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Well let me ask you this, then: what do you recommend as a way to learn Mathematica? I feel like I should know more about it. My interest is less in the details of using the software and more in its computational model and its approach to language design. Any suggestions on what a natural approach would be?

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Start with Mathematica's builtin documentation. The overview pages are actually quite good. They are also online [1].

Of course, given that the early drafts were written by Wolfram himself, you'll have to ignore absurd statements like this one: "Long viewed as an important theoretical idea, functional programming finally became truly convenient and practical with the introduction of Mathematica's symbolic language." [2]