Montaigne

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I was introduced to Montaigne when I was 18. I hope to sometime run into the man who introduced me to this work. He sold the stories very well to me, and I devoured the book. He comments in his essays about the absolute range of human endeavour, from frienship, to torture, to sex, to the Education of Children.
The American philosopher Eric Hoffer employed Montaigne both stylistically and in thought. In Hoffer’s memoir, Truth Imagined, he said of Montaigne, “He was writing about me. He knew my innermost thoughts.”
The reason I bring up Montaigne, is I came across the following Book review in the LA Times addressing a new biography of the great man.
Montaigne is certainly the author an amazing ‘commonplace’ .
Blogs are certainly the modern interpretation of the common place idea. Hence the purpose of writing this blog, may be to help me to learn how to live better. Which is a very human idea.

But, of course, the Montaigne party has still not ended, and it is the measure of Bakewell’s book that she makes it seem like the hottest ticket in town. As she reminds us,

the best reason to read Montaigne is the one he would most approve: because it helps us contemplate ourselves

. Surely we, too, need to think about Liars and Idleness and Prognostications; about Solitude and Moderation; about Friendship and Age and Sleep. Surely we need to read about the Education of Children, and the Uncertainty of our Judgments, and the Inequality Among Us. Surely we, too, need to learn how to live. It is not that Montaigne’s essays contain an answer to that question. It is that they are an answer. How to live? Try.

It is said that great literature is timeless. What strikes me most about Mathematics and Culture (or anything done very very well) is the absolute timelessness of it. Montaigne should be read by us now, and by our descendants,

In sum, this book, like its subject, is expansive, genre-defying, and preposterously smart.

Works of Genius truly do last for ever. And perhaps the next book I re-read will be Montaigne. It sounds confusing to some but a book like that truly changes lives. And in some sense I don’t think we read Montaigne, like with Wittgenstein, Nietzsche, Shakespeare, Goethe or Poincare we meet them.
And like all the wonderful people we meet in our lives, they change us.

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Is Democracy the right answer?

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It is as I think Orwell observed worrying if we see democracy as ‘the’ answer.
how voters decide.

Indeed, voters constantly complain that initiatives are too complicated. Two out of three told the PPIC poll in December that the wording of the initiatives was confusing. But over-complex language is only one worry in a process where every statute and constitutional amendment interacts with every other to shape policy in this huge state. It raises the more general question of how large, diverse and dispersed populations filter information and arrive at decisions. “If those most likely to think they have a grasp on political information are in fact wrong,” says Ms Nalder, there may be a need to “think twice about the wisdom of direct democracy”.

The world is getting more complicated. Those who worship democracy are perhaps not dealing with how it works in practice.
So what is the solution? Well I’m not exactly clear, but the persuasion by advertisers/ political campaigners makes making a thorough educated and rational decision very difficult.
As Johann Hari wrote in his well-written article about Ed Miliband, most people don’t think too much about politics. This does leave us open to manipulation.
One of my favourite quotes is the following:

Let’s start with the words themselves. The best research indicates the average person in Britain spends two minutes a day talking or thinking about politics. It’s not because they are “thick” or “apathetic”, as some people haughtily assume. They have stretched lives. They have a lot to do. When they catch snatches of politics, it has to be clear and plainly expressed. They aren’t going to go away and google the terms.

We vote emotively not rationally, which means that most political analysis is an entire waste of time.
Democracy may be merely an approximation to a better solution, but I’m not sure what that is.
Jonah Lehrer also chimes in on the ‘ignorance of voters’.
I particularly like the phrase we are ‘rationalizing voters’ not ‘rational voters’. In recent debates I’ve had with people about Nuclear Power, I’ve seen this one occur quite a lot.
The Ignorance of voters
We truly are ridiculous creatures.

