Formal Indecision

A formal system in mathematics is a system which contains a set of axioms (unquestionable statements of truth), and then adds to these a set of production rules by which new true statements can be generated. The process of production can go on forever allowing a never-ending list of truths to be derived.

A famous result called Gödel’s incompleteness theorem shows that in such a system, there are some true statements that are nevertheless un-provable (undecidable) using the rules of the system. It shows that a self-consistent Continue reading

The Raven and the Shoe

The hypothesis that all ravens are black is logically equivalent to the statement that all non black things are non ravens, and this is supported by the observation of a white shoe.

This is a paraphrasing of a famous paradox due to Hempel. There is a lot of fairly impenetrable discussion on the Wikipedia page about this paradox, some of which I believe to be incorrect, and so I include a readable resolution based on a Bayesian perspective here, and I also relate this issue to our ability to know the truth Continue reading

Trees, Fractals and Solar Power

The Wall Street Journal recently ran a story about Aidan Dwyer, a 13 year old, who has created a stir with his science project which purported to show that if solar cells are arranged in space like leaves on a tree rather than on a flat surface, they collect more solar energy. His project write-up can be found here. The internet response has ranged from praise and offers of business capital to aggressive attacks and charges that his experiment is pseudoscience.

Unfortunately Aidan did not measure the right quantity (power) when performing his experiments, so in the original form they are not conclusive. What aspects of his ideas Continue reading

My Infinity is Bigger than Yours

I have been reading a lot about numbers, Georg Cantor, and infinite set theory recently and thought I would share some of the more interesting things here. Mostly this concerns properties of the integers (whole numbers such as 0, 1, -1, 1032, …), the rationals (numbers that can be written as a/b, such as 1/2, 7/8, 22/7, …), and the irrational numbers, which are things like √2 or $\pi$ which cannot be written as a/b. All these together make up the real numbers. Conventionally we write real numbers using decimal sequences so that 1.25 is a way of writing 5/4, but actually is a representation of the series 1 + 2/10 + 5/100. It is important to Continue reading