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Did anyone argue that mathematical concepts and principles are neither true or false? I am not sure if I am crazy, but I feel like they are neither true or false just like a word is not true or false. Is the word "dog" true or false? Is the statement "dog is walking" true or false? The statement can refer to anything and we cannot know if it's true or false unless we tell what exactly we mean by that.

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  • If mathematical propositions are neither true nor false, what is the difference between the proposition "1+1=2" and the proposition "1+1=3"? Commented Jul 31, 2021 at 1:22
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    See Fictionalism in the Philosophy of Mathematics Commented Jul 31, 2021 at 10:12
  • @DavidGudeman The theory of Fictionalism, noted by Mauro holds that the system that includes the former preserves truth. But it may not contain any. It proceeds from a delusion - either the existence of the Platonic world, or the trust in certain common human intuitions - that humans cannot manage to shake. Despite being a fiction, it is a very useful one. Our idea of the fictional world where our intuitions are true helps us because we are going to inadvertently keep falling back on those intuitions, anyway, so we might as well institutionalize them. Commented Jul 31, 2021 at 18:56
  • @hide_in_plain_sight, although modern philosophers who claim that abstract objects exist are often called platonists, they don't believe in anything remotely like the Platonic world of true forms. Commented Jul 31, 2021 at 19:54

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There is an approach to the philosophy of mathematics called formalism. In fact, it is a whole family of related positions. In one of the more extreme versions of formalism, mathematical sentences are not considered to be propositions that state truths or falsehoods, but are just strings of symbols that are manipulated according to a set of rules that we invent. Some manipulations are allowed and some are not. An analogy would be with a game such as chess: some moves are allowed and some are not, but a move is not 'true' or 'false'. On this view, manipulating symbols can be useful, in that such manipulations can be used to solve problems, but mathematical sentences do not state truths or carry any ontological commitment.

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The short answer is yes — the status of mathematical truth is sometimes described as “hypothetical” or even “fictive” given the objects it seems to properly apply to are ideal and so inexistent. Like theology it stands entirely suspended from its tautologies — that is, the “matter of fact” with respect to its universe of discourse seems tied to aspects of our beliefs in it.

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It all depends. Whether a particular mathematical theory is true depends on whether its axioms are true. Arithmetic and Euclid's geometry are good candidates as examples of true theories because most people believe their basic axioms are true. However, if Einstein's General Relativity is correct and the geometry of space is curved, then the axioms of Euclid's geometry are false. The curved geometry used in General Relativity, however, might well be a true mathematical theory.

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Platoonism or idealism presupposes a metaphysical domain where mathematical entities exist without them being true or false. They just exist.

Plato thought we have access to this world only by means of the material shadow the cast here on Earth. We will never know the exact nature of the ideal world but math comes close iby describing it.

Some people, like Max Tegmark and Deutsch, think the world we live in is the metaphysical world of Plato.. We have direct acces instead the "dirty" access mediated by the physical world which clouds a direct view. Also in this view we can't say that there are wrong or right objects, only our thoughts and interpretations can be right or wrong (1+1=7 doesn't have to be wrong).

Of course we can say that some things are true or false when comparing certain objects. But the entities themselves are not wrong or right. They simply exist.

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  • Tegmark's proposals (e.g. MUH, CUH, ...) are pure nonsense, as is obvious to anyone who knows logic well enough.
    – user21820
    Commented Feb 15, 2022 at 22:39

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