Re: [Haskell-cafe] why the name lambda calculus?
It's always been my understanding that calculi were systems that defined particular symbols and the legal methods of their manipulation in the context of a particular calculus. The point, generally (har har), seems to be abstraction. The lambda calculus describes computation without actually implementing it, the predicate/propositional calculi describe logic without necessarily containing any explicit logical statements. Algebras, on the other hand, are structures whose properties are defined by a (usually) small number of properties and axioms. A Boolean algebra is a 6-tuple (A, ∧, ∨, ¬, ⊥, ⊤) such that for all a, b, c in A, associativity, commutativity, absorption, distributivity, and complement axioms all hold. An algebra over a field describes a vector space with a bilinear vector product. The other axioms that must hold depend on the particular vector space, though. Jack Henahan jhena...@uvm.edu == Computer science is no more about computers than astronomy is about telescopes. -- Edsger Dijkstra == 398E692F.gpg Description: application/apple-msg-attachment On Aug 24, 2011, at 9:20 AM, Dominic Mulligan wrote: > On Wed, 2011-08-24 at 14:01 +0100, Tony Finch wrote: >> Ezra Cooper wrote: >>> >>> I believe this to be a general trait of things described as >>> "calculi"--that they have some form of name-binders, but I have never >>> seen that observation written down. >> >> Combinator calculi are a counter-example. > > As is the propositional calculus. I seem to remember Joe Wells once > asking Wilfrid Hodges what he thought the definition of a calculus was. > He didn't provide a convincing definition. > > > > ___ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
On Wed, 2011-08-24 at 14:01 +0100, Tony Finch wrote: > Ezra Cooper wrote: > > > > I believe this to be a general trait of things described as > > "calculi"--that they have some form of name-binders, but I have never > > seen that observation written down. > > Combinator calculi are a counter-example. As is the propositional calculus. I seem to remember Joe Wells once asking Wilfrid Hodges what he thought the definition of a calculus was. He didn't provide a convincing definition. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
Ezra Cooper wrote: > > I believe this to be a general trait of things described as > "calculi"--that they have some form of name-binders, but I have never > seen that observation written down. Combinator calculi are a counter-example. Tony. -- f.anthony.n.finchhttp://dotat.at/ Biscay, FitzRoy: Mainly westerly or southwesterly 4 or 5, occasionally 6 in Fitzroy, becoming variable 3 at times in south. Slight or moderate, becoming rough or very rough in northwest Fitzroy. Rain or showers, thundery at times. Good, occasionally poor. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
On Tue, Aug 23, 2011 at 10:58 AM, Ezra Cooper wrote: > An algebra is a specific kind of structure which is itself formalized > mathematically. I've never seen a formalization of the notion of "a > calculus" and I believe it to be a looser term, as KC defined it. > > Specifically, an algebra consists of a set (or several "sorts" of sets) and > operations that reduce pairs of elements from that set (or the pairs can be > triples, etc.) back into the set. Usually that set corresponds to the > "semantics" of the algebra, and syntactic equations like xy = yx exist in a > different realm from the operations and their actions. > > Lambda calculus differs from an algebra by having a construct (lambda > abstraction) that only makes sense if you know the syntactic structure of > the term it applies to. That is, it has a binding construct. You could > define lambda calculus as an algebra by taking the underlying set to be the > syntax of the calculus itself, but that would require infinitely many > operations (a lambda-binder for each variable) and equations, so perhaps > that would be awkward. > > Pi calculus, like lambda calculus, has binders, while "process algebras" > are usually defined via operations on processes. I believe this to be a > general trait of things described as "calculi"--that they have some form of > name-binders, but I have never seen that observation written down. > > I'm sure that an algebraist could give a more definite answer about this. > Aside from "the calculus", a calculus is just a language with syntactic rules of inference/deduction. Indeed, "the calculus" was a calculus in this sense way back when, before Riemann formalized the subject with deltas and epsilons (that is, arguments about sequences and limits). I am referring to the syntactic rules for manipulating derivatives and integrals which Liebniz invented, such as "formal cancellation of derivatives": dy/dx * dz/dy = dz/dx An algebra is a model for a language that quantifies over objects and function symbols on objects. For example, the "symmetric group S4" is an algebra that models the symmetric group axioms. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
