Questions tagged [mathematical-philosophy]
Philosophical aspects of logic and set theory; truth status of mathematical axioms; Philosophy of Mathematics; philosophical aspects of mathematics in general; relation of mathematics to philosophy; etc. Consider also posting at http://philosophy.stackexchange.com/, where philosophy-of-mathematics is one of the most popular tags.
375 questions
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Multiverse view of set theory and finitary statements
I think almost all mathematicians would agree that the finitary statements (like those expressed in PRA) have defnite realist objective truth values. E.g. either there is a finite string of bits (an ...
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Would Hofstadter's critique of the theory of types apply to modern type theories as well?
Douglas Hofstadter, the author of Gödel, Escher, Bach and I am a Strange Loop, critiqued Principia Mathematica in both books. This quote is from the latter one:
In order to convey the fatal nature of ...
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Why and how do (classical) reverse mathematics and intuitionistic reverse mathematics relate?
Broadly speaking, the idea of “reverse mathematics” is to find equivalents to various standard mathematical statements over a weak base theory, in order to gauge the strength of theories (sets of ...
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$p$-adic concepts having no analogue in the real case
I've always been fascinated how among completions of $\mathbb{Q}$, the real field $\mathbb{R}$, althrough historically more "ancient", seems to be the odd one out. Many interesting concepts ...
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Metamathematical/philosophical understanding of the smallest aleph
Forgive me for asking a perhaps low level question here but I suspect that the answer may be somewhat subtle and my confusions around it are profound.
Working over ZFC, say (assuming the well-ordering ...
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Why is it that higher-order logic and geometric logic are so tightly connected?
By higher order logic, I mean logic which quantifies freely over types including propositions and functions (thus predicates, predicates of predicates, etc).
We have a direct connection between higher-...
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Ordinary mathematics intrinsically requiring unbounded replacement/specification?
Encouraged by some users on MO, I'm going to ask this question that I have had for years. I have always felt that the iterative conception of sets makes some sense for justifying BZFC (i.e. ZFC with ...
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Is there an “opposite” hypothesis to the (Generalized) Continuum Hypothesis?
There are many questions on this site about the (Generalized) Continuum Hypothesis, its philosophical or epistemological justifications, and various attempts at “solving” it. Because one such ...
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Why do some mathematicians think Gödel and Cohen have not closed the Continuum Hypothesis?
Gödel, in 1938, showed that CH is consistent with ZFC. In 1963, Paul Cohen proved the opposite: CH can be false in some models of ZFC. Together, these results mean that within ZFC alone, CH can’t be ...
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Dependent Choice and Epistemological Justification: Munchausen's Trilemma in Not-DC Models?
My question overlaps set theory and epistemology; I'm hoping for references to anyone discussing the problem of "justification" (especially Munchausen's Trilemma) with the Axiom of Dependent ...
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How do mathematical developments make their way into the mass media?
This question may seem off topic, and feel free to express that, but first let me say why I think it belongs here.
Every so often there are articles in mainstream newspapers, and also in popular ...
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Can Singleton augmented Mereology provide a truth argument for set theory?
Let "$ \phi \text { is one-to-one between } \pi, \psi $", stands for meeting both of: $$ \forall x \pi(x) \exists!y \psi(y): \phi(x,y) \\ \forall y \psi(y) \exists!x \pi(x): \phi(x,y) $...
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Is it possible to add a new axiom schema to classical propositional logic?
Let IPL mean the intuitionist propositional calculus. One can add a great diversity of axiom schemas to obtain intermediate logics between IPL and CPL, where CPL is the classical propositional ...
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Are Category and Measure Special?
In logic, and I expect in mathematics more broadly, it seems like there is a special role played by notions like measure and (baire) category (as in meeting/avoiding dense sets). Obviously, these ...
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Poincaré and the principle of induction
In his popularization book "Science and Method" (1905),
Henri Poincaré argues that
mathematics cannot be reduced to logic or set theory
and that there is always the need to appeal to ...
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