I noticed that not a lot of authors of math books don't use the $\leadsto$ to denote the next step. As far as I understand it, that is supposed to mean "the next step of the proof is". Instead, I notice that they use $=$ to mean the same thing, or also $\implies$ or $\iff$. Which symbol is the most appropriate for stating the next step? Can we use $\implies$ and $\leadsto$ interchangeably? Is this accepted in formal proofs?
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@kimchilover partially, yes, but it's not clear if we can interchangeably use $\leadsto$ with $\implies$...it seems to suggest that $\leadsto$ can be used, but also $\implies$ provided that it is used with a condition and somewhere else in the proof that the condition exists. I'd still like to know what would be much clearer to a reader who reads proofs – Paco G Mar 05 '20 at 00:03
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Mathematics is about understanding, not loads of symbols. Use words. Most often just placing one equation after the other, perhaps with a few well-placed comments interspersed, is much easier to read.
Don't introduce new notation if you can help it. Yes, it might sometimes be worthwhile to introduce new notation, for a rather complex concept that repeats often enough (e.g. Landau's $O()$, later extended by Knuth to $\Omega()$, $\Theta()$, and so on). "Next step is..." just doesn't cut it.
vonbrand
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I don't completely agree this in the sense that using words doesn't necessarily imply better understanding. The problem with words and phrases is that they are also used in everyday English and their meaning in that context is close to but not equivalent to their meaning in math. This undermines the need to be logically rigorous in math. My current thought (untested) is that using symbols initially forces students to learn exact meanings of mathematical sentences. There does have to be a transition to using words, but I'm not sure it is what we should start with. – Deane Jan 06 '24 at 17:18