One dimensionless number is more familiar from everyday life than any others except the integers. That number is (named ``pi''). The integers arise in the most elementary of ways, because they describe things that can be clearly separately identified and counted, like apples in a barrell or fingers on a hand. requires slightly more sophistication, because it refers to continuous things, and so is most easily represented as a ratio of dimensionful numbers, like lengths.

For instance, on a flat table we define a circle as the pencil mark made when a pencil is tied to the end of a stretched string, then moved while the other end of the string is pinned to a fixed place, keeping the string stretched. In that way, the distance from the pinning-point, the center, to any part of the pencil mark, on the circle itself is the same. But the circle itself, like a line that could be drawn from the center to any point on it, has a length, and that length can be measured with a piece of string just like the distance from the center to the edge was ``measured'' with the piece of string used to draw the circle. We find that the ratio of those two lengths, no matter how big the actual circle, is always the same. In particular

where C is the length around the circle and r the distance from
the center. It is still simplest to *define* a circle by
its qualitative trait of having all the points on the circle at
a single distance from the center, because that makes no
particular reference to how big the circle is, which we
recognize as being an additional detail that is peripheral to
what we recognize as the essence of circles. But this
dimensionless ratio, C/r = 2 , at least on flat tables, is
just as characteristic of circles and as independent of their
distances.

But describes more than circles. If, in flat empty space (an idea we will define later; for now, the naive notion of ``normal'' empty space represents what we intend), we do the same kind of thing to draw a sphere, that sphere has an area that can be related to little square tiles that we can lay side by side to measure it, and is related to the radius by

Or, on the table, we could have measured the area of the disk enclosed by the circle, in which case we would have found

or the volume of the ball enclosed by the sphere, to find

What we measure, even in its dimensional content, changes from case to case, as do the combinations of integers, (2,4,1,4/3, etc) that we find, but the number is common to all of them.

Because Euclid's work on the basic ideas underlying geometry was
done so long ago and has been used for so much since then, there
is a temptation to think of as somehow a number proceeding
from ``pure reason'' (to use a senseless expression from Kant),
or some kind of *a priori* logic. But this view is almost
certainly not a sensible one.

Consider: Even the notion of a length is not something that
proceeds from anything internal to the logical landscape of a
person's mind. It comes about as a discovery of a certain kind
of repeatable interchangeability, though admittedly one first
recognized a long time ago. And yet in a sense, the entire
observability of comes about because two lengths
consistently have a certain ratio, such as the lengths of
circumferences and radii of circles on a flat table, and so *we see it at all through the vehicle of another discovered
quantity*, the ``abstractability'' of length. Now it is
possible to go further and create mind-models that well mimic
all of the properties of flat tables, and especially those
characteristics like dimensionless ratios, that even in the
measured world have no reference to a particular scale. These
abstractions can tend to lose all suggestion of having been
discovered, and come to look like ``primary elements'' of the
mind model, along with ideas like the abstractability of the
notion of a point as the idealization of a precisely
recognizable place or of a line as a very fine string stretched
between two such places.

But that is not how appears in measurements of the world.
Our qualitative definition of a circle, as something that can be
drawn with a pencil and a stretched string, works perfectly well
on a globe, or even on a potato if we have a way to keep the
string bound against the surface as it is stretched. Yet the
ratios of circumference to radius (the length of the string that
was used to draw the circle) will not be in general such
instances, or even anything very close to that. Thus the
scale-independence of this ratio for circles on the table tells
us not only something about circles, but also something about
the table, namely that it is flat, and so very different from
the surface of a globe or of a potato. These *assumed
properties of the table* are incorporated into the basic objects
in Euclid's geometry, and carried along with them then is pi.

The previous examples will be of more concern to us later when
we discuss how to extend Euclid's geometry beyond the case of
flat tables, as we will need to do to understand Einstein's
gravity. But it could be objected that here they are not
especially persuasive of anything, because even from a potato or
a globe we can recover by looking at the *limiting
behavior* of the ratio as we draw very small circles, say around
the north pole of the globe. In that case, the whole surface on
which we are drawing the circles becomes indistinguishable from
the flat top of a table, and again becomes the important
and fundamental number. Thus we could claim that the fact that
does not describe *all* C/r ratios on curved surfaces
does not even strongly encourage us to abandon the notion of
as an object of ``pure reason'', because it is still there
as the limiting behavior of small circles.

But perhaps in this objection we see at work a limitation of the
human imagination. From the earliest experience with tables, we
might have asserted that is an artifact of pure reason, as
evidenced by the fact that it applies to all circles,
independently of their size, to their areas and circumferences
in ways we can predict, as well as to spheres, etc. But
realizing that we can still *define* circles perfectly well
on globes, but that does not then play the same role *independently of size of the circles* we have to relinquish the
claim of size-independence as evidence of anything. So we can
no longer claim that is independent of measurements
because it is obtained in the same way from the same
measurements *at all sizes*, though a weaker version of the
``pure reason'' claim is that exists independently of
measurements because it can be obtained as a limit from small
measurements, which is independent of any of them in particular.

