A function is something that takes a bunch of inputs and returns one or more values. If the returned values are entirely determined by the inputs, and the function doesn't have any side effects (logging, perhaps, or causing state changes outside itself), then it's called a pure function.
A procedure is a function that doesn't return a value. In particular, this means that a procedure can only cause side effects. (That might include mutating an input parameter!)
A method is a function that closes over a set of variables, that is, a closure. It takes zero or more input parameters, has access to this set of variables, and returns zero or more values. In OO languages these methods are attached to objects or classes.
In most mainstream OO languages, those closed-over variables are called the member fields, or instance variables, of an object. A method can be a pure function, an impure function or a procedure.
practically speaking, there's really no difference, with the slight exception that "method" usually refers to a subroutine associated with an object in OO languages.
The terms "procedure, function, subroutine, subprogram, and method" all really mean the same thing: a callable sub-program within a larger program. But it's difficult to come up with a definition that captures all variant usages of these terms, because they are not used consistently across programming languages or paradigms.
You might say a function returns a value. Well, the following C function doesn't return a value:
void f() { return; }
...but I doubt you'd find anyone who would call it a procedure.
Sure, in Pascal, procedures don't return values and functions return values, but that's merely a reflection of how Pascal was designed. In Fortran, a function returns a value, and a subroutine returns multiple values. Yet none of this really allows us to come up with a "universal" definition for these terms.
In fact, the term "procedural programming" refers to a whole class of languages, including C, Fortran and Pascal, only one of which actually uses the term "procedure" to mean anything.
So none of this is really consistent. The only exception is probably "method", which seems to be used almost entirely with OO languages, referring to a function that is associated with an object. Although, even this is not always consistent. C++, for example, usually uses the term "member function" rather than method, (even though the term "method" has crept into the C++ vernacular among programmers.)
The point is, none of this is really consistent. It simply reflects the terminology employed by whatever languages are en vogue at the time.
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