Very interesting historical document, though I don't have that much confidence in the precision of the explanation of the terms.

Related to this: does anyone know if there's any document that delves into how Church landed on Church numerals in particular? I get how they work, etc, but at least the papers I saw from him seem to just drop the definition out of thin air.

Were church numerals capturing some canonical representation of naturals in logic that was just known in the domain at the time? Are there any notes or the like that provide more insight?

▲ mutkach a day ago | parent | next [-]

Before Church there was Peano, and before Peano there was Grassmann

> It is rather well-known, through Peano's own acknowledgement, that Peano […] made extensive use of Grassmann's work in his development of the axioms. It is not so well-known that Grassmann had essentially the characterization of the set of all integers, now customary in texts of modern algebra, that it forms an ordered integral domain in which each set of positive elements has a least member. […] [Grassmann's book] was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis.

▲ viftodi a day ago | parent | prev | next [-]

While I don't know much about Church numbers or the theory how lambda calculus works, taking a glance at the definitions on wikipedia they seem to be the math idea of how numbers works (at the meta level)

I forgot the name of this, but they seem the equivalent of successors in math In the low level math theory you represent numbers as sequences of successors from 0 (or 1 I forgot)

Basically you have one then sucessor of one which is two, sucessor of two and so on So a number n is n successor operations from one

To me it seems Church numbers replace this sucessor operation with a function but it's the same idea

▲ rtpg a day ago | parent [-]

Church ends up defining zero as the identity function, and N as "apply a function to a zero-unit N times"

While defining numbers in terms of their successors is decently doable, this logical jump (that works super well all things considered!) to making numbers take _both_ the successor _and_ the zero just feels like a great idea, and it's a shame to me that the papers I read from Church didn't intuit how to get there.

After the fact, with all the CS reflexes we have, it might be ... easier to reach this definition if you start off "knowing" you could implement everything using just functions and with some idea of not having access to a zero, but even then I think most people would expect these objects to be some sort of structure rather than a process.

There is, of course, the other possibility which is just that I, personally, lack imagination and am not as smart as Alonzo Church. That's why I want to know the thought process!

	▲	itishappy 11 hours ago \| parent [-]
		> Church ends up defining zero as the identity function Zero is not the identity function. Zero takes a function and calls it zero times on a second function. The end result of this is that it returns the identity function. In Haskell it would be `const id` instead of `id`. `zero := λf.λx.x one := λf.λx.fx two := λf.λx.ffx id := λx.x` I suspect that this minor misconception may lead you to an answer to your original question! Why isn't the identity function zero? Given that everything in lambda calculus is a function, and the identity function is the simplest function possible, it would make sense to at least try! If you try, I suspect you'll quickly find that it starts to break down, particularly when you start trying to treat your numerals as functions (which is, after all, their intended purpose). Church numerals are a minimal encoding. They are as simple as it possibly gets. This may not speak to Church's exact thought process, but I think it does highlight that there exists a clear process that anyone might follow in order to get Church's results. In other words, I suspect that his discover was largely mechanical, rather than a moment of particularly deep insight. (And I don't think this detracts from Church's brilliance at all!)

▲ measurablefunc a day ago | parent | prev [-]

Their structural properties are similar to Peano's definition in terms of 0 and successor operation. ChatGPT does a pretty good job of spelling out the formal structural connection¹ but I doubt anyone knows how exactly he came up with the definition other than Church.

¹https://chatgpt.com/share/693f575d-0824-8009-bdca-bf3440a195...

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rtpg a day ago | parent [-]

Yeah I've been meaning to send a request to Princeton's libraries with his notes but don't know what a good request looks like

The jump from "there is a successor operator" to "numbers take a successor operator" is interesting to me. I wonder if it was the first computer science-y "oh I can use this single thing for two things" moment! Obviously not the first in all of science/math/whatever but it's a very good idea

	▲	black_knight a day ago \| parent \| next [-]
		The idea of Church numerals is quite similar to induction. An induction proof extends a method of treating the zero case and the successor case, to a treatment of all naturals. Or one can see it as defining the naturals as the numbers reachable by this process. The leap to Church numerals is not too big from this.
	▲	measurablefunc a day ago \| parent \| prev [-]
		Probably not possible unless you have academic credentials to back up your request like being a historian writing a book on the history of logic & computability.