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Mathematicians since antiquity have been thinking about length and angle, including doing things with straight-edges, rulers, compasses, and protractors. Fast-forward to modern physics, and you'll see everything from the Euclidean distance in Cartesian coordinates to pseudo-metrics on various smooth manifolds.

But I am wondering if there is a sense that mathematical notions of distance are distinct from empirical distance. When I take a ruler and place it between two places I have an empirical experience of length by the way it looks. I don't have to propose anything formal to notice this experience that some things are further apart than others.

Are formal notions our best models of that experience of what we see with a ruler, or can we go further by saying that they are the same thing? My intuition is the former, but when I talk to mathematicians they have sometimes given me an impression of the latter being correct.

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  • I originally asked this on physics.se but By Symmetry told me it would be better suited for this site.
    – Galen
    May 19, 2022 at 18:04
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    I'll note that mathematical distances can describe "empirical" distances other than the usual "as the crow flies" distance. Imagine a spiral path of stepping stones in a lake - although the distance from edge to center is relatively short, the "useful" distance of your walking path is much longer, measured along the spiral manifold. Is the empirical distance here how far away the center looks, or how long it takes you walk there? You don't need mathematical theory to experience either measure, but the theory does provide ways to formalize different ways of experiencing distance.
    – Nuclear Hoagie
    May 19, 2022 at 21:14

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Are formal notions our best models of that experience of what we see with a ruler, or can we go further by saying that they are the same thing?

Many people used to believe that empirical distance corresponded to the mathematical notation of Euclidean distance. Modern physicists believe that distance is more complicated than that; they have separate notions of:

  • proper length
  • observed length
  • spacetime interval (the "distance" through spacetime between two events)

which describe different things. The spacetime interval is the most objective, proper length is what you'd see if you taped a ruler to the object, and observed length is what you'd see in a photo. (Which of these best matches your idea of "empirical distance" is up to you.)

Many physicists believe that we don't have a true understanding of space and distance yet, and that it might work slightly differently in a future, more accurate model of physics. Some philosophers of science believe that no model of physics can ever be true, so there will always be a difference between empirical and mathematical distances (though it may be a very small difference).

Mathematicians have created all sorts of different notions of distance, too. Far, far too many to list here. Of note:

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The mathematical concept of distance is an abstraction of the physical experience of distance. As such, we can use the mathematical concept to discuss 'distance' between things that are non-physical as well as physical: e.g. we can create metrics of distance between (say) political orientations, cultural groups, emotional states, etc. As such the mathematical notion does not depend on physical representations; physical representations are merely the most salient manifestation of the abstract principle.

Put another way, the concept of 'distance' is a mental construct attached to human values. We only think about the 'distance' of something when we want it to be closer or farther away, or when we have some use for the space between us and it. This desire then manifests as a physical, emotional, attitudinal, or conceptual sense of distance. Because it is first and foremost a concept, it is naturally abstract and only becomes concrete when we apply it to a particular case. The ancient Greeks were geometers who worked mainly by drawing lines in sand, but the lines in sand were themselves irrelevant except where they served to point out abstractions that could be applied elsewhere. And the Greeks were not at all averse to applying those abstractions to intangibles, such as music (with scales and chords constructed from geometric principles) or art (where the golden ratio became an important aspect of aesthetics).

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  • This seems to be talking about metaphorical distance rather than physical distance. But maybe your point is that everything is effectively metaphorical, since it's all abstracted by the mind.
    – Barmar
    May 20, 2022 at 13:57
  • @Barmar: More or less, yeah.
    – Ted Wrigley
    May 20, 2022 at 14:13
  • @Barmar It is precisely the argument of cognitive semanticists that conceptual metaphors are the primary means of extending literal language.
    – J D
    May 20, 2022 at 19:10
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    @JD Quite true, this is explored by Hofstadter and Sander in Surfaces and Essenses
    – Barmar
    May 20, 2022 at 19:15
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    @Barmar: It remains the case that, even in an entirely physical setting, there can be more than one reasonable, physical interpretation of "distance," and so the question's notion of "empirical distance" as a single, unambiguous concept, is questionable.
    – Kevin
    May 21, 2022 at 3:48
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Short Answer

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.

