In this post I want to ask about the implication used in mathematical proofs. Well, as a far as I know, in our known logical model that is used in mathematics, implications are not causal, and thus statements like "moon revolves round the earth implies dinosaurs don't exist" or "moon is made of green cheese implies Socrates is a man" are true. But these are formal implications, not causal ones. Now let us look at mathematical proofs. For example suppose a simple proof that $4$ is even should run as follows:
\begin{align}4=2×2&\implies \text{there exists an integer}\ n=2\ \text{such that}\ 4=2n\\&\implies 4\ \text{is even}\end{align}
If we assume that the implications used are material conditionals, then we can introduce further steps like:
\begin{align}4=2×2&\implies \text{there exists an integer}\ n=2\ \text{such that}\ 4=2n\\&\implies \text{the earth goes round the sun}\\&\implies \text{the sun rises in the east}\\&\implies 4\ \text{is even}\end{align}
Doesn't look like a healthy proof right? The implications used in the first proof satisfy the conditions of the material conditional, but still some intuition goes in our mind regarding the causal relationships between the successive steps that prevents us from writing the unnecessary steps. My question is whether we can formalize and incorporate this notion of causality in our existing material condition.
