...a companion blog to "Math-Frolic," specifically for interviews, book reviews, weekly-linkfests, and longer posts or commentary than usually found at the Math-Frolic site.

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." ---Bertrand Russell (1907) Rob Gluck

"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." ---Bertrand Russell (1957)

******************************************************************** Rob Gluck

Friday, August 19, 2016

Wrapping Up the Week


1)  "Phantom" traffic jams:

2)  There is no shortage of proposals of what problem the Polymath Project should tackle next:

3)  Mathematics joins in on the parodies of Alexander Hamilton:

4)  Deborah Mayo again on p-values:

5)  An old, delicious and deceptive math Olympiad problem that got some attention on Twitter this week, starts on Numberphile here:

6)  Searching for math geniuses:

7)  The latest "Carnival of Math":

8)  Ben Orlin's round-faced buddy explains logarithms:

9)  Appropriate to an election year, James Propp tackles "Bertrand's Ballot Problem":

Potpourri BONUS! (extra NON-mathematical links of interest):

1)  I got a kick out of Futility Closet's post on an analysis of "stupid people":

2)  Nice Guardian piece on Daniel Kahneman:

Friday, August 12, 2016

Math Potpourri Served Up

End-of-week mishmash of math miscellany:

1)  Medical tests... false positives and negatives:

2)  Exploring Mersenne numbers a bit:

3)  Donald Knuth wins the John von Neumann Lecture Prize:
4)  Evelyn Lamb lists a few keen educational math blogs (for different levels) here:

5)  Deborah Mayo on Popperian falsification etc.:

6)  A review (pdf) of the latest volume from Raymond Smullyan, "Reflections: the magic, music, and mathematics of Raymond Smullyan":

7)  Speaking of booksI wrote this week about an older volume from Keith Devlin that, if you can find it, is worth having:

8)  And of course if you need more math for your weekly fix, you can always visit "Mike's Math Page":

Potpourri BONUS! (extra NON-mathematical links of interest):

1)  One physicist's interesting take on the LHC results:

2)  ICYMI, "60 Minutes" Anderson Cooper met playful Bonobos in the Congo last weekend:

...and lastly, perhaps my favorite tweet from the week (from @AdamSacks):
"The party of Lincoln is now the party of John Wilkes Booth."

Monday, August 8, 2016

"Making the Invisible Visible"

Some years ago I mentioned stumbling across Keith Devlin’s older book (1997), “Goodbye Descartes” and really enjoying it. Dr. Devlin’s interests in cognitive topics, as represented in that volume, overlap my own, and often aren't broached by math writers. Cognitive psychology was a major focus in college, and to this day I find the linkages between mathematics, language, music, and semantics fascinating — despite all that has been written about such topics, I suspect they remain little deeply understood. (And some of it even links back in my own mind to the “General Semantics” of Alfred Korzybski, and the way words and meaning govern/manipulate human thought/behavior).

Anyway, last week I stumbled upon another similar, vintage Devlin volume, “The Language of Mathematics: Making the Invisible Visible” (1998)… and am yet again enthralled by Keith’s beautifully-straightforward exposition! The book obviously can’t qualify for my end-of-year list of best popular math books of 2016, but may nonetheless turn out to be my favorite read of this entire year! And while most of you won’t go searching for an almost 20-year-old volume, that’s what I’d urge you to do, if you've not already read it!.

Keith articulates his theme of math as the science of patterns, while offering excellent introductions to math foundations, history, logic, set theory, calculus, geometry, symmetry, knot theory, topology, probability, and some physics, while admitting there is much he is leaving out. As he writes in a postscript, rather than 'serving up a vast smorgasbord of topics, each one allotted a couple of pages' (as other books often do) he has tried to show that the "mathematical study of any one phenomena has many similarities to a mathematical study of any other." Where so many introductory books to mathematics focus on the logic and procedures of math, Keith emphasizes here the abstract and interwoven nature of real mathematics.
Even though the book's age means a few passages are out-of-date, for the up-and-coming young person interested in math, I don't know a better, clearer overview of what the field is all about.

This extended quote from the volume's Prologue I think captures an essence Dr. Devlin often addresses:
The use of a symbol such as a letter, a word, or a picture to denote an abstract entity goes hand in hand with the recognition of that entity as an entity. The use of the numeral ‘7’ to denote the number 7 requires that the number 7 be recognized as an entity; the use of the letter m to denote an arbitrary whole number requires that the concept of a whole number be recognized. Having the symbol makes it possible to think about and manipulate the concept.
 “This linguistic aspect of mathematics is often overlooked, especially in our modern culture, with its emphasis on the procedural, computational aspects of mathematics. Indeed, one often hears the complaint that mathematics would be much easier if it weren’t for all that abstract notation, which is rather like saying that Shakespeare would be much easier to understand if it were written in simpler language.
"Sadly, the level of abstraction in mathematics, and the consequent need for notation that can cope with that abstraction, means that many, perhaps most, parts of mathematics will remain forever hidden from the nonmathematician; and even the more accessible parts — the parts described in books such as this one — may be at best dimly perceived, with much of their inner beauty locked away from view. 
In turn, all of that reminds me, oddly enough, of a favorite passage (I’ve used here before) from controversial David Berlinski in his book about Euclid, “The King of Infinite Space.”  Sometimes I flip-flop between thinking that this odd passage is just wordplay sophistry, and feeling that it is actually a rather profound statement about the nature of proof or certainty (...and I hope Dr. Devlin won't mind that something he has written reminds me of words from Berlinski ;-):
"Like any other mathematician, Euclid took a good deal for granted that he never noticed.  In order to say anything at all, we must suppose the world stable enough so that some things stay the same, even as other things change. This idea of general stability is self-referential. In order to express what it says, one must assume what it means. 
"Euclid expressed himself in Greek; I am writing in English. Neither Euclid's Greek nor my English says of itself that it is Greek or English. It is hardly helpful to be told that a book is written in English if one must also be told that written in English is written in English. Whatever the language, its identification is a part of the background. This particular background must necessarily remain in the back, any effort to move it forward leading to an infinite regress, assurances requiring assurances in turn. 
"These examples suggest what is at work in any attempt to describe once and for all the beliefs 'on which all men base their proofs.' It suggests something about the ever-receding landscape of demonstration and so ratifies the fact that even the most impeccable of proofs is an artifact."
[I do loves me some recursion! ;-) ]

Anyways, there have been many wonderful popular math books in recent years, but sometimes you can't improve much on older volumes, when it comes to a timeless subject like mathematics. And Keith's writing is always as lucid as there is.
Finally, for an older review of Dr. Devlin's book, see here:

Friday, August 5, 2016

From the Past Week...

Friday potpourri is back... for now (no telling when politics and pickleball may intervene again to postpone it):

1)  Evelyn Lamb on higher dimension sphere-packing:

...And Dr. Lamb again, explaining why the Menger Sponge is one of her "favorite spaces":

2)  Nice interview with mathematician/Ramanujan-scholar Ken Ono:

3)  Followup to Mochizuki's ABC conjecture proof:

4)  Ben Orlin and his round-faced buddies on Graham's Number:

5)  Data analysis of a disgusting political candidate:

6)  Patrick Honner continues his analysis of the NY State Regent's math examination:

7)  John Cook explains continuity, such as it is, in the real world:

8)  The great writing at Quanta never stops... Kevin Hartnett on "A Unified Theory of Randomness":

9)  Read about Miranda Cheng's amazing journey to sipping the umbral moonshine and working on string theory physics:

10)  Mathematician, computer scientist, and pioneering educator Seymour Papert died last weekend:

...also passing this week, Canadian mathematician Jonathan Borwein:

11)  And I got back to regular blogging at Math-Frolic this week (after 2 weeks of slacking-off) with posts via Kurt Gödel, Keith Devlin, James Maynard, and Battlebots!

Potpourri BONUS! (extra NON-mathematical links of interest):

1)  "Active surveillance" of cancer diagnoses becoming more common:

2)  More on China's cr-r-r-razy, jaw-dropping traffic-straddling bus:

...and lastly, this from Stephen King on his latest macabre subject:

Friday, July 15, 2016

Friday Update...

Been busy with a lot of summer stuff this week (luckily, fun stuff) and won’t have a Friday math potpourri this week… in fact, they could be sporadic over the next month-or-two(?), including next week when I’ll be 'entertained' watching the GOP Convention — I bought a television this year, having gone without for SEVEN years, just to watch this upcoming 3-ring circus, though I expect it may be one of the more depressing things I’ve ever seen on American TV! :-( 

Blog posts may be a little light in the interim as well, but still plenty of great math out there, and will be back to regular Friday wrap-ups at some point. (For now though, got to go play pickleball! ;-)

Monday, July 11, 2016

A Few Books....

