There's the nub. It looks circular...

For me this quote is most indicative of the relativism I have opposed:

Take all the points Euclidean distance 1 from the point (0,0) in the Euclidean plane. Then delete the point (0,0) from the plane. Is that set still a circle? Looks like it, but they're no longer equidistant from a point in the space. Since the point they were equidistant from has been deleted. — fdrake

For fdrake it would seem that when we see a shape he has drawn on a piece of paper, which looks like a circle, we must ask him if he "deleted the point at the center" before drawing the conclusion that it is a circle. Apparently in order to identify a circle, formally or materially, we must worry about whether the center point has been "deleted." This is taking the subjectivism and relativism a bit far.

(Like points, apparently planes can also be "deleted.")

(Like points, apparently planes can also be "deleted.") — Leontiskos

Yes! The set {1,2,3} can have the element 3 deleted, giving the subset {1,2}. Is what I meant. The plane without the origin. This is a perfectly cromulent thing to do with sets.

Yeah I was only thinking about the point being away from the plane, no other fiddling. If I've ever considered that, it was so long ago I've forgotten.

It's just a curiosity that talking about the center of a circle is a little over-committal. It's the center, coplanar, only under a particular projection onto the plane of the circle. But under other projections, the "center" lands elsewhere, which for some reason seems really cool and even useful to me.

- My contention would be that there is no such thing as coplanar points without a plane, and that the cross-section of a hollow sphere is a collection of coplanar points.

My contention would be that there is no such thing as coplanar points without a plane, and that the cross-section of a hollow sphere is a collection of coplanar points. — Leontiskos

I suppose that means the great circle isn't a circle, since there's no coplanar points on it... Since there's no way to form a plane out of the points on a sphere's surface when you're only allowed to consider those.

But if indeed you can form a cross section, allowing yourself the exuberance of 3-space, then they are indeed coplanar and form a circle.

I suppose it's then an odd question why the same set of points can be considered a circle or not depending upon whether you consider them as part of a larger space.

Regardless though, there's no word for "coplanar" in Euclid's definition of a circle either. So we've needed to go beyond Euclid regardless. It would be odd if Euclid ever had need of the word, considering his is the geometry of the plane.

But under other projections, the "center" lands elsewhere, which for some reason seems really cool and even useful to me. — Srap Tasmaner

Can you show me one please?

I suppose that means the great circle isn't a circle, since there's no coplanar points on it... Since there's no way to form a plane out of the points on a circle's surface when you're only allowed to consider those. — fdrake

It seems that we mean different things with the words "point" and "plane." On my view you have reified abstract realities, making them, among other things, delete-able.

Regardless though, there's no word for "coplanar" in Euclid's definition of a circle either. So we've needed extra concepts from Euclid regardless. It would be odd if Euclid ever conceived of the word, considering his is the geometry of the plane. — fdrake

These objections are too subtle, such as supposing that I meant to confine myself to Euclid in an especially strict manner, or that the cross-section of an abstract sphere cannot be an an abstract circle.

On my view you have reified abstract realities, making them, among other things delete-able. — Leontiskos

Deletion is shorthand for considering different sets - or using the set division operation. The sets I'm referring to were $\mathbb{R}^2$ and $\mathbb{R}^2/\{0\}$.

Are you not used to this sort of maths?

Are you not used to this sort of maths? — fdrake

It's been too long to do much more than mildly jog the memory.

It's been too long to do much more than mildly jog the memory. — Leontiskos

Fairy muff.

I suppose it's then an odd question why whether the same set of points can be considered a circle or not depends upon whether you consider them as part of a larger space. — fdrake

As I understand it, the "plane" in the definition of a circle is not a space, at least in the sense that your term "larger space" indicates. The cross-section of a sphere conceived as two-dimensional is planar in one sense and non-planar in another.

So is there some impediment to taking the basic definition of a circle given and saying that the cross-section of a sphere conforms to this? I don't see any real impediment. Any three-dimensional translation that occurs will not be contentious. If we interpret the abstract space presupposed by the definition of a circle to be incommensurable with the abstract space presupposed by the cross-section of a sphere, then there is clearly an impediment, but this sort of exclusion is less plausible than the alternative. How exactly do the three-dimensional points of a sphere translate to the two-dimensional points of its cross-section? I don't know, but it doesn't strike me as a great problem.

