I have to say, I love the cheekiness of the cover. — Count Timothy von Icarus
This is the principle that animates all living beings, from the most simple up to and including humans. it is why, for instance, all of the cells in a living body develop so as to serve the overall purpose of the organism. So the 'one-ness' of individual beings is like a microcosmic instantiation of 'the One'. — Wayfarer
a stipulated logical monist of a certain sort, that there is only one entailment relation which all of these logics ape. — fdrake
No True Scotsman doesn't admit of an easy formalisation in terms of predicate logic — fdrake
I imagine monists are generally going to just deny this, because monism is about logical consequence relative to some non-arbitrary context — Count Timothy von Icarus
I'd also want to liken the relationship of formalisms to their intended objects, or intended conceptual content — fdrake
My intuition is also that there are other principles that set up relations between the practice of mathematics and logic and how stuff (including mathematics) works, which is where the metaphysics and epistemology comes in. But I would be very suspicious if someone started from a basis of metaphysics in order to inform the conceptual content of their formalisms, and then started deciding which logics are good or bad on that basis. That seems like losing your keys in a dark street and only looking for them under street lamps. — fdrake
Unum in the same sense as in non-dualism, advaita, non divided. — Wayfarer
In your other thread we touched on the Scholastic transcendentals or convertibles. Another transcendental besides being and truth is oneness (unum). — Leontiskos
There was consensus among the scholastics on both the convertibility of being and unity, and on the meaning of this ‘unity’—in all cases, it was taken to mean an entity’s intrinsic indivision or undividedness. [19] In this, the tradition was continuing and affirming a definition first proposed by Aristotle in the Metaphysics. [20] This undividedness, in the words of Aquinas in his Commentary on the Sentences, is said to lie “closest to being.”[21] For the most part, ens and unum were distinguished by these thinkers only logically or conceptually—unum adding nothing real to being, or more properly, adding only negation, only a privation of actual division.[22] It was common practice in medieval philosophy to distinguish the transcendental sense of unum, running through all of the categories, from the mathematical sense of unum, restricted to the category of quantity. These two ‘ones’ are each in their own way opposed to ‘multiplicity.’[23] Aquinas offers a succinct account of this in his Summa Theologiae (Ia. q. 11, art. 2).[24] The ‘one’ of quantity is the principle of number; it is that which, by being repeated, comprises the sum (the multiple).[25] Aquinas says that there is a direct opposition between ‘one’ and ‘many’ arithmetically, because they stand as measure to thing measured, as just-one to many-ones. Likewise, transcendental unity is opposed to multiplicity, but in this case not directly. Its opposition is not to the many-ones per se, but rather to the division essentially presupposed in and formal with respect to the multiplication of actual multiplicity. This tracks with a consistent distinction in Aquinas between division and plurality in which division is seen as ontologically and logically prior.[26] Transcendental unity then, has a certain priority to its predicamental counterpart.
We will return below to the consequences for contemporary ontology that follow upon this fact that, in its developed form, it was division, not plurality, that was taken by the classical tradition to be the precise contrary to transcendental unity. . . — Being without One, by Lucas Carroll, 121-2
If your prayers are answered you assume it was God who did the answering. — Metaphysician Undercover
Your understanding of each of the positions seems to make them trivial rather than controversial. — Count Timothy von Icarus
There are two questions with this pluralism/monism debate: What the heck is the thesis supposed to be, and Who has the burden of proof in addressing it? The answers seem to be, respectively, "Who knows?" and "The other guy!" :lol: — Leontiskos
I would agree that every quantification is into a domain, and I don't think there are context independent utterances. I do not think it follows that there is no metaphysics. I'm rather fond of it in fact, but the perspective I take on it is more like modelling than spelling out the Truth of Being. I think of metaphysics as, roughly, a manner of producing narratives that has the same relation to nonfiction that writing fanfiction has to fiction. You say stuff to get a better understanding of how things work in the abstract. That might be by clarifying how mental states work, how social structures work, or doing weird concept engineering like Deleuze does. It could even include coming up with systems that relate lots of ideas together into coherent wholes! Which it does in practice obv. — fdrake
And this sounds a lot like Srap's approach. I was encouraging him to write a new thread on the topic.