He who refuses to do arithmetic is doomed to talk nonsense

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Dice


The above slogan came to my attention once again from the excellent Cosma Shalizi comments on Krugmans book ‘Pop Internationalism’. In the book review
The great Computer Scientist has a slogan which Shalizi extends from merely arithmetic to algebra.
This short essay will be looking at this issue. It is often said by politicians for instance that Mathematical literacy is important. I can’t disagree with this comment, yet Mathematics is a deep subject and I doubt everyone needs to know for instance ‘Sheaf Theory’.,
The question could be stated: ‘What is essential knowledge from Mathematics so that one doesn’t speak nonsense’.
I can only propose a provisional list, and clearly its worthwhile writing some articles on this.
Probability I feel is essential, since so many things in our world involve calculation probabilities. For instance it is more likely that Mary is an investment banker than Mary is an investment banker AND a Feminist.
Rather than reinvent the wheel, allow me to link to Steve Strogatz writing about probabilties and conditional probabilities:

The comments on the cognitive problems involved in calculating probabilities (our brains are bad at dealing with such things) is fascinating. I think its excellent reading on the limits of our cognition, and how Mathematics can be used to stop us talking nonsense, and to better understand the risks involved in our every day lives. From Cancer diagnosis, to catching those who have broken the law, to accessing the risks involved in building Nuclear Power Plants, if we fail to learn probability we are doomed to talk nonsense.
So Algebra, Arithmetic AND Probability, are all essential skills. Would anyone care to disagree?

Procepts in Mathematics education

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Todays blog post is a bit different to some of my views on politics, or current affairs. Its back closer to my student life as a Mathematics Masters student.
Inevitably when one enjoy mathematics and physics one gets involved in giving tutorial classes or helping friends and family. Mathematics is something that we all agree is very important, yet no one knows why.
Anyway today I want to talk about a ‘procept’, which I think reminds me of some of David Hestenes work on ‘modelling’ theory.

I came across the following paper in Mathematics Education which introduces the concept of a procept.
David Tall’s seminal paper on procepts
PROCEPT: An elementary procept is the amalgam of three components: a process which produces a mathematical object, and a symbol which is used to represent either process or object.
A procept consists of a collection of elementary procepts which have the same object.

“A proceptual known fact should be distinguished from a rote learned fact by virtue of its rich
inner structure which may be decomposed and recomposed to produce derived facts. For
instance, faced with 4+5, a child might see 5 as “one more than 4” and might know the double
4+4=8 to derive the fact that 4+5 is “one more”, namely 9. For the proceptual thinker this gives
a powerful feedback loop which uses proceptual known facts to derive new known facts”

Take when you learn Noetherian Rings in Commutative Algebra, well its a struggle at first. Yet eventually you learn what this is, you accept the definition and you may think for instance that a Noetherian ring is a finitely generated ideal, or one with an ascending chain of ideals which eventually become stationary. There is a lot locked up in that short statement, knowing what an ideal is etc takes a long time. Yet its amazing that one eventually gets to the point when one can discuss very complicated topics, such as ‘categories’ and ‘functors’ and one is happy to have a procept understanding of these. A functor is both a morphism between two categories and an object in itself. There are also various types of functors – but this is for another discussion.
Some of that may be too technical for some readers, yet I see it in most areas of expertise. A problem in exams and for most disciplines is ‘passing the test’ versus true understanding.

On tricks for exams versus rich cognitive units.
Another article by David Tall Plenary talk
“Consider, for example, the notion of ‘linear relationship’ between
two variables. This might be
expressed in a variety of ways
• an equation in the form y=mx+c,
• a linear relation Ax+By+C=0,
• a line through two given points,
• a line with given slope through a given point,
• a straight-line graph,
• a table of values,
and so on. Crowley (2000) (reported in Crowley & Tall, 1999) reveals
how successful students
develop the idea of ‘linear relationship’ as a rich cognitive unit
encompassing most of these links as
a single entity, whilst the less successful simply carry around a
‘cognitive kit-bag’ of isolated tricks
to carry out specific algorithms. The cognitive kit-bag may get the
student through the examination,
but it is too diffuse to build on in later courses and students may
soon reach a point where the ideas
they are handling place too great a cognitive burden, leading
inexorably to failure.”

