So a commonality is that they are mappings from some space to the real line. But what matters - what determines the meaning of the entropy is both what the inputs to the entropy function are and how they are combined to produce a number. To speak of entropy in general is to let the what and the how vary with the implicit context of the conversation; it destroys the meaning of individual entropies by attempting to unify them, the unification has poor construct validity precisely because it doesn't allow the what and the how of the mapping to influence the meaning. — fdrake

And yet something still ties all this variety back to some general intuition. The usual response is "disorder".

As I have said, at the metaphysical level, we can only approach proper definitions by way of a dialectical or dichotomistic argument. We have to identify the two complementary extremes that are mutually exclusive and jointly exhaustive. So a metaphysical-strength discussion of entropy has to follow that form. It has to be entropy as "opposed to what?". Your concern is that there seem multiple ways to quantify "entropy". My response has been that a metaphysical-strength definition would be qualitative in this precise fashion. It would lead us to a suitable dichotomy.

Hence why I keep trying to return the conversation to entropy~information as the candidate dichotomy that has emerged in recent times. It seems a stronger statement that entropy~negentropy, or disorder~order, as mere negation is a weak kind of dichotomy, not a strong one.

Likewise constraints~degrees of freedom slice across the debate from another direction. How the two dichotomies of entropy~information and constraints~degrees of freedom might relate is a further important question.

So a qualitative approach here isn't just a hand-waving, anything goes, exercise in speculation. Metaphysics does have a method for clarifying its ideas about reality. And that approach involves discovering a reciprocal relation that connects two opposed limits on Being.

As I said, information and entropy capture a duality in terms of relative surprisingness. An entropy-maximising configuration is reciprocally the least-informational. No need to count microstates even if every one is different. You only need to measure a macrostate to completely characterise the aspect of the system that is of interest.

By contrast, a surprising state of the world is the most informational. It is high on negentropy. And now just one "microstate" is the feature that appears to characterise the whole situation - to the degree that is the aspect of interest.

So "disorder" is just an application of the principle of indifference. A messy system is one in which the details don't matter. Gone to equilibrium, the system will be generic or typical.

However no system can in fact maximise its entropy, reach equilibrium, except that it has stable boundary conditions. So - dichotomously - there is negentropy or order in the fact of boundary stability, in the fact of being closed for causality.

Thus is it common to sum up entropy as about a statistical propensity to be disordered - to be in a system's most typical state. But that "first law of thermodynamics" state of closure (well, it includes the necessity of the third law to put a lower bound on things to match the first's upper bound) is only half the story. Somewhere along the line, the system had to be globally parameterised. The general holonomic constraints or boundary conditions had to form somehow.

So here we see a good example of why dichotomies or symmetry-breakings are basic to metaphysical-strength analysis. They alert us to the other half of the holistic story that reductionists are wont to overlook. A full story of entropy can't just presume stable boundary conditions. Those too must be part of what develops (along with the degrees of freedom that global constraints produce, or parameterise).

To the degree your discussions here are not dialectically explicit, they simply fail the test of being adequately metaphysical. Your obsession about quantitative methods is blinding you to the qualitative discussion where the hunt is on for the right way to frame information~entropy, or degrees of freedom~constraints, as the fundamental dichotomy of a developmental Cosmos.

Maybe I'd be more comfortable with what you're saying if you used scarequotes like I do. — fdrake

Your comfort is definitely my number one priority. I mean "number one priority".

I'm not suggesting you 'look at the formulas and find a master one', the thing I cared about was that the measures of entropy in terms of ascendency and relative abundance meant different things - they summarise different aspects of the behaviour of the ecosystem. — fdrake

What else do you expect if you take the attitude that we are free to construct metrics which are valid in terms of our own particular interests?

I agree we can do just that. We can describe the world in terms that pick up on some characteristic of interest. I just say that is not a deep approach. What we really want is to discover the patterns by which nature organises itself. And to do that, we need some notion about what nature actually desires.

This is where thermodynamics comes in. This is what is unifying science at a foundational level now. Both biology and physics are reflecting that emergent metaphysical project. And thermodynamics itself is becoming semiotic in recognising the duality of entropy and information.

So you are down among the weeds. I'm talking about the big picture. Again, it is fine if your own interests are narrow. But I've made different choices.

What does this measure? The diversity of flows within a network. How? It looks at the proportion of each flow in the total, then computes a quantification of how that particular flow incorporates information from other flows - then scales back to the total flow in the system. It means that the diversity is influenced not just by the number of flows, but their relative strength. For example, having a network that consisted of 1 huge flow and the rest are negligible would give an ascendency much closer to a single flow network than another measure - incorporating an idea of functional diversity as well as numerical biodiversity. Having 1 incredibly dominating flow means 0 functional diversity. — fdrake

You seem terribly concerned by things that don't seem a big issue from the thermodynamic point of view.

A lot of your focus seems to be on how to we specify richness or complexity or healthy biodiversity. And you instinctively think in terms of counting niches or something else "structually concrete".

But the flow view of an ecosystem would see a hierarchy of flow just happening naturally. You would get complexity arising in a way that is essentially "meaningless".

So think of a scalefree network or other fractal growth model. A powerlaw hierarchy of connectivity will just arise "randomly". It doesn't need evolutionary semiosis or natural selection to create it. The complexity of the flow is not something that needs a designing hand to happen. It is the natural structure of the flow.

Check out Adrian Bejan's constructal law. He is pretty strong on this issue.

So a reductionist would think of a richly organised hierarchy of relations as a surprising and delicate state of affairs. But the switch is now to see this as the inevitable and robust equilibrium state of a freely growing dissipative structure, like an ecosystem. Scalefree order comes for free.

