too many facts

Ludwig Wittgenstein began to write his Tractatus almost exactly hundred years ago and the little book would become quite important for the development of positivism and analytic philosophy.
It begins with proposition 1: "Die Welt ist alles was der Fall ist."
This is usually translated as "The world is all that is the case."
followed by proposition 1.1: "The world is the totality of facts, not of things."

But I think there is a problem with this proposition.
Let us begin with a simple fact, e.g. F1 = "I am."
This is a truly basic fact and I can immediately add another one: F2 = "I know that I am, i.e. I know F1."
But there is now another fact: F3 = "There is more than 1 fact."
We can now construct an infinite number of facts from these three facts: Fn = "There are more than n-2 facts." with n=4,5,...
An interesting question is if we can count all facts and I think the answer is No.
Let us assume that all facts form a countable set {F1, F2, F3, ...}
We then have another fact for each subset; e.g. F' = "{F1, F2, F3} is a subset of all facts."
But the power set of a set S (the set of all subsets of S) always has higher cardinality than S itself.
Therefore one cannot count all facts and in fact (!) they do not even form a well-defined set.

The set of all finite English sentences is countable and so is the set of all Turing machines. In other words, neither mathematics nor the English language is capable to handle the world of all facts as envisioned by Ludwig Wittgenstein (*).
Unfortunately, he never really discusses this.

I am aware, that almost all facts stated above are statements about other facts without reference to much else. We never really got beyond "I am." and "I know that I am." and "I know that I know ... that I am."
It would seem that something like Russell's type theory would be needed here.
But the Tractatus does not do that, so I recommend to jump straight to proposition 7 and skip all the stuff in the middle.


(*) A while ago I showed that a seemingly straightforward definition of "possible worlds" leads to something that cannot be handled by set theory, i.e. math as we know it.
I think it is amazing that already all the facts that follow from two basic facts are too many to be properly handled by set theory; notice that none of the facts I considered is self-referential or otherwise ambiguous.


9 comments:

Lee said...

It seems like Wittgenstein would have been aware of the problem since, Russell, König, Burali-Forti, et. al. discovered their paradoxes well before he started his Tractatus. As far as I know he was a pretty well informed guy so I wonder what he had in mind when he said "totality?"

wolfgang said...

I was wondering the same thing; he must have been aware of Russell's work, at least to some extent. But either he did not understand that his approach to philosophy was running into similar problems, or he assumed somehow that it was not an important problem (in the sense that one can practice naive set theory and ignore type theory for most practical purposes).

Lee said...

I think I remember reading somewhere that he didn't think Gödel's work was applicable to his own work either. It could be an illusory remembrance though.

wolfgang said...

As I know it, there is/was some debate if Wittgenstein understood (the implications of) Gödel's theorem. Obviously, in 1917/1918 he would not have known about it 8-)
But he should have known that the "set of all sets" leads to problems, because he met Russell six years earlier. So it would not have been a big stretch to suspect that "the totality of all facts" might lead to problems as well.

Lee said...

In 1930 Tobias Dantzig wrote a book entitled "Number." I thought I remembered something in the last chapter of the book that was germane to your post. I didn't really find what I was looking for but I think you might enjoy the quote below.

"A principle of relativity is an act of resignation, and a philosophical principle of relativity would consist in the frank admission of the insolubility of the old dilemma: has the universe an existence per se or does it exist only in the mind of man? To the man of science, the acceptance of the one hypothesis or the other is not at all a question of "to be or not to be"; for from the stand point of logic either hypothesis is tenable, and from the standpoint of experience neither is demonstrable. So the choice will forever remain a matter of expediency and convenience. The man of science will act as if this world were an absolute whole controlled by laws independent of his own thoughts or acts; but whenever he discovers a law of striking simplicity or one of sweeping universality or one which points to a perfect harmony in the cosmos, he will be wise to wonder what role his mind has played in the discovery, and whether the beautiful image he sees in the pool of eternity reveals the nature of this eternity, or is but a reflection of his own mind."

wolfgang said...

>> has the universe an existence per se or does it exist only in the mind of man?
I guess this is a variant of the interpretation problem ... in Copenhagen (classical) reality is only in in the mind of the observer.

Lee said...

After reading the rest of what he says in that chapter and elsewhere in the book fairly closely, I think he is arguing that the underlying reality is whatever humans (I assume humans who think about such things) say (think) it is at a given point in time. The final two sentences of the last appendix are:

"In this, then, modern science differs from its classical predecessor: it has recognized the anthropomorphic origin and the nature of human knowledge. Be it determinism or rationality, empiricism or the mathematical method, it has recognized that man is the measure of all things, and that there is no other measure."

Again sorry about being off topic.

Arun said...

Reminds me of one of Zeno's paradoxes, how does Achilles overtake the tortoise?

Let us start with facts as points in a topological space. A neighborhood of F consists of F and the facts that use F, such as the fact F1 = "{F} is a subset of facts". Perhaps we can answer questions about this topological space, e.g., is it separable? (i.e., does it contain a countable dense subset?) If it is, then the main objection about Wittgenstein might evaporate: "The world is a countable dense subset of the topological space of facts".

wolfgang said...

I am afraid if facts could be arranged as a set S of e.g. points in a topological space, then there would be a new fact for each subset of S; as simple as e.g. "there is a subset ...".
But the powerset of all such subsets cannot be in S.