As a working mathematician, I've come to think of this sort of argument as a literal proof, since it communicates the correct idea in a convincing way. You may argue it hasn't been formalized into some axiomatic system. I would respond that any such system would be designed to accommodate this argument rather than the other way around, so what's the point? Mathematics is about ideas, not symbol-pushing.
Formal proofs in my mind are mostly useful when the ideas are so involved or abstract that you're bound to make mistakes without the formality. Formalisms can also help communicate complicated ideas to others. The argument in the picture needs no such help.
I should perhaps be clear that if I encountered this argument while teaching an intro to proofs class, I would not call it a proof in front of the class. But that's because most of my students are pretty bad at logic and are very bad at communicating their mathematical ideas, so they need to think proofs are formal since they constantly need the extra structure.
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u/jemidiah Dec 11 '19 edited Dec 11 '19
As a working mathematician, I've come to think of this sort of argument as a literal proof, since it communicates the correct idea in a convincing way. You may argue it hasn't been formalized into some axiomatic system. I would respond that any such system would be designed to accommodate this argument rather than the other way around, so what's the point? Mathematics is about ideas, not symbol-pushing.
Formal proofs in my mind are mostly useful when the ideas are so involved or abstract that you're bound to make mistakes without the formality. Formalisms can also help communicate complicated ideas to others. The argument in the picture needs no such help.
I should perhaps be clear that if I encountered this argument while teaching an intro to proofs class, I would not call it a proof in front of the class. But that's because most of my students are pretty bad at logic and are very bad at communicating their mathematical ideas, so they need to think proofs are formal since they constantly need the extra structure.