Just a few quick book blurbs today....

1)  Geometry is being emphasized less these days in secondary school curriculums, even disappearing in some cases. As someone who loved geometry that seems ashamed, though I understand the reasons for it. If only we lived in a Euclidian universe the importance of geometry would remain. But we don't, and your interest in Alfred Posamentier's (with Robert Geretschlager) latest book, "The Circle," will largely depend on your enamorment with geometry (I have a review copy of it; not due for general release 'til latter August).

A few years ago Posamentier was out with "The Secrets of Triangles," a fantastic overview of the geometry of that ubiquitous figure, and now he follows that with this treatment for the triangle's curvier cousin, the circle, in all its glory (triangles make plenty of appearances in this volume as well).

In his usual style, Posamentier sends a firehose of interesting ideas, examples, theorems, problems, etc. at the reader. You can virtually start with any chapter and be assured an interesting ride... as long as you have a yen for geometry. The book contains constructions, "art and architecture," history, and paradoxes, along with an abundance of plane geometry facts/problems/theorems.

2)  Avner Ash and Robert Gross's "Summing It Up" is a much more advanced lesson in math, specifically number theory. 
Like the Posamentier book this one abounds with examples and specific cases, but that's the only similarity. This is actually the third book from Ash and Gross, and it is focused on modular forms, a very hot topic since Wiles' proof of Fermat's Last Theorem as well as increased interest from the Langlands Program. Though it starts at a more elementary level the book builds toward ideas in number theory that will be difficult for the casual math reader to follow in the second half of the volume. As such it is not so much a "popular" read as an academic-like treatment of, or introduction to, some advanced math. 200 pages of rich, dense reading for the uninitiated. Certainly worth it if number theory is your particular area of interest or you've been wanting an intro to modular forms.

This MAA review of the book offers more details and will give you a better sense of whether or not you're prepared to tackle the subject matter involved:

3)  Lastly, and I'm waaaaay late to the party on this one, but just now finishing Nate Silver's 2012 bestseller, "The Signal and the Noise." (Somehow, every time I thought of reading it, other books got in the way.)  Anyway, I love this volume; far more interesting and wide-ranging than I realized it would be.  A good introduction to Bayesian statistics, but covers a lot of other ground as well. So if by any chance you too passed it up along the way for some reason, I recommend it, especially since this is an election year and of course parts of it pertain especially to election/voter analysis.

Three completely different books, new, old, and forthcoming, that may satisfy very different tastes and levels of math background.

We are passed the halfway point for this year, and usually by now I already have an idea what my favorite math-read of the entire year may turn out to be. This year no single volume stands out for me though above others, so we'll see what the second half of the year brings along.

Friday, July 8, 2016

Friday Math-mix

Miscellany from the week:

1)  Fantastic primer on Bayesian statistics (h/t Gary Davis):

2)  Feynman on Fermat's Last Theorem:

3)  More on infinity from Quanta:

4)  This is a few weeks old but Steve Strogatz just pointed it out this week... one teacher's decision to leave the field early (...after 29 years)... nothing that others haven't said, but still sad to read:
5) John Baez continues his mind-blowing exploration of the exploration of the exploration ;-) of infinity in these two posts:

6)  James Grime and Numberphile on anti-prime numbers (or "highly composite numbers" via Ramanujan):

7)  Dave Richeson explained the math behind a fantastic optical illusion:

8)  I'm not in the loop of primary/secondary education, so only follow a small number of bloggers/tweeters who cover that area. And there are so many math-problem sites on the Web at this point I don't try keeping track of them any longer. But with all that said, I'll put in a plug for this site that looked interesting (it's not new, but I only came across it this week):

9)  More physics than math... any story on Richard Feynman from a physicist is usually wonderful, including this one from Frank Wilczek:
10)  Finally, if you missed the Math-Frolic post yesterday, I'm asking to hear stories from folks about their earliest memories of being drawn to numbers and mathematics:

Potpourri BONUS! (extra NON-mathematical links of interest):

