Do mathematicians use logic when adding things up?

Question

I asked a similar question recently, and it was closed. So, bear with me.

When two things are materially equivalent we don't add anything to work out how much we have of both together, right? If sentient beings are materially equivalent to human beings, then to work out how much we have of both together we just take the number of human beings there are.

But we do when things aren't materially equivalent. e.g. when neither is a condition of the other, then we just add the amount of both. Gibbons are not great apes, so we have as many hominoids as we have gibbons and great apes.

So how do mathematicians express material equivalence, and do they need logic to express states like the above? They seem very basic to applied mathematics.

Answer 1

We do not care about material equivalence either way, when we add things as objects we disregard anything aside from them being objects, be they sentient beings, human beings, gibbons, or apples and oranges. This uses abstraction, and hence, I suppose, logic in the old sense of the word.

In both cases we can either count directly, or count separately and then add the numbers. Whether addition of integers is logical or not is controversial. Kant famously held that it is not, and that in fact what we do in adding is mentally put the stand-ins for numbers together, and then count them up.

"One must go beyond these concepts [of seven and five], seeking assistance in the intuition that corresponds to one of the two, one's five fingers, say… and one after another add the units of the five given in the intuition to the concept of seven… and thus see the number 12 arise"

In contrast, Frege thought that such operations are reducible to doing logic (although this does not necessarily mean that our minds actually do the counting logically). Modern cognitive psychology seems to side with Kant more than with Frege, mathematicians tend to rely more on intuitive models when they reason than on drawing logical inferences, see How We Reason by Johnson-Laird. But this presumably changes when they prepare papers for publication.

Answer 2

The Inclusion-Exclusion Principle

Sometimes you have two collections $A$ and $B$ of objects. They can either:

1. Be nested ($A$ = dogs and $B$ = animals)

2. Be nested the other way ($A$ = fruits and $B$ = bananas)

3. Be identical ($A$ = humans and $B$ = people)

4. Be disjoint ($A$ = snooker tables and $B$ = rowboats)

5. Intersect ($A$ = men and $B$ = actors)

In any case you can combine the two collections to get a larger collection $A \cup B$. For example in case (5) the new collection contains all men AND all actors AND all men who are actors. The Inclusion-exclusion principle says you can calculate the size of $|A \cup B|$ of $A \cup B$ using the formula

$|A \cup B| = |A| + |B| - |A \cap B|$

where $A \cap B$ is the intersection of the two starting classes. In this example it contains all male actors.

The formula means you counted all the men once. Then you counted all the actors once. But that means you counted all the male actors two times. So you have to uncount all those people once each by subtracting $|A \cap B|$ from the total.