I asked a question over on math.SE and as part of an exchange someone said:

Morally the function is csc φ in the limit for the reason you mention.

...a pretty funny thing to say. I asked them about it and they said that,

It is a figure of speech sometimes used in mathematics and physics

and that one interpretation might be

What follows may be wrong in detail, or not contain enough detail, but it gives the right intuition.

Does anyone know about how this started, or have any other understanding of what it means? Are there any parallels elsewhere?

Even if used figuratively it betrays something quite strong about one's philosophical attitude, or that of one's community. For this reason I am quite interested in this turn of phrase. Does it reflect a lack of seriousness or respect for moral questions, or does it simply acknowledge that mathematical and social realities are different?

Anything that can help me understand would be appreciated.


To be clear, I would like to know if it has an implication regarding the value of moral reasoning. If it has the implication that moral reasoning is less rigorous, then it is certainly a possibility. Please do not take my question as an assertion of what it does mean.

Although some might find asking this outside the community in which it is used a little strange, there is one reason, and one excuse for this:

  1. The word is probably used elsewhere in a similar way.
  2. There's probably some mathematicians here.


A coincidental discovery: There is such a thing as a moral graph, which seemingly has a rather esoteric etymology.

  • 1
    It means that 1+1=3 for large values of 1.
    – MetaEd
    Jun 16, 2013 at 6:29
  • 2
    @MετάEd ... so how do I downvote a comment? ;)
    – Lucas
    Jun 16, 2013 at 6:36
  • Interesting question Lucas. I'd rather hear from a mathematician describing the use of the word instead of others postulating on what sentiment it might entail.
    – Pete855217
    Jun 16, 2013 at 11:08
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    It is a bit metaphorical and provocative. It is a recognition that though the thrust of mathematical presentation is rigor and exactness, communicating about mathematical ideas can sometimes be the artistic truth. Some of Euler's proofs had 'moral' certainty despite missing some rigor. 'Moral' is shorthand for 'If you ignore certain unimportant details, this -must- be right, and you should be able to recognize that.'
    – Mitch
    Jun 16, 2013 at 19:54
  • 2
    "Sometimes 2+2=4, Winston. Sometimes they are five. Sometimes they are three. Sometimes they are all of them at once. You must try harder. It is not easy to become sane." Perhaps this was Orwell's take on moral certainty.
    – tylerharms
    Jun 17, 2013 at 8:08

3 Answers 3


The phrases morally certain and moral certainty are not restricted to the math and physics communities; they been current in English since the 17th century, and I have heard and read them all my life in literary and colloquial contexts. OED 1 gives s.v. Moral, 11.

Used to designate that kind of of probable evidence that rests on a knowledge of the general tendencies of human nature, or of the character of particular individuals or classes of men ; often in looser use, applied to all evidence which is merely probable and not demonstrative. [my emphasis] Moral certainty : a practical certainty resulting from moral evidence; a degree of probability so great as to admit of no reasonable doubt ; also, something which is morally certain. […]
 This use of the word is prob. ultimately connected with Aristotle’s ἠθική πίστις, which means the effect of the known personal character of an orator in producing conviction.
 The currency of the terms certitudo, evidentia moralis appears to be due to the Cartesian logicians of the 17th c.

A typical early citation there is this, from about 1677:

HALE Prim. Orig. Man, II. i. 128 Though the evidence be still in its own nature but moral, and not simply demonstrative or infallible.

Morally in this sense is thus grounded in the Classical use of the Latin term moral (which, like the corresponding Greek term ethical derives from a word meaning ‘local custom’) to designate what rests on practical rather than logical judgments.


In mathematical contexts, I would use "Morally such-and-such is the case" to mean "Such-and-such ought to be the case". In somewhat more detail, it would mean that I expect there to be a theorem whose main content is "such-and-such" but which might require some hypotheses or some exceptions that I haven't thought about (or that I have thought about and will tell you but only after I've given you the main idea, "such-and-such").

I believe this mathematical usage of "morally" is somewhat different from the notion of "moral certainty" described in StoneyB's answer. The latter seems to refer to a situation where we have (or think we have) all the details but insufficient evidence for complete certainty. The mathematical notion usually refers to a situation where we don't have all the details yet (or at least we're not confident that we have them all). Incidentally, I believe the phrase "moral certainty" is also used in moral theology to mean a level of certainty sufficient to support moral decisions and judgments.


