Your understanding of each of the positions seems to make them trivial rather than controversial. — Count Timothy von Icarus

Great posts. :up:

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

It seems to me that Sider's thread is the better place for this, but what you describe here doesn't really sound like metaphysics at all. The only point that sounds like metaphysics is the fanfiction metaphor, but if the fanfiction cannot be good or bad then one cannot be doing metaphysics, and are you willing to say that the fanfiction can be good or bad?

- Good to know.

- Good points.

One of the great things about producing formalisms is that they're coordinative. — fdrake

Coordination, cooperation, intersubjective agreement, etc., really tends to be the goal and limit of contemporary thinking. I think such things are useful, but I also think that at some point we have to venture out beyond the bay and into the open sea.

Seems right.

There is also a really odd thing that happens constantly on TPF (and it usually happens with SEP). Someone will champion a position like logical pluralism or dialetheism or something like that, but when it comes down to the question of what exactly they are promoting they are at a loss for words. They don't have any clear definition of, say, logical pluralism.

So we go to a secondary source like SEP or Griffiths and Paseau. But as soon as the content of the position is being taken from SEP and not from the TPFer we are no longer engaging/arguing with that TPFer. The TPFer had superficially identified with logical pluralism without being able to say what logical pluralism is, or what they mean by it, and when one flies over to SEP they have overlooked the crucial nature of this conundrum. SEP is not going to tell us what the TPFer thinks; it is only going to tell us what the author of SEP thinks. The thread becomes the discussion of a position that no one in particular holds, and that no one in particular has a stake in. In my opinion this outlines one of many misuses of SEP on TPF. Yet there is a fascination in our contemporary culture with labeling and labels!

...And to be specific, after this thread was necro-bumped Banno did the thing, "Yay for logical pluralism! Boo for Leontiskos and his logical monism!" What did Banno mean by logical pluralism? He had no real idea. Why did he think I was committed to logical monism? Again, probably no idea, although everyone took him at his word (!). It was a half-baked thought meant to stir up controversy, and that is the heart of the problem. Bringing in something like SEP is not going to make that initial move impressive or substantive.

- So apparently if you didn't get a good look at the guy who hit you, you would just assume it was Tyson. I still don't see how you would write him a letter if you don't believe he exists.

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

Right. As I said earlier, a basic challenge for the pluralist is to show which logics are acceptable/correct and which are not. I haven't seen anyone in the thread attempt such a feat, and if that can't be done then I'm not sure a serious position is being put forward. The same could be said for nihilism or monism, but no one has claimed such positions.

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

I gave that option before giving Euclid's. You are the one who brought up Euclid in the first place, but I really don't see the two descriptions as competing.

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

Pretheoretical or intuitive reasoning need not be quantified, does it? In making that comment I was making the point that pretheoretical reasoning represents the same basic idea as the calculus definition you gave. "...In calculus [consistent roundness] cashes out as a derivative, but folks do not need calculus to understand circles. Calculus just provides one way of conceptualizing a circle."

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

Well then I would ask whether the intuitive concept that is the intended concept is a mathematical concept. When a child learns to place circle-shaped blocks in circle-shaped holes they are not involved in formal geometry.

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

Or rather, producing a thing that can produce discursive knowledge. And knowing a true logical system is a kind of knowledge, which is probably discursive. I think that's right. But they are prepackaged in a very relevant sense, particularly for those of us who are not their inventors.

But I also don't think a logic like Frege's is merely a model, nor that it could be. To invent a logical system is to attempt to capture a (or the?) bridge to discursive knowledge, and I don't know that any success or failure is complete.

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

Rather, if the context is different then the geometrical response is different, and I have no dog in the fight over the question of "family resemblances" as applied to geometrical abstractions. I have claimed that there are not square circles, not that "circle" can only ever be utilized within a single context.

There will be Euclid circles in that space which are not Aristotle circles too, I believe. — fdrake

Sure.

I believe "A closed curve of constant positive curvature" is the one the differential geometry man from above would've said — fdrake

Yes, or:

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

But what is the "intended concept"? Presumably it is an intuitive concept, and are intuitive concepts mathematical formalisms? I wouldn't think so. So:

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

Why think that the intended concept is a formalism, a mathematical equation? Similarly, why think that logic is a formalism, a logical system? Perhaps logic is as I've said: that which produces discursive knowledge. It is a natural or anthropological reality, not a prepackaged formalism.

- Fair enough. Or I suppose the person could respond to the quibbler, "If the center was deleted—per impossibile—then there would only be an Aristotelian Circle."

