The Culture of Mathematics

"The idea that there is a single body of knowledge or a single way of thinking that we call 'mathematics' is a myth."

Maneuvering the path between Platonism and non-Platonism... Keith Devlin discusses the "culture" of mathematics (with interesting set of questions toward the end):

https://www.mathvalues.org/masterblog/of-course-2-2-4-is-cultural-that-doesnt-mean-the-sum-could-be-anything-else

Pickover on Platonism…

Some extended discourse via Cliff Pickover today from his volume, "A Passion For Mathematics" (one of my favorite Pickover offerings):

"I think that mathematics is a process of discovery. Mathematicians are like archaeologists. The physicist Roger Penrose felt the same way about fractal geometry. In his book The Emperor's New Mind, he says that fractals (for example, intricate patterns such as the Julia set or the Mandelbrot set) are out there waiting to be found:

'It would seem that the Mandelbrot set is not just part of our minds, but it has a reality of its own… The computer is being used essentially the same way that an experimental physicist uses a piece of experimental apparatus to explore the structure of the physical world. The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there.'

I think we are uncovering truths and ideas independently of the computer or mathematical tools we've invented. Penrose went a step further about fractals in The Emperor's New Mind: 'When one sees a mathematical truth, one's consciousness breaks through into this world of ideas… One may take the view that in such cases the mathematicians have stumbled upon works of God.'

Anthony Tromba, the coauthor of Vector Calculus, said in a July 2003 University of California press release, 'When you discover mathematical structures that you believe correspond to the world around you, you feel you are seeing something mystical, something profound. You are communicating with the universe, seeing beautiful and deep structures and patterns that no one without your training can see. The mathematics  is there, it's leading you, and you are discovering it.'

"Other mathematicians disagree with my philosophy and believe that mathematics is a marvelous invention of the human mind. One reviewer of my book The Zen of Magic Squares used poetry as an analogy when 'objecting' to my philosophy. He wrote,

'Did Shakespeare 'discover' his sonnets? Surely all finite sequences of English words 'exist,' and Shakespeare simply chose a few that he liked. I think most people would find the argument incorrect and hold Shakespeare created his sonnets. In the same way, mathematicians create their concepts, theorems, and proofs. Just as not all grammatical sentences are theorems. But theorems are human creations no less than sonnets.'

Similarly, the molecular neurobiologist Jean-Pierre Changeux believes that mathematics is invented: 'For me [mathematical axioms] are expressions of cognitive facilities, which themselves are a function of certain facilities connected with human language.'"


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]

JMM and MoMath

Brian Hayes of "bit-player" offers a wonderful account of his experience at the recent Joint Math Meetings in Baltimore:

http://bit-player.org/2014/notes-from-the-jmm

… really, a must-read; especially interesting stuff on Yitang Zhang, of twin-prime fame (and apparently now tackling the Goldbach conjecture), including this quote from one of his former Purdue colleagues T.T. Moh:

"When I looked into his [Zhang's] eyes, I found a disturbing soul, a burning bush, an explorer who wanted to reach the north pole, a mountaineer who determined to scale Mt. Everest, and a traveler who would brave thunders and lightnings to reach his destination."

Meanwhile, Peter Woit reports on a "cagematch" between Ed Frenkel (Platonist) and Jim Holt (non-Platonist) held at the Museum of Mathematics in New York recently; he calls it "a no-holds-barred discussion of Platonism and mathematics in front of a standing-room-only crowd"... sounds like it was quite entertaining:

http://www.math.columbia.edu/~woit/wordpress/?p=6609

Don't know if MoMath will eventually upload the session to their YouTube channel, but if you wish to watch for it, or explore some of their other events, the channel is here:

http://www.youtube.com/museumofmathematics

Math... Is That All There Is?

"So nature is clearly giving us hints that the universe is mathematical. I’ve taken it to the extreme by proposing that our entire physical reality isn’t just described by math, but that it is a mathematical structure, having no properties besides mathematical properties."  -- Max Tegmark (physicist)

 "There are still some unanswered questions. For example, would the Higgs boson exist if there wasn't the mathematics to describe it? Perhaps this is a question best solved after a few drinks."  -- Brian Butterworth (neuroscientist)

Interesting discussion (among 2 physicists and 2 neuroscientists) from the Kavli Foundation on the 'origins of math' (the word "Platonism" barely even arises, and yet it is once again largely a debate of two different views, Platonist versus non-Platonist). Is mathematics all-and-only in our heads, or is the Universe completely a mathematical structure, regardless of our human presence to recognize it?:

http://www.kavlifoundation.org/science-spotlights/kavli-origins-of-math

(edited transcript)

The Atlantic Asks... "Does Math Exist?"