1)  this segment ("Past Life Detective") of NPR's "Snap Judgment" from last week was oddly fascinating:

2)  Science writer George Johnson, interesting as always, this time on the subject of consciousness:

Saturday, July 2, 2016

Brian Hayes... Award-winning Science Writer

Math-Frolic Interview #38

If you ever lie awake pondering the complexities of the universe, you may have a soul buddy in Brian Hayes.” 
― Justin Mullins, New Scientist

I read a LOT of math blog entries. In the last year-or-so though, two people, above all others, have consistently blown me away with their postings. The two (among so many solid writers), are James Propp and Brian Hayes. So after interviewing James in the last interview here, I'm especially thrilled today to present Brian, who's blog "Bit-player" is quirky (not limited to math), exceedingly well-written, wide-ranging, and always much anticipated.
I really knew very little about Brian, who has been called, "one of the most accomplished essayists active today," so I especially love how much new information there is for me in these answers.  And I'm glad to introduce him to any readers not already familiar with his work.

His fantastic blog, Bit-player, is here: http://bit-player.org/
He tweets as: @bit_player
...and many of his prior prolific essays/book-reviews are here:

Without further adieu...:


1) Your blog posts aren't frequent, but are always interesting, unpredictable, and thought-provoking. How do you choose your topics and how long does it take you to complete a typical post?

The world is chock full of stories I’d love to tell, but I can’t do them all. So I try to choose topics on which I feel I have something distrinctive to say, something to contribute to a wider conversation. I also try to choose stories that aren’t already getting a heap of attention from other writers.

As you point out (very tactfully!), I’m a mighty slow writer. American Scientist comes out every other month, so in all my years there I was on a 60-day schedule. I don’t recall ever finishing an article ahead of deadline. When I write for bit-player.org, I often spend a week or two on a piece, or more if there’s a substantial amount of programming needed, or elaborate illustrations.

2)  I didn't find much at all about your academic or personal past online; can you say a little about your path to becoming a popular writer on math/computer science?

In my youth I was seized by a fierce impatience. I couldn’t wait to get on with life, and so I skipped all the usual preparatory steps — like a college education — and plunged directly into a career as a writer and editor. (Advice to young people: Don’t be in such a hurry. The world will still be there waiting for you when you graduate.) 

While still in high school I began reviewing books for Saturday Review (a literary weekly, long defunct) and later for the New York Times Book Review. A couple years later I was the book review editor of the Baltimore Sunday Sun. I was also deeply interested in the sciences, technology, and mathematics, but I didn’t have much chance to connect my literary and scientific passions. Then a miracle happened: I was hired as an editor at Scientific American. The miracle worker was Dennis Flanagan, the editor in chief, who took a huge chance on a kid with very skimpy credentials. Dennis became my mentor and dearest friend. The dozen years I spent at Scientific American were my education, both in the craft of writing and in the fields I was writing about.

The staff editors in those days were not really supposed to specialize, but I did most of the high-energy physics for a few years, and a lot of molecular biology. There were also occasional pieces on mathematics and computer science, but my deepening interest in those subjects was something I pursued on my own time. This was in the late seventies and early eighties, the moment when personal computers first came into bloom. Eventually I got my hands on some hardware and learned how to do stuff with it. This was another transformative experience. I gradually came to understand that in computational science I could do more than just read about what other people were doing. I could repeat the experiment, or try one of my own. All the instruments I needed were sitting on my desk.

3)  You worked a dozen years at Scientific American, and your writing style is very reminiscent of Martin Gardner. I'm curious if you feel you have your own writing style, or have you consciously tried to emulate Gardner, who's writings were of course immensely popular?

Like so many mathy/nerdy folks of a certain age, I grew up devouring Martin Gardner’s Mathematical Games column. I was 13 or 14 when I started reading him regularly, in the early 1960s. I soon found a disreputable shop near the waterfront in Philadelphia where I could buy old issues of the magazine for 10 cents apiece, thereby catching up with some of the columns I had missed from earlier years.