But a more natural way to look at the claim of ``pure reason''
is simply as a wrong claim that we are permitted to get away
with as long as we fail to notice the correct counterexamples.
Thus, the case of circles on a table is so very structured that
it allows us to assume all sorts of properties of that are
in fact partly properties of the flatness of the table. We
deviate from the case of the table by only a little, to the case
of the potato, and are forced to give up the scale-invariance
property as evidence of an *a priori* nature of pi. It
would be more than slightly naive to expect that, even if the
``pure reason'' claim is totally wrong, our first counterexample
would be enough to show is that every aspect of it is wrong.
After all, we could only have fallen into the ``pure reason''
claim in the first place because our first flat-table example
allowed us to get away with it. So it is hardly surprising that
our next attempt, which differs only slightly from the case of
the table, will still allow many of the same aspects of the
claim to be made.

A better thing to do is to look at more counterexamples and see
how they differ, which properties of they force us to give
up, and thus what is telling us *in the measureable
world* about the cases where it applies. Viewed in this light,
the ``pure reason'' arguments become visible as claims that, to
the human mind, it is natural for something to seem
``inevitable'' as long as the mind has not imagined a variation
which can exist and still make sense without it. Thus the
``pure reason'' argument is a comment on the power of
imagination of its user.

From a great many cautions of this type, mathematicians have
come to develop very powerful imaginations. And in the process
they have come to invent many useful counterexamples that serve
to show what is different between one ``way the world could
conceivably be'' and another. The first case in point was
precisely this extension of the assumptions of geometry away
from the flat tables described by Euclid to curved but still
smooth surfaces, like potatoes, which were first successfully
described by the mathematics of B. Riemann and E. Cartan. This
showed that the scale-invariance of ratios like those for the
circle are the criterion by which one can distinguish what is
**flat**,
like tables, from what is non-flat, like potatoes,
without needing anything more than pencil lines and stretched
strings that can be entirely kept on the surfaces. This
mathematics turns out to be precisely what is needed to
understand gravity, which turns out to be a kind of curvature of
the real world of empty space through all of its history, though
we have a hard time ``standing outside that history'' in order
to describe it.

So the criterion of flatness, identified by a kind of scale
invariance, is thus seen to be a property of *large-scale
relations of smooth objects*. But now we realize that we have
required another assumption even to claim that can be
recovered as a limit of measurements of small circles. It is
the requirement that the circles be drawn on something that is
*smooth*. This holds true for the surfaces of potatoes, but
not around the points of cones. So we see that it makes a lot
of sense to regard *as a measured quantity*, and the
fact that it can be consistently obtained from measurements as
evidence that *on the scales of the measurement*,
independently of whether it is flat, at least the space in which
the measurement is made *seems to be smooth*. In other
words, this criterion of smoothness is a way to represent that
the *smaller-scale* geometry of the space is still a lot
like *any* of the geometry of planes (remember that the
geometry of planes has no scale, so there is no distinction of
small from large), even if on larger scales the geometry of
space differs from that of planes.

The point of this whole development, which may seem somewhat obscure, is that, having developed more experience and better imaginations, people can now consider even things that are not smooth, and for which is not the relevant dimensionless ratio of a particular variety of measurements, like those of circles, whose difinition does not intrinsically require it. Thus it becomes clear that our observation of this dimensionless ratio is telling us about the repeatable property of the fairly small-scale behavior of the geometry of empty space, which we then name ``smooth'' and encode in the axioms called Euclidean geometry. It took a lot of work to develop the mathematics to describe counterexamples that are not ``smooth'' in this sense (referring here, of course, to fractals), and perhaps a certain amount of intellectual maturity to recognize that even in those cases where we cannot yet generate a counterexample, that does not constitute any real evidence that a dimensionless number like fails to tell us something important about the physical world from which we learned of its existence.

It is probably fair to represent the attitude ``characteristic'' of modern physics as one through which people take dimensionless ratios, like and also like the ratios in the hierarchy problem, as presumably indicating something about the structure of the physical world, not at all necessarily like anything we currently understand, and look for a set of organizing principles, like those of the ``smoothness'' (local equivalence to Euclidean geometry) of spacetime which make them intelligible. That is what is taken to be the essence of ``explaining'' or ``accounting for'' something.

The rôle of geometry in producing dimensionless ratios is such a seductive paradigm, that it has been an ongoing and recurrent practice for physicists to try to find an explanation of the dimesionless ratios of the standard model (and even to predict those beyond it) in some kind of geometric language. But these, more than everything else mentioned so far, lie at the frontier between understanding and ignorance, and have a context only after much more has been said than can be done here. But, it is good to know where people are trying to go

Thu Aug 31 12:01:42 CDT 1995