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In wikipedia, the bottom type is simply defined as "the type that has no values". However, if b is this empty type, then the product type (b,b) has no values either, but seems different from b. I agree bottom is uninhabited, but I don't think this property suffices to define it.
By the Curry-Howard correspondence, bottom is associated to mathematical falsity. Now there is a logical principle stating that from False follows any proposition. By Curry-Howard, that means the type forall a. bottom -> a is inhabited, ie there exists a family of functions f :: forall a. bottom -> a.
What are those functions f ? Do they help define bottom, maybe as the infinite product of all types forall a. a ?
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Bottom-up language learning is a teaching strategy that starts with basic language units and moves on to more complex structures. This is the most common technique used to teach students their native language or introduce a new foreign language.
The mathematical symbol for bottom is '⊥'. That's Unicode character 22A5 hex = 8869 decimal. Also available in HTML as '⊥' and in LaTeX as '\bot' (within math mode).
The bottom() function ranks the time series described by the expression, and then uses that ranking to represent each time series as either the constant 1 or the constant 0: 1 is returned for each time series that is in the specified number of series at the bottom of the ranking.
Bottom is a constant of propositional logic interpreted to mean the canonical, undoubted contradiction whose falsehood nobody could possibly ever question. The symbol used is ⊥.
In Math
Bottom is a type that has no value. That is : any empty type can play the bottom role.
Those f :: forall a . Bottom -> a functions are the empty functions. "empty" in the set theoretical definition of functions.
In Programming
Dedicating a concrete empty type to have it as bottom by a programming language base library is for convenience. Readability and compatibility of code benefits from everyone using the same empty type as bottom.
In Haskell
Let us refer to them with the more friendly names "Bottom" -> "Void", "f" -> "absurd".
{-# LANGUAGE EmptyDataDecls #-}
data Void
This definition does not contain any constructors => an instance of it can not be created => it is empty.
absurd :: Bottom -> a
absurd = \ case {}
In the case expression we do not have to handle any cases because there are none.
They are already defined in package base
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