When I met the man himself a few years later, it was an unsettling experience, like walking into a room and being told that the kindly, white-haired fellow in the far corner is Carl Friedrich Gauss. I was never Martin’s editor at Scientific American (he didn’t need much editing anyway), but I did have a chance to get to know him. He seldom came to the office, but for a few years I lived two blocks away from his home (on Euclid Avenue!) in Hastings-on-Hudson, New York. I was recruited as courier to carry manuscripts back and forth. Those visits always involved a glass of iced tea (Martin’s wife Charlotte would accept no refusals) and a puzzle or a magic trick or a bit of hot mathematical news from Martin.

Some years later, when Martin had retired and I began writing a column of my own, he became a generous advisor and guide, introducing me to his friends, suggesting topics, gently calling me out when I goofed. We remained in touch until shortly before his death in 2010.

I never consciously modeled my prose on Martin’s, but if you see a resemblance, I consider that a compliment. 

[Great to hear about this relationship, and I do think there's a similarity in the succinct precision of your prose and his.]

4)  If any of my readers aren't familiar with you could you point them to a few favorite (Web-accessible) mathematical essays they might enjoy?

All of my American Scientist columns and most of my other writings are freely available on the web. There’s a list with links at http://bit-player.org/publications-by-brian-hayes. (And of course the bit-player.org site itself is also open to everyone.)

Which pieces would I suggest as starting points? Here are a few that seem ripe today. Ask me tomorrow and I’ll have a different list.
5)  Are there certain current popular math writers you especially enjoy reading?

The community of mathematics writers is not large, and most of them are my friends. I can’t name favorites among them without snubbing the rest. So I’ll just mention one math writer I’ve never met: Sherman K. Stein. I particularly recommend his book How the Other Half Thinks (McGraw-Hill, 2001). 

[Totally new author to me... I'll have to check this out!]

6)  Any new books coming from you in the near future?

Yes! I’m working on a collection of essays that MIT Press will bring out next spring. Title: Foolproof, and Other Mathematical Meditations.

[This is super news to hear, and even without seeing the contents I love the title.]

7)  You're a bit of a polymath in your writings... is there some math/science subject though that you'd love to study more intensely and write about, but just haven't had the time or chance to cover yet?

Having missed my first chance at getting a proper education, I am highly susceptible to the daydream of going back to school. But I probably won’t do it. There’s too much on the agenda for my remaining years.

8)  You've worked at American Scientist magazine, which I always think of as one of the finest under-the-radar publications out there. But I know, like so many print publications these days, it's been through some rough times. Will just give you a chance to put in a promotional plug for the magazine if you care to tell folks anything about it….

Science writers often serve as intermediaries, standing between the scientist and the public, performing a sort of simultaneous translation. American Scientist has a different model: they foster direct communication between the scientist and a wider audience, helping the scientist tell his or her own story with the assistance of professional editors and illustrators. Both approaches have their virtues, but the direct channel of communication is becoming rare and thus all the more precious.

I no longer write for American Scientist on a regular basis, but I remain a consulting editor — and a huge fan. 


Great answers Brian; I'm glad to know more about you, where you've come from, and to learn there is a new book on its way from you.
Brian's last book by the way, was "Group Theory In the Bedroom, and Other Mathematical Diversions," another superb collection of mathematical essays.

Friday, July 1, 2016


Of course you want to read some math over the July 4th holiday, don't you...:

1)  A simple, but interesting function:

2)  If you're not tired of reading explanations of Bayesian stats, here's a simple post from Brian Clegg:

3)  Deborah Mayo re-blogged a post from a few years ago:
4) The truth is out there... Samantha Oestreicher speculates on alien life:

5)  In an imperfect world you have (thus far) 49 perfect numbers to choose from:
6)  Nice essay on the Axiom of Choice (h/t John Golden):
7)  To retract or not to retract? a longish post from Andrew Gelman:
8)  Jordan Ellenberg on this week's Numberplay (NY Times); includes interesting "chat" with him at end:
9)  Very good (and fun)... Adam Kucharski talking about gambling and math (video: 36-min. talk + question period):
10)  Great, long John Baez post on "large countable ordinals" (h/t John Golden):

11)  A new Twitter account, @Team_Maths1 started "for sharing maths teaching resources, developments, research, blog posts, and ideas." 

12)  A little portrait of "puzzle master" Scott Kim:

13)  Winners of the "best Illusion of the year 2016":

p.s. -- I wrote a couple days ago here about my on-off relationship with Gödel over the years, and sometime this weekend I'll have a new interview up here... with one of the finest science writers around.

