Don't fear the Monad

Universality:You can convert any function into a table lookup... , and the function is just data (start at 355)

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Monad in terms of Functions, Monoids, Monads...(starting at 455)

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Monoid is the way to build complexity from simplicity...(start at 1010)

Simple in Brian's terms means small. I assume this means a smaller vocabulary, in the monoidal case, only one type, so in the size of type vocabulary, it is small. {{#ev:youtube|ZhuHCtR3xq8|||| |start=1010&end=1080}}

Monoid helps to guarantee you to build a software with one and only one type ...(start at 1126)

Create a generic compatibility operator, that allows you to create other functions through composition, ... if one follows this discipline (of being monorail), that you cannot make mistake of creating types that are not the same. {{#ev:youtube|ZhuHCtR3xq8|||| |start=1126&end=1230}}

Monoid helps to guarantee you to build a software with one and only one type ...(start at 1126)

Create a generic compatibility operator, that allows you to create anything in your set of types, ... that are guaranteed to be in the same type, ... functions through composition, ... if one follows this discipline (of being monorail), that you cannot make mistake of creating types that are not the same. {{#ev:youtube|ZhuHCtR3xq8|||| |start=1126&end=1230}}

A Monoid is a collection of things, plus a rule for combining the things, where that (one) rule, obeys some (custom defined) rules ...(start at 1265)

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The Punch Line: Functions under composition form a Monoid...(start at 1542)

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As long as types line up, functional composition make sense...(start at 1570)

You don't need to know Category Theory to be fully conversant in this (Monoidal) language of functional composition. {{#ev:youtube|ZhuHCtR3xq8|||| |start=1570&end=1637}}

The Slogan: Compositionality is the way to control complexity...(start at 2095)

1. Side effects are complicated. Examples include: Input/Output, Concurrency, Shared Memory, Continuation, Exceptions, Interactions with SQL Databases, Exceptions, ... yuck... If I can get you to associativity...
2. LINQ designed by Eric Meijer, is based on Monad.
3. Functions under composition is a Monoid, therefore puts you in a position where you cannot make mistake.

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A New Operator: Haskell calls this Bind, I call it Shove (start at 2210)

It can be expressed as follow:

\a -> (f a) >>= \a -> (g a)

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Broken symmetry causes confusion (with Monad) (start at 2400)

The following definition shows a structure asymmetry...

M a >>= a -> M a

The LHS and RHS across the >>= operator has two different types. In order to restore the symmetry, we need to put a $\lambda$ expression (\a -> M a) in front of the overall structure, to get back the world of compositions. We just have to put one little $\lambda$ from the outside, the entire expression is back to our compositional universe. It is almost exactly the same as a Monoid.

\a -> ( M a >>= a -> M a )

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The Definition of Monad: (start at 2442)

The functions \a -> M a live in a Monoid and this data: M a, live in a Monad, that's the definition. ... (because) we want compositionality, for the same reasons we mentioned before, we want our functions to live in a Monoid, but we need this extra data, so that we can do concurrency, side effects, I/O, whatever else ... (to make programs useful). {{#ev:youtube|ZhuHCtR3xq8|||| |start=2442&end=2477}}

Why Monad?: Allow functions to be combined in arbitrary ways, while obeying meta-rules (start at 2620)

The bind operator >>= has to obey the meta-rule. It has to obey associativity. (Associativity is also a order-preserving rule, which is related to the notion of time, sequence, and other time-like, entropy-increasing. additive kinds of properties.) {{#ev:youtube|ZhuHCtR3xq8|||| |start=2620&end=2662}}

Industry Applications of Monad?: LINQ passes functions as data in C# (start at 3665)

F# has the performance profile of C#, yet has immediately comfortable to people who are familiar with Haskell... {{#ev:youtube|ZhuHCtR3xq8|||| |start=3665&end=3705}}