Perhaps you can see my complaint. Given that the sort of mathematics we are engaged in is in an important sense limited only by our imaginations, so too quibbles are limited only by our imaginations. For example:

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

The flip side of this is that mathematical concepts seem to become purely stipulative and imaginary when viewed in such a way. In that case the ground rules for something like propositional logic lose all coherence and plausibility—as do all concepts—once we have dispensed with the notion of the true or useful. It then becomes nothing more than Banno's "symbol manipulation." That's why I keep asking things like this:

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

Fair enough. :wink:

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

Yes, I didn't really understand it, and it seems like neither you nor I have a firm grasp on what it means for something to be a paraconsistent logic. Like probably everyone on TPF, I have read about paraconsistent logic as I read about animals in a far off land, but I have never worked with it or made use of it. They seem to be used mostly in the way that Aldous Huxley used his encyclopedia entries.

Are you asking me whether I think that accepting both paraconsistent and explosive logic results in the robust kind of logical pluralism? My guess is that I would answer 'no.' Paraconsistency does not entail Dialetheism. And paraconsistent logic is often used informally in everyday life (if that counts). I also haven't seen anyone in this thread who favors logical pluralism embrace Dialetheism - other than yourself, of course. They seem to be mostly Augustinians, "Lord, give me logical pluralism, but not yet!"

Because every Aristotle Circle can be shown to be a Euclid Circle and vice versa. — fdrake

Suppose the quibbler has "deleted" the center, and therefore it can only be shown to be an Aristotle Circle?

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

Okay, so R^3 is a Euclidean space and R^2 is the place where all of Euclid's mathematics lives. I mean, your early insistence on locating Euclidean circles in R^2 is why I am thinking of R^2 as Euclidean space. Apparently you are making the "...ean" of Euclidean do a lot of work here.

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?

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

None of this matters much to me. I only took Euclid's definition as a point of departure or something I would be comfortable with. But I view Euclid's definition as describing a relative property of a continuous curved line that forms an enclosed shape, which is probably why I don't think the center can be "deleted."

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

But why couldn't a quibbler say that their definitions disagree on account of the formal differences between them?

- If you write a letter to Mike Tyson asking him to punch you in the face, and the next day a random guy on the street punches you in the face, has your petition been granted? Would you still await a response from Tyson?

The restricted sense of "pray" is just an accident of contemporary English. The concept traditionally has to do with petition:

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

And as such, prayer is not restricted to God, worship (latria) is.

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

I agree, but that's why I would not say that an incline plane in a Euclidean space is definitely a Euclidean plane. I don't see that there are incline planes in Euclidean space.

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

But here too, I would say that you are confusing a "flat" plane with a Euclidean plane. A Euclidean plane is not a "0 gradient plane," it is a plane without any gradient dimension whatsoever. I have been overlooking these sorts of errors, but if you are going to be persnickety about what you see as my errors then I suppose I should return the favor, especially given that you haven't shown interest in trying to mete out the question of why/when we should disagree.

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

Yep, I sympathize with him.

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

People really will say that they have four of the same tires.

But the same question about Euclid's Circle vs. Aristotle's Circle is arising here. If there is no right answer to these questions then there are no real questions, and in that case I don't know why we're arguing.

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

I'm being flippant, but not "just." :wink:

But no, I take it that your "correctly assertible" means something like "justifiably assertible," and on that reading I think it is correctly assertible that the cross-section contains a Euclidean circle. At the same time, I think the phrase "correctly assertible" is only a placeholder for further explication, because justification doesn't have food to eat unless there is a truth of the matter, at least on the horizon.

Do we get a referendum on what topics we have a referendum on? — unenlightened

:lol:

Are mathematical truths necessary if mathematics is grounded in the contingency of the world?

So where are abstractions taken from? I suggest "the world" is a sensible answer, and one that explains the "mystery" rather well. — unenlightened

But you said:

If you have 23 objects you have already mathematicised them by counting — unenlightened

Apparently you should have said, "If you have mathematized objects you have already had recourse to the 'pre-mathematical' world."

If the abstraction of mathematics is derived from the world, then the indivisibility of the 23 is more than a merely mathematical fact.

- You have often ignored my inquiry about whether it is possible to delete a point as rhetorical or unworthy, but I don't think it is. In Aristotelian terms you are conflating a description with a definition. There are different ways to describe a circle, but where each description overlaps the object in question is identical, at least according to an Aristotelian frame. That is to say, whether we draw a circle with a compass or with Aristotle's method, we still arrive at a circle. The method of drawing is not itself the definition of a circle.

You seem to identify different mathematical representations with the definition of a circle in a curious way. This strikes me as odd, but I don't mean to imply that a consensus of mathematicians would favor my view. So to nail it down a bit:

• EC (Euclid's Circle): The set of points equidistant from a single point.
• AC (Aristotle's Circle): "The locus of points formed by taking lines in a given ratio (not 1 : 1) from two given points constitute a circle."

(We are implicitly talking about a plane figure.)

Do Euclid and Aristotle disagree on what a circle is? That sort of question is what I think lurks behind much of our disagreement, such as the deletion of points. If two people draw something differently, can they both have drawn a circle?

- I was trying to use your own verbiage there, as I had been using the word "contains." For example:

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

So suppose we are talking about the cross-section of a sphere, which is what I originally thought you were pointing at. Is that something like a circumscribed inclined plane? It is certainly a set of coplanar points. Now you say, "The incline plane can be reduced to a flat plane without distortion." This captures what I said by, "an inclined plane is [...] reducible to a Euclidean plane." "Qua circles," meant to indicate the idea that an inclined cross-section of a sphere could be reduced to a Euclidean circle or else a flat circle." Or to use my own language, the inclined cross-section of a sphere "contains" a Euclidean circle.

Now does such a cross-section really contain a Euclidean circle? Trying to gain a great deal of precision on the answer to this question seems futile, but it seems to me that it is "correctly assertible" that it does (whatever your "correctly assertible" is exactly meant to mean :razz:).

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

Yes, good. First I want to say that every metaphysics is going to be a little bit like a regenerating hydra by definition. This is because the metaphysics provides a scientific paradigm, and to falsify a paradigm is more difficult than to falsify a theory. Paradigm shifts are unwieldy. Nevertheless, if a paradigm shift is made impossible by the ambiguity of the metaphysics then there is something wrong with the metaphysics. A metaphysics should be durable but not invincible.

Second, there is a significant difference between an explicit metaphysics and an implicit metaphysics. In some ways those on my side of the aisle want to say that the methodological naturalist should get explicit about his implicit metaphysics. I think this is clearly right, at least to the very limited extent that the methodological naturalist needs to explicitly admit that he has an implicit metaphysics. Does he need to go further and "make it explicit"? Not necessarily. That may not be the job of the scientist, and it may be imprudent for him to try if he is not up to the task. If his metaphysics is a fuzzy background to his theories, then it may be better to leave it fuzzy rather than try to explicate what is not clear. I want to say that the scientist only needs to muster his metaphysics when he is challenged on that front, but that for the most part he should leave it alone.

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

Sure, but if science is the grass and metaphysics is the soil then I would want to talk about the kind of soil/metaphysics required, namely rich or fertile soil. If that is the right analogy, then we would never talk about soil encroachment in general.

I was just told about a new book in this area: Spencer Klavan's, "Light of the Mind, Light of the World: Illuminating Science through Faith." Apparently he makes a case that the (religious) metaphysics of the West birthed science.

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

:up:

- There's no point in continuing if it is the same thing over and over. I have tried to move it away from the great circle into questions about disagreement in general, but if you only want to keep bringing it back to the great circle without introducing any new arguments regarding the great circle, then it will be the same thing over and over. In that case I agree that we should not continue.

Though I can see you're not having it. — Moliere

I'm not having it because you keep begging the question. You say there is a sentence/claim but you won't say what the sentence is.

It's not much different to say, "Suppose there is a sentence that is true and false. Therefore the PNC fails."

Or else, "Suppose there is a sentence that is true if it is false and false if it is true. Therefore the PNC fails." But that's not an argument. It's, "Suppose the PNC fails; therefore the PNC fails." In order to make an argument you would actually have to identify such a sentence, and I have already pointed out the problems with the "Liar's sentence."

-

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

"If false doesn't mean 'false', but instead means 'fake', then <This duck is false> succeeds even though 'this duck' is not an assertion or claim."

Do you see how silly this is? You redefined falsity as something other than falsity in order to try to make a substantive point about falsity. Do you see why I feel that I am wasting my time? These are the sort of moves that so-called "Dialetheists" routinely engage in, at least on TPF.

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

So what do you think is false? <This sentence>, or <This sentence is false>? "This sentence" cannot refer to both at the same time. You have to pick one.

How can you insist that one is more correct than another? — fdrake

I think I've been pretty clear that I don't think one is more than correct than another, at least in the face of a skepticism or a univocity like your own. For instance:

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

In common usage there are no square circles, but if we redefine either one then there could be. I've said this many times now.

