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My great uncle Sean passed away in '94 after a long battle with illness. Not many people came to his funeral; he was 105 when he died, and had outlived all of his friends and colleagues.

I'm told he was once a respected mathematician, but had since faded into obscurity. Even his most important work is now remembered by another's name. The last forty years of his life were spent as a recluse.

When he died, he had few possessions. There was no will and not much in the way of inheritance anyway; what little money there was went to help pay for the funeral. But he did will me his notebooks.

I'm not sure why. I suppose he had taken a liking to me, saw in me a kindred spirit. Most of his notebooks were full of mathematical formulae that were well beyond my capability at the time. The rest seemed to be records of his various musings: these seemed to grow increasingly incoherent and mystical as they progressed toward the end of his life.

To my young mind, it was all gibberish; two decades later, most of it still is. But one passage has always stuck out. It was toward the end of a notebook filled mostly with mathematics, but seems to belong more to the mystical writings.

I reproduce it below:

The primal UNITY is that which is neither two, nor three, nor even truly one.

It reflects upon itself and discovers its IDENTITY.

It reflects upon this identity: to know that it is such is to know it is no other. To claim to be other would be FALSE; a new discovery.

When it reflects upon the false, could it do other than arrive at the TRUE?

What remains to be learned from reflection? The APPLICATION of these principles, through which all things were made.

The program is complete, though its end is unknowable. All reduces to one, not two, for truth and its application are inseparable. ABH

I've pondered these words for quite some time. They seem significant, but what could they mean?

What is he talking about?

Bonus: Can you guess my uncle's last name?


Hint 1

The title is significant.

Hint 2

It may help to go back to Church.

Hint 3

The year '94 is not significant, but the year my uncle was born is significant for the bonus.

Hint 4

My uncle was fond of birds.

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He's talking about

combinatory logic, which is a sort of "point-free" version of the lambda calculus.

The primal UNITY is that which is neither two, nor three, nor even truly one.

This is the iota combinator, which I'll write as Q because I is already taken. It can be defined in terms of the famous S and K combinators by the relation Q x = x S K. [EDITED to add:] I'd expected the "two ... three ... one" bit to be some sort of reference to Church numerals, but in fact OP has clarified in chat that he intended this as an allusion to the way that the iota combinator's behaviour can't be specified "equationally" as e.g. that of the S combinator is when one writes S x y z = x z (y z), but has to be written in terms of other combinators like S and K, or in raw lambda calculus.

It reflects upon itself and discovers its IDENTITY.

Let's apply Q to itself. We get Q Q = Q S K = S S K K. Now apply the defining property of S: S S K K = S K (K K). What does this do to, say, x? S K (K K) x = K x (K K x) = x. So indeed Q Q is the identity.

It reflects upon this identity: to know that it is such is to know it is no other. To claim to be other would be FALSE; a new discovery.

The wording here seems wilfully unclear, but what I believe it means is that applying Q to the identity gives (Church's version of) boolean FALSE, which I'll write as F. Its defining property is that F x y = y. (In terms of the usual SKI basis, it's the same as K I.) So, let's give it a go. Q I x y = I S K x y = S K x y = K y (x y) = y. Yup!

When it reflects upon the false, could it do other than arrive at the TRUE?

So now let's apply Q to F and verify that we get Church's boolean TRUE, which I could write as T and which needs to have T x y = x ... but that has a more common name, namely K which I've used several times already. Anyway. Q F x y = F S K x y = K x y = x, as required.

What remains to be learned from reflection? The APPLICATION of these principles, through which all things were made.

Combinatory logic is all about (function/combinator) application. [EDITED to add:] OP says in comments that this was actually meant to indicate another specific combinator, so here we go. Applying Q to the K combinator we just derived gives Q K = K S K = S, and the S combinator is kinda-sorta doing application.

The program is complete, though its end is unknowable. All reduces to one, not two, for truth and its application are inseparable.

Combinatory logic is (Turing-)complete: any computation we know how to formalize has a combinatory-logic equivalent. Now that we've derived all three of S,K,I (which are well known to be a basis for combinatory logic -- given these you can make equivalents for anything else), we have shown that the iota combinator itself is a basis: "all reduces to one". ((Why "not two"? Perhaps because a more commonly used basis is S plus K.)

"Its end is unknowable" is presumably a reference to the undecidability of the halting problem: informally, given a combinator expression, there is no guaranteed-to-terminate procedure that will tell you whether the sort of recursive expansion process we've seen many examples of above will ever come to an end.

I'm not sure whether "truth and its application are inseparable" means anything very precise. Perhaps it's alluding to how Church booleans work -- if b is one, you use it by applying it to two arguments, one of which is the result you want if it's true and one of which is the result you want if it's false.

[EDITED to add:] OP has clarified in chat that "t.a.i.a.a.i." is about the fact that the iota combinator is usually defined in terms of K ("truth") and S ("application").

ABH

This is not ay-bee-aitch, it's alpha-beta-eta, and these are the names of the three basic kinds of transformation permitted in the lambda calculus. Alpha-conversion is variable renaming; beta-reduction is substitution of a function's argument for the parameter it's bound to; eta-conversion is the elimination of redundant function applications. (Hmm, so maybe I want lambda calculus rather than combinatory logic as such? Still thinking about that.)

Hint 1: The title is significant.

Yes, there is some logic in the mysticism.

Hint 2: It may help to go back to Church.

That would be Alonzo Church.

Hint 3: The year '94 is not significant, but the year my uncle was born is significant for the bonus.

That year must be either 1888 or 1889. 1889 was Schoenfinkel's year of birth, so I guess your uncle's surname is Finkel or maybe Finkle.

Hint 4: My uncle was fond of birds.

Perhaps a reference to Raymond Smullyan's book about combinatory logic, To Mock a Mockingbird, in which the roles of the combinators are taken by various kinds of birds.

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  • $\begingroup$ You're very close! But you've misidentified the UNITY, so the pattern doesn't work. You did get the bonus correctly. (As for the sequence of letters… yeah, they're all Greek to me, too. 😉) $\endgroup$ – James Jensen Jul 19 '18 at 12:08
  • $\begingroup$ Oh, they're reductions. D'oh. OK, thinking some more about the UNITY now. $\endgroup$ – Gareth McCaughan Jul 19 '18 at 18:18
  • $\begingroup$ Aha, got it. Editing now. [Done.] $\endgroup$ – Gareth McCaughan Jul 19 '18 at 19:41
  • $\begingroup$ Excellent! Yes, ABH isn't directly about the subject, per se (though in working with combinators, one can certainly carry out these reductions). But it's yet another hint about the general conceptual area. (Also, it makes the whole passage feel a little bit more like an early-20th-century esoteric text, with the odd AUM inserted at seemingly-random places.) $\endgroup$ – James Jensen Jul 19 '18 at 22:04
  • $\begingroup$ I've accepted your answer, because you identified the core elements, but there are a couple of things you missed: there's one more combinator corresponding to APPLICATION, and a few other references to related mathematical concepts. $\endgroup$ – James Jensen Jul 19 '18 at 22:08

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