-1

I recently started studying math, one of the things that really interested me about it was the idea that math, unlike science, depended solely on deductive proof. With this in mind, I've went back to arithmetic in search of deductive proof for "laws" (hope that's the right word) such as the commutative law of multiplication. However, I'm having trouble locating a deductive proof of even this most basic "law". Can anyone point me in the right direction?

Thanks!

joht
  • 1

1 Answers1

0

Using the Peano axioms, you can prove that all of the "laws" for addition and multiplication hold in the natural numbers (i.e. the non-negative integers). From there, we normally define the integers, rational numbers, and real numbers as incremental extensions of the natural numbers, and part of that development is showing that the corresponding definitions of addition and multiplication in those sets still preserve the associative, commutative, distributive, and any other laws that you're familiar with.

There are any number of elementary math courses and textbooks that cover this "program", like introductory set theory, real analysis, discrete math, or a first proof course.

  • Hi, first, thanks for the answer. Second, when you write "and part of that development is showing that the corresponding definitions of addition and multiplication in those sets still preserve the associative, commutative, distributive, and any other laws that you're familiar with." I'm concerned that preservation, on its face, is inductive. Am I misapprehending something? Third, when you speak of sets, should I take that as a reference to set theory? If so, how can one rigorously account for the use of a theory in deduction? – joht Dec 15 '20 at 14:56
  • And yes, while surely this subject is elementary, it isn't obvious that most introductory texts are concerned with deductive proofs in the way that Euclid was. Indeed, from what I can ascertain, the focus in modernity is more on application than deduction from the very beginning throughout every proceeding step. – joht Dec 15 '20 at 14:57