TITLE: grid "proof" for commutativity of multiplication
QUESTION [0 upvotes]: In this write-up by Tim Gowers on why multiplication is commutative,
https://www.dpmms.cam.ac.uk/~wtg10/commutative.html
he gives a physical grid model to which multiplication corresponds and says - "This argument, compelling as it is, doesn't quite qualify as a mathematical proof"
Why, precisely, doesn't this qualify as a mathematical proof ?

REPLY [0 votes]: From a semi-formal point of view, it doesn't qualify as a mathematical proof because:

it is unclear whether it is speaking of the single identity $395\times 428=428\times 395$ or whether the author claims of having given a general argument;

it hasn't been shown how the properties of being a rectangular grid and rotating by ninety degrees work together towards the result of preserving the cardinality of a (the?) rectangular grid.


It should be pointed out that modern mathematical formalism generally demands that, with a few notable classes of exceptions, a proof should be founded in the language of some set theory, the most common being ZFC, and that the argument of interest does not immediately qualify as such.
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