# Mathematical Philosophy Question, trying to understand a discussion on Principia Mathematica

I will admit that this questions isn't exactly mathematics, but it involves mathematical logic so I thought I would try here. If I am in the wrong place just let me know.

I am reading 'Landmark Writings in Western Mathematics 1640 - 1940' by I. Grattan-Guinness. While reading the chapter on Russell and Whitehead's 'Principia Mathematica' (PM) I came across a portion that doesn't make sense to me. When discussing the clarity issues of PM it says: "The difficulty concerns incoherence of expression; it largely sprang from the inherited Peanist belief that logic was an absolutely general discipline, so that (as we now say) there is no room to talk about it."

It is the last part of the sentence that I don't understand. What is a "general discipline"? I tried looking this up and found nothing, so I am assuming it is not a term of mathematical philosophy. There is also, obviously some sort of insiders comment there when they say 'there is no room to talk', can you explain that as well? A part of me reads this and thinks they might just be implying that there are almost no descriptions for any of the approx. 500 definitions in the book, but that seems too simple a meaning. If you have any ideas about this I would love the help. Thanks!

• The quote alludes to the well-known fact that the Principia system has no meta-logic: the book does not ask (and has no answers for) questions about interpretation, consistency, complete essere of the logical system. Apr 16, 2022 at 13:29

The whole passage from the book quoted in the question reads:

The opening Part dealt with the propositional and predicate calculi, including both individual and functional (and relational) quantification. It is the best known and least clear portion of PM. The difficulty concerns incoherence of expression; it largely sprang from the inherited Peanist belief that logic was an absolutely general discipline, so that (as we now say) there was no room to talk about it. For example, the LEM [law of excluded middle], which is (and was) normally construed as a metalogical principle, was taken instead to be the proposition ‘p ∨ not-p’ in the calculus. (p.787, 789)

The penultimate sentence illustrates by an example the claim ‚the inherited Peanist belief that logic was an absolutely general discipline, so that (as we now say) there was no room to talk about it.‘:

The law of excluded middle was not considered part of metalogic – as usual – but as a proposition, i.e. a genuine part of logic.

The example also illustrates @Mauro’s comment.

Sometimes knowledge of logic in itself is thought of as (the most) generalized kind of knowledge, even moreso than mathematics. Kant said as much (with endless qualifiers about reason's limits) when he demarcated general from transcendental logic. (His conclusion: general logical laws, like non-contradiction, hold everywhere and forever, even for the an sich or noumena, but we are forbidden from trying to use those laws to prove more substantive things (about noumenal things in themselves).)

One phrasing of this supposition is topic neutrality:

Logic, it seems, is not about anything in particular; relatedly, it is applicable everywhere, no matter what we are reasoning about. ... Unfortunately, the notion of topic neutrality is too vague to be of much help when it comes to the hard cases for which we need a principle of demarcation. Take arithmetic, for instance. Is it topic-neutral? Well, yes: anything can be counted, so the theorems of arithmetic will be useful in any field of inquiry. But then again, no: arithmetic has its own special subject matter, the natural numbers and the arithmetical relations that hold between them. The same can be said about set theory: on the one hand, anything we can reason about can be grouped into sets; on the other hand, set theory seems to be about a particular corner of the universe—the sets—and thus to have its own special “topic.” The general problem of which these two cases are instances might be called the antinomy of topic-neutrality. As George Boolos points out, the antinomy can be pressed all the way to paradigm cases of logical constants: “it might be said that logic is not so ‘topic-neutral’ as it is often made out to be: it can easily be said to be about the notions of negation, conjunction, identity, and the notions expressed by ‘all’ and ‘some’, among others …” (1975, 517)

I can't say for sure, but the thing about "no room to talk" might be a reference to Tractarian Wittgenstein {Tractgenstein} and his ladder allegory, in association with the quote, "Whereof one cannot speak, thereof one must remain silent." This notion is caught up in the positivist (or at least specifically Carnapian) interpretation of "logical truth," which allowed for an impressive, surprising amount of pluralism; and then, in the Quinean evolutionary offshoot of positivism, we have Quine's own "change the logic, change the subject" mantra.

• Now I'm wondering if model theory somehow throws a wrench into this. One could argue, for example, that Peano arithmetic is "supposed to be" about the standard model of arithmetic (in which e.g. Goodstein's theorem is true), and not about any of the nonstandard models - but they all satisfy the same axioms, and they're all "arithmetic" of a sort. OTOH, under standard model theory, logic itself doesn't really have models per se, but only interpretations. Apr 17, 2022 at 0:35