Showing posts with label math. Show all posts
Showing posts with label math. Show all posts

Friday, November 28, 2014

Scientific Computing with Hy and IPython

This blog post is a bit different than other technical posts I've done in the past in that the majority of the content is not on the blog in or gists; instead, it is in an IPython notebook. Having adored Mathematica back in the 90s, you can imagine how much I love the IPython Notebook app. I'll have more to say on that at a future date.

I've been doing a great deal of NumPy and matplotlib again lately, every day for hours a day. In conjunction with the new features in Python 3, this has been quite a lot of fun -- the most fun I've had with Python in years (thanks Guido, et al!). As you might have guessed, I'm also using it with Erlang (specifically, LFE), but that too is for a post yet to come.

With all this matplotlib and numpy work in standard Python, I've been going through Lisp withdrawals and needed to work with it from a fresh perspective. Needless to say, I had an enormous amount of fun doing this. Naturally, I decided to share with folks how one can do the latest and greatest with the tools of Python scientific computing, but in the syntax of the Python community's best kept secret: Clojure-Flavoured Python (Github, Twitter, Wikipedia).

Spoiler: observed data and
polynomial curve fitting
Looking about for ideas, I decided to see what Clojure's Incanter project had for tutorials, and immediately found what I was looking for: Linear regression with higher-order terms, a 2009 post by David Edgar Liebke.

Nearly every cell in the tutorial notebook is in Hy, and for that we owe a huge thanks to yardsale8 for his Hy IPython magics code. For those that love Python and Lisp equally, who are familiar with the ecosystems' tools, Hy offers a wonderful option for being highly productive with a language supporting Lisp- and Clojure-style macros. You can get your work done, have a great time doing it, and let that inner code artist out!

(In fact, I've started writing a macro for one of the examples in the tutorial, offering a more Lisp-like syntax for creating class methods. We'll see what Paul Tagliamonte has to say about it when it's done ... !)

If you want to check out the notebook code and run it locally, just do the following:

This will do the following:

  • Create a virtualenv using Python 3
  • Download all the dependencies, and then
  • Start up the notebook using a local IPython HTTP server

If you just want to read along, you're more than welcome to do that as well, thanks to the IPython NBViewer service. Here's the link: Scientific Computing with Hy: Linear Regressions.

One thing I couldn't get working was the community-provided code for generating tables of contents in IPython notebooks. If you have any expertise in this area, I'd love to get your feedback to see how I need to configure the custom ihy IPython profile for this tutorial.

Without that, I've opted for the manual approach and have provided a table of contents here:
If all goes well, you will enjoy that as much as I did :-)

More soon ...


Thursday, July 03, 2014

Uncovering Inherent Structures in Organizations

Vladimir Levenshtein
This post should have a subtitle: "Using Team Analysis and Levenshtein Distance to Reveal said Structure." It's the first part of that subtitle that is the secret, though: being able to correctly analyze and classify individual teams. Without that, using something clever like Levenshtein distance isn't going to be very much help.

But that's coming in towards the end of the story. Let's start at the beginning.

What You're Going to See

This post is a bit long. Here are the sections I've divided it into:

  • What You're Going to See
  • Premise
  • Introducing ACME
  • Categorizing Teams
  • Category Example
  • Calculating the Levenshtein Distance of Teams
  • Sorting and Interpretation
  • Conclusion

However, you don't need to read the whole thing to obtain the main benefits. You can get the Cliff Notes version by reading the Premise, Categorizing Teams, Interpretation, and the Conclusion.

Premise

Companies grow. Teams expand. If you're well-placed in your industry and providing in-demand services or products, this is happening to you. Individuals and small teams tend to deal with this sort of change pretty well. At an organizational level, however, this sort of change tends to have an impact that can bring a group down, or rocket it up to the next level.

Of the many issues faced by growing companies (or rapidly growing orgs within large companies), the structuring one can be most problematic: "Our old structures, though comfortable, won't scale well with all these new teams and all the new hires joining our existing teams. How do we reorganize? Where do we put folks? Are there natural lines along which we can provide better management (and vision!) structure?"

The answer, of course, is "yes" -- but! It requires careful analysis and a deep understanding every team in your org.

