An outcropping of jargon occurred while my back was turned, and frequent commenter Noetica has called my attention to it in this thread:

I have been trying to sort it out for several years, now. All OED has for modulo is this:

With respect to a modulus of. Also attrib., = modular.

What does it mean though, exactly, in the sentence above? From the Wikipedia article, after its mention of the first meaning in mathematics:

Ever since however, “modulo” has gained many meanings, some exact and some imprecise.

Tell me about it!

What he doesn’t know is that the OED entry was revised in March 2003, and after the literal definition he quotes there is now the following:

b. In extended use. (a) With respect to an equivalence defined by (some feature), disregarding differences indicated by (some unimportant feature); (b) taking into account (a particular consideration, aspect, assumption, etc.).
1953 W. AMBROSE Let. in S. Nasar Beautiful Mind (1998) xx. 155 [John Nash] proceeded to announce that he had solved it, module [sic] details. 1960 Jrnl. Philos. 59 776 Which we choose is entirely arbitrary, but (modulo the assumption that any run covers a line segment) it determines how we answer the question [etc.]. 1973 C. C. CHANG & H. J. KEISLER Model Theory 7 The language is determined uniquely, modulo the connectives, by the sentence symbols. 1992 Stud. Eng. Lit.: Eng. Number (Tokyo) 161 The Navajo underlying structure is identical, modulo word order, to the one found in all the languages studied in Ch. 3.

Anyone who wants more examples can visit Noetica’s comment in the thread I linked above; he cites a passel of ’em. My initial reaction was that I didn’t like it: what’s wrong with “with respect to” or “taking into account”? Of course modulo is shorter and I can see how it would be convenient, but it still strikes me as one of those bits of verbiage that serve mainly to show that you’re one of the gang.
I was curious about the “etc.” in the second citation—I wondered how long that sentence ran in its original setting. So I googled it, and thanks to the wonders of JSTOR I found it: “…the question ‘Given that he disappeared at 1, did he occupy 1?'” This was a pretty disappointing payoff for the effort of trawling through a 20-page philosophy article that made my eyes glaze over so thoroughly that I missed the quote the first time through and only caught it on the reverse journey (yes, I was an idiot for not thinking of looking on p. 776, where the OED said it would be), but I now have the following burning question: why does the OED cite this as Jrnl. Philos. 1960 when 1) it’s from an article with an author and title (Paul Benacerraf, “Tasks, Super-Tasks, and the Modern Eleatics”) and 2) Vol. 59, No. 24 of The Journal of Philosophy is from November 1962?


  1. Peter Maydell says

    I note that the OED’s (a) and (b) actually define opposite meanings! (a) is the one I’m more used to, being approximately synonymous with ‘ignoring’ or ‘disregarding’, ie “not taking into account”. Presumably (b) (“taking into account”) is a further development by people unaware of the mathematical jargon origins of the word…

  2. Damien Warman says

    On the (in)equivalence of (a) and (b): I suppose it depends on whether the details are ignorable, so we can speak of equivalence modulo those (hinted-at) details, or whether the details of the equivalence are not immediate, and so we specify the relationship. This satisfies me for now, modulo examples.

  3. Hurray, for the first time ever a Languagehat question I am qualified to answer!
    The origin here is modular arithmetic: one says 5+6 = 1 (modulo 10) to mean that 5+6 = 2 is correct, as long as one considers any two integers equal when their difference is a multiple of 10. (Thus 10 is the modulus, whose multiples are to be thought of as negligible or disregardable in this setting.) The sense a) in the OED is in direct analogy with this usage. It is quite handy, and not at all synonymous with “with respect to” — for instance. A typical usage would be
    “I’ll be there at four, modulo traffic.”
    “With respect to” doesn’t even substitute here: “Taking into account” does, but would mean something like “If there were no traffic, I’d get there at 3:15, but I estimate the traffic will add 45 minutes.” Whereas “I’ll be there at four, modulo traffic” means more like “My best estimate of when I’ll be there is four — this should be correct as long as we disregard the unpredictable variations in trafic, which I expect in this context to be negligible.”
    I’m not familiar with sense b); what are examples?
    Anyway, this word is definitely part of my idiolect, so I think I can judge the grammaticality or non- of any sentence containing it, if it helps to pin down what it actually means!

