EXPLAIN SET THEORY TO ME OR I'LL FUCKING KILL YOU! DON'T DUMB IT DOWN INTO SOME VAGUE SHIT! EXPLAIN SET THEORY TO ME RIGHT NOW OR I'LL LITERALLY FUCKING KILL YOU! WHAT THE FUCK IS A SET AND WHY ARE THEY CONSIDERED TO BE LOGICALLY PRIMARY ENTITIES CAPABLE OF SUSTAINING A LOGICAL FOUNDATIONALISM? WHAT THE FUCK ARE VON NEUMANN SETS? DON'T DUMB IT DOWN OR I'LL FUCKING KILL YOU.
EXPLAIN SET THEORY TO ME OR I'LL FUCKING KILL YOU! DON'T DUMB IT DOWN INTO SOME VAGUE SHIT! EXPLAIN SET THEORY TO ME...
Kayden White
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Julian Roberts
Bump
Aiden Hernandez
Watch this.
So there exist objects, presumably. And these objects can be put into collections. These collections are sets. Naively.
Are you still with me? Hold on, things are going to get crazy.
So we also have nothing. Now, a set might be a collection of objects. But it could also contain nothing. It is the set with nothing in it.
{ }.
Nolan Cruz
Sorry, I meant 'union' not intersect. I should also note that 0 ∪ {0} = {0} precisely because the set 0 contains nothing, so its union with anything is just that other thing.
Adrian Cook
Pretty neat desu. I still like literature a lot more but at one point I took logic for fun. Don't see a point anymore though. It's really helped me in my writing, I will say that.
Ian Gonzalez
>when the schroeder-berenstein hits
Isaac Lewis
i am not being pedantic or hostile i swear to god, i genuinely want to understand the epistemology of logicists
>So there exist objects
is this just meant heuristically? it's not a naive realism or platonist realism, right? this is basically shorthand for, "Conventionally, logic needs to refer to 'entities' of some kind." but then: doesn't that convention itself run deeper (i.e., require axiological justification at a level PRIOR to), the axioms of set theory? is "there exist objects" just something taken on faith and understood linguistically & non-logically, here?
>these objects can be put into collections.
similar question here: is this shorthand for, "A convenient way to talk and think about these things, linguistically and conventionally, and therefore pre-logically, is in terms of 'sets'"?
>[the actual derivation of the integers, etc., from the ground-set of nothing]
this is interesting, but unless it's justified post hoc by its (heuristic) utility in allowing for mathematical formalism (this is the case, right?), isn't it arbitrary again? this may be really naive, but: what is the ontological/epistemological status of "nothing" or "the set" or "the set of nothing" in this case? again, are these just linguistic/conventional entities we take on faith as being decently well agreed upon prior to starting the logic proper?
or is there some appeal to self-evidence and intuitive certainty built into the foundations of set theory?
Alexander Reed
You're mostly right.
en.wikipedia.org
"To a non-mathematician it often comes as a surprise that it is impossible to define explicitly all the terms which are used. This is not a superficial problem but lies at the root of all knowledge; it is necessary to begin somewhere, and to make progress one must clearly state those elements and relations which are undefined and those properties which are taken for granted."
Luis Turner
This.
Frege's work explores further consequences logic and theory, though. Cf. "Sense and denotation", for example. Of course this section of his oeuvre has been discarded by mathematicians, but it's still relevant for analytic philosophy of language (thanks, in part, to the developments done by Kripke and others).
Sorry for the shitty English. Great thread and interactions!
Jayden Rogers
are mathematicians and analytic philosophers mostly self-aware about this?
it's always been intuitively difficult for me to grasp how anyone ever thought OTHERWISE than this. to me, wittgensteinian and quinean things about the fundamental undecidability of all definitions/meanings, the irreducible "embeddedness" of all logical arguments in pre-axiomatic, taken-for-granted "grammar" (here "primitive notions"), the fact that all logical "proofs" are universally true only insofar as some individual, contingent, finite mathematician guy goes "Yeah, seems true to me" -- all this seems like it would be the more basic assumption, the starting point. any kind of strong, platonist logicism seems like it would come in much later, not be an intuitive assumption.
not that i'm unsympathetic to the ideal of completely clarified terms in a perfectly leibnizian universal language, locking together in harmony. but it seems to me that anyone interested in such a thing would have to begin pretty self-consciously with a rather strong platonism (actually various esoterics do this - fabre d'olivet tried to do it with hebrew).
so i often can't tell whether math/analytic philosophy people take this shit for granted, or whether they are outstandingly naive/platonic realists without even noticing it. it often seems like the most exaggerated mathematical formalism is underpinned by a more basic assumption of intuitionism or realism.
Austin Gomez
>These threads actually have serious answers
I thought these were jsut meme threads...
>Frege's work
>but it's still relevant for analytic philosophy
This very much so.
Incredibly enciteful even if you are already aware of certain inherent flaws.