-

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

I don't know where you're getting these ideas. This started with an offhand comment to frank about "square circles lurking just around the corner," and then you launched into an extended argument in favor of square circles. Early on I asked about your motivations, and you said something in favor of "shit-testing" and then tried to repair that idea in favor of "counterexamples based on accurate close reading." But it is not coincidental that shit-testing is something like the opposite of close reading, and that your posts haven't engaged in much close reading at all.

I mean, what would a university math professor think if they saw someone arguing that they can delete the point in the center of a circle and make it a non-circle? I think they would call it sophistry. They might say something like, "Technically one can redefine the set of points in the domain under consideration, but doing this in an ad hoc manner to try to score points in an argument is really just sophistry, not mathematics."

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

It is petitio principii to simply insist that, say, an inclined plane is not reducible to a Euclidean plane qua circles. You haven't offered anything more than arguments from your own authority for such premises. Beyond that, I see misreading, not close reading. I have said things like this many times:

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

But how do you know that when I talk about a circle I am restricting myself to a very strictly interpreted Euclidean conception, such that an inclined plane is not reducible to a Euclidean plane? You are the one who is insisting that there is a right answer to questions like these, not me.

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

But it's odd to talk about an "object" here. As you go on to say, you don't even know if the "object" exists. You're just attempting to solve a problem or create a model.

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

J's new thread seems on point.


The interesting question I see here is something like, "Why should we disagree?" What is a sufficient reason to disagree with someone? You seem to have fallen into the odd trap of claiming that mathematics is all arbitrary and that I have nevertheless committed some grievous sin by supposing that an inclined plane can be reduced to a Euclidean plane. If all mathematics is arbitrary, then there are no grievous sins. There is just ignorance of stipulations (such as the "great circle"). So then perhaps I am ignorant of the precise properties of a commonly-known stipulation in the math world (i.e. a "great circle"). But is that really a problem? Does someone really need to have a Masters in mathematics and understand the stipulated metaproperties of great circles in order to claim that there are no square circles lurking around the corner? I really doubt it.

Granted, I realize you think some mathematical constructs are more applicable than others, but I won't press you on that unless you somehow think that it bears on this question of the great circle.

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

Shit-testing? I think you're just pulling shit out of your ass out of desperation at this point. You're a few inches away from Amadeus', "I'm right because I'm right, and you're wrong because I said so!" ...Which is ironic given that you meant to demonstrate that being right about math is not as easy as one supposes. Have you succeeded, then?

I've had plenty of university math. You strike me as someone who is so sunk in axiomatic stipulations that you can no longer tell left from right, and when you realize that you've left yourself no rational recourse, you resort to mockery in lieu of argument.

Of course you can. If someone tells you that modus ponens doesn't work in propositional logic, they're wrong. — fdrake

Maybe "propositional logic" is as slippery as "circle."

More normative. It's not correct to assert that modus ponens fails in propositional logic because how propositional logic works has been established. — fdrake

"Established"? A bit like, "verbatim"? All you mean is, "If you mean what I mean then you will conclude what I have concluded." You vacillate on the question of whether one should or does mean what you mean, and that's a pretty serious problem. It seems like you haven't thought about these issues as much as you thought you had.

they're norms of comprehension, and intimately tied up with what it means to correctly understand those objects. — fdrake

So are there rational norms or aren't there? What does it mean to "correctly understand a stipulated object"? One minute you're all about sublanguages and quantification requiring formal contexts, and the next minute you are strongly implying that there is some reason to reject some sublanguages and accept others. I suggest ironing that out.

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

The problem is that if you hold that mathematics has no unconditional or "unquantified" relevance, then you can't give a top-level mathematical critique. You say the point at the center of a circle can be "deleted" and I say it can't, but you presuppose that there is no way of adjudicating this question. You want to be right while also holding that there is no right or wrong in such things. Hence the bluster.

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

So you use phrases like that in conversation?

To use ↪Srap Tasmaner 's division, this example is in (1). A child can understand the sentence. — Moliere

Bollocks. It is absurd to claim that such a sentence pertains to, "everyday language use and reasoning," or that a child could understand it.

"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

Well, 2+3+4+5 doesn't seem to be a noun, but okay.

The pronoun in "This sentence is false" points to itself, which is a statement. — Moliere

You haven't managed to address the argument. Let's set it out again:

1. The clause "...is false" presupposes an assertion or claim.
2. "This sentence" is not an assertion or claim.
3. Therefore, "This sentence is false," does not supply "...is false" with an assertion or claim.

