[catnaroek]

> because computation is not (classical) math

Of course, computation is more foundational. It's mathematics that's just applied computation.

> as it does not preserve equality under substitution

You just need to stop using broken models.

> but the recognition that their entire justification is not mathematical but pragmatic

I don't see a distinction. To me, nothing is more pragmatic to use than a reliable mathematical model.

> the (leaky) abstraction of those models

Other than the finiteness of real computers, what else is leaky? Mind you, abstracting over the finiteness of the computer is an idea that even... uh... “less mathematically gifted” languages (such as Java) acknowledge as good.

> such as empirical results showing a certain "affinity" to human cognition

Experience shows that humans are incapable of understanding computation at all. But computation is here to stay, so the best we can do is rise to the challenge. Denying the nature of computation is denying reality itself.

[pron]

> You just need to stop using broken models.

No computation preserves equality under substitution. If your model assumes that equality, it is a useful, but leaky abstraction.

> Other than the finiteness of real computers, what else is leaky?

The assumption of equality between 2 + 2 and 4, which is true in classical math but false in computation (if 2+2 were equal to 4, then there would be no such thing as computation, whose entire work is to get from 2 + 2 to 4; also, getting from 2+2 to 4 does not imply the ability to get from 4 to 2+2).

> Experience shows that humans are incapable of understanding computation at all.

Experience shows that humans are capable of creating very impressive software (the most impressive exemplars are almost all in C, Java etc., BTW).

[catnaroek]

> The assumption of equality between 2 + 2 and 4, which is true in classical math but false in computation

Using Lisp syntax, you are wrongly conflating `(+ 2 2)`, which is equal to `4`, with `(quote (+ 2 2))`, which is obviously different from `(quote 4)`. Obviously, a term rewriting approach to computation involves replacing syntax objects with syntactically different ones, but in a pure language, they will semantically denote the same value.

Incidentally:

0. This conflation between object and meta language rôles is an eternal source of confusion and pain in Lisp.

1. Types help clarify the distinction. `(+ 2 2)` has type `integer`, but `(quote (+ 2 2))` has type `abstract-syntax-tree`.

> very impressive software

For its lack of conceptual clarity. And for its bugs. I'm reduced to being a very conservative user of software. I wouldn't dare try any program's most advanced options, for fear of having to deal with complex functionality implemented wrong.

[pron]

> Using Lisp syntax, you are wrongly conflating `(+ 2 2)`, which is equal to `4`

It is not equal to 4; it computes to 4. Substituting (+ 2 2) for 4 everywhere yields a different computation with a different complexity.

> but in a pure language, they will semantically denote the same value.

The same value means equal in classical math; not in computation. Otherwise (sort '(4 2 3 1)) would be the same as '(1 2 3 4), and if so, what does computation do? We wouldn't need a computer if that were so, and we certainly wouldn't need to power it with so much energy or need to wait long for it to solve the traveling salesman problem.

> For its lack of conceptual clarity. And for its bugs.

That's a very glass-half-empty view. I for one think that IBM's Watson and self-driving cars are quite the achievements. But even beyond algorithmic achievements and looking at systems, software systems that are successfully (and continuously) maintained for at least a decade or two are quite common. I spent about a decade of my career working on defense software, and that just was what we did.

[catnaroek]

If you can't distinguish object from meta language, I'm afraid we can't have a reasonable discussion about computing. This distinction is crucial. Go get an education.

[pron]

If you don't understand what I'm saying -- and that could be entirely my fault -- you can just ask. If you (mistakenly) assume that by 2 + 2 I mean the expression "2 + 2" rather than the computation 2 + 2, why not assume that you may have missed something (which is the actual case) rather than assume that I don't understand the basics (which is not)?

Since I don't wish to discuss this topic further with rude people, but I do wish to explain my point to other readers, I'll note that the entire concept of computational complexity, which is probably the most important concept in all of computer science (and is at the very core of computation itself -- there can be no computation without computational complexity), is predicated on the axiom that in computation 2+2 does not equal 4 (in the sense that they are "the same"), but is computed to be 4. If 2+2 were actually 4, there would be no computational complexity (and so no computation).

As a matter of fact, an entire model, or definition of computation (another is the Turing Machine) called lambda calculus is entirely based on the concept that substitution is not equality in the theory of computation, by defining computation to be the process of substitution (which is what lambda calculus calls reductions). If 4 and 2+2 were the same (as they are in classical math), there would be no process, and the lambda calculus would not have been a model of computation but simply a bunch of trivial (classical) mathematical formulas.

Indeed, some people confuse the LC notation with classical mathematical notation (which it resembles), and mistakenly believe that 2+2 equals 4 in LC in the same sense that it does in math (I assume because the same reductions preserve equality in math). This is wrong (in LC reductions do not preserve "sameness" but induce -- or rather, are -- computation). To their defense, LC does make this fundamental distinction easy to miss in hiding 100% of what it is meant to define -- namely, computation -- in operations that classical mathematicians associate with equality[1], and in itself does not have a useful formulation of complexity[2]. Nevertheless, those people might ignore computational complexity, which is the same as ignoring computation itself, and while they may turn out to be great mathematicians, you would not want them specifying or writing your traffic signal or air-traffic control software.

[1]: Although I believe most notations take care to not separate consecutive reductions with the equal sign but with an arrow or a new line, precisely to signify that reduction is not equality. Also, unlike in math, LC reductions are directional, and some substitutions can't be reversed. In this way, LC does directly represent one property of time: it's directionality.

[2]: The challenge complexity poses to LC is great, and only in 2014 was it proven that it is not just a model of computation but one of a "reasonable machine": http://arxiv.org/abs/1405.3311

[tel]

Computation is something different. Models like call by push value make this very clear. LC does as well, though, but LC tends to be joined up with an equality semantics which intentionally sweeps computation under the rug for simplicity.

This is a big hairy problem in untyped LC, though, since untyped LC has non-termination and therefore is not confluent. This is what I mean by taking non-termination seriously is one way to force "time" and "computation" back into models. It means that LC has no beta-equivalence the same way that, say, simply typed LC does.

So anyway, you're wrong to say that LC has no notion of complexity—people count reduction steps all the time—but right to say that often this is intentionally ignored to provide simpler value semantics. It's foolish to think of this as equivalent to LC, though.

This paper is interesting. I think what they prove was at least folk belief for a long time, but I've never seen a proof.