Some families have secrets: stories that aren't told, events never brought up once they have passed, or members whose names are not to be uttered. My family has its share, I suppose, but we also have something else: not just *family secrets*, but *secret family*.

Don't misunderstand me. We don't keep dotty aunts locked in hidden towers, or uncooperative children chained to basement walls. At least, not that I've been told. No, my family's secret seems to be names.

My mother has three brothers (Ralph, Jerome, and Mac) and two sisters (Pat and Mary Helen). That's how I've known them all my life. They're my aunts and uncles, after all.

So imagine my surprise when I found out that two of those names aren't their given names. Aunt Pat's real name is Marian Patricia. She was called Pat rather than Marian because Marian is too similar to Mary Helen. It was a surprise to find this out when I was in my twenties.

Not too long after that, I discovered that my Uncle Mac's name isn't Mac. It's George. I have no idea why he uses Mac. (My mother told me, soon after writing this, that Mac is a shortened version of his middle name.)

All this has got me wondering: what else about don't I know about my mother's siblings. It's as if I had this secret family the whole time I was growing up but never knew about. What other mysteries lie waiting to be revealed?

There's a metamathematical theorem that provides a good metaphor for this situation. The Löwenheim-Skolem theorem states that there are an infinite number of interpretations of a given mathematical system. For example, you might think you've created a system that models, say, simple arithmetic. The theorem says that there are an infinite number of systems of which your simple arithmetic is a part. There are more potential meanings than you can ever know.

That's how I feel about my family sometimes. It seems to be a simple system, with just a few axioms and postulates. Yet for years I believed one simple theorem (Aunt Pat's first name is Patricia), and the family mathematics worked just fine. Now I know that Aunt Pat's first name *isn't* Patricia, and the mathematics of this family still works. Two interpretations of the same system, both useful. How many other interpretations are there. What other secret family awaits discovery?

Information about the Löwenheim-Skolem theorem comes from Morris Kline's *Mathematics: The Loss of Certainty*.

Last updated 3 June 2000

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