  4. I think the thing to note in this instance is that all of these quotations are from scientific or quasi-scientific contexts. That’s natural, of course, as it’s a pretty technical word to begin with, but for this specific meaning—the classic theoretician’s “it’s all there but for the implementation” handwaving—it seems to me not a bad guess that the early cases of its extended usage probably involve a tongue-in-cheek or humorous light. Especially given mathematicians’ and computer scientists’ known propensity to using technical jargon in mundane(r) circumstances, for humorous effect.

  5. For future reference, when you search for something on JSTOR, each search result is followed by a row of gray-text links, one of which is “Page of First Match”, which is even better than it sounds: not only does it take you directly to the first page in the article containing (one of) the search term(s), it actually highlights the search term(s) in yellow, so you don’t need to skim even one whole page.
    Very convenient.

  6. Prepositions in English are funny: they look like a closed class, but it’s a pretty big closed class (more than 300 members, if you include the commonly used compound prepositions), and every so often the class gets pried open and a new one inserted.
    The Jargon File definition is rather simpler: “Except for. An overgeneralization of mathematical terminology; one can consider saying that 4 equals 22 except for the 9s (4 = 22 mod 9). ‘Well, LISP seems to work okay now, modulo that GC bug.’ ‘I feel fine today modulo a slight headache.'”

  7. Definition b) is probably just sloppily written, and meant to say “taking into account X and discarding it”. None of the examples show the contradictory sense.

  8. So, use or non- use of ‘modulo’ is a difference that makes no difference, modulo modulo… ?

  9. Peter Maydell says

    The usage in the Journal of Philosophy article doesn’t look much like sense (a) to me; it might or might not be sense (b) or something even closer to meaninglessness. Can’t really tell without more context, though (and I don’t have JSTOR access).
    Definition (a) could be better as well, too; it doesn’t say whether in the phrase “modulo X” X specifies the “(some feature)” or the “(some unimportant feature)”…

  10. JSE: Thanks very much! I think I understand its basic use, if your “I’ll be there at four, modulo traffic” is representative. Hell, it may even slip into my own usage if I get to feeling jargony.
    Ran: I wasn’t searching on “modulo” but on the title of the article, and I’d love to know how to search within an article once you’ve gotten to it.
    Peter: For you and anyone else curious about the JPhil usage, here’s (some of) the context:
    Let t0 be the time at which the genie started from 0, and, where applicable, let ti be the time at which he is at i. [The 0 and i should be subscript — LH.] The question then becomes: Does this imply that at t1 he occupies 1? […] If the genie has carried out my instructions, at t1 he cannot be at 1, because at 1 he is no more. To be sure, he vanishes at a point: 1. But what does this mean? In particular, does this mean that 1 is the last point he occupied? Of course not.[…]
    To illustrate, we draw two lines L1 and L2. […] We may view each line in two different ways, corresponding to the ways in which each point may be seen as dividing its line into two disjoint and jointly exhaustive sets of points: any point may be seen as dividing its line either into (a) the set of points to the right of and including it, and the set of points to the left of it; or into (b) the set of points to the right of it and the set of points to the left of and including it. That is, we may assimilate each point to its right-hand segment (a) or to its left-hand segment (b). Which we choose is entirely arbitrary, but (modulo the assumption that any run covers a line segment) it determines how we answer the question “Given that he disappeared at 1, did he occupy 1?”

  11. I’ll second everything JSE said. Except the 5+6=2 modulo 10 part — if he’s who I think he is, he’s usually very good at arithmetic.

  12. LH, thank you! Of course I should have checked OED online, to which I do have access. I relied on my CD-ROM version (installed on my hard-drive, of course), which I was zealous enough to buy before 2003.
    OED divides the extended senses into two, as you show:

    (a) With respect to an equivalence defined by (some feature), disregarding differences indicated by (some unimportant feature);


    (b) taking into account (a particular consideration, aspect, assumption, etc.)

    But why should a be taken as a single unitary sense? With respect to an equivalence defined by is quite different from disregarding differences indicated by. So that makes three senses, by my lights.
    Still, OED has done it, at last! How well it has done it is open to question.
    I still think modulo is absurdly unruly, and best avoided. A word that has three senses, so that the need to differentiate by context is an annoying distraction at best (for those who know the senses), and impossible at worst (for many highly competent users of English who, like LH yesterday, have not come across it before).

