3

Interesting question. I think one part of the answer is that logicians and mathematicians such as Frege were concerned to remove 'psychologism' from their subjects. Psychologism in its various form is a set of views : about the relationship between psychology and logic, but its traditional form holds that the laws of logic are grounded in psychological ...


2

You are confusing two uses of the word argument. In one sense, an argument is an extended discourse with limited aims such as education or persuasion. In the second sense, argument is a synonym for the technical term inference which is the process by which a single proposition can be constructed from a collection of premises (sometimes unstated). So, in ...


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On the referenced page 29, Tim Bayne is making an argument by analogy both for and against al-Ghazali's view that the universe is not eternal. He claims that just as people become confused about the cardinality of infinite sets so "al-Ghazali's argument appears to succumb to this confusion". Bayne's analogical argument may be weak, but making it does not ...


2

It is worthwhile pointing out that there are set theories having a set of all sets, hence the issue with the non-existent set of all sets not containing themselves is not about "size". Russell's paradox comes from unrestricted comprehension, and only becomes related to "size" if we keep set-restricted comprehension around.


2

At the time of the Tractatus (to which Anscombe and Weir refer), Wittgenstein was a logicist of sorts, so there was no principled difference for him between tautologies and mathematical propositions (with mathematics reduced to a fragment of arithmetic). So 4.461's sinnlos applies to both. "4.461 The proposition shows what it says, the tautology and the ...


1

To pile on Arno's answer, non-containability really has nothing to do with size. Having members of every size is a simple test, but it is not the basic reason any set is not containable. It is actually quite an indirect one. Getting from the one fact to the other formally involves sets having cardinalities, cardinals being ordinals and the Burali-Forti ...


1

According to Tim Button the reason Russell's paradox is a problem in set theory is because set theory relies on classical first-order logic and one can express that paradox there. First he considers the paradox from the perspective of naive set theory: (page 109) In part II, we worked with a naïve set theory. But according to a very naïve conception, ...


1

I like Geoffrey Thomas's answer, and would add. Mathematicians and rationalists more generally have always chased after certainty of reason. Plato was perhaps the first mathematical foundationalist who attempted to deal with infinite regress by creating a distinct, objective reality for abstraction with his theory of forms. For over two thousand years, ...


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The question presupposes that there is anything approaching any kind of universally standardized criterion for being a tenured professor. De facto, the main criteria are how many articles you published and (at top schools like the one you mentioned) how many potential top graduate students you will attract as an adviser. At smaller universities they will ...


1

Logical entailments of what, in particular? Clearly Newtonian physics -- the set of logical entailments of Netwon's principles, is not included. Newtonian definitions clearly involve guessing and observation and are definitely not apriori. Logic may be how synthetic apriori content has to be related internally, since the illogical is not really "known". ...


1

IMO one should not play off one against the other, i.e. geometry against arithmetic or vice versa. At the time of Poincaré both disciplines were separated. At the base of Grothendieck‘s revolutionary view onto Algebraic Geometry lies the concept of the spectrum: Introducing the spectrum Spec R of a commutative ring R means to consider the algebraic object ...


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