Category Archives: The more technical stuff

The symmetry between Completeness and Vagueness

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The (Problem of the) Heap of Sand is the classical representation of vagueness, where you create an inference: if you take away a grain from the heap then it remains a heap. This is a good inference for a long time, but eventually breaks down due to the vagueness of what a heap is (eventually you will have taken so many grains away that it wont resemble a heap anymore). And you can’t even tell when the inference will break down, exactly. So, as you can see, vagueness has a very precise and non-vague formulation, something similar to this strange property of vagueness (that it is a clear and distinct concept, even though it is a concept about indistinctness) is common for mathematical concepts. That is, mathematical concepts are chosen and defined so that they contain the terms necessary to get round their own difficulties—their faults or cracks, as all concepts are imperfect. Example: “Completeness”.

The difference between vagueness and completeness, is:

1) that the term completeness is in direct opposition to the claim that every concept has a “fault” or “crack”, in this it is a “perfect” term. The definition of completeness, however, must be vague, because the terms used to define completeness all have cracks.

2) The definition of vagueness, by contrast, is perfect, but the term refers to an essential fault of all terms, and is always perceived. (perception is always vague) Vagueness is defined as how the if, then fails— The “heart of logic” (the paragon of precision) and the most reliable way to preserve truth. It is clear and distinct in its description of how the if, then fails. Since each of the terms in the definition of vagueness have cracks, the definition is precise.

The term vagueness refers perfectly to its referent because it is well defined as an indistinctness, the definition of completeness is vague because the term refers to something that can’t be perceived, a perfect whole.

Stuff, and Higher-Order Vagueness

30 Friday Jun 2017

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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.