Lambda the Ultimate

It would be great even with the language barrier. We can always setup a Wiki and find people willing to translate the chapters. I can do some of the technical work and setup things, but Hebrew is a complete mistery to me.

Well, if someone volunteers to do the translation, I am willing to try to dig up what I wrote. Some of the stuff was written on very obscure Hebrew word processors so just handling the file formats is going to be an issue. As far as I remember some chapters were complete, where as others are only stubs, so maybe people here will want to improve them...

I agree completely. The removal of long division from the classroom completely misses its actual educational value.

(a) what you wrote is interesting and makes me think that we have a layer cake of computation and pattern matching. one learns to recognize the characters 0..9 via pattern matching. then one learns to compute that 3 oranges + 2 oranges = 5 oranges. then one learns to pattern match e.g. by memorizing multiplication tables. so then one can compute something more advanced, but not pattern match it until much practice has happened. etc.

(b) there can be different approaches to computation. a calculator might from an external perspective easily calculate either of what you wrote. but a human given enough time can as well. it just isn't as fast in each case. that doesn't prove no computation. in fact to me it proves computation because we are failing to pattern match the latter in some sense.

(c) the fact that given enough time we can compute the answer to 123456789 + 987654321 means that we can do computation.

?

That's why we can easily compute 1000000 + 1000000 but not 123456789 + 987654321, for example.

I think that's just an artifact of the base 10 number system we use. Given another base, 123456789 + 987654321 might be just as easy for us to add as 1000000 + 1000000 in base 10.

The references escape me, but there has been quite a deal of experimental and modeling research into this concrete question of addition of big, hairy numbers in the cognitive science community. The essential point probably does not veer far from what you do in practice: pick fence posts (big number, really big number, etc) and put those together. I think James Anderson has written a bit about it as part of his Ersatz project.

123456789 + 987654321 is not much harder than 1000000 + 1000000. Even if it was, that doesn't mean that the brain does not compute. It just means that the brain has different primitive operations.

... but we can compute 123456789 + 987654321 quite easily. All you have to do is notice that each pair of digits sums to 10 and the answer has to be 11111111110.

But it is not obvious whether that supports or contradicts your point. If you take sequences of pattern matching, where we can choose what to do next based on the previous pattern then don't we have a simple model of computation?

...though I doubt Endo's going to spring fully-formed from our brains any time soon!

Repeatedly matching and rewriting is a form of computation.

What about complex adaptive systems? Swarm intelligence? Biological computation? These approaches are a huge part of computational "thinking" in the natural world. They should be studied by anyone interested in computation, especially beginners.

An example I first heard from Chris Langton: ask some young children to make a circle on the playground around a point. How do they "compute" the outline of the circle? Perhaps by holding hands and stepping back until theirs arms are fully stretched. That's a pretty good solution compared to use of graphs, trig, or complex roots.

With the coming "problem" of too many CPU cores and no way to program them, these alternative computational strategies will be even more important.

By a unifying theory I was thinking of something that works for computations in any form, whether analog or digital, biological or mechanical. Dynamic programming actually has a history of doing that. In its early days it was applied mainly to control systems, and now days, judging from the wikipedia, it is used in the more familiar computing with numbers setting.

In my thinking at least it is a fundamental law, that helps to derive various models of physicality, and thereby constrain the computation to familiar semantics.

Edit, Clarification: Prograqmming is often not so much about the program but the algorithm. Algorithm design is an overlooked part of the programming puzzle that has been studied extensively.