I've got to love it when the popular press addresses the issue of mathematical Platonism:

http://www.theatlantic.com/technology/archive/2013/06/does-math-exist/276680/

It includes this fun PBS video:

Remembering... Gardner

"It’s quite astonishing and I still don’t understand it, having been a mathematician all my life. How can things be there without actually being there? There’s no doubt that 2 is there or 3 or the square root of omega. They’re very real things. I still don’t know the sense in which mathematical objects exist, but they do. Of course, it’s hard to say in what sense a cat is there, too, but we know it is, very definitely. Cats have a stubborn reality but maybe numbers are stubborner still. You can’t push a cat in a direction it doesn’t want to go. You can’t do it with a number either. I’m only using the word number because you’ll have a vague idea in your head as to what I mean. The objects that a mathematician studies are more abstract than numbers but very real.
"I often think of cats. I think of trees. I think of dogs occasionally but I don’t think of them all that much because dogs are agreeable. They do what you want them to do to some extent. Some people believe that mathematics is what we think it is and it’s created by our thoughts. I don’t. I’m a Platonist at heart, although I know there are very great difficulties in that view." -- John Conway

Not to take anything away from our Veterans, but this is a math blog, and I'll use the opportunity of Memorial Day to once again remember Martin Gardner, whose death just over 3 years ago inspired me to start this endeavor (with no idea it would still be up-and-running 3 years later!!).

The above quote from John Conway, one of the most creative, productive mathematicians around, is taken from a book review Gardner wrote for a 2009 volume by Mariana Cook, covering 92 mathematicians, entitled "Mathematicians: An Outer View of the Inner World."
It's not a book I've personally seen, but read Gardner's review here:

http://www.newcriterion.com/articles.cfm/Still-four-4349

Gardner (a vocal math Platonist) uses the review to go off on the topic of Platonist vs. non-Platonist viewpoints, writing at one point, "I suspect that almost every mathematician in the book is a Platonic realist, one who believes that mathematical theorems are forever true in all possible worlds and are independent of human culture," before continuing on to offer the above quote from Conway. If anything, the Platonism divide has only deepened since 2009, with brilliant adherents on both sides -- I can't help but think some of it is little more than muddy semantics, while also recognizing that there does exist a core of real (and perhaps non-resolvable) disagreement.


On a side note, I see that Martin Gardner's forthcoming autobiography "Undiluted Hocus-Pocus: The Autobiography of Martin Gardner" is already listed on Amazon, and I suspect will include more of his Platonist evangelism (available in September):

http://www.amazon.com/Undiluted-Hocus-Pocus-Autobiography-Martin-Gardner/dp/0691159912

….I almost wish I didn't know it was on the way... because the 4-month wait will now be excruciating!! :-/

Lastly, if in the mood for some more memories of Martin see here:

http://www.ams.org/notices/201103/rtx110300418p.pdf

also, this great 2005 AMS interview with Martin:

http://www.ams.org/notices/200506/fea-gardner.pdf 

Saturday Potpourri...

A few sundry items for your artful attention and possible perusal ;-):

1) Lance Fortnow, computer scientist and author of "The Golden Ticket," (which I reviewed a bit ago), all about P vs. NP, is Sol Lederman's latest podcast guest at Wild About Math:

http://wildaboutmath.com/2013/04/12/lance-fortnow-inspired-by-math-28/

2) A nice little primer on the nature of real numbers and pi from physicist Matt Springer here:

http://scienceblogs.com/builtonfacts/2013/04/12/everything-in-pi-maybe/

3) An interesting-looking list here of 24 video lectures in number theory:

http://www.infocobuild.com/education/learn-through-videos/mathematics/introduction-to-number-theory.html

4) Just a heads-up that E.O. Wilson is scheduled to be on NPR's Sunday "Weekend Edition" (tomorrow). I assume there will be some discussion of his recent much-debated commentary asserting that scientists need not know advanced mathematics to be successful.

5) Finally, a site I only recently learned of called "Ideas Roadshow" which looks interesting and includes this recent 5-minute clip by philosopher James R. Brown on Platonism in mathematics:

http://www.ideasroadshow.com/issues/james-robert-brown-2013-04-12

Mathematical Truth...

