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I came across this sentence in the American Heritage Dictionary, but still do not understand it.
This proposal is the best so far, modulo the fact that parts of it need modification.
The relevant definition of modulo provided by AHD is correcting or adjusting for something, as by leaving something out of account.
Please elaborate on the meaning of modulo the fact. Does it mean the same as the following?
This proposal is the best so far, but note what parts of it still need to be modified.
The pattern
X modulo Y
is an informal but common parlance in technical, especially mathematically, oriented talk. It is used to mean informally 'X, ignoring Y'. For example,
"The rocket design was flawless, modulo the toxic waste produced by its fuel."
The meaning is inspired by, but not perfectly corresponding to, the arithmetic modulo function (for example, clock-time addition) which when suitably abstracted involves 'collapsing' all items of a set into the special items of the set, so that the full set does not need to be dealt with (this is where the associated meaning of 'ignoring' comes from).
In your interpretation "note what parts of it still need to be modified", the 'modified' part is irrelevant. 'Modulo' is pragmatically "I'm telling you about the most important part (the X), but remarking on the existence of some part that might be important for other reasons but under the current context we want to ignore (the Y)".
I only see this usage from academics with a background that includes England. It is used to mean the opposite of except - some part of the set is included, not excluded, and you're saying that you're including it even though some people might not. It isn't a substitute for but or except, because those would be about excluding something from the set. It might be closer to even though.
In math, modulo is the remainder after dividing, so 5 mod 2 is 1. In words, it's something like even after accounting for. However, it has been heard by generations of people who aren't sure what it means, don't want to ask, and feel that smart people use it. Those people tend to use it as except or but, meaning that you probably can't be entirely sure any more what someone means when they use it.
The OED3 has mod as a preposition dating from 1854, and modulo as a preposition dating from 1887 — but those are the more purely mathematical senses, not the extended senses that seem to crop up in the 1950s.
However, programmers and perhaps others regularly use mod or modulo in its extended sense to mean “save/except for”, or “without”, or “minus”. It’s to exclude something. This isn’t a mathematical use, although it may be a form of shop jargon.
Again, it is by no means uncommon in programmer circles, although I don’t know that I’ve myself used it in formal writing.
Modulo is useful in computers for getting a number that can be no bigger than the modulus. So if it were me, I'd take "modulo" in this sentence to mean "limited by".
Here's a use of 'modulo' by a mathematician working on the four-colour theorem quoted in Msrk Walters, "It Appears That Four Colors Suffice: A Historical Overview of the Four-Color Theorem (2004):
Shortly after testing the final configuration for reducibility, Appel celebrated the success by etching the statement ‘Modulo careful checking, it appears that four colors suffice’ onto the department’s blackboard.
This follows the general sense 'A mod B' being 'A seems generally true but for B' but B is not necessarily an exception, but something to bear in mind.
The example that The American Heritage Dictionary uses for modulo, "This proposal is the best so far, modulo the fact that parts of it need modification," is a confusing and poor usage of the phrase. Unless a reader has a strong analytical mind and understanding of the phrase, they would likely interpret the sentence to mean that the proposal is the best as-is, and it would be even better if it was modified.
On the other hand if the sentence is correctly interpreted, the sentence is meaningless, for a couple reasons. First, a “fact” does not change the quality of the proposal, and so has no impact upon its ranking (i.e. whether it is “best” or not). In general the phrase “modulo the fact” is a poor usage of the word modulo. Second, even if the reader imagines modifying the proposal to be the best, the sentence is still meaningless, because every proposal could be modified to be the best.
Mitch’s example is far better:
"The rocket design was flawless, modulo the toxic waste produced by its fuel."
The best replacement word that I can come up with is “discounting” or "not accounting", so
"This proposal is the best so far, discounting that parts of it need modification."
(Still meaningless for the second reason stated above.)
Or, in Mitch’s example,
"The rocket design was flawless, discounting the toxic waste produced by its fuel."
Like zero or orthogonal, the word modulo has a precise technical meaning in mathematics. This makes it useful in some limited non-mathematical situations. But most people just say it to sound smart, and miss the point.
In mathematics, "A = B modulo C" means roughly that A and B are the same thing, except for differences of type C.
Sometimes people use "modulo" as a synonym for "except for." Example: "All mammals, modulo the monotremes, give birth to live young." This example is arguably correct, but we have far better words for this (like except).
Sometimes people use the word modulo to mean "almost." This is almost as bad as saying "literally" when you mean "figuratively." The word "modulo" has nothing to do with closeness. In math, -1 and 999999999999 are equal modulo 1000000000000.
This proposal is the best so far, modulo the fact that parts of it need modification.
If the author knows what "modulo" really means, this should mean
This proposal would be the best so far if parts of it were modified.
The dictionary author could have found a better example.
The basic meaning is if you say something, and then I give a potentially but not necessarily disruptive piece of information, modulo that means your information is understood taking into consideration the information I just gave
The primary meaning of the word is for math but colloquially people use it this way too
The intention of saying "understood modulo that" is basically to say "I'll do it and you don't have to address me again unless that new information mattered to your thinking"
If anyone is offering a definition that is a synonym for “notwithstanding”, then just use “notwithstanding”.
“Modulo” has a distinct usefulness that is to do with remainders (i.e. what is left over after you put items into groups), and contrary to many other opinions here, it is useful beyond mathematics.
Politicians vote along party lines, modulo party membership
means
Politicians vote along party lines, split into parties and as groups, with a few leftover (sometimes) because they don’t fit into parties
and not simply
Politicians vote along party lines, unless they don’t have a party
along with the missing implication that most have them always, and sometimes all have them.
This is the correct usage taken beyond mathematics without the grandstanding. If the grandstanding form is synonymous with “notwithstanding”, maybe the correct form is synonymous with “after sorting into every”; but the next phrase must be a noun, not a circumstance or condition.
It is at this point we have to face up to the unsatisfactory use of the word “notwithstanding”, where some people follow that word with a precondition and others do the opposite, and give an exclusion. Others still just give the noun without stating inclusion or exclusion, but if you follow a literal logic, it should be an exclusion to the preceding statement.