Potpourri BONUS! (extra NON-mathematical links of interest):

1)  "Slate Money podcast" is one of my favorite podcasts outside of NPR (I've even grown fond of that Felix Salmon fellow, or at least his accent). If you're not familiar with it, last week's "Brexit" edition is as good as any place to start:

2) ...and, I just listened to the new Invisibilia episode ("The Problem With the Solution"); another gem: http://www.npr.org/podcasts/510307/invisibilia

Wednesday, June 29, 2016

Me and Gödel

Siobhan Roberts has a piece in the current New Yorker on Kurt Gödel:

This all reminded me of my own experience with Gödelian thought decades ago...

When I was around 20 years-old someone showed me Kurt Gödel's "proof of the existence of God" (a simplified version of course), and explained that Gödel, who I'd never heard of, was regarded as one of the most brilliant men of the century. I couldn't believe it!  I looked over the proof, thought it was one of the most ridiculous academic pieces I'd ever seen, refutable by any intelligent 8th grader, and promptly filed the name "Gödel" away in my mind as a silly individual to be ignored.

During those same years I was, on my own, playing around with ideas about self-reference in language, and what I later found to be known of as "recursion." Intuitively, I felt these were crucial concepts for comprehending the way the brain worked -- but ironically it also implied that we COULDN'T understand how the brain worked, because that would require the brain to analyze itself, and the point was, that I didn't think such a recursive process was possible -- a device cannot fully explore its own processes. My ideas were purely intuitive though, and I couldn't find a way to verbalize or refine them. So they just sort of hovered there loosely in the back of my mind.

Some years passed and, in general, I deliberately ignored Gödel whenever his name arose, but at some point was reading a popular math volume by someone I respected, and it included a chapter on Gödel and this thing I'd not heard of called "the incompleteness theorem." Reluctantly, I read what this author I regarded highly, had to say about Gödel's work. And, I was blown away! Kurt Gödel was saying, and indeed being hailed for rigorously proving, what I could only barely sense, but never put into empirical words, that certain knowledge could never be proven using human logic, that any system of thought can only be fully known by another system outside or beyond that system. It was a tour de force. I suddenly wanted to read much more about Gödel, the man I'd earlier discarded to the intellectual junkbin. Gödel's life was of course fascinating, and he did have many somewhat 'crazy' thoughts (and I STILL regard his God proof as one of them!**), but simultaneously he generated some of the most profound ideas ever derived by human brains, changing the face of philosophy and mathematics forever, as few have done.

Then, in 1979, a young whippersnapper named Doug Hofstadter came out of nowhere to write "Gödel, Escher, Bach," one of the most acclaimed nonfiction books of the 20th century, winning numerous awards, and setting its youthful author on the road to fame and... well, more fame. And again I thought, "WOW!" someone figured out a clever, creative, insightful way to take all this self-reference/recursion stuff and put it into a book form that boggles readers' minds. (I had always hoped that one of Hofstadter's followups, "I Am a Strange Loop," might really blow the doors wide open on this subject area, but unfortunately it did not.)

There are so many good reads on Gödel and his work out there these days that I won't even favor any by singling them out; you can find plenty on him just by googling.  And we've already had fine movies about John Nash, Alan Turing, and Ramanujan. Perhaps a film version of Gödel is not far off. 
I will close out however with this quote (I've used before as a 'Sunday reflection') from Freeman Dyson in "The Scientist As Rebel":
"Gödel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.
"It is a curious paradox that several of the greatest and most creative spirits in science, after achieving important discoveries by following their unfettered imaginations, were in their later years obsessed with reductionist philosophy and as a result became sterile. Hilbert was a prime example of this paradox. Einstein was another…

"Science in its everyday practice is much closer to art than to philosophy. When I look at Gödel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as Chartres Cathedral… The proof is a great work of art. It is a construction, not a reduction. It destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas. Gödel proved that in mathematics the whole is always greater than the sum of the parts. Every formalization of mathematics raises questions that reach beyond the limits of the formalization into unexplored territory."

**  [just to be clear, my own view of God aligns with Martin Gardner's "fideist" view (a form of theism), as a concept so far beyond human comprehension or definition, let alone "proof," that it almost defies discussion, which too often turns God into what I call the 'Santa Claus' version that predominates Western religion (...even while people often try denying that is their version)]