Vignette:
So a guy walks into a bar and says, “Barkeep, did you know that there are no such things as paradoxes”?
The barkeeper says, “How so”?
The guy says, “Well, like most paradoxes, the sentence now named the Russell Paradox appears to be paradoxical but Wittgenstein, in Tractatus 3.333 shows us that it is simply a linguistic confusion”.
The barkeeper says, “Wow! Now that’s funny”.
The guy says, “How so?”
The barkeeper says, “There are many counter examples to Tractatus 3.333, ranging from the identity function on the Rationals, permutations on n elements, Fractals, and even self referential coding in Computer Science”.
The guy says, “You ignorant fool! How can you dispute the greatest philosophical genius of the 20th century? Wittgenstein wrote: ‘one cannot speak of something that one does not know’”.
The guy leaves the bar.
The barkeeper thinks to himself, “Moron, I should have kept my mouth shut; I might have gotten more tip money”.
End Vignette:
The above dialog was a fictionalization of a real event; only the names and places have been changed to protect the guilty.
Ludwig Wittgenstein wrote the bulk Tractatus Logico-Philosophicus during the war years of 1914 through 1918.
In 1929 Wittgenstein submitted Tractatus as his doctoral thesis, which Russell, Moore and possibly others accepted.
Subjective Nonsense:
The acceptance of Tractatus as a doctorial thesis, it seems to me, was an amazing comedy/tragedy: a combination of the good ole boys club, fear of Wittgenstein’s bullying, the appeal of celebrity, and an ignorance of the nature of axioms.
You can read the Wiki bio, and form your own opinion, at:
en.wikipedia.org/wiki/Ludwig_Wittgenstein
It is an interesting read.
As far as an ignorance of the nature of axioms goes, many of the leading intellectuals of the day along with David Hilbert, someone whom I admire, all seemed to think that axiomatic objects were arbitrary. Hilbert thought that primitive geometric objects could be thought of as beer mugs. In real life, for an axiomatic system to be valid, one requires a “Domain of Discourse”.** Basically we need some background information. This lack of understanding fed the sense that it was all right not to understand the Tractatus. Therefore, for those responsible for granting Wittgenstein his doctorate, it was OK to neglect their due diligence.
End Subjective Nonsense:
(Well OK, who am I to judge?)
Some background on the Tractatus 3.333:
It should be noted that despite the fact that Wittgenstein later rejected much of Tractatus he always believed in this part of the document*.
Wittgenstein, who visited Frege and was referred by Frege to Russell, studied under Russell and was familiar with the Russell paradox.
The Russell paradox is derived from the statement:
Let R be the set of all sets that are not members of themselves.
Russell pointed out that R could not be a member of itself by definition and yet if it were not a member of itself then it would be a member of itself again by definition.
Wittgenstein wrote:
Any mathematician, physicist, or engineer that reads 3.333 out of context should, nearly immediately, recognize that it is wrong. In fact anyone that has taken a first year calculus course should recognize that this statement is wrong. This is because one of the first things one learns is the derivative of f(g(x)), where critically!, f and g are functions that map the Reals to the Reals. Specifically f(g(x)) must exist for these statements to be true. Even more trivially, there is no reason why f and g can not be the same functions.
I actually timed a friend of mine to see how long it would take for him to find the errors. It was nearly instantaneous. Certainly less than 1 second.
There are at least two reasons that one should not read 3.333 out of context.
One reason is that, in context, a function might not represent a mathematical function.
The second reason is that some interpreters think that Wittgenstein requires the functions to have functions as variables rather than simple variables as functions.
Mathematical versus Logical functions:
The first reason has some merit because, in the context of Tractatus, we are talking about logical propositions. At some point in time, someone decided to call a logical proposition with a variable a function.
An example of a logical proposition using a variable would be:
All x are mammals. This sentence would be true if we substituted horses for x.
Now let’s see what happens if we look at the Russell’s paradox and check it against Wittgenstein’s 3.333 exclusion.
Consider the sentence:
“Let R be the set of all sets that are not members of themselves.”
We can write:
Let R = the set of all sets that are not members of (sets that are not members of (themselves)).
Now substituting “sets that are not members of” for F and “themselves” for u. we can write:
Let R = the set of all F(F(u)).
This is actually a simple definition and not a proposition!!
We could make it into a proposition if we write:
If R = the set of all F(F(u)), then R is well defined. Proposition I ***
Well defined means that F(F(u)) exists for all u (suitably defined) and F(F(u)) is unique.
It is important to note that Wittgenstein does not actually require F to be a logical proposition. It can simply be an “expression” or in common parlance a variable such as above. He develops his thinking and definitions in 3.31 through 3.313.
Assuming 3.333 is true, Wittgenstein can claim that Proposition I is simply a linguistic confusion, because F(F(u)) does not exist. This means that with Wittgenstein’s restriction (3.333) we might get rid of all those pesky little self referential paradoxes.
So we now know that Wittgenstein did not intend F to be a proposition with a variable. It could simply be any propositional variable.
Counter Examples to 3.333 where F is a function of simple variables:
As an example let F(u) be the identity function F(u) = u where u is an element of the Rationals. Then, by substituting u for F(u) inside the outer brackets F(F(u)) = F(u). Again substituting u for F(u) on the right side of the equation we get F(F(u)) = u.
There is no linguistic or ontological confusion here because both u and F(u) are elements of the same set.
In order to see that F(F(u)) is well defined we will let F(F(u)) = v[size=85]1[/size] and F(F(u)) = v[size=85]2[/size]. Since F(F(u)) = u, we must have u = v[size=85]1[/size] and u = v[size=85]2[/size]. Therefore v[size=85]1[/size] = v[size=85]2[/size].
