Mathematics proposes numbers to measure real things. There are notches corresponding to numbers on the measuring tape, but even if the notches succeed in referring to that real position. (although they remain a sign of the real object), gaps are still on the measuring tape with no notch and no number to describe the intermediate positions.The real number system attempts to fill the gaps that most numbers leave when describing something real, removing the need for metaphor. “Metaphorical language is language proper to the extent that it is related to the need for making up for gaps of language”(Giuliani, 1972, p. 131). The system “covers the gaps” and does the job of describing physical reality (and more) without metaphor. But how do real numbers go about covering the gaps?

The work of covering the gaps and freeing real numbers from metaphor is done with The Axiom of Completeness:

A bounded increasing sequence has a least upper bound (that is a real number)

Why would the axiom of completeness cover all the gaps of a real line?

A good example is in the act of measuring a plank with a straight-looking side. One compares the plank with a measuring tape and measures the whole meters, but there is still some plank left to measure. (The number of whole meters is the first number (position)in the sequence.) So one counts the number of decimeters left (the resulting position is the second number in the sequence), but there still remains more plank after the largest marker for decimeters. The process continues until the precision of the measuring tape is exhausted, eyesight fails, or the measurer loses interest. Even though one must fail in measuring the exact length of the plank, the axiom of completeness provides assurances that there exists a real number for the “actual” length of the plank (and that there is an “actual” length of the plank). But the process cannot take the full measure of the plank, and so we remain in the poetic world of metaphor, “a process, not a definitive act; it is an inquiry, a thinking on” (Hejinian, 2000).

We want to talk about something real, something as simple and straightforward as the length of a plank. We have an apparatus of controlled inquiry, tools and will-more than the casual use of words, but we still fail.

We must admit that the measurements (words) we have used remain metaphorical and the actual measure of the plank (object) ultimately falls into the gaps of language. The words (measurements) we started with in our task of measuring the plank are no less metaphorical than the measurement we have when we stop. How can we wake up from metaphor?

(PDF) Many Roads from the Axiom of Completeness. Available from: https://www.researchgate.net/publication/327227248_Many_Roads_from_the_Axiom_of_Completeness [accessed Sep 28 2024].