaborate with them over time, or is it more of a strictly arms-length business relationship?

Many of my closest friends are mathematicians.  In a way I have grown up with them and my son is the same age of many of their children. In addition to our love of math and great books, we share this common bond of raising kids, sending them to college, and watching them find their way. When I began my career in publishing in 1977 I did not know anyone. There was a lot more competition than there is now and there were many seasoned editors who had built a core of authors who always published with them. I decided that the best thing to do was to contact the people at the top of their careers (all the big prize winners) and ask them about their brightest students. These are the people I contacted and talked to about what books they needed or would have been helpful when they were studying math.  Since these were the rising stars they were soon in a position to write books and they remembered me when they were considering a publisher for their book. Now they are the prize winners and I am still publishing them and they are now recommending their students. Every book I have published has been special in one way or another. I have had the privilege to meet the most honored and famous mathematicians of our time. I also have had the honor to meet some of the greatest teachers around the world. You don’t have to win a lot of prizes to write a good book. You do have to be creative and be passionate about your subject. The trick to publishing great books is finding these people. Sometimes an author comes to me with a completed manuscript that is almost ready to go. The books that are the most fun, however, are the ones that we design together from the seed of an idea to a finished product that is widely read. This can take several years which is plenty of time to establish a lasting friendship.

 [....Sounds like a dream job!! ;-)]

3) Princeton University Press puts out some of the most consistently excellent, interesting, well-designed math books of any publisher! So I'm curious how that selection process works so successfully. Can you describe a little of how things proceed from the time a writer approaches PUP to the time a book is accepted for publication and finally produced, and what is your role along the way?

Princeton University Press cares a great deal about its authors and the books it publishes. Each book is carefully selected to ensure that it is accurate, fits a specific audience and is pleasant to look at and read. Sometimes authors come to me with an idea and sometimes I think of a topic I think will be great for a book and seek an author who would be perfect to write it. This can take a long time so you have to be patient. I have given up on a topic at times when I can’t find the right author. The first step in our process is to put together a proposal for a book. I then present it to my colleagues who help me decide if the topic fits our list and if we can promote the book effectively. If so, I have the proposal reviewed and if the readers are positive, we offer a contract. We might work on the development of the project over many years or it might come together quickly. Once the final manuscript is complete, it is sent out for a final review. If the book is for the general reader or an undergraduate textbook, I read though it as well and give the author advice on changes to consider. If the reader reports suggest more work, then the author revises and we send it back to the readers. If the suggestions are minor, I take the book to our editorial board for final approval. The board consists of five Princeton University professors across all disciplines who approve books for publication based on my recommendation and those of the readers. They ensure that the book reflects the mission of the University as well as the Press.

The production process is a careful one. We copyedit all of our books and redraw art where necessary. We have designers who look at the book to make sure that the manuscript will be laid out it the most user friendly way. They also are responsible for designing a cover that is attractive and reflects the content of the book. During production, our publicity, marketing, and sales departments are all preparing materials and contacting people to make sure that our newly published books will be as noticeable as possible and will get into the hands of readers.  During this time I am solving any problems that arise and working on getting endorsements which will go on the cover of the book. I no longer have to write solution’s manuals but I make sure that the authors are. I also help authors come up with ideas for ancillary material that they might want to put on the webpage for their book. We are also developing Facebook and Twitter accounts for each book during this time.

[Very interesting to hear about the whole process! Needless to say I think PUP achieves its goals well -- your math books are always very readable, informative, AND very attractive to look at!]

4) Roughly speaking, of proposals you get for math fare, what percentage might PUP generally end up publishing? And is it possible to generalize about what the most common reason for rejecting a proposal is?

The sciences are different from the humanities and social sciences where it is imperative to write a book or two in order to get tenure. The editors in these areas are deluged with proposals. In math, the opposite is true. They are writing and publishing research papers to get tenure. Most of the proposals I get are from direct recommendations from someone I know, an author I have already published, or someone I have approached so I don’t get a lot of unsolicited proposals. I do get a few and look at each one carefully before deciding what to do. Many are rejected for various reasons and others get published. The most common reason for rejecting a proposal is that it is totally wacky or not prepared properly. Sending an editor a proposal with hand drawn figures and no coherent description of the book or who you are writing for is a good sign the book will not be worth publishing. Just as you would never think about sending in a resume that is loaded with typos for a job application, you should check your proposal carefully to make sure it states what the book is about, why it is important, who it is for, what the reader will gain from reading the book,  and what the competition includes. Oh yes, and check for typos!

I publish a very small percentage of the unsolicited proposal I receive. However, there is a very high publication rate of those that are recommended to me or that I go looking for.

5) Obviously, any book you choose to publish, you believe is well-done and will have an audience, but are there any examples of Princeton math books that especially surprised you with the volume of their sales?

As I said, each book I work on is special in some way. It may be that it sells only 600 copies but the readers use the information inside to solve some great problem or advance a new area of math in some way. Others might sell tens of thousands and get high school students excited about math. All are important in my mind. Some books get great reviews and just don’t live up to expectations. Others get little notice in the media but find their way and outsell our expectations.

6) On the other side of the coin, have you ever been involved in rejecting a book for Princeton, only to see it become a major seller for another publisher, and thought, 'ohhh man, why did we let that one slip away!'…?

On several occasions I have rejected books that I know will sell well but which didn’t meet our mission in one way or another. Every publisher wants their books to sell as many copies as possible but not at the risk of getting negative reviews and possibly damaging a reputation that has long been established. I actually can’t think of a book that I was sorry I rejected for these reasons.

7) Can you tell us anything about some of the math titles/topics/authors we have to look forward to coming down the pike shortly? 

