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Consider a person reading a mathematical proof, then each syllogism from it's antecedent maybe understood by that person, yet they may find it difficult to understand the whole proof. At times however, a person maybe able to understand the proof as a whole, and in that case, provided they know the proper fundamental deductions, can deduce individual steps of the proof from it.

I hope the reader relates to this example I gave. In which case, I ask, why exactly is it difficult to move from understanding of the fragements to understanding of the whole? What is the missing piece?

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    People do not understand a proof by dissecting deductive steps of its formalization, at best, this only constitutes its verification. Semantic structures that are essential to human understanding of a proof, such as key ideas, their relation to known results and established techniques, etc., are often suppressed and obscured by formalization. See SEP, Mathematical explanations within mathematics.
    – Conifold
    Apr 18 at 6:58
  • What happens in maths is that there are many very precise definitions and prior theorems to keep in mind, so usually a good memory is required. And as Conifold says, sometimes some of those prior definitions and theorems may not be stated explicitly in a given proof, but assumed known by the reader, which compounds the effect.
    – Frank
    Apr 18 at 13:51
  • I found it very difficult to answer this question in the context of mathematics. The physical analogy is "How does understanding atoms differ from understanding water?
    – user64314
    Apr 18 at 15:30

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