In any case we are very far from demonstrating square circles, which was the original topic.

I'll draw if I have to, but I think I can clarify it verbally.

1. Pick a point and a length.

These together determine a bunch of circles in 3-space.

2. Pick one.

If you picked one that isn't coplanar, there's a projection of the "measuring point" onto the plane the circle is in that preserves the property of being equidistant from points on the circle, in fact preserves it as you move the point toward the plane, shrinking your originally chosen length until it's the radius of the circle.

But there are other projections where that original point will land off-center, or on the circle, or outside it.

If you want to go backwards, you need an additional constraint**, because there's a whole line of possible "measuring points" through the center of a circle, perpendicular to its plane, like an axel. Your measuring point could be projected to anywhere in the plane, and any point in the plane could be projected to anywhere on that axel line.

You could also play with projecting the circle and the point onto yet another plane.

It's just curious that you can separate the point that generates the circle from its center, that those are two different properties, and there are projections that will separate them in a plane.

** The original length gives you two, I think

I guess once you have the "axel" in mind, you could say that choosing the point where that line intersects the plane of the circle as the point that "determines" the circle is natural and convenient, but just a convention. The radius and center and plane of a circle determine it, but so would an infinite number of pairs of points and distances.

*** If you think of the determining point as the vertex of a cone, there are an infinite number of cones, all sharing an axis, the circle is a section of.

I would quite like you to draw this. I don't think I am imagining it accurately.

*** If you think of the determining point as the vertex of a cone, there are an infinite number of cones, all sharing an axis, the circle is a section of. — Srap Tasmaner

I was imagining a cone, yeah. But now the variability makes sense given that there's an infinity of them. Am I right in thinking that the "correct" visualisation regarding picking the vertex is also equivalent to picking the gradient of the lines bounding the cone? Insofar as it constraints the circle in the plane's radius anyway.

I would quite like you to draw this — fdrake

I'm glad you came back to this, and I'm going to draw some pictures. I had decided last night there was nothing here and I don't know why I was going on about it, but I have an idea now!

Sure, if by "pure" we mean "ignoring the content and purpose of logic." But even nihilists and deflationists don't totally ignore content and the use case of logic. If you do this, you just have the study of completely arbitrary systems, and there are infinitely many such systems and no way to vet which are worth investigating. To say that some systems are "useful" is to already make an appeal to something outside the bare formalism of the systems themselves. "Pure logic" as you describe it could never get off the ground because it would be the study of an infinite multitude of systems with absolutely no grounds for organizing said study. — Count Timothy von Icarus

The difference I intend between pure (as such) logic and applied (transcendental) logic is that we can do logic without addressing questions of being, whereas the latter gets into the weeds of various philosophical questions (but simultaneously presupposes a logic to get there). Logic is an epistemic endeavor dealing with validity whereas the question of the relationship of logic to being is getting more into metaphysics rather than logic.

One might push back on Aristotle's categories sure, but science certainly uses categories. The exact categories are less important than the derived insights about the organization of the sciences. And the organization of the sciences follows Artistotle's prescription that delineations should be based on per se predication (intrinsic) as opposed to per accidens down to this day....

That said, if all categories are entirely arbitrary, the result of infinitely malleable social conventions, without relation to being, then what is the case against organizing a "socialist feminist biology" and a "biology for winter months," etc ?

They certainly wouldn't be useful, but that simply leads to the question "why aren't they useful?" I can't think of a simpler answer than that some predicates are accidental and thus poor ways to organize inquiry. — Count Timothy von Icarus

And to highlight why this is difference -- this line of questioning you're exploring here will be an interesting question whether we are logical monists, logical pluralists, or logical nihilists. Deciding the first question doesn't necessitate a relationship between logic, the mind, being, and knowledge. We could be logical monists on the basis that there is one true logic, but we don't know what that one true logic is yet -- inferred from the conflicting accounts of logical laws -- but retain the notion that there must be One Logic to Rule them All (or, that, in fact, one logic does rule them all, if you just incorporate this already implicit Lemma....)