Plato's phrase, "carving nature at it's joints," seems appropriate here. I would say more but in this I would prefer a new or different thread (in the Kimhi thread I proposed resuscitating the QV/Sider thread if we didn't make a new one). I don't find the OP of this thread helpful as a context for these discussions touching on metaphysics. — Leontiskos
One of the great things about producing formalisms is that they're coordinative. — fdrake
But again, virtually no one wants to claim that truth should be both deflated and allowed to be defined arbitrarily. So we still have the question (even in the permissive case of Shapiro) about what constitutes a "correct logic." The orthodox position is that this question is answered in terms of the preservation of "actual truth." But we also see it defined in terms of "being interesting" (e.g. Shapiro). Either way, we are right back to an ambiguous metric for determining "correct logics," hence to common appeals to popular opinion in these papers. — Count Timothy von Icarus
That reads disingenuously to me. Your use of "roundness" previously read as a completely discursive notion. If you would've said "I think of a circle as a closed curve of constant curvature" when prompted for a definition, and didn't give Euclid's inequivalent definition, we would've had a much different discussion. I just don't get why you'd throw out Euclid's if you actually thought of the intrinsic curvature definition... It seems much more likely to me that you're equating the definition with your previous thought now that you've seen it. — fdrake
The latter of which is fair, but that isn't a point in the favour of pretheoretical reasoning, because constant roundness isn't a concept applicable to a circle in Euclid's geometry, is it? Roundness isn't quantified... — fdrake
Mathematical concepts tend to be expressible as mathematical formalisms, yeah. And if they can't, it's odd to even think of them as mathematical concepts. It would be like thinking of addition without the possibility of representing it as +. — fdrake
And therein lies a relevant distinction. Formalisms aren't prepackaged at all. In fact I believe you can think of producing formalisms as producing discursive knowledge! — fdrake
But you also seem to think the context you have in mind for any question that arises is the only context it can possibly arise in. — Srap Tasmaner
There will be Euclid circles in that space which are not Aristotle circles too, I believe. — fdrake
I believe "A closed curve of constant positive curvature" is the one the differential geometry man from above would've said — fdrake
We could say that a circle is a [closed] figure whose roundness is perfectly consistent.* There is no part of it which is more or less round than any other. — Leontiskos
The discussion about capturing the intended concept is relevant here. The interplay between coming up with formal criteria to count as a circle and ensuring that the criteria created count the right things as the circle. That will tell us what a circle is - or in my terms, what's correctly assertible of circles (simpliciter).
That's the kind of quibble we've been having, right? Which of these definitions captures the intended object of a circle... And honestly none of the ones we've talked about work generically. I believe "A closed curve of constant positive curvature" is the one the differential geometry man from above would've said, but that doesn't let you tell "placements" of the circle apart - which might be a feature rather than a bug. — fdrake
It might not be a confusion, it could be an insistence on a unified metalanguage having a single truth concept in it which sublanguages, formal or informal, necessarily ape. — fdrake
Edit: And why can't a quibbler say that R^3 and even R^2 spaces are not Euclidean? What's to stop him? When is a disagreement more than a quibble? — Leontiskos
But the deeper issue is that I don't see you driving anywhere. I don't particularly care whether the great circle is a Euclidean circle. If you have some property in your mind, some definition of "great circle" which excludes Euclidean circles, then your definition of a great circle excludes Euclidean circles. Who cares? Where is this getting us? — Leontiskos
Gonna call it for tonight and rethink stuff, though obviously not in your favor :D — Moliere
I'd appreciate you answering my question about whether or not paraconsistent logic would count as a plural logic insofar that we accept both paraconsistent logic and classical logic. — Moliere
Because every Aristotle Circle can be shown to be a Euclid Circle and vice versa. — fdrake
Then we're using Euclidean space differently. To me a Euclidean space is a space like R^3, or R^2. If you push me, I might also say that their interpoint distances must obey the Euclidean metric too. Neither of these are Euclid's definition of the plane. "A surface which lies evenly with straight lines upon itself" - R^2 isn't exactly a surface, it's an infinite expanse... But it's nice to think of it as the place all of Euclid's maths lives in. R^3 definitely is not a surface, but it is a Euclidean space. — fdrake
You also disagree with him strongly if you like Euclid or Aristotle's definition of a circle. I actually prefer his, since you can think of the car wheels as its own manifold, and the one he would give works for the great circle on a hollow sphere too. I think in that respect the one he would give is the best circle definition I know. Even though it individuates circles differently from Aristotle and Euclid. — fdrake
I'm not familiar with Aristotle's definition of a circle at all. I might not even understand it. Though, if I understand it, I think the two definitions are equivalent in the plane. So there's no disagreement between them. Which one's right? Well, is it right to pronounce tomato as tomato or tomato? — fdrake
early 13c., preien, "ask earnestly, beg (someone)," also (c. 1300) in a religious sense, "pray to a god or saint," from Old French preier "to pray" (c. 900, Modern French prier), from Vulgar Latin *precare (also source of Italian pregare), from Latin precari "ask earnestly, beg, entreat," from *prex (plural preces, genitive precis) "prayer, request, entreaty," from PIE root *prek- "to ask, request, entreat."
