Has it been fixed? The "sophist" would say no, and can quibble endlessly. They might ask you to specify what exactly "I am Gillian" means; what 'I' means; what a name is; what the predication of amness means (all difficult questions). They might splice (1) and (2) into different contexts, pointing out that (1) is a third-person description and (2) is a first-person description, and that it is not clear that these two discrete contexts can produce a conclusion that bridges them. "Shit-testing" seems to have no limits and no measure. — Leontiskos

Those are quite different I believe. There's no attempt to change the verbatim meanings of argument terms in @Count Timothy von Icarus's repair, in fact there's an insistence on representing the conceptual content of what's said in spite of the means of its representation (predicate logic vs "I"). In effect, Timothy's takes the truth of the argument for granted and treats the inability of the verbatim machinery of propositional logic to reflect that truth as a failing of the logic... thus repairing the argument by explicitly spelling out the context sensitivity of "I".

Whereas your examples do not insist on taking the conceptual content of what's said for granted, indeed they're attempting to distort it. Allegorically, the logic of shit testing is that of a particularly sadistic genie - taking someone at their word but exactly at their word, using whatever pretheoretical concepts they have. The logic of your sophist is closer to doubting the presuppositions which are necessary for the original problem to be stated to begin with.

Our dispute was similar to the former - we both have the same pretheoretical intuitions about what a circle is. Agreeing on Euclid's and on the great circle's satisfaction of it. And we'd probably agree on the weird examples containing deleted points too, they would not be circles even though if you drew them they'd look exactly like circles. The issue we were having is that Euclid's definition clearly did not accurately represent our (mostly) shared pretheoretical intuition regarding what a circle was - what it looked like -, and I kept asking you to repair it.

Remember even Euclid saw fit to define a circle axiomatically. And his works exactly as planned in the plane. Just circles also live outside the plane, and thus are not bound by Euclid's plane figure definition of them verbatim.

"For all circles in the plane... (Euclid's theorems follow)" - another example which could've been in Russell's paper.

So a logical nihilist might say "Aha, "this sentence is false" disproves LEM!, we cannot use propositional logic". and Russell invites us to say: "I'm going to use propositional logic only for sentences we know satisfy LEM". The latter constrains the range of stuff you can sensibly throw into the collection of models of the logic, and so you end up filling up the semantic entailment relation again in the system by artfully removing the counterexamples.

In effect the nihilist doubt machine gets going by noticing that there's arbitrary degrees of contextual variation, and throws every available piece of crap against the expectations of logical form a universalist has (like @Leontiskos and I's discussion earlier), when ultimately only the universalist need read the nihilist doubt machine as nihilist - it's just a doubt machine, you can tell it to sod off by specifying the exact mess you're in.

Per Russell it is "the claim that there are no laws of logic, i.e., no pairs of premise sets and conclusions such that premises logically entail the conclusion." — Count Timothy von Icarus

But... P & P => Q entails Q in propositional logic, who is denying this? It does not seem Russell is:

like thinning, cut, and the sequent forms of conjunction elimination. The
reason is this: a natural interpretation of the claim that there is no logic is that
the extension of the relation of logical consequence is empty; there is no pairing
of premises and conclusion such that the second is a logical consequence of the
first. This would make any claim of the form Γ |= φ false, but it would not
prevent there from being correct conditional principles.10

And footnote ten:

A note about vocabulary: arguments are often said to be neither true nor false, but
rather valid or invalid. This is correct as far as it goes, but a principle containing a turnstile
as its main predicate can be regarded as a sentence making claim about the relevant argument.
Such a claim will be true if the argument is valid, false if it is not. Hence the nihilist can be
said to believe that there are no true atomic claims attributing logical consequence.

The logical consequence relation is preserved, even if the intended objects it's supposed to refer to can be taken as counter models. Like "This sentence is false" might be taken as a countermodel for the law of excluded middle, or the great circle might be taken as a countermodel for Euclid's definition of a circle.

Consider Russell's proof and refinement of LEM:

Either φ is true in a model M, or it is false. In the first case, φ∨¬φ is true in M because of the truth-clauses for ∨. In the second case, ¬φ is true in M because of the truth-clause for negation, and
so again φ ∨ ¬φ is true in M. So either way it is true in the model, and—since M was arbitrary—it is true in all models. So φ ∨ ¬φ is a logical truth...