The remainder of this post will set up a scenario and then figure out how to do a re-org. I use a software engineering org as an example, but that's just because I have a long and intimate knowledge of them and understand the ways in which one can classify such teams. These same methods could be applied a Sales group, Marketing groups, etc., as long as you know the characteristics that define the teams of which these orgs are comprised.



Introducing ACME

ACME Corporation is the leading producer of some of the most innovative products of the 20th century. The CTO had previously tasked you, the VP of Software Development to bring this product line into the digital age -- and you did! Your great ideas for the updated suite are the new hottness that everyone is clamouring for. Subsequently, the growth of your teams has been fast, and dare we say, exponential.

More details on the scenario: your Software Development Group has several teams of engineers, all working on different products or services, each of which supports ACME Corporation in different ways. In the past 2 years, you've built up your org by an order of magnitude in size. You've started promoting and hiring more managers and directors to help organize these teams into sensible encapsulating structures. These larger groups, once identified, would comprise the whole Development Group.

Ideally, the new groups would represent some aspect of the company, software development, engineering, and product vision -- in other words, some sensible clustering of teams doing related work. How would you group the teams in the most natural way?

Simply dividing along language or platform lines may seem like the obvious answer, but is it the best choice? There are some questions that can help guide you in figuring this out:
  • How do these teams interact with other parts of the company? 
  • Who are the stakeholders in feature development? 
  • Which sorts of customers does each team primarily serve?
There are many more questions you could ask (some are implicit in the analysis data linked below), but this should give a taste.

ACME Software Development has grown the following teams, some of which focus on products, some on infrastructure, some on services, etc.:
  • Digital Anvil Product Team
  • Giant Rubber Band App Team
  • Digital Iron Carrot Team
  • Jet Propelled Unicycle Service Team
  • Jet Propelled Pogo Stick Service Team
  • Ultimatum Dispatcher API Team
  • Virtual Rocket Powered Roller Skates Team
  • Operations (release management, deployments, production maintenance)
  • QA (testing infrastructure, CI/CD)
  • Community Team (documentation, examples, community engagement, meetups, etc.)

Early SW Dev team hacking the ENIAC

Categorizing Teams

Each of those teams started with 2-4 devs hacking on small skunkworks projects. They've now blossomed to the extent that each team has significant sub-teams working on new features and prototyping for the product they support. These large teams now need to be characterized using a method that will allow them to be easily compared. We need the ability to see how closely related one team is to another, across many different variables. (In the scheme outlined below, we end up examining 50 bits of information for each team.)

Keep in mind that each category should be chosen such that it would make sense for teams categorized similarly to be grouped together. A counter example might be "Team Size"; you don't necessarily want all large teams together in one group, and all small teams in a different group. As such, "Team Size" is probably a poor category choice.

Here are the categories which we will use for the ACME Software Development Group:
  • Language
  • Syntax
  • Platform
  • Implementation Focus
  • Supported OS
  • Deployment Type
  • Product?
  • Service?
  • License Type
  • Industry Segment
  • Stakeholders
  • Customer Type
  • Corporate Priority
Each category may be either single-valued or multi-valued. For instance, the categories ending in question marks will be booleans. In contrast, multiple languages might be used by the same team, so the "Language" category will sometimes have several entries.

Category Example

(Things are going to get a bit more technical at this point; for those who care more about the outcomes than the methods used, feel free to skip to the section at the end: Sorting and Interpretation.)

In all cases, we will encode these values as binary digits -- this allows us to very easily compare teams using Levenshtein distance, since the total of all characteristics we are filtering on can be represented as a string value. An example should illustrate this well.

(The Levenshtein distance between two words is the minimum number of single-character edits -- such as insertions, deletions or substitutions -- required to change one word into the other. It is named after Vladimir Levenshtein, who defined this "distance" in 1965 when exploring the possibility of correcting deletions, insertions, and reversals in binary codes.)

