Can residents of Flatland really see the edges of polygons?

In the philosophico-mathematical novella by Edwin A. Abbot, titled “Flatland,” Abbot imagines what life would be like in a two-dimensional world, filled with intelligent beings of the same two dimensions.

It is said that Flatlanders would only see lines, or edges, of polygons. They would not have the privileged “God’s-eye” view that we might imagine of their two-dimensions, from our three-dimensional frame of reference.

But we can’t imagine what a Flatlander would see as an edge without imposing our own concept of width onto it. Even if we imagine an extremely narrow line, that line still has to have some width in our imagination to be conceivable.

We might always imagine, in our mind’s eye, the line in question becoming narrower and narrower, and yet, however so narrow we make it, the concept of line cannot be thought without imposing an infinitesimal amount of width onto its form in our mind.

The closest we could get to imagining an *actual* Euclidean line in 2D space would be to imagine the line with a width such that in our imagination, the width is always approaching zero.

But something which approaches zero is not yet a zero, and we already know that anything with a `value > 0` *is not* actually equal to zero, so that however narrow we might imagine this line to be, it cannot be stolen away from some amount of borrowed conceptual width. (A borrowed concept, that is, from 3D space, of “width.”)

Therefore, lines in Euclidean absolute-space, (though they are pragmatically applicable in our world as useful approximations to reality, the precision of which can be indefinitely extended to suit a multitude of occupations) do not, in fact, exist outside of our thought-structures, such structures themselves only being able to give a rough approximation of this impossible ideal.

Math’s most basic postulates are imagined, but we are not taught to imagine the postulates, we are taught to hear and obey and scratch symbols on paper, and that these operations are actually happening somewhere in some other world, or indeed in our own world. Then we are told we should accept these things on faith, not by great prophets, but by average people just like ourselves, who apparently “get” Math, and can accept it without questioning its most basic assumptions.


Is it any wonder so many of us are mortally afraid of Math?


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