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Specifically, the associative, distributive, and commutative laws of addition and multiplication.

Was it Peano?

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    $\begingroup$ Peano and Dedekind deduced them from arithmetical axioms, but prior "partial" proofs were already produced by Leibniz and Grassmann. $\endgroup$ – Mauro ALLEGRANZA Jun 21 '17 at 6:09
  • $\begingroup$ @MauroALLEGRANZA, did Leibniz use the inductive principle? $\endgroup$ – Ameet Sharma Jun 21 '17 at 6:14
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    $\begingroup$ NO, and he did not gave complete proof; e proved that e.g. $2+2=4$ and in this proof he relied on associativity (that he assumed). But the idea that we can prove "obvious" facts of elementary arithmetic "popped-up". $\endgroup$ – Mauro ALLEGRANZA Jun 21 '17 at 6:44
  • $\begingroup$ Which arithmetical axioms? Many axiomatizations of arithmetic take those laws as axioms. And if we are talking about Peano axioms specifically then of course it would have to be him. But it is dubious that Peano axioms are any more obvious than elementary laws of arithmetic, induction definitely seems less so, so his "proofs" are only meaningful within his particular formalization. $\endgroup$ – Conifold Jun 23 '17 at 0:21
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    $\begingroup$ Euclid proves the commutativity of multiplication of whole numbers $> 1$ in Elements VII 16. $\endgroup$ – Marius Kempe Jun 23 '17 at 11:30

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