Just as a photon is either a wave or particle depending on how it is measured, it seems like these difference in math philosophy may all be neither wrong nor right - it depends on how the topic is approached. — EricH
For example, certain principles or properties involving collections of sets may be more naturally expressed in SOL.
Second-order logic plays a significant role in model theory, which is the branch of mathematical logic that deals with the study of mathematical structures (such as sets) using formal languages.
Pretty much. So mathematical expressions are true only if there is a proof-path that shows it to be true. There are, one concludes, mathematical expressions that are neither true nor false. — Banno
This is opposed to Platonism — Banno
But then again, prima facie there is nothing necessary about the idea of cats, protons, or communism. It could be that numbers are innate ideas, being then "world-independent".
But then the claim "it is not the case that this proof-path pre-exists our construction of it", the syntax being the proof-path, and in our case being the FOL that we see in things such as ZFC, did we really construe relations such as ∧ and →? If so, it would then bring up "how did we"? — Lionino
I'm not following what you say here. — Banno
This would be impossible if the contradictory theory was erected on a logical foundation containing the Boolean principle Ex Contradictione Quodlibet ECQ, from a contradiction everything follows. So ECQ has to be abandoned, but fortunetely that proves possible, indeed mathematically straightforward. What remains is a rich field, of novel mathematical applications interesting in their own right, which sidestep the vexing questions of which foundational principles to adopt, by developing contradictions in areas of mathematics such as number theory or analysis which are far from foundations. — SEP's Inconsistent Mathematics
In a parallel with the above remarks on rehabilitating logicism, Meyer argued that these arithmetical theories provide the basis for a revived Hilbert Program. Hilbert’s program was the project of rigorously formalising mathematics and proving its consistency by simple finitary/inductive procedures. It was widely held to have been seriously damaged by Gödel’s Second Incompleteness Theorem, according to which the consistency of arithmetic was unprovable within arithmetic itself. But a consequence of Meyer’s construction was that within his arithmetic R# it was demonstrable by finitary means that whatever contradictions there might happen to be, they could not adversely affect any numerical calculations. Hence Hilbert’s goal of conclusively demonstrating that mathematics is trouble-free proves largely achievable as long as inconsistency-tolerant logics are used.
The arithmetical models used by Meyer and Mortensen later proved to allow inconsistent representation of the truth predicate. They also permit representation of structures beyond natural number arithmetic, such as rings and fields, including their order properties. Axiomatisations were also provided. — SEP's Inconsistent Mathematics
It should be emphasised again that these structures do not in any way challenge or repudiate existing mathematics, but rather extend our conception of what is mathematically possible. This, in turn, sharpens the issue of Mathematical Pluralism; — SEP's Inconsistent Mathematics
Various authors have different versions of mathematical pluralism, but it is something along the lines that incompatible mathematical theories can be equally true. The case for mathematical pluralism rests on the observation that there are different mathematical “universes” in which different, indeed incompatible, mathematical theorems or laws hold. Well-known examples are the incompatibility between classical mathematics and intuitionist mathematics, and the incompatibility between ZF-like universes of sets respectively with, and without, the Axiom of Choice. It seems absurd to say that ZF with Choice is true mathematics and ZF without Choice is false mathematics, if they are both legitimate examples of mathematically well-behaved theories. — SEP's Inconsistent Mathematics
Shapiro’s distinctive position has other ingredients: mathematics as the science of structure, and mathematical pluralism implying logical pluralism (on logical pluralism see also Beall and Restall 2006); but we do not take these up here. — SEP's Inconsistent Mathematics
The primacy of theories fits, too, with the natural observation that the epistemology of mathematics is deductive proof. It is only if one takes as a starting point the primacy of the mathematical object as the truth-maker of theories, that one has to worry about how their objects manage to co-exist. — SEP's Inconsistent Mathematics
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