  13. I put modulo in the same mental cubbyhole as mutatis mutandis, i.e., a cool-sounding classicism that you can throw around when comparing two roughly similar situations. I rarely remember to use either of them, though.

  14. Um, crap, I have to apologize for my arithmetic. I originally had 5+6 = 2 (mod 3) but then decided that mod 10 arithmetic was a more natural example since it corresponds to addition of digits. But then I decided not to say anything about digits. And then I forgot to change one of the 2’s back to a 1. As Jim says, I am usually very good at arithmetic, modulo silly mistakes.

  15. Why not just say, “I’ll be there at four, barring traffic”? This seems to mean the same thing, and underscores that relative uncertainty is already built into our notion of traffic, rather than something conveyed by the use of “modulo”. “Modulo”, on the other hand, seems like it would need an ad hoc definition whenever it appears. This is reinforced by the quoted definitions (a) and (b), which some have described as contradictory. I don’t think they really are — it’s just that the actual impact of the item being moduloed isn’t explicit in the syntactic relationship, but instead is left as an exercise for the reader. (Here, you have to know something about traffic to figure out what the speaker’s trying to tell you with the word “modulo”, in a way you don’t for the word “on” in the phrase ‘the zorf is on the biblob’.) If that’s the case, perhaps modulo doesn’t really mean anything at all, but is just filler to keep the sentence from sounding stilted or ungrammatical. “I’ll be there at four… traffic.”

  16. Because “I’ll be there at four, barring traffic” means “I’ll be there at four if there is no traffic, but later if there is traffic, and if there’s traffic, who knows, it might be much later.” Whereas implicit in “I’ll be there at four modulo traffic” is that a) four is actually the expected time of arrival; and b) I am telling you that the uncertain effect of traffic is something that can be disregarded in the present context.
    The two sentences are not wildly different, but they’re not the same.

  17. Whereas implicit in “I’ll be there at four modulo traffic” is that a) four is actually the expected time of arrival; and b) I am telling you that the uncertain effect of traffic is something that can be disregarded in the present context.
    Is that right, JSE? Hmmm. Even with this supposedly straightforward example, I’m not so sure. You seem to want modulo to mean something like despite, notwithstanding; but does that accord precisely with any one of the OED senses? Not with disregarding differences indicated by (some unimportant feature), I think. That OED sense allows that there might well be differences that one might sometimes take into account, but that we take them to be tangential to our present concerns (“unimportant”). On the other hand, you want four to be expected time of arrival, and the delaying effects of traffic, though they are likely to be active, not to make any difference to that expected time. (Is that what you mean? I really can’t be sure!) I had thought that the OED “differences indicated” must be orthogonal to the differences of interest, and that is why we can disregard them.
    Anyway, you might mean what you say you mean, suitably interpreted: but it’s interesting that you have to explain it to us, and that it’s not yet in any dictionary, not even OED.
    I would be very interested if anyone has seen coverage of these meanings of modulo in any other dictionary, in fact. So far I have checked Chambers, Macquarie (to which I am tempted to affix the fixed epithet wretched), Collins, Penguin, some variants of AHD, Merriam-Webster’s Online, and Webster’s 3rd International (which is way too early for this, really).

  18. Peter Maydell says says the American Heritage Dictionary 4th Ed defines this sense:
    Correcting or adjusting for something, as by leaving something out of account: This proposal is the best so far, modulo the fact that parts of it need modification.

  19. 1. I can’t say I agree completely with JSE’s last comment. I think I use “modulo” synonymously with “disregarding” or “disregarding the effects of”. If I used the phrase “I’ll be there at four, modulo traffic”, I would mean that if I’m late it will be only by the time I spend in traffic. I think you could reasonably infer JSE’s a) and b), but only in some secondary sense, like how if someone says they’ll come over after four you can be confident that it won’t be two weeks from now, even though it’s not literally ruled out.
    Maybe I’m sticking a little to close to the original, formal sense here, but even if the word is used beyond mathematics (and mathematicians), I doubt it really has an independent life. Even “exponentially”, which is often used synonymously with “very”, hasn’t quite broken free of its precise meaning. And it’s much more common than “modulo”.
    2. Noetica, I don’t see the distinction between the two senses in the OED’s (a). Can you give an example where it’s clear?
    3. I’ve never heard it used in German, but I would guess that it originated there, and was then picked up in English when the German mathematics community exploded before the Second World War.
    4. Finally, I’ll stand up and say that IMHO the OED’s sense (b) is plain wrong, as others have suggested above, though a good example could convince me otherwise.

