1.6. Abstraction 9 x 5 is preferable to 5 x, despite the logical equivalence of the two statements. This is why analysts make a significant distinction between “upper bounds” and “lower bounds” the two are not symmetric, because in both cases one is bounding an unknown quantity by a known quantity. In a similar spirit, another convention in analysis holds that it is preferable to bound non-negative quantities rather than non-positive ones. Continuing the above example, if the known bound Y is itself a sum of several terms, e.g. Y1+Y2+Y3, then it is customary to put the “main term” first and the “error terms” later thus, for instance, x 1 + ε is preferable to x ε + 1. By adhering to this standard convention, one conveys useful cues as to which terms are considered main terms and which ones considered error terms. 1.6. Abstraction It is somewhat unintuitive, but many fields of mathematics derive their power from strategically ignoring (or abstracting away) various aspects of the problems they study, in order to better focus on the key features of such problems. For instance: • Analysis often ignores the exact value of numerical quantities, and instead focuses on their order of magnitude. • Geometry often ignores explicit coordinate systems or other de- scriptions of spaces, and instead focuses on their intrinsic proper- ties. • Probability studies the effects of randomness, but deliberately ig- nores the mechanics of how random variables are actually gener- ated. (This is in contrast to measure theory, which takes the com- plementary point of view see [Ta2011c, §1.1] for further discus- sion.) • Algebra often ignores how objects are constructed or what they are, but focus instead on what operations can be performed on them, and what identities these operations enjoy. (This is in contrast to representation theory, which takes the complementary point of view.) • Partial differential equations often ignore the underlying physics (or other branches of science) that gives rise to various systems of interest, and instead only focuses on the differential equations and boundary conditions of that system itself. (This is in contrast to, well, physics.)

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