So something like the reason for the occurrence of niches over all scales is not something in need of some "contextualised" metric to index it. We don't have to find external information - some historic accident - which specifies the fact of a hierarchical order. That kind of order is already a natural pattern or attractor. It would take external accidents of history to push it away from this natural balance.

Where every pipi is the proportion of the i-th species of the total. This is a numerical comparison of the relative abundance of each species present in the ecosystem. This obtains a maximum value when each species has equal relative abundance, and is then equal to the number of species in the ecosystem. Look at the case with 2 species each having 2 animals. p is constant along i, being 0.5, then the Shannon Biodiversity is -2*0.5*log(0.5) = log2, so its exponential is 2. — fdrake

So here, aren't you assuming that we can just count species and have no need to consider the scale of action they might represent? One might be a bacterium, the other an elephant. Both might be matched in overall trophic throughput. We would expect their relative abundance to directly reflect that fact rather than a species count having much useful to say about an ecosystem's healthy biodiversity or state of entropic balance.

Of course, if you've read this far, you will say 'the middle state is the one furthest from order so of course it has the highest degrees of freedom', which suggests the opposite intuition from removal of dominant energy flows 'raining degrees of freedom' down onto the system. This just supports the idea that your notion of entropy has poor construct validity. — fdrake

Exergy? I mean you seemed to agree that it is about quality of the entropy. So a big tree dropping leaf litter is a rather different story to a forest clearing being blasted by direct sunlight again.

This is why what we're talking about has almost no relation to the OP. — fdrake

But the OP was about extracting a political analogy from a lifecycle understanding of ecosystems. I simply responded by saying Salthe's infodynamic perspective gives you a self-explanatory three stage take on that.

You then got shirty about my use of infodynamic or dissipative theory jargon. I'm happy to explain my use of any terminology. And I'm happy to defend the fact that I do indeed have an over-arching worldview. That is way more than nearly anyone else does around these here parts.

Going from ATP being used to fuel an organism straight to a 'global' sense of infodynamics and signals/signs in pansemiosis. It works only when you wave your hands and don't focus on the specifics. When what before was concrete becomes metaphorical, then what was metaphorical becomes concrete. — fdrake

Its not metaphorical if infodynamics/semiosis is generalisable to the material dissipative structures in general.

Again, you might have to actually read the literature - Salthe for instance. But the metaphysical ambition is clear enough.

The information is separate from the dynamics via the epistemic cut in biosemiotic systems - organisms that are living and mindful. A organisation mediated by signs is perfectly concrete.

What still counts as speculative is then generalising that concrete description of life/mind so that it is seen to be a concrete theory of the Cosmos. The Universe would be understood as a dissipative system organised by a sign relation. The biological understanding of the duality of information and entropy would prove to apply to the whole of existence as its scientific theory.

So it is not me waving my hands. It is you demonstrating a deaf ear to context. I am careful to distinguish between the part of what I say which is "normal science" and the part that is "speculative metaphysics". And the speculative part is not merely metaphor because the project would be to cash it out as concrete theory, capable of prediction and measurement.

I agree that may also be a tall order. But still, it is the metaphysical project that interests me. The fact that you repeatedly make these ad hom criticisms shows that you simply wish not to be moved out of your own particular comfort zone. You don't want to be forced to actually have to think.

Shannon's strictly broader than Boltzmann since it allows for non-equidistribution. — fdrake

Does that remain the case now that information theory has been tied to the actual world via holographic theory?

Boltzmann's k turned out to be physically derived from the dimensionless constants of the Planck scale. And Shannon likewise now represents a fundamental Planckian limit. The two are united via the basic physical limits that encode the Cosmos.

The volume of a spacetime defines some entropic content. The surface area of that volume represents that content as information. And there is a duality or reciprocality in the relation. There can't be more entropy inside than there are questions or uncertainties that can be defined on a 4 to 1 surface area measure.

It is about the biggest result of the last 30 years in fundamental physics.

The theoretical links between Shannon's original entropy, thermodynamical entropy, representational complexity can promote a vast deluge of 'i can see through time' like moments when you discover or grok things about their relation. BUT, and this is the major point of my post:

Playing fast and loose with what goes into each of the entropies and their context makes you lose a lot. They only mean the same things when they're indexed to the same context. The same applies for degrees of freedom.

I think this is why most of the discussions I've read including you as a major contributor are attempts to square things with your metaphysical system, but described in abstract rather than instantiated terms. — fdrake

I'm baffled that you say Shannon entropy came before Boltzmann's entropy.

But anyway, again my interest is to generalise across the different contextual instantiations of the measurement habits which science might employ. I am indeed interested in what they could have in common. So I don't need to defend that as if it were some problem.

And as I have pointed out, when it comes to semiosis and its application to the world, we can see that there is a whole level of irreducible complexity that the standard reductionist approach to constructing indices of information/entropy/degrees of freedom just misses out.

It is fine that science does create simpler indexes. I've no problem with that as a natural pragmatic strategy. But also, with Shannon and Boltzmann, it became clear that informational uncertainty (or configurational degrees of freedom) and entropic material degrees of freedom (or countable microstates) are two sides of the same coin. The mathematics does unite them in a general way at a more abstract level.

And then when it come to biosemiosis, information and entropy become two sides of a mechanically engineered epistemic cut. We are talking about something at a level above the brute physical realm imagined by the physical discourse that gives us Shannon uncertainty and Boltzmann entropy. It thus needs its own suitable system of measurement.

That is the work in progress I see in literature. That is the particular story I am tracking here.

You can keep re-stating that a proper scientist would use the proper tools. You can reel off the many kinds of metrics that reflect the simpler ontology of the reductionist. You can continue to imply that I am somehow being unscholarly in seeking to consider the whole issue at a more holistic level - one that can encompass physicalist phenomena like life and mind. And indeed, even culture, politics, economics, morality and aesthetics.