So we examine our simple proof and realise that our assumption that the sentence could only be true or false is violated by the monster*. Hence our culprit is the assumption that sentences can
only be true and false. Still, perhaps there are some sentences which can only be true or false—sentences in the language of arithmetic might be like—and our result would hold for these. Our new theorem reads: for any φ which can only be true or false, φ ∨ ¬φ is a logical truth. Just as the geometry teacher dubs polyhedra which satisfy the stretchability lemma simple, so we could give a name to sentences which meet our assumption. Perhaps bivalent would be suitable. Then we can retain the proof above as a proof of:

For all bivalent φ, φ=>φv~φ

I underlined "bivalent" in the final bit, since you produced a similar repair to the argument:

1 ) Gillian is in Banf
2) Therefore, I am in Banf.

by understanding "I" as "Gillian", then adding this as a specification in the argument:

1 ) Gillian is in Banf
2 ) I am Gillian.
3 ) Therefore, I am in Banf.

Your repair could well have read "For all I-s who are Gillian", just like Russell's repair of LEM reads "for all bivalent φ".

It's also worth noting that Russell's countermodels, monsters and context specifying information (eg "for all bivalent") aren't necessarily in the object language in question. EG propositional logic just
assumes bivalent φ, so LEM applies, so you couldn't formulate a "neither" valued statement in its standard operation.

And since her countermodel of a statement which evaluates to "neither" does not have an interpretation in terms of standard propositional logic, she expands what ought to clearly be the scope of any logic of propositions to include that statement, goes "bleh, any logic worth its salt should account for this...". marks down on the page "eh, propositional logic as is works fine for bivalent φ" and then moves onto new pastures of polyvalent φ.

Russell's approach is largely telling logical nihilists not to throw the baby out with the bathwater, just because they expect logical laws to behave like The One Law To Rule Them All, a kind of context invariant divine providence.... and when they don't, why not just say they work when they work and find out where they work?

That makes sense. Equivalence classes of pre-images of projections under some relation seems like a cool idea.

Sphere, not ball. The surface. The 2-manifold. — Srap Tasmaner

With you. Yeah.

Am I getting something wrong here? — Srap Tasmaner

Nah it looks fine. I'm just confused, it's doing away with the centre by providing an equivalent construction of the centre. Which is also fine, I just want to see what you're seeing in it.

those three points determine a unique plane, — Srap Tasmaner

Are those points in the interior of the sphere or on its surface?

Do they? Isn't the question one of the questions at issue whether anything follows from anything else? — Count Timothy von Icarus

I don't think that's an issue at stake at all. If no principle holds in complete generality, they may still hold in certain well understood and well demarcated cases and contexts. Such as modus ponens in propositional logic.

The idea that (nothing follows from anything else in virtue of a valid argument) if (there are no principles which hold in complete generality) is ultimately not a premise in the OP's argument. It could be, and I believe Gillian Russel lectures as if, there are valid arguments even if there are no principles which hold in complete generality. Because she specifies what context she's speaking in. It then remains to be seen if a sense of complete generality can be restored by supplanting restricted statements - like Euclid's definition of a circle - with disambiguating phrases - like "in plane geometry".

Do they? Isn't the question one of the questions at issue whether anything follows from anything else? — Count Timothy von Icarus

Does everyone who understands a system and a proof in it believe the conclusion if the proof is correct and understood? Yes. Everyone agrees that P & P=>Q allows you to derive Q in propositional calculus. That's less about there being rules which cover everything, and more about there being followable rules. That's a followable, derivable rule.

However, it seems to be something quite different to claim that all[/ claims are true only relative to stipulated systems and that none are more true than any other. — Count Timothy von Icarus

Are any of these more true than any other? — Count Timothy von Icarus

Indeed. It doesn't seem meaningful to claim that the axiomatic systems are true or false in toto. But nevertheless, if there is a single unifying, bivalent truth concept, and two systems have incompatible theories, we should be able to say which is true and which is false. If we did not need to, we'd have to suspend that some claims are not evaluable as true or false in principle - and thus jettison bivalence by destroying the assignment mechanism of statements to truth values. And if we did need to, we'd have to claim that some systems are... false.... somehow, even when they seem to adequately represent concepts in precisely the same manner as others, just different concepts.

claims are true only relative to stipulated systems and that none are more true than any other. — Count Timothy von Icarus

I don't think logical pluralists are committed to that. Everyone agrees what follows from what stipulations. So it's true to say that "not every group is abelian". You can think of stipulations as disambiguations - which is what lemma incorporation works like.