From early 14c. as "to invite." The deferential parenthetical expression I pray you, "please, if you will," attested from late 14c. (from c. 1300 as I pray thee), was contracted to pray in 16c. Related: Prayed; praying. — Pray Etymology
An incline plane in a Euclidean space is definitely a Euclidean plane. An incline plane can't contain a circle just rawdogging Euclid's definition of a circle, since an incline plane is in a relevant sense 3D object - it varies over x and y and z coordinates - and is thus subsets of it are not 'planar figure's in some sense. — fdrake
However, for a clarified definition of plane that lets you treat a plane that is at an incline as a standard flat 0 gradient plane, the "clearly a circle" thing you draw in it would be a circle. — fdrake
I have had a similar experience to this. It was a discussion about rotating an object 90 degrees in space, and having to consider it as a different object in some respects because it is described by a different equation. One of the people I spoke about it with got quite frustrated, rightly, because their conception of shape was based on intrinsic properties in differential geometry. I believe their exact words were "they're only different if you've not gotten rid of the ridiculous idea of an embedding space". IE, this mathematician was so ascended that everything they imagine to be an object is defined without reference to coordinates. So for him, circles didn't even need centres. If you drop a hoop on the ground in the NW corner of a room, or the SE, they're the same circle, since they'd be the same hoop, even though they have different centres. — fdrake
Which might mean that a car has a single wheel, since shapes aren't individuated if they are isomorphic, but what do I know. Perhaps the set of four identical wheels is a different, nonconnected, manifold. — fdrake
I can't tell if you're just being flippant here (which is fine, I enjoyed the razz), or if you actually believe that something really being the case is impossible to demonstrate in maths (or logic). Because that would go against how I've been reading you all thread. — fdrake
So where are abstractions taken from? I suggest "the world" is a sensible answer, and one that explains the "mystery" rather well. — unenlightened
If you have 23 objects you have already mathematicised them by counting — unenlightened
The incline plane does let you see something important though, you might need to supplement Euclid's theory with something that tells you whether the object you're on is a plane. Which is similar to something from Russell's paper... "For all bivalent...", vs "For any geometry which can be reduced to a plane somehow without distortion...". The incline plane can be reduced to a flat plane without distortion, the surface of the sphere can't - so I chose the incline plane as another counterexample since it would have had the same endpoint. But you get at it through "repairs" rather than marking the "exterior" of the concept of Euclid's circles. Understanding from within rather than without. — fdrake
I think this is a good line of argument. I had thought of physicalism, also metaphorically, as kind of a snake pit where whenever one snake pops its head up and you cut it off, another one simply reappears in its place, reflecting the adaptive ability of physicalism to proliferate new versions of itself in response to new objections. This overall amorphism seems highly suspect in the context of scientifc endeavour. But then the question arises, as you and others have pointed out, is it really realistic to presume you can entirely rid yourself of that type of problem and "just do" science under the guidance of methodological naturalism or some other supposedly more neutral framework? Aren't there snakes everywhere? Aren't there metaphysical commitments inherent in making your job philosophically coherent as an enterprise?
