The categoricity of an axiom system means that its non-logical symbols have, up to isomorphism, only one possible interpretation. The first axiomatizations of mathematical theories such as number theory and analysis by Dedekind, Hilbert, Huntington, Peano and Veblen were indeed categorical.
These were all second order axiomatisations, suffering from what many consider a weakness, namely dependence on a strong metatheory, casting a shadow over these celebrated categoricity results. In finer analysis a new form of categoricity has emerged. It is called internal categoricity because it is perfectly meaningful without any reference to a metatheory, and it is now known that the classical theories, surprisingly even in their first order formulation, can be shown to be internally categorical.
Gödel's incompleteness theorems pose a problem for Carnapian conventionalism: if our conventions are recursively axiomatizable, then the ensuing system is (arithmetically) incomplete. However, it does not immediately follow that the (arithmetical) concepts used in that system are imprecise. This is because internal categoricity allows incompleteness to sit alongside (mathematical) determinacy.
Mathematicians and philosophers have appealed to categoricity arguments in a surprisingly varied range of contexts. One familiar example calls on second-order categoricity in an attempt to show that the Continuum Hypothesis, despite its formal independence, has a determinate truth value, but this does not exhaust the uses of categoricity even in set theory, not to mention its appearance in various roles in discussions of arithmetic. (https://arxiv.org/abs/2204.13754)
These were all second order axiomatisations, suffering from what many consider a weakness, namely dependence on a strong metatheory, casting a shadow over these celebrated categoricity results. In finer analysis a new form of categoricity has emerged. It is called internal categoricity because it is perfectly meaningful without any reference to a metatheory, and it is now known that the classical theories, surprisingly even in their first order formulation, can be shown to be internally categorical.
Gödel's incompleteness theorems pose a problem for Carnapian conventionalism: if our conventions are recursively axiomatizable, then the ensuing system is (arithmetically) incomplete. However, it does not immediately follow that the (arithmetical) concepts used in that system are imprecise. This is because internal categoricity allows incompleteness to sit alongside (mathematical) determinacy.
Mathematicians and philosophers have appealed to categoricity arguments in a surprisingly varied range of contexts. One familiar example calls on second-order categoricity in an attempt to show that the Continuum Hypothesis, despite its formal independence, has a determinate truth value, but this does not exhaust the uses of categoricity even in set theory, not to mention its appearance in various roles in discussions of arithmetic. (https://arxiv.org/abs/2204.13754)
жаль только никто не хочет платить за пары по аналитической или формальной философии
I assume the reader has some basic knowledge of category theory, such as can be obtained from [ML98] or [Awo06], but little or no experience with formal logic or set theory.
давай наоборот
давай наоборот
Summer School on Mathematical Philosophy for Female Students:
We welcome applications from all students who identify as women
We welcome applications from all students who identify as women
In 1967, Friedman was listed in the Guinness Book of World Records for being the world's youngest professor when he taught at Stanford University at age 18 as an assistant professor of philosophy.
He has also been a professor of mathematics and a professor of music.
He has also been a professor of mathematics and a professor of music.
пхд в MIT в 19 лет, а еще у него ютуб канал где он бетховена на пианино играет
Empty Name
In 1967, Friedman was listed in the Guinness Book of World Records for being the world's youngest professor when he taught at Stanford University at age 18 as an assistant professor of philosophy. He has also been a professor of mathematics and a professor…
параллельно троллит всех в дискуссиях мэйлинг листа по основаниям математики
In order to formulate the Leibnizian law of Identity of Indiscernibles, and examine its validity, we need higher order logic.
Empty Name
connectionism? sucks
всю радикальность этого тезиса поймут только пацаны из cognitive science
Итак, в данной работе нами должны руководить только опыт и наблюдение. Они имеются в бесчисленном количестве в дневниках врачей, бывших в то же время философами, но их нет у философов, которые не были врачами.
Первые прошли по лабиринту человека, осветив его; только они одни сняли покровы с пружин, спрятанных под оболочкой, скрывающей от наших глаз столько чудес; только они, спокойно созерцая нашу душу, тысячу раз наблюдали ее как в ее низменных проявлениях, так и в ее величии, не презирая ее в первом из этих состояний и не преклоняясь перед нею во втором.
Повторяю, вот единственные ученые, которые имеют здесь право голоса. Что могут сказать другие, в особенности богословы?
Первые прошли по лабиринту человека, осветив его; только они одни сняли покровы с пружин, спрятанных под оболочкой, скрывающей от наших глаз столько чудес; только они, спокойно созерцая нашу душу, тысячу раз наблюдали ее как в ее низменных проявлениях, так и в ее величии, не презирая ее в первом из этих состояний и не преклоняясь перед нею во втором.
Повторяю, вот единственные ученые, которые имеют здесь право голоса. Что могут сказать другие, в особенности богословы?