  20. Thanks Peter. That’s a useful one.

  21. I have to agree with James: it seems to me the OED screwed this one up. Presumably the editors do not have intimate acquaintance with either mathematics or philosophy, and their valiant attempts to pull order out of apparent chaos didn’t quite work. (They should hire James and JSE as consultants.)

  22. James:
    You don’t see a difference between the two elements of OED’s sense a? Let’s label them separately:
    a1: with respect to an equivalence defined by (some feature)
    a2: disregarding differences indicated by (some unimportant feature)
    I freely admit that I remain at a loss to find clarity in any of this, but it does seem to me that there is a plain difference here, at least. Let’s suppose, as seems reasonable, that a1 is illustrated in this, from OED’s examples:

    The Navajo underlying structure is identical, modulo word order, to the one found in all the languages studied in Ch. 3.

    Here the feature would be word order, and the equivalence would be defined by it (somehow! don’t ask me!). Now, try a direct substitution of with respect to an equivalence defined by for modulo:

    The Navajo underlying structure is identical, with respect to an equivalence defined by word order, to the one found in all the languages studied in Ch. 3.

    This suggests that we set word order up as something salient, dominant, or determining, when underlying structure is in our sights, yes? But then try a parallel substitution using a2:

    The Navajo underlying structure is identical, disregarding differences indicated by word order, to the one found in all the languages studied in Ch. 3.

    It just seems to me obvious that this substitution suggests the reverse: that word order is not a matter of great moment, when our interest is in underlying structure. So a1 and a2 can hardly be equivalent – or even similar enough to warrant their association in one single sense, OED’s sense a.
    Or again, if any sense is illustrated in the following example, I think it must be a2:

    [John Nash] proceeded to announce that he had solved it, [modulo] details.

    So substitute for modulo, using a2, but making charitable adjustments to the wording:

    [John Nash] proceeded to announce that he had solved it, disregarding some matters of detail.

    But now do the same sort of charitable substitution with a1 instead:

    [John Nash] proceeded to announce that he had solved it, with respect to an equivalence defined by certain details.

    The first substitution seems to preserve the intended meaning well enough. But not this second substitution: not the same meaning, in any case.
    Let’s look at sense b:
    (b) taking into account (a particular consideration, aspect, assumption, etc.)
    I don’t particularly like this, any more than you do. But my own objection is that, while it seems to reflect some observed uses, it is far from clear which of the OED examples are connected with it (as opposed to sense a, or to a1 or a2), and precisely how they are connected with it.
    I am at a loss to make any more of this, right now! Too late at night, in Oz. That’s my excuse anyway. I am heartened to see that my difficulties with modulo are shared by others, and are revealed as “enlightened” difficulties – modulo the capacities and talents of those sharing them with me.

  23. Noetica’s comment is very well thought out. Let me explain why a1 and a2 are actually the same here (though perhaps this is due to a mathematically flavored understanding of the phrases “with respect to” and “equivalence.”) When you say “with respect to equivalence defined by word order”, what you mean by equivalence is:
    “We take two underlying structures to be _equivalent_ if and only if they differ only as regards word order.”
    In other words, by adopting this notion of equivalence we are indeed declaring that (for our present purposes) issues of word order are not our concern. And you see that the meaning of your a1 sentence is now just the same as the meaning of your a2 sentence. And, by the way, I also agree that this use of “modulo” is perfectly in conformity with the way I would use it in everyday speech.
    As for the second sentence: The sentence “I have proved this theorem modulo details” unpacks to something like “We take two arguments in favor of theorem X to be equivalent when the difference between them consists of the presence or absence of details which, while important, are routine and could be verified by any competent mathematician without the need for substantial new ideas. Under this definition of equivalence, I have written down an argument for this theorem which is equivalent to a proof.” So yes, in this case, it would be the same to say “I have all but proved this theorem — what remains is only a pile of routine details.”
    The special power of “modulo” is that it captures the important idea that there may be features of a situation which are important in one context but negligible in another; “modulo” says that “whatever follows the modulo is something which I’m declaring to be unimportant, but only within the limited scope of this sentence.” I.E. if someone asks John Nash “Should my Ph.D. student work on this theorem?” his response would be “No, I have proved the theorem modulo details.” But if you asked him “Have you submitted the paper to a journal?” he would say “No, although I have proved the theorem modulo details.”

