Arithmetic and the Real World - An Imperfect Fit
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Consider this question: How much is one apple plus one apple?

A simple pre-school question that prepares the student for both the world of arithmetic (and then mathematics) and science (which draws upon the worlds of arithmetic and mathematics in general).

The answer to this question is considered fundamental: two apples?

But is this true?

Only in the domain of mathematics wherein apple is treated as though it was a variable, an x, that itself is unambiguous. That is, an apple is an apple is an apple.

But in the domain of reality, an apple is never another apple. In fact, an apple doesn't even stay an apple. By that, I mean that even as time passes, the apple that existed at any given instance is not the same apple exactly that exists at any other instance.

This might seem as though I am being pedantic or overly precise. I am being neither.

Ask any cook if one apple plus any other apple is the same as two apples.

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The problem is right at the beginning.

What makes mathematics so unassailable (to the limited degree that it is so) are the limited number of postulates upon which it is founded.

And among these postulates, you will find a fairly limited number of definitions that use the phrase "well-defined" in combination with some word or phrase that means "self-evident".

These last two phrases are essential for any discourse because it allows discourse to begin upward (that is, creating constructs out of fundamental words and phrases) rather than remaining paralyzed in the downward spiral of definitions and theorems.

For nonmathematicians, what I am talking about is this: all linguists would like the ultimate dictionary, one in which there are fundamental words that need no definition because they are self-defined, and which can be used to define all other words. But no such words exist. Not a single one. Discourse between two people rely on what mathematicians would call "postulates': that is, phrases such as "you know what I mean."

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(c) Michael J. Carroll [March, 2018]