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Sunday, April 23, 2017

Truth, Certainty, Explanation... and Mathematics

Physicist David Deutsch reflecting on mathematics (from his "The Fabric of Reality"):
[There is] "...an ancient and widespread confusion between the methods of mathematics and its subject-matter. Let me explain. Unlike the relationships between physical entities, relationships between abstract entities are independent of any contingent facts and any laws of physics. They are determined absolutely and objectively by the autonomous properties of the abstract entities themselves. Mathematics, the study of these relationships and properties, is therefore the study of absolutely necessary truths. In other words, the truths that mathematics studies are absolutely certain. But that does not mean that our knowledge of those necessary truths is itself certain, nor does it mean that the methods of mathematics confer necessary truth on their conclusions. After all, mathematics also studies falsehoods and paradoxes. And that does not mean that the  conclusions of such study are necessarily false or paradoxical.
"Necessary truth is merely the subject-matter of mathematics, not the reward we get for doing  mathematics. The objective of mathematics is not, and cannot be, mathematical certainty. It is not even mathematical truth, certain or otherwise. It is, and must be mathematical explanation."

Sunday, April 16, 2017

"Whence Certainty?"

Sunday reflection... from Rebecca Goldstein in "Incompleteness: the proof and paradox of Kurt Gödel":
"So the question is: Whence certainty? What is our source for mathematical certainty? The bedrock of empirical knowledge consists of sense perceptions: what I am directly given to know -- or at least to think -- of the external world through my senses of sight and hearing and touch and smell. Sense perception allows us to make contact with what's out there in physical reality. What is the bedrock of mathematical knowledge? Is there something like sense perception in mathematics? Do mathematical intuitions constitute this bedrock? Is our faculty for intuition the means for making contact with what's out there in mathematical reality? Or is there just no 'there'?"

Tuesday, April 11, 2017

For Your Funnybone…

 At a recent large used book sale I headed to the math section and picked up a few older volumes, including John Allen Paulos’ “Once Upon a Number” from 1998. Of course mathematics is timeless, but I was pleasantly surprised to see how much of the less-mathematical content of this volume is still relevant today (heck, maybe even more so since Nov. 8, 2016); much of it concerning logic, reasoning, meaning, information, clear/critical thinking and the like.
Anyway, I’ll put all that aside to only pass along this non-math joke Paulos tosses in at one point (the book is sprinkled with his typical humor):

A young man is on vacation and calls home to speak to his brother.

 “How’s Oscar the cat?”

 “The cat’s dead, died this morning.”

 “That’s terrible. You know how attached I was to him. Couldn’t you have broken the news more gently?”


“You could’ve said that he’s on the roof. Then the next time I called you could have said that you haven’t been able to get him down, and gradually like this you could’ve broken the news."

‘Okay, I see. Sorry.”

 “Anyway, how’s Mom?”

 “She’s on the roof.”

Sunday, April 9, 2017

Science As Uncertainty Reduction

Sunday reflection:

“It feels like there are two opposite things that the public thinks about science: that it’s a magic wand that turns everything it touches to truth, or that it’s all bullshit because what we used to think has changed… The truth is in between. Science is a process of uncertainty reduction. If you don’t show that uncertainty is part of the process, you allow doubt-makers to take genuine uncertainty and use it to undermine things…
“And it’s absolutely crucial that we continue to call out bad science. If this environment forces scientists to be more rigorous, that’s not a bad thing.”

— Christie Aschwanden (of FiveThirtyEight )

Wednesday, April 5, 2017

Riemann In the News...

Lots of interesting mathy stuff out there this week, but hey, you can’t go wrong with the crown jewel of number theory, so I’ll direct you to two pieces on the Riemann Hypothesis, if you’ve not seen them:
First, a brief interview with Barry Mazur and William Stein, authors of “Prime Numbers and the Riemann Hypothesis” (one of my favorite 2016 books):
…and then the incomparable Natalie Wolchover summarizing the latest intriguing approach from physicists to Riemann’s 150+ year-old, million-dollar conundrum:
The actual (physics) work was published last year but is just now being widely disseminated on popular media:
There are a great many other introductions to RH on the internet, including some video ones such as these:
From Numberphile:

…and from 3 Blue1Brown:

Sunday, April 2, 2017

Beyond the Boundary of Logic

For a beautiful Sunday reflection, the ending words from Eugenia Cheng in "Beyond Infinity":
"The most beautiful things to me are the things just beyond that boundary of logic. It's the things we can get quite a long way towrd explaining, but then in the end they just elude us. I can get quite a long way toward explaining why a certain piece of music makes me cry, but after a certain point there's something my analysis can't explain. The same goes for why looking at the sea makes me so ecstatic. Or why love is so glorious. Or why infinity is so fascinating. There are things we can't even get close to explaining, in the realm far from the logical center of our universe of ideas. But for me all the beauty is right there on that boundary. As we move more and more things into the realm of logic, the sphere of logic grows, and so its surface grows. That interface between the inside and the outside grows, and so we actually have access to more and more beauty. That, for me, is what this is all about.
"In life and in mathematics there is often a trade-off between beauty and practicality, along with a contrast between dreams and reality, between the explicable and the inexplicable. Infinity is a beautiful dream, inside the beautiful dream that is mathematics."

Thursday, March 30, 2017

For Mathematics Lovers

A great read for anyone who wants to learn what math really is, no prerequisites required. And those of us in the field are reminded of what first drew us to it."    Maria Chudnovsky, Princeton University and 2012 MacArthur Fellow
An elegant sampler of many beautiful and interesting mathematical topics. This could become one of the best books available for a popular audience interested in what mathematics really is.”  —Jayadev Athreya, University of Washington

Apologies for my slightly redundant book blurbs lately, but I again want to re-mention a new volume I already praised. The above publicity blurbs are for “The Mathematics Lover’s Companion” by Ed Scheinerman, and it is so far, my favorite book of this young year. I’m a sucker for volumes that offer up a buffet of interesting math topics without lingering too long on any one. You can view the Table of Contents (and some of the content) here:
I especially like Part 3 of the volume on “Uncertainty” (though it is the shortest section), but the other two parts, covering a nice selection of algebraic and geometry topics, are very good as well.
The reason for even bothering to note this book again, is my disappointment at how little “buzz” I see it getting in cyberspace (I do see it regularly in my local Barnes & Noble outlets, so I know it’s being well-distributed). My only guess is that the publisher, Yale University Press, just doesn’t put as much energy into promotion as some of its counterparts. This is a fabulous book, especially for a lay audience, but also for folks farther along their mathematics journey, so I'd hate to see it ignored... especially if you already love mathematics!

Wednesday, March 29, 2017

Fun From Fibonacci (via Keith Devlin)

Keith Devlin’s new volume, “Finding Fibonacci” is more historiography than math, but there is some simple fun math sprinkled along the way deriving from Fibonacci’s writings. One old problem (translated from Fibonacci’s “Liber abbaci”) that Keith quotes runs as follows [I’m re-wording it in updated English]:
A man buys 30 birds composed of partridges, pigeons, and sparrows, for 30 denari. A partridge costs 3 denari, one pigeon costs 2 denari, and 2 sparrows cost 1 denaro, or 1/2 denaro/each. How many birds of each kind has the man purchased?
Of course this looks like a classic multi-equation problem, except there are only 2 equations, yet 3 unknowns:
1)  x + y + z = 30  (number of birds; x partridges, y pigeons, z sparrows)
2)  3x + 2y + z/2 = 30 (total price)
You can solve it, or look at the answer below…
Keith notes, there is a third hidden piece of knowledge buried in the problem: 
namely, that x, y, and z must be positive integers (because birds don’t come in fractions!)
Thus, the two equations above are easily reduced to:
5x + 3y = 30 (where both x and y must be whole positive numbers)

Keith notes the first and third terms are divisible by 5 and so the second term (3y) must also be divisible by 5. In turn, that means y must equal 5, 10, 15 etc… but 10 or more is too large to work in the equation, so only 5 can be the correct answer (and x = 3, and z = 22).
As I indicated previously, the book may be more appealing to math history buffs than for mathematicians themselves (I found the last few chapters, containing more math bits, the most interesting of the book), though, with warm weather approaching, I'm tempted to call it a good beach read for the nerdier among us. In it, Dr. Devlin makes a case that "Liber abbaci," from the under-appreciated Leonardo Bonacci ("Fibonacci") is "a book that changed the course of Western civilization," and seeing him build that case (including making an analogy between Fibonacci and Steve Jobs) is interesting in its own right.

Monday, March 27, 2017

Re-playing Testosterone

To start the week, a blast from the past….

