Did early Wittgenstein view mathematics as "sense-less" or "non-sensical"?

Question

G. E. M Anscombe makes the following distinction between Wittgenstein's use of sense-less (sinnlos) and nonsense (unsinnig): (page 163)

We must distinguish in the theory of the Tractatus between logical truths and the things that are 'shewn'; logical truths, whose character we have already discussed, are the 'tautologies', and are 'sense-less' propositions (lacking TF poles), their negations being 'contradictions'; attempts to say what is 'shewn' produce 'non-sensical' formulations of words - i.e. sentence-like formations whose constituents turn out not to have any meaning in those forms of sentences - e.g. one uses a formal concept like 'concept' as if it were a proper concept.... Here the attempt to express what one sees breaks down.

Given that logical truths have no true-false poles because they are "sense-less" (not contingent facts of the world), I assumed Wittgenstein would also claim mathematical well-formed formula to be the same. However, Alan Weir suggests that "mathematical utterances" for Wittgenstein may be "non-sensical":

Care must be taken, however. Wittgenstein distinguishes utterances which are sinnlos, which lack sense (including logical tautologies and contradictions here) from those which are unsinnig, nonsensical; it is not clear into which class mathematical utterances fall. One might well think that the game formalist should treat mathematical utterances, on that view just strings of meaningless marks, as unsinnig, not just sinnlos.

This leads to my question whether Wittgenstein views "mathematical utterances" as sinnlos (sense-less) or unsinnig (nonsense)?

Any reference to someone who has considered this question, besides Anscombe or Weir, is all I expect as an answer. I assume Wittgenstein views mathematical utterances as mathematical truths much like he views logical tautologies as logical truths and hence they are sinnlos because they are always true and so cannot have a false pole.

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Anscombe, G. E. M. An Introduction to Wittgenstein's Tractatus. (1971). St Augustine's Press.

Weir, Alan, "Formalism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/fall2019/entries/formalism-mathematics/.

Answer 1

At the time of the Tractatus (to which Anscombe and Weir refer), Wittgenstein was a logicist of sorts, so there was no principled difference for him between tautologies and mathematical propositions (with mathematics reduced to a fragment of arithmetic). So 4.461's sinnlos applies to both.

"4.461 The proposition shows what it says, the tautology and the contradiction that they say nothing. The tautology has no truth-conditions, for it is unconditionally true; and the contradiction is on no condition true. Tautology and contradiction are without sense [sinnlos].

4.4611 Tautology and contradiction are, however, not nonsensical [unsinnig]; they are part of the symbolism, in the same way that "0" is part of the symbolism of Arithmetic."

"Sense" (Sinn) here is Frege's term, and it is linked to truth conditions in Frege's semantics, which is Wittgenstein's point of departure. Here is Friedman's characterization of the Tractarian logicism from Logical Truth and Analyticity in Carnap's "Logical Syntax of Language" (p.85):

"For Wittgenstein, there can be only one language the single interconnected system of propositions within which everything that can be said must ultimately find a place; and there is no way to get "outside" this system so as to state or describe its logical structure: there can be no syntactic metalanguage. Hence logic and all "formal concepts" must remain ineffable in the Tractatus... Of course, the Tractatus is itself quite clear on the restricted scope of its conception of logic and mathematics in comparison with Frege's (and Russell's) conception. Wittgenstein's response to this difficulty is also all too clear: so much the worse for classical mathematics and set theory".

Weir's phrasing is unnecessarily convoluted and confusing, but right after the OP quote he writes:

"One clear difference from game formalism however is this: for Wittgenstein mathematics should not be conceived of as a calculus separate from other uses of language".

In other words, "game formalist" does not apply to Wittgenstein, and it is not really "not clear" where mathematical propositions fall for Wittgenstein. At least, for the Tractarian Wittgenstein, and at least, in his narrow range of "mathematics". Those parts for which "so much the worse" might not have been so lucky. On the evolution of Wittgestein's "sense" and "nonsense", as they apply to mathematics, see Mathematical Sense: Wittgenstein’s Syntactical Structuralism by Rodych.