  24. Having studied math and worked with the people who wrote the jargon file, I’ve probably been known to say things like, “modulo a few bugs,” in the right company.
    And I agree that “disregarding X” really does mean “with respect to an equivalence defined by X” in a certain way. I also agree that there are clearer ways of explaining that way.

  25. Anyone who doesn’t have JSTOR can look at an anthology called Zeno’s Paradoxes, ed. Wesley P. Salmon (1970). The anthology includes the woefully-execrated Thomson’s remarks on Benacerraf’s paper. Analytic philosophy is a nasty business, and Benacerraf’s article is a perfect example of this.

  26. OK, I’m persuaded; “modulo” is an affectation that mathematicians might use where others might use plain English: traffic permitting, save for a few bugs, and so on. But “mutatis mutandis” really is useful, especially as part of “vice versa mutatis mutandis”.

  27. Noetica: Thanks, and greetings from the ACT. Now I understand what you meant. JSE already explained everything perfectly, so I can’t really add anything.
    Dearieme: I agree completely. I usually try not to use it, especially outside the clubhouse.

  28. So then, JSE (and James: doxographically equivalent, I take it, for present purposes), what do you make of the examples I cite in that other thread (Anthimeria)? In particular, is the following “canonic”, according to you?

    Fundamentally motion is possession of virtus, something that is ultimately non-spatial (modulo its interpretation as primitive force limited by collision).

    What are we to make of OED’s sense b being, as you put it, James, “plain wrong”? Perhaps the example just cited is evidence for the word’s use in that sense. You want a “good example” to convince you, but what could count as a good example? Very plausibly, the present example means just this:

    Fundamentally motion is possession of virtus, something that is ultimately non-spatial (taking into account its interpretation as primitive force limited by collision).

    Whatever its plausible meanings, they do not appear to include …(with respect to an equivalence defined by, or disregarding differences indicated by, its interpretation as primitive force limited by collision). I may be wrong, of course! But the point is that the authors have not succeeded in conveying their message, given that most of us here – who are virtuoso users of the language – simply can’t work it out!
    I agree with LH that OED has goes a poor job this time, as it often does with highly specialised jargon. (I am thinking right now of diatonic as an example, which OED and most other “official” sources handle in a way that utterly fails to reflect the confusing diversity in historical and current usage.) And I understand how modulo is used in mathematics, and why it is therefore also used by logicians, and therefore by a natural sort of expansion throughout philosophy (whence the far-flung use in an article headed Egalitarianism, cited in the other thread) – often without respecting the original well-regimented meaning or meanings. I also understand how the term can be useful in a limited circle using the same dialect; but I deplore careless uses beyond such cliques. When I teach philosophy, I make it a point to forewarn students about such jargon (along with relatively elementary special senses of common words and phrases like valid, sound, imply, and just in case). When I edit the work of philosophers, I counsel them to strike out such high-priestly affectations, which will often be interpreted, rightly or wrongly, as marking insecurity, and will nearly always glaze the eyes of those who are not already of the inner circle.
    With best wishes from Melbourne, the Alexandria of the South.

  29. Michael Brazier says

    Noetica: I can’t see any difference in meaning between the two versions of your last sentence. In the “John Nash” pair, substituting “taking into account” for “modulo” would mean Nash’s proof had the details, while “disregarding” would mean it lacked them. But in this sentence, either substitution would convey the same meaning: that motion has an interpretation as […] but in all other ways is possession of virtus, and we are to forget about the interpretation for the present. What shade of difference do you find between them?

  30. Michael:
    You assume, I think, that the intended meaning of the sentence is this:

    Fundamentally motion is possession of virtus, something that is ultimately non-spatial (setting aside its possible interpretation as primitive force limited by collision, by which interpretation things might go differently).

