Field Notes: Observations on Counting Physical Objects
Researcher: E. Peano, Fruit Counting Laboratory

Observation 1: Sequential Counting

When counting apples in a basket, we point to each apple and assign it the
next number. The first apple makes one, the second makes two, and so on.
Every count is reached by adding one to the previous count. An empty basket
contains zero apples.

Observation 2: Combining Collections

When we pour one basket into another, the resulting count equals the sum of
both counts. Combining 3 apples with 2 apples always gives 5 apples.
We repeated this trial forty times with identical results.

Observation 3: Order Independence

The order of combining does not matter — putting basket A into basket B gives
the same count as putting basket B into basket A. We confirmed this by
swapping the pour direction across all forty trials.

Observation 4: Adding Nothing

Adding nothing to a basket does not change the count. A basket of 7 apples
still has 7 apples after we pretend to pour an empty basket into it.

Observation 5: Comparing Collections

The basket with more apples is always heavier. If basket A has 5 and basket B
has 3, then basket A has more. The difference is found by subtracting the
smaller count from the larger.

Observation 6: Swap Invariance

Swapping all apples between two equal baskets does not change either count.
If both baskets have 4 apples, after swapping they still each have 4.

Observation 7: Repeated Addition

Having 3 baskets of 4 apples each is the same as adding 4 three times.
Multiplication is a shortcut for counting equal groups.