The underlying issue seems to be that everyone can agree that eg groups have certain properties, but if you stipulate the definitions differently you change the properties. But you don't change the properties of the intended object when that object is the group, you perhaps change what the intended object is tout court.

I think that is in the direction of the intended thread topic. Because the ability to stipulate lemmas that make an axiomatic system better track an intended object's properties thereby lets you make more universal judgements about more precisely demarcated structures. Everyone will agree that Euclid's definition of circle captures plane circles, but not all pre-theoretically intuited circles are plane circles.

In effect this is a way of massaging the "complete generality" predicate in the OP's argument. You can restore a sense of "complete generality" by using lemmas, by speaking about something ultra specific and formalised you can guarantee that it works in that way for that system, the latter applies without exception. Applies without exception in the sense that "fdrake is sitting drinking tea now" is true at time of writing, and thus applies at that time without exception forever. Only "now" for those refined systems is a new lemma, allowing them to better specify their intended conceptual content.

If it is the case that different "correct (truth preserving) logics" contradict one another, what exactly are they preserving? — Count Timothy von Icarus

Jack: I don't know. We know a member when we see one... except lots of people disagree about membership. — Count Timothy von Icarus

It breaks down ambiguities in a concept, attempts to clarify and resolve them, and if the resolutions contradict each other they are presented with their merits and drawbacks. That seems like standard flavour "conceptual analysis" to me. There's just no presupposition that there's one right way of doing things, even if there is a presupposition that people can come to understand the same things with sufficient thought and chatting.

And regarding truth, truth as a concept applies to both.

Gillian is in New York
Therefore, I am in New York.

will have true premises and conclusion when and only when "I" and "Gillian" refer to the same entity. It's thus not a valid argument in the standard sense, as it can be false (the author need not be understood to be Gillian).

vs

Gillian is in New York
I am Gillian.
Therefore, I am in New York.

will be valid, as you've plugged the hole in the previous argument.

If it is the case that different "correct (truth preserving) logics" contradict one another, what exactly are they preserving? — Count Timothy von Icarus

Referencing the above, what they preserve is truth of conclusion given true premises. That is just what truth preservation means. Stipulate what you like, see what follows from it.

Whether you have true premises is a different issue. When you stipulate axioms, you treat them as true. Are they true? Upon what basis can they be considered as such?

Whether you have a true axiomatic system is a different issue again, and I don't really know what it means. How would you compare Peano Arithmetic and Robinson Arithmetic, for example? Which one is true? Is one "more true" than another? What about propositional logic and predicate calculus? These aren't rhetorical questions btw.

I would posit that axioms can be considered to be correct when they entail the intended theorems about the object you've conceived. That is, when they reflect the imagination. For example with @Leontiskos using Euclid's characterisation of circles as a plane figure, it would entail that a great circle on a sphere surface is not a circle... whereas it seems to be "contained in the intended concept" (scarequotes) of a circle. Which might lead you to reject the axioms, or insist upon them... Hence the method adopted in the paper and my dialogue with Leontiskos.

And there is a formalistic definition of truth, a statement is true in a theory when that statement holds in every model of that theory. Like "swans are birds" is true because there are no swans which are not birds, but "swans are white" is false because there are swans which are not white. Every collection of swans is a model of the term "swan", and all you need is one collection with a black swan in it to show the latter is false. Similarly if you wrote down the axioms of a group, something would be true of groups when it is true of every model of the theory induced by group axioms - the sets the groups are made of, and the set operations the group mappings use.

You can think of the latter as related to my dialogue with Leon in the following way - the intuition of a circle makes you want to put the great circle into every theory of circles, everything which describes what circles are, so if you think it should be in the theory, you have to reject (or repair) Euclid's definition.

Edit: or, my preferred option, acknowledge that "circle" is an imprecise concept in natural language and also that there are lots of different useful ways of fleshing it out.

If I'm doing something dumb, it's okay to just say that. — Srap Tasmaner

It's just a question of understanding the detail for me.

And you might then think of the center of the circle as a projection of the center of the sphere. And it is, but it's entirely optional. That projection comes after we already have the circle. It's the canonical projection alright, but you could also project that point to any point on the plane, because this projection is just a thing you're doing ― the circle doesn't need it, isn't waiting for this projection, you see? — Srap Tasmaner

1) So I pick a point A in 3 space A={0,0,10} as {x,y,z} coords.
2) I place a plane cutting the point {0,0,0} with unit normal vector {0,0,1} (that's the xy plane). The axel is parallel to the z-axis, it points in the direction of the unit normal.
4) I then pick a circle in the x-y plane, let's just say it's centred at the origin O={0,0,0} with radius r=sqrt(10), which I think is the appropriate distance to make your construction with the cone work.

