Heribert Schütz

@tomstafford@mastodon.online Reconstructed this one from memory

@johncarlosbaez

A lot of [...] prize winning scientists (typically old white men) start acting like every thought they have on every topic is worth telling the world.

I love your sense of humor!

@sibrosan@mastodon.social @divbyzero@mathstodon.xyz The precise formula here is actually not important. The important observation is that this volume depends only on the *area* of P. It depends neither on the shape of P nor on its position on the outer sphere. So if you do steps 1 to 4 for two slices P and P' of the outer sphere with the same area, the resulting "rings" will have the same volume. So, does this convince you?

@johncarlosbaez@mathstodon.xyz

But Dawkins is saying Claude has consciousness, and that the Turing test is a test for consciousness.

With another look at the article I didn't find these points so explicitly stated by Dawkins. But your English is better than mine, and maybe this lets you understand him that way or draw those conclusions.

And yes, Dawkins is somewhat sloppy, but aren't we all? After all, this UnHerd magazine doesn't appear to be a science journal.

@johncarlosbaez 4/4 I essentially agree with Dawkins. If consciousness can develop from genetic code + learning, why shouldn't it be reachable for machines as well? Yes, there's still some way to go from today's LLMs to something that can fully mimic humans. But the latter would anyway only be needed for a Turing-style test scenario, not for practical purposes.

@henryseg Yes, you're right. Even a construction with 14 "black" gears (6 at the cube faces + 8 at the cube corners) should work without locking: Each of the resulting 24 meshes would consist of 4 gears.

@Eigenraum @holothuroid @Harald Noch eine Klarstellung, nur für den Fall, dass hier das Missverständnis liegen sollte: Wenn in der theoretischen Informatik die Frage gestellt wird, ob die Mitgliedschaft in einer Menge entscheidbar ist oder nicht, dann geht es um Mengen im mathematischen Sinn, nicht um (typischerweise endliche) Instanzen irgendeines "Set"-Datentyps.

@rauschma An old Armenian Radio/Radio Yerevan joke: Q: Given the ongoing advances in technology, is it true that a war can soon be fought by robots? A: Yes, but it still takes some humans to start it off.

@Eigenraum @holothuroid @Harald Natürlich kann man auch ein Orakel entscheiden lassen, ob ein Ding Element einer Menge ist oder nicht. Für mich klingt "entscheiden" aber nach einem Vorgang / Algorithmus. Liegt vielleicht an meinem Informatik/Logik-Hintergrund, kann aber auch von anderen so (miss)verstanden werden. Über Mengenlehre und "∈" wollte ich mit meinem Haarspalter-Tröt nichts aussagen.

@Eigenraum @holothuroid @Harald Nun kann man noch diskutieren, ob das Wort "entscheiden" unbedingt im Sinne der theoretischen Informatik verstanden werden muss. Muss es nicht, kann es aber. Um Missverständnisse zu vermeiden, ist es wohl tatsächlich besser, das Wort "entscheiden" in einer Erklärung des Begriffs "Menge" nicht zu verwenden. Mir fällt aber gerade nichts besseres ein.

@johncarlosbaez 2/4 Several participants in the discussion here seem to base their argumentation on an axiom that anything artificial cannot be conscious (or maybe not even intelligent). Apparently Dawkins does not believe in such an axiom.

@johncarlosbaez 3/4 As an evolutionary biologist, Dawkins sees consciousness not as a yes/no property but something that has developed gradually. And he asks what survival advantage ("competence") came with consciousness, or if there is any advantage at all. He gives some possible answers but leaves the question open. (Unfortunately you did not quote that last part of the article.)

@sibrosan@mastodon.social @divbyzero@mathstodon.xyz

1. Construct a solid S consisting of all the points between C and some point of P. We get:

volume(S) := area(P) * R / 3.

1. Construct another solid s by contracting S towards C by the factor r/R. We get:

volume(s) = (r/R)^3 * volume(S)

@sibrosan@mastodon.social @divbyzero@mathstodon.xyz 4. Notice that s is contained in S. If we remove s from S, the remaining solid is the part of the shell between P and the corresponding piece of the inner sphere. It has the volume

volume(S) - volume(s) = volume(S) * (1 - (r/R)^3) = area(P) * R / 3 * (1 - (r/R)^3)

@sibrosan@mastodon.social @divbyzero@mathstodon.xyz And when I called the cut slopes "negligible" for thin shells, this was not about a particular amount or percentage of difference I'm willing to neglect, but it was an informal/sloppy way to say that the difference between planar and conic cuts decreases faster than linearly if we make the shell thinner. So even the relative difference approaches zero by making the shell sufficiently thin.