    Am I right? If this is your interpretation, on what do you base it: your take on some presumed stable understanding of the word modulo, or on the immediate context? If the first, that stability is the very matter disputed in this thread; if the second, then you are a better physicist and historian of ideas than I am – and, I venture to say, than almost all of LH’s regulars. The sentence comes from an article in a copious and useful general encyclopaedia of philosophy; if it is to communicate effectively beyond a clique circumscribed by particular use of jargon, it needs to avoid this unsettled word modulo.
    The expanded version of the sentence that I give above is a detailing of the version with taking into account. I don’t see how it it can be equivalent to second of my versions, which uses with respect to an equivalence defined by, or disregarding differences indicated by. Where is the onus of proof, here? I can see that on certain narrow and partisan applications of an extension of mathematical usage, an equivalence might be conjured up. But this is about clear communication in plain prose – and not about twisting the neck of language (as Verlaine advised the poets to do).
    But perhaps we have dwelt long enough on this, and should turn instead to a consideration of bronchial affricates in early Proto-Nostratic.

  31. Michael Brazier says

    No, I wasn’t assuming anything like that; indeed, the reading you now suggest never occurred to me. But then, I wouldn’t read “X is Y, taking into account Z” to mean that Y and Z are alternative descriptions of X, either; to me, that phrase would convey that Y is a partial description of X, and Z is what’s left over. Hence my difficulty.

  32. ?
    QED, more or less.

  33. Five more years of hindsight, and I know that modulo is what the computerniks say; mathematicians say up to, as in “These are the same up to the integers”, or even better “up to isomorphism”.

  34. Jonathan D says

    Two years further, I can say that mathematicians as a whole definitely use both modulo and up to. I’m sure there are differences in usage, based on meaning and/or types of mathematician, but I wouldn’t like to put my finger on exactly what the differences are.

    Beyond that, I’m hesitant to call it an affectation. It’s definitely jargon, and I wouldn’t use it “outside the clubhouse”, but at least in some areas that way of framing things is ubiquitous when thinking/talking about actual maths, and it’s fairly natural to use the same wording when doing the same thing with more mundane topics.

  35. Thanks, that’s a useful approach.

  36. “modulo M” versus “up to E”, from a non-mathematician: I think there’s a syntax difference, which is that “up to E” takes E the equivalence relation itself, and “modulo M” takes M the element that generates the equivalence relation. So it would be “3 is the same as 13, modulo 10”; “3 is the same as 13, up to equality of the ones digit”. The modulus M generates an equivalence relation, “x and y shall satisfy the relation if adding and subtracting multiples of M can get from x to y”. If I knew how to say this less technically I would.

    The two phrases have what I think is overlap in the middle, though. You wouldn’t say *”up to 10″, and I think *”modulo isomorphism” is somewhat unidiomatic, but “rotation” goes with both. “These patterns are the same modulo rotation”, or “up to rotation”, both okay. I think in unnaturally strict usage you would have to say “modulo this generating set of rotations”, or “up to equality after rotation”, but nobody wants to say either. To be painfully precise about the syntactic point, it’s that “rotation” is neither a generating element (or elements) of some type X, nor an equivalence whose type is the relation function “given two X values, deem them related or not”. It’s a transform function, “given one X value, spit out another”. I think “modulo rotation” is a little bit of a metaphor, while “up to rotation” is just skimming over a technical detail of “given a transform, define an equivalence”.

    Googlecounting sees “up to” 90x more common than “modulo” for “rotation”, 150x for “isomorphism”, which I don’t know if I want to hang anything on, so maybe I’m wrong about ?”modulo isomorphism”.

    Now, in the social side of usage, I’d say “up to” marks a mathematician, or someone who wants to say they are, variously seriously; “modulo” could be a mathematician, or it could be computer person or general techie usage.

    My sense is that the usage “modulo isomorphism” says not a mathematician, but one of the other users, but a quick scan of web hits suggests I’m wrong.

  37. Just found this in a linguistics article on relative clauses: “[appositives] do have a fixed position adjacent to the antecedent (modulo right-extraposition to the edge of the containing clause), contrary to independent parentheticals.”

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