Now here's what you have to do to address the argument. You have to argue against one of the premises or the inference. So pick one and have a go.

-

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.

---

Edit:

"This sentence is false" — Moliere

Or if you like, why is it false, whatever "it" is supposed to be? How do we know that it is false? Is it because you said so? But you saying so does not make a thing false, so that's a dead end. Even Wittgenstein understood that a sentence cannot prove or show its own truth or falsity.

It is as interesting to say, "2+2=4 is false." Have we thus proved Dialetheism? That 2+2=4 is both true and false? Of course not. :roll:
In both cases the only takeaway is that the speaker is confused.

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

We seem to think about mathematics very differently. You think that a point can be deleted; that a set of coplanar points might not lie on a plane, etc. Those strike me as the more crucial disagreements. Whether something can be "reduced to" a Euclidean plane or "contains" a Euclidean plane seems less crucial and more arbitrary.

At the heart of this thread seems to be the question of whether we can actually say that someone is wrong. In mathematics the point becomes protracted. For example, you might say that I am wrong about the great circle only if I am determined to bind myself to purely Euclidean constraints. Your notion of "correctly assertible" seems to be something like a subjective consistency condition, in the sense that it only examines whether someone is subjectively consistent with their own views and intentions. For example, given that someone says something contradictory, on this theory one can only say that they are wrong and disagree if there is good reason to believe that the person accepts the PNC. If there is no good reason to believe that the person accepts the PNC, then one cannot call them wrong or disagree. The logical monist, among others, will say that someone can be wrong for contradicting themselves even if they don't subjectively claim to accept the PNC.

As I have noted many times, whether the great circle is a circle seems to be a mere matter of names, or stipulated definitions. Not so with the PNC. We can't just change a name and resolve that conflict.

A paper that I often return to in this regard is Kevin Flannery's, "Anscombe and Aristotle on Corrupt Minds," although this paper is about practical reason, not speculative reason.

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

Okay, but I still don't understand why you are calling this "shit testing." Why does it have that name? It sounds like you want to give counterexamples that highlight subjective inconsistencies. Fine, but why is it called "shit testing?"

If you are just trying to give good counterexamples, then my critique of Cartesianism does not hold, but in that case I have no idea why it would be called "shit testing."


(The other possibility here is that someone's counterexample is more method than argument. For example the ancient Skeptics would argue with everyone who made a strong claim in order to try to demonstrate that strong claims cannot ultimately be made. That is apparently part of what is going on here, for the great circle has no direct bearing on square circles, but if one can generate a strong enough skepticism about circles then all claims about circles become mush, including claims about square circles.)

- I agree. Theism and evolution are both examples of unified theories. Theism is a case where mind and matter are said to come from mind; evolution is a case where mind and matter are said to come from matter. In both cases one side is given a primacy, even if the explananda are only thought to be virtually or implicitly present in the explanans. Of course there are also more robust dualistic theories than the brute fact theory noted.

Good shit testing requires accurate close reading. This is how you come up with genuine counterexamples. — fdrake

I am considering making a new thread on a related topic, but I am wondering what you actually mean by "shit testing"? Originally I thought you meant something like, "Throwing all the shit you can think of at a wall and seeing if anything sticks. Submitting an idea to a shitstorm of objections and seeing if it is still standing in the end." Yet now as you refine the idea we seem to be getting further and further from that idea, even to the point that I am wondering whether "shit testing" is an appropriate name.

(I suppose you might have meant, "Testing an idea to see if it is shit," except that that is much too far away from the quibbling that I complained of.)

- No, you're to blame for trying to reframe the issue around bogeys of "authoritarianism" and "closed-mindedness." You're a joke.

- Prayer is just a special form of impetration or petition. I suppose one could send a petition to no one in particular—a kind of message in a bottle addressed to the universe at large—but that's really not what the word means. So if person X does not exist, you do not ask person X to do something for you.

And thus the moralistic undercurrents driving this silliness have finally become fully explicit. It's hard to put so much effort into defending an undefined thesis without this sort of moralistic self-righteousness. But of course it was there all along.

- And here I was under the impression that Jamal invented TPF. :smile:

- That is closer to the foundational discussion between Srap and I, but still different. I think 's post is quite good.

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:

By rephrasing it in terms of the puzzle of the Meno and the possibility of discursive knowledge I sought to avoid such swamps, and I did that before this thread was necrobumped. The problem with this thread is that Banno and G. Russell want to say something controversial and novel and are therefore always moving between their motte and their bailey. The first question is to ask what the thesis is supposed to be, and what 'logic' means for the person proposing a thesis.

