Good Math Bad Math

Yearly Archives: 2007

The Evil, Imprecise, Confusing Metric System

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Yet another alert reader just sent me a link to this extremely
humorous blog. It’s not recent, but it’s silly enough that it’s worth pointing
out even now. I’m not one hundred percent sure that this isn’t a parody. Looking at
the blog as a whole, I think it’s serious. Pathetic, but serious.

According to scientists, The French / Eurotrash “metric” system is in a state of crisis today, as scientists discovered that the weight of the “kilogram” is flip-flopping faster than John McCain at an episcopalian rally.

The Kilo – a unit with an identity crisis: This is one of the approximately 30 standard kilograms stored in a French castle. The polished metallic pellet a the center of this contraption is made of an alloy of precious metals, provided at government expense. Only in Europe would government subsidize such folly!

Since Europe converted to the metric system a few years back, this could have a grave impact on that continent’s faltering economy. Imagine a situation where customers wishing to buy some meat, or fill their car with gas literally have no idea how much their goods will cost, simply because nobody knows the exact definition of a weight.

Jesus, the humble carpenter no doubt used a ruler to measure his wood every day, however the Bible clearly states that he had no need of the metric system.

How much simpler life would be if we only adopted the standard units of weights and measures mentioned in the King James Bible? Biblical measures are authorized by God, and like God are not tied down to fickle matter. This is yet another example of how Science only really makes sense when it is rooted in the Bible.

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Building up more: from Groups to Rings

Group Theory

If you’re looking at groups, you’re looking at an abstraction of the idea of numbers, to try to reduce it to minimal properties. As I’ve already explained, a group is a set of values with one operation, and which satisfies several simple properties. From that simple structure comes the
basic mathematical concept of symmetry.

Once you understand some of the basics of groups and symmetry, you can move in two directions. You can ask “What happens if I add something?”; or you can ask “What happens if I remove something?”.

You can either add operations – which can lead you to a two-operation structure called a ring; or you can add properties – in which the simplest step leads you to something called an abelian group. When it comes to removing, you can remove properties, which leads you to a simpler structure called a groupoid. Eventually, I’m going to follow both the upward and the downward paths. For now, we’ll start with the upward path, since it’s easier.

Building up from groups, we can progress to rings. A group captures one simple property of
a set of number-like objects. A ring brings us closer to capturing the structure of the system
of numbers. The way that it does this is by adding a second operation. A group has one operation
with symmetric properties; a ring adds a second symmetric operation, with a well-defined relationship
between the two operations.

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The Meaning of Division: Quotient Groups

Group Theory

After that nasty diversion into economics and politics, we now return to your
regularly scheduled math blogging. And what a relief! In celebration, today I’ll give
you something short, sweet, and beautiful: quotient groups. To me, this is a shining
example of the beauty of abstract algebra. We’ve abstracted away from numbers to these
crazy group things, and one reward is that we can see what division really means. It’s
more than just a simple bit of arithmetic: division is a way of describing a fundamental
structural relationship that pervades mathematics.

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Insurance: Why it sucks

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Look folks, I don’t want to become an economics blogger! Stop sending me economics questions. I hate to disappoint my readers and not answer their questions, but this economics stuff is almost terminally dull to me.

The mortgage posts have gotten an insane amount of traffic, which has in turn brought in a huge number of questions. Most of them are about details of the whole mortgage situation – and honestly, I can’t answer those. I don’t know the details, only the basics, and I can’t explain what I don’t know.

On the other hand, a lot of people have used my down-to-earth explanation of the mortgages as
a springboard to ask for a similar explanation of another issue that’s been getting a lot of attention in America – insurance. Why is health insurance such a big problem?

I’m going to focus specifically on the insurance part. There are plenty of things wrong with the
American medical care system. But I’m going to focus specifically on how insurance works, and
how simple min/max calculations led to the current situation.

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The Total Stupidity of Crowds: Bad Mortgages and Circular Solutions

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Reading the news lately, I’ve come across an amazing example of how ubiquitous bad math can be used. Most of you have probably heard about what’s been called “the sub-prime crisis”. Despite a lot of media hand-wringing about how complicated it all is, the sub-prime crisis is really a very simple phenomenon: basically, you’ve got a lot of banks that have loaned out money without worrying about whether or not it could get paid back, and now those loans aren’t getting paid back, which is causing all sorts of grief to people who invested in them.

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Macros: why they're evil

Programming

I’ve gotten both some comments and some e-mail from people in response to my mini-rant about Erlang’s macros. I started replying in comments, but it got long enough that I thought it made sense to promote it up to a top-level post. It’s not really an Erlang issue, but a general language issue, and my opinions concerning it end up getting into some programming language design and philosophy issues. If, like me, you think that things like that are fun, then read on!

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Records in Erlang

Erlang

One of the things I discovered since writing part one of my Erlang introduction is that Erlang has grown a lot over the last few years. For example, the idiom of tagged tuple as a way of creating a record-like structure has been coded into the language. There is, unfortunately, a bit of a catch. They aren’t really added to the language. Instead, there’s a pre-processor in Erlang, and records
are defined by translation in the pre-processor. This to me typifies one of the less attractive
attributes of Erlang: much of Erlang has a very ad-hoc flavor to it. There are important high-level features – like record data and modules – which aren’t really entirely part of the language. Instead, they’re just sort of spliced in however it was easiest for the implementors to cram ’em in. And there are other things that were added to later versions of the language that, while first class, are awkward – like the “fun” prefix for first-class functions.

The effect of that kind of thing is mainly syntactic. The implementation is good enough that
even though things like records are really pre-processor constructs, they feel almost as
if they’re built-ins.

Anyway – let’s take a look at records.

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Symmetric Groups and Group Actions

Group Theory

In my last post on group theory, I screwed up a bit in presenting an example. The example was using a pentagram as an illustration of something called a permutation group. Of course, in
my attempt to simplify it so that I wouldn’t need to spend a lot of time explaining it, I messed up. Today I’ll remedy that, by explaining what permutation groups – and their more important cousins, the symmetry groups are, and then using that to describe what a group action is, and how the group-theory definition of symmetry can be applied to things that aren’t groups.

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