Leo Evans
Most teachers don’t even know this. Which is why they hated me and I hated them. When you really get to the bottom of it, it’s all agreement. Rules of a game. The consequences are real, the activity is magical, and it’s beautiful how some abstract math can come up hundreds of years later to solve some real life problem, but at its core the people involved with math don’t really know what they are doing. I know the subject of genesis and structure has been the topic of all who are interested in a firm foundation in regards to geometry, but the fact is the actual activity is itself mysterious to most if not all engaged with it. The only actual concrete explanation of the activity of mathematics I ever got that made sense was from Hegel
Ian Johnson
>but unless it's justified post hoc by its (heuristic) utility in allowing for mathematical formalism (this is the case, right?), isn't it arbitrary again?
Yes, absolutely.
Set theory, specifically it's Axioms (meaning what sets are allowed to exist) were created very deliberately to result in what mathematicians want to be true. This also means that that there are mathematicians who do not accept set theory, in part or as a whole.
>are these just linguistic/conventional entities we take on faith as being decently well agreed upon prior to starting the logic proper?
The properties of these objects are entirely defined and their names are just there to intuitively reflect the properties.
There are actually equivalent ways to talk about sets which use different names for the same thing, namely trees.
>or is there some appeal to self-evidence and intuitive certainty built into the foundations of set theory?
You want to get to the natural numbers, that is what almost all mathematicians agree on as the starting point and this demand for the natural numbers is the motivation for why things are defined the way they are.
There are (seemingly) arbitrary choices made, but they are justified more by the desireded goals, then by appeals to intuition.
Logan Reyes
I don't know what to tell you guys. It's just the way things are. Set theory is not an exercise in producing some kind of all-encompassing theory of everything, its moreso about clarifying the commonly accepted practice of mathematics. Even then it fails in many respects, as per Godel. At least, that's from my perspective as someone who has studied pure mathematics (but not philosophy)
Gabriel Myers
>Set theory is not an exercise in producing some kind of all-encompassing theory of everything
Well, that might not have been the intent, but as pretty much all knowledge about the physical world is defined mathematically and pretty much all mathematics is defined through set theory. It really is the furthest we have got to a theory of everything.
Christian Campbell
You should note that there is only a single primitive notion associated with ZFC (the highly formalized version of set theory used today). It's /almost/ there so to speak
Anthony Davis
I believe that he was trying to say that set theory is full of ad-hoc modifications for it to be consistent with the already known mathematics.
I remember my first course of uni that touched this topic. My teacher used a Jonathan Swift's quote that talked about an architect that builds a house starting by the roof and then moving down. It was supposed to ilustrate the way the foundations of mathematics originated.
Hudson Williams
Sure, set theory was constructed such that it would lead to the right results, my point was just that it *has become* the closest thing to a theory of everything.
Angel Foster
Also, when I say 'associated with', I mean formalized by. The entire point of the theory is to formalize the primitive notion using a number of logically consistent and intuitive axioms.
Liam Rivera
There is some truth to the idea you always need axioms but also the poster isnt being autistically formal.
Maybe a bit more formal would be to reason about strings of characters.
So then set theory wouldn't really talk about "sets" but about whether such and such a string of characters fits certain properties which only incidentally seem to relate to our notions of sets.
Just if I plug this string of characters into my algorithm will it return a yes or no result.
Whether and how it all actually maps to reality is completely different question.
Not that it really helps much.
A bit further reasoning along these lines leads to the conclusion no possible formal system will work as a logical foundation at all.
Anthony White
thank you for this, like i said sorry if i come across as belligerent.
so it has a real utility in clarifying existing practice? it's not just a case of people feeling discomfort at the "it just works" nature of math (say, their refusal to just take the natural numbers for granted, and not bothering to justify them).
that would explain what i've always wondered, namely: if godel blew out the possibility of foundationalism, why bother continuing with pseudo-foundationalism at all? why not just shrug and smile and keep doing math? but it makes sense that set-theory provides an internally consistent formalism that is general enough to group lots of domains together etc., there's a reason for it.
>It really is the furthest we have got to a theory of everything.
isn't it just a really specialized language for permitting quantification and measurement of manifolds, ultimately? how would that amount to a theory of anything? it is measuring real reality, but it's still only measuring it, and translating the measurements into human-comprehensible language. it doesn't answer the question, for instance, of what language is in the first place.
John Morris
>permitting quantification and measurement of manifolds
Do you meant that mathematically? Measuring manifolds doesn't seem all that interesting.
Pretty much anything we know about reality is expressed mathematically which is again expressed in set theory, but of course it doesn't tell us what the physical things actually are.
You can describe language in set theory, with a bit of effort you might be able to write down "the set of all English texts", but of course that doesn't explain what language is.
Set theory allows you to abstract pretty much everything to something extremely basic, but it doesn't tell you about the thing in itself.