I just presented my fall 2013 list to our sales reps and there are some great books on that list. They include:

Undiluted Hocus Pocus: The Autobiography of Martin Gardner -- This is one of the last things that he wrote before he died. He was a very private person and even his closest friends have learned a lot from reading the manuscript.   

Beautiful Geometry by Eli Maor and Eugen Jost is an illustrated guide to some of the major ideas in geometry. It includes proofs, history and art designed just for this book.  

Wizards, Aliens, and Starships by Charles Adler is all about the math and physics in fantasy and science fiction. Which cool things could actually happen and which are impossible?  

Will You be Alive Ten Years From Now is a book of probability puzzlers by Paul Nahin who is a perennial favorite. He has published many books with us and has a very loyal following. 

 [Ohhh Wow, these sound FANTASTIC!!! And an autobiography from Martin Gardner... I'm almost drooling over the keyboard thinking about it... who knew there would be yet more Martin to enjoy 3 years after his demise. I'm not familiar with Adler, but Maor and Nahin are other favorites. THANKS so much for letting us know about these ahead-of-time!]

8) When you're not editing math books, what are some of your other main interests/hobbies/activities?  

I like to tutor kids who are struggling with math and I volunteer for a pet rescue. I also like to read and solve logic problems.

9) Any parting words, not covered above, you'd want to pass along to an audience of math readers and enthusiasts?

Every day I try to find someone who does not like math (or thinks they don’t) or thinks it is hard and convince them that math is fun and is not really that hard (at least on some level). Appreciating the power of math and what it has and can do for us is really important. It isn’t just a lot of numbers, it is about people and applications and improving everything we care about. From sports to medicine to ensuring we are safe, math plays a large part. If every reader could convert another person every day, we soon would have a hard time finding people who don’t like math.

...A great thought to end with!

THANKS, Vickie, for giving us an inside look at how the books we enjoy so much end up in our hands.

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And if you want to hear her voice, Vickie was also interviewed last year as part of Sol Lederman's podcast series here:

http://wildaboutmath.com/2012/06/03/vickie-kearn-inspired-by-math-8/

From Whence...?

"The more the universe seems comprehensible the more it also seems pointless."
-- physicist Steven Weinberg as famously quoted in Jim Holt's "Why Does the World Exist?"


I'm currently finishing Jim Holt's bestseller (and one of the NY Times' Top 10 nonfiction picks for 2012), "Why Does the World Exist?" (now out in paperback). Won't write a full review since it's more philosophy than math or science, but will recommend it to those with a philosophical bent, or who have enjoyed any of the other recent books on that most fundamental of questions, 'why is there something instead of nothing?"

The first half of Holt's book, while good, takes a little bit of time to gain traction, and the second half is especially good and invigorating. Holt keeps the sometimes deep philosophical and theoretical discussion not only accessible, but also moving along at a pace which doesn't get too bogged down with any one set of arguments or thinker. If you are someone who scoffs at the very idea of reading an entire book on a question that essentially can't be answered, then you'll want to pass on this volume, but if you enjoy seeing the wide variety of cerebral exercises major thinkers have employed to approach this basic conundrum than Holt takes you on a good ride, tossing in personal anecdotes along the way.

My favorite chapter (not too surprisingly) is chapter 10 on Platonism, where, in addition to Plato, the likes of Kurt Gödel, Roger Penrose, Max Tegmark, Bertrand Russell, and Hartry Field are among those making appearances in the debate over whether there is an independent platonic realm of mathematics (apart from human consciousness), or is mathematics merely a human construction. Both sides have very astute and brilliant proponents.

The mini-portraits of the many fascinating individuals Holt discusses or holds court with in this book are just as interesting as the ideas they put forth. John Updike fans will find a chapter toward the end (#13) with their literary hero. Heidegger, Quine, Wittgenstein, Leibniz, Richard Swinburne, David Deutsch, Adolf Grunbaum, Derek Parfit, John Leslie, Thomas Nagel, Robert Nozick, are among the intriguing panoply of players (living and dead) who are aired in these pages (many individuals who I was not previously familiar with at all).  Some of the book's philosophical discussion is a bit muddied in semantics (as could be expected), and I prefer the discussions with scientists, but Holt deftly works his way through all of it, be it religion, cosmology, or quantum mechanics. And at the end comes a moving chapter on the death of his own mother.

For some extended reviews of the book here are two (of several) from the Web:

http://bnreview.barnesandnoble.com/t5/Reviews-Essays/Why-Does-the-World-Exist/ba-p/8475

http://articles.washingtonpost.com/2013-02-08/opinions/36990295_1_wrong-question-answer-universe

Of course the volume reaches no final resolution and there is a bit of predictability in so much as certain issues keep recurring as sticking points, yet the intellectual exercise remains entertaining. Holt notes early on in the volume that the question of why the world exists is "so simple that it would occur only to a child." Maybe that explains why, despite its intractability,  we find it such an irresistible inquiry... it makes us all feel like children again in this great big farfetched universe of ours (...or, multiverse).

One last thing:
The quotation I've long-used from Bertrand Russell heading my Math-Frolic blog comes from early in his career when his optimism about mathematics and empiricism was strong. Holt's book (again the chapter on Platonism) introduced me to another quote, I was unfamiliar with, from much later in Russell's storied life. I love the quote, and change-of-heart it expresses, so much so that I've added it to the Math-Frolic blog heading, and will leave you with it here:

"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal."

[NOTE: I've actually since moved the paired quotes over to the MathTango heading.]

[...This all reminds me of yet another brand new book now showing up in bookstores: not meaning to stray too far from mathematics, but computer scientist/keen-thinker Douglas Hofstadter's latest tome is, "Surfaces and Essences" (about the role of 'analogy' in human cognition), and I suspect it is must-reading for anyone interested in cognition. (Hofstadter was author of "Gödel, Escher, Bach," one of the most acclaimed works of nonfiction of the last half-century, and several works since. He also took over Martin Gardner's column at Scientific American for a few years back when Gardner retired.)]

