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Relating mathematics to XML -- using properties for enablingunderstanding
• From: "Costello, Roger L." <costello@mitre.org>
• To: "xml-dev@lists.xml.org" <xml-dev@lists.xml.org>
• Date: Wed, 25 Jul 2012 13:00:15 +0000

```Hi Folks,

I think mathematics is fascinating.

I learned recently that in mathematics it is not the particular symbols that matter, but rather that the symbols (whatever they may be) exhibit the desired properties such as commutativity and associativity.

For example, the plus sign ( + ) is usually the symbol used to express addition, but other symbols could be used. In fact, in mathematics "terminology and notation are individual and often idiosyncratic." [1]

Despite no standardization of symbols, mathematics is considered the universal language.

Wow!

Why is that? What enables understanding even in the midst of different terminology and notation?

I think it's the properties -- such as commutativity, associativity, and identity -- that enable understanding. If we use different symbols but our properties are the same, then we are probably talking about the same thing.

For example, suppose I tell you that I have an operator @ that has these properties:

a @ b = b @ a                                for any a, b

a @ (b @ c) = (a @ b) @ c            for any a, b, c

a @ 0 = 0 @ a = a                          for any a

then you probably understand that I am talking about the addition operation.

Now let's see how this applies to XML.

Here is the title of a (fabulous) book:

Category Theory for Computing Science

I may mark up that title like so:

<Title>Category Theory for Computing Science</Title>

But clearly with markup "terminology and notation are individual and often idiosyncratic." That is, you may mark up the same data like this:

<Name>Category Theory for Computing Science</Name>

How shall we understand each other, given our different terminology?

Perhaps there is a lesson from mathematics -- use properties for enabling understanding.

But does markup have properties? What properties might there be for the above example?

/Roger

[1] "Category Theory for Computing Science" by Michael Barr and Charles Wells, p. xvi.
```

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