This is a helpful OP.

Q1. Why is the number 23 not divisible (evenly) by 3?
Q2. Why are 23 objects not evenly divisible into three collections of whole and unbroken objects? — J

In your other thread we touched on the Scholastic transcendentals or convertibles. Another transcendental besides being and truth is oneness (unum).

For Aristotle mathematics is the study of what belongs to quantity in various different ways. For example, arithmetic is the branch of mathematics that studies discrete quantity.

Now is it a causal fact that reality is bound up with oneness? Not really. Oneness is metaphysically foundational to reality, and is convertible with other foundational rational aspects of reality. Usually when we think of a causal reality we think of something that is limited to some subset of reality or some subset of substances. For example, reproduction via pair mating is a causal reality because it is differentiable from other kinds of reproduction and from other kinds of causes. To call the transcendental of unum "causal" would seem to be mistaken given its extreme ubiquity. Nevertheless, we need not say that it is necessary in some super-metaphysical (mathematical?) sense. So if the only categories are thought to be the category of the causal and the category of the mathematically necessary, then we would be out of luck. A universal metaphysical property of all reality, such as unum, is neither.

This idea is bound up with Platonism: that there are universal forms in which all of reality participates, and in which the human mind participates in a special way through studies like mathematics. In that way I would want to say that mathematics is not prior to reality and reality is not prior to mathematics—which is perhaps an Aristotelian variant of the Platonism. But whether we think of Plato or Aristotle, in either case there must be some tertium quid in which both reality and human knowing participate.

What we really want is an explanatory structure that preserves both of the seemingly ineluctable realities – of logic and of being. Kimhi has his views about how we might get there. A theistic argument might posit a “perfect match” because creation is deliberately thus. Or – using a metaphor from Banno – we find ourselves with a Phillips-head screw and a screwdriver that matches, so let’s leave a designed creation out of it and try to work on the problem in evolutionary terms. (I don’t think such an approach will take us far enough, but it’s certainly respectable.) — J

To say that the alignment between screwdriver and screw is an opaque and brute fact is to have abandoned the search for an overarching explanatory structure. If there is an explanatory structure that preserves both, then that explanation must encompass both the mind that knows reality and reality itself. I don't see how one could arrive at an explanatory structure such as you desire without this overarching aitia.

You forgot that Euclid specifies a circle as a plane figure. — fdrake

No I didn't.

I realise you're not going to accept that a great circle is not a Euclid circle, or that a circle in a plane at an angle isn't a Euclid circle without a repair of his definition — fdrake

See:

Yet perhaps it is not a torus but is nevertheless a set of coplanar points, falling on an implicit plane which possesses a spatial orientation. Is it a circle then? Not strictly speaking, because two-dimensional planes do have not a spatial orientation. — Leontiskos

I've been using the word "verbatim" to try to mean a couple of things:
A ) At face value.
B ) Using only the resources at hand in a symbolic system.

Thus Euclid's definition of a circle, verbatim, would exclude the great circle. — fdrake

But it is here illustrative that I am not familiar with the concept "great circle," especially as to its specific geometrical properties, and I did query you about the picture you posted. You thought there was a verbatim sense of "great circle," but you were mistaken. You would have to explain what you mean by it in order to achieve your contradiction, because "great circle" says very little, verbatim.

And if you want to just talk about your intuitions without recourse to formalism, I don't know if this topic of debate is even something you should concern yourself with. — fdrake

I think you're moving too fast. Formalisms have limits. What are the specific properties of lines, points, circles, great circles, two-dimensional planes, three-dimensional planes, etc.? How do they relate to each other? For example, can points be deleted or not? Is the great circle a torus, and if not is it three-dimensional at all? You're making a bunch of assumptions in all of this and drawing a fast conclusion.

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?

If you actually want my perspective on things, rather than trying to illustrate points from the paper: I'm very pragmatist toward truth. I prefer correct assertion as a concept over truth (in most circumstances) because different styles of description tend to evaluate claims differently. As a practical example, when I used to work studying people's eye movements, I would look at a pattern of fixation points on an image - places people were recorded to have rested their eyes for some time, and I would think "they saw this", and it would be correctly assertible. But I would also know that some subjects would not have had the focus of their vision on some single fixation points that I'd studied, and instead would have formed a coherent image over multiple ones, in which case they would not have "seen" the area associated with the fixation point principally, they would've seen some synthesis of it and neighbouring (in space and time) areas associated with fixation points (and other eye movements). So did they see it or didn't they?

