Tag Archives: math

On Godel and Game Theory (what I find interesting in math and sciences, May 2022)

Math education: here’s some good pieces on that topic. First up, Susan Fowler (now Rigetti) on how you can learn more math. One way to learn is to see how others do it. Here’s Fields medalist Tim Gowers working through math problems on YouTube. Also this is a good resource on YouTube: math channel.

Speaking of learning math, here’s a piece on how to fix math class.

Math theory: For fans of game theory: Game theory and Cuban Missile crisis. Here’s two more good theory pieces, one on Godel’s Incompleteness Theorems and one on Fermat’s last theorem.

Philosophy: Here’s two good pieces on math and philosophy: How Julia Robinson helped define the limits of mathematical knowledge and another on how math and philosophy need each other.

Physics and space: this is cool, Visualizing Black Holes with General Relativistic Ray Tracing. So is this: SpaceX’s Starship and NASA’s SLS Could Supercharge Space Science. This is odd,  NASA Will Test Gigantic Centrifuge for Hurling Objects Into Space. This is fun, measuring the earth using the traceroute command! This is fascinating, what I learned as a hired consultant to autodidact physicists.

Brain: here’s two good pieces on the brain. First, Your brain expands and shrinks over time. Second, Deep sleep drives brain fluid oscillations.

Here’s how to know when to quit, using math.


Knowing when to quit is difficult. Fortunately, there is science and math to help us, as this article in plus.maths.org explains:

Knowing when to quit is one of life’s great dilemmas, whether to persist in the face of diminishing rewards, or to quit now in the hope of finding richer rewards further afield. For every gold mine, eventually there comes a point when the amount of gold extracted can no longer justify the cost of keeping the mine open; once that point has been reached, it is time to quit, and start looking for a new mine.

Similarly, for a bird feeding on caterpillars in a bush, there comes a point when the calories gained can no longer justify the energy expended in searching for more caterpillars; once that point has been reached, it is time to quit, and move to another bush. And, for a honeybee, there comes a time when the weight of pollen collected can no longer justify the energy required to carry that pollen; at that point it is time to quit collecting, and fly back to the hive.

Fortunately, there is a mathematical recipe, embodied in the marginal value theorem (developed by the American ecologist Eric Charnov in 1976), which specifies when to quit in order to maximise rewards. More importantly, the marginal value theorem has an enormously wide range of applications, from its origins in optimal foraging theory to how brains process information. In essence, the marginal value theorem provides a general strategy for maximising the bang per buck, irrespective of the nature of the bang and the buck under consideration.

Wait you say: it’s Friday afternoon! You know when to quit (at least for this week). Ok, that works too. But if you need more general guidance, read the article.

On Emmy Noether

Recently the Perimeter Institute held a contest called “Physics Frenzy – Battle of the Equations Championship”. I was sure James Maxwell and his equations were going to win. It turns out he lost (in the final, mind you) to Emmy Noether and her theorem!

Afterwards I wondered if people knew much about her. Besides being arguably the greatest woman mathematician, she was a significant contributor to math and physics in the 20th century. You can read a bio of her here.

Fun aside, the good people at the Perimeter Institute have a series of initiatives centered around Emmy Noether. They also have material where you could learn more about her.  Read about them, here.

 

What I find interesting in math and sciences, March 2022

Here are two pieces on physics that really are also about philosophy. First, in the realm of the very small: On quantum theory and reality. Second, in the realm of the very large: On space and time.

Of all the sciences, those having to do with space are the ones I enjoy reading about the most. I got interested in Lagrange points due to the James Webb Telescope. Here’s something on them: Lagrange points.

I thought this was a good reminder to me of any images we get regarding black holes (something I find fascinating): The Images of Black Holes are Not Photos.

COVID has made me think more and more about biology these days. That’s a good thing. It has me reading things like this: Disease moves like ripples on a pond.

One of my fundamental beliefs is that we don’t think with our brains but that we think with our entire bodies. This makes me interested in anything that makes that connection. This is such a piece: Our nervous system is connected to our organs and shapes our thoughts and our memories.

There was some discussion on racism in mathematics over the last while. I think this is a good discussion to have. Here are two pieces on the topic: Racism in our curriculums isn’t limited to history. It’s in math too and Opinion: Is math racist? Wrong question.

I continue to learn math in an amateurish way. I enjoyed these sources of information on it: Guide to Tensors and The Best Math Books and Statistics Books.

(Photo by Roman Mager on Unsplash)

Can math prevent gerrymandering? (or what I find interesting in math, Dec 2021)

Here are seven good links to pieces on math I thought were good:

  1. These are cool:  cool alternative numbers.
  2. This is a worthwhile project:  About Project Euler.
  3. For fan of   Godel’s Theorems 
  4. This was somewhat amazing:  Fermat’s Library : Magic : The Gathering is Turing Complete annotated/explained version.
  5. This was a good intro into a form of math I wasn’t aware of:  Maths in a Minute: Category theory
  6. This is fascinating:  The 26 000-Year Astronomical Monument Hidden in Plain Sight.
  7. Some practical math:  Virginia wants to prevent gerrymandering. Can a mathematician help?.

A great introduction to Bayes’ Theorem and how it relates to COVID-19

You may have heard references to Bayes’ Theorem in light of the pandemic and wondered how it is relevant. Well I am here to help. First off, here’s a great guide to Bayes’ Theorem from the website MathIsFun.com. Even if you are math phobic, I think you will be able to read that piece and understand it. Secondly, check out the site Varsity and how it explains how Bayes’ Theorem and COVID-19 testing are related. Both are well worth a read.