But I know what I'm about so I'm only going to respond to your critique to the degree it throws light on the connecting commonality, the linkages to that more holistic worldview.

The Shannon Entropy is related to the Boltzmann entropy in thermodynamics in a few ways I don't understand very well. — fdrake

What's wrong with a reciprocal relation? If Shannon entropy is the degree of surprise to be found in some system, then the Boltzmann entropy is the degree to which that system is in its least surprising state.

So if a system is constrained and is thus composed of some set of independent elements, states, or events, its arrangement can be described somewhere on a spectrum between maximally surprising and minimally surprising. An unsurprising arrangement requires the least amount of information to specify it. And it thus represents the most entropic arrangement.

Up to you. It's a nice day outside. I doubt I will tick off every point you raised.

What does 'dichotomous to constraints' mean?

There are lots of different manifestations of the degrees of freedom concept. I generally think of it as the dimension of a vector space - maybe calling a vector space an 'array of states' is enough to suggest the right meaning. If you take all the vectors in the plane, you have a 2 dimensional vector space. If you constrain the vectors to be such that their sum is specified, you lose a degree of freedom, and you have a 1 dimensional vector space. This also applies without much modification to random variables and random vectors, only the vector spaces are defined in terms of random variables instead of numbers. — fdrake

In mechanics, degrees of freedom are a count of the number of independent parameters needed to define the configuration of a system. So your understanding is correct.

And they are dichotomous to constraints as they are what are left over as a result of a configuration being thus limited. Constraint suppresses freedoms. What constraint doesn't suppress then remains to become some countable degree of freedom for that system.

Then from an infodynamic or pansemiotic point of view, constraints become the informational part of the equation, degrees of freedom are the dynamics. In the real material world, the configuration can be treated as the knowledge, the structure, that the organismic system seeks to impose on its world. The constraints are imposed by a mind with a purpose and a design. The degrees of freedom are then the entropy, the dynamics, that flow through the organism.

So a structure has to be imposed on the flow to in fact create a flow composed of some set of degrees of freedoms. A bath of hot water will simply cool by using its surrounds as a sink. An organism wants to build a machinery that stands inbetween such a source and sink so as to extract work along the way.

That is why I suggest ATP as a good way to count degrees of freedom in biology. It is the cell's meaningful unit of currency. It places a standard cost on every kind of work. It defines the dynamical actions of which a cell is composed in a way that connects the informational to the entropic aspects of life. An ATP molecule could be spent for any purpose. So that is a real non-physical freedom the cell has built for itself.

An ATP molecule can be used to make a kinesin "walker" transport molecule take another step, or spin the spindle on ATP-ase. But then the spending of that ATP has an actual entropic cost as well. It does get used up and turned into waste heat (after the work is done).

So degrees of freedom are what constraints produce. And in living organisms, they are about the actions that produce units of work. The constraints are then the informational structure that regulates the flow of material entropy, channelling some source to a sink in a way that it spins the wheels of a cellular economy along the way.

A cooling bath of hot water lacks any interesting informational structure apart from perhaps some self-organised convection currents. Like a Benard cell, it might have its thermodynamic flow improved by emergent constraints producing the organised currents that are now some countable set of degrees of freedom. A more chaotic path from hot to cold has had its own vaguer collection of degrees of freedom suppressed so the flow is optimised by a global structure.

But life has genes, membranes, pores, switches, and a host of molecular machinery that can represent the remembered habits of life - some negentropic or informational content - that produces a quite intentional structure of constraints, a deliberately organised set of degrees of freedom, designed to extract self-sustaining work from any available entropy gradient.

So I suppose I should talk about configurational entropy. — fdrake

Yep. But note that biosemiosis is about how life has the memory to be in control of its physical configuration. It uses a potential gradient to do the work of constructing itself.

So that brings in the informational aspect of the deal - Pattee's epistemic cut. The organism first insulates itself from dynamics/entropy by creating its own informational degrees of freedom. It does this by using a genetic code. But also, it does it foundationally down at the level of the dynamics itself in having "a unit of work" in an ATP molecule that can be used "to do anything a cell might want".

What gets configured is not just some spatial geometry or thermal landscape. The material world is actually being inscribed by an organism's desires. The dynamics is not merely self-organising. It is being organised by a proper self.

It is this situation which your notions of degrees of freedom don't really cover. You are not accounting for the epistemic cut which is the added semiotic feature of this material world now. If you are going to demand quantitative measures, the measures have to span the epistemic cut in some fashion. You are trying to make measurements that only deal with one of the sides of the equation.

Yeah. But just have a go. Let's see what you could come up with. It truly might help to make sense of your attacks on mine.

If instead you really want to say that entropy is simply whatever act of measurement we care to construct as its instrumental definition - that there is no common thread of thought which justifies the construct - then how could you even begin to have an intelligent discussion with me here?

Entropy is absolutely well defined. — fdrake

What's your single sentence definition then? I mean, just for fun.

So already we agree that the notion is ill-defined? It is a fast and loose term in fact. Just like entropy. Or information. Maybe this is why I am right in my attempt to be clear about the high-level qualitative definition and not pretend it has some fixed low-level quantitative measure.

But I'll keep waiting until you do connect with what I've already posted.

Hmm. Just not convincingly butch coming from you. And more importantly it has no sting. You've got to be able to find a real weakness to pick at here. Calling me buttercup once again ends up saying more about your life experience than mine.

Sousing? You really do have a tin ear when it comes to your ad homs. It absolutely spoils the effect when you come across as the hyperventilating class nerd.

Why the sudden interest in Nick Lane and Peter Hoffmann? Couldn't possibly be anything I said.

I'll find time to respond to your post later. But it is a shame that you bypass the content of my posts to jump straight back to the world from your point of view.

