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This is a bit of a soft question, but are undergraduate math courses and textbooks more rigorous than graduate math courses and textbooks? In my experience so far, undergraduate textbooks prove every detail, or at least more detail than graduate textbooks. For example, an undergraduate real analysis textbook might prove that for any two distinct real numbers, there is a real number strictly between them, but in a graduate real analysis textbook, this and similar details would be taken for granted. So, then, are undergraduate math courses and books more rigorous than graduate courses and books? I apologize if this is not the right stack exchange for this question. If not, it could be moved to an appropriate SE, like maybe Academia SE.

user107952
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    One must take in to account the fact that if every single little detail is proven, this will clutter and obscure the main ideas. Since graduate students are expected to have some level of mathematical maturity, many little details are "left to the reader" to verify so that the text does not get bogged down with minutia and rather focuses on the "more important" ideas. – morrowmh Feb 17 '22 at 01:40
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    This blogpost of Prof Tao answers your question (or at least, a closely related one) far better than I could https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ – DanLewis3264 Feb 17 '22 at 01:42
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    Different textbooks are written for different audiences. I know many grad level textbooks filled to the brim with proofs, and I know many undergrad textbooks that have no proofs at all. – Spencer Kraisler Feb 17 '22 at 01:43
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    I think you're confusing the ideas of "more detailed" and "more rigorous". It's entirely possible for a text to be detailed but not rigorous (see certain notes in algebraic topology, for instance, which leave a lot of proofs to geometric intuition, albeit with lots of description) or to be entirely rigorous, yet light on details (see many books on category theory, which leave a lot of "diagram chases" as exercises for the reader). – HallaSurvivor Feb 17 '22 at 01:46
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    Yeah, undergrad books know you are new to proofs, and are as much teaching you the nature of proofs in general, not just the subject at hand. So an undergrad proof won’t shorthand stuff. Grad student books assume you know the usual shorthands and how to fill in steps. – Thomas Andrews Feb 17 '22 at 02:07

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"Rigor" is not synonymous with "provide every detail in every proof".

Rigor should be sufficient to convince a student who has the maturity to be studying this material of the correctness of the arguments. When the reader is properly prepared, more details can be left to the reader.

Ethan Bolker
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