But Wittgenstein soon abandoned his logicism, and this minimalistic conception of mathematics along with it. Mathematical propositions he later characterizes as "grammatical" (in his peculiar extended sense of "grammar" that covers any regimentation of use). Wittgenstein's mature view of mathematics is described in RFM (Remarks on the Foundations of mathematics) , and in IV.2 we read:

"I want to say: it is essential to mathematics that its signs are also employed in mufti. It is the use outside mathematics, and so the meaning of the signs, that makes the sign-game into mathematics. Just as it is not logical inference either, for me to make a change from one formation to another... if these arrangements have not a linguistic function apart from this transformation."

So again, while "sign-games" might be unsinnig, that is only because they are not "proper" mathematics (interestingly, Husserl said something very similar, contra Hilbert, in late 1920s). But the classical mathematics is now squarely sinnig by the mufti criterion. So much for game formalism. Wittgenstein did have some bones to pick, especially with the set theory, but if the label applies to it it would be because it stands apart from mathematics. He saw it as engaged in the same "dangerous, deceptive thing" as the unsinnig in philosophy generally.

Answer 2

I have Philosophical Investigations translated by Anscombe, Hacker, and Schulte, and it is bilingual, und auch kann ich ein Bißchen sprechen und lesen!

Unsinn, unsinnig - 39,40,79,13,134,197,246,252,282,464,448,512,524,540,PPF19,53,309,328 sinnlos - 71,157,247,358,361,500,554,PPF9,80,310

I've taken a look at the usage in these passages, and my amateur sleuthing suggests that Unsinn und unsinnig are used with negative connotation as exclamation or deprecation... Das ist Unsinn! (That's nonsense!) but in some places like words which have grammatical expression but are without rational meaning. (Time existed before time.) For sinloss, the translation seems to be more along the lines of a remaining connected with the intuitive meaning, which to me comes In PFF 9, the word is followed by the statement that there is something right about the "disintegration of the sense" (Colorless green ideas sleep furiously.) Note how Chomsky's sentence feels right even though it is contradictory.

The two words are used back to back in PFF 309 and 310, and the former with the phrase philosophers' nonsense (which I take to be contemptuous of metaphysical speculation), and the later referring to doubting which is without the realm of logical function as it is intentional, which seems to recognize that sentences that aren't strictly declarative are still attitudinal propositions.

My interpretation is that LW seems to recognize the locutionary-illocutionary-perlocutionary distinction, and that he recognizes that there are positive propositions as well as attitudinal ones. In this way, mathematical truths are not of the attitudinal sort, and do not have full meaning accept when invoked within a specific illocutionary context, and that is from the interplay between the illocutionary and the perlocutionary that words derive their strongest meaning and relevance in language games.

$0.02

Answer 3

Clearly, we don't want to put tautological truths and contradictions on the same footing semantically speaking.

By default, in mathematical formulas (wff), variables don't refer, and so, in this sense, they don't have a sense. They are senseless.

However, if ever they were used for representing some physical thing, they would thereby be given a sense.

The tautology A ⇔ A for example is true whether A has a sense or not, and so it would remain true if A was used to mean any real thing, for example that "All electrons have a negative charge". In this sense, a tautology is consistent with the whole of reality, and, in this sense, it is not nonsensical, even though it is not necessarily not senseless, and even though it is indeed senseless as we usually think of them.

A tautology would also be consistent in this way with any alternative reality, one for example where "No electron has an electric charge". However, and because of that, it is consistent, also, with the actual reality.

A contradiction is false and would remain false whatever its variables would be used to represent, including alternative realities.

Thus, there is a distinction between tautologies which are usually senseless but not nonsensical because they could be given a sense, and contradictions which are nonsensical because they cannot possibly be given any sense. They cannot make sense. They are nonsensical.