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Stream: deprecated: mathematics

Topic: on a remark by Rovelli

Peiyuan Zhu (Feb 22 2023 at 18:09):

These two are from Carlo Rovelli's pop science book the Order of Time footnotes of Chapter 8: Dynamics as Relation:
image.png
image.png

Is he talking about how dynamics of a physical system can instead of being seen as time evolution $\frac{dx}{dt}=f(x,y),\frac{dy}{dt}=g(x,y)$ it can be written as $\{(x,y):C(x,y)=0\}$ so there's no independent time variable involved?

Morgan Rogers (he/him) (Feb 23 2023 at 08:56):

Yes.

Peiyuan Zhu (Feb 23 2023 at 19:40):

Is there good mathematics textbooks that talks about something like this? I think this notion is rather unfamiliar to me except the idea of a graph come up here and there occasionally.

Matteo Capucci (he/him) (Feb 23 2023 at 20:11):

Uhm I'm sure any book on multivariate analysis treats implicit representation of curves

David Egolf (Feb 23 2023 at 20:20):

The book "Ordinary Differential Equations: Basics and Beyond" (by Schaeffer and Cain) is a book I have not read, but exercise 3 on page 27 has to do with the elimination of time. It's possible surrounding material in that book might be of interest to you.

Peiyuan Zhu (Feb 23 2023 at 21:47):

(deleted)

Peiyuan Zhu (Feb 23 2023 at 21:59):

(deleted)

Peiyuan Zhu (Feb 26 2023 at 06:26):

For physicists and mathematicians what does "event" mean? From what Rovelli says it sounds like it always involve at least two objects interacting with one another. Also it sounds like the states of the objects have to change.

Jean-Baptiste Vienney (Feb 26 2023 at 07:21):

As far as I know it's a point $(x,y,z,t) \in \mathbb{R}^4$ of the spacetime once you chose a coordinate system.

John Baez (Feb 26 2023 at 16:08):

Jean-Baptiste is giving the standard definition of event in special relativity, but in diffeomorphism-invariant physics (which Rovelli is studying) we need another definition of 'event', and nobody knows what the right definition is.

John Baez (Feb 26 2023 at 16:08):

It should capture the intuitive concept of 'something happens somewhere'.

David Egolf (Feb 26 2023 at 16:50):

I've never heard of "diffeomorphism-invariant physics" before, and I find my curiosity piqued. Looking at Wikipedia, I see that a diffeomorphism is an isomorphism of smooth manifolds. In a physics context, what would we be using smooth manifolds for? And why would we want to consider isomorphisms between them?

My first guess would be that we can model regions of space as smooth manifolds, and that some diffeomorphisms might conceivably include rotating/flipping or translating. I think I remember reading that some conservation laws in physics can be arrived at by requiring the equations that govern certain phenomena to have the same solutions even after these transformations are applied (?). I have no idea if this the kind of thing "diffeomorphism-invariant physics" is concerned with though.

John Baez (Feb 26 2023 at 17:33):

In a physics context, what would we be using smooth manifolds for?

Lots of things, but in diffeomorphism-invariant physics we'd be using them to describe spacetime.

John Baez (Feb 26 2023 at 17:34):

And why would we want to consider isomorphisms between them?

Because we can't physically tell the difference between living in some spacetime manifold $M$ and some isomorphic spacetime manifold $M'$.

John Baez (Feb 26 2023 at 17:35):

Einstein was one of the first to understand this principle, in a somewhat confused way. He called it "general covariance":

• Wikipedia, General covariance.

John Baez (Feb 26 2023 at 17:36):

and physicists like Rovelli (and once upon a time me) who work on quantum gravity sometimes spend a lot of time trying to reconcile quantum theory with general covariance, or diffeomorphism-invariance.

John Baez (Feb 26 2023 at 17:38):

It's a tricky subject, as the "century of confusion" quote in Wikipedia explains.

Peiyuan Zhu (Feb 26 2023 at 17:50):

What about the idea of event that Glynn Winskel's talking about? https://www.cl.cam.ac.uk/~gw104/

Peiyuan Zhu (Feb 26 2023 at 17:50):

This would be in the context of computer science

John Baez (Feb 26 2023 at 17:51):

I don't answer vague questions like "what about X?"

