To be a philosopher is, among other things, to be haunted by problems.
A philosophical problem is that to which one is always returning, a burning in the mind that refuses to be quenched, an attractor in one’s cognitive meanderings—until one day (or night) it is replaced by (or evolves into) another.
A philosophical problem is that to which one is always returning, a burning in the mind that refuses to be quenched, an attractor in one’s cognitive meanderings—until one day (or night) it is replaced by (or evolves into) another.
сука приехал в лицей вести пару, подготовил материал, а у них репетиция последнего звонка
звоню в полицию, обращение регистрируют и говорят «приезжайте в отдел, но…
оперативник на весь район у нас один, он сейчас выехал по делу о минировании всех университетов москвы, а потом у него еще дело по мошенничеству на миллион рублей, так что через 5 часов сможет заняться вашим делом, но вы приезжайте»
оперативник на весь район у нас один, он сейчас выехал по делу о минировании всех университетов москвы, а потом у него еще дело по мошенничеству на миллион рублей, так что через 5 часов сможет заняться вашим делом, но вы приезжайте»
While I confess to being an occasional user of hashish and marijuana during the early 1960s, I cannot recommend these drugs as a secret to success in physics.
(c) Sheldon Lee Glashow, Nobel Prize-winning American theoretical physicist
(c) Sheldon Lee Glashow, Nobel Prize-winning American theoretical physicist
в следующем году на этой страничке должно появиться, что я стажер в LLFP (и опубликованная статья)
В пустяке содержится крупица высокого, как в капле росы отражается солнце. Мы можем обнаружить это в инвенциях Баха, прелюдиях Шопена, где видим ту же высоту, что и в больших композициях, но сконцентрированную в мгновении.
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)