And simultaneously hold that there is no relationship between logic and being -- i.e. that the One True Logic is the result of the structure of knowledge requiring this or that axiom, but could still be anti-realist projections which have no relationship to being.

The purpose and scope of logic is certainly being considered by logicians, it's just that these are different questions. (also -- I, for one, am all for a socialist feminist biology for the winter months :D )

What if in place of Kant’s Transcendental categories we substituted normative social practices? Doesn’t that stay true to Kant’s insight concerning the inseparable role of subjectivity in the construction of meaning while avoiding a solipsistic idealism? Don’t we need to think in terms of normative social practices in order to make sense of science? — Joshs

That's a lot closer to home to my way of thinking -- and why I like Feyerabend's deconstruction of Popper as a kind of object lesson for all philosophies of science which try to encapsulate the whole within some system: what I'd call totalizing.

Though at that point we would be kind of in the realm of both Hegel and Marx -- the historical a priori looks a lot like those big theories of history to me. And that's getting close to a similar totalizing project, at least on its face.

Though at that point we would be kind of in the realm of both Hegel and Marx -- the historical a priori looks a lot like those big theories of history to me. And that's getting close to a similar totalizing project, at least on its face — Moliere

That’s what pragmatist-hermeneutical and poststructural models of practice are for. For Hegel and Marx the dialectic totalizes historical becoming. In these latter models cultural becoming is contextually situated and non-totalizable. It is normativity all the way down.

But these are so far from counterexamples to Aristotle that they are all things he explicitly takes up. — Leontiskos

Do they need to be counterexamples to Aristotle?

I don't think so. I think that I'd simply have to want to utilize some other logic -- and there are some good reasons for putting Aristotle aside in these cases. First and foremost because we're not strictly utilizing Aristotle's logic here. The Logical nihilist or pluralist or monist isn't putting together All/Some statements into the classical forms -- The Background here has incorporated parts of Aristotle (classical logic is still taught!), but isn't appealing to Aristotle's commonsensical intuition about the logic of objects.

But I don't think statements behave exactly like objects do (and I am terribly allergic to commonsense -- it's not that I don't get it, but if the appeal is to commonsense then one need not study logic in the first place. There are far more lucrative and stable careers than academia)

Basically we don't need to explicitly refute Aristotle in how we do logic. We are free insofar that we create something interesting.

Every time I have seen someone try to defend a claim like this they fall apart very quickly. The "Liar's paradox" seems to me exceptionally silly as a putative case for a standing contradiction. For example, the pages of <this thread> where I was posting showed most everyone in agreement that there are deep problems with the idea that the "Liar's paradox" demonstrates some kind of standing contradiction. — Leontiskos

(1) is false. (1)

Read that as (1) being the name of the sentence so that the sentence references itself like we can do in plain English.

At face value it's clear to see that if 1 is false then it is true. And if it is true then it is false. If we combine this with the law of the excluded middle we must conclude that (1) is both true and false.



This is the notion of a dialethia. I went for a review before posting here and want to reference the SEP bit on paraconsistent logic in the liar's paradox article because just below it has an entry on dialetheism.

Priest (1984, 2006) has been one of the leading voices in advocating a paraconsistent approach to solving the Liar paradox. He has proposed a paraconsistent (and non-paracomplete) logic now known as LP (for Logic of Paradox), which retains LEM, but not EFQ.[10] It has the distinctive feature of allowing true contradictions. This is what Priest calls the dialetheic approach to truth.

He has some interesting examples, but this would take us very far astray.

It's more that here seems a reasonable approach to the liar's paradox that produces interesting and novel results in logic.

Then it seems we're more or less in agreement. :up:

I would also tend to suppose that there may indeed be many ways to "skin a cat," different systems that are equally good for x purpose, but then these systems will have similarities, mappings to one another.

A good example of how re-thinking how we phrase the apparent paradox can provide new insight. We have "This sentence is false". It seems we must assign either "true" or "false" to the Liar – with all sorts of amusing consequences.

