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The debate is what is commonly referred to as the Foundational Crisis of the early XX century (https://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis). A well-known example of one of the issues from which this crisis stemmed is Russel's paradox in Frege's foundations for mathematics. Hilbert's position on this is well explained in https:...


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The key term is "ultrafinitism." In one sense it's not hard to whip up such a system. Here's a straightforward, if naive, development of a semantics for arithmetic which basically follows your first sentence: We pick some number, say x=10^80, to be our limit on the "actual natural numbers." With x in hand, given a sentence p in the ...


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Contrary to @Mauro ALLEGRANZA's reply, there are cases where mathematicians rely on implication where the premise is false. This occurs in certain theorems where either the statement, or the proof, can be rendered 'cleaner' (in the sense of having fewer assumptions in the statement, or better proof structure) by exploiting this feature of implication. A ...


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I'm not sure the fully understand your question, but I'm trying to answer, reading the title-question: Do mathematicians care about implications where the hypothesis is always false? as regarding the well-known issues about the truth-functional definition of the "if..., then..." connective, and specifically regarding the fact that "if P, ...


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Axioms are not in actual practice/history, chosen at the beginning of mathematising, but are part of a specific type of project, to work backwards to find logical grounding. It began with geometry, as any "proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., ...


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The following passage that I am quoting at length from Dag Prawitz ("Intuitionistic Logic: A Philosophical Challenge" in Logic and Philosophy edited by G. H. von Wright, Hague, Martinus Nijhoff Publishers, pp. 8-9) may be more illuminating: Intuitionistic philosophers sometimes use true as synonymous with the truth as known, but this is clearly a ...


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Physics, and in general science, is all about developing "models" that we hope fit to physical reality as much as possible. Fitting criteria are explanation of current observation (that is not necessarily the pure reality in its essence as the observation might well be restricted by not only the instruments but also the current paradigm and ...


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Pure mathematics can and does stand alone as its own science. It is the objective of pure mathematics to systematically express its content in a culture-free manner. If its truth content depends on the cultural context of its practitioners then it is assuredly not pure mathematics; in fact, if that is true then it does not qualify as mathematics at all. ...


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