You make very little effort to engage with my qualitative argument. Well none at all. So it feels as though I'm wasting my breath if you won't spell out what you might object to and thus show if there is any proper metaphysics motivating your view, or whether you just want to win by arguing me into some standard textbook position on the various familiar approaches to measuring entropy.

Perhaps I'll let you finish first.

However, that is not my experience, and I would challenge the idea that a city-dweller who has never seen or heard of the mountains would experience no unusual psychological response from teleporting to the top of Mont Blanc on a clear summer day. — TJO

I'd say from experience it goes both ways. These days I live in a small city surrounded by awesome mountains. So taking a trip back to one of the world's big cities is pretty awesome as a contrast.

Psychologically, I think you are only talking about the sense of arousal we get from something "high contrast", something that is out of scale with the familiar. The arousal can be read as frightening or exhilarating depending on our mindset. It depends upon whether we are judging the situation as something to approach or avoid.

So there is no necessity that the top of a mountain, or the busy centre of London, be either frightening or awesome. That bit of the feeling is down to some further judgement. But if the environment has a high contrast with your familiar environment, it is going to be arousing in some way. You will be feeling like wanting to make some sense of its novelty.

Another quick point about the aesthetics of nature. I would argue we are also aesthetically tuned to the recognition of symmetry. We like highly regular shapes that approach some ideal limit. But that explains the appeal of cubes and spheres, not the roughness and jaggedness of your typical spectacular landscape.

However a spectacular landscape does have a perfect fractal symmetry. It has an ideal balance in its self-similarity or scale-free shape. So a tree or fern is lovely because it has a self-similar branching structure - completely regular in the irregular way it forks. Same with river networks or eroding mountains and coastlines.

We can of course appreciate this natural symmetry in a fern or tree quite easily. But to see the fractal nature of a landscape, we do have to have a big vista. We have to step back far enough to see nature over a lot of the scales all at once. Hence another reason why mountain climbing is an aesthetic experience.

First up, I'm not bothered if my arguments are merely qualitative in your eyes. I am only "merely" doing metaphysics in the first place. So a lot of the time, my concern is about what the usual rush to quantification is missing. I'm not looking to add to science's reductionist kitset of simple models. I'm looking to highlight the backdrop holistic metaphysics that those kinds of models are usually collapsing.

And then a lot of your questions seem to revolve around your definition of degrees of freedom vs mine. It would be helpful if you explained what your definition actually is.

My definition is a metaphysically general one. So it is a little fuzzy, or broad, as you say.

To help you understand, I define degrees of freedom as dichotomous to constraints. So this is a systems science or hierarchy theory definition. I make the point that degrees of freedom are contextual. They are the definite directions of action that still remain for a system after the constraints of that system have suppressed or subtracted away all other possibilities.

So the normal reductionist metaphysical position is that degrees of freedom are just brute atomistic facts of some kind. But I seek to explain their existence. They are the definite possibilities for "actions in directions" that are left after constraints have had their effect. So degrees of freedom are local elements shaped by some global context, some backdrop history of a system's development.

Thus I have an actual metaphysical theory about degrees of freedom. Or rather, I think this to be the way that holists and hierarchy theorists think about them generally. Peirce would be the philosopher who really got it with his triadic system of semiosis. Degrees of freedom equate to his Secondness.

A second distinctive point is that I also follow semiotic thinkers in recognising an essential connection between Boltzmann entropy and Shannon uncertainty - the infodynamic view which Salthe expresses so well. So this is now a quantification of the qualitative argument I just gave. Now biosemiotics is moving towards the possibility of actual science.

Theoretical biologists and hierarchy theorists like Howard Pattee in particular have already created a general systems understanding of the mechanism by which life uses codes to harness entropy gradients. So the story of how information and dynamics relates via an "epistemic cut" has been around since the 1970s. It is the qualitative picture that led to evo-devo. And it is the devo aspect - the Prigogine-inspired self-organising story of dissipative structures - that has become cashed out in an abundance of quantitative models over the past 30 years. I assume you know all about dissipative structure theory.

So what we have is a view of life and mind that now is becoming firmly rooted in thermodynamics. Plus the "trick" that is semiotics, or the modelling relation.

The physico-chemical realm already wants to self-organise to dissipate energy flows more effectively. That in itself has been a small revolution in physical science. What you call configuration entropy would seem to be what I would call negentropy, or the degrees of freedom spent to create flow channelling structure - some system of constraints. And in the infodynamic (or pansemiotic) view, the negentropy is information. It is a habit of interpretance, to use Peirce's lingo. So we have the duality of entropy and information, or a sustaining flow of degrees of freedom and set of structuring constraints, at the heart of our most general thermodynamical description of nature.

Reductionist thinking usually just wants to talk about degrees of freedom and ignore the issue of how boundary conditions arise. The thermodynamics is basically already dead, gone to equilibrium, by the time anything is quantified. So the boundary conditions are taken as a given, not themselves emergently developed. For example, an ideal gas is contained in a rigid flask and sitting in a constant heat sink. Nothing can change or evolve in regard to the constraints that define the setting in which some bunch of non-interacting particles are free to blunder about like Newtonian billiard balls. But the dissipative structure view is all about how constraints can spontaneously self-organise. Order gets paid for if it is more effective at lowering the temperature of a system.

So thermodynamics itself is moving towards an entropy+information metaphysics. The mental shift I argue for is to see dissipative structure as not just merely a curiosity or exception to the rule, but instead the basic ontological story. As Layzer argues, the whole Big Bang universe is best understood as a dissipative structure. It is the "gone to equilibrium" Boltzmann statistical mechanics, the ideal gas story, that is the outlier so far as the real physical world is concerned. The focus of thermodynamics has to shift to one which sees the whole of a system developing. Just talking about the already developed system - the system that has ceased to change - is to miss what is actually core.

