TITLE: Is there any rigorous mathematical definition of "Algorithm"?
QUESTION [5 upvotes]: I know the term "Algorithm" is used in "Computer science" and "Computer programming" as any set of operations done in an ordered way to solve a problem. But this definition seems not a rigorous mathematical definition.
Is there any rigorous definition of the concept of "Algorithm" in the scope of "Pure mathematics"?

REPLY [6 votes]: Speaking as a computer scientist here, I think the important thing to understand is that in practice the word "algorithm" is generally not used to denote any rigorous technical concept.
When people speak about "an algorithm" what they're referring to in 95% of cases is a fuzzy concept of the idea behind how a computer program does what it does. (In this context a "program" is generally imagined to be an mainframe-era batch-mode thing that takes some input and produces some output, not one of those newfangled 'applications' that you interact with). In this context the program itself is a precise formal object, but when we speak about "which algorithm does this program use for such-and-such" or "this program and that one use the same algorithm", what we're speaking about is ideas in the programmer's head, not a formal thing that can be written down and compared for equality.
At least in computer science (and practical software engineering) this usage is ubiquitous and non-controversial. And it looks very much to me like it is also this way in mathematics.
In mathematics and theoretical computer science we do make claims such as "there is an algorithm for such-and-such" (or "there is no algorithm ...") as if they are precise technical statements that are subject to proof or disproof. And they are, by a kind of terminological sleight of hand, namely that such claims are supposed to be interpreted as "there is a program that behaves like such-and-such". Or "there is a Turing machine", or "there is a set of mutually recursive definitions", or whichever formalism for computations we prefer to work in.
The underappreciated reason why this sleight-of-hand works in practice is that it doesn't really matter what you take a "program" to mean in this case, as long as it's a halfway reasonable one. The tasks that there are Turing machines that solve are exactly the same tasks that there are Haskell programs or rewrite systems or lambda terms to solve -- this is Church's thesis and is why we can get away with just saying "there is an algorithm" without bothering to explain up front what the exact kind of thing we assert exists is.
Saying "there is an algorithm ..." is useful in practice because it allows us to communicate the interesting content of the claim without being sidetracked by which precise formalism the speaker prefers working with. At least if both the speaker and listener both know that all of the formalisms are equivalent anyway.
There the many concrete formalisations of what it means for an algorithm to exist, each of which can be defined mathematically with all the rigor you want. Turing machines is one of them, and there's a wide selection of other similar "machine" concepts that make different trade-offs between definitional simplicity, ease of programming, and "realism": counter machines, random access machines, and so forth. Then there are various rigorously defined programming languages, including the lambda calculus which also counts as an example of general rewrite systems that can express computation. Finally there are more purely number theoretic formalizations, such as Gödel's concept of primitive recursive functions (appropriately extended to give all the computable functions), and so forth.
Again, each of these particular formalism can be rigorously defined. If I say "I'm thinking of an algorithm for such-and-such", what I have in mind is a mental concept that can be realized (with greater or less ease) in each of all of these formalisms, but I usually haven't committed to use any particular one among them.