Peiyuan Zhu (Feb 26 2023 at 17:51):

The question is still "what is event" but in the context of computer science.

John Baez (Feb 26 2023 at 17:53):

Okay. I don't know what an event is in the context of computer science. I imagine it's a somewhat vague and ill-defined notion. I'll let computer scientists decide whether they want to try to make it precise.

Martti Karvonen (Feb 27 2023 at 13:41):

Winskel talks about something called event structures, and uses them to model concurrency. In this setting the word "event" has a precise meaning as an element of an event structure. I wouldn't expect any connection to the general topic just because the word "event" is used in both cases, and I suspect that concurrency theory has very little to do with physics (both now and probably in the future, although I'd be happy to be proven wrong).

Peiyuan Zhu (Feb 27 2023 at 18:55):

Concurrency seems to relate directly to what Rovelli's talking about as event vs objects in book The Order of Time Chapter 6. The World is made of Events, Not Things

Peiyuan Zhu (Feb 27 2023 at 18:57):

“A family is not a thing, it is a collection of relations, occurrences, feelings. And a human being? Of course it’s not a thing; like the cloud above the mountain, it’s a complex process, where food, information, light, words, and so on enter and exit. . . . A knot of knots in a network of social relations, in a network of chemical processes, in a network of emotions exchanged with its own kind.”

Peiyuan Zhu (Feb 27 2023 at 19:00):

They seem to write in a way explaining the causal structures of events are never static, they're constantly changing.

Peiyuan Zhu (Feb 27 2023 at 19:04):

Or what Sarah Walker called "state-dependency" in her work The Descent of Math that information has causal power:
"An example of why any of this matters may be in order. Consider the set of states of the world corresponding to Earth plus its satellites. This set may be partitioned into two subsets: states where some physical systems have knowledge of Newton’s law5 (i.e., humans) and states where there is no such knowledge6. One example of the latter is Earth with one natural satellite – the Moon, which was the state of Earth in our history prior to 1957. It is entirely possible (and by that I mean allowed by the laws of physics) that Earth might have also had no Moon, two captured asteroids for Moons (as is the case for Mars) or a potential host of alternative smallish rocky bodies orbiting it. All are equally viable states of the world consistent with known physics. It is also entirely possible (again I mean allowed by the laws of physics) that Earth have any number of artificial satellites or space junk. However, the latter set of states, while possible, is not accessible (by this I mean not encoded in the initial conditions) without human or human-­‐like technology. Thus, when comparing the size of the state space of Earth plus satellites, the state space is much larger if artificial satellites are included, i.e., if laws of gravitation are known. Thus when considering the number of states of the world consisting of Earth plus satellites, very many more states are potentially accessible from states of the world in which some physical systems have knowledge of the laws of gravitation than from a node where there is no such information. Thus, in a network of all possible states of the world, nodes with knowledge of Newton’s law are more highly connected to other nodes within the space of all possible configurations of Earth plus satellites."

David Egolf (Feb 27 2023 at 20:08):

On the topic of dependence, you might find parts of "Duoidal Structures for Compositional Dependence" (by Shapiro and Spivak) to be of interest. I've only looked at the start of it, but it talks about categories with two ways to combine pairs of objects: one that is used when the two objects model independent events, and one that is used when one event depends on the other. Apparently this relates to something called a "duoidal category".

dusko (Feb 27 2023 at 20:31):

Peiyuan Zhu said:

Concurrency seems to relate directly to what Rovelli's talking about as event vs objects in book The Order of Time Chapter 6. The World is made of Events, Not Things

hi peiyuan. trying to understand carlo rovelli's ideas about events and objects, and trying to understand the meanings of words in general, is a noble goal. the general activity of mining the meanings of words is close to what they did in ancient greece at one point. aristotle generalized it to the point of asking what is the meaning of "to be". he studied that in a book called metaphysics.
mathematics begins when people stop expecting that words carry their meaning with them. they assign a different meaning to $x$ in every story. the word "event" is a variable. glynn winskel used it to denote an element of an algebraic lattice equipped with an inconsistency relation. he used the word "event" to suggest a vague intuition, and because it had not been used in the same context before. the same word means another thing in special relativity and yet another thing in general relativity. it is a variable.
it becomes complicated when people who have been doing math all their lives, and used variables, get old, and propose that the meaning that they assigned to a word in this or that exercise should be nailed to that word, imagining that their thoughts might live longer that way. don't fall for that. words are just words.

