The two fundamental operations: Multiplication and Addition.

Not until I got to graduate school did I learn the concept of a Ring. It seems a bit abstract until I realized how fundamental it is-- more on that in a second.

A Ring has two fundamental operations, multiplication and addition. These are two separate fundamental operations on numbers, and mathematicians don't fully understand their relationship (See the links in this post on the abc conjecture). However, we are taught in school that multiplication is repeated addition: 5 times 3 is just 5 added 3 times. While this is true, it's only true in the special case of multiplying whole numbers. 16.4 times pi is not calculated by repeated addition.

I'm writing this now because I recently read this excellent blog post on exactly this point. I then read a very interesting response by a math teacher, who of course himself learned multiplication as repeated addition. Finally there is a follow-up written by the first blog post author.

One of my favorite quotes from Devlin, the original author, is "No wonder so many people end up thinking mathematics is just a bunch of arbitrary, illogical rules that cannot be figured out but simply have to be learned - only for them to have the rug pulled from under them when the rule they just learned is replaced by some other (seemingly) arbitrary, illogical rule."

I think that we should definitely avoid this approach to teaching math. Of course, perhaps some of my peers and I would have loved to learn abstract algebra in grade school-- but presumably, not everyone wants to. Up until high school, I do think that we should focus on teaching what is most practical. But it would be nice if in topics like this, we could teach the more fundamental way, and so both practically-minded students and academic-minded students would do better as they continue to learn more math.

For those who are interested, a Ring is an abstract concept of a set of objects with two operations: addition and multiplication. The set of objects must be "closed under those operations," meaning that if you add or multiply two objects, the result is also in the set. So the integers are a ring. If you add or multiply any two integers, the result is an integer. Also, to be a ring, the addition operator must satisfy associativity, commutativity, distributivity, and the multiplication operator must be associative as well. There must be an additive identity (a+0=a) and an additive inverse for every set element (-a+a=0). These are all basic concepts to which we are introduced in grade school. It's pretty neat to see that they are fundamental principles on which all of mathematics is built.

open access

Thank you to PHD comics for making this nice animation along with a voiceover describing open access publishing, starting with the current state of scholarly publishing and why it needs to change. I admit I am part of the problem-- I do submit to closed journals with a desire to get one of their names on my CV. But with the arxiv and other ways to put our content online, there is now the opportunity share every preprint with open access, and I hope that researchers continue to move in that direction. Hopefully before too long the most prestigious journals will be ones that were invented now as open access.

Air Pollution Sampling

Reading this article from yesterday about "microsampling air pollution" made me hopeful for my submitted NSF proposal on data methods for air pollution sensing. Just as in every area where "bigdata" is making a difference, sensor data collection could make a big impact on understanding our environment and the role it plays in our health.

the social process of the proof

Proofs of mathematical theorems each have their own story.

This essay by journalist Caroline Chen describes the ABC conjecture and its story, including the fact that mathematician Shinichi Mochizuki posted a proof not quite a year ago, and the mathematical community has yet to confirm or even understand it.

Three years ago when the P ≠ NP conjecture was purportedly proven, I blogged about it and was excited for the social process of checking the proof. Unfortunately it was pretty quickly decided that there were large holes in the reasoning and it wasn't even a proof, though it did give some new ideas on how to solve the problem.

And then just last week, it was announced that Yitang Zhan proved the "bounded gaps" conjecture for prime numbers. The paper was submitted to the Annals of Mathematics in April, and was accepted 2 days ago, meaning that the proof has passed peer-review.

What will happen with the ABC conjecture? No one has found holes in Mochizuki's proof, but no one can understand it either. My favorite quote from Chen's essay:

"[In mathematics,] Colleagues check each other’s work, spending hours upon hours verifying that a peer got it right. This behavior is not just altruistic, but also necessary: unlike in medical science, where you know you’re right if the patient is cured, or in engineering, where the rocket either launches or it doesn’t, theoretical math, better known as 'pure' math, has no physical, visible standard. It is entirely based on logic. To know you’re right means you need someone else, preferably many other people, to walk in your footsteps and confirm that every step was made on solid ground."

SF Bay Area Maker Faire

The Maker Faire is this weekend! And ladies, be sure to stop by, because you can be engineers too :) My colleague (and dear friend) Dr. Angi Chau will be there with her students from Castilleja School, go say hello!

two things that made me laugh today

#1. Google "recursion."

#2. From the New Yorker April 29, 2013:

We got used to it.

"It takes time for an acorn to turn into an oak; but the oak is already implied in the acorn."

I hope you enjoy this absolutely lovely video as much as I did this evening.

"For example, people used to believe ... people who lived in the antiquities would fall off. And that was scary. But then when somebody sailed round the world, and we all got used to it, and we travel around in jet planes and everything... We have no problem thinking that the earth is globular. None whatever. We got used to it."

I have faith in mankind, that someday we will get used to a lot of the scientific ideas that are controversial today. Pretty neat to think about it.