So physics itself is entropy+information in some deep way it is now exploring. And then biology is zeroing in on the actual semiotic machinery that both separates and connects the two to create the even more complex phenomenon of life and mind. So now we are talking about the epistemic cut, the creation of codes that symbolise information, capture it and remember it, so as to be able to construct the constraints needed to channel entropy flows. Rivers just carve channels in landscapes. Organisms can build paths using captured and internalised information.

Only recently, I believe the biosemiotic approach has made another huge step towards a quantitative understanding - one which I explained in detail here: https://thephilosophyforum.com/discussion/comment/105999#Post_105999

So just as physics has focused on the Planck-scale as the way to unify entropy+information - find the one coin that measures both at a fundamental level - so biology might also have its own natural fundamental scale at the quasi-classical nanoscale (in a watery world). If you want to know what a biological degree of freedom looks like, it comes down to the unit of work that an ATP molecule can achieve as part of a cell's structural machinery.

To sum up, no doubt we have vastly different interests. You seem to be concerned with adding useful modelling tools to your reductionist kitbag. And so you view everything I might say through that lens.

But my motivation is far more general. I am interested in the qualitative arguments with which holism takes on reductionism. I am interested in the metaphysics that grounds the science. And where I seek to make contact with the quantitative is on the very issue of what counts as a proper act of measurement.

So yes, I am happy to talk loosely about degrees of freedom. It is a familiar enough term. And then I would define it more precisely in the spirit of systems science. I would point to how a local degree of freedom is contextually formed and so dichotomous to its "other" of some set of global constraints. Then further, I would point to the critical duality which now connects entropy and information as the two views of "a degree of freedom". So that step then brings life and its epistemic cut, its coding machinery, into the thermodynamics-based picture.

And then now I would highlight how biophysics is getting down to the business of cashing out the notion of a proper biological degree of freedom in some fundamental quantitative way. An ATP molecule as the cell's universal currency of work looks a good bet.

I'm sure you can already see in a hand-waving way how we might understand a rainforest's exergy in terms of the number of ATP molecules it can charge up per solar day. A mature forest would extract ATP even from the tiniest crumbs dropping off the table. A weedy forest clearing would not have the same digestive efficiency.

So I've tried to answer your questions carefully and plainly even though your questions were not particularly well posed. I hope you can respond in kind. And especially, accept that I just might not have the same research goals as you. To the degree my accounts are metaphysical and qualitative, I'm absolutely fine about that.

Your questions seem off the point so I’m struggling to know what you actually want.

If you have a professional interest, then there is a big literature. Maybe start with https://www.jameskay.ca/about/thermo.html

Rod Dewar and Rod Swenson also. I’ve mention Stan Salthe and Robert Ulanowicz. Charlie Lineweaver is another. Adrian Bejan might be the strongest in terms of generic models.

I’ve not been close to the research for 20 years and I was always only really interested in the qualitative arguments. Also the quantitative support wasn’t exactly slam dunk. Measuring ecosystems is not easy.

But for instance, one line of research involved thermal imaging of rainforests and other ecosystems. The hypothesis was that more complex ecologies would stick out by having a cooler surface temperature. They would extract more work from the solar gradient.

Is that the kind of experiment you have in mind?

Here’s a presentation with references at the end as well as charts of data - https://hyspiri.jpl.nasa.gov/downloads/2011_Symposium/day1/luvall%20hyspiri%20ecological%20thermodynamics%20may%202011%20final.pdf

https://en.m.wikipedia.org/wiki/Ascendency

What do you mean? Either we do blow ourselves up, or we do find a long-run ecological balance.

Well, I was just trying to cheer you up. I realise there is in fact a third option where human ingenuity does get used to keep the game going in ever more extravagant fashion. Rather than changing ourselves to fit nature, many people will quite happily go along with changing nature to fit us.

This is the anthropocene. Once we have artificial meat, 3D printed vegetables made from powdered seaweed, an AI labour force and nuclear fusion, who cares about rain forests and coral reefs? Rent your self some VR goggles and live out of that old time stuff if you are sentimental. Meanwhile here is an immersive game universe where you can go hunting centaurs and unicorns.

So probably bad luck. We likely have enough informational degrees of freedom to beat nature at its own game.

I am basing a lot of my claims on another thread debating why 0.999... is 1, not just infinitesimally close to it. It was explained by someone who knows their stuff far better than I. — noAxioms

Hah. There certainly is an official position on this. But it is more about what has to be agreed to make the maths come out right than one based on force of metaphysical argument.

And I'm not complaining. Maths needs to secure its constructs. It needs to be axiomatic.

I'm just reminding that this is what happens and so maths isn't in a position to tell metaphysics "what is really going on" due to what if finds works. Maths can act as a powerful constraint on free metaphysical speculation, and also serve as a powerful inspiration to further inquiry. But it isn't how metaphysical truth is discovered.

So the infinite and the infinitesimal speak to the need to establish limits. Constructive actions like counting need to have constraints to bound them as well. It is in fact the same dialectical issue which is at the root of metaphysical reasoning. For something to be the change, something complementary must be made the bit that stands still.

The number line has to be both continuous yet discrete at the same time. It must be a line composed of points. So of course some fancy mathematical machinery must be added to negotiate what must be a tricky change-over going on somewhere. What connects the points? What permits an exact cut?

The official answer works. But it is also pervaded by a spirit of "OK guys, shut your eyes for a moment, don't ask any annoying questions, as we do this bit of tricky surgery".

Cheer up Schop. Take the long view. Either humanity will work out what it is about or your wish will be granted. You can wait 50 years surely?

Canopy succession is an example. Once a mighty oak has grown to fill a gap, it shades out the competition. So possibilities get removed. The mighty oak then itself becomes a stable context for a host of smaller stable niches. The crumbs off its feeding table are a rain of degrees of freedom that can be spent by the fleas that live on the fleas.

