# Turing's bridging argument of conflating mathematical logic and the philosophy of mind?

So I read this paper and I'll quote the relevant parts:

'Turing's machines are humans who calculate

On Computable Numbers' thus took on the aspect of a hybrid paper: an attempt to integrate what should be regarded as independent issues in mathematical logic and the philosophy of mind. And it was precisely Turing's bridging argument which concerned Wittgenstein.

(But why not argue): Turing's humans are really machines that calculate?

What he was trying to bring out is that the mathematical concept of calculation — as opposed to the empirical concept of counting — cannot be separated from its essential normativity.

*In other words the idea there are 2 distinct concepts here one is the concept of mathematics and the empirical concept that pertains to the physical world in Wittgenstein's eyes.

The basic problem is that it is completely misleading to speak of a 'mechanical deduction'; for the two notions operating here, inference and sign manipulation, cannot be used together. One can speak of comparing the orthography, size, etc., of meaningless marks, but not of deducing one string of meaningless marks from another. To grasp that q follows from/? just is to understand the nature of the conceptual relationship between the meaning of q and p: to know that p entails q.

But let us assume otherwise: One cannot separate these concepts. The question becomes does this lead to any sort of contradiction? Consider Wigner's friend extended. Roughly speaking, those assumptions are

(Q): Quantum theory is correct.

(C): Agent's predictions are information-theoretically consistent.

(S): A measurement yields only one single outcome.

Again an attempt is made assuming mathematical logic and philosophy of the mind can be unified. However, this time there is a clear contradiction that emerges (if one assumes the problems stem from C).

Assumption (C) invokes a consistency among different agents' statements in the following manner: The statement "I know (by the theory) that they know (by the same theory) that x" is equivalent to "I know that x".

Notice, the same but more advanced version of the conflation happens here:

1. know (by the theory): a mathematical concept
2. They (are in superposition): An empirical concept
3. know (by the same theory) that x: A mathematical concept

This is again a conflation of 2 distinct categories. Though it is self referential something interesting happens in the realms of mathematics: a proof by contradiction. Note, this assumes a mechanical deduction (inference and sign manipulation, are used together) when going from point 2 to point 3. In other words it is presumed: Turing's humans are really machines that calculate

Is it a fair argument to say that if one naively conflates the mathematical logic and the philosophy of mind and model's the universe one ends up with contradictions as one would expect if they were indeed distinct phenomena?

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Apr 7 at 23:22
• Interesting papers, but I cannot understand what Wittgenstein's issue with the bridging argument, which was based on his idea of rule-following requiring practiced communal norms, has to do with the Frauchiger-Renner's no-go theorem, which is a typical consequence of self-reference. Turing was very familiar with the analogous Godel's result and the halting problem, which are more relevant to the Turing machine because it is classical. The Lucas-Penrose argument along these lines did not get very far with that. So why would the bridging argument, specifically, "take another hit"? Apr 8 at 18:30
• @Conifold I've tried to make my stance clearer. Hopefully this makes more sense Apr 8 at 20:56
• Entailment is mental causation. To say that q follows from p, is to say that the state where a mind initially believes p, causes the mind to later believe q, when a suitable proof is presented to the mind. Apr 9 at 1:56