I will continue puzzling over the meaning of "if...then..." in mathematics. Currently I think there is the 2 question: Why certain relations are transitive and: How mathematical implication is an invented umbrella term. Arguable this invention is at the heart of mathematical statements.
Let's talk about the latter. What "A implies B" (or "if A then B") means is that we can describe steps that take us from A to B, each of which is its own little "if...then". By saying "A implies B" we brush all those sub-steps together and ignore the details deliberately.
Here is a simple example: "If you count from 1 to 10 then you pass 5". To clarify this statement, you refer to counting and (perhaps even counting from 1 to 10 to demonstrate) show that passing 5 is inherent in the definition of counting from 1 to 10. The "truth" of the statement relies on physical intuitions that have become abstracted. The childhood intuitions of counting and putting things in boxes is very important for many of these little steps. As a result there are all kinds of real world intuitions that start as empirical and become abstracted.
On the other hand, there are relations that are transitive by definition, like 'isKindOf'; and possibly 'isPartOf'. These definitions can often be restated as implications. By the time we have enough word definitions, the intuitions that gave rise to them are no longer obvious, but the definitions contain the needed necessities.
Update: I finally decided that "A implies B" is just mathematical shorthand for either a definition or a more complex: "I can prove it" - which could be quite involved, including definitions as well as intuitions developed from games, and who knows what else. For example: "if you count from 1 to 10 then you will pass 5".
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