I think to an extent there are. And an associated problem is even finding generally accepted definitions of the concepts in question, so that hard lines can be drawn. Perhaps the scientific method, methodological naturalism, metaphysical naturalism (including physicalism) can be placed on a kind of spectrum of increased commitment and perhaps even that modest enterprise has its complications. — Baden
But I still think its useful to try to get out Occam's razor and try to do what we can, especially when one finds oneself defending science against ideological and metaphysical encroachment in general. — Baden
But qualitative studies do play a part in science and the soft sciences are absolutely drenched in philosophical commitments, particularly structuralist ones. Though, again, there is some kind of division envisioned between methodologies and metaphysics, it's very hard to see where that line really is. That's probably a conversation that's too broad for the scope of this thread, though I won't deny its relevancy. — Baden
Though I can see you're not having it. — Moliere
I'll start with your first premise. "...is false" presupposes no such thing as an assertion or claim -- like I noted earlier "This duck is false" could mean "This duck is fake", right? — Moliere
Note too that, "This sentence is false," is different from, "This sentence is false is false," or more clearly, " 'This sentence is false' is false. " Be clear on what you are trying to say, if you really think you are saying something intelligible at all. Be clear about what you think is false. — Leontiskos
"This sentence is false" is all I need. — Moliere
How can you insist that one is more correct than another? — fdrake
If they are different theories then they define different things, i.e. different "circles." The monist can have Euclidean circles and non-Euclidean circles. He is in no way forced to say that the token "circle" can be attached to only one concept. — Leontiskos
Alright. It just surprises me that you survived all of these different things to do with maths concepts with a strong intuition remaining that there's ultimately one right way of doing things in maths and in logic, and that understanding is baked right into the true metaphysics of the world. — fdrake
Neither of us disagree on what Euclidean, taxicab or great circles are at this point, I think. So they're not "slippery", their norms of use are well understood. The thing which is not understood is how they relate to the, well I suppose your, intuition of a circle. — fdrake
But the deeper issue is that I don't see you driving anywhere. I don't particularly care whether the great circle is a Euclidean circle. If you have some property in your mind, some definition of "great circle" which excludes Euclidean circles, then your definition of a great circle excludes Euclidean circles. Who cares? Where is this getting us? — Leontiskos
I would say that someone correctly understands a mathematical object when they can tell you roughly... — fdrake
I wouldn't say I understand the object well yet, nor what theorems it needs to satisfy, but I have a series of mental images and operations which I'm trying to be able to capture with a formalism. — fdrake
I also don't want to say that all objects are "merely" stipulated, like a differential equation has a physical interpretation, so some objects seem to have a privileged flavour of relation to how things are, even if there's no unique way of writing that down and generating predictions. — fdrake
I don't know what to tell you other than you learn that stuff in final year highschool or first year university maths. If you're not willing to take that you can do those things for granted I don't know if we're even talking about maths.
Maybe we're talking about Leontiskos-maths, a new system. How does this one work? :P — fdrake
Of course you can. If someone tells you that modus ponens doesn't work in propositional logic, they're wrong. — fdrake
More normative. It's not correct to assert that modus ponens fails in propositional logic because how propositional logic works has been established. — fdrake
they're norms of comprehension, and intimately tied up with what it means to correctly understand those objects. — fdrake
Someone who was familiar with the weirdness of sphere surfaces, eg Srap Tasmaner, will have seen the highlighted great circle, said something like "goddamnit, yeah" — fdrake
Yes
Here I am using it, no? Its use-case is philosophical, rather than pragmatic, but I don't think that makes it meaningless. — Moliere
To use ↪Srap Tasmaner 's division, this example is in (1). A child can understand the sentence. — Moliere
"Duck is false" and "2+3+4+5 is false" don't work because "Duck" and "2+3+4+5" are not assertions at all, but nouns. — Moliere
The pronoun in "This sentence is false" points to itself, which is a statement. — Moliere
"This sentence is false" — Moliere
That isn't strictly speaking true, it's just that the generalisation of the concept of planar figure which applies to circles is so vast it doesn't resemble Euclid's one at all. You can associate planes with infinitely small regions of the sphere - the tangent plane just touching the sphere surface at a point. And your proofs about sphere properties can include vanishingly small planar figures so long as they're confined to the same vanishingly small region around a point. — fdrake
What I was calling shit testing is the process of finding good counterexamples. And a good counterexample derives from a thorough understanding of a theory. It can sharpen your understanding of a theory by demarcating its content - like the great circle counterexample serves to distinguish Euclid's theory of circles from generic circles. — fdrake
Good shit testing requires accurate close reading. This is how you come up with genuine counterexamples. — fdrake