Within a different framework, say that of binary numeral system, 1 + 1 = 10 — RussellA
But whatever we take mathematics to be talking about, at least we may speak of abstractions "as if" they are things or objects.
— TonesInDeepFreeze
Is this an example of Putnam's Modalism, the assertion that an object exists is equivalent to the assertion that it possibly exists? — RussellA
What does "it, the knight on a chess board, refer to? — RussellA
"It" must refer in part to a physical object that exist in the world and in part to rules that exist in the world. — RussellA
Innatism — RussellA
For me, the value and wisdom of philosophy is not in the determination of facts, but rather in providing rich, thoughtful, and creative conceptual frameworks for making sense of the relations among facts.
— TonesInDeepFreeze
But how can there be wisdom in the absence of facts. — RussellA
There was developing an interesting discussion on the law of identity and (non-ordered) sets. — jgill
The logic is not merely supposed to be rigorous. It is rigorous in these senses: (1) The axioms and rules of inference are recursive, thus, for a purported proof given in full formality, it is mechanical to check whether it is indeed a proof, i.e., merely an application of the inference rules to the axioms. (2) It is proven that the logic is sound, i.e. that a formula is is provable from a given set of formulas only if the formulas is entailed from the set of formulas. — TonesInDeepFreeze
mathematics, in ordinary context, 'x=y' is true if and only if x and y are the same object, which is to say 'x=y' is true if and only if what 'x' stands for is the same as what 'y' stands for. The claim that there are no such objects is not properly given as an objection to the fact that '=' stands for identity, since we would still have '=' standing for identity if the objects were physical, concrete, fictional, hypothetical, 'as if', abstract, platonic, etc. — TonesInDeepFreeze
* Sets are not determined by an order in which the members happen to be mentioned. If I say, "What are the members of the set of books on your desk", then if you say, the set of books on my desk is all and only the books 'The Maltese Falcon', 'Light In August' and 'The Stranger', then no one could say "No, that's wrong, the set of books on your desk is actually all and only the books 'Light In August', 'The Stranger' and 'The Maltese Falcon'!" — TonesInDeepFreeze
No law of identity is violated there. — TonesInDeepFreeze
Nobody says that the set of items on a desk is different depending on the order you list them. — TonesInDeepFreeze
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