Posts Tagged ‘Quotes’

“Among all positive integers, the integer 1 is the largest.”

Here’s a funny anecdote about the subtleties of logic. The isoperimetric theorem is the statement that, among all planar figures of a fixed perimeter, the circle (of that circumference) encloses the greatest area.

Steiner gave five proofs of the isoperimetric theorem. Lovely as they are, he left one point open to attack: all proofs assume the existence of a solution (his strategy is always to take a figure that is not a circle and show that its area can be improved). This did not go unpunished. The analyst vultures can smell an existence assumption from miles away. […] Perron at least jokes about it:

Theorem. Among all curves of a given length, the circle encloses the greatest area.
Proof. For any curve that is not a circle, there is a method (given by Steiner) by which one finds a curve that encloses greater area. Therefore the circle has the greatest area.

Theorem. Among all positive integers, the integer 1 is the largest.
Proof. For any integer that is not 1, there is a method (to take the square) by which one finds a larger positive integer. Therefore 1 is the largest integer.

— Viktor BlåsjöThe evolution of the isoperimetric problem (2005), quoting Oskar PerronZur Existenzfrage eines Maximums oder Minimums (1913)

“The abstract structure that all progressions have in common”

Hello! I’ve been terribly caught up with research lately, but the next several posts on categorical logic are in the pipeline, and will surface sooner or later. Until then, here’s a nice quote about representation independence in mathematics.

Therefore, numbers are not objects at all, because in giving the properties (that is, necessary and sufficient) of numbers you merely characterize an abstract structure—and the distinction lies in the fact that the “elements” of the structure have no properties other than those relating them to other “elements” of the same structure. […] To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by 4, 5, and so forth. And to be the number 4 is no more and no less than to be preceded by 3, 2, 1, and possibly 0, and to be followed by…Any object can play the role of 3; that is, any object can be the third element in some progression. […]

Arithmetic is therefore the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions. It is not a science concerned with particular objects—the numbers. The search for which independently identifiable particular objects the numbers really are (sets? Julius Caesars?) is a misguided one.

— Paul BenacerrafWhat numbers could not be (1965)