Robots, Humans, Ants

 Not precisely mathematics, but this week's Sunday reflection by physicist Max Tegmark on why we need to be careful when it comes to programming artificial intelligence:

"If you're walking on the sidewalk and there's an ant there, would you actively go and stomp on it just for kicks? (Me: 'No.')
"Now, suppose you're in charge of this big hydroelectric plant that's gonna bring green energy to a large region of the U.S. And just before you turn the water on, you discover there's an anthill right in the middle of the flood zone. What are you gonna do? It's too bad for the ants, right? It's not that you hate ants. It's not that you're an evil ant-killer. It's just that your goals weren't aligned with the goals of the ants, and you were more powerful than the ants. Tough luck for the ants. We want to design AI in the future so that we don't end up being those ants."

From 6th Grade to Advanced Physics...

Just a couple of links that hint at the breadth of mathematics, to start the week with:

a)  Brian Hayes, who is not a 6th-grader ;-) is off-and-running with an analysis of an arithmetic problem Fawn Nguyen previously posed to her 6th graders (...both delightful and interesting):
http://bit-player.org/2015/mrs-nguyens-prestidigitation

b)  Despite David Hilbert famously chastising the detractors of Cantor's "infinity," that "no one shall drive us from the paradise which Cantor has created for us," physicist Max Tegmark argues for finding "the true laws of physics" by finally getting rid of infinity:
http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/#.VOs2kcZDTLX

Sunday Morning With Max and Martin

"The Mathematical Universe Hypothesis offers a radical solution to this problem: at the bottom level, reality is a mathematical structure, so its parts have no intrinsic properties at all! In other words, the Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks. The external physical reality is therefore more than the sum of its parts, in the sense that it can have many interesting properties while its parts have no intrinsic properties at all."

-- Max Tegmark from "Our Mathematical Universe"

…and in a footnote on the same page Max pushes the idea further, writing:

"Our brain may provide another example of where properties stem mainly from relations. According to the so-called concept cell hypothesis in neuroscience, particular firing patterns in different groups of neurons correspond to different concepts. The main difference between the concept cells for 'red,' 'fly,' and 'Angelina Jolie,' clearly don't lie in the types of neurons involved, but in their relations (connections) to other neurons."

Martin Gardner almost seemed to pre-sage Tegmark when he wrote the following more than a decade earlier (in his essay, "Is Mathematics 'Out There'?"):
 

"No mathematician, Roger Penrose has observed, probing deeper into the intricate structure of the Mandelbrot set, can imagine he is not exploring a pattern as much 'out there,' independent of his little mind and his culture, as an astronaut exploring the surface of Mars.
"In the light of today's physics the entire universe has dissolved into pure mathematics. The cosmos is made of molecules, in turn made of atoms, in turn made of particles which in turn may be made of superstrings. On the pre-atomic level the basic particles and fields are not made of anything. They can be described only as pure mathematical structures. If a photon or quark or superstring isn't made of mathematics, pray tell me what it is made of?"

Or similarly, this from another Gardner piece, "In Defense of Platonic Realism":

"In a curious way, numbers may be more real than pebbles. Matter first dissolved into molecules, then into atoms, then into particles, which are now dissolving into tiny vibrating strings or maybe into Penrose's twistors. And what are strings and twistors made of? They are not made of anything except numbers. If so, the numbers are as much 'out there' as molecules. They could be the only things out there. As a friend once said, the universe seems to be made of nothing, yet somehow manages to exist. As Ron Graham remarked, mathematical structure may be the fundamental reality."

A Weekend Potpourri… (including Tegmark, Riemann, JMM14)

1) Not being an educator, I often ignore tweets appearing in my math feeds which seem geared strictly for teachers, but a recent one was getting RE-tweeted so often I finally caved and had to look at the link being passed along to see what it was. Indeed it was a delightful, simple idea most any primary school could employ… check it out if you've not seen it:

https://twitter.com/wterral/status/423755360512790528/photo/1

2) Below, a quick note on the packed talk covering recent twin-prime gap work at the Joint Math Meetings in Baltimore, MD.:

http://blogs.ams.org/jmm2014/2014/01/17/mind-the-gap/

3) I'm currently about half-way through Max Tegmark's new book, "Our Mathematical Universe."  Probably won't do a full review of it here since a) it's far more physics than mathematics (despite the title) and b) there are already many reviews of it around, so no need to add to the cacophony of publicity (good and bad) it's receiving.  Having said that, I will say I'm immensely enjoying it as a popular science volume -- in fact, it's one of the BEST cosmology reads I've yet come across (and I've read many)… which is not an endorsement of its views but simply of its readability… as cosmology books go, I'd call it a lively romp! For a more critical look though, see multiverse-phobic ;-) Peter Woit's review here, and be sure to peruse the comments:

http://www.math.columbia.edu/~woit/wordpress/?p=6551

A New Scientist review below (and Dr. Tegmark has been all over the Web promoting the volume as well, just google the book's title):

http://tinyurl.com/jwdfjbv

and more quick takes here:
http://space.mit.edu/home/tegmark/mathematical.html

4) Finally, a fantastic overview (in under 20 mins.) of the Riemann Hypothesis from that gregarious explainer James Grime (if anyone can make the Riemann Hypothesis fun, James can!):



That should fill your weekend for awhile.

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)

Weekend Potpourri

So many mathy reads to choose from this weekend, if you've missed them (really enough for a couple of weekends!)....

1) Brand new from Sol Lederman, a wonderful podcast with mathematician/writer Erica Klarreich:

http://www.buzzsprout.com/5316/78729-erica-klarreich-inspired-by-math-22

I've previously mentioned my belief that interviewing lesser-known math figures is almost more enticing than interviewing the titans in the field because readers/listeners already know so much about the 'big names' out there that much of what they say may seem repetitious (even if important) of things they've voiced before. Lesser-known folks have a fresher appeal as there is so much new to learn about them, and I think this podcast demonstrates my point. I was fascinated hearing Erica's views and experiences on a wide range of topics. See if you don't agree.
And below, some of Erica's prior writings for the Simons Foundation:

https://simonsfoundation.org/?s=erica+klarreich&submit

2) MIT physicist Max Tegmark is famous for his belief that the Universe is "built" of mathematics. He expresses his viewpoint in this straightforward interview from the ScienceNow site… and attracts a lot of comments in the process (including cynical ones):

http://news.sciencemag.org/sciencenow/2013/02/do-we-live-inside-a-mathematical.html

3) Julie Rehmeyer, who I just posted about a few days ago, has a new piece on Fermat's Last Theorem and its axiomatic basis, in ScienceNews here:

http://tinyurl.com/a4ujnx5

4) Patrick Honner again laments a question from a NY State Regents Math exam that entails unstated assumptions and in so doing short-circuits deep mathematical thinking:

http://mrhonner.com/2013/02/21/regents-recap-january-2013-unstated-assumptions/

5) Some bloke named Keith Devlin has a fantastic longread on math games in American Scientist, leading up to release of his own company's new animated game for math learning. He sets forth the criteria or qualities a video needs to possess to be successful as a math instruction tool (and explains why MOST games FAIL):

http://www.americanscientist.org/issues/pub/the-music-of-math-games

From the description, it sounds to me as if the prospective player (young person) of these new games will learn math in a manner reminiscent of the original Karate Kid (the movie) learning karate without ever knowing it from his master Mr. Miyagi. Keith's discussion of the "symbol barrier" and symbol manipulation in math is especially enlightening, but the entire piece is GREAT. He employs a music (piano) metaphor to explain what a successful math game should be like.

6) The brilliant Barry Mazur, recent recipient of the National Medal of Science, gives us a rich (and philosophical) piece called "Shadows of Evidence" on what constitutes "evidence" in the realm of mathematics. The essay ends with a quote from Chris Anderson essentially arguing that "modeling," as traditionally used in science, is becoming obsolete (because ALL models are, technically, flawed), and that with the advent of computer number-crunching ability, only "correlation" derived from huge data sets will be needed. To which Mazur responds that, "correlation alone will never replace the explanatory power of mathematics.":

http://tinyurl.com/bhzrjgf

This essay in turn leads to an even longer Mazur read entitled "What Is Plausible" here (pdf):

http://www.math.harvard.edu/~mazur/papers/Plausibility.Notes.3.pdf

7) Finally, perhaps appropriately after all of the above, I'll end with 50 varied quotations just put up by Guillermo Bautista on what mathematics is:

http://mathandmultimedia.com/2013/02/23/50-mathematics-quotes/

a few of my favorites:

"Mathematics is no more computation than typing is literature." – John Allen Paulos

"Mathematics, in the common lay view, is a static discipline based on formulas…But outside the public view, mathematics continues to grow at a rapid rate…the guide to this growth is not calculation and formulas, but an open ended search for pattern."  -- Lynn A. Steen

"Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them." — Joseph Fourier 

.... ADDENDUM:  Experimenting with shorter time segments, Sol Lederman has already put up another podcast interview (22 mins.), this time with Jason Ermer, creator of the "Collaborative Mathematics" project (who I referenced a bit ago):

http://www.buzzsprout.com/5316/78788-jason-ermer-inspired-by-math-23

Mathematicians have been at the forefront of bringing productive Web collaboration to academic subjects, and now Jason is attempting it at lower levels. Check it out!

One Physicist's Slant on Math

Physicist Max Tegmark muses on math and reality: