6 1. Logic and foundations statements about non-arithmetical functions (e.g., the logarithm function), and so forth. When trying to read and understand a long and complicated proof, one useful thing to do is to take a look at the strength of various key statements inside the argument, and focus on those portions of the argument where the strength of the statements increases significantly (e.g., if statements that were only known for a few values of a variable x, somehow became amplified into statements that were true for many instances of x). Such amplifications often contain an essential trick or idea which powers the entire argument, and understanding those crucial steps often brings one much closer to un- derstanding the argument as a whole. By the same token, if the proof ends up being flawed, it is quite likely that at least one of the flaws will be associated with a step where the statements being made became unexpectedly stronger by a suspiciously large amount, and so one can use the strength of such statements as a way to quickly locate flaws in a dubious argument. The notion of the strength of a statement need not be absolute, but may depend on the context. For instance, suppose one is trying to read a convoluted argument that is claiming a statement which is true in all dimensions d. If the argument proceeds by induction on the dimension d, then it is useful to adopt the perspective that any statement in dimension d+1 should be considered “stronger” than a statement in dimension d,even if this latter statement would ordinarily be viewed as a stronger statement than the former if the dimensions were equal. With this perspective, one is then motivated to look for the passages in the argument in which statements in dimension d are somehow converted to statements in dimension d+1 and these passages are often the key to understanding the overall strategy of the argument. See also the blog post [Go2008] of Gowers for further discussion of this topic. 1.4. Stable implications A large part of high school algebra is focused on establishing implications which are of the form “If A = B, then C = D”, or some variant thereof. (Example: “If x2 − 5x + 6 = 0, then x = 2 or x = 3.”) In analysis, though, one is often more interested in a stability version of such implications, e.g., “If A is close to B, then C is close to D.” Further- more, one often wants quantitative bounds on how close C is to D, in terms of how close A is to B. (A typical example: if |x2 − 5x + 6| ≤ , how close must x be to 2 or 3?)

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