@Count Timothy von Icarus

Still hunting for a solid example, but in the meantime there's

(1) You only won because you cheated.

The sequence from here is most likely

(2) So you admit I won.

(3) It's not a proper game if you cheated.

And then the argument shifts from whether I got more points than you (or whatever) to whether the rules were followed.

But "the rules" can be surprising. Casinos universally have a rule against "card counting," which amounts to a rule against being too good at playing cards.

Sometimes among children using greater skill or knowledge is treated as cheating. It's clear they have an intuition about what fair competition is, but they mistakenly treat every advantage as unfair, or every unfair advantage as cheating.

What's happening here, broadly, is that the competition continues "by other means." The losing party, in one sense, grants that they lost, but continues in the competitive spirit, which means they have to shift ground from whether they "officially" or "technically" lost to whether that was a "real" loss, or whether there had a been a "real" competition in the first place.

So we have two versions here:

(4) If you won because you cheated, then you didn't win.
(5) If you "won" because I cheated, then you lost.

'degenerate' in a non-pejorative sense as often in mathematics — TonesInDeepFreeze

Yes, that was my meaning, as with the boat example. I think, though, we can allow a somewhat negative connotation because reliance in argumentation on degenerate cases is often inadvertent or deceptive. "There are a number of people voting for me for President on Tuesday [and that number happens to be 0]."

importance in Boolean logic used along the way in switching theory, computation, etc. — TonesInDeepFreeze

Absolutely. I mentioned automated reasoning projects when you need classical logic in full generality with P → P and all. Of course.

But that is not the only sort of application of logic, and as I noted reducing someone's argument to P → P is pointing out that they are "begging the question," generally considered a fatal problem for an argument. That conditional is legitimate in form, and is generally a theorem, but it is fatal if relied on to make a substantive point or demonstrate a claim. It will only happen inadvertently ― in which case, a good-faith discussant will admit their error ― or with an intent to mislead by sophistry.

Logic is a vast field of study, including all kinds of formal and informal contexts. I would not so sweepingly declare certain formulations otiose merely because one is not personally aware of its uses. — TonesInDeepFreeze

A fair point. Let's say, I'm only pointing out forms, or uses of particular schemata, that might raise our suspicions. "Heads I win, tails you lose" may in some cases demonstrate that I'm necessarily right and you necessarily wrong. Hurray! But in some cases it might amount to me stacking the deck against you.

Fundamentally, all we're talking about in this case is arguing from a set a premises which are inconsistent ― in fact, here, necessarily inconsistent. As I keep saying, that's either inadvertent or deceptive. Very often on this forum people attempt to show that a set of premises is inconsistent precisely by making correct inferences that make the inconsistency obvious. But people arguing from inconsistent premises often make inferences that, while in themselves correct, continue to hide their inconsistency. (Sometimes this is because not all premises are explicitly stated, and the inconsistency is in what is "assumed".) In such cases, it is not uncommon for people to insist on the correctness of their inferences. But it's not validity we usually disagree over, but soundness, and inconsistent premises make valid inferences unsound.

None of this news to you, I'm sure.

I don't see how it would not be natural to take you as first claiming that my remarks were non-cooperative and abusive. — TonesInDeepFreeze

Agreed, a natural reading, but my target was really someone who might present an argument in the OP's schema, as a perfectly respectable modus ponens. It's MP alright, but it's a degenerate case.

Similarly for the disjunction. My point was that disjunctions that amount to "heads I win, tails you lose" are disjunctive in form only, and we expect something more substantive.

In short, no attack was intended on you or any other poster, but only on the illicit use someone might make of legitimate argument form. To the extent that I was offering criticism, it was to say that we are not helpless when confronted with correct inference in form only, and can choose to block such deviant uses if we like.

And we can leave formal logic alone, as a study in its own right, but not import it wholesale when all we really need is the convenience of schematizing arguments.

A related example would be various attempts to deal with what many people find counterintuitive about the material conditional. There are several ways to block troublesome cases.

Hence my casual suggestion that we have very little practical use for "If grass is green then grass is green" or "If grass is green then grass is not green."

opprobrium — TonesInDeepFreeze

I'll just say that no opprobrium was intended. I too have gained, I believe, from my study of logic and mathematics, and I have found formal methods intensely interesting, so neither am I disparaging that as an interest.

Some people are interested in formal logic full-stop. Some people are only interested in formal logic as a help meet to argumentation and analysis. For the latter, results about logic ― the completeness theorems and such ― are not only of less interest, but less everyday use. All I was saying.

Nor was I accusing you or the original poster, or anyone in this thread, of engaging in sophistry, if that's what you thought. Maybe it seemed to you I was engaging in the current controversy on the other side, but I was not. You are, of course, correct about the formal question.

I was looking at the argument schema presented in the OP. If you imagine this as the formal representation of a substantive argument, you would have to have serious doubts about what was going on in that argument. This was the "veneer" of logic I was talking about. Any argument that could be formalized in the schema presented would instantiate an accepted form in a deeply questionable way. Hence "sophistry". That wasn't intended to refer to you, to your explanations, to anyone in this thread, but to a hypothetical argument that would fit the schema under discussion. ― You'll note that @TonesInDeepFreeze and I were trying to think of a genuine example of such an argument, and I'll now be getting back to that.

I hope my "position" is clearer now.

these threads — Count Timothy von Icarus

FTFY

can you demonstrate that the resulting calculus will be complete? — Banno

If you are interested in the basics of ordinary formal logic, then it would be a question that would naturally occur to you. But I don't see why you couldn't study other branches of philosophy without understanding the completeness of the propositional calculus. — TonesInDeepFreeze

Quite. I worked through some of the usual metatheorems years ago when I was studying formal logic. If you're interested in the properties of these formal systems, such results are just what you're interested in. And I'm sure there are issues that come up in philosophy that depend substantively on such results.

But for the everyday use of logic just to schematize and clarify arguments, you get a lot more mileage out of de Morgan's laws, contrapositives, a solid understanding of quantifiers, and such. The cash value of completeness for such applications is nil.

Can you prove A→A, for example? — Banno

As Tones suggested, it might be necessary for proving certain metatheorems, but of course in real applications of logic ― such as on TPF ― "A→A" usually only appears as an accusation of question-begging. It's not something anyone would have any reason to argue for, and it's not a premise anyone would intentionally rely on. ― Hence my suggestion that we could usually get along without it.

But what do you mean by 'abusive'? — TonesInDeepFreeze

The basic idea is "formally correct but misleading". Akin to sophistry. Or to non-cooperative implicature, like saying "Everyone on the boat is okay" when it's only true because no one is left on the boat and all the dead and injured are in the water.

In this case, for instance, it is suggested that we conclude ~A by modus ponens. The form is indeed instantiated ― I'm not contesting that ― but the first premise is materially equivalent to ~A. People worry over the sense in which the conclusion of a deductive argument is "contained" in the premises ― here it is one of the premises. Who needed modus ponens?

(Besides, you are effectively arguing from the set {A, ~A}. You could as well conclude A from that ― or any B you like ― so in what sense should this count as a "demonstration" that ~A? In what sense is the relationship of A and ~A revealed or clarified? It may be modus ponens in form, but hardly in spirit.)

And if we step back and look at the offending premise, we get to ~A by noting that A→~A is materially equivalent to ~A v ~A. Now what kind of disjunction is that? It's a well-formed-formula ― no one can deny that ― but it's hardly what we usually have in mind as a disjunction. It's "heads I win, tails you lose." That's abusive.

There is, in this case, a veneer of logic over what could scarcely be considered rational argumentation. If this appearance of rationality serves any purpose, it must be to mislead, hence abusive, eristic, sophistical, non-cooperative. ― Again, I am only talking about how logic is used as an aid to ordinary philosophizing, not what people get up to in a logic lab.

Well for a start you would no longer be dealing with a complete version of propositional calculus... — Banno

1. Meaning what exactly?
2. Is the answer to (1) something I should care about?

Do you, Srap, agree that the argument in the OP is valid? — Banno

I don't really care. It's abusive.

I cannot think of a way to frame this as a real example — Count Timothy von Icarus

I'll come up with one. I think you see it around the forum and elsewhere in the wild pretty regularly. Informally, we're looking for a case where you try to disagree with me, but that attempt misfires because in stating your position you have to tacitly agree with me.

So you claim A, but I show that still leads, by some chain of reasoning or argument to my claim ~A. Introducing A⊃~A is just icing on the cake, because it's still just an extravagant way of saying ~A.

It seems folk think A → ~A is a contradiction. It isn't. — Banno

But (A->~A) & A is a contradiction.

If you assert A->~A, and then go on to assert A, then you have contradicted yourself.

The set {A->~A, A} is not a contradiction because it is not a formula, but a set. It is, however, inconsistent.

Would there be any harm in requiring that the conditional in a modus ponens have fresh variables on the right hand side? We would lose, in effect, only this one and A->A, which is either useless or the LEM, and thus innocent or tendentious, depending on how you look at it.

I mean, mathematicians always prefer the greatest generality, at the minor, to them, cost of letting in the degenerate case. If your project is automated reasoning, you'll go with the usual. But for doing philosophy, we don't have to let the mathematicians have the last word.

If the only question here is "How does formal logic work?" we know the answer to that. But around here we're more interested in the practical use of logic, and it seems to me letting mathematical logic have the last word is the tail wagging the dog.

Only line 1 is not, ~A. It's A→~A. — Banno

I mean, I get that MP requires a "->", but (A->~A)<->~A, so I'm puzzled by insisting on this nicety. In classical logic it's materially equivalent to the disjunctive syllogism, isn't it?

She seemed to be interacting with her environment to be sure, but that is not langauge. — Hanover

I still think there's a divide here.

If you look at formal approaches to language -- Frege, Tarski, Montague, that sort of thing -- language is a system for representing your environment. That could, conceivably, be just for you. A language of thought.

And it is only because you can put the world, or some part of it, into language, that it is useful for communication. When you communicate, you put part of the world into words (or claim to) and pass those words to someone else. Language as descriptor of the world underlies language as means of communication.

I happen to think that's wrong, for various reasons, but I think it's a fairly common view, maybe more common among philosophers but maybe not.

Here "conventional" does come apart from "arbitrary." — Leontiskos

Yeah I think Sausaure's phrase is a little misleading.

On the one hand, as I said, it refers to the conventional nature of the linkage between signifier and signified.

On the other, it's also intended to convey that there's nothing special about the signifier that makes it the right signifier for the job. The structuralist approach is to see the signifiers as forming a system, the whole group of them, and what's important is just that they can be and are distinguished from each other, a "system of differences" .

This is especially clear in phonology, I think, where you can draw up a table of possible phonemes. What you also get from phonology though is that this perfect system of phonemes is not quite real: in practice we accept a considerable range of allophones as a specific phoneme, and we rely on context to make the assignment.

The cartographic symbol is conventional but not arbitrary. Is it natural? — Leontiskos

Not natural, no.

On the typical road maps I look at, towns and cities are indicated by circles, filled circles of different sizes and stars (for capitals).

There's no resemblance there, not even stylized "iconic" resemblance. It's "arbitrary," if you like. We might have used squares or triangles or whatever.

But in general I think "arbitrary" carries the wrong connotation. What matters is that it's one of many available Nash equilibria, so it's a possible solution to a coordination problem, i.e., a possible convention. Why it's this signifier rather than another is usually a matter of chance, of history.

Also, for what it's worth, Baggs used to speak, went through something like normal language acquisition, got all the way to college before they started to lose their speech, I think. So we're not dealing with a "feral child" situation. And they continued to write even after they stopped speaking. It's complicated.

conventional but non-linguistic meaning — Srap Tasmaner

The obvious example was right in front of me: cartographic symbols. While there is obviously structure in the way these are placed on the map, that structure is not grammatical.

This is similar to your premise that if a picture is being used as a picture then it must not be being used as a sign. — Leontiskos

Maybe my next reply covered that.

Is a picture already a sign, albeit a non-arbitrary sign? — Leontiskos

Generally, although I think children need to learn to see pictures as pictures of things. It doesn't seem to be quite automatic and may be more culture-bound than we think.

"leaf" and a picture of a leaf are both signs of a leaf; one arbitrary and one non-arbitrary. — Leontiskos

And the shadow a leaf casts is also a sign of a leaf.

The arbitrariness of the sign, per Saussure, refers to the conventional nature of the linkage between the signifier and the signified, iirc. But there are some famous studies suggesting that might be overstated a bit (bouba/kiki for starters).

Is it not language unless the meaning relation is conventional rather than natural? The traditional answer is obviously "yes" but I'm not so sure. Especially if you wonder how language could get started in the first place.

If it's not absolutely essential, then what's the relation here? Is it the other way? That is, conventional meanings as a subset of linguistic meaning? That looks to be the story with writing. (Or with the use of natural gestures, like folding your arms, to indicate an attitude.) Are there counterexamples? Any cases of conventional but non-linguistic meaning? --- I'm having trouble coming up with something, but I'm not even sure what the criteria would be.

Further reply with example.

Sometimes maps for children will have little pictures. At Paris, a little Eiffel Tower; at South Dakota, a little Mount Rushmore. Here the picture is a straightforward representation of a thing, but used by a sort of metonymy to mean the whole place where that thing is. So in such a case, both.

specifically — Leontiskos

Whether the picture is being used as a picture or a sign. Writing, for example, seems to begin pictorially, but then become simplified, stylized, and conventionalized. Even alphabetic letters are just special little pictures that are not intended or expected to resemble anything. It's not part of their use.

Is that the kind of answer you were looking for?

If someone thinks in pictures is their thought process therefore non-linguistic? — Leontiskos

I mean, it depends, right? But I think Wittgenstein is right, that it doesn't just depend on the thinker's intention. There's some obvious wiggle room here, which you can see clearly with novel uses of words. There you already have in place the linguistic habits of a community, and novel uses rely on that. Some novel uses catch on and some don't; some are intended to catch on and some are just mistakes, and some mistakes catch on. But I think we're by and large right to reject Humpty-Dumpty-ism as a theory of language or language use. It's not enough.

you can't make people follow standard word usages — Baden

Or stipulated, scientific definitions.

I like what Baggs does and I find it interesting from an artistic perspective. But I wanted to clarify it's not language. — Baden

I agree, but I'm curious about some people's strong intuition to call this behavior linguistic. On the one hand, I'm biased against that usage, but on the other I can almost understand it. (Some people would no doubt object to my somewhat wide use of "thinking" nearby.) There's a part of me that would love to call this "language" but I can't figure out why I would do that.

The whole thread is just littered with equivocation between the literal and metaphorical meanings of "language". — Baden

I don't think that's it really. I think the disagreement is between (1) those of us who think language is primarily and originally for mediating the connection of one mind to another (communication); and (2) those who think language primarily mediates the connection of a mind to the world. Each camp allows that the other function exists, but it's treated as secondary or derivative or parasitic.

(I framed (1) as I did for contrast, but better would be something like, connecting one mind to another with regard to the world.)

If you spend a lot of time thinking about poetry, in particular, it's hard not to think of language as this quasi-magical means of reaching out to the world, to things themselves, as the man says. The social-use-first view feels a bit deflationary by comparison.

I would call this "language" it is just not as prominent and familiar to many because we are told what "language" is and what "grammar" is. I can absolutely think without words and form ideas and images in my head that play out without any need for worded thought. — I like sushi

What puzzles me is that you seem to be offering the second sentence as a reason supporting the first. My inclination is just to say that thought need not be linguistic, but I get the impression you want to claim thought without words or grammar is still linguistic, so what's left? Conceptualization? Logic? Is what's left inherently linguistic?

People can communicate extremely complex ideas in other forms than worded language. — I like sushi

Same question. Why not just say not all communication is linguistic?

To what extent is an immediate relationship with our non-human surroundings a language? — Joshs

I lean toward "not" and honestly I'm not sure why the word "language" gets used here. Some of the places where there's a sort of extended use of the word ― a filmmaker's language, a painter's language ― there's still communication, and what we're talking about is something like a repertoire or a toolbox. I think people rightly perceive that a lexicon is a kind of repertoire, as are other elements of language use ― prosody, rhetorical constructions, and so on. But that doesn't make a repertoire or a toolbox linguistic, because it's the other way around.

Anyway, the part of the film ― and thank you for posting it ― I found most interesting was the point that if she doesn't pay attention to the right things and ignore the right things, people assume she's not thinking. That's gold. It's clear that her perception of the affordances in her environment is very different from a neurotypical adult's, but I think it's also clear that a good chunk of that neurotypical perception and behavior is due to enculturation. Children and (non-human) animals have their own sometimes quite different ways of interacting with their environment. (I remember having considerable difficulty convincing a toddler to look out the window at an airport, to see the airplanes. They were more interested in the window.) Animals we can only guess at, and children we mostly treat as imperfect, unfinished adults. Baggs is something else again, a type of mind I doubt I can really understand.

One further thought: all the sorts of minds I've mentioned are related, and there don't seem to be boundaries between them, just patterns and tendencies. Even typical enculturated adults run their hands across the fabric of clothes when they're shopping, or sharing ― here, feel this! A shower isn't always just a minimal and efficient body cleaning, but a chance to stand for a few moments, eyes closed, feeling the water running over you. I think there's probably almost nothing in her video that I haven't done myself, even though I do other things she doesn't, and the place in my life of what we have in common is different.

And it's plain to me this is thinking behavior we're talking about. When I gaze up at the night sky, I'm surely engaging with what I see thinkingly, but it's not always accompanied by thoughts in words, or even by specific feelings. Sometimes there's a definite "sense of wonder," but sometimes I just look and it doesn't have to be anything else, but it's still a sort of thinking.

At first I didn't like the way you were using the word "productive", but here's another way you might stick with that.

Every moment you're making the next version of you.

There are people out there who have ideas about what they would like you to be -- a pair of eyeballs connected to a bank account, maybe.

But also your parents have ideas about what you should be. Your friends do. Your teachers or your bosses do. Those ideas aren't all bad, probably. You should always be willing to learn from other people.

But the thing is, they can't make you anything. They need to get you to make yourself into what they think you should be.

And that's what going on when you're chilling on YouTube or whatever. You're making the next version of you, but you're making a you that someone else wants, not a version of you that you've thought about and chosen.

If x is a prayer answered by y, then x is a prayer, and y is a prayer answerer.

Axy -> (Px & Gy)

That's a real argument. Other versions are abusive.

three kids who want to play with the same toys — Leontiskos

Common solutions: we introduce other toys so that everyone gets something (not an option in our example); no one gets it (not allowed in our example); they each get the whole thing because they will play with it together (not helpful for consumables, as in our example, which is why we split them); we divvy up not the toy but the time playing with it, take turns, and we can even measure the duration of those and make them equal-ish.

(My youngest son used to come to me brandishing minor premises like this: "Dad, don't you think Andy has been on the computer a long time?" Andy is his older brother. Just look at the beautiful deviousness of this, the command of rhetoric ― how young they master the dark arts of persuasion.)

I go to the car dealership and I am offered whole cars. They don't let me buy a half car for half the money. — Leontiskos

You know what would be cool? What if you could give them part of the price and someone else, say a bank, gave them the rest, and you got to take the whole car right now. Then you could have an arrangement with the bank where you pay them back over time, a little each month, and the bank could charge interest to compensate them for the risk they're taking. I think that could catch on! I think that could end up being the way almost everyone buys a car.

See also.

what is your alternative? — Leontiskos

In an artificially bounded task like this, with artificial bounds on the means by which we may complete it, there are no options. Life is not like that. This task would arise as a potential option in furtherance of another goal. Either we find a creative way to complete this subtask (making do with rough equality ― 7 or 8 each, cutting the strawberries, if that's an option, or switching measures, say from units to weight, and so on) or we mark this path off in the search and backtrack until we find a path.

Either way the math/reality constrains our options. — Leontiskos

Reality, sure, but mathematics is how we conceptualize our situation and can inform both our choice of action and our method. Mathematics is adverbial.

Anywhere you want to look, it is plain as can be that thinking and acting mathematically is empowering for humans, not some implacable constraint. Gravity is a constraint, but once we conceptualize it and understand it mathematically we put ourselves in a position to work around it, or to put it to use.

there is very little difference between the less numerate and the more numerate. The less numerate just takes a few more minutes — Leontiskos

For a toy problem like this, maybe, maybe not. I'm not sure you're entitled to this assumption.

But for the general distinction in approaches, which this little problem illustrates, the entire business world disagrees with you, the natural sciences disagree with you, the various branches of engineering disagree with you.

chickens — J

I think what's strange about this problem is that the setup makes human beings helpless before the implacable necessity of mathematics, and that's the wrong story to tell.

In so many cases, it is the use of mathematics that enables us to identify problems, clarify them, and solve them. And there is immense creativity here ― which is why I gestured at the invention of rational numbers.

In real life, a case like this is more likely to play out this way: you've got these 23 thingamabobs, and there's talk of splitting them three ways. You say, "Won't work," and someone less numerate than you says, "Well, let's just try." As they fail, with a puzzled look, they say, "Wait, I messed up somewhere. Let me start over." You will want to explain to them that it's impossible, because 23 is not only not a multiple of 3, it's a frickin' prime.

What's of primary interest here is that you, because of your relative expertise in mathematics, understand the situation better than the person who, even after trying and failing several times, still believes it might be possible.

But I'll address what the philosophers want to say. First, of course this scenario only makes sense given the relative durability of the thingamabobs across the sort of time scales we're interested in, and they have to be such that we can reliably distinguish them and count them. Our faculties must persist too. On and on. Absolutely there are prosaic physical qualities of the situation assumed.

All of those physical factors are also presumed in the case where we have 24 thingamabobs. In that case, if the divider-up failed, you would be the one to say, "You must have messed up, because it definitely can be done."

There's a disjunction in there right? In the first case, with 23, it was "Either I messed up, or it's impossible"; in the second, with 24, it's "Either it's impossible, or I messed up." Same thing, but with a different expectation. For the first sort of situation, it is sometimes much easier to determine that the task was impossible, than to confirm that no mistake was made in any attempt. In the second case we rely, again, on it being easier to confirm the possibility (of evenly dividing 24 by 3) than to figure out where you went wrong. Same thing again!

But because we short-circuit the disjunction differently, we're actually using it in slightly different ways. In the first case, you're discharged of responsibility for your performance because the task is impossible ― for all we know or care, you did mess up, but that's not why you failed. In the second case, we know it's not impossible, so you must have messed up; here we do judge your performance, and your mistake is why you failed. (We need a little more here actually: some guarantee that an algorithm exists, some cap on its complexity, our ability to implement it, and so on. You might still be off the hook.)

There are several options we pass by in such reasoning: we say, it's impossible, thus you needn't or shouldn't try, not that you cannot try or must not try; we say, it's possible, thus you can succeed and maybe ought to succeed, not that you must succeed.

Now come back to "why". Given 23 thingamabobs, does mathematics guarantee failure? No. It guarantees only the conditional, if you try then you will fail. (As Simpson noted, "Can't win, don't try.") Given 24 thingamabobs, does mathematics guarantee success? No, of course not, not even if you try.

So mathematics cannot compel you to succeed or to fail, but it does play a role in how we judge performance.

square circles — Leontiskos

As for the analytic/synthetic distinction, I'm not sure we really have to go there. Perhaps it won't do as a description of the difference between logic and facts-about-the-world, but wouldn't you agree that Jha et al. are pointing to something that can be talked about, and represents a genuine question? — J

Doesn't this remind you of "Two Dogmas of Empiricism"? Quine's target was the atomic proposition, and he intended to show that there's no hope of teasing out which parts of our beliefs were analytic and which synthetic.

Here we have explanations. Which part of the explanation is pure math and which part involves facts about the world?

Sound familiar? Even the form of the argument struck me as similar but I haven't made a side-by-side comparison.

The result, as before, is that you cannot tease out any supposedly pure math part, roughly.

So I assume, in the great sweep of things, Lange is fighting Quine, and Jha defending Quine, roughly.

It's understandable. Quine himself had very mixed feelings about whether the laws of logic were subject to revision. I think his final answer was yes, but it's a last resort, and they are very insulated, resistant to revision.

I could be way off here, but that was my impression as I was trying to decide whether to really dig into the paper.

The DLE problem assumes a linguistic stipulation that is uncontroversial or at least agreed-upon, or at the very least accepted for the purposes of discussion. With this stipulation in place, we then go on to ask about the relation between logic-or-math-as-necessity and the alleged necessity of events in the world, such as the 23 objects that can’t be evenly divided, or the cat that can’t be here and in Paris too. — J

I understand the point you're making, but I want to push back on this a little.

The most interesting thing going on here is the creative leap of enlarging your mathematics to include fractions as well as integers.

And their use in solving problems of sharing is probably the most common way fractions are introduced to children. Banning them "for the sake of argument" is depressing, and gives the whole problem an air of artificiality I suspect it deserves.

Sorry, @J, I would like to be more enthusiastic ― I've even made a first pass at the paper ― but surely this is all just beating the analytic/synthetic horse to death yet again. Compare that to the invention of fractions, and which is more impressive?

Do we really expect explanations of anything to be relative to nothing , not even the whole universe?

Ho ho, comes the answer, mathematics is eternal and unchanging and not relative to our universe! Logic too! And some other stuff. A complete explanation is relative only to this meta-universe, which, being necessarily as it is, needs no explanation. (Which is lucky.)

Yeah.

A Great Circle is defined by only two points on the surface. It can do this becasue it is a straight line. So as on a plane, a line can be defined by two points and a circle by three. — Banno

Something's not right here, which is just sloppiness and rustiness on my part.

In general, three non-colinear points in 3-space determine a unique plane, a unique triangle, and a unique circle. And then it takes a fourth point, not in the plane of the first three, to pick out a unique sphere.

When I was talking about sectioning a sphere ― after I realized that using a non-coplanar point to determine a circle could be thought of this way ― I reached for three points to pick out the sectioning plane out of habit, thinking that the section is guaranteed to be a circle because it is (a) planar and (b) a subset of a sphere.

Which is super super dumb. What the sphere guarantees is that the points selected are non-colinear, which hadn't even occurred to me.

All this sphere business ran roughshod over my original thinking, which was very cone oriented, as the drawings show.

Sheesh.

Way back when we started, what interested me was decoupling the point with reference to which the circle is constructed from the plane within which it is constructed.

Then I noticed you can decouple the point used to construct the circle from the (in-plane) center of the circle, because that's a projection, but it's a projection you don't need to do to construct the circle. Which means you can project the circle's originating point anywhere in the circle's plane.

I guess it would be better, and simpler, to say we can decouple the projection onto the plane of the originating point from the center of the circle.

And I thought there might be something interesting there, just in the geometry, but then realized the model I was creating was suggestive of stuff I've been thinking about a lot. That happens to me all the time.

Sphere, not ball. The surface. The 2-manifold.

I think this is the simplest version of what I was thinking.

Given a sphere centered about A,
pick any three points in the sphere,
those three points determine a unique plane,
the intersection of that plane and the sphere is a circle.

We're just taking a section of the sphere, without any further reference to the point A, which has already done everything needed to guarantee that its coplanar subsets are circles. In particular, we did not need to project A onto the plane that sections the sphere. (We can project it onto that plane, using the obvious orthogonal projection, or anything we like.)

