Saying and Showing
Witttgenstein, Tractatus Logico-Philosophicus, proposition 3.331:
“Von dieser Bemerkung sehen wir in Russells ‘Theory of types’ hinüber: Der Irrtum Russells zeigt sich darin, daß er bei der Aufstellung der Zeichenregeln von der Bedeutung der Zeichen reden mußte.”
The Ramsey/Ogden translation (1922):
“From this observation we get a further view—into Russell’s Theory of Types. Russell’s error is shown by the fact that in drawing up his symbolic rules he has to speak about the things his signs mean.”
This remark is of significance. It has a background that shaped Wittgenstein’s thought for the remainder of his life.
Part of the background was formed by Frege’s attempt to reduce mathematics to logic, and also by Russell’s Paradox which evolved from this attempt: the paradox (and Russell’s attempt to solve it through the Theory of Types) led Wittgenstein to introduce a distinction between “saying” (sagen) and “showing” (zeigen). By this distinction, Wittgenstein came to believe he had resolved (that is: not solved, but shown as empty) Russell’s Paradox, as he proclaimed in proposition 3.333:
“[…] Nehmen wir nämlich an, die Funktion F (fx) könnte ihr eigenes Argument sein; dann gäbe es also einen Satz: „F (F (fx))“ und in diesem müssen die äussere Funktion F und die innere Funktion F verschiedene Bedeutungen haben, denn die innere hat die Form ϕ(fx), die äussere, die Form ψ(ϕ(fx)). Gemeinsam ist den beiden Funktionen nur der Buchstabe „F “, der aber allein nichts bezeichnet. / Dies wird sofort klar, wenn wir statt „F (F (u))“ schreiben „(∃ϕ) : F (ϕu) . ϕu = Fu“. / Hiermit erledigt sich Russell’s Paradox.”
The Ramsey/Ogden translation (1922):
“[…] If, for example, we suppose that the function F(fx) could be its own argument, then there would be a proposition “F(F(fx))”, and in this the outer function F and the inner function F must have different meanings; for the inner has the form ϕ(fx), the outer the form ψ(ϕ(fx)). Common to both functions is only the letter “F”, which by itself signifies nothing. / This is at once clear, if instead of “F(F(u))” we write “(∃ϕ):F(ϕu).ϕu=Fu”. / Herewith Russell’s paradox vanishes.”
To this day, logicians criticize Wittgenstein for proposition 3.333, claiming that what they see as his “solution” (i.e. writing “(∃ϕ):F(ϕu).ϕu=Fu” for “F(F(u))”) presents no solution at all. This is right and wrong, as far as I am concerned. For considering it wrong, I see the following two reasons: (1) we should stop focusing on “(∃ϕ):F(ϕu).ϕu=Fu” and shift weight to “F(F(u))” as the more important expression of the two (no logician does that!): (2) we should be looking for what “F(F(u))” shows (zeigt), not for what it says (sagt); however, the latter is what logicians aim their attention at.
How should we look for what “F(F(u))” shows?
Here is a hint: “(∃ϕ):F(ϕu).ϕu=Fu” is a “picture” (Bild, see Tractatus 2.12) of the “world” (Welt, see Tractatus proposition 1). “F(F(u))”, on the other hand, makes no sense as a picture of the world. That is the point. “F(F(u))” only makes sense in, for example, a prayer; take St. Augustine’s “An nun opus habes, ut quoquam continearis, qui contines omnia, quoniam quae imples continendo imples?” (Confessiones I.3). Phrased as such, “F(F(u))” shows something.
What does it show? Tractatus proposition 6.45: “Die Welt als begrenztes Ganzes” (Ramsey/Ogden: “the world as a limited whole”).
This—the world as a limited whole—shows as a “feeling”, Wittgenstein says in the same proposition (Gefühl, ancient Greek: πάθος, pathos; this pathos is the beginning of philosophy, the ἀρχὴ φιλοσοφίας, archê philosophias: Plato, Theaetetus 155D; and so, proposition 6.45 forms the very heart of the Tractatus).
How frustrating it must have been for Wittgenstein that nobody ever saw what he meant by rewriting “F(F(u))” as “(∃ϕ):F(ϕu).ϕu=Fu”. It had the opposite effect from what he had intended: it blinded people rather than allowing them to see. In that sense, the critics of proposition 3.333 were right.
But it’s a wrong way of being right.
Note:
The image shows the interior of Wittgenstein’s (recently reconstructed) house in Skjolden, Norway. Wittgenstein wrote much of the Tractatus in Skjolden in 1913-14, before the house was finished, in the form of a manuscript called Logik.