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I both agree and disagree. On one hand I disagree because mathematics is abstraction built on top of abstraction many times over. It can be incredibly dense wi

by vlasev 12y ago

I both agree and disagree. On one hand I disagree because mathematics is abstraction built on top of abstraction many times over. It can be incredibly dense with information. An example comes to mind - Stokes' theorem. It's an amazingly elegant piece of math. It can be stated very neatly with 6 symbols - the integral symbol, omega, d omega, Omega, del Omega and an equal sign. If you were to unravel what each of the symbols meant using the closest terms related to them and then did the same for those terms and you continued some more, you'll have a small book. This kind of thing is paralled in programming where you rarely want to just push bits around. I imagine you as a person who specializes in assembly complaining about the lack of rigor in more abstract languages. However I agree with you because the above stuff is sometimes just an excuse for writing terribly. Firstly, there are practical considerations about the sheer length of text. Then you have problems of explaining. A professor writing a textbook can be far detached from that past when they first learned the stuff. My own teachers always told me to never write "obviously" unless I also explain why it is obvious (It can be very useful to explain to the student what needs to be obvious and why). The best solution I can think of is writing with the online world in mind. A mathematical paper does not need to be short and concise. I envision two versions - the long online version and the shorted published version. There are after all practical and physical considerations in the latter while the former shouldn't be a big deal. In this way whoever is interested in more than the overview can access the longer online version and learn from it. The biggest problem I see in this scheme is that of laziness - not everyone will want to write long texts.

grovulent 12y ago
Nicely said. Bear in mind though that logic books and their proofs generally rely on a 'stack' as well. You prove a theorem and then you're able to use that theorem in your proofs going forward - with the theorem itself being used as the justification. From a pedagogical standpoint - the act of writing that justification is enormously valuable since the book will also have an index where you can quickly look up a theorem if you've forgotten the basics of it. Perhaps the difference is more in the fact that maths textbooks will often rely on theorems proved in other mathematical contexts (where it doesn't make sense to provide the proof in the context currently being explored). The problem being that there can be no reliable way to index that information. Logic textbooks, however, present systems that are much more self-contained (if not completely).
- mikevm 12y ago
  When I took Logic for CS we were taught the Natural Deduction proof calculus, which mirrors the kind of proofs we do in maths. We had an exercise during the course to write out which ND inference rules were used in every step of some natural language math proof, and it really shows you how one sentence can pack 10 or more inference rules if you try to unpack it! He also put a similar question on the test which had you first prove a theorem using natural language, and then you had to find the use of ND inference rules in your proof (that wasn't easy... even though you wrote the proof yourself).

by vlasev 12y ago

grovulent 12y ago

Nicely said. Bear in mind though that logic books and their proofs generally rely on a 'stack' as well. You prove a theorem and then you're able to use that theorem in your proofs going forward - with the theorem itself being used as the justification. From a pedagogical standpoint - the act of writing that justification is enormously valuable since the book will also have an index where you can quickly look up a theorem if you've forgotten the basics of it. Perhaps the difference is more in the fact that maths textbooks will often rely on theorems proved in other mathematical contexts (where it doesn't make sense to provide the proof in the context currently being explored). The problem being that there can be no reliable way to index that information. Logic textbooks, however, present systems that are much more self-contained (if not completely).

mikevm 12y ago
When I took Logic for CS we were taught the Natural Deduction proof calculus, which mirrors the kind of proofs we do in maths. We had an exercise during the course to write out which ND inference rules were used in every step of some natural language math proof, and it really shows you how one sentence can pack 10 or more inference rules if you try to unpack it! He also put a similar question on the test which had you first prove a theorem using natural language, and then you had to find the use of ND inference rules in your proof (that wasn't easy... even though you wrote the proof yourself).