Beholding Mathematics...

“Beauty is the first test: there is no permanent place in this world for ugly mathematics.” -- G.H. Hardy


Just a little Sunday rumination on mathematical beauty today….

Math bloggers come in a variety of flavors, sometimes with few specific overlapping interests, but what they often hold in common is an overarching desire to share math's beauty and wonder with others.
"The beauty of mathematics" is a phrase most math bloggers have probably used at some point… it rolls off our tongue (or keyboard), so obvious is the "beauty" of mathematical thought and application we see, despite its frequent lack of resonance with other lay folks.

This article about young Princeton University professor Manjul Bhargava well captures his enthrallment with the "beauty" of math:

http://www.livescience.com/28508-numbers-manjul-bhargava-nsf-bts.html

In it Bhargava says: "When you discover things about numbers, it's very beautiful. When mathematicians are thinking about their problems, we're not thinking about their various applications, but rather are pursuing beauty. That's how pure mathematicians think."

Bhargava is of Indian origin... same as the remarkable Ramanujan who also found incredible, and uncanny, beauty in mathematics, which he felt was communicated to him directly in visions from an Indian goddess. 

And most anyone who reads science on the Web has probably seen this Richard Feynman clip which goes semi-viral from time to time. In it he describes, touchingly, how he, as a scientist, is capable of perceiving beauty, just as much as one of his artist friends can do… perhaps even more-so. As he essentially says toward the end, 'How does science detract from the ability to perceive beauty… it only adds, it only ADDS!!' …and so too mathematics, one could say:



So why do people universally comprehend the notion of "beauty" when applied to art or music or fashion or even food, but not when it comes to mathematics?
Usually beauty is beheld directly through the senses, vision, hearing, taste, touch… but math is more abstract, beheld at a different cerebral level where we discern patterns or connections mentally, whether or not they are physically evident.
Certainly not all math is beautiful, and it is indeed interesting how some theorems or proofs can be viewed as elegant or beautiful, while others simply are not (EVEN though they may be equally true or valid). As Paul Erdös was fond of saying some proofs are "from the Book" (God's creation manual).

An article from Smithsonian Magazine notes "...two of the essential requirements for mathematical beauty. First, it is surprising... Second, it is simple."

Read more:
 http://www.smithsonianmag.com/science-nature/the-natural-beauty-of-math-174842751.html#ixzz2KS3JBicz

Meanwhile S.Lang, in "The Beauty of Doing Mathematics," addressing those who view mathematics as dry and dull, writes "Last time, I asked: 'What does mathematics mean to you?' And some people answered: 'The manipulation of numbers, the manipulation of structures.' And if I had asked what music means to you, would you have answered: 'The manipulation of notes?' "

Robert Krulwich, Marcus du Sautoy, and Simon Singh explore the topic in this short clip from the World Science Festival:



And below a longer piece from the same Festival (including, at the 76-minute point, comments on Platonism vs. non-Platonism in math):

http://worldsciencefestival.com/videos/mysteries_of_the_mathematical_universe

There the debate continues over whether mathematics is discovered or invented; whether it exists in some outer 'real' world or only within the processes of the human brain.
And one may reflect on and on and on about mathematical 'beauty,' but surely it at least (like all beauty), ultimately exists in the… mind… of the beholder (be that beholder Man or God).

Of P and NP

"The Golden Ticket" by Lance Fortnow


Who'd a thunk it!? …that somebody could write an engaging, fascinating account of the P vs. NP Millennium Problem for a mass audience? Moreover to have done so without ever too-technically defining either P or NP, nor introduced much of the jargon or mathematics one might expect such a treatise to require! Hats off to Scott Fortnow (a blogger at "Computational Complexity") for doing it!

P vs. NP is one of the seven famous Clay Institute Millennium Problems for which one earns a cool $1 million for simply finding a solution (… but of course "simply" is NOT the operative word! -- thus far only one of the problems, the Poincare Conjecture, has been solved… and, ironically, the solver, Grigori Perelman, refused the prize money!).

Fortnow's book has been well-praised by reviewers, and oddly it reminds me a bit of Jason Rosenhouse's "The Monty Hall Problem." Both books are uncannily the same size and shape (and about the same number of pages), but more importantly, just as Rosenhouse did a wonderful, meticulous, intricate job of explaining the Monty Hall Problem (and its many nuances) to a lay audience, Fortnow does a similar job for P vs. NP, with a book that is, surprisingly, a bit of a page-turner.

I suspect I could read Fortnow's work 3-4 more times and each time take away a little more knowledge and understanding of the depth of P vs. NP, such is the richness of the volume. The content and implications of the book crosses boundaries of computer science, philosophy, math, logic, complexity, and number theory.

For any who don't know, P problems are those that can essentially be solved relatively quickly; i.e. "efficiently" (via a computer at least), while NP problems cannot be solved in any reasonable amount of time, even with the aid of computers, even though proposed answers CAN BE checked quite quickly. And the conundrum is to discover if these two classes of problems are in fact one-and-the-same, i.e. P = NP… or, as MOST researchers believe, do P and NP problems represent distinctly separate categories. To the uninitiated it may seem like a question of hugely abstract or narrow interest, but as Fortnow notes, "Determining whether P = NP is the most important question in computer science and perhaps in all mathematics." [bold added] Again, most believe P ≠ NP, but this too is exceedingly difficult to PROVE.

