Mathematics of the tractatus logico philosophicus

Wittgenstein

There was a general reading that covered the ontological part of the tractatus but it didn't cover the mathematical part of it, or the logical one to speak on a broader spectrum. There are many points addressed in tractatus regarding mathematics but l can only afford to discuss a few here without losing any depth on them.

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. 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.

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 here

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.)

This worked fine for equalities but Ramsey pointed out later to Wittgenstein that there will be a problem with regards to inequalities as we can substitute signs that are not equal. I don't know if wittgenstein resolved this.

There is also another interesting statement with regards to probability

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 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.
It is an interesting observation regardless of that.

I will discuss other statements in tractatus if we can get over these ones

alcontali

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))'

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. There is nothing inconsistent in the practice of function iteration, i.e. a Picard sequence. Furthermore, the remark puts into question the entire field of studying fixed points.

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)).

Wittgenstein sees a problem where there isn't one ...

ssu

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.

I think this statement makes total sense. You will get circularity otherwise.

alcontali

You will get circularity otherwise. — ssu

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.

Wittgenstein

↪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. There is nothing inconsistent in the practice of function iteration,

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 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, but it is hard to defend his viewpoint. Most philosophers reject his solution and pass over it casually.

The definition:

n>1: n! = n * (n-1)!
n=1: n! = 1

How about for n=0 : n!=1.
We can factorial using the gamma function too.
Gamma(n+1)=n! , interestingly we can use the integral to approximate values of (3.5) ! Or (5.7) ! but the real interesting part is how they extend the factorial.

What's your take on this statement in the tractatus

5.153 In itself, a proposition is neither probable nor improbable. Either an event occurs or it does not: there is no middle way

Wittgenstein

↪ssu

I think this statement makes total sense. You will get circularity otherwise.

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.

alcontali

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

Ludwig Wittgenstein was apparently one of Bertrand Russell's favourite students:

His teacher, Bertrand Russell, described Wittgenstein as "perhaps the most perfect example I have ever known of genius as traditionally conceived; passionate, profound, intense, and dominating".

In his Principia Mathematica, Russell had done some really important work, especially with his ramified type theory. It is the overly favourable opinion of a real grandee such as 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.

5.153 In itself, a proposition is neither probable nor improbable. Either an event occurs or it does not: there is no middle way

Yes, of course, but that is not what it is about. The observer does matter. Probability is about the relationship between observer and event, and not about the event "an sich". As Immanuel Kant famously wrote, "Das Ding an sich ist ein Unbekänntes".

Immanuel Kant. Prolegomena, § 32. And we indeed, rightly considering objects of sense as mere appearances, confess thereby that they are based upon a thing in itself, though we know not this thing as it is in itself, but only know its appearances, viz., the way in which our senses are affected by this unknown something.

Again, Ludwig Wittgenstein, who in my opinion is very overrated, was clearly late in the game to start fretting over this problem.

Wittgenstein

↪alcontali

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.

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:

Again, Ludwig Wittgenstein, who in my opinion is very overrated, was clearly late in the game to start fretting over this problematic

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.

sime

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.

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

The requirement of having a base case, is a well-known problem:

In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties:

1. A simple base case (or cases)—a terminating scenario that does not use recursion to produce an answer
2. A set of rules that reduces all other cases toward the base case

The way in which Wittgenstein formulated his objection does not allow for providing such base case. Therefore, I think that I have to reject his objection.

Wittgenstein

↪sime

Is this explanation valid or have l got it wrong. It can be hard to interpret wittgenstein sometimes

....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

ssu

is indeed somehow circular, but that is the essence of recursion. It works absolutely fine. — alcontali

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...

The standard argument against to this is typically that we can have working self-referring models, dynamic models or similar things, yet that is not the point: self reference opens the door to negative self reference. And with negative self reference you get all the troubles.

Like try to write an answer in this Forum that you don't write in this Forum. Such answers exist of course, answers one doesn't write, but naturally due to the negative self reference you cannot write such answers to this Forum. Yet self reference itself isn't the problem, as you surely can refer to what you have written earlier.

alcontali

The function itself cannot be input as then it does open up the for paradoxes and the circularity that Wittgenstein opposes. — ssu

To tell you the truth, I somehow suspect that I do not _really_ understand the objection voiced by Wittgenstein in 3.333. Is it related to Curry's paradox?

In the 1930s, Curry's Paradox and the related Kleene–Rosser paradox played a major role in showing that formal logic systems based on self-recursive expressions are inconsistent. These include some versions of lambda calculus and combinatory logic.