@sibrosan@mastodon.social @divbyzero@mathstodon.xyz In more formal "Weierstraß style": For *whatever* relative difference ε > 0 we might be willing to neglect, there is a thickness δ > 0 such that |vol_planar - vol_conic| / vol_conic < ε for all shells with thickness δ or thinner, that is, for any r with R-δ <= r < R.

@holothuroid @Harald @Eigenraum Da kann ich folgende Haarspalterei beitragen: Es gibt sogar Mengen, für die man nicht einmal "entscheiden [kann], ob eine Sache drin ist". Standard-Beispiel: die Menge aller terminierenden Programme. Man kann zwar für viele Programme entscheiden, ob sie terminieren oder nicht. Es kann aber keinen Algorithmus geben, der das für alle Programme korrekt entscheiden kann. https://de.wikipedia.org/wiki/Entscheidbarkeit https://de.wikipedia.org/wiki/Halteproblem

@henryseg Cool! I was wondering if the 6 black gears at the cube faces/octahedron vertices could be replaced by 8 gears at the cube vertices/octahedron faces. I think it should work since each "mesh" again consists of an even number of gears: 6 instead of the current 8. 6 of the 12 edges would then rotate in the opposite direction.

@sibrosan@mastodon.social @divbyzero@mathstodon.xyz With a thin shell the slopes of the slicing surfaces are negligible. With a thicker shell (outer diameter D and inner diameter d) such a surface should have a slope such that the outer edges have vertical distance D/n and the inner edges have vertical distance d/n. Thus a slicing surface is a part of a cone with its apex at the center of the spheres. This way the slices should have the same weight again. Edit: Ok, I was too slow. You already wrote essentially the same.

@sibrosan@mastodon.social @divbyzero@mathstodon.xyz I think I did get your point. There are differences in weight/volume with planar cuts. (And the diffs are small for thin shells and get bigger for thick shells.) But these differences go away *completely* with the conic cuts.

@sibrosan@mastodon.social @divbyzero@mathstodon.xyz Well, I'm not a mathematician, but I can prove it. I don't know into how much detail a proof must go to convince you, but here is an attempt:

We have a shell delimited by an outer sphere with radius R and an inner sphere with radius r. The spheres have the same center C. (In your example R = 150 mm and r = 1 mm, but we need not restrict ourselves to these values.)

1. Select some piece P of the outer sphere.

@sibrosan@mastodon.social Good that we agree on the conic cuts now. But sorry, I can't resist to discuss the words "wrong" and "negligible" a bit. I wouldn't call @divbyzero@mathstodon.xyz's initial statement "wrong". The conic cutting can be seen as a less interesting detail not worth mentioning in that context. (It might even have confused some readers.) But yes, it can be misunderstood easily if one implicitly transfers the planes from Archimedes' statement on 2D surfaces to the 3D shells.

@sibrosan@mastodon.social The post you were replying to was about the missing explicit requirement for conic cuts. With conic cuts one doesn't need any thickness limits or approximation as we have seen. (Or have you mixed up my remarks on "wrong" with my remarks on "negligible"?)

If you are not able or not willing to distinguish between

• plain wrong statements and
• oversimplified and thus technically incorrect, but easily repairable statements, then we will not find a common ground.

Have a nice week!

@sibrosan@mastodon.social Nobody criticizes your initial response from Saturday~7am (CET). divbyzero even acknowledged your point on Saturday~3:30pm when he realized that you were assuming planar cuts. But the end of your Sunday~8am post sounds to me like finger-pointing at divbyzero, just because he didn't mention the conic cuts initially. As a matter of taste, I prefer cooperative and productive discussions where people try to understand each other instead of finger-pointing at (repairable) inaccuracies.

@sibrosan@mastodon.social No need to be sorry. But it's good if you learned that there is more than just true and false in mathematics.

@johncarlosbaez@mathstodon.xyz

Of course consciousness is a matter of degrees, and of course it's in principle obtainable by machines.

These points were not meant to contradict you. Maybe I should have replied to the appropriate posts. OTOH I didn't want to repeat these points several times and so I appended them to the end of the discussion.