At the heart of real analysis and the study of real numbers is a confusion between points and “stuff,” or how points can perfectly describe distance/space/extension. Put the other way around, how can extension be reduced to points? Or how can we know the extension (with points) of a length/width of an object?

This question is caught up in measurement, and the relationship between word and object. I have seen people give (real) number a privileged position between word and object, but I would argue that number suffers from the same flaws that language suffers from, including vagueness. One reason is there is vagueness in what to call “one” (see the classical “heap of sand” in a previous post), further there is vagueness in when something is countable or uncountable. Electron cloud size to water or human height is uncountable, but what about popcorn or puffed rice? These could be counted, but should we be counting the grains? Should we count water molecules? Should we go back to a particle model for electrons so we can count them? This should comes more pointed when we consider the ancient belief that counting/taking measurements about humans directly endangers them (Feyerabend 1999). Why was it believed that being unsure which side of the microscope you are on endangers?

Vagueness runs right through these issues from the finer points of graduate school mathematics to the ethical issue raised above. And of course it does, since we would like number to draw a clear line between word and object, point and “stuff”. Strangely, insight into these distinctions can be garnered by understanding just how troubling (and what) vagueness is.

Enter higher-order vagueness. Now the question has been put to me “Well the vagueness between, say, “high up” and “not high up”, (this can easily be pictured on a cartesian graph with a line gradually going down from left to right) can be dealt with by adding and third “uncertain” value/region. And the trouble here is, as is well known, that adding a region uncertain only adds two new borderline cases between “high up” and “uncertain”, and another borderline case between “uncertain” and “not-high up”, so that new “higher order” uncertainty regions have to be added for these borderline cases. And now we have introduced new sources of vagueness, etc. Ultimately the pursuit of conquering higher-order vagueness by exchanging borderlines(points) with intervals(stuff/extension) is a vagueness between points and intervals.

As that sinks in, realize that a vagueness between points and intervals is a general problem reproducible anytime vagueness rears its ugly head. If we draw the connections from points to words and from intervals to the “stuff” of objects, we find that the line between words and objects is vague, which is also well known. What is new here is that we found this well known vagueness by investigating vagueness itself in a general way.

To summarize I would say that the vagueness between word and object is an essential or “stock” vagueness that crops up anytime we are in vague territory, and is the heart of analysis of “real” numbers. In this sense, vagueness is an important ultimate concept for mathematics, and it ought to be mentioned in analysis text books that vagueness is what the book is about.