But if the oak gets knocked down in a storm or eaten away eventually by disease, that creates an opening for faster-footed weed species. We are back to a simpler immature ecosystem where the growth is dominated by the strong entropic gradient - the direct sunlight, the actual rainfall and raw nutrient in the soil.

The immature ecology doesn't support the same hierarchy of life able to live off "crumbs" - the weak gradients that highly specialised lifeforms can become adapted to. It doesn't have the same kind of symbiotic machinery which can trap and recycle nutrients, provide more of its own water, like the leaf litter and the forest humidity.

So the degrees of freedom are the system's entropy. It is the through-put spinning the wheels.

An immature ecology is dependent on standing under a gushing faucet of entropy. It needs direct sunlight and lots of raw material just happening to come its way. It feeds on this bounty messily, without much care for the long-term. Entropy flows through it in a way that leaves much of it undigested.

But a senescent ecology has build up the complexity that can internalise a great measure of control over its inputs. A tropical forest can depend on the sun. But it builds up a lot of machinery to recycle its nutrients. It fills every niche so that while the big trees grab the strongest available gradients, all the weak ones, the crumbs, get degraded too.

So the degrees of freedom refers both to the informational and entropic aspects of what is going on. Salthe is explicit about this in his infodynamic account of hierarchical complexity - the forerunner of what seems to have become biosemiosis.

The degrees of freedom fall out of the sky as a rain of entropy, an energy source that would welcome being guided towards some suitable sink. Life then provides that negentropic or informational structure. It becomes the organised path by which sunlight of about 6000 degrees is cooled to dull infra-red.

Then switching to that informational or negentropic side of the deal - the tale of life's dissipative structure - the degrees of freedom become the energy available to divert into orderly growth. It is the work that can be done to make adaptive changes if circumstances change.

A weed can sprout freely to fill a space. It is green and soft, not woody and hard. It remains full of choices in its short life.

An oak wins by trading that plasticity for more permanent structure. It grows as high and strong as it can. It invests in a lot of structure that is really just dead supporting wood.

So degrees of freedom have a double meaning here - which is all part of the infodynamic view of life as dissipative structure.

Life is spending nature's degrees of freedom in entropifying ambient energy gradients. And it spends its own degrees of freedom in terms of the work it can extract from that entropification - the growth choices that it can make in its ongoing efforts to optimise this entropy flow.

So there is the spending of degrees of freedom as in Boltzmann entropy production. Turning energy stores into waste heat. And also in terms of Shannon informational uncertainty. Making the choices that remove structural alternatives.

An immature system is quick, clever and clumsy. A senescent system is slow, wise and careful. An immature system spends energy freely and so always seems to have lots of choices available. A senescent system is economic with its energy spending, being optimised enough to be mostly in a mode of steady-state maintenance. And in knowing what it is about, it doesn't need to retain a youthful capacity to learn. Its degrees of freedom are already invested in what worked.

Which then brings us back to perturbations. Shit happens. The environment injects some unpredicted blast of entropy - a fresh rain of entropic degrees of freedom - into the system. The oak gets blown down and the weeds get their chance again.

a conservative ecology would be precisely a senescent one, one that, yes, acknowledges the need for 'community' and so on, but that doesn't valorize the changes that such community fosters (correlatively, a philosophy of individualism lies on the other side of the spectrum). The 'best' ecosystems are precisely those perched halfway between immaturity and senescene, insofar as they can accommodate change in the best way. — StreetlightX

Senescent is probably a bad word choice by Salthe as he means to stress that a climax ecology has become too well adapted to some particular set of environmental parameters. It has spent all its degrees of freedom to create a perfect fit, and so that makes it vulnerable to small steady fine-scale changes in those parameters outside its control - something like coral reefs collapsing as we cause changes in ocean temperature and acidity. Or else the perturbations can come from the other end of the scale - single epic events such as an asteroid strike or super-volcano.

So evolution drives an ecology to produce the most entropy possible. A senescent ecology is the fittest as it has built up so much internal complexity. It is a story of fleas, upon fleas, upon fleas. There are a hosts of specialists so that entropification is complete. Every crumb falling off the table is feeding someone. As an ecology, it is an intricate hierarchy of accumulated habit, interlocking negentropic structure. And then in being so wedded to its life, it becomes brittle. It loses the capacity to respond to the unpredictable - like those either very fine-grain progressive parameter changes or the out of the blue epic events.

So senescence isn't some sad decaying state. It is being so perfectly adapted to a set of parameters that a sudden breakdown becomes almost inevitable. Coz in nature, shit always happens. And then you struggle if you are stuck with some complicated set of habits and have lost too much youthful freedom to learn some new tricks.

One can easily draw economic and political parallels from this canonical lifecycle model. And it seems you want to make it so that conservatives equal the clapped out old farts, neoliberal individualists equal the reckless and immature, then the greeny/lefties are the good guys in the middle with the Goldilocks balance. They are the proper mature types, the grown-ups.

Well I'd sort of like to agree with that, but it is simplistic. :)

The left certainly values the co-operative aspect of the human ecosystem, while the greens value the spatiotemporal scope of its actions.

So conservatives certainly also value the co-operative aspects of society, but have a more rigid or institutionalised view. The rules have become fixed habits and so senescent even if a good fit to a current state. The left would instinctively take a sloppier approach as it would seem life is still changing and you need to still have a capacity for learning and adaptation in your structures of co-operativity.

Then conservatives also value the longer view of the parameters which constrain a social ecology. Like greens, they are concerned for the long-term view - one that includes the welfare of their great grand children, their estates, their livestock. But while greenies would be looking anxiously to a future of consequences, conservatives - in this caricature - are so set in their ways by a history of achieving a fit that the long-view is more of the past. It is what was traditionally right that still looks to point to any future.