A great, older interview with Rebecca Goldstein on Gödel, Platonism, and everything in-between, from 2005:

http://edge.org/conversation/godel-and-the-nature-of-mathematical-truth

an excerpt:

 "Gödel mistrusted our ability to communicate. Natural language, he thought, was imprecise, and we usually don't understand each other. Gödel wanted to prove a mathematical theorem that would have all the precision of mathematics—the only language with any claims to precision—but with the sweep of philosophy. He wanted a mathematical theorem that would speak to the issues of meta-mathematics. And two extraordinary things happened. One is that he actually did produce such a theorem. The other is that it was interpreted by the jazzier parts of the intellectual culture as saying, philosophically exactly the opposite of what he had been intending to say with it. Gödel had intended to show that our knowledge of mathematics exceeds our formal proofs. He hadn't meant to subvert the notion that we have objective mathematical knowledge or claim that there is no mathematical proof—quite the contrary. He believed that we do have access to an independent mathematical reality. Our formal systems are incomplete because there's more to mathematical reality than can be contained in any of our formal systems. More precisely, what he showed is that all of our formal systems strong enough for arithmetic are either inconsistent or incomplete. Now an inconsistent system is completely worthless since inconsistent systems allow you to derive contradictions. And once you have a contradiction then you can prove anything at all."

"Does Math Really Exist"

"The bottom line is that human beings have brains capable of counting to high numbers and manipulating them, so we use mathematics as a useful tool to describe the world around us. But numbers and math themselves are no more real than the color blue – ‘blue’ is just what we tag a certain wavelength of light because of the way we perceive that wavelength. An alien intelligence that is blind has no use for the color blue. It might learn about light and the wavelengths of light and translate those concepts completely differently than we do."

The above comes from a Forbes magazine piece, of all places, that touches on (I think pretty weakly) the whole Platonist/Non-Platonist debate within mathematics... (does the human mind discover mathematics or create it?):

http://www.forbes.com/sites/alexknapp/2012/01/21/does-math-really-exist/

Long-term readers here know that I am fascinated with this very basic (philosophical) question of whether mathematics is a real part of the extant world, or merely a human cognitive construction. (see Web Platonism articles here or here.)

I have to believe that most of us with math inclinations grow up essentially as Platonists -- perceiving mathematics as real, whether or not any humans exist to explore it. But somewhere along the line, a lot of mathematicians step back to explore the question more openly, objectively, and casting biases aside, and end up swayed to the Non-Platonist side… I'm always impressed by some of these folks who not only see it that way, but seem to think it is rather obviously so (a mere human construction). While still leaning toward mathematical Platonism myself, each year I feel less certain of it. ...Circles, lines, prime numbers, etc. do not exist in the same sense that stars, hydrogen, or atoms do, but does that really make them less real than the latter??? -- they may not represent "real" things, but do they not represent real "relationships"?

One of the ideas that makes non-Platonism at least slightly more palatable is the notion, popular in many physics quarters these days, of a "Multiverse" -- the concept that the Universe we humans have long studied may be only one of many (or even an infinite number of) separate universes that exist. The "laws" and order we discover operating throughout 'our' universe, may simply not be operative in other unseen, separately-evolved universes (whose mathematics might therefore be hugely different). And then there is also (physicist) Max Tegmark's view that the entire Universe is nothing but mathematics ( “there is only mathematics; that is all that exists”).

The Platonist debate includes some purely semantic elements, and is less black-and-white than can be fleshed out here -- there are several possible non-Platonist stances, and the Platonist view itself comes in stronger and weaker forms -- certainly too complex an issue to ever resolve here... yet, a topic I'm continually drawn back to from time to time....

Lastly, a bit of an aside -- in the process of composing this post I chanced upon a couple of wonderful, past Martin Gardner (a well-known Platonist) book reviews bearing on the subject -- but you can read them for the sheer enjoyment of Gardner's prose apart from any philosophical content!:

http://www.newcriterion.com/articles.cfm/Larger-than-proof-2299

http://www.newcriterion.com/articles.cfm/Still-four-4349

Mystical Path... Mystical Math?

Popularizer Clifford Pickover often writes about the mystery and even mysticism of numbers. Paul Erdos was famous for saying certain (beautiful) mathematical proofs must come from 'God's book.' Lover of numbers, Martin Gardner. regarded himself as a "Mysterian" (and also a theist/fideist) who believed, despite the reality of numbers, humans could never fully comprehend the workings of their own minds. Cantor was deeply religious, writing proofs for the existence of God, which never gained the traction his proofs involving infinity did.

In short, I've always found fascinating the link many sense between math or numbers, and the mystical or Godly realm of existence. Math is often perceived, more than any other science, to somehow be associated with a deeper reality than we can otherwise be in touch with directly.
And yet, a different school of math, views math as little more than a creation or construct of the human mind; not so much existing in the 'world out there' so much as constrained to the world inside our heads.
Such basic, fundamental notions, yet leading to such divergent, unresolved thoughts.

Here's an old Julie Rehmeyer posting that touches on the subject (in which she quotes British mathematician Brian Davies as saying that Platonism “has more in common with mystical religions than with modern science"):

http://tinyurl.com/ycsn2bl

And lastly, if you have the time, Ben Vitale recently put up this hour+ long YouTube roundtable video on "Mathematics and Religion":


http://www.quora.com/Benjamin-Vitale/Mathematics-and-Religion