Tarski and Mentalese

This was originally posted on Friday, August 30, 2013

Blogging gives me a chance to unload many years of unpublishable thinking.  One of the things that I have thought long and hard about is semantics in the sense of Tarski’s definition of truth and its relationship to Platonist vs. formalist philosophies of mathematics.  Here I will try to reconcile Platonist and formalist philosophies and relate both to the the notion of semantics for natural language.We are faced with two positions on the nature of mathematics both of which have powerful arguments.

Platonism: Gödel and Tarski each took a Platonic approach to meta-mathematics (mathematical logic). This is perhaps clearest when discussing the natural numbers.  The formulas of natural numbers can be defined by the following grammar.
t ::= 0 \;|\; 1 \;|\; x \;|\; t_1 + t_2 \;|\; t_1 \times t_2
\Phi ::= t_1 = t_2 \;|\; \neg \Phi \;|\; \Phi_1 \vee \Phi_2 \;|\; \Phi_1 \wedge \Phi_2 \;|\; \forall x \;\Phi[x] \;|\; \exists x \;\Phi[x]
If we let \rho denote an assignment of a natural number to each variable then Tarski defined the meaning or value {\cal V}\llbracket t \rrbracket\rho and {\cal V}\llbracket \Phi \rrbracket \rho recursively by equations such as
{\cal V}\llbracket x \rrbracket \rho = \rho(x)
{\cal V} \llbracket t_1 + t_2 \rrbracket \rho = {\cal V} \llbracket t_1 \rrbracket \rho + {\cal V} \llbracket t_2 \rrbracket \rho
{\cal V} \llbracket \forall x \;\Phi[x] \rrbracket \rho = T if for all n we have {\cal V} \llbracket \Phi[x] \rrbracket \rho[x := n] = T.
Here we have a clear distinction between syntax and semantics.  The syntax consists of the formal expressions and the semantics is the relationship between the formal expressions and the actual natural numbers.
These definitions of semantic values are Platonic in the sense that we, the mathematicians, are taking the natural numbers to actually exist and are taking English statements such as “for every natural number n we have …” to be meaningful.  Of course Platonism extends beyond natural numbers — working mathematicians adopt a Platonic stance toward all of the objects of mathematics.
Platonism seems essential to the clear-headed thinking of Gödel and Tarski both of whom proved fundamental (true) theorems in mathematical logic such as the statement that no finite axiom system for the numbers can be both sound and complete.  Such theorems cannot even be stated without reference to the actual, Platonic, natural numbers.  But neither can any normal arithmetic statement, such as Fermatt’s last theorem.  Fermatt’s last theorem is, after all, about the actual numbers.
Formalism:  I firmly believe that my brain performs classical (possibly stochastic) computation (I do not believe that my brain exploits quantum computing in any significant way).  Hence, whatever access I have to the actual natural numbers must be explainable by a computational process taking place in my brain.  The best model I know for what such a computation might look like is the manipulation of symbolic expressions.  It seems to be true that the axioms of Zermelo-Fraenkel set theory with the axiom of choice (ZFC) characterize the set of theorems that Platonic mathematicians accept as true.  So at some level the formalist position — that mathematics is ultimately just the manipulation of formal expressions — must be true.
Synthesis: Many years ago I was discussing Tarskian semantics with Marvin Minsky and he remarked that it all seemed silly to him — Tarski is just saying that the symbol string “snow is white” means that snow is white.  Paraphrasing Minsky — Tarskian semantics is just a way of removing the quotation marks.  Once the quotation marks are removed we become Platonists — with the quotation marks removed we are talking in the normal (Platonic) way.
This can be made a little clearer, I think, by considering a mathematician robot who, by construction, thinks by formal inference in a formal rule system.  Now, I argue, one can still imagine both Platonist and formalist Robots.  When thinking about arithmetic the Platonist robot manipulates formulas which are “about” numbers such as
There do not exist natural numbers x,y,z,w with w greater than 2 and x^w+y^w = z^w.
I have written an English sentence, but there is no reason an English sentence can’t be a formula (see my previous post on Wittgenstein’s public language hypothesis).  The formalist robot thinks about manipulating formulas.  The formalist robot manipulates formulas such as
If \Phi is derivable and \Phi \rightarrow \Psi is derivable then \Psi is derivable.
Again this is an English sentence just like the Platonic formula except that this sentence (this formula) is about formulas rather than about numbers.  A statement of Tarskian semantics is simply another formula (in English) defining a value function which maps terms to numbers and formulas to truth values.
Bottom Line: My position is that Platonic reasoning is simply formal reasoning taking place in a native mentalese.   Following Wittgenstein, I think that mentalese is closely related to the surface form of natural language.
When I think about the vector space \mathbb{R}^{100} there is no magical or mystical connection between my brain and an actual 100 dimensional vector space (Roger Penrose seems to think otherwise).  When I think about my mother there is no magical or mystical connection between my brain and my mother (many people think otherwise). Tarskian denotational semantics is a process of erasing quotation marks — of mapping external formal symbol strings into the language of mentalese.  This said, the Platonic treatment of formal logic is essential and central to the understanding of logic and Marvin should not be calling it silly.  (Marvin, if you read this please forgive me for my 30 year old possibly misremembered quotation).
When we talk about semantics from an NLP perspective we should be talking about a translation into some kind of mentalese and we should be striving for an understanding of that mentalese.
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