But then conservatives may be the ones that don't rush into the future immaturely. The stability of their social mores may actually encode a long-run adaptiveness as the result of surviving past perturbations. Lefties and greenies can often seem the ones who are in the immature stage of being too eager for the turmoil of reforms, too quick to experiment in ways that are mostly going to pan-out as maladaptive, too much the promoters of a pluralist liberal individualism, too quick to throw history and hierarchy in the bin.

So as usual, the science of systems - of which ecology is a prime one - could really inform our political and economic thinking. It is the correct framework for making sense of humans as social creatures.

But once we start projecting the old dichotomous stereotypes - left vs right, liberal vs conservative - then that misses the fact a system is always a balancing act of those kinds of oppositional tensions.

And we also have to keep track of what is actually different about an ecology and a society. An ecology lacks any real anticipatory ability. It only reacts to what is happening right now as best it can, using either its store of developed habits to cope, or spending the capital of its reserve of degrees of freedom to develop the necessary habits.

But a human society can of course aspire to be anticipatory. It can model the future and plan accordingly. It can remember the past clearly enough to warn it of challenges it might have to face. It can change course so as to avoid perturbations that become predictable due to long-range planning.

And the jury is actually out on how an intelligent society ought to respond. On climate change, the conservatives were at first the ones worried about its potential to disrupt the story of human progress. Then the neoliberal attitude took over where the strategy for coping with the future is to rely on human ingenuity and adaptability.

One view says tone everything down as we are too near the limit. The other says if shit is always going to happen - if not global warming, then the next overdue super-volcano ice age - the imperative is to go faster, generate more degrees of freedom. The planet is just not ever going to be stable, so to flourish, planned immaturity beats planned senescence.

Both views makes sense. As does the further view that of course there must be a balance of these two extremes. The right path must be inbetween.

As far as your snark, it's not necessary. — fishfry

Give snark and you get snark. Fair enough?

Philosophically it's perfectly valid that you have alternative ideas about math and don't accept parts of standard modern math. But physics is most definitely based on standard modern math; and to the extent that your outlook diverges from that, you are introducing confusion into the conversation. — fishfry

Even in philosophy of maths, these are routine debates. And physics doesn't base itself on the "correctness" of maths. So you may feel confused by conversations that veer of the beaten textbook track, but there is no particular reason to think the secrets of reality are already written in those textbooks. The relationship between maths and physics is much more subtle than that.

So look, I am interested in what you have to say about standard views within mathematics. It is good that you can explain the structures of justification upon which certain mathematical positions are constructed. But if you just want to lecture on the correct methods of the academy - the stuff that will get you a pass in class - I don't really call that a conversation in the context of a philosophy forum where the thread is about the material reality of multiverses.

For the record I do understand that you have your own private notation and that you reject the standard notation of modern math, on which physics is based. Is that a fair assessment? — fishfry

Hardly. I'm quite happy to accept more than one way of doing anything. And happy also that the standard way will have proven itself to be the most pragmatic in finding the simplest path - the one that disposes of the most metaphysical baggage.

But when we are talking about how existence itself comes to be, as we are with multiverses, then that is when the axioms upon which a reductionist simplicity is founded come into metaphysical question. That is when we have to rewind and see what was being left out, or being assumed, when starting down that path.

But since we're doing physics, it's important to make sure we get the math right. — fishfry

Err, no. It is important to get the physics right and then find the right maths to make those ideas precise enough to measure.

I'm not denying that physics and maths enjoy a remarkably fruitful relation. But I do deny your understanding of how science works. The maths does not lead, it follows. And the fact that the maths then works as the precise description of the scientific claims merely shows that the maths has been suitably fitted to the task in hand. The maths was convenient.

But that's exactly wrong. You need a finite die with a very large number of faces. One for each admissible state of all the particles in some bounded region of space. — fishfry

So why did you introduce the blather about binary coin flips where you either got Earth or Mars in the course of some infinite sequence? That was the misstep I was addressing.

Of course I understood the ergodic argument that motivates the talk of repeating outcomes in a multiverse of infinite spatial extent. You were the one who had to go look it up.

And of course I understood the ridiculousness of the idea of a spherical die that could be said to "land on 1 of its infinite number of sides". It was your coin-flipping nonsense I was lampooning. Exactly which infinitesimal point are we suppose to say a sphere has landed?

Now you are talking about a finite sided die with enough degrees of freedom to represent every thermal microstate in a bounded space. As if this fixes either your blather about Earth/Mars coin-flipping sequences, or deals with my actual objections to the simplicities of the correlation-less ergodic view of nature.

Imagine your finite side die had dependencies between the faces so that all the Earth-like combos are clustered together on one neighbourhood of the die, not represented randomly across the die surface.

Perhaps now you can start to see the mathematical presumptions you are failing to take account of in attempting to construct some physical model of a "multiverse generator"? A fair die would scramble the microstates randomly. But a die more properly representing the material world, with its dependencies and entangled history, would already be built in a way that was pre-loaded. It would either fall somewhere in the neighbour where the many Earth-like outcomes were clustered - or mostly somewhere not anything like the Earth at all.

That should be blindingly obvious. Yet it's not because the idea of modelling probability spaces with correlations is not so "mainstream".

But I have zero expectation you will pick up on such a point and argue it through. You are way too intent on finding an ever higher horse from which to look down your nose at any "non-mathematician".

Why don't we agree not to interact? I was really surprised earlier that you directly replied to something I said. I don't think our interactions are productive. — fishfry

I'm amused and entertained. I'm finding the dispute productive. What more could I wish for? I'm not complaining.

As it happens I've spent the afternoon chasing down ergodicity. — fishfry

Hmm. "Ergodic" in this context is really talking about Poincare reccurence -
https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem

In an ideal gas, the particles will recross their initial conditions in the long run.