The so-called "Travelling Salesman Problem" is likely the most well-known of the NP problems… given say a few hundred cities to visit, find the shortest, most efficient route for visiting them all (…might seem to some like a straightforward problem for a modern-day computer to solve… but, it ISN'T! there simply is no algorithm, nor brute force technique, to solve it in a human time-frame.
The initial chapters of the book employ made-up examples to communicate the nature of P vs. NP. Often I prefer the use of 'real-life' type examples over invented ones to get across mathematical or scientific ideas, but Fortnow's concocted versions are so good and entertaining that they serve his purposes well. Chapters 4 and 5 go into some of the history of P vs. NP… normally, a book might start off with the historical part, but I think Fortnow succeeds in drawing you into the whole subject with the introductory chapters, and only then putting forth the history which may be a little dry or tedious and not the best lead-in to the topic.

Chapter 7 touches on Gödelian thought as well as the "halting problem." Most folks are familiar with the paradoxical, self-referential sentence, "This sentence is not true." Fortnow introduces a variation, "There is no proof that this sentence is true," to help explain how Gödelian self-reference demonstrates that there exist sentences which may be known to be true, yet are unprovable (if the prior sentence is false, then there IS a proof of the sentence, but if there IS a proof that the sentence is true, then it CANNOT be false! And vice-versa IF the sentence IS true, then, by its own admission, there is NO PROOF of its truth). In the end though, Fortnow notes that "this paradoxical approach to P versus NP seems doomed to fail, at least as a direct attempt at showing P ≠ NP."

Chapter 8 is on cryptography, one of the most significant real-life areas that would be affected if P were ever shown to be equal to NP (in which case code breaking, including breaking RSA encryption, could be done with ease).

In Chapter 10 we learn about NC, 'Nick's Class' of problems, another category within complexity theory, that may or may not be equal to the NP category of problems. Such ideas may be old-hat for those well-steeped in computer science, but are a new arena for the rest of us.

One update I learned from the book is that Vinay Deolalikar's claimed 2010 proof that P ≠ NP is apparently no longer in play. Last I'd heard (but I haven't followed the story that closely) Deolalikar was still trying/hoping to patch the various objections that critics voiced concerning his 'proof.' Apparently, though there were (not surprisingly) fatal flaws he could not overcome.
In fact, according to Fortnow, "We are further away from proving P ≠ NP then we ever were. Not literally, but in the sense that there is no longer any obvious path, no known line of reasoning that could lead to a proof in the near future."
Fortnow does mention that the field of "algebraic geometry" is currently the best candidate for possibly making progress on the problem. And he does indicate toward the end that he believes eventually P will be shown to not equal NP "…but it might take twenty or two hundred or two thousand years."
This is yet another splendid book from Princeton University Press, on a topic rarely treated in book-length form for the layperson, and for most of us a valuable, mind-stretching offering.

(Princeton Univ. Press has a transcribed interview with Fortnow about his book here: http://press.princeton.edu/releases/m9937.html )

ADDENDUM: Scott Aaronson, who is far more expert than I to review the Fortnow book, now has his own very positive review up over at his blog here:

http://www.scottaaronson.com/blog/?p=1293

Clifford Pickover... One-Man Carnival

Math-Frolic Interview #13


"I do feel that there are facets of the universe we can never understand, just as a monkey can never understand calculus, black holes, symbolic logic, and poetry. There are thoughts we can never think, visions we can only glimpse. It is at this filmy, veiled interface between human reality and a reality beyond, that we may find the numinous, which some may liken to God." -- CP

Cliff Pickover, computer scientist/mathematician/prolific-writer/creative-thinker/curiosity-seeker with a wide-and-wild-ranging array of interests... hopefully needs no introduction here (because I wouldn't even know where to start!).
Dr. Pickover answered a few questions I sent him recently (hope he doesn't mind being interview #13...):

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1) Most (but not all) of the folks I interview are active math bloggers; you are quite active on the internet with webpages and on Twitter (which some people call "microblogging"), but you don't have a conventional blog. Have you ever considered doing one, or does that not fit your interests or time constraints?

Shecky, wonderful to hear from you. Your math blogs are great. [hmmm... I should interview this guy more often! ;-)] You raise an excellent question about blogging. I regularly tweet here, and I update my Reality Carnival blog every day. Also, Pickover.com and its various sublinks receive millions of visits. However, I don’t regularly post essays at any web page. Perhaps most of my creative writing is redirected to the various books I write.


2) With your recent Math, Physics, and Medicine books you've developed an almost formulaic presentation mode that has wide appeal… I assume you have another such volume in the works, just can't guess what the next topic might be! (several possibilities). Can you say what you have in store for readers and fans in the near future? And do you work on multiple books at once, or one-at-a-time?

I believe that The Math Book, The Physics Book, and The Medical Book trilogy of books -- which cover key milestones, breakthroughs, and curiosities -- are among my most popular books.  I really enjoyed trying to condense a vast amount of information into page-sized nuggets, each accompanied by a photo or artwork.  I am currently working on the next book in this series.  Perhaps I can tell you more about the project in a few months.  It follows the same kind of colorful format, but the book is more eclectic in terms of science, history, art, and culture.
To answer your other question, whenever possible, I try to work on only one book at a time.

[Okay, I'll take a wild stab-in-the-dark and guess that Cliff's next book is on "Music" ...maybe we'll hear in a few months.]

3) How many hours a day do you spend writing or doing writing-related work/research? And seriously, how many hours a day/night do you sleep!? (And, how much coffee do you drink... no, just kidding on that one).

I started drinking coffee about three years ago. Writing is not too difficult if one is willing to give up other activities. Some people play golf on the weekends. I would prefer to write.

["three years ago"... you got a LOTTA coffee-drinkin'-catching-up to do fella! How many more books would you have written by now, if only...]

4) What were some of your main interests as a child growing up that led to the career you now have?

