Earlier quoted context omitted.
The algorithm is the abstract class, or possibly the spec, and software is one possible implementation. They’re at different levels of abstraction.
Both software and algorithms are an emergent property of the rules of the systems we create. They are essentially akin to prime numbers, in a direct and provable sense. (1) All computable programs can be expressed as a Universal Turing machine. (2) There are countably many Universal Turing machines (3) In an abstract sense, we can thus assign a unique identifier to each possible computer program. I can tell you "run…
Second, even for algorithms which can be expressed as Turing Machines, you are giving the computer science equivalent of saying that a house and it's architecture drawing are the same. One is a thing which exists, one is a description of the thing. You cannot run an unimplemented algorithm without first doing the work of implementing it.