Learn Bayes’ Theorem. It’s good to help you understand many things in life, including what is happening during the pandemic.

P.S. This related piece at FT.com  explains why you should expect to see vaccinated people in the hospital with covid despite high vaccination rates.

What I find interesting in math, summer 2021

I read many ofpieces on mathematics: here are some pieces I have found worthwhile. Most of them are readable by folks who are not mathematicians.

Here’s a number of good pieces

Finally, This 3 700-Year-Old Tablet is the Oldest Example of Applied Geometry. Cool.

(Photo by Saad Ahmad on Unsplash )

Is math political? Does it need to be? A few thoughts

You would think so, if you read this: Sneering at Ontario’s anti-racist math curriculum reveals a straight line to what people value in The Star

What has the columnist angry was the removal of several passages of progressive political text that went with the update to the recent changes of the math curriculum. I can see why that removal would anger some people with progressive political values.

I can also imagine how many conservatives would have been angry if there was text like this removed from a new curriculum: “recognize the ways in which mathematics can be used as a tool to uncover, explore, analyse, and promote actions to address greater productivity and growth within our economy and to lead Canada to a strong future of wealth and opportunity”, or if the government removed anything to do with teachers creating “pro-capitalist and pro-business teaching and learning opportunities.” Any group that tries to explicitly frame a curriculum and then have that framing removed will be upset.

Mathematics itself is not political, but it is always taught within a political and historical context. For example, I have math texts that make it seem that the only worthwhile math came from European men, while I have others that show mathematics has roots all over the world. I have math textbooks that mention 0 women, while other texts show the role women have played in mathematics and delve into why women had a hard time making more of a contribution.

Whatever context you want to frame a curriculum, I think that emphasizing politics and history with regards to teaching mathematics will not achieve some of the goals that progressive thinkers hope it will achieve. I think the new changes in the curriculum with regards to things such as streaming will help achieve those goals, as I wrote here.

Additionally, I think there are other things that can be done outside the curriculum that could help students that are disadvantaged when it comes to education in math. I am thinking of the work done by organizations like BlackGirlsCode. We could use more organizations like that who can provide specialized programs not just to help kids who are struggling with math, but to uplift kids that excel in math. Organizations that can support the next Maryam Mirzakhani, wherever she is. The kids who are struggling with math need more help than what the schools can provide: the same is true for kids that excel in math.

(Photo by ThisisEngineering RAEng on Unsplash)

The best book to learn calculus from

This may be the best book to learn calculus from: Calculus Made Easy.

I like it for two reasons. One, it’s free. Two, it does not take itself seriously nor does it take calculus seriously. To see what I mean, here’s a clip from the beginning of the e-book:

Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks. Some calculus-tricks are quite easy. Some are enormously difficult.

The fools who write the textbooks of advanced mathematics—and they are mostly clever fools—seldom take the trouble to show you how easy
the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.

Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.

So if you want to learn calculus but are struggling, give that book a look. Sure it’s an old book, but calculus is an old subject. It may suit you just fine.

(Photo by Jeswin Thomas on Unsplash)

Great insights on mathematics you should read about even if you don’t think you want to


I suspect many people will not want to read this article containing great insights on mathematics by Steve Strogatz. That’s a shame, because it is really approachable by anyone of any mathematical ability. It’s especially good for people with limited math skills, because he does a good job of showing the value and benefits to be gained from thinking mathematically. I highly recommend it if you read it.

For example, one thing I found fascinating is his discussion of the Prisoner’s Dilemma by comparing it to religion. You should read it, but in short, it’s been shown that one approach to succeeding in playing several rounds of the Prisoner’s Dilemma is to use a Tit-for-Tat strategy. This is highly effective and is similar to Old Testament Eye-for-an-Eye morality. However that can also go wrong on occasion, leading to long lasting feuds that never get resolved. Then he gets into a discussion of New Testament morality and how that can avoid some of the problems of Old Testament morality. It’s a great discussion, and one of the many great discussions in the article.

Photo by Erol Ahmed on Unsplash

Quote

9 Numbers That Are Cooler Than Pi


Fun! For all you number theory fans out there: 9 Numbers That Are Cooler Than Pi

I still think Pi is pretty cool. But so are these other numbers.

Quote

Why math is great, and other interesting ideas, from Steven Strogatz

It’s hard to say why this interview with Strogatz is so good, other than to say he covers much ground on a variety of interesting topics and speaks lively on them. (Ok, I find game theory, “elegant” math, math education, etc, interesting, but you likely will too).

If you enjoyed this interview, he has a recent book out, “Joy of X: A Guided Tour of Math, from One to Infinity.” Worth a look. 

Interview is here: Steven Strogatz interview on math education and other related topics

A math lover’s post

I knew that Paris had streets named after politicians and historical figures, but I didn’t know how many Paris streets are named after mathmaticians. Apparently quite a few! Amazing. One more reason to love Paris.

Dover publishers have the best books when it comes to math. If you want to see some of their better ones, see this list.

Good stuff.

Published on 9/29 at 9:29 🙂

 

My favourite mathematics proof.

Is this:

A nice use of computing to refute an old conjecture from no less than Euler.

Found thanks to a tweet from Aatish Bhatia 

Is your day boring? You need to think bigger thoughts. Here’s a short post on Infinity

This is a great introduction to the topic of Infinity. I think even people who struggle with math will get this and enjoy it.

Unless you studied mathematics, you likely didn’t know that about infinity. It is fascinating stuff, I find.

Found here: Infinity is bigger than you think – Numberphile – YouTube via @anitaleirfall on twitter.