The center circle O is equivalently determined by the distance sqrt(10), the point A and the choice of the x-y plane.

That connotes a more general construction.

1) I form the sphere of radius R around A.
2) I pick a projection P and a point A. I constrain the projection P that it projects onto a plane whose normal vector is parallel to the sphere radius and that norm(PA)<=R. Intuitively, you travel along a sphere radius and blow up a plane orthogonal to the radius from a point on the radius.
3) I apply P to A, producing PA.
4) I collect the points this plane intersects the sphere's surface together, this will be a circle of radius... sqrt(d(A,PA)^2 + d(PA,intersection point of plane with sphere surface)^2)
5) I have more than enough degrees of freedom in the distance expression in 4, when I can pick A and P and R, to define any circle centred at any point.

Was that the construction?

Or does the truth and validity depend on the system being used? — Count Timothy von Icarus

The validity of Russel's argument depends upon the interpretation mechanism you apply to the sentences in it, and their terms. The first formalisation of it is:

1) Gillian is in Banf,
2) Therefore, I am in Banf.

In standard predicate logic, there would be nothing saying that Gillian=I, because all you can do is assign symbols based on what's in the argument. "I" and "Gillian" are distinct referential symbols, therefore they must be parsed as different entities. In standard predicate logic, something being red does not imply that it is coloured.

If you're thinking "that's nuts", because the argument clearly is "valid" in some sense, you need to come up with a reason why. And you did just that, you mapped the argument as presented to another argument:

1) Gillian is in Banf
2) I am Gillian
3) Therefore, I am in Banf — Count Timothy von Icarus

Which is clearly valid in the original predicate logic. However the mapping between the arguments:

1) Gillian is in Banf.
2) Therefore, I am in Banf.

to

1) Gillian is in Banf
2) I am Gillian
3) Therefore, I am in Banf

is not an operation available to you in original predicate logic. It's an extra logical operation to map argument to argument like that, through the means of natural language comprehension. In effect you've supplemented the original predicate logic with an extra rule, in which you resolve coreference classes of each denoting term in the argument's sentences prior to evaluating whether the premises can be true and the conclusion nevertheless false.

You could then prove a meta-theorem that states that any argument of the first form is valid so long as it's valid in your new logic that resolves the coreference classes - any one where "I" and "Gillian" co-denote.

That is, you stipulate a set of equivalent denoting terms prior to evaluating it - in this case, you would stipulate that "I" would denote the same entity as "Gillian", which makes sense since Gillian was understood to be author.

In another interpretation of that same argument, the argument would be invalid, since when fdrake writes the argument, fdrake is the author, so we don't belong in the same coreference class.

There's a considerable ambiguity in natural language terms and concepts, which gives them a kind of cohesion through fuzzy boundaries, which can then be interpreted as a coherent unity, which seems to be @Leontiskos's method of argument in this thread, to my reckoning.

This is still circular logic. What makes one collection of cells and protoplasm a member of the human species? It is not merely the presence of a particular set of genes/chromosomes - there must be something else. — EricH

@Bob Ross

I want to "yes, and" Eric's comment. Even if you grant that a being is a member of the human species, that does not mean they count as a person or as a moral agent.

Everyday intuitions about moral agency are also limited by the status of a person. Children (especially) and young adults are treated with more lenience for behaving in a socially unacceptable manner and for committing moral wrongs, children's legal status is also different. People's status as an agent may change if they go into a permanent coma, we have next of kin rules, waivers, and even (arguably) the ability to extend our capacity for consent after our death with organ donation and wills. Moreover, unfertilised gametes and severed limbs are recognisably of the species homo sapiens and are not treated as moral persons - unless one is willing to admit that shagging, the normal functioning of fertilised ovums, menstruation and masturbation are each a peculiar brand of industrial slaughter.



To summarise, each of those entities counts as a member of the species homo sapiens, but they are not a moral agent. Some of them count as persons and have restricted or removed moral agency, some of them don't count as persons but nevertheless are treated as moral agents.

