In our scientific writing we often make statements that could be interpreted as ambiguous. Just as operator overloading (using the symbol "+" both for the addition of numbers and of matrices, for example) simplifies and actually clarifies writing, leaving some parts of our statements not spelled out in detail can sometimes improve writing. It presents ideas at a higher level, preventing the reader from being side-tracked by minor points.
However there is an underlying assumption, that the reader is at the level of maturity that will permit them to reconstruct the unsaid without effort. The writer wants to convey something very specific, and, if both reader and writer took the time to spell out the missing pieces in full detail, they would both come up with the same interpretation. I cannot think of any instance where it is desirable that the reader come up with several interpretations of what the writer really meant. There is no such freedom. The text and its meaning belong to the writer, and the reader is not at liberty to go at variance with what the writer intended to convey.
That is not true in literature. For example, "The Raven" poem by Poe. What is that bird really about? Why does it say "Nevermore" repeatedly? The answers can vary from reader to reader. Once the poem is released, it escapes from the poet, and other people are free to give it meanings that Poe never intended. They read the poem in the context of their own experience, and interpret it in radically different ways. The multivalence is not a fault but a richness. The stand-alone poem is incomplete, and it only achieves meaning when the reader's perspective is incorporated into it. The poem is ambiguous in a way that a mathematical text never is.
The writer wants to convey something very specific, and, if both reader and writer took the time to spell out the missing pieces in full detail, they would both come up with the same interpretation.
ReplyDeleteEven for scientific literature, this is not necessarily true for things such as motivation. If you ask different people what is interesting about a certain paper, they will all usually give you different answers. I routinely review papers where the authors have missed highlighting the most interesting (to me!) part of the story!
I had in mind primarily the scientific content of the paper, but even for motivation, I think that the author has something specific in mind when he is writing, that he or she is trying to convey to the reader. It could be that they missed something that you would have liked to see, but not that they half-said something, not quite knowing how to complete it and hoping that you would find a way to interpret their statement. Or at least, if that happens, it's not good writing, right?
ReplyDeleteI think there may be ambiguity at a "higher level", even more aesthetically pleasing than the ambiguity in literature (at least to me :)). Many times, you have a notion which directly emerges from the definition of the problem, and then suddenly when you view the notion from a different aspect, it acquires a whole different meaning and sheds new light on the original problem! For example, hardness vs. randomness, or proofs of classical results using quantum ideas, or communication complexity for proving streaming lower bounds, etc.
ReplyDeleteMaybe the difference from literature is that we can actually take advantage of mysterious similarities to produce interesting results instead of just hinting at them?
Arnab, that's an interesting idea. Do you think that for the same result, the way in which we write can help or hinder such insights?
ReplyDeleteI try to cast my scientific literature net a bit wider than my interests: sometimes you can read other problems into the science of a paper. The facts of a scientific endeavor often seem to be an instantiations of more fundamental insights you can try to tease out (and those insights are ambiguous).
ReplyDeleteI'm also reminded of Einstein's discovery of special relativity. The pieces were just about all there before his work. The key equations are named for Lorenz. But he was able to detect to coherent theory among other people's equations, insights and experiments.
In a sense, I think the ambiguity appropriate to scientific literature is one of paucity. A good paper should express its ideas as generally as possible, and avoid forcing unnecessary interpretations in. Returning to relativity, a lot of the previous work just took Newtonian physics as unquestionable, and so sought to justify any deviation in those terms. If you let the equations "speak for themselves" people can arrive at different ways of connecting them back to the world, be it interpretations or other problems you could substitute in for the identifies of the variables.
Regarding "unsaid" parts in proofs, those "left as an exercise to the reader" - very often, you actually do not care how the reader proves the fact (there are often several equally valid easy proofs). Actually, it is a mistake to leave "unsaid" a proof fragment whose details are important for the following text, because the reader may not have the same thing in mind as you!
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