The algebraic ellipsis and its consequences have been a disaster for the human race

The algebraic ellipsis (" . . . ") and its consequences have been a disaster for the human race. They have greatly increased the convenience of those of us who write in “simple” notation, but they have deformalized mathematics, have made algebra unfulfilling, have subjected math students to indignities, have led to widespread academic non-rigorousness (in the algebra II classroom to learning deficiencies as well) and have inflicted severe damage on the mathematical world. The continued use of ellipses will worsen the situation. The algebraic ellipsis has no formal definition. It is a tool Satan uses to tempt mathematicians to simplify their proofs, at the cost of selling their souls for non-rigorousness. The more one finds convenience in using the ellipsis, the more damning its use becomes: surely one would know that the more complicated the algebra, the more important it is to showcase it in its entirety?

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Lets examine some cases where one may use an algebraic ellipsis, and how it should have been written.


Here one is using the ellipsis to imply that there is a sum of numbers in a sequence. What is that sequence? Obviously its just the sequence of natural numbers, but that does not make it mathematically rigorous. One should assume a computer with no pattern recognition is reading the math. Here's how one should showcase this series properly:

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This one is even more disgusting. Repulsive, even. First, one must come to assume the bottom sequence is the sequence of primes. Second, one must pick up on the fact that the index is just the natural numbers. Look how much more beautiful product notation makes this:

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Okay—this is a bit of a strawman. Most would opt to represent this set using exponents. Still, we were blessed with set-builder notation, and we ought to use it.

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So, why do we still use the algebraic ellipsis? For convenience? When has math ever been about convenience? Aside all of applied "math", that has never been the case. We do these things not because they are easy, but because they are hard. For we have an obligation to bring about the truth in all things: an obligation to burn away each and every shadow in this labyrinth that is the universe. And if it be necessary to do away with the ellipsis to achieve this, so be it.

-PL9050