Is there "empirical" distance without "mathematical" distance?
Broadly construed, yes. Distance as related by the senses, that which is perceptual, is different than distance understood by logical and mathematical terms, that which is conceptual. The act of measurement pairs the two sorts of distances together. Doing so has been quite a challenge, with intuitive notions of physical continuity of a curve driving development of such rigorous mathematical theories as real analysis.
An interesting fact about the Ancient Greeks and their mathematics is that they dealt in lengths, but didn't measure in the same way you and I might, with a ruler. Greek geometry, which didn't have the same sort of notation that we have today, was done with straight edges and compasses, as opposed to rulers and protractors. Mathematics actually went a revolution of sorts when Rene Descartes introduced the notion of analytic geometry, which is essentially the imposition of coordinates on Euclidean space. This underlines a basic difference between distance as understood from a phenomenological perspective, and one based more on analytic foundations. In fact, one is tempted to see geometry as a theory that deals with perceptual distance, and arithmetic as one that deals with a conceptual one.
Almost everyone is familiar with the experience of looking out over a distance and trying to gauge relative lengths. Who is closer? About how far away? We have an set of intuitive notions about distance that come to us from our sensory apparatus. In a naturalized epistemology, we can invoke the visual cortex, for instance, as a means by which our embodied intelligence creates visual, stereoscopic representations (SEP) of the world. In fact, some philosophers of artificial intelligence have proposed a term for this basic intelligence, called naive physics. The cognitive scientists George Lakoff and Raphael Núñez propose that one of the four basic primitive neural computations our mind can draw from is a Metaphor of Motion, which might be understood as a motion of an object along a smooth path. And this makes sense from a perspective of evolutionary psychology. Predators and prey need to be able to traverse space and time; measuring lengths isn't a survival skill on the plains of the Serengeti!
But mathematicians take distance to a more precise place, and have a mathematical construct called the metric space which requires:
- the distance from A to B B is zero if and only if A and B B are the same point,
- the distance between two distinct points is positive,
- the distance from A to B is the same as the distance from B to A, and
- the distance from A to B is less than or equal to the distance from A to B via any third point C.
From Mendelson's Introduction to Topology:
A metric space is a set of points and a prescribed quantitative measure of the degree of closeness of pairs of points.
And, that's just the beginning. According to the article 'Measurement and Measurement Theory' in the Encyclopedia of Philosophy:
Metrology in general and measurement theory in particular, have grown from various roots in fields of great diversity in the natural and social sciences, engineering, commerce, and medicine. Informally, and its widest empirical sense, a measurement of a property, exhibited by stereotype objects in variable degrees or amounts, is an objective process of assigning numbers to the objects in such a way that the order-structure of the numbers faithfully, reflects that of degrees or amounts of the measured property... Abstractly, a particular way of assigning numbers as measures of extents of a property in objects is called a quantity scale.
So, in a way, distance-as-space is empirical in the philosophical sense: derived from the bodily senses. Whereas distance-as-measure is more rooted in the rational practice of counting and subsequent arithmetical theory. Ultimately, metrology and measurement theory, which seeks to marry the two types of distances, impacts everything from statistical analyses of materials to the manufacture of fasteners which in high-tech devices demands a rigorous accounting of engineering tolerances.