Straight lines on spheres? That's interesting too. — creativesoul

Yep! It turned out a property that uniquely characterised straight lines in our normal kind of space also applied to spheres, and it makes great circles. It's the taxicab circle thing again. Straight lines are only the things we expect in Euclidean ("flat") space. But that's an artificial restriction.

Edit: even flatness. The volume in the room you're in is flat.

This was cool. I would need to sit down with some algebra to understand it properly though. Regarding the projection - there will be a lot of degrees of freedom if you get to choose an arbitrary projection onto the plane, so I suppose picking a specific projection to the centre point in the plane and looking at its preimage under that projection is the idea you had in mind?

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.

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.

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

Fairy muff.

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?

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?

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.

(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.

I'm glad someone looked at the Russell article. — Banno

I had comments I really wanted to make about the original article but considering that a Proofs and Refutations style chat about square circles was right there it seemed a better opportunity to illustrate the intuitions behind lemma incorporation.

We could also define a circle as the cross-section of a sphere, but I was only saying that every (planar) cross-section of a sphere will in fact fulfill the definition I already set out. — Leontiskos

It would if you give yourself the liberty of hammering the cross section down onto a flat plane. Which is an exercise of the imagination, and not something set out in Euclid's axioms. Is the point. You end up having to mathematise all the stuff you do to make it work. The operative distinction is you're relying on a lot of extra-mathematical intuition and not putting in the work to make it precise. Which is mostly fine, it's just in such imprecision where lots of allegedly undesirable plurality can hide.

Do trust me that the counterexamples work verbatim though!

―― I don't know why I'm participating in this. — Srap Tasmaner

I'm gonna bugger off now too.

Does that point need also to be coplanar? Is there a counterexample I'm missing? — Srap Tasmaner

I was imagining putting the point away from the plane and bending the underlying surface we're trying to draw the circle on. I'm pretty sure we'd end up with some other shapes possible if we inclined the plane, never mind if we corrugated the fucker.

But I suppose that would also apply if we chose the coplanar point far away from the candidate point set... I wish I knew what circles were.

But don't you need to specify coplanar? If we're in 3d space, you've defined a sphere, in 4th I guess some sort of hypersphere, I don't know, blah blah blah. — Srap Tasmaner

Yeah you're right. Circle, n-sphere, all the same thing in my head. Coplanarity works. A set of coplanar points equidistant from a point in their plane of coplanarity. Thanks!

We took our definition from Euclid, and the term there means a figure that lies entirely on a flat plane. — Leontiskos

Do you think the "great circle" (which you have yet to define) lies in three dimensional space rather than two dimensional space? That ambiguity is why I asked you to be more clear about what you were depicting in the first place. — Leontiskos

Fair enough. There's two things though:

Either you consider the sphere as embedded in 3-space, and the cross section plane isn't "flat" in some sense - it's at an incline. Or you consider the surface as a 2-dimensional object, in which case there's not even a plane to think about. Pick your poison. The latter is the original counter example and is much stronger, the former is easier to remedy.

If we define "distance" in the commonly accepted way, then there aren't. Are we disagreeing on something more profound than that? — Leontiskos

You're behaving like you know what these things are so well you've got them baked into your cerebellum. But clearly that's not true, as the definition you provided doesn't match something you clearly recognised as a circle! So yes, we could insist on your pretheoretical intuition, but it's no longer Euclid's... so I'm wondering what's wrong with it? How will you parry my counterexample?

I'm enjoying this discussion. — Banno

It is a lot like something from Proofs and Refutations.

I think it does. You've only asserted otherwise, you haven't shown it. — Leontiskos

Well I can tell you what I think a plane figure is.

A plane figure is closed curve which is inside a subset of $\mathbb{R}^2$. By that definition the great circle is not a plane figure, as it's not inside a subset of $\mathbb{R}^2$ - that circle instead would be a closed curve inside a subset of $\mathbb{R}^3$, or with extra precision the surface of the sphere.

What do you think a plane figure is?

Why are you doing this sort of thing? — Leontiskos

You do this sort of thing because stipulating a definition and then shit-testing it is standard mathematical practice.

I showed you the great circle on the surface of a sphere because I expected you would see it as a circle - it is - but it does not satisfy Euclid's definition of one verbatim, which you were clearly inspired by. And with maths words, verbatim is all anyone has. That's how you test the boundaries of your definitions and the consequences of ideas.

In picking out the great circle as a circle, you in fact sided with the example over the definition you stipulated. Which is the right thing to do, I think. You could also have ardently insisted that indeed, the great circle was not a great circle because it was not a plane figure. But you did not.

So now that you've abandoned Euclid's verbatim definition of a circle, you've got work to do in telling us what you mean by one.

As for me, I mean a set of points equidistant from a point. And by the by that also makes the great circle a circle. Score one for the thing which includes the taxicab circle over Euclid!

Exactly.

The cross-section of a sphere is a circle. — Leontiskos

Well who said anything about cross sections? I was talking about the sphere's surface. You chided me before about extraneous points and operations, and now you've given yourself the liberty of splitting a shape in two, taking an infinitely small cut of it, how exuberant. I just gave you a sphere's surface, not a cross section so...

You'll now need to tell me in what circumstances can you take a cross section of a volume and have it work to produce a circle. Let's assume that you can take any volume and any cross section and that will produce a circle...



Therefore those squares and rectangles are circles. Which is absurd. So your principle must have caveats. What are they, you've got some explaining to do!

Why do you think this? — Leontiskos

Read the definition:

A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre. — Circle | Wikipedia

A circle is a plane figure... so something which is not a plane figure cannot be a circle.

And what is "the great circle"? — Leontiskos

The great circle is the circle I've highlighted on the surface of the sphere. Since the circle is confined to the surface of the sphere, and the surface of the sphere is not a plane, it is not a plane figure.

I hope I'm not the only one who recognizes that you are more interested in this conversation than me. :grin: — Leontiskos

Aye.

A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre. — Circle | Wikipedia

Euclid says: not a circle. The great circle is not a plane figure.

And why is this? Is it not because of what those things actually are? — Count Timothy von Icarus

I think so, relative to tasks.

Sure. So with the "raindrop" addition example, isn't the appropriateness of the system determined by the real properties of rain drops? — Count Timothy von Icarus

Yes.

I am all on board with the idea that the tools will vary with the job, but it seems to me that to explain why some tools are better for some jobs than others requires including properties of "things in the world." — Count Timothy von Icarus

Yeah that's a hard one. I don't know if there's a hard and fast answer for systems generically! This seems to be a root level epistemological issue - what it means for a description to be adequate.

Even when we speak of "concepts," it seems to me that there is plenty of evidence to support the claim that our cognitive apparatus is shaped by natural selection, and this in turn means our thinking and our preferences, relate to "how the world is." — Count Timothy von Icarus

Indeed. Though there are lots of ways what we create can model, describe or explain stuff. Maybe even mirroring different aspects of stuff. Maybe it doesn't need to do any of these things to still be important.

But they are. You have an odd assumption that points are stipulative, as if we could delete a point or as if a point could have spatial extension. The set of points is still equidistant from a point. This idea of "deleting" points mixes up reality with imagination. — Leontiskos

Let's change track. You tell me exactly what you mean by a circle with an intensional definition, and we'll go with that. Then do the same for roundness and pointy!

But is our preference for systems arbitrary? — Count Timothy von Icarus

I don't think it is.

But we don't pick systems arbitrarily. — Count Timothy von Icarus

I agree. They are picked to reflect, capture or illustrate certain ideas. If you came up with a system of arithmetic that couldn't prove 1+1=2, it'd be a shitty system of arithmetic.

It's not the case that the Earth, baseballs, and basketballs are all just as triangular as they are spherical just because it is possible to define a system where this is so.

I agree. The everyday conceptual content of Earth (the concept), baseballs (the concept) and basketballs (the concept) are that they are round.

To affirm that would be to default on the idea that any statement about the world having priority over any.

I disagree. I think you missed the case that priority can also be seen as purpose and context relative. Here's a series of examples regarding roundness and sphericality.

I prioritise the notion of roundness when considering the Earth on an everyday basis, and I might while calculating its surface area - fuckit it's a sphere and that'll do. But on a day to day basis, my body treats the Earth by and large as flat. And that has priority over a merely intellectual commitment to its roundness as far as my feet are concerned. If I'm trying to stand on a bosu ball, now that fucker is round.

If I were studying variations in the acceleration due to gravity on the Earth's surface, I couldn't treat the Earth as a sphere - since it's roughly oblate, it's a spheroid. And crap like Mt Everest sticks out of it, so it's pointy. If we go by @Leontiskos intuition that round things cannot be pointy in any context, well the Earth is in trouble.

More generically, the role specifying a system has might be thought of as setting out some concept for some purpose. That allows you to see whether the system specification is fit for task.

How do you decide whether it's fit for task? Well I suppose you decide on a task by task basis. Thinking of Earth strictly as a sphere, with the assumption that a sphere is like a circle where every point on its surface is equidistant from its centre... That doesn't work as soon as your legs move. So that's not fit for walking.

But it is fit for a quick and dirty calculation of volume. Or an explanation for how it attained its shape due to gravity.

Here are more abstract examples.

Those tasks are quite concrete - there are harder ones. Like how might we consider fitness for task of a concept of logic in the context of arguing with a salesperson? Their responses aren't going to follow propositional logic... So something informal is required, they're definitely trying to persuade you. Emotional appeals? Reframing? Motivational speech? We could speak of a "logic of sales" that consists of such chicanery. And it would be nuts to think of the salesperson's behaviour solely terms of syllogisms and propositions.

How might we consider the laws of addition when considered from the perspective of raindrops? Well one raindrop alongside another raindrop might be two raindrops, but it could be one larger raindrop depending upon the distance between them. So "raindrop addition" might be way more complicated than adding discrete units of things...

Here's what I think is the general principle.

The rough trick is the same in each case, you have some conceptual content you want to specify, you try to set out a collection of rules that specify the conceptual content, then you shit test the rules to see if you got anything wrong. Or you can maybe prove all and only the results that you want - or solve all your problems - then you've succeeded beyond your wildest dreams.

"An interesting question arrises". There are two values for the limit - 2 and √2. So the space is not smooth, unless we re-define "smooth". — Banno

We had a related discussion here.

My explanation for the weirdness of the staircase paradox. The tl;dr of it is that the length you get by placing a measuring tape along a curve doesn't respect the process of infinitely refining shapes. So it's nothing to do with the shape, it's to do with the concepts of length and limit.

I honestly don't have the maths to try to think about volume and rate concepts in taxicab geometry. Other than my intuition that they're the same as the Euclidean ones... even though the length is different.

Importantly, doing this would not be wrong, as such. It's just one approach amongst many. — Banno

Indeed.

Mathematics papers absolutely call taxicab-circles circles. I just wouldn't call them circles to my students learning shapes.

A circle is, by definition, a set of points Euclidean equidistant from one central point. — fdrake

@Leontiskos

As an aside, here are some possible counterexamples.

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.

Another one. Take the circle with radius 5 centred at the point (0,0). Then remove all points in the space which have coordinates which are both natural numbers - like (1,2), (7,8). Removing all those points removes the point (3,4), which lays upon that circle (since 3^2+4^2=5^2). That doesn't do anything to change the smoothness of the circle either, since every point on it is the same as before. So it's still smooth, no corners, all points equidistance... It's just missing a point. So, all points in that space which are Euclidean distance 5 from the origin are in the set - so is it a circle?

These would mean you have to come up with some constraint on how hole filled the space, or the circle, could be, and think about holiness itself in order to restore the fact both are clearly circles... Or maybe they're not circle at all at this point. Or neither of them are real counterexamples - it could be my specification's shite.

See what I mean?

Well, your post would appear obtuse to the layman, and maybe it just is. — Leontiskos

It is obtuse, but I don't think it just is.

A metric is a way of assigning distances to pairs of points. When you consider a space, it has a metric. The usual distance people think of is called the Euclidean distance, and it's the one you're thinking of and measure with a ruler on a piece of paper.

The thing is that the choice of metric is just that, a choice, and you can write down various other spaces with various other metrics. One of those other metrics is called the taxicab metric. Contrasting that to the Euclidean metric:

Imagine you start at a point, and you go 1 step north and 1 step northeast
The taxicab metric says you've travelled 2 total units - you add the steps.
The euclidean metric says you've travelled sqrt(2) total units - you measure the line.

Because a metric defines the concept of an interpoint distance, circles in taxicab geometry are different from circles in euclidean geometry. A circle in taxicab geometry, a set of points defined as equidistant from a single point, looks a lot like a square in euclidean space. 4 corners, 4 right angles, 4 equal sides.

So it is a circle, if a circle is defined by the property of being equidistant from a point. But perhaps it is not a circle, because... well, like you, you could insist that we're not talking about a circle when we're talking about sets equidistant from a point in the taxicab metric. So for you, you'd have to do something to block what we're talking about as a circle in taxicab geometry being a "real" circle.

