Wednesday, February 01, 2006

Limits vs. Limit Cases

I just finished reading Barry Miller's The Fullness of Being, a very stimulating book. Miller argues against Russell, Frege, Quine, et al. that existence can be predicated of individuals and develops an interesting metaphysical account of the relation between an individual and its existence.

Anyway, at one point Miller argues for a sharp distinction between limits and limit cases. Both are endpoints of a series, but the difference is that limits are themselves part of the series, whereas limit cases stand outside the series.

For example, according to physics, the maximum speed is the speed of light, 300,000 km/s. That is a limit. We have a series of speeds in which the uppermost speed, the speed of light, is itself part of the series. On the other end of the spectrum, we have 0 m/s. Now we might think that this is the minimum speed, but we would be wrong. 0 mph is not a speed at all, but the absence of speed. It's a limit case, not a limit. (Miller, p. 137)

Consider again a series of regular polygons with an increasing number of sides: triangle (3), square (4), pentagon (5), hexagon (6), ... , dodecagon (12), ..., chiliagon (1000), ..., etc. The triangle is the lower limit of the series, because there can be no polygons with fewer sides. And, of course, the triangle is itself part of the series. On the upper end, however, there is no limit, but there is a limit case: the circle. Circles are not polygons, and so they lie outside the series. (Miller, pp. 138-139)

One last example. Consider a series of lines of decreasing length: 1 in, 0.5 in, 0.1 in, ..., etc. There is no lower limit here because for any line we can always cut it in two to make a shorter line. But there is a limit case: a point, which is not a line at all. (Miller, p. 139)

Mathematicians, of course, like to use the word "limit" to cover both limits and limit cases, but I think Miller's made an interesting and potentially useful distinction nonetheless.


At 2/14/2006 3:15 PM, Blogger Jeremy Pierce said...

The fact that they do distinguish between open and closed intervals shows that mathematicians sometimes do care about differences like this, so it's a little surprising that they don't distinguish these. Surely there must be some mathematical contexts in which this distinction makes all the difference.

At 2/14/2006 4:22 PM, Blogger Alan Rhoda said...

Hi Jeremy,

Good point. Mathematicians do distinguish between one-sided and two-sided limits. Perhaps that's enough to distinguish between open and closed intervals. If memory serves me correctly, a function is continuous at a point iff it has the same limit from both sides at that point. Anyway, the boundary of an open interval can be defined by a one-sided limit. As for the boundary of a closed interval, I imagine that could be defined as the complement of a one-sided limit from the other direction, since every closed boundary stands opposite an open boundary.

But, hey, what do I know. I'm only an amateur mathematician!


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