Long Answer

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.

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Is there empirical distance without mathematical distance?

Yes. A toddler might have the experience of distance (e.g. Goo, goo... so this is what movement feels like!), without having created the corresponding arithmetical (or mathematical) knowledge (Pee, coo... what is an unbounded surface?).

Also, animals can run fast or slow because they have the sensitive experience of distance: a dog will not run fast towards a close wall, because it risks hurting itself, so it has empirical knowledge of distances. But animals lack of mathematical knowledge.

On his Critique of Pure Reason, Immanuel Kant proposes that geometric objects are produced in the the Transcendental Aesthetics stage of knowledge development. Concretely, space and time do not exist out of our heads, our of our minds, but are metaphysical (not directly dependent of experience) forms of intuition (in the Kantian jargon: intuition == representation). Space is a subjective construct, and geometrical objects are synthetic a priori constructs. Consider that synthetic a priori does not mean completely independent of sense experience, but moreover indirectly based on sense experience.

That is, first, we develop the intuitions of space and time, and later, we create abstract geometric objects, like lines, triangles, distances, etc.

So, yes, empirical distance without mathematical distance is possible. What is impossible is mathematical distance without empirical distance. A person that absolutely lacks of sensitive experience (for whatever reason) cannot know space, and cannot develop geometric concepts, given that sense experience is mandatory for such development.

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  • +1 Kudos on Kantian forms of space and time!
    – J D
    May 19, 2022 at 21:00
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    Nitpick in your conclusion: just because something's how baseline humans develop (i.e., sufficient), doesn't mean it's necessary. A person who absolutely lacks any experience of space or spacial distance could develop the mathematical concept of a metric, since metrics crop up in many places other than geometry. (A person who lacks any experience of space could also develop geometry, though to them it'd be quite an exotic field of mathematics. Real-world mathematicians have done far more impressive things.)
    – wizzwizz4
    May 21, 2022 at 1:48
  • @wizzwizz4 that would be the accepted view in 1700.
    – RodolfoAP
    May 22, 2022 at 2:03
  • I was agreeing with you until the final paragraph, which I consider incorrect. Mathematicians have the notion of a metric space, which is basically a set on which distances are defined, subject to certain restrictions. A metric space may or may not have any relationship whatsoever to the real world. In other words, it's a notion of mathematical distance without any corresponding empirical distance.
    – Dawood ibn Kareem
    May 22, 2022 at 9:01
  • @DawoodIbnKareem: notion of "metric space" without notion of space... exactly like the notion of blue whale without notion of whale. Same philosophical view of 1700s rationalism.
    – RodolfoAP
    May 22, 2022 at 9:10
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Is there "empirical" distance without "mathematical" distance ?

Empirical distance lives in the physical world - it is the observed outcome of one or more physical measurements. Mathematical distance lives in whatever realm of abstraction is occupied by mathematical objects - but certainly not in the physical world. You can definitely have one without the other.

The question is like asking "Can I count without numbers ?" - to which the answer is "Yes - use a tally stick".

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  • +1 In some cultures, counting doesn't even use a representation like a tally. It consists of pairing until there is excess if any. Counting is not metaphysically necessary for the extension of space. :D
    – J D
    May 20, 2022 at 19:09
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Is there "empirical" distance without "mathematical" distance?

Absolutely. I would argue that "empirical" distance has been the primary metric for nearly all of the millions of years of human evolution and "mathematical" distance is the new kid on the block. Ask a friend to "think fast" and throw your keys. The moment they are caught, ask your friend the following:

  1. How far away was I when I threw the keys
  2. How fast were the keys moving when they were caught
  3. How far was your hand from from the center of your eyes when the keys were caught.
  4. Etc...

Unless your friend is a savant, no exact numerical answers regarding, position, distance, velocity, or time will be given and the keys will have been caught without the benefit of any math or "mathematical" distances.

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