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