So I like correctly assertible because it connotes there being norms to truth-telling, rather than truth being something the world just rawdogs into sentences regardless of how they're made. "There are 20kg of dust total in my house's carpet"... the world has apparently decided whether that's true or false already, and I find that odd. Because it's like I'm gambling when I whip that sentence out. — fdrake

Okay, thanks. And I agree with this. I am interested in knowledge—including justification—as opposed to just truth. Very often justified knowledge is precisely that which has been (correctly) logically inferred. I would define logic as that thing that gets you to (discursive) knowledge, or at least to justified assertion.

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.

I would have thought it clear how it relates to logical pluralism. If you model circles in Euclid's geometry, you don't see the great circle. But if you look for models of the statement "a collection of all coplanar points equidistant around a chosen point", you'll see great circles on balls (ie spheres, if you don't limit your entire geometry to the points on the sphere surface). They thus disagree on whether the great circles on balls are circles.

If you agree that both are adequate formalisations of circlehood in different circumstances, this is a clear case of logical pluralism. — fdrake

So:

Let's suppose it is a countermodel. How does the logical pluralism arise? I can only see it arising if we say that a "circle" means both Euclid's definition and the great circle countermodel, and that these two models are incompatible. Is that what you hold? — Leontiskos

For the univocalist the two definitions are incommensurably different. For the analogical thinker there is an analogy between a great circle and a circle. I think both adhere to the definition, "A set of coplanar points equidistant around a single point," but this also involves analogical equivocity between 2D planes and 3D planes.

That also lines up just fine with my view of logic. If logical pluralism means there are incommensurably different logics which are true/correct, then I disagree. If it means there are analogically similar logics which are true/correct, then I agree. But I don't think that all true logics are isomorphic. "Incommensurably" is meant as strong incommensurability, in the sense of excluding analogical equivocity.

The taxicab example is designed as a counterexample to the circle definition "a collection of all coplanar points equidistant around a chosen point", since the points on the edge of the square in Euclidean space are equidistant in the taxicab metric on that Euclidean space. It isn't so much an equivocation as highlighting an inherent ambiguity in a definition. — fdrake

Again, I think there is an equivocation on "distant." Equidistant qua circularity pertains to straight lines. The taxicab circle is premised on an extreme redefinition of "distance" - an equivocation.

The extensional difference between all of these different formalisms are the scope of what counts as a circle. A pluralist could claim that some definitions work for some purposes but not others, a monist could not. — fdrake

Although I don't hold to logical monism, this doesn't seem right. You are claiming that for the logical monist a token such as 'circle' can mean only one thing. I don't think that's right.

The Analytic dispute between logical pluralism and monism strikes me as a superficial dispute. The deeper question is univocal vs. analogical predication. That source abandons the more interesting question as soon as it limits itself to, a "model-theoretic definition." Pluralism looks like a poor man's analogicity, like trying to draw a perfect circle with pixels. My guess is that most versions of soft pluralism and monism are not even differentiable, unless there is some precise concept of "equally correct" logics or arguments (which I highly doubt).

To put it in super blunt terms, Euclid's theory would have as a consequence that the great circle on a ball is not a circle. The equidistant coplanar criterion would prove that the great circle on a ball is a circle. Those are two different theories - consequence sets - of meaningful statements. A pluralist would get to go "wow, cool!" and choose whatever suits their purposes, a monist would not. — 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.

@Count Timothy von Icarus

You might not even be a logical monist in the OP's sense, since the kind of logic it's talking about is formal? — fdrake

Just pulling this for context. The OP is three years old. The recent discussion is not about the OP. After frank bumped the thread Banno brought in an external conversation, and pigeon-holed the discussion into one of those interminable, internecine Analytic disputes (Pluralism vs. Monism).

The external conversation revolves around this post from Srap:

So we have (1) the primary phenomena, everyday language use and reasoning.

Then there's (2) the way logic schematizes these.

And there's the further claim that in carrying out (2), we see (3) the deep structure of everyday language and reasoning, the underlying logical form.

My claim was that we can talk about (2), whether (3) is true or not, and even without considering whether (3) is true or not.

It's the same thing I've been saying all along, that (2) doesn't entail (3). — Srap Tasmaner

This was Srap's attempt to frame it, but we went on to ask whether that framing was neutral or not.

I tried to continue the conversation in that thread, but Banno insisted on bringing it here. If Srap had continued the conversation in that thread I would have simply ignored Banno's transplant, given how insubstantial it was bound to become.