Here is a branch on this tree. We might decide that instead of only "true" or "false" we could assign some third value to the Liar - "neither true nor false" or "buggered if I know" or some such. And we can develop paraconsitent logic.

Here's another branch. We might recognise that the Liar is about itself, and notice that this is also true of similar paradoxes - Russell's, in particular. We can avoid these sentences by introducing ways of avoiding having sentences talk about themselves. This leads to set theory, for Russell's paradox, and to Kripke's theory of truth, for the Liar.

Again, we change the way we talk about the paradox, and the results are interesting.

And again, rejecting an apparent rule leads to innovation. — Banno

Right!

And far from rejecting classical logic it seems to me to give clarity to its underlying intuitions. These extensions of logic aren't so much an Undermining of All Thought, but in the critical tradition which explores terra incognita.

Super cool stuff.

Super cool stuff. — Moliere

Yeah, I agree. Links form here to a whole lot of other stuff.

The difference I intend between pure (as such) logic and applied (transcendental) logic is that we can do logic without addressing questions of being, whereas the latter gets into the weeds of various philosophical questions (but simultaneously presupposes a logic to get there). Logic is an epistemic endeavor dealing with validity whereas the question of the relationship of logic to being is getting more into metaphysics rather than logic.

I agree that you can study logic in total abstraction from content.

I am not sure if you can have an "epistemic endeavour," that is unrelated to being though. What is our knowledge of in this case? Non-being?

I also don't think we can have such an abstract study without the concepts provided by experience and sense awareness. For if we had no experience of the world, of encountering falsity, how would we even know what terms like "truth-preserving" meant? Likewise, how does one even have a concept of existential quantification without a concept of existence? That is, we can only abstract away the world so much.

Which is a good thing IMO. If we totally leave the world behind we'd have an infinite number of systems and no way to judge between them vis-á-vis which are deserving of study.

And simultaneously hold that there is no relationship between logic and being -- i.e. that the One True Logic is the result of the structure of knowledge requiring this or that axiom, but could still be anti-realist projections which have no relationship to being.

Suppose we had a formal system that answered all our questions about physics, or maybe some area of it like fluid dynamics. How could it have "no relation" to being? At the very least, it would have a relation to our experiences, which are surely part of being.

The purpose and scope of logic is certainly being considered by logicians, it's just that these are different questions. (also -- I, for one, am all for a socialist feminist biology for the winter months :D )

I want to do leap year physics. You get a nice three year break.

different systems that are equally good for x purpose, but then these systems will have similarities, mappings to one another. — Count Timothy von Icarus

Yes. Like Hamiltonians and Lagrangians. Do the same thing differently.

If we totally leave the world behind we'd have an infinite number of systems and no way to judge between them vis-á-vis which are deserving of study. — Count Timothy von Icarus

You mean if we leave the world behind after discovering the systems? :wink:

Do they need to be counterexamples to Aristotle? — Moliere

They are supposed to be objections to Aristotle, so yes, of course they do. You might as well have objected to Mr. Rogers by telling us that you prefer people who put on shoes. Mr. Rogers puts on shoes in every episode.

(1) is false. (1)

Read that as (1) being the name of the sentence so that the sentence references itself like we can do in plain English. — Moliere

As has been pointed out numerous times, this is just gibberish. What do you mean by (1)? What are the conditions of its truth or falsity? What does it mean to say that it is true or false? All you've done is said, "This is false," without telling us what "this" refers to. If you don't know what it refers to, then you obviously can't say that it is false. You've strung a few words together, but you haven't yet said anything that makes sense.

The 'great circle' looks elliptical to me. "Circle" is being used in the same argument in two different senses.

Hence @fdrake's pointing out the inadequacy of @Leontiskos' definition.

A great circle is the longest possible straight line on a sphere. No midpoint and diameter in that definition.

Hence fdrake's pointing out the inadequacy of @Leontiskos' definition.

A great circle is the longest possible straight line on a sphere. No midpoint and diameter in that definition. — Banno

Ah. Understood. I need to read more carefully. Thanks. I appreciatcha!