Sean at Cosmic Variance wonders why we can’t visualize more than three dimensions. I find it both hard to imagine how you could visualize four dimensions and hard to imagine what biological feature of our brains prevents us from doing so.

That mathematics is dangerous stuff. According to the article, a single formula — the Gaussian copula — killed Wall Street. But according to this article, it’s all the fault of physicists. The only explanation I can come up with that makes sense of these two conflicting versions of events? Mathematics is dangerous only when it falls into the wrong hands.

John Baez alerts us to a a bill before Congress that would overturn the National Institute of Health’s open access policy. The NIH, one of the major sources of funding for medical research, requires that any papers that result from its funding must be made publicly available within a year of publication.

I couldn’t remember how to construct an example of a two closed subspaces of a Banach space such that their sum is not closed, so I searched online. I found a discussion of an example at physicsforum, and a paper by Schochetman, Smith, and Tsui that characterizes when the sum of two closed subspaces of a Hilbert space will be closed.

A couple of weeks ago, John Armstrong posted an interesting story about Abraham Lincoln. Lincoln carried three books with him when he traveled: the Bible, the works of Shakespeare, and Euclid’s Elements. For Lincoln, the Elements represented proof beyond the possibility of doubt.

The Elements once loomed large in the imagination a way no mathematical work does today. The influence can be seen from Newton’s Principia to Spinoza’s Ethics. The popular experience of mathematics and the experience of the practitioners of pure mathematics have diverged since Lincoln’s day. The difference between the Elements and a modern monograph is only one of style and sophistication (and accessibility). Pure mathematicians follow the same axiomatic method as Euclid. But the modern high school and early college curriculum concentrates on subjects useful in the sciences, which are mostly computational rather than deductive. (I had plane geometry in high school, but it was an elective.) It’s an interesting development.

I promised some posts about the significance of Polish spaces in probability. I thought I would start with a philosophical point about the interpretation of probability.

Probability has a strange dual nature. Ask a mathematician, and you’ll get an answer in terms of measure theory. Ask someone who applies probability like a physicist or a statistician, and you’ll get an answer in terms of random draws generated by some physical process. But the two notions are the same, right? Not quite.

The measure-theoretic axioms of probability do not fully capture the folk intuition for continuous random variables. Measure-theoretic probability is not just more general, but it is missing one ingredient in what we mean by probability. That missing ingredient is supplied by the setting of complete metric spaces.

For continuous random variables, not all measure zero events are created equal. Suppose you have a random variable that is uniformly distributed between 0 and 1. Mathematically, the probability that the variable takes on the value of 0.5 and the value of 2 are the same: zero. But conceptually, when you draw from this random variable, some value between 0 and 1 actually happens (even though any specific value is very unlikely), and 2 never happens.

So what’s the difference? You can’t reach 0.5 itself with positive probability, but you do reach every neighborhood of 0.5 with positive probability. On the other hand, small enough neighborhoods of 2 occur with probability zero.

This points us to a method for interpreting the notion of drawing a point from a complete metric space. Imagine that after a random draw, we can ask for each open ball in the space whether an event occurred in that ball. To find out if a specific point occurred we check each open ball around that point to see if that ball occurred. To find out which point has occurred, we just need to find a sequence of open balls that contain the event whose radius go to zero. (By completeness, the intersection of these open balls describe a unique point.)

If the metric space is separable, we can extend this to give a method for simulating draws on a computer. For a fixed radius, we can cover the space by a countable number of open balls of that radius. (This claim isn’t completely obvious, but a standard result of point-set topology is that for a separable metric space every open cover of the space has a countable subcover. This is known as the Lindelöf property.) We randomly draw one of these open balls. Then, using that open ball as our new space, we repeat the process with a new ball of half the radius. After enough steps of this process, we have specified – up to an arbitrarily small error – a point from the space.

So separable metric spaces are a natural setting for probability, one that bridges the gap between the abstract notion of a probability space, and the concrete notion of physically taking a random draw. In a future post I will talk about some of the mathematical implications of this setting.

William Fulton has made his book, Algebraic Curves, available for download. Fulton’s book is a classic exposition of algebraic geometry at the level of an advanced undergraduate. It is one of the few undergraduate texts that cover the Riemann-Roch Theorem.

Bousquet, Boucheron, and Lugosi have authored a short introduction to statistical learning theory.

Found here.

Finally, something we’re not number one at. In empirical research at its finest, this paper has investigated ratings at RateMyProfessors.com, and determined that mathematicians are not number one, either in easiness, or in hotness. (We’re also not 36th out of 36 in either — this honor is reserved for chemists in both categories.)

Here’s an interesting little tidbit I spotted today. Polynesians once traveled throughout the South Pacific by canoe. Apparently, one way they navigated was by stick charts, which were bits of wood woven together to represent both the locations of islands and the direction of ocean waves. The charts themselves look like abstract sculptures rather than anything we would recognize today as maps.

More, including pictures, here.