My childhood fascination with science was partly due to my parents' emphasis of science as a topic of study, as well as my budding interest in science fiction.  I fondly recall watching the old black-and-white Outer Limits and Chiller Theater shows on Saturday-night TV, which featured such notables as Attack of the Fifty Foot Woman and Attack of the Crab Monsters. Wow!  I can still remember the scenes in which scientists discover a pair of giant crabs mutated by atomic tests on a remote island. Many scientists and science popularizers got kick-started in life by reading science fiction.


5) What is the most interesting aspect of mathematics for you?

At first glance, some math classes may appear to involve a long catalogue of isolated concepts.  But as we learn more, we begin to see many linkages.  Obviously, the final goal of scientists and mathematicians is not simply the accumulation of facts and lists of formulas, but rather they seek to understand the patterns, organizing principles, and relationships between these facts to form theorems and entirely new branches of human thought. For me, mathematics cultivates a perpetual state of wonder about the nature of mind, the limits of thoughts, and our place in this vast cosmos.


6) Out of all the books you've written do you have a personal favorite? Also, so far as I'm aware all your writing has been nonfiction… have you ever thought of writing a piece of fiction (…or correct me if you already have)?

In addition to The Math Book, The Physics Book, and The Medical Book trilogy, I really enjoyed writing Sex, Drugs Einstein and Elves.  In this nontraditional book, I explore the borderlands of science. Part memoir and part surrealistic perspective on culture, I attempt to give readers a glimpse of new ways of thinking and of other worlds as we reach across cultures and peers beyond our ordinary reality.
I have written a few novels. Your readers can always visit my page that lists most of my 45+ books to learn more.  The titles of the novels include: Liquid Earth, The Lobotomy Club, Sushi Never Sleeps, Egg Drop Soup, The Heaven Virus, Spider Legs, and Jews in Hyperspace.

[I'm embarrassed that I wasn't aware of all this fiction... I don't read novels so it's somewhat understandable, but still surprised (and impressed)!]


7) Who are some of today's math authors/popularizers that you most enjoy reading? Other than math & science, what is some of your other favorite reading?

I enjoy all of the famous math popularizes, including, and not limited to, Ian Stewart, Ivars Peterson, Steven Strogatz, Rudy Rucker, Keith Devlin, and so many more.  Of course, Martin Gardner’s books still provide a goldmine of joy and information.  For other favorite reading beyond science, I enjoy reading and rereading the multivolume graphic-novel treatments of Proust's Remembrance of Things Past by Stephane Heuet, and some of the later novels of Robert Heinlein, including The Number of the Beast. In this novel, the protagonists can access 10,314,424,798,490,535,546,171,949,056 universes.


8) You're often referred to as a "polymath" because of your wide interests, and in fact I once said that "poly-polymath" was perhaps a more apt description! I can't help but think there's a strong genetic component to your far-flung talents/interests. Going back again to childhood, do you have a sense of where those mind-stretching proclivities stem from -- were your parents so inclined, and do you have any similarly-predisposed siblings?


While growing up in New Jersey, my bedroom featured plastic anatomical models of the heart, brain, and eye; posters of the human circulatory system; and trilobite fossils, science-fiction books, and Ugly Stickers displaying wild-eyed, grinning creatures with names like “Bob, “Sandy,” and “Iris.”  My childhood interest in science also was stimulated by my desire to learn how the world works and from my passion for science fiction. Today, my books are motivated by my interest in bringing science, mathematics and creativity to the broader public.
To answer your other question, I have one brother.  He’s a medical doctor.  He doesn’t write books, but we have similar interests.


9) I think it's accurate(?) to say you have very strong interests in both science and religion. Of course, a lot of scientists (vociferously) feel the two topics are mutually exclusive of one another. Can you say a little about how you hold both of them together in your own thinking and reflection? Do they mix together in some interactive way, or are they more like separate compartments?

Sometimes readers of my books ask me why I have also written on God, strange realities, and religious subjects. I tend to be skeptical about many claims of the paranormal. However, I do feel that there are facets of the universe we can never understand, just as a monkey can never understand calculus, black holes, symbolic logic, and poetry. There are thoughts we can never think, visions we can only glimpse. It is at this filmy, veiled interface between human reality and a reality beyond, that we may find the numinous, which some may liken to God.


10) You have several interviews and/or videos on the Web… If folks want to get to know you still better is there a specific one you would point people to, to learn more about you?


Visit Pickover.com and RealityCarnival.com.  Also, your readers can easily read my latest tweets, for free, here.

[There was a lengthy, wide-ranging audio interview with Cliff I heard on the Web not too long ago, which was what prompted me to ask this question, but unfortunately I can't re-locate it now.] 

11) Any parting words, not covered above, you'd care to pass along to a math-oriented audience?

Yes, the following spooky-looking number is a prime number.  Enjoy: 1000000000000066600000000000001
Belphegor's Prime

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Thanks for taking time to participate here, Dr. Pickover, and may you keep exploring that 'filmy, veiled interface between human reality and a reality beyond,' and reporting back to us! (Ohh, and I'll recommend this coffee for you.)
You can check out all Dr. Pickover's books at Amazon here:
http://www.amazon.com/Clifford-A.-Pickover/e/B000AQ13EG 

Taking Math To Court

A book blurb today...:

"Math On Trial" by Leila Schneps and Coralie Colmez, is a recent volume out from a mother/daughter mathematics team. It piggybacks on the current trendy interest in statistical-and-probability analyses of human affairs. Anyone with a penchant both for the 'true crime' genre of reporting, and for math, will likely enjoy this volume, which combines a look at various famous true crimes with the mathematical evidentiary material brought to bear in such cases...