Is it about the use of the Y-combinator to allow anonymous functions to refer to themselves?

The heart of Curry's paradox is that untyped lambda calculus is unsound as a deductive system, and the Y combinator demonstrates that by allowing an anonymous expression to represent zero, or even many values. This is inconsistent in mathematical logic.

You can actually implement the Y-combinator in a run-off-the-mill scripting language such as Javascript:

Y = f => (x => x(x))(x => f(y => x(x)(y)))

So, in order to avoid self-reference for the purpose of recursion, you can use the Y-combinator instead:

(f => (x => x(x))(x => f(y => x(x)(y))))(
 f => (n => ((n === 0) ? 1 : n * f(n - 1))))(5)
//returns 120

It is theoretically unsound in ways, but it practically really works. That is indeed a strange thing.

Is that the problem that Wittgenstein has with self-reference?
His objection in 3.333 does not properly explain what exactly his problem is ...

ssu

To tell you the truth, I somehow suspect that I do not _really_ understand the objection voiced by Wittgenstein in 3.333. — alcontali

Your honest modesty here has a grain of wisdom in it.

Wittgenstein's whole book is written in a style that seeks to be simple and short as possible, but really isn't at all. When you lack the simple examples of "what you are talking about", it really leaves huge areas for interpretation.

And obviously this issue wasn't so clear to Wittgenstein or to others. Once Wittgenstein met Alan Turing, who tried explain his findings. The two giants of intellect didn't understand each others points and the counter wasn't fruitful.

Robert Durkacz

↪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

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? An example of a function which accepts a function as an argument would be something that integrates the supplied argument function over some interval and returns the result. Made up example: Int (cos) integrates cos() over a quarter cycle from 0 to pi/2 and returns 1. But in this case the integrating function is nothing like the functions which it takes as an argument - you could not think of operating the integrating function with itself as an argument, ie Int(Int). There is no difficulty in coming up with functions that can be composed with themselves, F(F(x)) where x is supplied, but very hard to think of a function that operates itself, and we would write it as F(F) because we are not supplying any x value.

TonesInDeepFreeze

Theorem: There is no S such that S is in the domain of S.

Proof: Use axiom of regularity.

jgill

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

As a retired mathematician I read the OP and thought, This is utter rubbish. I have worked with infinite compositions of functions for years. But some degree of clarification followed. Thinking of a function as a collection of ordered pairs helps.

Robert Durkacz

If we could go back right to the start, this is paragraph 3.333 of Ludwig Wittgenstein's Tractatus Logico-philosophicus (in the original post the Greek and mathematical characters were lost):

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.
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 φ(fx) and the outer one has the form ψ(φ(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 ‘(∃φ): F(φu).φu = Fu’.
That disposes of Russell’s paradox.

This statement looks like it should be plain enough, but it seems that no-one in the world really knows what the point of the second last line is. That is how I got here. Some clues that I have picked up: Although it seems to be a sensible discussion about mathematical functions it may be more particularly aimed at logical propositions expressed as functions or set theory propositions expressed as functions. The notation seems to be non-standard but guessable, however it supposedly goes back to Principia Mathematica as described in [url=https:// plato.stanford.edu/entries/pm-notation/] plato.stanford.edu/entries/pm-notation/[/url]

Raymond

Isn't the function of the function a functional?

Robert Durkacz

↪Raymond

yes I think that is the correct term. But then what is a function of a functional - a functionalal?

Raymond

yes I think that is the correct term. But then what is a function of a functional - a functionalal? — Robert Durkacz

Haha! Well, the integral, for example, is a functional. It's a function of a function. Every function returns a value (considering bound, continuous functions only). The area between the function and the domain axis, say x (say there is an x domain only). How would the integral of the integral look like? You calculate the area first. Of an arbitrary function (well, bounded, continuous, and one variable). Say you calculate the integral of f=x. So F(f(x))= int(f(x), say between 0 and 2. So the integral is 1/2x^2+c, put in the values 2 and 0, and subtract. So, you get two. Now what about F(F(f(x))? Is it 1/6x^3+cx+b? The integral of the function returns a number, the area. If you apply the integration to this number, what do you get? If you integrate f(x)=x between 0 and 2, how is the integral of this integral defined?
Note that differentiation is not the inverse of integration. If you see differentiation as an operator, the operator is applied not to the whole function, but only pointwise to the range points and the domain points (df(x)/dx). The function can be seen as an operator on the. If the operator is to multiply by one, then f(x)=x and f(f(x))=x also. The integral operator takes the whole function as argument. All points of a function are involved, the domain as well as the range, like in the integration (dx and f(x)). Differentiation is no functional. The process of finding a primitive function is the inverse of finding the derivative function though.

jgill

Infinite Compositions of Analytic Functions

If F is a functional operating on a function f, then F(f) makes sense, but F(F(f)) does not. Functionals map functions to real or complex numbers and not to other functions. Operators can take functions to functions.

