How is this approach different from theory of types which wittgenstein thought was not good enough.It appears that he wants to have a different hierarchy of functions. So we cannot express express statements of second order in first order functions. Does this only work in propositional logic since in maths, f(x) can be used an argument to f(x) and ff(x) is well defined as a mapping, but wittgenstein despised set theory later on despite its success. Russell's paradox was resolved by not allowing statements like , "the set of all sets " and correcting axioms that allow us to generate such statements. Wittgenstein viewed maths as a form of logic, a manipulation of symbols that can be replaced as he writes here3.333 The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition 'F(F(fx))', in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(do) : F(Ou) . Ou = Fu'. That disposes of Russell's paradox.
4.241 When I use two signs with one and the same meaning, I express this by putting the sign '=' between them. So 'a = b' means that the sign 'b' can be substituted for the sign 'a'. (If I use an equation to introduce a new
sign 'b', laying down that it shall serve as a substitute for a sign a that is already known, then, like Russell, I write the equation-- definition--in the form 'a = b Def.' A definition is a rule dealing with signs.)
In a sense this is true but we will have a problem assigning truth values to statements regarding the future, for example if l say, " it will rain tomorrow ", the states of affairs at present do not provide any concrete information regarding the statement but it is nevertheless a sensible statement as it falls under the possible states of affairs.5.153 In itself, a proposition is neither probable nor improbable. Either an event occurs or it does not: there is no middle way.
For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition 'F(F(fx))'
in that case there would be a proposition 'F(F(fx))', in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)).
Yes, it is quite often used in mathematics and computer science, like the iterative function f(x)=x , hence f(f(x))=x and so on. I don't think wittgenstein defined function in set theoretic terms and a function was more or less considered to be a transformation , so f(x) was a propositional functions of the following statementF(F(x)) is allowed only if the co-domain is equal to or a subset of the domain of F(x). Beyond that, I don't see what the problem is with the repeated application of functions. There is nothing inconsistent in the practice of function iteration,
The definition:
n>1: n! = n * (n-1)!
n=1: n! = 1
5.153 In itself, a proposition is neither probable nor improbable. Either an event occurs or it does not: there is no middle way
In my opinion, it has got to do with the semantics as in propositional logic , inserting f(x) into f(x) can cause us to have meaningless statements which may look paradoxical but aren't really. I believe wittgenstein made a an error here as the solution undermines the real reason behind the paradox.I think this statement makes total sense. You will get circularity otherwise.
He was trying to show that it was a problem of semantics and I think this was a little of what wittgenstein was getting at, but it is hard to defend his viewpoint. — Wittgenstein
5.153 In itself, a proposition is neither probable nor improbable. Either an event occurs or it does not: there is no middle way
The influence Wittgenstein had on Russell was partially due to the how wittgenstein approached problems. Russell wanted philosophy to be build into some kind of a grand theory and his logicism too for maths where we can have analytic tools to study problems and correct them but his famous student was trying to draw boundaries and to throw out any systematic attempts to build one. It also has to do with the environment around cambridge at that time maybe, since Hardy a close friend of Russell had found a genius in such a romantic way, Ramanujhan who was clearly on par with the brilliance of euler and gauss. Maybe that prompted Russell to declare his genius successor :wink:Bertrand Russell on Wittgenstein that is so misleading. Seriously, I really do not see what exactly would be so inspiring about Wittgenstein's own work.
Wittgenstein was influenced by Kant undoubtedly as the latter tried to draw boundaries on metaphysics by describing the limits of the mind and the former tried to draw boundaries based on logic and language.Again, Ludwig Wittgenstein, who in my opinion is very overrated, was clearly late in the game to start fretting over this problematic
The definition:
n>1: n! = n * (n-1)!
n=1: n! = 1
is indeed somehow circular, but that is the essence of recursion. It works absolutely fine. Wittgenstein does not seem to handle that. — alcontali
Wittgenstein is rather attacking the heuristic semantic notion of "self reference" in relation to the iterative evaluation of a sequence of expressions via recursive substitution. Unless the iteration eventually halts, the resulting sequence isn't even sentence, never mind a proposition. Yet if the iteration is halted, each resulting sub-expression has non-equivalent arguments. — sime
....Yes, it is quite often used in mathematics and computer science, like the iterative function f(x)=x , hence f(f(x))=x and so on. I don't think wittgenstein defined function in set theoretic terms and a function was more or less considered to be a transformation , so f(x) was a propositional function of the following statement
f(x) = x belongs to a set A, let x be any natural number.
f(f(x))= f(x) belongs to set A, but f(x) isn't a natural number. He was trying to show that it was a problem of semantics and I think this was a little of what wittgenstein was getting at
Recursion isn't the thing here and creates confusion, because we indeed use models with 'self reference' all the time. Still, with recursion we have a starting point, a base case, from which function then goes on. Yet this is a different issue from a far more simple issue that I think Wittgenstein is talking about. A function, a 1-to-1 mapping, is where each input has a single output and you have the function as a 'black box' in between to get from input to output. The function itself cannot be input as then it does open up the for paradoxes and the circularity that Wittgenstein opposes. Hopefully people understand here the difference between a recursive function that starts and evolves. And once you have that black swan there...is indeed somehow circular, but that is the essence of recursion. It works absolutely fine. — alcontali
The function itself cannot be input as then it does open up the for paradoxes and the circularity that Wittgenstein opposes. — ssu
Y = f => (x => x(x))(x => f(y => x(x)(y)))
(f => (x => x(x))(x => f(y => x(x)(y))))( f => (n => ((n === 0) ? 1 : n * f(n - 1))))(5) //returns 120
Your honest modesty here has a grain of wisdom in it.To tell you the truth, I somehow suspect that I do not _really_ understand the objection voiced by Wittgenstein in 3.333. — alcontali
F(F(x)) is allowed only if the co-domain is equal to or a subset of the domain of F(x). Beyond that, I don't see what the problem is with the repeated application of functions. — alcontali
3.333 The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself.
alcontali is taking the expression F(F(x)) as function composition, ie compute x1 = F(x) and then compute F(x1) , and x could be an integer and F(x) returns an integer. That is different from the question considered by Wittgentstein, which is could there be a function which takes the function itself (the mapping) as an argument? — Robert Durkacz
yes I think that is the correct term. But then what is a function of a functional - a functionalal? — Robert Durkacz
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