Archive for June, 2010

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Avoiding Epistemic Circularity

Finance, business, and indeed many fields share commonality with mathematics in that there is a need to separate fact from fiction. However, there is much formal terminology that is subtly but significantly different in meaning. What follows is a small sample of the formal terms often used by academics and well-educated practitioners:

Axiom - an axiom or postulate is a proposition that is not proved or demonstrated but taken for granted. Therefore, axioms often serve as a starting point for deducing and inferring other truths.

Theorem - In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.

Conjecture - In mathematics, a conjecture is a proposition that is unproven but appears correct and has not been disproved.

Lemma - In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself. There is no formal distinction between a lemma and a theorem, only one of usage and convention.

Corollary - A corollary is a statement of fact that follows readily from a previous statement, usually a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective.

Axiomatization - the act or process of reducing something to a system of axioms. This happens a lot in quantitative finance but not everyone approves of this.

Hypothesis - A hypothesis is a proposed explanation, that can be tested, for an observable phenomenon. More specifically, in the mathematical sense statistical hypothesis testing is used to evaluate the hypothesis. Thus, the test is a method of making decisions about the certainty of the stated hypotheses using experimental data. In statistical hypothesis testing a result is called statistically significant if it is unlikely to have occurred by chance.

Epistemic Circularity - An epistemically circular argument defends the reliability of a source of belief by relying on premises that are themselves based on the source. You wont sound convincing if you put forward epistemically circular arguments! NB: this is used in philosophy more so than mathematics (hence the photo of Rene Descartes) but non-the-less a useful term to know.

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Warren Buffet, the Innovators, the Imitators and the Idiots

Legendary investment sage, Warren Buffer, was speaking to a journalist about the financial crisis of 2008 when he came out with a fairly accurate summary of how good ideas go bad. Buffet said... there's a "natural progression" to how good new ideas go wrong. He called this progression the "three I's." First come the innovators, who see opportunities that others don't. Then come the imitators, who copy what the innovators have done. And then come the idiots, whose avarice undoes the very innovations they are trying to use to get rich.

I reckon that cycle applies to many new ideas not just those dreamed up by traders at investment banks.

You can read more on the Harvard Business Review website.

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Explaining Eulerian Paths

A popular tool of creative programmers and mathematicians is graph theory. The father of graph theory is arguably Leonard Euler who studied a now famous problem called the 7 Bridges of Konigsberg. The problem involved crossing each of 7 bridges exactly once without any backtracking or swimming across the rivers - aptly named a Eulerian path. Here's is a map of the seven bridges.

Can you see a path that crosses each of these bridges exactly once? No, me either. Euler in fact proved that one does not exist in this case and one wont exist in any case in which the number of vertices with odd degrees is not zero or two. Let me explain... First lets reduce the above picture into something more simplistic. When you cross a bridge you travel from one land mass to another, so lets reduce all the relevant land masses to "nodes". These are shown in red dots below. Note that we are only really interested in land masses that are on either side of one of the seven bridges.

Now we need to add the different routes that you can take over the bridges. If there are multiple bridges that span the river between two land masses then we can use multiple paths/edges on our "graph". In the image below, the black lines indicate possible bridge-crossing paths.

Admittedly this looks a little messy so here a nicer version of the same thing without the background imagery and having the "vertices" labelled with letters for easier reference.

This picture represents an undirected graph that has 4 vertices and 7 edges between them. If you count the number of edges that each vertex "connects to" you will note A, C and D all have 3, whilst B has 5. The number of edge connections a node/vertex has is called the degree (or less often, the valency) of the vertex. It is a common measure of centrality in graph theory applications - e.g finding the most influential person in a social graph could be the person with the most connections. Here, we are trying to find a path in the graph that traverses each edge exactly once. (This is in contrast to a Hamiltonian path in which you attempt to visit every node/vertex exactly once.)

So back to Euler's theorem: it's provable that there is NOT a Eulerian path here because you need exactly zero or two vertices with odd degrees for such a path to exist and we have 4 nodes with odd degrees. So now you know. It may sound a little academic and irrelevant but it does have a habit of cropping up in discussion of many other famous graph theory problems.

For example, suppose that a postman has to deliver letters to the residents in all the streets of a village. Assume that the village is small enough for the postman to be assigned this task everyday. If the undirected graph that represents the street network of the village has a Eulerian path then this path gives rise to a handy route that the postman can use to deliver letters going through every street exactly once. Clearly, this is a desirable for the postman. Alternately, if no such Euler path exists then the postman may have to repeat some of the streets. This problem is called the Chinese postman problem in honour of a Chinese mathematician Meigu Guan who proposed this problem.

If that's piqued your curiosity here a puzzle for you whose solution depends on existence or absence of a Hamiltonian path.