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:

http://tinyurl.com/mb4rd8q

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!

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…

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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.

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:

https://www.thisamericanlife.org/radio-archives/episode/220/Testosterone?act=2#play

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]

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."

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.

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")

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)

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.] 

"Experimental Math"... never-ending explorations

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

https://blogs.scientificamerican.com/cross-check/polymath-stephen-wolfram-defends-his-computational-theory-of-everything/?

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):

http://blog.stephenwolfram.com/2017/03/two-hours-of-experimental-mathematics/

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

https://mikesmathpage.wordpress.com/2017/03/07/sharing-stephen-wolframs-momath-talk-with-kids/

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."

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:

http://tinyletter.com/evelynjlamb/letters/stuff-evelyn-wants-you-to-read-2

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