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I read in the Wikipedia article about Vladimir Arnold, that:

"While a school student, Arnold once asked his father on the reason why the multiplication of two negative numbers yielded a positive number, and his father provided an answer involving the field properties of real numbers and the preservation of the distributive property. Arnold was deeply disappointed with this answer, and developed an aversion to the axiomatic method that lasted through his life."

There are 41 answers to young Arnold's question here, but excluding the informal ones, the formal ones all seem to be ultimately equivalent to his father's answer (they invoke the distributive property). I think there is no escape to this, but I may be wrong.

P.S. According to this, Igor Arnold, the father of Vladimir Arnold, learned abstract algebra from Emmy Noether (one of the founders of abstract algebra)!

My main question:

Isn't Igor Arnold (standard) explanation to his son the only rigorous/correct way to understand why the multiplication of two negative numbers yields a positive number? It seems to me, that Igor Arnold gave the only mathematically correct explanation there is for the reason why, but it was, despite that, insatisfactory to his son (who would later become a notable mathematician).

If that is not the only rigorous/correct way, what are the alternatives? Maybe there are other rigorous alternatives I don't know? (I'm not talking about heuristics like finite induction generalizations from examples (they are valid for intuitive understanding, but they are not rigorous). I mean mathematically rigorous/correct alternative explanations.

P.P.S. I think it's very relevant to ask whether rigor is really appropriate for children learning basic arithmetic!!!

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    $\begingroup$ P.P.P.S. Rather than cast aspersions on the appropriateness of Arnold Sr.'s explanation, it seems to me that what this story tells us is that Arnold Sr. perhaps understood quite well the immense mathematical capacity of his son Arnold Jr. $\endgroup$ Commented Oct 1, 2024 at 12:17
  • $\begingroup$ Have you read through the responses from the other eighteen times this question has been discussed on this website, Viktor? $\endgroup$ Commented Oct 1, 2024 at 13:48
  • $\begingroup$ @GerryMyerson: No. I'm not aware this is a duplicate. The other question I've read is linked at my OP. $\endgroup$ Commented Oct 1, 2024 at 13:54
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    $\begingroup$ As I explained in an answer there, every nondegenerate theorem of rings necessarily uses the distributive law since it is the only ring law connecting the additive and multiplicative structure (here nondegenerate theorem means the theorem doesn't belong to the underlying additive group or multiplicative monoid). In particular this holds true for said law of signs. $\endgroup$ Commented Oct 1, 2024 at 17:23
  • $\begingroup$ @BillDubuque: Thank you! It makes sense to me now! There is no escape, one must use distributivity. If you post your comment as an answer here I'll accept it! $\endgroup$ Commented Oct 1, 2024 at 17:33

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"I think it's very relevant to ask whether rigor is really appropriate for children learning basic arithmetic"

You nearly had it right there. I explain it to students using the number line. The default direction for adding is to the right (i.e. positive numbers), but each time we have a minus sign we have to change direction. i.e. to the left in the first instance, which has us pointing in the direction of the negative numbers. This means every second minus sign has us pointing back in the positive direction again. Done. :-)

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    $\begingroup$ This does not answer the OPs question, which asks for a rigorous argument that does not use the distributive law. The intuitive one you suggest is well known. $\endgroup$ Commented Oct 1, 2024 at 12:13
  • $\begingroup$ @EthanBolker It says "rigorous/correct", so my answer is a correct one that doesn't use distributivity, as requested. $\endgroup$ Commented Oct 1, 2024 at 13:09

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