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I've personally tripped up on a few concepts that came down to an abuse of notation, and I've read of plenty more on stack exchange. It seems to all be forgiven with a wave of the hand. Why do we tolerate it at all?

I understand if later on in one's studies if things are assumed to be in place, but there are plenty of textbooks out there assuming certain things are known before teaching them. This is a very soft question, but I think it ought to be asked.

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    Perhaps you can elaborate on how you define "abuse of notation": is it when notation is introduced, but not explained or defined (i.e., assumed to be understood)? or do you mean when unconventional notation is used in place of what is standard? Or both. Examples would help.2012-12-24
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    Sometimes, good notation doesn't exist; I've even heard it said that in some cases, simply coming up with good notation for something can be an important mathematical advance. Alas I can't find a reference.2012-12-24
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    @Hurkyl Maybe this: "The invention of the symbol $\equiv$ by Gauss affords a striking example of the advantages which may be derived from an appropriate notation, and marks an epoch in the development of the science of arithmetic."? (G. B. Matthews in "Theory of Numbers", 1892)2012-12-24
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    We accept it because no one is a notation-Lincoln to free the notation from the horrible context they live in which allows us to abuse it endlessly to our great pleasure!2012-12-24
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    One abuse that obfuscates, serves no one and should be eradicated immediately is the awful use of ${\cal L}\{f(t)\}$ for the Laplace transform. Just write ${\cal L}f$.2012-12-25
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    Oh I remember how as a freshman I was taking notes on math related lectures and was trying to purify math so that I have no words at all there except titles, names and some minor comments and it should be as compact as possible. Now I can make a 10 page theorem proof in to one page so that I can see a full picture just with one glance, it helps a lot.2012-12-25
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    @copper.hat Agree, but then we need to rewrite the tables of transforms so that they have $\mathcal{L}\{t\mapsto \sin t\}$ instead of $\mathcal{L}\{\sin t\}$ etc.2012-12-26
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    @PavelM: Or just $\mathcal{L}\{\sin\}$ :-). It is too entrenched to change, but if I was to pick one notational abuse that I have seen students stumble over, it is the distinction (or lack thereof) between a function and its evaluation.2012-12-26
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    @copper.hat Neat, but would not work for $\mathcal{L}\{1/(t^2+1)\}$. Yes, throughout the calculus/differential equations sequence students are held back by insufficient understanding of the concept of a **function**. Having notation that blends functions and algebraic expressions together does not help. Maybe this is where computer algebra systems could actually help, because they are less tolerant to notational abuse. In Maple, y:=x^2 and y:=x->x^2 are different things.2012-12-26
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    @Korgan I have been in the same boat as you are and have tried to use completely correct and well-defined notation as long as possible. Believe me, in a course as simple as basic analysis, I was not able to proceed much further than 6-7 sections without having to write stuff too much tediously. I instead gave up and adopted the loose notations.2013-05-10
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    Note that different people have different levels of tolerance for abuses of notation. For example, I'm a fan of leaving off the domain of quantification, but including the quantifier. E.g. writing $\forall x.\sin^2(x)+\cos^2(x)=1$ (slightly abusive) to mean that $\forall x \in \mathbb{R}.\sin^2(x)+\cos^2(x)=1$ (formal), which is often just written $\sin^2(x)+\cos^2(x)=1$ (maximally abusive).2014-06-05
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    Ha. Just write $\sin^{2} + \cos^{2} = \mathbf{1}$.2018-07-06
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    It reminds me of this answer: https://math.stackexchange.com/questions/1093696/is-arrow-notation-for-vectors-not-mathematically-mature/1093725#10937252018-11-29

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