Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

58 marcus giaquintoof them. In addition to a geometrical concept of extensionless points we have aperceptual concept of points as tiny dots. Perceptual points do have extension;some parts of a perceptual point on the line would be nearer the beginningof the line than others, so a perceptual point does not lie at exactly onedistance from the beginning. For this reason no line of juxtaposed perceptualpoints could have the structure of [0, 1] under the ‘less than’ ordering. So wemust make do with geometrical points. But thinking of a line as composedof geometrical points leads to numerous paradoxes. For instance, the parts ofa line either side of a given point would have to be both separated (as thegiven point lies between them) and touching (as there is no distance betweenthe two parts, the given point being extensionless). A related puzzle is thata line segment must have positive extension; but as all its components havezero extension, the line segment must also have zero extension. Yet anotherpuzzle: the unit interval has symmetric left and right halves of equal length;but symmetry is contradicted by the fact that just one of the two parts hastwo endpoints—either the left part has a right endpoint or the right part hasa left endpoint, but not both, otherwise there would be two points with noneintervening. These puzzles show that the visuo-spatial idea of a continuousline cannot be coherently combined with the analytical idea of a line as ageometrical-point set of a certain kind.¹⁴Setting aside these puzzles, thinking of an interval of real numbers as a setof points composing a line segment cannot reveal a crucial structural feature,Dedekind-continuity (completeness): for every set of real numbers, if it isbounded above it has a least upper bound, and if it is bounded below it hasa greatest lower bound. So this visual way of thinking of an interval of realnumbers is unrevealing about that feature which distinguishes it structurallyfrom an interval of rational numbers. Hence we cannot know the structure Uby thinking of it visually in terms of the points in a line segment.Perhaps there are other visual ways of thinking of the closed unit interval.But none that I can think of fares any better. One possibility is the set of allbranches of the infinite binary tree under the following relation of precedence:x precedes y if and only if, when the branches x and y first diverge, x goes to theleft and y goes to the right.We can identify branches with infinite sequences of 0s and 1s. The leftsuccessor of each node is assigned 0, the right successor is assigned 1; nothingis assigned to the initial node. The branch up to a given node is identified¹⁴ This is not to deny that we can usefully flip back and forth between the two conceptions inpractice.