That places a burden on you to study the concepts of circularity and square-iness, and to say why the first blocks the latter and vice versa. Which is what I did in the post. I'll go through it for nonmathematicians.

For something to count as a square, it needs to have:

S 1) Four sides of equal length.
S 2) Each side meets exactly two other sides at right angles.

Let's just take that as a given, that is what a square is. Now we need to think about a circle. What's a circle?

C 1 ) A circle is a set of points equidistant from one point.

If ( C 1 ) is the only defining property, the taxicab circle is indeed a circle, it's just a circle in taxicab space. Clearly you don't want it to be a circle, so you need to stipulate a restriction. I could also insist that it is a circle, and how are we to decide between your preference and my preference? Anyway, onwards:

What is round is not pointy — Leontiskos

You specified such a restriction with "what is round is not pointy", which is something similar to what I formalised with the idea of smoothness. The "corners" form the "pointy bits" of the square because the function that defines a square is not smooth at the exact corner point.

There is an ambiguity regarding pointiness, which is similar to the above ambiguity regarding equidistance. In thinking about the corners of the square thing (the taxicab circle) in taxicab space as pointy in the above sense, that requires specifying the roundness concept in terms of the measure of size - smoothness is typically characterised with respect to a measure of size.

Something is differentiable when its derivative exists at every point.
The derivative of a curve exists at a point if and only if at that point the limit of the ratio of the function evaluated at the endpoints of an arbitrarily small interval divided by the length of that interval exists (IE it becomes just a number).
A curve is smooth if you can apply the procedure above to it arbitrarily many times.

The concepts of "interval" and "length" there are also doing a lot of work, since they're distance and size flavoured. And should we expect them to work as our prior Euclidean flavour intuitions would in taxicab geometry? What gives us the right to insist that we think of smoothness as we would in a Euclidean space and transfer it onto smoothness in a taxicab space?

Clearly you would want to insist that they do, my intuitions also run that way. But my intuitions can also side with circles not necessarily being smooth since I'm used to dealing with this stuff!

Where we can agree, though, is with lemma incorporation. In which we specify a set of properties that say exactly what counts as a circle (in your sense) and why it can't be a square.

So for you:

A circle is, by definition, a set of points Euclidean equidistant from one central point.

And thus we've revealed what sneaky hidden presumption you had through lemma incorporation. What we haven't done is decided why that must be accepted as the definition of a circle.

If you want to join in with this exercise of lemma incorporation, I invite you to stipulate a definition of pointy! And we will see where it goes.

Here's a Proofs and Refutations - the source of Lakatos' concept of lemma incorporation - inspired investigation into square circles.

It's the corners that screw you up in trying to come up with such a square circle object, I think. For something to be a corner, two lines must meet at a right angle. Two lines meeting at a right angle doesn't produce a differentiable function (along the shape the lines meet in) regardless of how you rotate the shape or embed it in another one's surface, so you've got to choose between jagged edge to allow corners, and roundness.

You use the above, and the taxicab thing in my previous post (quoted below), to stipulate the following:

The properties that define circles make shapes that appear as squares in taxicab space. But the geometry jettisons our concept of roundness, unfortunately. — fdrake

I could guess the principle: every circle with corners is not round. Specifying

1) A circle is shape resulting from constant distance around a point.
2) A corner is a meeting of two lines at a right angle.
3) A round shape is smooth along its curve.

And hope to prove that there's no such shape. But I could've misspecified the underlying concepts. I imagine there's something odd about "corner" and "smooth", because "corner" relies upon "right angle", and "right angle" depends upon "angle", which depends upon the concept of an inner product, and the privileged connection between inner product and metric is something we get from usual Euclidean space. Moreover, "smooth" could also be generalised to reference a different metric.

So perhaps there is some space that has a metric related to an inner product in which there are round circles with corners, but I've not thought of such a counterexample myself.

Me going through the maths there isn't an attempt to side with over @Banno, because being able to explore the conceptual content of the allegedly logically impossible should tell you that logical impossibility isn't all it's cracked up to be. You do have to ask "which logic and system?", and "what concept am I not formalising right?" or "what concept is making the weird shit I'm imagining weird?".

What would we get if we just assumed a perfectly round square circle with four corners? — Banno

The properties that define circles make shapes that appear as squares in taxicab space. But the geometry jettisons our concept of roundness, unfortunately.

Thought it might have been him. — Srap Tasmaner

We even had a long discussion about it 7 years ago.