Very interesting historical document, though I don't have that much confidence in the precision of the explanation of the terms.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?
viftodi•59m ago
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•33m ago
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!
measurablefunc•48m ago
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.
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
veqq•57m ago
The Shen project is quite fascinating - and tedious to work with, as evidenced by this book of images across different pages etc.
pyrolistical•56m ago
Oh god. Where is the pdf. This format is horrible to read from
tonypapousek•42s ago
> Note that this material is here for reference and for sampling. Reading all of TBoS through this website is possible but not the intention. If you like the material and want to read the book conveniently then do buy either the latest hardcopy or e-version.
Unfortunately, I don’t think one can be linked given the author’s note.
rtpg•1h ago
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?
viftodi•59m ago
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•33m ago
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!
measurablefunc•48m ago
¹https://chatgpt.com/share/693f575d-0824-8009-bdca-bf3440a195...
rtpg•28m ago
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