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.

  • also wondered if it makes sense to say that one just doesn't add up things that are materially equivalent. that question is too vague tho... – confused Nov 13 '18 at 0:13
  • i guess one might object that philosophy isn't and philosophers don't answer really simple questions about other subjects haha – confused Nov 13 '18 at 0:17
  • If the number of sentient beings is S and the number of human beings is H, then if we say that sentient beings are human beings then whatever we got as the count for S should be the same as the count for H. But perhaps I am misunderstanding. If you have a text that illustrates the problem this may help. – Frank Hubeny Nov 13 '18 at 1:02
  • i'm not sure if you're misunderstanding me or just not answering the question! yes your calculation was what i meant @FrankHubeny – confused Nov 13 '18 at 1:10
  • I probably don't understand the question. – Frank Hubeny Nov 13 '18 at 1:27

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.

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    your answer is fine, the only issue i had with it was the segue between the 1st and 2nd sentences of the 2nd paragraph. it seemed to jar – confused Nov 13 '18 at 2:28
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    @confused Sorry, I was in a hurry. Is it better now? – Conifold Nov 13 '18 at 18:35

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


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.

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