ROADWAYS & HORIZONTALS
THE PEANO POSTULATES
The Laws of Sequence
1. Zero is a number.
2. Every number n has exactly one successor n'.
3. Zero is not the successor of any number.
4. No two numbers have the same successor.
5. If n belonging to a set of numbers called S implies that its successor n'
also belongs to S, and if 0 is also in S, then S contains all the numbers.
Formulaire de Mathématiques (1889)
Any progression may be taken as the basis of pure mathematics: we
give the name 0 to its first term, the name number to the
whole set of its terms, and the name successor to the next in the
progression. The progression need not be composed of numbers: it
may be composed of points in space, or moments of time, or any
other terms of which there is an infinite supply. Each different
progression will give rise to a different interpretation of all
the propositions of traditional pure mathematics; all these
possible interpretations will be equally true.
Introduction to Mathematical Philosophy (1919)
"Well now, would you like to hear of a race-course, that most
people fancy they can get to the end of in two or three steps,
while it really consists of an infinite number of distances, each
one longer than the previous one?"
What the Tortoise Said to Achilles (1895)
He sat leaning on his cane and thinking that with a Knight's move
of this lime tree standing on a sunlit slope one could take that
telegraph pole over there, and simultaneously he tried to
remember what exactly he had just been talking about.
The Defense (1964)
-W. A. MOZART
Symphony 40 in G minor: Movement I (1788)