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I have a number of true statements. Each of these statements is a case where I have difficulty seeing how (assuming physicalism) the statement could correspond to a state of affairs. My question is: how do correspondence theories account for statements like these?

  1. There are infinitely many prime numbers.

  2. A unit circle has circumference 2π.

  3. There is no Turing machine which solves the halting problem.

  4. If 2+2=3, then there is a number larger than every prime number.

  5. If a purple unicorn materialized in my room ten minutes from now, I would be surprised.

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    This would be like saying the mountain is disobeying the map. Mathematics and logic are models, or maps. They do not always predict what is found. Sometimes the territory has not been fully mapped or modeled and are used old maps or maps of other territories. Sometimes they are approximations. – Annotations Apr 15 '13 at 13:11
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    @RicardoBevilaqua: I think at issue is whether or not these statements, which would conventionally be taken as true, can correspond to any 'facts' per se. (There is a notorious problem in exhibiting facts corresponding to universal negative statements, for example.) – Niel de Beaudrap Apr 15 '13 at 13:32
  • @NieldeBeaudrap There is no theory-independent way to reconstruct phrases like "really there", each theory has its own ontology. Convergence to the truth scientific progress seems to be impossible, if ontologies change with theories observations, and ontologies are relative to theories. Many past theories were not approximately true or truthlike. Ptolemy's geocentric theory was rejected in the Copernican revolution, not retained in the form “approximately Ptolemy”. – Annotations Apr 15 '13 at 14:43
  • @NieldeBeaudrap What is truth in Science? the progressive steps from Ptolemy to Copernicus or from Newton to Einstein are not only matters of improved precision but involve changes in theoretical postulates and laws. Science is progressive only on values other than the truth, such as simplicity, predictive accuracy, comprehensiveness, and requirements for consistency. Scientific theories are hypothetical and always corrigible in principle. They may happen to be true, but we cannot know this for certain in any particular case.What is "fact" in science? – Annotations Apr 15 '13 at 14:46
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    Statements 1-4 are mathematical propositions. Can you provide a source where physicalism is claimed to apply to mathematics? Otherwise, the question addresses a red herring. – DBK Apr 15 '13 at 21:12
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In general correspondence theory has a problem dealing with mathematical and tautological sentences, and the strategy to dealing with these has generally been either: 1) to struggle to find amendments to correspondence theory that provide the intuitive truth-assessments of various mathematical claims--and I have the sense that the general opinion among analytic philosophers is that this project has not yet accomplished its goal--or alternately, 2) to claim that correspondence theory is meant only to apply to sentences whose truth-values are determined entirely by the existence and properties of physical objects. It is not entirely uncommon for correspondence theorists to talk of correspondences with abstract objects like mathematical objects, which suggests a way out, but it violates your condition of physicalism unless they find some way of naturalizing the talk of abstract objects.

As for the last sentence that you used, this requires first an analysis of conditional sentences, which is its own bag of worms. I refer you here to Bennett's Philosophical Guide to Conditionals.

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Mathematical statements can be taken to be predictions about the state of the world. You never expect to experience a time where a program-enumerating Turing Machine will output a program that solves the halting problem in general, you will expect circular objects to have circumferences that relate to their radii by a factor approaching 2π and so on.

Do keep in mind that modern science tells us that all maths corresponds to neurons firing in certain ways in certain parts of your brain. There's your correspondence in any case.

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