My position has never been logical monism's program of a single true formalization. That's just something Banno falsely pinned on me. For example:

Each time you state the problem in terms of artifice or invention you fail to capture a neutral (2). Do you see this? To call logic an invention of artifice, or a schematization or formalization, is to have begged the question. If that's all logic is then the answer to (3) is foreclosed. — Leontiskos

Fair enough. Part of the issue here is whether pluralism can be set out clearly. As the SEP article sets out, the issue is as relevant to monism as for pluralism. The question is how the various logics relate. It remains that monism must give an account of which logic is correct. — Banno

No, not really. You really ought to read Rombout on the way that Frege and Wittgenstein mean different things by "logic." Your whole frame is mistaken. I am not a "logical monist," and I don't think Timothy is either. If every logic is on the same level, then pluralism must be true. Logical monism and logical pluralism strike me as equally silly.

You've made it plain that you don't accept Dialetheism, and will give no reason, so the point is moot. — Banno

You've made it plain that you won't offer any arguments, only assertions. Moliere tried and I answered his.

It's like "This sentence has six words" in some ways — Banno

"In some ways."

Unlike "...is false," "...has six words" does not require an assertion/claim.

(Moliere and yourself are doing what I would call Dialetheist apologetics. You've heard objections to the "Liar's paradox" and you are responding to those objections, regardless of the fact that my objection is quite different.)

- So for Griffiths and Paseau "logical monism" holds that there is one true formalization. I have not seen anyone on TPF hold this theory, and I certainly do not. He is also talking about consequence rather than inference. "Logical monism" does not look at all like the classical view.

Again, for Aristotle logic is the solution to the problem of the Meno. It is how discursive knowledge is achieved. It is primarily a matter of inference. Aristotle was quite clear that his formalization was not identical to logic in this fundamental sense.

If someone wants to argue for logical pluralism I would want to know exactly what they mean by that term, because it has been unhelpfully ambiguous all throughout this thread.

Point well-made and taken. That should have been further qualified as all spherical lines of circumference. That's what I meant. That's what I was thinking. Evidently a few synapses misfired. — creativesoul

Well, one might accept it. I don't see any of these objections as straightforward. I don't think there is a "verbatim" meaning, to use @fdrake's word.

Does the circumference of a (Euclidean) circle encircle space? Yes, two-dimensional space. But then does the great circle's encompassing space make it a non-circle? Apparently not. Unless what we mean is that the great circle encompasses three-dimensional space, in which case this does make it a non-circle.

Just wondering if I've understood something. — creativesoul

Fair enough, and I meant to ask in a broader way and include fdrake.

My interest was piqued by the claim that a line of circumference around a sphere was a circle. — creativesoul

I am quite fine with that claim. Apparently I think the coplanar points of the great circle contain a circle (and a two-dimensional plane).

fdrake effectively puts words in my mouth in declaring victory, "Ah, when you say 'great circle' you mean something which does not contain a two-dimensional plane, therefore when you say 'great circle' you don't mean a Euclidean circle." But I never assented to any of these sorts of interpretations.

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My position was that there are circumstances in which it makes sense to say there are square circles, perhaps even that there are circumstances in which one can correctly assert that there are square circles, not "there are square circles" with an unrestricted quantification in "there are". — fdrake

So you are ("perhaps") willing to say that there are circumstances in which one can correctly assert that there are square circles, but you won't commit yourself to there being square circles. This is odd.

The idea behind this sort of thinking seems to be that every utterance is limited by an implicit context, and that there are no context-independent utterances. There is no unrestricted quantification. There is no metaphysics. I take it that this is not an uncontroversial theory. Here is an example of a statement with no implicit formal context, "There are no Euclidean square circles." You would presumably agree. But then to be wary of the claim that there are no square circles, you are apparently only wary of ambiguity in the terms. You might say, "Well, maybe someone would say that without thinking of Euclidean geometry." But we both know that there is no verbatim meaning of "square" and "circle," at least when subjected to this level of skepticism. This is a nominal dispute, but it won't touch on things like logical pluralism, for that question has to do with concepts and not just names. A new definition of "circle" will not move the needle one way or another with respect to the question of logical pluralism. As noted, the taxicab case involves equivocation, not substantial contradiction.

I am still wondering:

I'm not really sure what you are arguing, fdrake. It doesn't sound like you hold to logical nihilism or logical pluralism in any strong or interesting sense. Am I wrong in that? — Leontiskos