EnPassant

Interestingly Pollard's Rho method uses this kind of iteration and waits for circularity to factor integers-

$x = 3$

$f(x) = x^2 + 1 (mod n)$

$f(x) = x_{1}$

$f(x_{1}) = x_{2}$

$f(x_{2}) = x_{3}$ etc

Raymond

↪EnPassant

What means the n in modn? The n in the xn? What is the goal of the method?

EnPassant

↪Raymond

n is the modulus. If pq = n the algorithm is designed to find p or q.

Mod n finds the remainder on division.

eg 114 = 14 (mod 100) because 14 is the remainder on division by 100.
14 o'clock = 2 o'clock (mod 12), on a 12 hour clock. Modular arithmetic is sometimes called clock arithmetic.

Search for Pollard's Rho Algorithm

Raymond

↪EnPassant

I mean, what's the value? Is it to be found?

Ah! It factors integers. All clear.

EnPassant

↪Raymond

n is known and its factors, p and q are unknown. p or q are to be found.

Robert Durkacz

Back to the original post again, this is paragraph 3.333 of Ludwig Wittgenstein's Tractatus Logico-philosophicus (in the original post the Greek and mathematical characters were lost):

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.
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 φ(fx) and the outer one has the form ψ(φ(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 ‘(∃φ): F(φu).φu = Fu’.
That disposes of Russell’s paradox.

You can take this at face value. No normal mathematical function could operate on itself as an argument. For example, considering a function from real numbers to real numbers like cosine. The type of the function is a real-to-real mapping, the type of the argument is real - two very different things. We are not talking about function composition -many of the comments have gone onto that but it is another subject entirely. (The book by Black mentioned below is clear on that.)

Wittgenstein's target is evidently Russell's paradox. Russell's paradox concerns sets that may include themselves. If we disallow sets from including themselves by making some analogy with functions then Russell's paradox could not arise. This is obvious in itself.

Some decent articles on the internet explain the importance of Russell's paradox to mathematical logic. It can be seen that Wittgenstein's suggestion made no impact in this field. He is known as a philosopher not a mathematician. It can also be understood how he came to be aware of it since he was close to Russell, who was both. In any case, the reference to Russell's paradox in the Tractatus just looks like an aside. It does not figure in anything that follows in the Tractatus.

The part that does not have an obvious interpretation is the second last line, which is a mathematical expression. Presumably this line would make the point clear and so we can turn to commentaries to explain it to us. I know of two commentaries on the Tractatus, the Routledge 'study guide' by Michael Morris and the Companion by Max Black (see below for links). Morris ignores the second last line entirely and Black offers a lame kind of comment that is not worth quoting. Neither of these commentaries otherwise advances us beyond the face value understanding that I have expressed.

The only conclusion is that the whole of the 3.333 paragraphs are a waste of words, being amateurish contributions on a specialised topic made to look more intelligent than they are by being obscure. In a way I hope I am wrong. I was expecting something else when I got around to reading Wittgenstein.

No longer giving W the benefit of the doubt, the talk about F have two unrelated meanings is also a waste of words. When one uses a mathematical symbol twice, the meaning is the same each time. W should simply say that it is being used incompatibly in the two occurrences.

Are the 3.333 statements as poor as they appear and if they are how can we take the rest of the Tractatus seriously, when we are forever trying to fill in the gaps in order to discern the meaning? Perhaps there is nothing there to be found.

Commentaries on the Tractatus

https://www.routledge.com/Routledge-Philosophy-GuideBook-to-Wittgenstein-and-the-Tractatus/Morris/p/book/9780415357227
https://www.google.com.au/books/edition/A_Companion_to_Wittgenstein_s_Tractatus/uzY9AAAAIAAJ?hl=en

Russell's Paradox

https://plato.stanford.edu/entries/russell-paradox/
https://www.britannica.com/topic/Russells-paradox , etc

jgill

A function defined in set theory as a certain kind of collection of ordered pairs - not necessarily real or complex numbers - might be connected to this topic. Other than that I've demonstrated that in a general iterative sense a functional has this property since a functional operates on a function and produces a real or complex number, not another function.

It's a waste of time.

Mathematics of the tractatus logico philosophicus

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