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I have lived my academic career thinking these two terms were interchangeable but something arose in me that said they're distinct. Are there any logical, formal definitions distinguishing between the two?

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    $\begingroup$ I would not classify numeric computation as symbolic manipulation. $\endgroup$
    – user16034
    Commented Aug 28, 2022 at 10:01

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I can only speak for me (and maybe my interpretation is also caused by a different understanding of the English language), but I would say

  1. Computation: That's numerical. Solving a problem by calculation with numbers.
  2. Symbolic manipulation: That's like analysis. Solving the problem by e.g. term transformation, if I say, I get the solution b, by transforming a + b = c to b = c - a.
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