Videos

So many great math-related videos coming out all the time. Passing a few along in the event you’ve missed them and have time to catch up:

From Numberphile this is actually an older video on the 'sum of three cubes' equation n = x³ + y³ + z³

...The video is suddenly very pertinent because a solution for the integer 33 has now been discovered, and here it is in all its glory!:

  33 = 8866128975287528³ + (-8778405442862239)³  + (-2736111468807040)³ 

From Matt Parker another pedagogical video with a mind-blowing ending:

Eugenia Cheng here in a TED talk applying pure, abstract math to our view of society:

If you can possibly find time for it (~90 mins.), phenomenal investor Jim Simons recounts his mesmerizing life in mathematics (and this is just the first of 3 discussions at MIT with Dr. Simons; I assume the others will also be uploaded):

...Lastly, I’ll just re-mention this fun note I tweeted out a couple days back: About a year ago I discovered a retired local math professor who gives free math talks that I attend, had previously worked with Tom Lehrer — I thought that was pretty cooool! Now just this weekend I learned he also previously worked with both Grothendieck and Ted Kaczynski. Wow, THAT'S quite an array of math folks! (and sets me thinking about my own  'six degrees of separation'). ;)

Category Theory via Eugenia Cheng

For Sunday reflection, Eugenia Cheng describing 'category theory':

"This is how category theory arose, as a new piece of math to study math. In a way, category theory is an ultimate abstraction. To study the world abstractly you use science; to study math abstractly you use category theory. Each step is a further level of abstraction. But to study category theory abstractly you use category theory."

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 toward 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."

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.