Now sometimes I cut corners and fail to develop the necessary
intuitions and range of techniques. I’m quite embarrassed sometimes at
the basic things that trick me.
Yet no one really carries around an infinite supply of answers, the
important thing is to think how to solve these sorts of problems.
The discussion in the article of the usage of computers got me
thinking about how they should be used to complement thinking rather
than replace them. I do however result to using Mathematica when I
come across a nasty integral. I’m not sure though if thats laziness or
a substitute for understanding.
When I now thinking of a derivative I for instance have a few
sets of ideas about them:
a) tangent to a curve or a surface
b) the standard analytic version of the limits etc
c) as part of an algebra of derivatives
d) rate of change
Obviously, there are other ways to define a derivative (I’ve been
told there are 167 ways – but don’t quote me on that).
The most important thing I took from that is the need for formal
theories. For instance thinking of Tensors as objects that change
under certain co-ordinate changes versus the following
Tensor. I’d take
the following, or perhaps more abstractly as an object satisfying the
universal mapping property.
Yet when writing notes on tensors I do also think in terms of ‘this
term eats the first r terms, and that term eats the rest’. So there is
a blend between intuition and formalism there.
The concept of a ‘procept’ is very useful, because it makes you understand how deeply you need to go into something to truly get an intuition for it. Definitions and vocab are needed, but so is an understanding (which is often developed through oral and visual explanation) of the concepts.
I’ll write some more about this, maybe thinking in terms of Physics too, but its a very power concept to have when learning mathematics. In some sense Marvin Minsky was correct when saying ‘to truly understand things, you need to know it in 2 or 3 different ways’

Online currencies

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While I interned in Shanghai, I did a lot of research on how mathematics (specifically graph theory) could be used in things like recommendation engines. A recommendation engine is simply some sort of addition to a social network which ‘recommends’ people, products or anything really. An example we’ve all seen is one in Facebook, the ‘recommended friends’.
I came across the following article The Bank of Facebook. It sounds a rather far fetched concept at first, but something I’ve often thought about is how we could attach weight or tangible value to social contacts. Money facilitates transactions, yet I’m suspicious of people who put a lot of emphasis on generating money. It doesn’t buy happiness, but it is the only form of capital that we as human beings have gotten used to. Social capital or cultural capital are more subjective things.
There are some rather scary things that people can infer from our habits on facebook, or our spending habits.
‘By analyzing even slices of this data, a wealth of information can be extracted and predicted about you. As a related example, Google vice-president Marissa Meyer was said to have claimed at this year’s SXSW festival that credit card companies can look at spending habits and predict with 98% accuracy, two years in advance, when a couple is going to divorce. Interesting. I wonder what Facebook is able to predict, and how that information can be served up to advertisers.’

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How to become an Astronaut or a Rhodes Scholar

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I’m a fan of Cal Newport’s blog. He reverse-engineers student success and also more recently looks at what constitutes professional success. His recent article on Rhodes Scholars is very interesting. When reading some of these bios, one can be overwhelmed. Yet Cal once again reminds us that these aren’t parallel achievements.
A friend of mine recently got nominated as an Astronaut, for a new ‘Scientists in Space’ initiative. When looking at his bio we hear ‘Astronaut, Diplomat and Physicist’ and we are overwhelmed. Chris Altman. Yet when speaking to him I got a sense of how long these things took. He worked on research in A.I. and Robotics for a few years at Starlab. Starlab unfortunately ran out of money, so he pursued other avenues. He did his MSc in Physics, and has produced a few papers for his PhD. His PhD has probably extended longer than normal due to his astronaut training which takes a long time to do. He leveraged his expertise in Quantum Computation to improve his chances of getting accepted to such a scheme, not to mention the amazing networking capabilities afforded to him by some of the people he has worked with. Not many of these things has Chris done parallel to each other. His diplomatic work was done on Scientific issues (once again leveraging his excellent Science knowledge and communication skills) a few years ago. And while he’s done numerous internships at prestigious labs, this has all come from his deep understanding of Computer Science and Physics. When you list some of these accomplishments, they sound out of this world. The same with Rhodes Scholars. But if you win ‘Best student of 2009’ it is highly likely you’ll get accepted to a prestigious Summer Scheme and so on.
We are all human certainly, and we shouldn’t forget that the pursuit of excellence needs to be a focused endeavor.
I end with a quote from Chris, which I’ll analyze for my readers.
“The nascent field of commercial spaceflight and the unique conditions afforded by space and microgravity environments offer exciting new opportunities to conduct novel experiments in quantum entanglement, fundamental tests of spacetime, and large-scale quantum coherence,” said Altman.
The fact is that to those outside of Physics, this sounds impressive. I must admit it is impressive that we could do experiments in space, especially quantum entanglement! Yet he’s got a lot of expertise in Quantum Physics, he regularly blogs about these topics and is a very engaging promoter of Quantum Computation research, this coupled with the rather unusual other hobby of having undergone Astronaut training means he is probably one of the few people in the world capable of doing such things.
It does prove however that some little boys do get to become Rhodes Scholars or Astronauts when they grow up.
Bottom line: Time, Focus and leveraging your skills are the keys to success.