Let's say the Software Development Group supports the following languages, with each one assigned a binary value:
  • LFE - #b0000000001
  • Erlang - #b0000000010
  • Elixir - #b0000000100
  • Ruby - #b0000001000
  • Python - #b0000010000
  • Hy - #b0000100000
  • Clojure - #b0001000000
  • Java - #b0010000000
  • JavaScript - #b0100000000
  • CoffeeScript - #b1000000000
A team that used LFE, Hy, and Clojure would obtain its "Language" category value by XOR'ing the three supported languages, and would thus be #b0001100001. In LFE, that could be done by entering the following code the REPL:


We could then compare this to a team that used just Hy and Clojure (#b0001100000), which has a Levenshtein distance of 1 with the previous language category value. A team that used Ruby and Elixir (#b0000001100) would have a Levenshtein distance of 5 with the LFE/Hy/Clojure team (which makes sense: a total of 5 languages between the two teams with no languages shared in common). 


Calculating the Levenshtein Distance of Teams

As a VP who is keen on deeply understanding your teams, you have put together a spreadsheet with a break-down of not only languages used in each team, but lots of other categories, too. For easy reference, you've put a "legend" for the individual category binary values is at the bottom of the linked spreadsheet.

In the third table on that sheet, all of the values for each column are combined into a single binary string. This (or a slight modification of this) is what will be the input to your calculations. Needless to say, as a complete fan of LFE, you will be writing some Lisp code :-)

Partial view of the spreadsheet's first page.
(If you would like to try the code out yourself while reading, and you have lfetool installed, simply create a new project and start up the REPL: $ lfetool new library ld; cd ld && make-shell
That will download and compile the dependencies for you. In particular, you will then have access to the lfe-utils project -- which contains the Levenshtein distance functions we'll be using. You should be able to copy-and-paste functions, vars, etc., into the REPL from the Github gists.)

Let's create a couple of data structures that will allow us to more easily work with the data you collected about your teams in the spreadsheet:


We can use a quick copy and paste into the LFE REPL for two of those numbers to do a sanity check on the distance between the Community Team and the Digital Iron Carrot Team:


That result doesn't seem unreasonable, given that at a quick glance we can see both of these strings have many differences in their respective character positions.

It looks like we're on solid ground, then, so let's define some utility functions to more easily work with our data structures:


Now we're ready to roll; let's try sorting the data based on a comparison with a one of the teams:


It may not be obvious at first glance, but what the levenshtein-sort function did for us is compare our "control" string to every other string in our data set, providing both the distance and the string that the control was compared to. The first entry in the results is the our control string, and we see what we would expect: the Levenshtein distance with itself is 0 :-)

The result above is not very easily read by most humans ... so let's define a custom sorter that will take human-readable text and then output the same, after doing a sort on the binary strings:


(If any of that doesn't make sense, please stop in and say "hello" on the LFE mail list -- ask us your questions! We're a friendly group that loves to chat about LFE and how to translate from Erlang, Common Lisp, or Clojure to LFE :-) )



Sorting and Interpretation

Before we try out our new function, we should ponder which team will be compared to all the others -- the sort results will change based on this choice. Looking at the spreadsheet, we see that the "Digital Iron Carrot Team" (DICT) has some interesting properties that make it a compelling choice:
  • it has stakeholders in Sales, Engineering, and Senior Leadership;
  • it has a "Corporate Priority" of "Business critical"; and
  • it has both internal and external customers.
Of all the products and services, it seems to be the biggest star. Let's try a sort now, using our new custom function -- inputting something that's human-readable: 


Here we're making the request "Show me the sorted results of each team's binary string compared to the binary string of the DICT." Here are the human-readable results:


For a better visual on this, take a look at the second tab of the shared spreadsheet. The results have been applied to the collected data there, and then colored by major groupings. The first group shares these things in common:
  • Lisp- and Python-heavy
  • Middleware running on BSD boxen
  • Mostly proprietary
  • Externally facing
  • Focus on apps and APIs
It would make sense to group these three together.

A sort (and thus grouping) by comparison to critical team.
Next on the list is Operations and QA -- often a natural pairing, and this process bears out such conventional wisdom. These two are good candidates for a second group.

Things get a little trickier at the end of the list. Depending upon the number of developers in the Java-heavy Giant Rubber Band App Team, they might make up their own group. However, both that one and the next team on the list have frontend components written in Angular.js. They both are used internally and have Engineering as a stakeholder in common, so let's go ahead and group them.

The next two are cloud-deployed Finance APIs running on the Erlang VM. These make a very natural pairing.

Which leaves us with the oddball: the Community Team. The Levenshtein distance for this team is the greatest for all the teams ... but don't be mislead. Because it has something in common with all teams (the Community Team supports every product with docs, example code, Sales and TAM support, evangelism for open source projects, etc.), it will have many differing bits with each team. This really should be in a group all its own so that structure represents reality: all teams depend upon the Community Team. A good case could also probably be made for having the manager of this team report directly up to you. 

The other groups should probably have directors that the team managers report to (keeping in mind that the teams have grown to anywhere from 20 to 40 per team). The director will be able to guide these teams according to your vision for the Software Group and the shared traits/common vision you have uncovered in the course of this analysis.

Let's go back to the Community Team. Perhaps in working with them, you have uncovered a hidden fact: the community interactions your devs have are seriously driving market adoption through some impressive and passionate service and open source docs+evangelism. You are curious how your teams might be grouped if sorted from the perspective of the Community Team.

Let's find out!


As one might expect, most of the teams remain grouped in the same way ... the notable exception being the split-up of the Anvil and Rubber Band teams. Mostly no surprises, though -- the same groupings persist in this model.

A sort (and thus grouping) by comparison to highly-connected team.
To be fair, if this is something you'd want to fully explore, you should bump the "Corporate Priority" for the Community Team much higher, recalculate it's overall bits, regenerate your data structures, and then resort. It may not change too much in this case, but you'd be applying consistent methods, and that's definitely the right thing to do :-) You might even see the Anvil and Rubber Band teams get back together (left as an exercise for the reader).

As a last example, let's throw caution and good sense to the wind and get crazy. You know, like the many times you've seen bizarre, anti-intuitive re-orgs done: let's do a sort that compares a team of middling importance and a relatively low corporate impact with the rest of the teams. What do we see then?


This ruins everything. Well, almost everything: the only group that doesn't get split up is the middleware product line (Jet Propelled and Iron Carrot). Everything else suffers from a bad re-org.

A sort (and thus grouping) by comparison to a non-critical team.

If you were to do this because a genuine change in priority had occurred, where the Giant Rubber Band App Team was now the corporate leader/darling, then you'd need to recompute the bit values and do re-sorts. Failing that, you'd just be falling into a trap that has beguiled many before you.


Conclusion

If there's one thing that this exercise should show you, it's this: applying tools and analyses from one field to fresh data in another -- completely unrelated -- field can provide pretty amazing results that turn mystery and guesswork into science and planning.

If we can get two things from this, the other might be: knowing the parts of the system may not necessarily reveal the whole (c.f. Complex Systems), but it may provide you with the data that lets you better predict emergent behaviours and identify patterns and structure where you didn't see them (or even think to look!) before.


Wednesday, April 10, 2013

The Lambda Calculus: A Quick Primer

The λ-Calculus Series
  1. A Brief History
  2. A Quick Primer for λ-Calculus
  3. Reduction Explained
  4. Church Numerals
  5. Arithmetic
  6. Logic
  7. Pairs and Lists
  8. Combinators
To the untrained eye, the notation used in λ-calculus can be a bit confusing. And by "untrained", I mean your average programmer. This is a travesty: reading the notation of λ-calculus should be as easy to do as recognizing that the following phrase demonstrates variable assignation:
x = 123
So how do we arrive at a state of familiarity and clarity from a starting state of confusion? Let's dive in with some examples, and take it step at a time :-) Once we've got our heads wrapped around Alonzo Church's notation, we'll be able to easily read it -- and thus convert it into code! (We will have lots of practice in the coming posts to do just that.)

A Quick Primer for λ-Calculus

Here's one of the simplest definitions in λ-calculus that you're going to see: the identity function:
λx.x
This reads as "Here is a function that takes x as an argument and returns x." Let's do some more:
λxy.x
"Here is a function that takes x and y as arguments and returns only x."
λx.λy.xy
"An outer function takes x as an argument and an inner function takes y as an argument, returning the x and the y." Note that this is exactly equivalent to the following (by convention):
λxy.xy
Let's up the ante with a function application:
λf.λx.f x
"Here is a function that takes a function f as its argument; the inner function takes x as its argument; return the result of the function f when given the argument x." For example, if we pass a function f which returns its input multiplied by 2, and we supplied a value for x as 6, then we would see an output of 12.

Let's take that a little further:
λf.λx.f (f (f x))
"Here is a function that takes a function f as its argument; the inner function takes x as its argument. Apply the function f to the argument x; take that result and apply f to it. Then do it a third time, returning that result." If we had the same function as the example above and passed the same value, our result this time would be 48 (i.e., 6 * 2 * 2 * 2).

That's most of what you need to read λ-calculus expressions. Next we'll take a peek into the murky waters of  λ-calculus reduction and find that it's quite drinkable, that we were just being fooled by the shadows.


Monday, April 08, 2013

The Lambda Calculus: A Brief History

Over this past weekend I took a lovely journey into the heart of the lambda calculus, and it was quite amazing. My explorations were made within the context of LFE. Needless to say, this was a romp of pure delight. In fact, it was so much fun and helped to clarify for me so many nooks and crannies of something that I had simply not explored very thoroughly in the past, that I had to share :-)

The work done over the past few days is on its way to becoming part of the documentation for LFE. However, this is also an excellent opportunity to share some clarity with a wider audience. As such, I will be writing a series of blog posts on λ-calculus from a very hands-on (almost practical!) perspective. There will be some overlap with the LFE documentation, but the medium is different and as such, the delivery will vary (sometimes considerably).

This series of posts will cover the following topics:
  1. A Brief History
  2. A Quick Primer for λ-Calculus
  3. Reduction Explained
  4. Church Numerals
  5. Arithmetic
  6. Logic
  7. Pairs and Lists
  8. Combinators
The point of these posts is not to expound upon that which has already been written about endlessly. Rather, the hope is to give a very clear demonstration of what the lambda calculus really is, and to do so with clear examples and concise prose. When the gentle reader is able see the lambda calculus in action, with lines of code that clearly show what is occurring, the mystery will disappear and an intuition for the subject matter will quite naturally begin to arise. This post is the first in the series; I hope you enjoy them as much as I did rediscovering λ-calculus :-)

Let us start at the beginning...

A Brief History

The roots of functional programming languages such as Lisp, ML, Erlang, Haskell and others, can be traced to the concept of recursion in general and λ-calculus in particular. In previous posts, I touched upon how we ended up with the lambda as a symbol for the anonymous function as well as how recursion came to be a going concern in modern mathematics and then computer science.

In both of those posts we saw Alonzo Church play a major role, but we didn't really spend time on what is quite probably considered his greatest contribution to computer science, if not mathematics itself: λ-calculus. Keep in mind that the Peano axioms made use of recursion, that Giuseppe Peano played a key role in Bertrand Russell’s development of the Principia, that Alonzo Church sought to make improvements on the Principia, and λ-calculus eventually arose from these efforts.

Invented in 1928, Alonzo didn't publish λ-calculus until 1932. When an inconsistency was discovered, he revised it in 1933 and republished. Furthermore, in this second paper, Church introduced a means of representing positive integers using lambda notation, now known as Church numerals. With Church and Turing both publishing papers on computability in 1936 (based respectively upon λ-calculus and the concept of Turing machines), they proposed solutions to the Entscheidungsproblem. Though Gödel preferred Turing's approach, Rosser suggested that they were equivalent definitions in 1939. A few years later, Kleene proposed the Church Thesis (1943) and then later formally demonstrated the equivalence between his teacher's and Turing's approaches giving the combination the name of the Church-Turing Thesis (1952, in his Introduction to Metamathematics). Within eight years, John McCarthy published his now-famous paper describing the work that he had started in 1958: "Recursive Functions of Symbolic Expressions and Their Computation by Machine". In this paper, McCarthy outlined his new programming language Lisp, citing Church's 77-page book  (1941, Calculi of Lambda Conversion), sending the world off in a whole new direction.

Since that time, there has been on-going research into λ-calculus. Indisputably, λ-calculus has had a tremendous impact on research into computability as well as the practical applications of programming languages. As programmers and software engineers, we feel its impact -- directly and indirectly -- on a regular, almost daily basis.


Tuesday, April 02, 2013

Maths and Programming: Whence Recursion?

As a manager in the software engineering industry, one of the things that I see on a regular basis is a general lack of knowledge from less experienced developers (not always "younger"!) with regard to the foundations of computing and the several related fields of mathematics. There is often a great deal of focus on what the hottest new thing is, or how the industry can be changed, or how we can innovate on the decades of profound research that has been done. All noble goals.

Notably, another trend I've recognized is that in a large group of devs, there are often a committed few who really know their field and its history. That is always so amazing to me and I have a great deal of admiration for the commitment and passion they have for their art. Let's have more of that :-)

As for myself, these days I have many fewer hours a week which I can dedicate to programming compared to what I had 10 years ago. This is not surprising, given my career path. However, what it has meant is that I have to be much more focused when I do get those precious few hours a night (and sometimes just a few per week!). I've managed this in an ad hoc manner by taking quick notes about fields of study that pique my curiosity. Over time, these get filtered and a few pop to the top that I really want to give more time.

One of the driving forces of this filtering process is my never-ending curiosity: "Why is it that way?" "How did this come to be?" "What is the history behind that convention?" I tend to keep these musings to myself, exploring them at my leisure, finding some answers, and then moving on to the next question (usually this takes several weeks!).

However, given the observations of the recent years, I thought it might be constructive to ponder aloud, as it were. To explore in a more public forum, to set an example that the vulnerability of curiosity and "not knowing" is quite okay, that even those of us with lots of time in the industry are constantly learning, constantly asking.

My latest curiosity has been around recursion: who first came up with it? How did it make it's way from abstract maths to programming languages? How did it enter the consciousness of so many software engineers (especially those who are at ease in functional programming)? It turns out that an answer to this is actually quite closely related to a previous post I wrote on the secret history of lambda. A short version goes something like this:

Giuseppe Peano wanted to establish a firm foundation for logic and maths in general. As part of this, he ended up creating consistent axioms around the hard-to-define natural numbers, counting, and arithmetic operations (which utilized recursion).  While visiting a conference in Europe, Bertrand Russell was deeply impressed by the dialectic talent of Peano and his unfailing clarity; he queried Peano as to his secret for success (Peano told him) and them asked for all of his published works. Russell proceeded to study these quite deeply. With this in his background, he eventually co-wrote the Principia Mathematica. Later, Alonzo Church (along with his grad students) sought to improve upon this, and in the process Alonzo Church ended up developing the lambda calculus. His student, John McCarthy, later created the first functional programming language, Lisp, utilizing concepts from the lambda calculus (recursion and function composition).

In the course of reading between 40-50 mathematics papers (including various histories) over the last week, I have learned far more than I had originally intended. So much so, in fact, that I'm currently working on a very fun recursion tutorial that not only covers the usual practical stuff, but steps the reader through programming implementations of the Peano axioms, arithmetic definitions, the Ackermann function, and parts of the lambda calculus.

I've got a few more blog post ideas cooking that dive into functions, their history and evolution. We'll see how those pan out. Even more exciting, though, was having found interesting papers discussing the evolution of functions and the birth of category theory from algebraic topology. This, needless to say, spawned a whole new trail of research, papers, and books... and I've got some great ideas for future blog posts/tutorials around this topic as well. (I've encountered category theory before, but watching it appear unsearched and unbidden in the midst of the other reading was quite delightful).

In closing, I enjoy reading not only the original papers (and correspondence between great thinkers of a previous era), but also the meanderings and rediscoveries of my peers. I've run across blog posts like this in the past, and they were quite enchanting. I hope that we continue to foster that in our industry, and that we see more examples of it in the future.

Keep on questing ;-)


Friday, December 28, 2012

The Secret History of Lambda

Being a bit of an origins nut (I always want to know how something came to be or why it is a certain way), one of the things that always bothered me with regard to Lisp was that no one seemed to talking about the origin of lambda in the lambda calculus. I suppose if I wasn't lazy, I'd have gone to a library and spent some time looking it up. But since I was lazy, I used Wikipedia. Sadly, I never got what I wanted: no history of lambda. [1] Well, certainly some information about the history of the lambda calculus, but not the actual character or term in that context.

