Haec est subintelligenda
Before I forget about this concept that I was thinking about relating to reading and doing mathematics, I wanted to "pen" down some of my thoughts relating to this concept of the "obvious", for lack of a better term. There is a frequently cited concept in written and Mathematics as it is often taught that certain principles or results are so "clearly obvious" or require simply "a moment's thought" that they simply require no further explanation. The author quite frankly shouldn't be bothered to stoop down to the level of the novice to explain such a concept. Put another way, it is a rhetorical gesture on the part of the author to incite the reader or student to action. The reader is implicitly asked by the author to make certain that this concept is obvious before moving on. (This is one of the reasons why "reading" mathematics is such a long and arguably exquisitely rewarding process since it practically self-enforces the habit of active reading ad nauseum.)
Setting aside the implicit nature of this call to action, my argument is that this seemingly harmless rhetoric in reality inculcates an inherent status of inferiority that places the reader/student in a lower position relative to the teacher/educator. Not to say that certain ideas are not to a certain extent "obvious" for the clearly learned professionals who report this as such from on high, but this concept of "it's so common sense that I don't need to explain it to you (implied, you idiot)" creates a certain moral sense of inadequacy for the student especially for those newly indoctrinated to the godspell of mathematics. On this latter point, for those simply trying to enjoy Mathematics for its own sake for those of us who suffered the American educational system version of canned, freeze-dried Mathematics concepts, I would argue that this rhetoric and intrinsic ideology of inadequacy actively prevents new learners from continuing to pursue mathematics for joy and leisure, especially since auto-didaction is such a common method of learning for adults attempting to (re)learn mathematics.
Now that all of this soap-boxy discourse is out of the way, ahem, this concept initially occurred to me while I was translating some of the introductory pages of C.F. Gauss's Disquisitiones Arithmeticae, one of his earlier written works on the topic of number theory. Gauss in his introductory concepts of the theory of modular numbers proposes that
Si plures numeri eidem numero secundum eundem modulum sunt congrui, inter se erunt congrui (secundum eundem modulum).
Haec modulorum identitas etiam in sequentibus est subintelligenda. (1)
The first sentence roughly translates as "If several numbers are congruent with respect to their second modulus for the same number, they will be congruent among themselves (the same second modulus)". This states that several numbers that share the same modulus relation also share that same modulus relation between themselves. (TODO, example). For example...
The second sentence is really the kicker. "This identity of moduli ought even to be obvious in the following (discussion)." Now, there is certainly some semantic room for argument here in that "Haec ... est subintelligenda" can perhaps be argued to be translated without a sarcastic kick of inherent inferiority, but I would like to argue that even Gauss here has unintentionally (or perhaps intentionally) propagated this harmful ideology to his readers. To be fair, he does suggest that this will be obvious "in sequentibus" (in the following), but notwithstanding this suggestion, it is still apparent that Gauss has explicitly stated that this concept only requires a "small amount of understanding" (subintelligenda).
This word subintelligenda is not the commonest of terms from Late Latin (or even classical Latin), and it is difficult to assert with utmost certainty that he means "little thought", but given the context of the passage being clearly an essay meant to help readers understand his argument, it is at least remotely within the realm of semantic possibility.
To break this idea down even more atomically, the word "subintelligenda" may be dissected into a few different components to fully parse this concept. "Sub" is a verb-prefix preposition meaning "under" quite literally (although often the prepositional prefix adds a certain "flavor" to the root verb rather than a stilted one-to-one idea like this fixed translation, but argumenti causa...). The verbal component "intelligenda" is a Latin gerundive of the feminine singular nominative variety, which often implies a sense of obligation or necessity. Thus, this can be translated quite literally as "This ought (and really should) be understood with little thought."
Now I don't know about you, but Gauss is in my estimations is one of the most lauded white men in the society of learned Mathematicians in the history of the western world, right up there with Newton, Leibniz, Bernoulli, Euler, and Le Grange. I mean, for crissake, he has his own cute little, apocryphal story about how to sum the integers from 1 to
As a result, we have a concise (even repeated several times in this one work) example from the history of mathematical technical composition that practically blatantly states that you are somehow below the threshold of requisite understanding for reading this book if you haven't figured this out yet. Is this the sort of rhetoric we want to engender on our readers and our audience when we write and talk about mathematics? Is this the sort of careful language that you want to be remembered by in your own mathematical scribbles? I'm not sure about you, but at least for me as admittedly a total novice in the vast realm of mathematics, I think we should be writing and thinking in a bit more of a positive and openly accepting manner rather than with this exclusive academic cipher of secrecy that only engenders inadequacy.