161–170 of 186 posts
"He is like the fox, who effaces his tracks in the sand with his tail" (Abel about Gauss) https://hsm.stackexchange.com/questions/3610/what-is-the-ori...
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The target audience point is fair. There are basic concepts that I don't bother defining in any paper I write, and I'm sure there are people who would make the same argument about them that I'm making here. But I'm not suggesting writing everything like "An Extremely Gentle And Slow Introduction to X, With Lots of Reassurances That You Can Do It". The question probably comes down to how accessible a paper should be.…
I’ve started to keep a collection of readable papers (mostly engineering/experimental science, but a little bit of more theoretical work as well) that cover what I consider to be pretty foundational concepts, because additional citations are ~free and it at least gives an entry point for readers that are on the edge of the target audience. As a consequence, I’ve read papers that were written anywhere from the late 19…
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> This may require “watering down” the results being described — stating corollaries or special cases instead of the full theorems in their maximal generality. Sometimes you may even need to leave out technical conditions required for the results to really be true. This is a trade off, don't you think? Without the marketing, the paper would be more complete and correct.
No, you skipped the context, which says that in the introduction you talk about special cases to build intuition, and then provide the general result in the technical part.
My (admittedly grumpy) gripe is more that the aim of the blog is that "dull" is bad and suggests to add/subtract candid mathematics with "heroes" and "conflict". If the paper isn't _clear_, that's one thing, but "trick"ing the reader to spend more time on your article than they would without the embellishment is patronizing at best and disingenuous at worst.
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No, your exercpt is not complete and correct. "So, the introduction to a math paper should set the scene as simply as possible" The introduction is not the whole paper.
If I have a salad and I water down just the dressing the whole salad is still worse, no? Unless you are saying the introduction isn't important, in which case why would you need to change it at all to make the paper more engaging?
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If academic papers were published in HTML with living links on the web it would help. Hypertext solves this but are they not allowed to use modern (1990s+) technology?
URLs die. A good citation can be interpreted by technology to search and locate the referenced object.
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As someone who routinely teaches (Non-STEM) university students practical applications of math I can assure you that none of the students that told me "I am bad at maths" went out of my class without an understanding of the topics we talked about. Yet I routinely hear a: "Wow, if they told it to me this way during school I might have cared!" or "It really made my head smoke, but it felt good." Step 1 is believing tha…
This is exactly how my education went. Oh you're not engaged with my shitty teaching and forcing you to do endless abstract nonsensical formulae with zero contextualization or application? Then you're the problem - you have cognitive defects, you are plain dumb and stupid, you are disinterested, you don't care about your education etc etc so I cannot help you. Now let me spend my time on "teaching" the kids who mostl…
No it didn't. I am very confident it did not. All students that make it past grade school go through nearly the exact same sequence and motivating examples beginning from geometry, high school algebra, and then some introduction to calculus if not calculus outright.
And the motivating example has always been how to calculate area, and if advanced enough, volume, for all kinds of shapes, regular or otherwise. This is not a recent development, its been the motivating example and the entire purpose for calculus having been invented.
I am so confident in this that I am here to call you a terrible student and that, yes, you likely were a terrible student, especially since your characterization is the exact standard excuse I hear from terrible students.
So now I have to drill - what was "nonsensical"? What had "zero context"? You immediately have lost the claim of "zero application", so we don't have to go there.
If your claim is beyond K-12 math and into university, then I would be too upset to even argue with you if that is your actual position. Making these claims after more than a decade of learning math tells me you have trained yourself to forget all you learned.
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culture war aside, there are many other more accurate ways to say "details omitted for brevity" than "obviously" and "it is easy to see that" this is also something that makes me want a more interactive publishing format, though i understand the good reasons to stick to the static quo. if it's easy to see, it shouldn't be too hard to write out in a collapsible sidebar for those interested
Like everything in mathematics, "obvious" is a term of art. Broadly speaking, it refers to a fact, proof, consequence, which is necessary for the proof to advance, but which is already established elsewhere, so it does not in itself aid in understanding the proof being presented. A proof is either providing a new result, or is proving an established result in a new way. Almost always, a proof will need other results,…
"Trivial" is a term of art (and doesn't mean exactly this) but I'm not sure that "obvious" is.
I think that something already proved is called an immediate corollary if it's a very small step, theorem if it's not that small but well known, a previous result (with a citation!) if not well known, or if you can't cite anything, at least say it's part of "mathematical folklore".
The author may well be on to something but personally I hate "story mode" in popular science books, and would really hate it in actual science papers, both the ones I read for work and the ones I read for fun. I want to go straight to the equations -- often I prefer them to the graphs. But (not joking here) this is a perfect opportunity for an LLM -- two opportunities, actually. LLM A takes a dry paper and gives it c…
I think John Baez agrees with you. At the conclusion he says,
> The ideas here take practice to implement well, and they should not be overdone. I’m certainly not saying that a good math paper should remind readers of a story. Ideally the tricks I’m suggesting here will be almost invisible, affecting readers in a subliminal way: they will merely feel that that paper is interesting, carrying them in a natural flow from the title to the conclusion.
The best math book I've ever read -- which I think can completely transform somebody's appreciation of math -- was William Dunham's "Journey Through Genius - The Great Theorems of Mathematics." What this book did was place mathematics in human and historical context. It starts with Hippocrates' Quadrature of the Lune, then moves on to Euclid's proof of the Pythagorean theorem, and moves along through history all the…
That's because it's the only book I know which is in a good sweet spot between being a true pop-math book, giving the history and context of math (kind of like, say, Fermat's Enigma by Simon Singh), but while also being a real math book, and actually teaching real maths and real proofs of all the theorems talked about. There are some similar books, but most don't get the mix right, and even the ones that do, are just not as good.
Such a wonderful wonderful book. Do you have any other recommendations for similar books?