Unfortunately, I'm not one of those people who takes much interest in crime reporting, so for me the volume did not hold undue interest. It covers ten notorious diverse cases (from murder to gender discrimination), but most of them are decades old, or even from over a century ago. A volume of more modern cases, say from the last dozen years, would have made for a more interesting and powerful read to me (...the Amanda Knox murder and Bernie Madoff swindle cases are among the more recent ones addressed in the volume). My own interests run more toward current events than history, and so the reaching back in time wasn't particularly effective, but other readers may find these historical accounts more keenly interesting. Each chapter focuses on a different kind of mathematical error (rather than the same sort of probability error occurring over and over) which is probably one reason for the somewhat odd selection of cases employed here.
At times the focus on math seems to oversimplify the multitude of legal issues and details involved in a given case, while at other times the mathematical material seems a bit buried under the legalities and details of a case. Still, the basic underlying point that math is often misused/misinterpreted in the courtroom can't be denied, and certainly anyone who doesn't already know that should read a book like this, because it is clearly a message worth spreading.

The book ends on the question of whether mathematical evidence should even be allowed in the courtroom, given how easily it is manipulated before a naive jury or judge. The final "Conclusion" chapter discusses Laurence Tribe's 40-year-old "beautifully argued, passionate" article on the subject, before noting that a lot has changed in the four decades hence, especially with the routine use of DNA forensics. Of course DNA evidence can bear problems as well, if not carefully administered, but is also a boon to those falsely-accused who may be exonerated through its use. Lastly, the authors acknowledge the more recent emphasis on Bayesian analysis and the attempt to set standards for its inclusion in the courtroom as well.

This volume didn't happen to grab me, but those of you more enamored of true-crime fare, may find it an interesting and worthwhile read, and its ultimate message that the public needs to be better educated as to the uses and abuses of statistics in legal matters is an important one.

Ian Stewart Delivers...

Overview of Ian Stewart's "Visions of Infinity"

"Mathematics is newer, and more diverse, than most of us imagine. At a rough estimate, the world's research mathematicians number about a hundred thousand, and they produce more than two million pages of new mathematics every year...
"When we think of mathematics, what springs to mind is endless pages of dense symbols and formulas. However, those two million pages generally contain more words than symbols... As the great Carl Friedrich Gauss remarked around 1800, the essence of mathematics is 'notions, not notations'. Ideas, not symbols."
                                   -- from Ian Stewart's Preface

Ian Stewart (or one of his clones… because I refuse to be duped into believing that one person can write all these books!) has a new volume out, "Visions of Infinity." There are several popular math books available that review a variety of the most famous, intriguing often-unsolved problems in mathematics (the exact selection can vary slightly from volume to volume, though certain standards almost always show up). Anyway, this is Stewart's contribution to the genre… and… as one might expect, it is excellent. If only it had been written first, some of the other attempts might even have been unnecessary.

To be clear though, while this volume covers some of the most interesting problems in all of mathematics it is NOT a book to draw your non-mathematical friends into the math arena. Even the non-math person who wishes they could like math, and who may have enjoyed Steven Strogatz's "Joy of X,"  I think will find this particular book too heavy-going.  But for the individual already enamored of the subject, and having some familiarity with math's deepest problems, this is a fantastic read. In fact, it's probably my favorite Stewart volume of all the ones I've read.

Stewart gives enough background and depth to each discussion to hold the attention, and exercise the brain, of most math buffs, without getting so technical as to lose them in the dust (though parts, toward the end especially, are definitely tough-going). I have no idea if he was able to produce this work largely based on his past writing and knowledge, or whether it required lots of research on his part to fill in all the detail, history, and richness that are on display here... either way the book is a great accomplishment.

Fermat's Last Theorem, the Riemann Hypothesis, the four-color theorem, the Goldbach conjecture, the Poincare' conjecture, and P vs. NP, are among the classics Stewart rolls out for scrutiny (the title may be a bit misleading in so much as "infinity" is not really the central topic). He does a great job of not only explaining these mathematical conundrums in an accessible way, but also of detailing their histories, context, and relevance (when it exists) to other matters.  The first few chapters tend to warm the reader up. From there the book moves on to more complex and difficult problems as it goes along. The last couple of conjectures dealt with, the Birch--Swinnerton-Dyer conjecture and the Hodge conjecture, are the densest to follow (...but Stewart notes of the Hodge conjecture that it "is arguably more representative of real mathematics, as done by mathematicians of the twentieth and twenty-first centuries, than any other topic in this book"). Soon after those chapters Stewart follows up with a lighter chapter titled "Twelve For the Future," briefly outlining 12 more unsolved problems, some fairly well-known. This section ends with the ABC conjecture, and was apparently written before last year's announcement from Shinichi Mochizuki that he had proven the conjecture, as there is no mention of the claimed proof (which has yet to be confirmed, and indeed some say may not be comprehensible to anyone except Mochizuki!).

I suppose my favorite chapter (pardon the bias) is on the Riemann Hypothesis, where Stewart seems to capture in a chapter what others have written whole books about. Another chapter that I found especially good was on the Mass Gap Hypothesis, and more generally on particle physics, but your own interests will determine which chapters you most enjoy. They are all good. This will likely be the initial 'go-to' volume on my shelf whenever I wish to check on something regarding one of these classic problems.

The book ends with a handy 11-page glossary, an interesting set of succinct "notes," and a brief bibliography for "further reading" -- on a side note, I was pleased to see that the brief bibliography cites Matthew Watkins' "The Mystery of the Prime Numbers," a book I've been touting for quite awhile, but which I'd not yet seen referenced by any prominent author (Watkins gets a very brief mention in the body of the book as well).
This is simply a fabulous book for most mathematics enthusiasts... no matter which clone of Stewart's wrote it!

