The sphere spectrum
BID, Lisboa 2017
1. Extensions of number systems
2. Negative numbers
3. Briefly on sheep
4. A trade deficit
5. Out in space
6. Ulla and Henriette live in RP∞
7. Negative sets
8. The sphere spectrum
9. Brave New World
The "natural" numbers N:
Good for counting things (e.g. sheep), and so been around throughout history.
It is true that it took a very long time before humans invented a symbol for zero. On the other hand, "no sheep" probably was a distressingly common concept for early man, and since it is synonymous with "zero sheep" I include 0 among the godgiven numbers.
At some point in history, we found it necessary to do basic accounting, not only of the sheep you might have, but also of those you owed the local strongman. Negative numbers turned out to be a splendid idea: