Russell’s Paradox (a version of the liar’s paradox), is not a paradox, it’s an ill formed statement (Grammatical error) because it failed the test of continuously recursive ambiguity – which is what ‘grammar’ means: rules of continuous recursive disambiguation. Nearly all seemingly challenging philosophical questions play on some variation of the verb to be. In the case of the liar’s paradox in all its forms, it’s not a paradox it’s constructed ambiguity. Words don’t mean things. People mean things. They use language well or not well to state their meaning – or their deceit.
A number is the name of a position, and beyond the base (glyphs) we use ‘Positional Naming’. We can name anything we choose with a position in an order just like we can name anything else. All that matters is that we all rely on the same names in the same order. Numbers exist as names. That’s it. Nothing else.
Mathematics is ill-grounded (vulnerable to grammatical errors) because of sets (platonic, ideal, verbal) rather than operations (gears and geometry). If you explain all mathematics using positional names, gears, and geometry (as it was invented) you do not expose yourself to grammatical errors.
The same is true of philosophical (verbal) statements. If you state all statements as promises, in operational prose, in complete sentences, without the ‘cheat’ (or lie) of the verb to be, you will have a very difficult time make grammatical errors.
So the entire analytic program (sets) was a failure. So was the attempt to discover a via-positiva scientific method. This is because all epistemology is falsificationary and adversarial, with surviving truth propositions competing in networks of paradigms themselves in falsificationary and adversarial competition.
Most of philosophy is little more than sophistry. (really)
Everything that isn’t sophistry is in the domain of science including that science we call ‘grammar’.
=== COMMENTS ===
—“Not quite, as Godel presented a mathematical model of this phenomenon. You cannot reduce this to mere positivistic linguistics. On which point, are you not assuming Chomsky’s universal grammar with your definition of grammar? If so, this has been shown to be unempirical.”—Rik Storey
I didn’t say anything like that. I’m saying that he’s correct.
I haven’t met anyone other than the author of the best book on the subject that understands the limit of Godel’s argument:
(a) we identify new constant relations (experiences)
(b) we invent new references
(c) we invent new paradigms
(d) we require grammars to talk about them
(e) we can make ungrammatical statements.
Godel said it. Turing said it. Kripke said it.
So there is no closure to logic without appeal to the operational, empirical, limits and completeness, and even then there is only closure on falsification not justification.
There is nothing positivistic in P. It’s purely falsificationary. Either it survives adverstarial competition by the terms stated in testimonialism or it doesn’t. If more than one does, then we just don’t know and nothing else can be said.
—“Oh very well. In that case, we must still follow Godel’s Platonism because of the assumptions we make in a purely sceptical and empirical worldview. That or nihilism are our two consistent options.”—Rik Storey
I can’t translate that into operational language. I don’t know what you mean.
“…we must still follow Godel’s Platonism…”
(Godel’s argument was operational, by applying the technique of pairing off (the foundation of mathematics in positional names) producing unique names for operations. Not all statements available in all grammar and vocabulary will be decidable within that grammar and vocabulary. And he did this for the special case of addition as an example, under the presumption the model would hold. But all he is saying is that no language is closed (other than first order logics maybe. Same is true even for math. We can write formulae that are descriptive but not deducible (we can’t write a proof)).
and how does that relate to:
“purely skeptical and empirical”
(Permanently contingent, uncertain, cannot abandon continuous learning and adaptation?)
and what do you mean by:
(means of understanding, predicting, decision making? paradigm?)
—“I made a similar argument on a Philosophy page. Russell’s paradox is just a domain error. A barber in a set of barbers or a tree in a forest. In the objective empirical world, it’s just a grammatical error. In the abstract world of numbers, a set of all sets must contain itself. “All” being transcendent can break the normal rules.”—Andrew M Gilmour