Am I getting something wrong here?

When I gave some arguments against square circles, I suggested that one could quibble with the arguments, but not oppose them in any way that goes beyond a quibble. I think that has turned out to be right. — Leontiskos

As you like.

It seems to me you think this is a question that can only ever be asked in one way and in one context, and therefore it only ever has one answer.

You can do that, and you can be right. Your response to a counterexample is "Well I didn't mean that, I meant this" and your honor is preserved. In the context you had in mind, you're still right. The counterexample isn't one.

Pick up a length of pipe. Look at it from the side and it's rectangular. Look at it straight on, it's circular. Done. "But I didn't mean that."

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. I tend to have less confidence in my own omniscience, but you do you.

I would need to sit down with some algebra to understand it properly though. — fdrake

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

Regarding the projection - there will be a lot of degrees of freedom if you get to choose an arbitrary projection onto the plane — fdrake

Yes exactly.

Here again is how I got here.

In school, we learn to think of circles this way:
1. You've got a plane.
2. Pick a point in the plane.
3. Find all the points in the plane equidistant from that point.
4. That set of points is your circle.
5. The point you picked in (2) is the center of your circle.

But it needn't be that way.

Your great circle example, or the conic sections we learn in Algebra II, are different.

1'. Pick a point in 3-space.
2'. Find all the points equidistant from that point.
3'. That set is a sphere, or a 2-sphere.
4'. Any coplanar subset of the points in (2') is a circle, or a 1-sphere.

If you now look at the plane of the the circle in (4'), there is a subtle difference from the plane in (4): the center is not marked. No point in the plane was used to generate the circle ― although, of course, the circle has a center you can find. But in the schoolboy's circle, you never have to go find the center ― you pick that point to start with.

(There's a direction-of-fit thing here: in one case, the center determines the circle; in the other, the circle determines the center.)

When you find the center, you might ask, is it related in any special way to the point in 3-space we picked (1')? And of course it is. There is exactly one line orthogonal to the plane that passes through that original generating point, and it passes through the plane at the center of the circle as well.

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?

There's nothing much to the geometry, but here's a picture to start with.



(There's other ways to look at this. You could of course go ahead and treat the "determining point" as a center and make a circle on a plane right there, then project that circle onto a parallel plane. Blah blah blah.)

Having separated the point that determines the circle from the center of the circle, it just occurred to me that you could treat it separately, do a lot of stuff with it. To start with, you don't have to project to the center of the circle in the plane, you don't have to use that orthogonal projection, but could send it (translate it) to any point A, B, or C, anywhere in the plane.

Then I thought there might be something interesting if you grouped these projections into buckets, those that send it into the circle, those that send it far away, and so on. And I thought there might be some interesting stuff there ― maybe allowing the axis to wobble a little, and see how stable your buckets were, and lots of other stuff.

But then it occurred to me what probably caught my eye about this.

If instead of thinking of the points A, B, and C as being projections of the "determining point", what if you went the other way, and thought of any point in the plane translating to the point off the plane that determines this circle.



Suddenly that cone looks like a field of vision, and the other points are other actors who are triangulating their view of ― in this case ― a tree (or whatever) with the red guy at the "determining point". (We'd probably want to move the red guy onto the plane with the A, B, and C, and create a new notional plane orthogonal to this one to represent Red's f.o.v., but whatever. At this point the whole setup is merely suggestive.)

And then it should be obvious there is a meaningful difference between being in the circle and outside it, because that determines whether you are also in Red's cone of vision.

It happens I've been reading about triangulation and joint and shared intentionality in apes and humans (Michael Tomasello), so it was probably on my mind, and that's why the whole arrangement, splitting one point into two (center/determiner), then splitting that second point into two as well (determiner/projected) ― it all suggested something to me, and this was probably it.

I wonder if there is something else interesting just to the geometry, but that's no doubt above my paygrade.

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!

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'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

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.