Peiyuan Zhu (Feb 27 2023 at 22:40):

dusko said:

Peiyuan Zhu said:

Concurrency seems to relate directly to what Rovelli's talking about as event vs objects in book The Order of Time Chapter 6. The World is made of Events, Not Things

hi peiyuan. trying to understand carlo rovelli's ideas about events and objects, and trying to understand the meanings of words in general, is a noble goal. the general activity of mining the meanings of words is close to what they did in ancient greece at one point. aristotle generalized it to the point of asking what is the meaning of "to be". he studied that in a book called metaphysics.
mathematics begins when people stop expecting that words carry their meaning with them. they assign a different meaning to $x$ in every story. the word "event" is a variable. glynn winskel used it to denote an element of an algebraic lattice equipped with an inconsistency relation. he used the word "event" to suggest a vague intuition, and because it had not been used in the same context before. the same word means another thing in special relativity and yet another thing in general relativity. it is a variable.
it becomes complicated when people who have been doing math all their lives, and used variables, get old, and propose that the meaning that they assigned to a word in this or that exercise should be nailed to that word, imagining that their thoughts might live longer that way. don't fall for that. words are just words.

Thanks for the encouragement. I think because mathematics is now divided into pure and applied, the meaning part is lost, although I think since mathematics concern the highest universality of human practices, the loss of meanings ends up putting up barricades on places where new development can happens.

My understanding now is that Rovelli's saying the world is not made of particles and reversible dynamical laws of the particles, it's made of relations between possible events, and events as some change that happens that may or may not be measured e.g. entanglement, therefore he puts uncertainty principle, irreversibility, non-equilibrium conditions, information loss from measurement at the centre of physics. The relation between events i.e. constraints is where complexity of an organism lies, which allows the organism to circumvent the dynamics causally, while the dynamics can be quite simple e.g. second law of thermodynamics. This is likewise the case of the inconsistency relations that Winkle is talking about, that the relation between uncertain environment and computer program(s), and between program and program(s). Same phenomena happen in complex system works such as Walker e.g. relation between satellites and Barrett e.g. relation between mental phenomena and physical phenomena. The relations are to be contrasted to functions of certitude and reversibility. Same phenomena happen in philosophy works such as James on pragmatic theory of Truth i.e. it's the value that's evolving towards higher complexity. Likewise in social sciences that has pragmatic orientations, such as those of Parsons.

Matteo Capucci (he/him) (Feb 28 2023 at 10:31):

dusko said:

mathematics begins when people stop expecting that words carry their meaning with them. they assign a different meaning to $x$ in every story. the word "event" is a variable. glynn winskel used it to denote an element of an algebraic lattice equipped with an inconsistency relation. he used the word "event" to suggest a vague intuition, and because it had not been used in the same context before. the same word means another thing in special relativity and yet another thing in general relativity. it is a variable.

This is a very interesting point regarding the praxis and the philosophy of mathematics. I strongly agree that the fundamental idea of mathematics is to strip form of substance and focus on that (abstraction), and it can be misleading to read more than a coincidence in a choice of terminology (or in any other recurrent formal pattern). In that sense, 'event' in Winskel's event structures and 'event' in GR are as related as 'x' in an equation and 'x' in another.

On the other hand, the reason Winskel and Einstein used the word 'event', and not some made up string, is to hint to an underlying intuition they would like to evoke. I argue this is at least equally fundamental to mathematics. Abstraction is something all people do and have been doing outside and before mathematics. The difference is mathematicians try to channel their intuitions into more precise language.
These underlying intuitions, producing formal coincidences across mathematics, can often be mined for interesting abstractions. Indeed, I would be very surprised if one couldn't meaningfully connect event structures and 'spacetime causality', even though the connection might not be fertile (one would need more motivation for sure).

As always, the answer lies in the synthesis of these two moments. I argue mathematics is powered by the meaning-form dialectic, and it's misleading to focus on either side only. There is no divide between applied and theoretical mathematics, there is a continuum, and ideas slowly move back and forth like air convecting in a room. So it's important to not mistake the finger for the moon (form for meaning), but it's important also to remember we are looking at the moon because the finger is pointing at it, and the finger is pointing at it because there is something there.

Peiyuan Zhu (Feb 28 2023 at 17:45):

So many good philosophical takes here.

Peiyuan Zhu (Feb 28 2023 at 18:07):

In category theory functors looks to me the most event-like structure. Any thoughts? The discussion here makes me want to go back to the Event as Basic Type chapter of Modal HoTT that I read a while ago but gave up because I didn't understand it.

Peiyuan Zhu (Feb 28 2023 at 18:09):

image.png

Peiyuan Zhu (Feb 28 2023 at 18:13):

I'm not sure how to read that dependent sum.

dusko (Feb 28 2023 at 20:30):

Matteo Capucci (he/him) said:

dusko said:

mathematics begins when people stop expecting that words carry their meaning with them. they assign a different meaning to $x$ in every story. the word "event" is a variable. glynn winskel used it to denote an element of an algebraic lattice equipped with an inconsistency relation. he used the word "event" to suggest a vague intuition, and because it had not been used in the same context before. the same word means another thing in special relativity and yet another thing in general relativity. it is a variable.

This is a very interesting point regarding the praxis and the philosophy of mathematics. I strongly agree that the fundamental idea of mathematics is to strip form of substance and focus on that (abstraction), and it can be misleading to read more than a coincidence in a choice of terminology (or in any other recurrent formal pattern). In that sense, 'event' in Winskel's event structures and 'event' in GR are as related as 'x' in an equation and 'x' in another.

On the other hand, the reason Winskel and Einstein used the word 'event', and not some made up string, is to hint to an underlying intuition they would like to evoke. I argue this is at least equally fundamental to mathematics. Abstraction is something all people do and have been doing outside and before mathematics. The difference is mathematicians try to channel their intuitions into more precise language.
These underlying intuitions, producing formal coincidences across mathematics, can often be mined for interesting abstractions. Indeed, I would be very surprised if one couldn't meaningfully connect event structures and 'spacetime causality', even though the connection might not be fertile (one would need more motivation for sure).

As always, the answer lies in the synthesis of these two moments. I argue mathematics is powered by the meaning-form dialectic, and it's misleading to focus on either side only. There is no divide between applied and theoretical mathematics, there is a continuum, and ideas slowly move back and forth like air convecting in a room. So it's important to not mistake the finger for the moon (form for meaning), but it's important also to remember we are looking at the moon because the finger is pointing at it, and the finger is pointing at it because there is something there.

you are right in the sense that i said it quickly and simply and it sounds like wittgenstein and i am definitely no wittgenstein.

what are we doing here, talking to each other, if all these words are just variables? at least :+1: means :+1:. when wittgenstein said consistency doesn't matter, turing said "but the bridges will fall if theories are inconsistent".

what is the difference between the finger and the moon? on the surface, they are two words. in the context, they are metaphors for some abstractions, which linguists would call signifiant and signifiee. oh, you said form-meaning. i agree. while there is, of course, the well-studied possibility that the moon could be made of cheese (stilton, i believe) it would be a hassle if we all engaged in cartesian meditations worrying how to be sure that the moon is there. so we agree that the moon is there. next thing you know, we also agree that lower taxes mean higher employment, since it really means a lot to me that we agree about that, and it would be hassle to go around disagreeng too much...

it's a hassle to disagree, and it's a hassle to agree. (before the "agreeing to disagree" paper, aumann constructed an example where two players with the same priors deduce from the same observations completely different models. he never ever mentioned that model again. he is a deeply religious man.)

exercise: express the relation of the finger and the moon as an adjunction.

PS the correlations are of course dialectical, and lawvere is a true giant, but he could have done even more if he didn't go around talking about dialectics so much, almost to the point of making himself into a sort of a wittgenstein of leninism. (or maoism a couple of years earlier.)

Peiyuan Zhu (Feb 28 2023 at 21:20):

Maybe it'll become Bogdanovism once the problem of history-dependency is resolved for these systems...?

Matteo Capucci (he/him) (Mar 01 2023 at 07:37):

dusko said:

you are right in the sense that i said it quickly and simply and it sounds like wittgenstein and i am definitely no wittgenstein.

Sure, I didn't mean to pick on you :) all you said was perfectly fine! I just took the opportunity to vent some 'philosophy'...

Naso (Mar 14 2023 at 01:02):

Martti Karvonen said:

I wouldn't expect any connection to the general topic just because the word "event" is used in both cases, and I suspect that concurrency theory has very little to do with physics (both now and probably in the future, although I'd be happy to be proven wrong).

Check out this article: "Causality in physics and computation" by Prakash Panangaden

Peiyuan Zhu (Mar 14 2023 at 21:11):

This is very interesting.

Peiyuan Zhu (Mar 15 2023 at 21:03):

In Sociology: http://grahamberrisford.com/AM%204%20System%20theory/SystemTheory/ChallengingSystemsThinkers/15%20Luhmann%27s%20autopoietic%20social%20system.htm

Peiyuan Zhu (Mar 15 2023 at 21:03):

"What makes Luhmann’s view so radical is that his social system contains nothing but communicative events.
It has no persistent state in the form of actors, objects, data or memory.
It not only depends on communicative events, it is communicative events.
It is the set of communicative events related to one theme or code such as justice/injustice."

Peiyuan Zhu (Mar 21 2023 at 22:57):

I'm seeing the Joy of Abstraction talks about relations in Chapter 7

Peiyuan Zhu (Mar 21 2023 at 23:09):

In category theory is there a specific structure that belongs to "events"?

Peiyuan Zhu (Mar 21 2023 at 23:10):

I was looking at identity of indiscernibles https://ncatlab.org/nlab/show/identity+of+indiscernibles. Is this a judgement based on events?

Peiyuan Zhu (Mar 21 2023 at 23:29):

Are the "objects" in category theory best understood as "events"?

John Baez (Mar 21 2023 at 23:55):

Peiyuan Zhu said:

In category theory is there a specific structure that belongs to "events"?

No.

John Baez (Mar 21 2023 at 23:56):

Peiyuan Zhu said:

Are the "objects" in category theory best understood as "events"?

No. For example an object in a category could be a set, a group, a vector space, a graph, a vertex of a graph, or many other things, depending on the category.

Peiyuan Zhu (Mar 21 2023 at 23:58):

I saw this in an article of Lee Smolin in Quantmagazine: " We propose as a principle of dynamics that each view should be unique. That idea comes from Leibniz’s principle of the identity of indiscernibles. Two events whose views are exactly mappable onto each other are the same event, by definition. " https://www.quantamagazine.org/were-stuck-inside-the-universe-lee-smolin-has-an-idea-for-how-to-study-it-anyway-20190627/

John Baez (Mar 21 2023 at 23:58):

Yup, sounds like Leibniz.

John Baez (Mar 22 2023 at 00:05):

Read his Monadology for more.

Peiyuan Zhu (Mar 22 2023 at 00:11):

But when people talk about relation they almost always refers to the arrows in some category?

John Baez (Mar 22 2023 at 00:17):

I don't know about people in general, but when mathematicians talk about a "relation" they most often mean a subset of $X \times Y$ where $X$ and $Y$ are sets. Try this:

• Wikipedia, Relation (mathematics): definition.

Peiyuan Zhu (Mar 22 2023 at 00:18):

But in this case of Yoneda Lemma "object is determined by its relations to other objects" it is referring to the arrows right? https://www.math3ma.com/blog/the-yoneda-perspective

John Baez (Mar 22 2023 at 00:22):

Here Tai-Danae is using the word "relation" in a loose, vague way trying to get you to understand the idea of the Yoneda lemma. "Relation" is one of those words that's often used vaguely. So don't think of this as some sort of definition of "relation" in math!

When we get precise we have to say "an object is determined by its representable presheaf". This representable presheaf involves the arrows from that object to other objects. But it's much more than just the set of arrows from that object to other objects. I'm sure if you read all 3 of Tai-Danae's articles on Yoneda she explains this.

Peiyuan Zhu (Mar 22 2023 at 00:51):

John Baez said:

Peiyuan Zhu said:

In category theory is there a specific structure that belongs to "events"?

No.

But it does look like it has something to do with homotopy type theory. In particular the univalence axiom. It even quoted you. So I guess HoTT is not necessarily considered a part of category theory.

John Baez (Mar 22 2023 at 04:56):

What is that "even quoted me"? What did I say?

Anyway, I don't think HoTT has a specific structure that's mainly used to describe "events". If you think there is, what is this structure?

But of course people can try to use category theory, homotopy type theory, and many other kinds of math to describe events.

Peiyuan Zhu (Mar 22 2023 at 14:57):

“There are several ways to think about the axiom of univalence. One can see it as a sophisticated updating of Leibniz’s principle of the identity of indiscernibles.” –John Baez nCafé

Morgan Rogers (he/him) (Mar 22 2023 at 15:17):

But the relation to "events" seems to have nothing to do with John.

Reid Barton (Mar 22 2023 at 15:18):

Or Leibniz, for that matter

Peiyuan Zhu (Mar 22 2023 at 16:29):

I suppose what's talked about in the univalence axiom can be considered as equivalence between events?

Josselin Poiret (Mar 22 2023 at 16:35):

Peiyuan Zhu said:

I suppose what's talked about in the univalence axiom can be considered as equivalence between events?

if your types are events, then yes. the univalence axiom is about isomorphisms of types and paths valued in types.

Josselin Poiret (Mar 22 2023 at 16:36):

Leibniz equality is not the same as the MLTT equality type though

Jason Erbele (Mar 22 2023 at 16:55):

Stated slightly differently, you may use the univalence axiom to reason about equivalence between events, but that doesn't mean the univalence axiom itself is about equivalence between events. The univalence axiom is not limited to talking about events.

Peiyuan Zhu (Mar 26 2023 at 15:49):

So Rovelli talked a lot about time is ignorance so how time depends on blurring. Does it mean we could have a different version of the second law of thermodyanmics such that there's no time variable? If we don't take temperature as the average but instead choose something else that discrimiate the states do we get some other kind of second law too?

Patrick Nicodemus (Mar 28 2023 at 16:50):

Josselin Poiret said:

Leibniz equality is not the same as the MLTT equality type though

What is your justification for this distinction? The induction principle for MLTT equality says something very close to: if a = b, then every property P which holds for a also holds for b. Conversely if every property which holds for property a also holds for b, then the property P(x) = "x = a" holds for b, thus a = b.

Josselin Poiret (Mar 28 2023 at 17:32):

Patrick Nicodemus said:

Josselin Poiret said:

Leibniz equality is not the same as the MLTT equality type though

What is your justification for this distinction? The induction principle for MLTT equality says something very close to: if a = b, then every property P which holds for a also holds for b. Conversely if every property which holds for property a also holds for b, then the property P(x) = "x = a" holds for b, thus a = b.

what your definition of property here? In some flavors of MLTT without UIP equality is not a property

Peiyuan Zhu (Apr 10 2023 at 02:34):

Peiyuan Zhu said:

These two are from Carlo Rovelli's pop science book the Order of Time footnotes of Chapter 8: Dynamics as Relation:
image.png
image.png

Is he talking about how dynamics of a physical system can instead of being seen as time evolution $\frac{dx}{dt}=f(x,y),\frac{dy}{dt}=g(x,y)$ it can be written as $\{(x,y):C(x,y)=0\}$ so there's no independent time variable involved?

How to then talk about symmetry with relations?