Morally speaking, 1+1=2
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.
Edit:
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:
- The word is probably used elsewhere in a similar way.
- There's probably some mathematicians here.
Edit:
A coincidental discovery: There is such a thing as a moral graph, which seemingly has a rather esoteric etymology.
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.