=ucubed.jpg)
Again u is defined over the Reals and F(F(u)) has a single value.
An example of the Fractals is the Mandelbrot Set which is defined:
Generally z[size=85]0[/size] is assumed to be 0
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By substituting for z[size=85]1[/size] in the last equation we get:
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Since this is true for all complex numbers, z[size=85]0[/size], we can write M in terms of
f(f(u)).
All functions that are defined by letting x[size=85]n+1[/size] = f(x[size=85]n[/size]) are called recursive functions and we can define f(f(u)).
Probably the most widely used recursive function is the logistics function which is defined as follows:
x[size=85]n+1[/size] = r(x[size=85]n/size) where r and xn are Real numbers and 0 < x[size=85]n[/size] < 1and r is any positive Real number.
This function has profound applications in biology and economics.
The reader might also consider the apparently self referential statements about genetic code. e.g. There exist objects in a biosphere that have a self replicating genetic form.
It is also widely reported that computer code is used self referentially in artificial intelligence programs. (I am not sure exactly how that works).
The Wiki article on self reference mentions a number of useful applications of self reference. It is located at:
en.wikipedia.org/wiki/Self-reference
Now for the second reason not to read 3.333 out of context:
In the book “A Companion to Wittgenstein’s Tractatus”, on page 149 Max Black writes Wittgenstein: “presumably [means] the variant concerning functions (rather than classes) that are not functions of themselves”.
It should be noted that the words classes and sets at the time of Wittgenstein’s writing were interchangeable. However, today at least in Mathematics, they have different meanings.
So perhaps Wittgenstein meant F to be a function of functions.
Let’s ask ourselves if we can think of any function of functions.
Again I offer an example from mathematics. Consider the length of a curved line. Basically there is a formula that depends on the function f and gives that function’s length. Therefore, generally, length is a function of functions.
The problem is that the length function maps functions to Real numbers. To avoid a linguistic or ontological confusion, we need functions which map functions to other functions.
Let’s consider rotations. These are relatively simple things that are studied in high school math. At least this is true in the US and since we are reportedly far behind many other countries I assume that this statement is mostly true.
This pair of equations represents a function f such that R X R (The Real plane) is mapped to R X R. It has the property that any set of points in on the Real plane will be rotated by the angle phi to another Real plane whose center coincides with the original center. (It also has the property that sets will retain the general shape, assuming that we are using a Euclidean metric to define distances).
This pair of equations can be represented by a 2 X 2 matrix as follows:
But we can multiply matrixes as follows:
Where w = ae + bg, x = af + bh, y = ce + dg, and z = cf + dh.
In order to map rotations to rotations we would need:
And we will assume that:
Now we let F =
Now we have F such that F maps rotations (functions) to rotations (functions).
Another example of a function that maps functions to functions is the Fourier transform
The Fourier transform F of f is defined as follows:
The exponent, in case you can not read it is:
Where f maps R to the complex numbers. (f also must be suitably restricted in order for the integral to exist under both integrations). Again F maps a function to a function and F is well defined (because the area under a curve is uniquely defined given any specified definition of integration).
And again F(F(f)) is well defined.
Analysis of Wittgenstein’s commentary:
Restatement:
Let’s start with the first sentence.
Wittgenstein is partially correct, mathematically, in saying F contains the prototype of its’ own argument. This is because functions require a predefined domain and this restricts how F may behave. However, the specific mapping can be completely free as long as the domain is mapped to a specified range. The additional restriction (it can not contain itself) is a bridge too far. There is no logical rule of inference**** to justify this reasoning. Wittgenstein probably considers an expression which contains itself a linguistic or ontological confusion, but he must prove that this is always the case.
While it is true that many times the image of F simply does not lie in the domain of F which would mean that F(F(u)) is simply nonsense (in other parlance a linguistic or ontological confusion), we can not generally draw this conclusion.
The second sentence.
The third sentence is:
Common to both functions is only the letter F, which by itself signifies nothing.
It is pretty clear that Wittgenstein never fully grasped the importance of the role that the domain and range of F play.
The concluding sentence:
One of the most interesting things about the last sentence is that Wittgenstein assumed that F(F(u)) existed and then arrived at what he thought was a contradiction. Basically, he thought he was providing a proof by contradiction.
The problem is that Wittgenstein was a very strict constructivist and rejected the excluded middle!
Therefore the perceived “proof” could not be right even by his own standards. It should be noted that Wittgenstein’s rejection of the excluded middle hardened later in his life.
You can read about Wittgenstein and the excluded middle at:
plato.stanford.edu/entries/wittg … thematics/
(The comments about Wittgenstein’s strong rejection of the excluded middle come in the paragraph just prior to section 2.2).
Both Russell and Wittgenstein knew a little mathematics. Frankly, it was a major blunder for them to let 3.333 go unchallenged. (Probably a bigger blunder for Russell who was watching his paradox become trivialized).
*Sparks Notes page 15
** en.wikipedia.org/wiki/Domain_of_discourse
***Wittgenstein makes a strong distinction between mathematical propositions and empirical propositions in other places in Tractatus and latter in his life, but it is clear that at 3.333 he was not entertaining that distinction. This is clear because the Russell paradox is dealing with the mathematical entity of sets. Much of this is discussed in the Stanford Encyclopedia of Philosophy article located at:
plato.stanford.edu/entries/wittg … thematics/
**** List of rules of inference:
en.wikipedia.org/wiki/List_of_rules_of_inference