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

Briefly on sheep

In "New Math" during the 60's and 70's, we learned that we should talk about sets, for instance the set consisting of Ulla, Henriette and Vignette (three random sheep that I came to think of). The farmer cares for his sheep, and does not think of them as "3 sheep", but - exactly - as "Ulla, Henriette and Vignette".

But, what happens with wonderful system (the integers) we've developed to do trade?

Is it so that this care for the individual animal is incompatible with trade?

Well, we can calculate with finite sets:

the union
of two sets acts like sum: The union of the sets
{Ulla, Henriette, Vignette} and {Steinfrid, Makronelle}
is the set
{Ulla, Henriette, Vignette, Steinfrid, Makronelle},
just as
3+2 = 5.

We can also multiply by using:

the (cartesian) product.
The product of the set
{Ulla, Henriette, Vignette} with {Steinfrid, Makronelle}
is the set of all pairs you can make, one sheep from each set:

 (Ulla, Steinfrid) (Henriette, Steinfrid) (Vignette, Steinfrid) (Ulla, Makronelle) (Henriette, Makronelle) (Vignette, Makronelle)
Just like 3 times 2 is 6.