This weekend’s “This American Life” episode was a replay of a crowd-favorite dating back 15 years to August 2002, on the subject of testosterone. The whole episode is wonderfully entertaining, but I always especially enjoyed “Act 2” with a transgender man, born as a female, but reporting on the results of undergoing years of testosterone treatments:

I’ve written about it here before because of one brief section that is fascinating (in a non-PC sort of way). I’ll simply re-quote from that earlier posting:
….After relating a lot of already interesting stories to host Ira Glass about how the change in gender affected him, the interviewee is asked by Ira if there are any other alterations due to testosterone he thinks worth mentioning. The individual responds that after taking testosterone he "became interested in science; I was never interested in science before." To this Ira can't help but chuckle and respond, "NO WAY!" adding that such a response is "setting us back 100 years." The individual goes on to insist that testosterone resulted in "understanding physics in a way I never did before."  (…the specific exchange occurs around the 22:30 point of the whole episode)
It is concerning, but also funny, to think of our abilities/skills being so subservient to our biochemistry, even if this is just one lone anecdotal case. Anyway, the entire segment is fascinating and worth a listen (assuming you enjoy the style of “This American Life;” however the part I’m noting above is the only bit that actually relates back to math or science; the rest is mostly social/psychological in nature).

In a quite long interview over at the Edge site a few years ago, Simon Baron-Cohen, a major researcher in this area, had this to say following a question from Marcus du Sautoy about the relationship between tendencies toward math, and human biology/testosterone:
“…you mentioned mathematicians, and I think you're right, that there are these areas of human activity, math is one of them, where we do see very disproportionate sex ratios. My understanding is that in mathematics, at university level, it's about 14 males for every one female sitting in the audience in those lectures. That's a very big difference. And there are other sciences, as we know, which used to be like that but which have changed dramatically. Medicine is a very good example. It used to be male dominated and it's now certainly 50-50, or if anything, it's gone beyond and there are now more female applicants and thankfully, successful applicants. If you look at the audience in medical lectures, the sexes are, if not equally represented, maybe even more women than men. But there remains this puzzle why mathematics, physics, computer science, engineering, the so-called STEM subjects, why they still remain very male biased. I’m the first to be open to anything we can do to change the selection processes at university, or change the way we teach science and technology at school level, high school level, to make it more friendly to females, to encourage more women to go into these fields. But there remains a puzzle as to why some sciences are attracting women at very healthy levels, and other sciences, including mathematics, remain much more biased towards males. Whether that's reflecting more than just environmental factors, and something about our biology, is something that I think we need to investigate.”

Controversy continues....

[...much of Baron-Cohen's research, by the way, studies the possible link between high pre-natal testosterone exposure and autism]

Sunday, March 26, 2017

Mathematical Values

This week's Sunday reflection, courtesy of Roger Penrose:
"How, in fact, does one decide which things in mathematics are important and which are not? Ultimately, the criteria have to be aesthetic ones. There are other values in mathematics, such as depth, generality, and utility. But these are not so much ends in themselves. Their significance would seem to rest on the values of the other things to which they relate. The ultimate values seem simply to be aesthetic; that is, artistic values such as one has in music or painting or any other art form."

Thursday, March 23, 2017

About Infinity...

I blurbed a little bit earlier about Eugenia Cheng’s new book “Beyond Infinity.” Very much enjoying it, now that I’m farther in (…but do realize it’s an entire book about infinity — so you need a significant interest in the topic to enjoy it; the typical popular math book might only have a chapter or two on infinity, touching a few highlights; this volume goes deeper).
For now just wanted to mention one small matter that came up:
Quite awhile back on Twitter I asked if there was any sort of “proof” that aleph-null must in fact be the ‘smallest’ infinity; i.e. infinity is full of so many counterintuitive outcomes, and the whole question of whether aleph1 really is the second infinity is so complicated, that I wondered how we could even be sure that the natural numbers, for example, represent the lowest degree of infinity.
The few replies I got implied that the minimalness of aleph-null was axiomatic or established by definition. BUT Dr. Cheng does offer a short form of something like a proof in her volume. Her basic argument is simply to indicate that there is no subset of the natural numbers that can be put into one-to-one correspondence with the natural numbers and have anything leftover (sort of a reversed diagonalization argument). Or as she concludes, “This means that every subset of natural numbers is either finite or has the same cardinality as the natural numbers. There is no infinity in between. So we have found the smallest possible infinity: it’s the size of the natural numbers.”
I don’t know if I’m quite fully convinced (that there is much more than tautology or definition at work here), but I was glad to at least see an argument put forth. Dr. Cheng herself admits “This is not quite a proof, but is the idea of a proof…” It’s at least better than saying that the natural numbers are the lowest infinity by edict ;)
A lot of the difficulty in wrapping one’s brain around infinity lies in our deep-seated entrenchment in one view of what “numbers” are. As Cheng writes at one point, “Infinity isn’t a natural number, an integer, a rational number, or a real number. Infinity is a cardinal number and an ordinal number. Cardinal and ordinal numbers do not have to obey all the rules that earlier types of number obey.” 
I still have several chapters to go, and they look like they will be quite good. As with her earlier work ("How To Bake Pi") Dr. Cheng writes in an off-hand, almost conversational style meant to draw readers in to sometimes difficult or abstract ideas. I don't think she is always successful, but admire her making the effort. And her own passion for her subject-matter is clear.

