What does the "stateless functional view of algorithms" mean?
We can think of an algorithm in the abstract as a description of a mechanism that takes certain inputs and produces certain outputs independent of the state of the world. If you have a mechanism called "sum" that takes in 2 and 3 and produces 5, it always produces 5 when given 2 and 3, no matter what the phase of the moon or who is the king of France today. There is no "state" that the algorithm must consult other than the values provided as arguments.
The distinction the author is drawing is that even in algorithms that are "pure" in this respect, it is often convenient to do some pre-processing, store that as state, and then use that state during the operation of an algorithm. Even if the result is an algorithm that appears to not depend on the state of the world, internally it is consuming or modifying its own private universe of state.
I think that the standard view of algorithms are based on states, correct?
The standard view of algorithms in OO or procedural languages is a sequence of mutations to state, yes. But that is not the standard functional view. The functional view is that algorithms have outputs that depend on inputs and there is no mutable state.
It is always possible to write a stateful algorithm as a stateless algorithm; you just make the state one of the inputs to the algorithm, and you make the outputs include a copy of "changes" to the state; nothing is actually mutated. But it is often inconvenient, inelegant or non-performant to do so in practice.
Of course the reverse is also true; many algorithms that are written as a sequence of mutations are vulnerable to nasty bugs where mutations are performed inconsistently or out of order. By rethinking an algorithm as a production of values from inputs with no mutations, you can often get significant performance wins (thanks to persistence, memoization, and so on) and remove sources of bugs.
