Straight lines on spheres? That's interesting too.

:yum:

Circles are straight lines. Squares are circles. Logic is just the manipulation of symbols. And there are no laws of logic. Really a brilliant thread, all around. Everyone here deserves a pat on the back. :wink:

I can't wait until tomorrow, when we show that 2+2=5.

It would appear obtuse to the layman, and maybe it just is. — Leontiskos

laws of logic — Leontiskos

I thought we agreed, formal logic is conventionalized ways of thinking :p. It can only be an approximation of our thinking, but not our thinking itself.

- The closer you get to the foundation, the surer it becomes. For example, modus ponens is arguably the most basic inference or law of propositional logic, and I don't see that it fails.

The closer you get to the foundation, the surer it becomes. For example, modus ponens is arguably the most basic inference or law of propositional logic, and I don't see that it fails. — Leontiskos

What's the "foundation" mean here?

Presumably, natural human reasoning, something akin to inferencing, let's say, is of an imprecise nature. It just needed to be "good enough". However, the kind of reasoning we developed- generally intertwined with linguistic capacity, and certain kinds of episodic memory, can get formalized culturally into more precise logical thinking. This is especially helped by the ability to write out the symbols.

From here, these more precise "crisp" arguments, might be said to have a foundation, perhaps Platonically (pace Frege and Plato). And thus, you might mean some kind of transcendental foundation (Platonic). Or, perhaps, like Kant, you think that it is internally a priori, and simply part of the human cognitive faculties. I challenge this, as evolutionary vagueness seems to be at play. Math is contingent on cultural preciseness, not internal preciseness. However, even math's preciseness and internal logic in its own system, doesn't necessarily have a foundation outside itself. Newton's Calculus system is not as accurate as Riemann's system, for example. And thus "foundation" can thus mean:

1) Human cognition- I challenge this usually works in vague approximations, not crisp exactitude.
2) Platonic transcendentalism- I am not sure what this would mean other than logical truths are somehow existent in some real way.
3) Naturally occurring patterns- this might be physical laws, for example. But this isn't really the logic itself. Logical systems, like mathematics, are applied to observable phenomenon, and "cashes out" in experiments and technological use.

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.

Just the shortest path between two points. So pick two points on a sphere, draw the shortest line, then extend that. The result is a great circle. The maths can be done intrinsically, without reference to some coordinate system in which the sphere sits.

- I only meant the foundation of the logical system. Frege's foundation is explicitly modus ponens, and many propositional systems similarly ground themselves in modus ponens. In fact we can think of modus ponens as the basis for the material conditional in propositional logic, where the modus ponens inference is more intentionally foundational to the system than the idiosyncratic behavior of the material conditional (which we are considering elsewhere). I tried to speak a bit to the odd foundationalness of modus ponens <here>.

If you want something more universally foundational, I would point to the principle of non-contradiction, and ultimately its unique character of being simultaneously subjective and objective, which Kimhi alludes to. A lot of the silliness in this thread is either a direct or indirect attack on the PNC.

What's the "foundation" mean here? — schopenhauer1

Those supposed foundations are addressed in the Russell article.

Few implementations of propositional logic start with modus ponens. It's most often just a theorem.

Earlier logicians had drawn up a number of rules of inference, rules for passing from one proposition to another. One of the best known was called modus ponens: ‘From ‘‘p’’ and ‘‘If p then q’’ infer ‘‘q’’ ’. In his system Frege claims to prove all the laws of logic using this as a single rule of inference. The other rules are either axioms of his system or theorems proved from them. — A New History of Western Philosophy, by Anthony Kenney, 155

Contemporary logicians like Enderton and Gensler begin the exact same way. Other starting points are possible, but they are not all on a par if one wants to do actual logic. Of course for metamathematics the starting point is arbitrary. Banno, under the spell of metamathematics, will be at a complete loss before your question about how true reasoning and logic interrelate. As Apokrisis has pointed out numerous times, Banno begins and ends with nothing more than a bit of posturing.

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?

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.