Haskell numbers

I just put the (still being worked on) slides for a talk in a few hours that I'm doing at the London Haskell User Group, on the subject of "Numbers".

Manhattan would be squarer if it had diagonal streets too

There are different ways of measuring distance - that's the abstraction of metric spaces.

The usual distance that people use looks like this:

Points at a certain distance form a circle around the point that you're measuring from.

Another metric is the Manhattan distance, where rather than moving in any direction you can only move along one axis at once. This leads to diamond contours instead of circles.

Those are both well known metrics. I wondered what the Manhattan distance would look like if you were allowed to move diagonally as well as along the axes. The distance seems to be max(abs(dx),abs(dy)), and that looks like this, all square:

roman numerals code

I made this roman numeral convert applet years ago (the RCS tag is $Id: Roman.java,v 1.11 2001/01/07 15:12:00 benc Exp $) and mostly left it untended since then. It accounts for 25% .. 50% of the hits on hawaga.org.uk and was brought to the front of my consciousness by someone asking if they could use the source in their school project. Now I feel all embarrassed about the clunky UI on that thing.

nominal, ordinal, interval, ratio

Decision by Objectives (DBO) classes applications of numbers into a hierarchy of four different types, each building on the properties of the previous one.

The concepts are not unfamiliar at all to a mathematician, but they are approaching from a management direction rather than from pure mathematics, and I think that is an interesting different perspective.

The first application of numbers is nominal. In this application, numbers are used to name things. DBO gives examples from horse racing: The number on a jockey or a horse is a nominal use. It identifies a particular jockey or horse; but it is meaningless to (for example) add two jockey numbers together, or divide one by another. In Maths, that means it is possible to permute the numbers in any way without losing any information. The numbers, ℕ, are used purely as a set of distinct objects without structure or relation. We could equally well use any other sufficiently large set of symbols (such as letters).

The next application of numbers is ordinal. In this application, numbers are used to order things. In the horse racing example, the finishing order of the horses is an ordinal application. Some horse finishes first, another finishes second, yet another finishes third. In Maths, there is a linear order between the numbers, and we have a relation < or ≤ that makes sense. We can compare numbers in this application, with a notion of bigger or faster or stronger). But it still doesn't make sense to (for example) add or subtract those numbers, or to make statements like "horse X finished 3 places ahead of horse y".

The third application of numbers is interval. It makes sense to subtract interval numbers from each other. The authors are a little vague in DSO about whether they think it makes sense to add interval numbers. I think in some applications it does (how much chocolate did you eat? how much chocolate did I eat? we can compute: how much chocolate did we eat together?) but in some it does not (what is the time now? what time did the meeting start? we can compute: how long has the meeting been going on (subtraction), but adding the time now and the meeting start time doesn't make sense). When it does make sense to add, the numbers form a group with operation +. When it only makes sense to take the difference, then the numbers form something like a torsor - a group without identity.

The final application is ratio. These is use of numbers where ratios make sense - for example, a horse might finish in half the time of another, meaning that the ratio of the time of the first horse to the second horse is 1/2. It makes sense to multiply and divide these numbers; and ratios can also be used to scale interval numbers. Mathematically, we have a (meaningful use of a) field, ℝ or ℚ.

The point of them making these distinctions in DSO was to point out that just because you have some information in your decision making represented as numbers, it doesn't always make sense to do things like taking the mean, or summing, or whatever.

seq

You(*) whisper to X, "the same package [GNU coreutils] which provides seq"
You(*) whisper to X, "the command line utility"
You(*) whisper to X, "also provides factor"
You(*) whisper to X, "which outputs prime factors"
You(*) whisper to X, "makes me want to start making fucked up shell scripts"
You(*) whisper to X, "that use goedel numbering"
You(*) whisper to X, "for some purpose."
You(*) whisper to X, "i have to wonder how a commandline utility for factoring primes is 'core' in any way, though"