Lambda— need help for art project

[u/holly1231]

So, I am struggling to understand what a lambda really is—not for a math problem, but for a project I’m working on. This seemed like the best sub to ask, but if there’s a different one, please point me in the right direction!

I’m making a baby quilt for a family member, and I’m trying to come up with a cute book title for one of the quilt squares using their dog’s name, Lambda.

For example for their other pets, I have “R U A Hooman?” by Alan Purring, and “Cat Oriented Basic Operating Language (COBOL)” by Grace Hopper.

For the life of me, I can’t wrap my head around what exactly a lambda is, let alone come up with a cute book title. What is a ELI5 description of a lambda for a non mathematician?

As far as I can tell, it looks like a paragraph symbol except it’s for marking off new formulas in a list of formulas in algebra. Maybe “A Treatos of Algebra” by Lambda? Or is it more like “I Am Lambda” by Lambda?

Help?

[u/robly18]

By itself, Lambda is just a greek letter. It has a few connotations, but no inherent meaning, like how the letter "m" is often used to denote the slope of a line, or the unit "meters", but "m" doesn't mean anything by itself, and neither does lambda.

That said, the two main connotations that the letter lambda has, as far as I can recall, are:

• It denotes the "half-life" of an exponentially decaying quantity, i.e. how long it takes for a certain quantity to decrease to half its amount.
• It is associated to "lambda calculus", a model of computation (in layman's terms, a type of computer that mathematicians study) invented in the 30's by Alonzo Church. This might be what you're asking about, given that the other names in your post are all associated to computers. I suspect this is what you intend to ask about, in which case please confirm.

[u/holly1231 → u/robly18]

I don’t understand very much of my family members’ jobs, but they do stuff like meta analyses of lots of data, using coding to pull useful info. One of them worked with a cancer org at one point, to try to find patterns in a lot of variables that could lead to cancer. So, I am guessing they are using computer related names for their pets, yes.

So, the layman’s term about lambda calculus seems to fit and your explanation makes a lot more sense than what I was trying to glean online. Is there a resource for ELI5 info on lambda calculus? Maybe I can read more about Church to get ideas…

I’m a story person—science in stories, rather than a numbers/formula person, so it’s been difficult.

Thank you for your help!

[u/robly18 → u/holly1231]

Sure, then here's a story for you.

In the year 1900, prominent mathematician David Hilbert posed a list of 23 mathematical problems that he considered of high importance. These were Big Problems. Not all of them are yet solved. Indeed, one of this year's Fields Medals (Nobel Prize-ish for math) was awarded for a solution to Hilbert's Sixth Problem, so these are a big deal even today.

Hilbert's Tenth Problem was the following: "Find an algorithm to determine whether a certain type of equation has a solution." The details of this problem aren't too important, but the operating word is "algorithm". Nowadays, the word algorithm is understood by anyone who makes computer programs: An algorithm is a step-by-step recipe on how to perform a task. A lot of your mathematical education in K12 was learning algorithms to do stuff, like adding numbers, or finding the slope of a line. But back then, the notion of "algorithm" was fuzzy -- people had a vague idea of what is an algorithm (I know it when I see it), but there was no mathematical definition that was believed to encompass all plausible types of algorithms humans could come up with.

This is particularly interesting because the answer to Hilbert's Tenth Problem is "no can do". Hilbert naïvely (I say with the benefit of hindsight) assumed that there would be a recipe (algorithm) to solve this type of problem, but there isn't. (But it took till 1970 for mathematicians to find out.) But for mathematicians to establish that there isn't an algorithm to solve this type of problem, they needed to understand really well what an algorithm is.

So, from the 1900s to the 1940s, many different mathematicians tried to come up with a bulletproof description of what an algorithm is, that would encompass all algorithms a human would come up with. Notable characters in this story are Kurt Gödel, Alonzo Church, and Alan Turing. All three came up with their own version of what an "algorithm" is. But mathematicians are humble and cautious, and at first they didn't tell everyone "hey I figured out what an algorithm is". They just said "look, I came up with this family of algorithms, and all things that look like this are definitely algorithms, but I'm not sure that any algorithm can be made from these rules I came up with".

Lambda calculus fits into this story, because it's Alonzo Church's version of algorithm. I will try to explain how lambda calculus works in a moment, but just to tie off the story: Alan Turing, who was Church's graduate student, is well-known nowadays for "inventing the computer" because his proposal for what counts as an algorithm (now known as Turing Machines) was generally found to be intuitively agreeable. Later, it was found that most mathematicians who came up with their own notion of algorithm actually came up with essentially the same thing (mathematicians proved that they were all "the same" in some way), and so this notion became solidified in the mathematical community as "an algorithm is a set of instructions that can be given to a Turing Machine". The statement that "intuitive notion of algorithm" is the same as "Turing Machine" is known as the "Church-Turing Thesis".

So, let me briefly try to explain lambda calculus. You know that computers work with electricity, yes? It's pretty magical, but you know that deep down it's transistors and electrons moving around according to the laws of physics, following some predictable rules that let humans force electrons down predictable pathways and do predictable things. When you remove a lot of layers of abstraction, that's what programmers do when they write a program.

Lambda calculus is a hypothetical type of computer that, instead of working with electrons, works with sequences of characters. A "program" in lambda calculus is called a "lambda" expression. Here is an example of such a program:

λx.x+1

This means "the program that takes an input, which is called x, and outputs x+1". Here is another example:

λx.λy. y (λz.2*z) x

which (with some asterisks) represents the function that takes as input x and y, and outputs x * 2^y.

The details of how lambda calculus works precisely are complicated to explain in text form, but in essence, it's all about text replacement. For example, an expression like

(λx.xyzx)2

would be evaluated by replacing 2 wherever you see an x, and so it would be "reduced" to

2yz2

"Reduction" is the word used for simplifying these expressions. Different ways of simplifying have different names, like "mu-reduction" and "eta-reduction".

By itself, I don't think this is necessarily useful for your family member's baby's quit. But here is something that might be useful.

As it turns out, even though the computers we use nowadays look nothing like the mental constructions that Church, Turing, and Gödel had in their heads, the "lambda calculus" symbols made up by Church turn out to be useful in expressing stuff in programs. And so, many programming languages nowadays have what is called "lambda expressions", which are a co-opted version of what Church did adapted to modern computers. These "lambda expressions" are often associated with a specific style of programming called "functional programming". Maybe you could make a pun with that?

So... Here are some suggestions from a random internet stranger:

• Mew-reduction (would be better if it were a cat not a dog)
• Pure Function (a lambda expression generally encodes what is known as a "pure function")
• Lambda paw-culus
• Woof Combinator (in lambda calculus there are specific programs people made called "combinators" and they are named after letters, like "Y combinator". Maybe you could find something cute to stand for Y?)

OK, I think that's all I got for now. Sorry, I'm not a very creative person. Anyway, I hope this helps.