Why lambda? Why not gamma or delta? Or Siddham ṇḍha?

To my great relief, this question was finally answered when I was reading one of the best Lisp books I've ever read: Peter Norvig's Paradigms of Artificial Intelligence Programming: Case Studies in Common Lisp. I'll save my discussion of that book for later; right now I'm going to focus on the paragraph at location 821 of my Kindle edition of the book. [2]

The story goes something like this:
  • Between 1910 and 1913, Alfred Whitehead and Bertrand Russell published three volumes of their Principia Mathematica, a work whose purpose was to derive all of mathematics from basic principles in logic. In these tomes, they cover two types of functions: the familiar descriptive functions (defined using relations), and then propositional functions. [3]
  • Within the context of propositional functions, the authors make a typographical distinction between free variables and bound variables or functions that have an actual name: bound variables use circumflex notation, e.g. x̂(x+x). [4]
  • Around 1928, Church (and then later, with his grad students Stephen Kleene and J. B. Rosser) started attempting to improve upon Russell and Whitehead regarding a foundation for logic. [5]
  • Reportedly, Church stated that the use of x̂ in the Principia was for class abstractions, and he needed to distinguish that from function abstractions, so he used ⋀x [6] or ^x [7] for the latter.
  • However, these proved to be awkward for different reasons, and an uppercase lambda was used: Λx. [8].
  • More awkwardness followed, as this was too easily confused with other symbols (perhaps uppercase delta? logical and?). Therefore, he substituted the lowercase λ. [9]
  • John McCarthy was a student of Alonzo Church and, as such, had inherited Church's notation for functions. When McCarthy invented Lisp in the late 1950s, he used the lambda notation for creating functions, though unlike Church, he spelled it out. [10] 
It seems that our beloved lambda [11], then, is an accident in typography more than anything else.

Somehow, this endears lambda to me even more ;-)



[1] As you can see from the rest of the footnotes, I've done some research since then and have found other references to this history of the lambda notation.

[2] Norvig, Peter (1991-10-15). Paradigms of Artificial Intelligence Programming: Case Studies in Common Lisp (Kindle Locations 821-829). Elsevier Science - A. Kindle Edition. The paragraph in question is quoted here:
The name lambda comes from the mathematician Alonzo Church’s notation for functions (Church 1941). Lisp usually prefers expressive names over terse Greek letters, but lambda is an exception. Abetter name would be make-function. Lambda derives from the notation in Russell and Whitehead’s Principia Mathematica, which used a caret over bound variables: x̂(x + x). Church wanted a one-dimensional string, so he moved the caret in front: ^x(x + x). The caret looked funny with nothing below it, so Church switched to the closest thing, an uppercase lambda, Λx(x + x). The Λ was easily confused with other symbols, so eventually the lowercase lambda was substituted: λx(x + x). John McCarthy was a student of Church’s at Princeton, so when McCarthy invented Lisp in 1958, he adopted the lambda notation. There were no Greek letters on the keypunches of that era, so McCarthy used (lambda (x) (+ x x)), and it has survived to this day.
[3] http://plato.stanford.edu/entries/pm-notation/#4

[4] Norvig, 1991, Location 821.

[5] History of Lambda-calculus and Combinatory Logic, page 7.

[6] Ibid.

[7] Norvig, 1991, Location 821.

[8] Ibid.

[9] Looking at Church's works online, he uses lambda notation in his 1932 paper A Set of Postulates for the Foundation of Logic. His preceding papers upon which the seminal 1932 is based On the Law of Excluded Middle (1928) and Alternatives to Zermelo's Assumption (1927), make no reference to lambda notation. As such, A Set of Postulates for the Foundation of Logic seems to be his first paper that references lambda.

[10] Norvig indicates that this is simply due to the limitations of the keypunches in the 1950s that did not have keys for Greek letters.

[11] Alex Martelli is not a fan of lambda in the context of Python, and though a good friend of Peter Norvig, I've heard Alex refer to lambda as an abomination :-) So, perhaps not beloved for everyone. In fact, Peter Norvig himself wrote (see above) that a better name would have been make-function.