22 October 2008

Conventions and rationality

One of the basic problems is the inability of many to recognise the difference between something that is fundamental and something that is a convention.

An example of a fundamental truth is the inverse square law of gravity. It is an empirical fact, based on observation, experiment, measurements; by accepting it, we build a theory upon which we can predict other facts, understand why it takes about 365 days for the earth to go around the sun, and build satellites, among other things. This is why the law of gravity is fundamental.

An example of a convention is that a week has seven days. It surprises many people that the concept of a week is not a concept, but a rather arbitray convention. It is difficult for many to digest this because the concept has been around for, well, some four thousand years. Of course, there are many a posteriori justifications. Why, say some, the week consists of seven days because the Bible says so. This is not a justification, even if the Bible was written by a God, as many contend.

The inability to distinguish between a fundamental truth and a convention is a major problem. It suddens me to have to point this out to my students (and others), in the post-enlightenement 21st century. But if we have to point out trivialities like this, we should do so if wesave the rapid decline that will lead us to the another medieval era.

I see this inability to distinguish between fundamentals and conventions each and every day in the university. Sadly, we all blame students. We blame poor elementary education. We blame politicians. But we should blame ourselves as teachers. I do encounter many professors who themselves cannot distingush between fundamentals and conventions. They pass on the wrong message to the students. And the result is catastrophic.

Let me give an example of poor teaching that cannot distinguish between fundamentals and conventions. Suppose a lecturer in a basic Statistics course tells students that sums of large numbers behave as if they come from a normal distribution without making the effort to explain (a) the assumptions or (b) why the normal distribution is unavoidable. I am well aware of the mathematical difficulties of such a concept for a beginning Statistics course, but I am also well aware of methods that can convey the message to the students without having to go through a stern mathematical demonstration. Such methods, unavoidably, will leave students with questions. But this is the point of education: students should be left with (the right) questions so that they will seek to fill in the gaps themselves or by attending more advanced courses. Instead, what is happening in many institutions of so-called higher education (and, indeed, in mathematical sciences departments), is that lazy (or ignorant?) lecturers will fill the students' minds with wrong impressions such as: Statistics means opening an excel programme, filling in the squares with data, and pressing a few buttons to get an answer. How boring! But how convenient if your utility function is to mazimize profit by having lots of students enrol in your classes. And how well it works for those students who choose the path of least action.

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant