Mindscape Ask Me Anything, Sean Carroll | August 2024

Blog post with audio player, show notes, and transcript:

Welcome to the August 2024 Ask Me Anything episode of Mindscape! These monthly excursions are funded by Patreon supporters (who are also the ones asking the questions). We take questions asked by Patreons, whittle them down to a more manageable number -- based primarily on whether I have anything interesting to say about them, not whether the questions themselves are good -- and sometimes group them together if they are about a similar topic. Enjoy!

#podcast #ideas #science #philosophy #culture

[akumar]

I never graduated and have always worked in manual jobs but the last four years, I've been absorbed by Science find the topic so exciting.

[BrianFedirko]

Sean's description of Belief and evidence here is genius, and calming. Alleviated so much frustration in my own worldview that I can't thank him enough. I value these talks every month. The AMA might even save me money on Dr. visits (oh, I guess he is a doctor at that). Thank you, Sean!!!. Gr8! Peace ☮💜Love

[jwvandegronden]

I woke up to the gauge field... Damn, electromagnetism isn't a fundamental law/element, further down the rabbit hole we go! What an amazing brain this speaker has! not only open to more fundamental components, but even able to express why and how, the curvature of the gauge field. I love science!

[GoatOfTheWoods]

2:13:00 Sean, I'm certain that me and the majority of the audience would gladly listen to more of you, even if you don't have a guest.
AMAs and Solos are great.

[PhilosophicalTrials]

21:55 What Tarski showed was that you cannot write down an open formula in the language of arithmetic which applies to all and only those natural numbers which encode arithmetical sentences which are true (in the standard model of arithmetic). Hence, arithmetical truth is not arithmetically definable: this is what logicians typically understand by Tarski’s Theorem. What was said in the video is different, but also wrong.

To give an idea of what it means for a numerical property to be arithmetically definable, let's consider the property of being an even number, which can be defined in the language of arithmetic as follows: ϕ(x) = ∃y(x=y+y). The only items from our domain of discourse, namely N={0, 1, 2, ...}, which satisfy this predicate are the even numbers (0, 2, 4, 6, ...) and nothing else. Coming back to Tarski, he has shown that you can't come up with a formula Ψ(x) of this kind which is only true of the (Gödel-)codes of true sentences, e.g. the code of ∀x∀y(x+y=y+x).

[Note for the next paragraph concerning notation: For any sentenceϕ, by <ϕ> I will mean the numeral of ϕ's code. Hence, if ϕ has code n, say, then its numeral is the successor function-symbol applied that many times to the constant 0, i.e. S(S(...S(0))...).]

The reason why you can't define arithmetical truth is because if a nice predicate T(x) that "disquotes" on all sentences was definable in our chosen language, then it would be impossible to assign a classical truth value to the "fixed-point" L of the related predicate ~T(x), often called "The Liar Sentence" of T. This is because the so-called Diagonal Lemma (which guarantees L's existence) tells you that L and ~T(<L>) must have matching truth values in the standard model, but since we also have the disquotational desideratum on T(x) for all sentences, L and T(x) must also have the same truth value as well. Since it is impossible to meet both of these constraints at once (via classical models), there can't be any defined formula like T(x) in the first place.

I want to stress that it is not correct that you cannot have a truth-predicate which doesn't disquote on any sentence whatsoever. For example, we can have "modest" arithmetical truth-predicates T'(x) which are restricted to, say, Σ1 sentences (and which *is* definable). So for any Σ1-sentence ϕ, a system like Peano Arithmetic (PA) *will have* the biconditional T'(<ϕ>) <-> ϕ as a theorem, i.e. PA ⊢ T'(<ϕ>) <-> ϕ. Likewise for other complexity classes. What we *cannot* have is a "global" truth-predicate T(x) which will do the job for all arithmetical sentences simultaneously, because of Liar-like phenomena.

Restricting ourselves to PA, this is connected to Gödel's Incompleteness Theorems, because the numerical property of encoding a PA-theorem is actually definable, whilst the property of being a true sentence formulated in PA's language is not definable. Hence, the set of provable arithmetical sentences and the set of true arithmetical sentences are not the same set. Since the former is a subset of the latter, it must be a proper subset, so there is at least one true sentence ϕ which is not a PA-theorem.

(Also, at 17:35, Russell's Paradox is not something that besets Principia Mathematica, which was explicitly designed to be immune from it.)

[bentationfunkiloglio]

If I may be gratuitously pedantic…

Well, actually, every computer program can be written in assembly. Yes, assembly, might not be the best choice, however.

To see this is true consider 2 things.

Modern compilers transform programs to an intermediate, assembly-like language before lowering code to machine language.

All compiled programs can be disassembled and analyzed in assembly.

[JohnChampagne]

A question from someone who didn't pay to ask a question: Do you have a favorite policy proposal for how to account for externalities in an efficient and fair way? I think it's the same question ass, "How can we end poverty and promote sustainability?". Maybe not the same question, but I think they have the same answer: Charge fees to industries proportional to pollution emitted, resources extracted or habitat disturbed or destroyed; Share fee proceeds to all people. We can raise fees if random polls show that most people think there's too much of this or that kind of impact. (We'd have a true democracy if conditions match what people want.) We should make this a global policy.

[ABARANOWSKISKI]

Hey Sean. Love the podcast! I heard you say on one of your podcasts that you're interested in talking to historians. Have you ever considered having a conversation with Bart Ehrman? Bart Ehrman is one of my favourite historians, and the two of you would have a lot in common, despite working in very different fields. It would be great to hear an episode of the show where you talk to him, the two of you are two of my favourite people, so I hope it happens one day!

[marierausku]

I love the answer to my question and so cute pronunciation of my surname 😊

[HunterYavitz]

Prof Carroll, I've enjoyed your work since the early days of your debates. I'd like to see more interactive content like this. Would you consider a call-in episode?

[PrmarySourc]

do we get the reading list for complexity class

[e-t-y]

I checked out your paper "Consciousness and the Laws of Physics, " after an interview in which you referenced it. In it, you're making the case (against) introducing mental concepts into fundamental ontology. In attempting to discount consciousness as fundamental, this seems to skip a key step. It is more "sensory experience" (not consciousness per se) that is being claimed as fundamental to all matter and energy, so therefore as fundamental. Mentation, actual consciousness and self-consciousness are evolving elaborations on this sensory experiencing nature, and can be thought of as a "meta sense, " that is the framework within which the senses function and elaborate. Even the sense-of-self itself is an elaboration of this meta-sensor, which goes all the way down in its origins to the fundamental particles, to energy, and to fields. So we don't get to leap straight to "mental concepts" as the framework of introducing consciousness into the ontology of the universe. It isn't mentation, it is discriminating interactions between all existing things and their environment, which evolved into elaborate forms of mental activity, including self-consciousness.

[savasgen]

Is it possible, that the 'elastic fabric' for spacetime is not a human mathematical constructor that deals with distance, time, matter, energy, but actually exists, and can we messure it, by how the time response differently in a 'standar' model like our solar system, combine the mass, size, weight and density of every planet?

[mraarone]

I’ve been wanting to ask you specifically that black hole question for at least a couple of months!

[savasgen]

Is it possible that in the bottom of black holes, after a supernova explotion, the matter that bent the spacetime it's Nickel, that doesn't have the oxydation of iron?

[kreuzritter]

You cana produce matter or energysources with specific constracted hologramms. No resource problems or energy insufficiens in the future of any kind. 😎 just use it responsible and for the good of all!

[e-t-y]

Information being lost to other human beings isn't the same as it actually being lost, right? Is it not there in the quantum fields once it is thought?

[dangerdingle]

If dark energy separates things and the strong force gets stronger as quarks separate, won't exponential dark energy inevitably start separating the quarks, leading to the production of more and more quarks fueled by the dark energy?

[lennyg]

So it's not "information", it's "matter"?

[jayvincent]

44:18 Why do people come up with every example except (Conserve Our Natural Resources) .. Over production of goods are the reason you get sale pricing at the grocery stores. Pump that product..