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Long ago, I have heard Itai Shafrir, one of my differential equations mathematics teachers, claim several times that a result or a conjecture should hold "moralement" in French ("morally" in English). Since then, I have observed the same gimmick in several conferences (in French and English as well), by many people like Haïm Brezis or Pierre-Louis Lions.

So far, I have heard this statement mostly in the mouth of mathematicians involved in partial differential equations (PDE). Nevertheless, occurrences of "Morally we have" can be found in several other subfields of mathematics.

Did someone else make the same observations? If so, where does "morality" come from, precisely, in mathematics?

I do understand the meaning ("stuff that should be, knowing what we know"), and would like to trace the source of this expression, in a domain where truth is more important than morality. Especially, is it more common in PDE or applied mathematics as I have observed? Are there occurrences in languages other that English or French?

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  • $\begingroup$ From what I can tell, this generally applies to the idea that our mathematical model should agree with what we would expect about reality. At least, that's how I've encountered it $\endgroup$
    – Alan
    Sep 13, 2015 at 20:16
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    $\begingroup$ Often a mathematician will use this phrase as a concise way of saying that he/she has good reason to believe the statement to be true. In other words, true for some intuitive reason. $\endgroup$
    – Tim Raczkowski
    Sep 13, 2015 at 20:19
  • $\begingroup$ I believe I understand that "moral" here has nothing to do with "behavior". I am wondering whether this gimmick originates from a mathematician. I used to believe Haïm Brézis was a common denominator of the PDE mathematicians I have heard. $\endgroup$
    – Laurent Duval
    Sep 13, 2015 at 20:38
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    $\begingroup$ Just out of pure interes I googled Pierre-Louis Lions and read something about his Viscosity Method. Can someone give some reference where this method is explained? $\endgroup$
    – MrYouMath
    Sep 13, 2015 at 20:56

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Moralement literally means "In a manner consistent with the rules of morality". It is commonly used by French mathematicians with the meaning "In a manner consistent with the rules of mathematics". Whether mathematics are moral or not is another story...

I do not know the origin of this expression, this is an interesting question.

Edit. Just found the following references from the Dictionnaire de L'Académie française, 1st Edition (1694):

On dit, Moralement parlant, pour dire, Vraysemblablement & selon toutes les apparences.

Dictionnaire de L'Académie française, 4th Edition (1762):

On dit, Moralement parlant, pour dire, Vraisemblablement & selon toutes les apparences. Cela est vrai moralement parlant.

This is very close to the mathematical use of this word.

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    $\begingroup$ In English, "morally certain" is a common expression meaning "so sure one is morally justified in acting upon the conviction", with earliest cite from 1645 in the OED. $\endgroup$
    – bof
    Sep 13, 2015 at 20:26

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