Sunday, March 19, 2017

The Universe as a Mathematically-designed Machine

"To the divine understanding, all phenomena are coexisting and are comprehended in one mathematical structure. The senses, however, recognize events one by one and regard some as the causes of others. We can understand now, said Descartes, why mathematical prediction of the future is possible; it is because the mathematical relationships are preexisting. The mathematical relationship is the clearest physical explanation of a relationship. In brief, the real world is the totality of mathematically expressible motions of objects in space and time, and the entire universe is a great, harmonious, and mathematically designed machine. Moreover, many philosophers, including Descartes, insisted that these mathematical laws are fixed because God had so designed the universe and God's will is invariable. Whether or not humans could decipher God's will or penetrate God's design, the world functioned according to law, and lawfulness was undeniable, at least until the 1800s."

-- Morris Kline (in "Mathematics and the Search For Knowledge")

Tuesday, March 14, 2017

Just For Fun

Just for fun today, some simple arithmetic….

Math Tricks” blog put up the somewhat classic '30-cows-in-a-field' riddle yesterday, and I’ll post another video of it here today for any not familiar with it:

(what I love about this is that it is simple, appropriate for all ages, and nicely demonstrates language/speech ambiguity)

Sunday, March 12, 2017

Atoms and Primes

 Beginning of Edward Scheinerman's new book, "The Mathematics Lover's Companion":
"The physicist Richard Feynman believed that if humanity were to be faced with the loss of all scientific knowledge but was able to pass on just one sentence about science to this postapocalyptic world, that sentence should describe how matter is composed of atoms. In that spirit, if we could pass on only one bit of mathematics to the next generation, it should be the solution to the problem: How many prime numbers are there?"
[...he goes on to describe some of the proofs for the infinity of primes, before embarking on a wide array of other topics throughout the book.] 

Wednesday, March 8, 2017

"Experimental Math"... never-ending explorations

ICYMI, John Horgan interviewed Stephen Wolfram recently at his blog:

That was followed up shortly by a long, interesting post from Wolfram himself on “experimental mathematics,” iteration, cellular automata, Mathematica, etc. (h/t to Mike Lawler):

...and then Mike Lawler followed that up incorporating some of Stephen's inventive ideas into his own "Family Math" series:

Sunday, March 5, 2017

Sunday With Hermann

Sunday reflection from a 2014 paper on mathematical neural correlates:
"Hermann Weyl is recorded as having said, 'My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.' Relevant here is the story of Weyl's mathematical formulations, which tried to reconcile electromagnetism with relativity. Rejected at first (by Einstein) because it was thought to conflict with experimental evidence, it came subsequently to be accepted but only after the advent of quantum mechanics, which led to a new interpretation of Weyl's equations. Hence the perceived beauty of his mathematical formulations ultimately predicted truths even before the full facts were known."

Wednesday, March 1, 2017

Evelyn Lamb Serves Up Her Mathiness From 'Hard-hitting' February

The 2nd edition of Evelyn Lamb's new newsletter is out... GREAT place to keep up with Dr. Lamb's writings (since she shows up in multiple outlets), as well as other things on her mind:


If you're not already a subscriber I encourage you to become one (so you don't have to rely on me pointing out each new issue):  https://tinyletter.com/evelynjlamb

Sunday, February 26, 2017

"replication is central to science"

For a Sunday reflection, this from Andrew Gelman in "The Best Writing on Mathematics 2016":
"To resolve the replication crisis in science, we may need to consider each individual study in the context of an implicit meta-analysis. And we need to move away from a simplistic, deterministic model of science with its paradigm of testing and sharp decisions: accept/reject the null hypothesis and do/don't publish the paper. To say that a claim should be replicated is not to criticize the original study; rather, a replication is central to science, and statistical methods should recognize this. We should not get stuck in the mode in which a 'data set' is analyzed in isolation, without consideration of other studies or relevant scientific knowledge. We must embrace variation and accept uncertainty."