List out of Lambda (opens in new tab)

I have a degree in mathematics from the University of Chicago. Want to talk about self-adjoint algebras of compactly supported continuous functions? I'm your man.

I also think teaching this using mathematical notation would be a mistake. People only learn things when they relate in part to something they already know. The original lambda calculus syntax is not only abstract, it was invented before programmable computers existed.

Unless someone is either already comfortable with mathematical notation or has a background in a programming language with similar concepts (Lisp, Haskell, etc.), it's going to be a stretch to teach someone the lambda calculus -- really teach them -- unless it's framed in terms of a language they already know.

JavaScript is a great language to use for this because it has C-like syntax, which virtually all programmers understand, and first-class functions. You could use Ruby, too, but the syntax would be even more opaque.

Here's someone building the lambda calculus using only Procs in Ruby: http://codon.com/programming-with-nothing

Then using Ruby 1.9's "stabbby lambda syntax" you get

    -> x { -> f { f[x] } }

Not sure that's any better. The key is explaining it in terms of something the ready already understands well.

If they're not already thinking mathematically a straight-forward introduction to the lambda calculus would be ineffective.

The point is not that you can represent it better. The point is that you can actually run it. You can't run the original notation (as such, we know languages that are close, of course).

I agree that the terser notation is better if you're just considering the mathematics in isolation. However, I also think that the true power of this kind of mathematics isn't made evident to people until you actually add some integers and make some lists.... from "nothing"... in a language you already know.

It's the same reason he doesn't use lisp or haskell, which are, obviously, vastly more appropriate. The idea is to illustrate something using something people have readily at hand. Once you have the concrete representation of the idea, you can work backwards to whatever notation you desire and forwards again using that notation, understanding that, yes, it has a basis in reality.

It's the same reason people draw 2d graphs to represent machine learning concepts instead of solely relying upon linear algebra notation. Humans are limited in that way: we have to gain some kind of appropriate intuition, and then we can work with the abstractions after that. No one is going to argue that js is an appropriate way to develop the mathematics -- it's just a way to make it more concrete for people, that's all.

Agreed. I can't understand how something like

    function(x) {return function(f) {return f(x);};}

can be considered easier to comprehend than

    λx. λf. f x

Losh doesn't use Church numerals, though, he represents numbers as lists. The fact that lists are also represented as functions is a diversion; his number three doesn't represent a function f -> x -> x that returns the threefold composition of its first argument applied to its second.

For anyone curious of the λ-calculus, and why the Y-Combinator plays an important role in the turing completeness of this, please check out the excellent Y-not series by Jim Weirich[0] [0] http://www.neo.com/2012/11/13/y-not-adventures-in-functional...

All you need is S and K - all the rest is syntactic sugar, even Y.

:-)

And for a pretty nice demonstration of (untyped) lambda calculus, http://codon.com/programming-with-nothing

> Turing-completeness, of course.

Lots of useful programs can be written without turing completeness. It can be useful, but only for a limited problem domain.

"some measure of support" like a way of interacting with the outside world. The pure SKI calculus is Turing complete, but no amount of library code (also written in the pure SKI calculus) or syntactic sugar for expressing your favorite paradigm will let you write cat(1) in it.

If you're soft real-time so that performance is a correctness issue you need more than Turing-completeness. I don't know off the top of my head if there's a name for a constant-slowdown universal computer or if such a thing is possible.

This celebrated discussion goes into some nice detail about object-closure equivalence:

http://people.csail.mit.edu/gregs/ll1-discuss-archive-html/m...

If you're too impatient, here's the punchline:

  The venerable master Qc Na was walking with his student, Anton.  Hoping to
  prompt the master into a discussion, Anton said "Master, I have heard that
  objects are a very good thing - is this true?"  Qc Na looked pityingly at
  his student and replied, "Foolish pupil - objects are merely a poor man's
  closures."

    Chastised, Anton took his leave from his master and returned to his cell,
  intent on studying closures.  He carefully read the entire "Lambda: The
  Ultimate..." series of papers and its cousins, and implemented a small
  Scheme interpreter with a closure-based object system.  He learned much, and
  looked forward to informing his master of his progress.

    On his next walk with Qc Na, Anton attempted to impress his master by
  saying "Master, I have diligently studied the matter, and now understand
  that objects are truly a poor man's closures."  Qc Na responded by hitting
  Anton with his stick, saying "When will you learn? Closures are a poor man's
  object."  At that moment, Anton became enlightened.

I'm having to reach back into the dustbin of my mind, but I seem to recall implementing an object system using only closures at one point. As I recall, it had all of the "normal" features of objects - inheritance, member variables, methods, etc. It was done with closures and the members were accessed in a message-passing style (in scheme, (myobject 'show) for example).

Based on that, I think that closures and objects are probably equivalent in their expressive ability. Could be wrong though, like I said it was a long time ago.

You could have run with a one-point basis: https://en.wikipedia.org/wiki/Combinatory_logic#One-point_ba...

Or just two integer variables, from which you can substract one, add one and compare to zero.