An algebra is a specific kind of structure which is itself formalized mathematically. I've never seen a formalization of the notion of "a calculus" and I believe it to be a looser term, as KC defined it. Specifically, an algebra consists of a set (or several "sorts" of sets) and operations that reduce pairs of elements from that set (or the pairs can be triples, etc.) back into the set. Usually that set corresponds to the "semantics" of the algebra, and syntactic equations like xy = yx exist in a different realm from the operations and their actions. Lambda calculus differs from an algebra by having a construct (lambda abstraction) that only makes sense if you know the syntactic structure of the term it applies to. That is, it has a binding construct. You could define lambda calculus as an algebra by taking the underlying set to be the syntax of the calculus itself, but that would require infinitely many operations (a lambda-binder for each variable) and equations, so perhaps that would be awkward. Pi calculus, like lambda calculus, has binders, while "process algebras" are usually defined via operations on processes. I believe this to be a general trait of things described as "calculi"--that they have some form of name-binders, but I have never seen that observation written down. I'm sure that an algebraist could give a more definite answer about this. Ezra On Aug 23, 2011, at 12:19 PM, Rajesh S R wrote: > Slight digression. Why not Lambda "Algebra"? > In particular, what is the criteria for a system to be calculus and how's it > different from algebra? > > On Mon, Aug 22, 2011 at 12:41 AM, Jack Henahan wrote: > The short answer is "because Church said so". But yes, it is basically > because λ is the abstraction operator in the calculus. > > Why not alpha or beta calculus? What would we call alpha and beta conversion, > then? :D > > On Aug 21, 2011, at 12:37 PM, C K Kashyap wrote: > > > Hi, > > Can someone please tell me what is the root of the name lambda calculus? Is > > it just because of the symbol lambda that is used? > > Why not alpha or beta calculus? > > Regards, > > Kashyap > > ___ > > Haskell-Cafe mailing list > > Haskell-Cafe@haskell.org > > http://www.haskell.org/mailman/listinfo/haskell-cafe > > Jack Henahan > jhena...@uvm.edu > == > Computer science is no more about computers than astronomy is about > telescopes. > -- Edsger Dijkstra > == > > > > ___ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe > > > > > -- > Rajesh S R > http://rajeshsr.co.cc/blogs/ > ___ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
See Serge Lang's "Algebra". 2011/8/23 Rajesh S R : > Slight digression. Why not Lambda "Algebra"? > In particular, what is the criteria for a system to be calculus and how's it > different from algebra? > -- -- Regards, KC ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
Slight digression. Why not Lambda "Algebra"? In particular, what is the criteria for a system to be calculus and how's it different from algebra? On Mon, Aug 22, 2011 at 12:41 AM, Jack Henahan wrote: > The short answer is "because Church said so". But yes, it is basically > because λ is the abstraction operator in the calculus. > > Why not alpha or beta calculus? What would we call alpha and beta > conversion, then? :D > > On Aug 21, 2011, at 12:37 PM, C K Kashyap wrote: > > > Hi, > > Can someone please tell me what is the root of the name lambda calculus? > Is it just because of the symbol lambda that is used? > > Why not alpha or beta calculus? > > Regards, > > Kashyap > > ___ > > Haskell-Cafe mailing list > > Haskell-Cafe@haskell.org > > http://www.haskell.org/mailman/listinfo/haskell-cafe > > Jack Henahan > jhena...@uvm.edu > == > Computer science is no more about computers than astronomy is about > telescopes. > -- Edsger Dijkstra > == > > > > ___ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe > > -- Rajesh S R http://rajeshsr.co.cc/blogs/ ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
I had thyroid cancer a few years ago; now I've lost my sense of tumour. :) -- -- Regards, KC ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
KC wrote: > > Lambda abstraction was probably chosen in case someone found better > abstractions; e.g. epsilon, delta, gamma, beta, alpha, ... :) http://www-maths.swan.ac.uk/staff/jrh/papers/JRHHislamWeb.pdf Page 7: By the way, why did Church choose the notation "λ"? In [an unpublished letter to Harald Dickson, Church] stated clearly that it came from the notation "x̂" used for class-abstraction by Whitehead and Russell, by first modifying "x̂" to "∧ x" to distinguish functionabstraction from class-abstraction, and then changing "∧ " to "λ" for ease of printing. This origin was also reported in [Rosser. Highlights of the history of the lambda calculus. Annals of the History of Computing]. On the other hand, in his later years Church told two enquirers that the choice was more accidental: a symbol was needed and "λ" just happened to be chosen.) Tony. -- f.anthony.n.finchhttp://dotat.at/ Tyne, Dogger: Northwest becoming variable then east, 3 or 4, increasing 5 or 6 later. Slight or moderate. Showers, thundery later. Moderate or good.___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