Evelyn Lamb... of Scientific American

Math-Frolic Interview #12

"If you already love math, welcome. If you were traumatized by a math class years ago, welcome. If your relationship with math is complicated, welcome. I hope there will be something here for everyone."
-- Evelyn Lamb, from her first blog post for "Roots of Unity"

Evelyn Lamb is a mathematics writer and post-doc who was a relatively recent addition to the stable of bloggers at Scientific American. Her blog "Roots of Unity" is fittingly subheaded, "Mathematics: learning it, doing it, celebrating it" and her SA profile is here.
If you're not familiar with her, you'll soon feel much more familiar after reading the wonderful, extensive answers she gave to questions I posed for her (and as usual, I've highlighted, in bold, parts of the responses). Enjoy:

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1) You mention in your blog bio that you didn't really get interested in math until your junior year of college… that is so contrary to what I usually hear from mathematicians (who often grow up with an unusual and inherent interest in math and numbers from early on). Can you describe a little further what that was like coming to math relatively late in your academic life (did it feel odd? did other math majors look a bit askance at you? was it a smooth or difficult transition?…)?

As a child, there were times that I learned about mathematical concepts that I found really fun and interesting, but I didn't realize they were real mathematics. My dad taught me about bases when I was in elementary school. I learned to count in binary on my fingers, and I loved figuring out how to represent numbers in different bases and the fact that decimal notation is a choice, not something intrinsic to numbers. But I didn't connect this concept to math. Math was where you learned the routine process of multiplying, not the fun process of representing your age in base eight. (And figuring out that no matter what your age, if you represented your age in base (your age), the answer was 10. Whoa!)

In general, I grew up thinking of math as just formulas: you look at a problem, there's one way to do it, you do it, and then you get the one and only answer. I was good at this, but it didn't excite me. I went to the Texas Academy of Math and Science for my last two years of high school. That is a program at the University of North Texas where high school students who are good at math and science take UNT classes during their last two years of high school. There, I went through the calculus sequence to multivariable calculus, so when I got to college, it made sense to me to work on a minor in math because I had so many math credits already going in. My junior year, I took a class that really changed the way I looked at math.

Coming to math so late in the game has been a challenge, and I still sometimes feel "behind." I was one of the top math students at my university, but at the time their math program was not what it is now. My first semester of grad school, I got a big dose of reality when I saw how difficult the qualifying exams were and how much better prepared everyone else seemed to be. My grad program was very small, and faculty and other grad students were very supportive of me. I felt out of my league a lot of the time, but it was internal. No one there tried to push me out or make me feel like I didn't belong.

2) You also mention that an "inquiry-based learning" class is what really turned you on to the beauty/creativity of math. Again, can you say a little more about what were some of the problems/examples that most moved you?

The class, basically "intro to proofs," was taught using the Moore method, or inquiry-based learning, meaning that our professor prepared sheets of definitions, examples, and theorems for us to work on outside of class. During class periods, students would present material for the whole class, and those who weren't presenting would ask questions or point out errors in the proofs. The professor worked very hard to make it a "safe space" for people so they wouldn't feel bad about being wrong. We pointed out errors in each other's work, but it was always done in a constructive, kind manner. That class, more than anything else, showed me that math is a creative discipline.

There were two mathematical concepts that stuck with me the most from this class. The first was proof by induction. Not only did we use this technique, but we also came up with a proof that the technique worked. That was the first time I had really thought about math as theorems and not formulas. I was used to learning a technique and trusting that it worked, not actually understanding why it worked. (My professors had presented proofs of theorems in calculus, differential equations, linear algebra, and so on, but I had always thought of those theorems as times to "tune out" until we got to the part where we used them to solve a problem.)

The second concept was proving that the rationals are countable and the real numbers aren't. I had worked for about a week on a proof that the irrationals were of a different cardinality than the rationals—my intuition said that n^2 just had to be fundamentally "bigger" than n, and there was some kind of "limit" argument that the irrationals had to be more numerous than the rationals because of this. Obviously, my proofs were not correct, and when another student presented a (basically) correct proof, I was very resistant to it. It took a while for me to learn to accept it, but that process of wrestling with the idea for so long was formative for me as a mathematician.

I'm not saying that inquiry-based learning is necessarily the best way to teach all topics, or the best way to teach all students, but I can say that there is a 99% chance I would not be in math today if I hadn't taken that inquiry-based learning class.

3)  How did the idea for your blog, "Roots of Unity," for Scientific American come about?

I had a fellowship at Scientific American during the summer of 2012, which is how I started writing about math and science. Through that, I met Bora Zivkovic, the blogs editor for Scientific American. Both he and my immediate supervisor Robin Lloyd recognize that Scientific American's audience would like to read more math stories, so they were very supportive of my coming on board with my own blog. I'm really grateful to both of them and my other colleagues at Scientific American for encouraging me and helping me find an audience by giving me the Scientific American platform.

[Bora has a knack for finding and promoting blogging talent! and you certainly seem a great fit for SA.]

4)  I would guess that anyone writing ongoing math pieces for Scientific American might feel the long shadow of Martin Gardner over their shoulder… is there some sense of that for you, or are your pieces different enough (and not restricted to recreational math) so that you don't feel that sort of connection?

I definitely feel the shadow of Martin Gardner! I do want to entertain people and help them engage with math, two things that he did very effectively. I do not know very much about recreational math, so I haven't ventured into that world very much yet. And there are already so many places where that is done so well! I will never be Martin Gardner, but I try to see myself as an entertainer at least as much as an educator. For the most part, I want my pieces to make people happy. If they learn something as well, that's a nice bonus. Sometimes I write about things that aren't happy, but I prefer the happy stuff.

5) What are your favorite aspects of mathematics to study or read about? And what is your math post-doc work related to?

