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Now we can see that Church was associated with the Simply Typed Lambda Calculus. Indeed, it seems he explained the Simply Typed Lambda Calculus in order to reduce misunderstanding about the Lambda Calculus.

Now when John McCarthy created Lisp - he based it on the Lambda Calculus. This is by his own admission when he published "Recursive functions of symbolic expressions and their computation by machine, Part I". You can read it here.

Now we know that at the core of Mathematica is a Lisp-like system, but instead of being based purely on the Lambda Calculus, it is based on a term-rewriting system.

Here the author states state:

Mathematica is fundamentally a term rewriting system... a more general concept than the Lambda Calculus behind Lisp.

My question is: Why did the Mathematica Language choose term rewriting instead of the Lambda Calculus as its basis?

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    $\begingroup$ I think you will have to ask Stephen Wolfram to get an answer that is not an opinion. $\endgroup$ – m_goldberg Jul 3 '16 at 15:38
  • $\begingroup$ @m_goldberg maybe Stephen Wolfram is a member of this site? $\endgroup$ – QuantumDot Jul 3 '16 at 16:32
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    $\begingroup$ I believe the reference to "terrm rewriting" is actually intended to be taken as an emulation of "infinite evaluation". It does not really mean rewriting in the sense that that is supported by replacement rules (which is an important part of Mathematica, but hardly qualifies as a single defining principle). $\endgroup$ – Daniel Lichtblau Jul 3 '16 at 16:55
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    $\begingroup$ I am a bit annoyed that it was closed; a link to somewhere Wolfram was explaining his design choice would have been an admissible answer. $\endgroup$ – J. M. will be back soon Jul 4 '16 at 3:36
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    $\begingroup$ Here is a related discussion: "Why Wolfram Mathematica did not use Lisp (2002)". $\endgroup$ – Anton Antonov Jul 7 '16 at 1:59
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The short answer is that the Mathematica Language did not choose anything. It was Stephen Wolfram back in 1979 when he started working on his own system after he reached the boundaries of Maxima which was his program of choice then.

There is a very interesting talk about this which is called How Mathematica, Wolfram|Alpha & the Wolfram Language Came to Be. In this talk, he described some of the reasons why he designed it the way he did. You might want to watch it from minute 24 when he talks about Algy - the algebraic manipulator which later became SMP and finally Mathematica.

Here is the probably most related part, freely transcribed by myself:

I knew most of the general-purpose Algol-like languages and as well as languages like Lisp and APL and so on at the time, but somehow they didn't seem to capture sort of the things that I wanted my system to do. So I guess what I did was what I learned to do in physics which was I tried to sort of drill down to find kind of the atoms; the primitives of what was going on in all these computations that I wanted to do.

I knew a certain amount about mathematical logic and the history of attempts to formulate things using logic and so on, even if my mother's textbook about philosophical logic didn't exist yet, but the history of all the effort of formalization I was quite aware of through Leibnitz, Hilbert, [...]

Back in 1979, I was sort of thinking about this kind of thing and that led me to design the design that I came up with that was based on the idea of symbolic expressions and doing transformations on symbolic expressions.

This all does not sound to me as it was an active decision to create a term rewriting system but rather, Wolfram wrote down the specifications of how he thought an expression manipulator should be designed. When we look at it now, it seems clear that it is of course a term-rewriting system, but maybe it wasn't so clear back then.

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    $\begingroup$ I wrote this answer because I remembered this specific part of the talk. It surely cannot be the only place where Wolfram says something about the creation of Mathematica. If there are other sources and you don't want to write up your own answer, feel free to edit this one. $\endgroup$ – halirutan Jul 3 '16 at 16:24
  • $\begingroup$ "Wikipedia seems to think these days that the most famous people from my class in elementary school that year are myself and an actor called Hugh Laurie". Interesting to know. $\endgroup$ – Emilio Pisanty Jul 3 '16 at 18:40
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A somewhat ahistorical or anachronisic take on "why" is given in Bruno Buchberge's article Mathematica as a Rewrite Language, which shows what can be done with term rewriting as developed within Mathematica. In particular,

... this gives a versatile and practically attractive potential for providing all the quantifiers one would want to see in a truly mathematical language.

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