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For Field, the following is 'perfectly obvious', but I would like confirmation that I understand it completely.

  • Let A be a nominalistically statable assertion.
  • Let A* be the assertion that results by restricting each quantifier of A with the formula not M(xi) (for the appropriate variable 'xi') (where M(x) is the predicate 'x is a mathematical entity').
  • Let S be any mathematical theory.

Then Principle C'' is as follows: let A be any nominalistically statable assertion. Then A* isn't a consequence of S unless it is logically true.

My understanding of what this is saying is that if we have a mathematical theory, from this we cannot then draw conclusions about statements regarding non-mathematical objects. Please correct me if I am wrong on that. I am also still slightly confused about the wording "unless logically true" and the meaning behind it.

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    – J D
    Commented Jan 5, 2020 at 20:39
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    Hartry Field explictly mentioned in SWN that he holds similar view as logical positivists' position on the analyticity of math including logic as mere tautological meaningless, so essentially he views any true statement other than scientific statements must come from math and logic alone. And you're right "if we have a mathematical theory, from this we cannot then draw conclusions about statements regarding non-mathematical objects". Non-mathematical statements can only be causally "entailed" by some body of nominalistically statable assertions... Commented Oct 10, 2021 at 0:52
  • He proved using Math that Math can't prove things outside of Math? Seems like a lot of people should take note of that.
    – Scott Rowe
    Commented May 25, 2022 at 15:05
  • To my reckoning Field, all due respect to his wisdom, isn't entirely correct or is possibly quite wrong. However, I don't contest the claim science sans mathematics. The problem lies elsewhere ... obviously.
    – Hudjefa
    Commented May 20, 2023 at 6:03

2 Answers 2

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See Principle C': Let A be any nominalistically statable assertion, and N any body of such assertions.Then A∗ isn’t a consequence of N∗ + S unless it is a consequence of N∗ alone.

If S is a mathematical theory, it "speaks about" mathematical objects while A∗ and N∗ do not.

Thus, IMO; we have to understand it as saying: in order to derive statements about the "nominalistic world" we have to use "nominalistic axioms" alone.

While this is equivalent to Principle C'': Let A be any nominalistically statable assertion. Then A∗ isn’t a consequence of S unless it is logically true ?

Because if we can prove it whitout nominalistic axioms at all, due to the fact that it cannot be consequence of the mathematical theory S, it must be "provable from logic alone".

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As appears in the OP, Hartry's argument is non liquet. Perhaps someone could edit-clarify Hartry's justification.

Speaking for myself, a wholly mathematical theory would require a number (arithmetic) and/or a shape (geometry) that corresponds to that-which-is-to-be-explained, in order to parse reality, the sum-total of all that needs explaining. If there exists an "object" thus not mathematizable, no wholly mathematical theory will be able to capture it in a way that could participate in our comprehension of reality.

P. S. I've restricted math to arithmetic & geometry. Math has grown in scope, making it rather implausible that nonmathematical assertions exist. Too, nonmathematical properties possibly are of zero importance, making any nonmathematical assertion/theory a triviality.

As for principle C, it seems to be saying, mathematics is subsumed by logic and is kinda sorta its lackey; no logic, no math. The universe is not in any way thus restricted (re Hume's problem of induction).

What of this 👉 "If you understand quantum mechanics, you don't understand quantum mechanics" ~ Feynman(?). Math has no issues computing quantum phenomena, but there's no consensus as to a theory (Copenhagen, Many Worlds, which is it?); also Schrodinger's rather confused cat.

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