Even if used figuratively it betrays something quite strong about one's philosophical attitude, or that of one's community. For this reason I am quite interested in this turn of phrase. Does it reflect a lack of seriousness or respect for moral questions, or does it simply acknowledge that mathematical and social realities are different?

I completely disagree with that sentiment. I don't think it shows a lack of seriousness for moral questions at all; rather, I think it demonstrates how much that community values rigorous proofs as opposed to unproven generalizations. Such a figure of speech may illustrate how much that community values methodically establishing the validity of assertions – but that doesn't necessarily devalue their views on ethics and morals.

I'm sure other examples abound. Consider a baker who says:

It would be a sin to use Splenda instead of sugar in my apple pie!

or a silversmith jeweler who playfully says,

Sorry, using pewter to make a necklace would be against my religion.

I'd interpret both of those statements to mean that these two have strong feelings about something in their realm of expertise – but I think it would be wrong to presume that the baker and silversmith must not regularly attend traditional religious ceremonies, or that they are people who don't most likely have strong moral principles. Only the most adamant fundamentalist would consider it sacrilegious to borrow from the language of religion in order to express zeal and fervor about something.

I could speculate as to how morally came to mean generally in the math and physics communities (it doesn't seem like a huge stretch to me, based on my interaction with mathematicians), but I'd rather someone more steeped in those sciences do that. Still, it doesn't seem like a stretch to me at all, and I can't see how it might reflect a lack of seriousness or respect toward moral questions.

EDIT: Such usages of words can be found in news articles, for example, a career advice columnist entitled a column 8 Things It Would Be A Sin To Skimp On when dishing out 8 nuggets of advice, and the U.K.'s Telegraph once proclaimed in a headline:

London 2012 Olympics: it would be a sin if Great Britain are excluded from the Games
It would be a sin, nothing less, if FIBA vote not to allow Great Britain to compete at the Olympics next year.

  • 1
    Please see my edit. Personally, if someone says "It would be a sin..." I think it says something about them and the community they live in. Moreover, I think it says something about what they think "sin" is. There is something shared between "sinning" and "using sugar in apple pie". My question is what is shared between "moral thinking" and "mathematical handwaving", there are different meanings depending on what is shared.
    – Lucas
    Jun 16, 2013 at 18:15
  • @Lucas: A theologian would define sinning as a moral lapse, or "missing the mark" in an endeavor to follow religious commandments. The baker might agree with that, despite calling it a "sin" to use an artificial sweetener. It's a lighthearted use of the word, one Macmillan supports in Def 2a (as opposed to Defs 1 & 2). In much the same way, I don't think using the word morally in the context you describe necessarily implies moral reasoning is somehow weaker or inferior to mathematical proofs – though opinions may vary.
    – J.R.
    Jun 16, 2013 at 19:01
  • I guess the reason I think there is a little more substance to it is that one is generally less inclined to be lighthearted about things one takes seriously - a priest may be disinclined to use the word "sin" in the 2a sense as it could suggest the matters of sin should be treated lightly. Another example, though quite an extreme one in comparison, is that of casual racism. If one wishes to be treated seriously as non-racist, it is better to avoid making lighthearted but racist jokes. Being lighthearted seems to me to indicate that one is not otherwise invested in it as a serious matter.
    – Lucas
    Jun 16, 2013 at 19:31
  • Maybe metaphorical is a better word than lighthearted. An engineer might say of his marriage, "Back to the drawing board," or a veterinarian might say, "It's raining cats and dogs," or a pitcher might tell a friend about how he "balked" at a sales pitch, or a judge might tell his children that he intends to "lay down the law" about curfew, or a fireman might speak of "putting out fires" at a town meeting. Such expressions don't mean any of those folks take their profession any less seriously. I don't see an "implication regarding the value of moral reasoning" – that seems a stretch.
    – J.R.
    Jun 17, 2013 at 2:15
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    @user867 I think the intention of this answer was to point out that I was being a little presumptive in the way I originally phrased the question (it hasn't changed except for where marked), J.R. was right about that even if I otherwise disagree with him. Although he didn't really answer the question, it was a constructive thing to add. I hope you will reconsider.
    – Lucas
    Jun 17, 2013 at 4:55

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