My field of study is hyperbolic geometry and Teichmüller theory, which is the study of spaces of hyperbolic surfaces. I use analytic techniques to attack geometric problems. I am a very visual person, and I love the visual aspect of hyperbolic geometry. I love reading (or looking at) books of mathematical art, particularly when it has to do with hyperbolic geometry. I have recently gotten more interested in math history, which I am actually woefully ignorant of. In my work for Scientific American, I have come across some biographies of famous mathematicians and books of ancient Chinese math, for example, that fascinate me. I do love to read popular books about math, in part as professional "research": how do more experienced math writers approach the subject, and what are their personal styles? I am still working on finding my voice and tone, so hearing the masters do it is very useful to me.

6) Who are some of your favorite mathematicians (living or deceased) to read?

I am an academic descendant of Bill Thurston, and his approach to geometry and mathematical thinking in general. He has a famous essay in a 1994 copy of the Bulletin of the American Mathematical society called "On Proof and Progress in Mathematics," which is an interesting read about the way mathematicians should approach mathematics. It has been circulating in my corner of the math research community a lot since Thurston's death in August. (My corner of the research community is probably at least half mathematical descendants of Thurston.)

I've also read two popular math books in the past few months that I really enjoyed: "Measurement" by Paul Lockhart and "The Joy of X" by Steven Strogatz. They are very different from each other, but both are worth a read. I read Measurement twice because I wrote two different book reviews of it, several months apart, and the second time through, I found it even more delightful. Lockhart reminds me of Thurston in some ways: both talk about math as a human activity in ways that speak to me.

One of my uncles gave me Strogatz's book "Sync" back when it was published, and I've tried to keep up with his writing since then. I was very excited that I got to meet him this past summer when I was in Ithaca for my brother's wedding. It turns out he was on my sister-in-law's oral exam committee. She is a grad student in physics at Cornell.

-- I thoroughly enjoyed and reviewed the Lockhart and Strogatz volumes as well, and Bill Thurston was certainly one of the most loved and respected members of the math community.

7) In an earlier piece you wrote that working at Scientific American made you feel like "two different people" --  being a reporter of sorts (educating the public) for SA, but also still wanting to be a discoverer, as an academic mathematician… Now that you're a regular blogger for SA I imagine that writing for the public becomes increasingly smooth and enjoyable over time. How strongly do you still feel pulled in two directions, or are they more integrated by now?

I very much feel pulled in two or more directions, but that is pretty normal for me. I have always had many divergent (sometimes competing) interests, and I feel like I've spent most of my life fighting to keep as many options open as possible. I am just the kind of person who wants to have lots of different mental stimuli over the course of a day or week. I have the luxury this year of establishing myself as a writer/blogger before starting my postdoc. Once I get there, I will have a whole new challenge of balancing my new professional responsibilities with my desire to write about math for a broad audience, but I know a change of scenery will also give me new inspiration.

My writing has given me a lot to think about when it comes to teaching, and I expect and hope that my work on my blog will help me become a more effective teacher.

8) How do you select the topics you post about on your blog?

My Valentine's Day post about continued fractions literally came to me in a dream, but that doesn't seem to be a consistent source of ideas. I don't have a magic formula. I do get some ideas from press releases and arxiv papers, but I also just notice math in my daily life and in popular culture. When you dig far enough, everything in the world is made of math, so it's just a matter of choosing where to dig and how deep. Right now I'm trying to formulate a blog post about why Project Runway is my favorite  math show on TV, for example, mostly because I want to be able to say I'm watching Project Runway for professional reasons. (Nobody steal my idea, OK?)  -- [just a guess, but I don't think you have to worry about that ;-) ]

So far this year, I've been to three large conferences, and I've gotten lots of ideas from them. But I don't have one tried and true way of finding stories yet, which means a lot of the time I put into my writing is "invisible" and just consists in figuring out story ideas and rolling them around in my head until they work. I am naturally attracted to stories that involve math and the arts because I find it easier to draw people into those stories and because I love the intersection, so I like to have an excuse to spend a lot of time thinking about them.

9) To round yourself out a bit, when you're not doing mathy things, what are some of your main interests/hobbies/activities?

I have way too many hobbies! My mom is a musician, and I have been involved in music since I was a toddler. Currently, I play viola with some amateur chamber music groups and sing shape-note music, a kind of early American folk hymn singing. I also sew, which I actually think of as applied mathematics, and I have been trying to learn to crochet. My husband and I live about a mile from Lake Michigan, and when the weather is nice, we like to walk down to the lake and collect beach glass. We also collect it on trips if we're near a shore. We have a few vases and bowls filled with it, and I've used it for some art projects. We also like riding bikes, doing yoga, and playing ultimate frisbee, although we haven't done a whole lot of that recently. We both do a lot of cooking, and we love traveling. Recently, most of our traveling has been to math conferences and to visit family, but we try to work in some personal vacations as well.

10) Any parting words, not covered above, you'd care to pass along to a math-oriented audience?

When you don't know what to do, do something. I think that's important both in mathematics and in life. When I am teaching or tutoring, I find that students get stuck the worst when they are afraid to write anything down before they're absolutely sure they have the right answer. People sometimes think mathematics happens when a smart loner has a sudden bolt of inspiration, but I think much more progress happens when people, often in collaboration, try lots of different approaches to the same problem until something works. At all levels of study, much of mathematics consists of getting used to techniques that you can get an instinctive feel for what technique might work for a new problem. In life, I try to use the same approach. I don't know what I'm doing with my writing exactly, but I'm doing something, and I'm getting a feel for what works and what doesn't, and what kind of math stories people want to read.

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THANKS so much, Evelyn, for these great responses and best of luck both with the blogging and your academic future.
Be sure to check out Evelyn's blog HERE, if you haven't already.