LINGUIST List 5.119

Thu 03 Feb 1994

Disc: The Last Phonological Rule

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1. John Goldsmith, Re: 5.118 The Last Phonological Rule: reply to Goldsmith
2. Bill Poser, infinite dimensions

Message 1: Re: 5.118 The Last Phonological Rule: reply to Goldsmith

The only matter that I'd like to reply to in Alex Monaghan's comments
on The Last Phonological Rule is the following one; if it weren't
of real importance to linguists -- at least to some linguists -- I'd
simply send him my remarks directly. But posting them here might
serve some good end.

The remark concerns what a finite-dimensional (vector) space is.

I'm going to offer a few remarks which are meant to explain; they're
pedagogical in character, and are not intended to be interpreted
literally, especially by people who already know what we're talking
about. To say that we're looking at a point in an n-dimensional space
means that we've specified n separate numbers (its coordinates).

Even if there were only one dimension to it, we'd be talking about
an infinite number of points (there are, after all, an infinite
number of points between 0 and 1, no matter how you slice it).

But this business of infinity is not as scary (or even as

overpopulated) as it might sound at first; a system (whether it's
mathematical or linguistic) which is continuous almost everywhere
will have the property that large chunks of regions in the space
lead to properties that are more or less identical throughout that
region.

This isn't mysterious in the slightest, though my formulation

might make it sound so. But think of the world that we navigate
in everyday: it's a 3-dimensional world which, if we forget about
quantum theory, contains an infinite number of points in it, even
right here in my office. But natural phenomena are continuous almost
everywhere, and so we recognize chunks and regions within this
space (like my body, the keyboard, etc.) which are recognizeable
subregions with specifiable properties.

Well! All that is to say that what's important in determining the
size of a space is its dimensionality, once we know that we're
dealing with a system that is continuous almost everywhere (I've
used that phrase three times now, and the reader will no doubt
have inferred, correctly, that it is a phrase with a technical
meaning). The fact that there are an infinite number of points
in an internal (like from 0 to 1) doesn't lead to any theoretical
problems.

In LINGUIST 5.118 Alex Monaghan writes:
>One further brief point. in his reply, Goldsmith mentions "a
>finite-dimensional space, the space of connection weights", but as far
>as I can tell this space is actually infinite: there is an infinite
>range of possible values for each connection weight...

There being infinitely many weights makes the system infinite in one
sense but it does not make it infinite-dimensional; it doesn't affect
dimensionality at all. For example, space-time is only four
dimensional even though all four dimensions are continuous. An
infinite-dimensional system is one described by infinitely many
parameters. The Fourier series representation of a signal, for
example, represents it in an infinite-dimensional vector space since
there are infinitely many frequencies at which there may be energy,
each such frequency consituting a dimension. A perhaps more familiar
and elementary example of an infinite-dimensional vector space is the
Taylor series representation of the functions of class c-infinity,
where each power of X defines a dimension.

Bill Poser