Definition of "calculus" a : a method of computation or calculation in a special notation (as of logic or symbolic logic) b : the mathematical methods comprising differential and integral calculus —often used with the So a "calculus" means more than differentiation and integration it can also mean a system of symbols that can be "mechanistically" modified; algorithmically modified if you prefer. Lambda abstraction was probably chosen in case someone found better abstractions; e.g. epsilon, delta, gamma, beta, alpha, ... :) Or Joel Spolsky's "leaky abstractions". :D On Sun, Aug 21, 2011 at 9:37 AM, C K Kashyap wrote: > Hi, > Can someone please tell me what is the root of the name lambda calculus? Is > it just because of the symbol lambda that is used? > Why not alpha or beta calculus? > Regards, > Kashyap > ___ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe > > -- -- Regards, KC ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
From Cardone, Hindley "History of Lambda-calculus and Combinatory Logic"[1]: "(By the way, why did Church choose the notation “λ”? In [Church, 1964, §2] he stated clearly that it came from the notation “ˆ x” used for class-abstraction by Whitehead and Russell, by first modifying “ˆ x” to “∧x” to distinguish function-abstraction from class-abstraction, and then changing “∧” to “λ” for ease of printing. This origin was also reported in [Rosser, 1984, p.338]. On the other hand, in his later years Church told two enquirers that the choice was more accidental: a symbol was needed and “λ” just happened to be chosen.)" [1] http://www-maths.swan.ac.uk/staff/jrh/papers/JRHHislamWeb.pdf -- Niklas 2011/8/21 Christopher Done : > IIRC Church found it easy to write on paper. > > On 21 August 2011 21:11, Jack Henahan wrote: >> The short answer is "because Church said so". But yes, it is basically >> because λ is the abstraction operator in the calculus. >> >> Why not alpha or beta calculus? What would we call alpha and beta >> conversion, then? :D >> >> On Aug 21, 2011, at 12:37 PM, C K Kashyap wrote: >> >>> Hi, >>> Can someone please tell me what is the root of the name lambda calculus? Is >>> it just because of the symbol lambda that is used? >>> Why not alpha or beta calculus? >>> Regards, >>> Kashyap >>> ___ >>> Haskell-Cafe mailing list >>> Haskell-Cafe@haskell.org >>> http://www.haskell.org/mailman/listinfo/haskell-cafe >> >> Jack Henahan >> jhena...@uvm.edu >> == >> Computer science is no more about computers than astronomy is about >> telescopes. >> -- Edsger Dijkstra >> == >> >> >> >> ___ >> Haskell-Cafe mailing list >> Haskell-Cafe@haskell.org >> http://www.haskell.org/mailman/listinfo/haskell-cafe >> >> > > ___ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe > ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
IIRC Church found it easy to write on paper. On 21 August 2011 21:11, Jack Henahan wrote: > The short answer is "because Church said so". But yes, it is basically > because λ is the abstraction operator in the calculus. > > Why not alpha or beta calculus? What would we call alpha and beta conversion, > then? :D > > On Aug 21, 2011, at 12:37 PM, C K Kashyap wrote: > >> Hi, >> Can someone please tell me what is the root of the name lambda calculus? Is >> it just because of the symbol lambda that is used? >> Why not alpha or beta calculus? >> Regards, >> Kashyap >> ___ >> Haskell-Cafe mailing list >> Haskell-Cafe@haskell.org >> http://www.haskell.org/mailman/listinfo/haskell-cafe > > Jack Henahan > jhena...@uvm.edu > == > Computer science is no more about computers than astronomy is about > telescopes. > -- Edsger Dijkstra > == > > > > ___ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe > > ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] why the name lambda calculus?
The short answer is "because Church said so". But yes, it is basically because λ is the abstraction operator in the calculus. Why not alpha or beta calculus? What would we call alpha and beta conversion, then? :D On Aug 21, 2011, at 12:37 PM, C K Kashyap wrote: > Hi, > Can someone please tell me what is the root of the name lambda calculus? Is > it just because of the symbol lambda that is used? > Why not alpha or beta calculus? > Regards, > Kashyap > ___ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe Jack Henahan jhena...@uvm.edu == Computer science is no more about computers than astronomy is about telescopes. -- Edsger Dijkstra == 398E692F.gpg Description: application/apple-msg-attachment ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] why the name lambda calculus?
Hi, Can someone please tell me what is the root of the name lambda calculus? Is it just because of the symbol lambda that is used? Why not alpha or beta calculus? Regards, Kashyap ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