But my counter has been that the universe is more like a box of gas with its lid off. The particles are all escaping as the Universe is expanding/cooling. The locations they might want to recross may well have moved over an event horizon forever.

And then even worse, the gas is far from ideal. All the particles have correlations or dependencies. At the least, gravity and other forces are in play. So a different maths is needed to make any multiverse extrapolations.

Think more of a chaotic attractor where the trajectories could go anywhere within the bounds of the phase space, yet also cluster ... because of the internal dependencies. The system is not random but emergently self-constrained - https://en.wikipedia.org/wiki/Attractor

There is no such thing as a probability of 1/infinity. — fishfry

You do get your knickers in a twist with great rapidity. As you know, that was Michael's terminology. I went along with it for the sake of discussion.

But also, 1/infinity is the proper definition of the infinitesimal as far as I'm concerned. Now you will get on your high horse and object no doubt. But I went along with Michael's terminology largely because I also like that sly implication. It is another way of getting across that the probability ain't actually zero even if it is almost surely zero when it comes to an infinite spatial universe producing replica earths with replica people doing replica things.

When people are talking about an infinitely-sided die, I assume they mean a countable infinity, which has no uniform probability distribution. — fishfry

If you remember, it was you who introduced the confusion. I was trying to sort it out for you by pointing out that those would be the kind "coins" you would need to be flipping...

If there are two states and infinitely many universes they could be 0, 1, 1, 1, 1, 1, ...

If 0 is the "earth" state, there is no other earth. 1 is maybe Mars. So Mars exists infinitely many times but not earth. If there are a trillion states, same argument. SOME state recurs infinitely many times, but not necessarily any particular state. Maybe there's only one earth even though there are infinitely many copies of Mars. It's perfectly possible. — fishfry

This still reads as nonsense to me. Maybe you agree now as you seem to have discovered ergodicity and moved on to a notion of a universe chopped up into sufficiently large but finite regions - the ensemble of microstates picture that I also have been at pains to criticise.

Nonsense? I think it really gets us to the heart of some really telling confusion.

This is a philosophy discussion group. When things seem unarguably right, that's when you know there must be the whole flip-side to the story. Dialectics always rules. So someone's nonsense is always the start of someone else's sense.

Having an infinite number of every planet is no more likely than having an infinite number of just one planet. — Michael

If the odds of earth being the case on any one roll are 1 in infinity, then the odds of earth being the case every time in an infinite series of rolls are 1 in an infinity of infinities.

If you just reduce this to a consideration of the statistics of a one off event - as is the case with the multiverse argument - then you must take a propensity view of the statistics. We should presume typicality for the outcome.

We are talking about a known outcome - the visible universe and the variety of planets if produces. We know what typicality looks like to a reasonable degree. You get gas giants, you get Mercuries, you get Earth-like planets. We are busy counting that variety around other stars now.

We also then have at least some grasp on the propensity for intelligent life arising on other planets. And we can keep tacking on propensity estimates for the history of an earth repeating such that it produces humans who are exactly, atom-for-atom, thought-for-though, like you and me, doing replica actions for as long into the future as this multiverse calculation requires. (Is it still a real multiverse if all the other replica Earths do a Boltzmann Brain disappearing or disintegrating act in the next split second? How long must that exact continuity of a history persist?)

So the actual situation for a multiverse "just one throw of a die" propensity calculations is that being "Earth-like" in the vague way astrobiologists have in mind is reasonably typical. There are many ways to be Earth-like. So it happens a lot. Even inside our visible universe.

Then harbouring Earth-like life is way less typical. How typical the biology of the Earth is - as an outcome of the physics and constraints of the universe - is an open question. Recent work - like Nick Lane's The Vital Question - is arguing that the ways life can biochemically develop are surprisingly limited. So the odds of Earth-like biology now seem much higher - if life develops on other planets at all.

Then we have the question of the typicality of a re-run of the Homo sapiens story down to the level of historic accident that produces you, me, and the rest of PF. The level of accident, the level of information discarded, argues for some extremely low propensity. I would say "almost never". Or probability = 0.

You thus run into a collision between two notions of the infinite. The combinatorial one says every possible combination simply gets realised. The constraints based one says the steady shrinking towards an infinite unlikelihood means you are headed towards almost never, a probability that is actually zero (fine print: for all practical purposes). That is, the principle of indifference kicks in to allow the state of infinite constraint to be achieved.

One notion of infinity operates on an already closed and bounded set. The other has to achieve that claimed closure.

So as I tried to point out, the simple minded combinatorial notion of infinity employed to motivate multiverse arguments is itself in question. It depends on the assumption of a bounded space with no internal correlations. That gives you one picture of what "infinity" means.

And then the more appropriate notion is a constraints-based infinity where the correlations get counted too. Restrictions on what is typical arise due to histories. Material accidents and formal necessities go into making those histories. The story in irreducibly complex and non-linear.

The propensities of Earth-like biology might be much higher than naive combinatrics would predict, if we buy Nick Lane's arguments about the biophysical constraints on life forming. But then the propensity of Homo sapien history being exactly repeated to the point of producing doppleganger you and me, is way less than naive combinatrics would predict.

But even if we put aside the difference between a combinatorial statistics and a constraints-based, negentropy-including, one, you are still only dealing with a one-off propensity story with the multiverse argument. Unless you can motivate the further idea of a multiverse of multiverses, we are only talking about a one time "roll of the dice" so far as there was a Big Bang that produce an infinite amount of spatial regions, all with the same propensity for star and planet formation.

The typicality is wired in just by observation. We have a sample size of the solar system, and now the solar systems of other stars. Already that is a constraint we can't just ignore (as good Bayesians).

But the very description of the die to be tossed - this spherical die with its infinity of marked faces - demands we assign a Bayesian propensity to the typicality of its outcomes. We already must "know" that it is going to generate the statistically typical, not the statistically infinitely unlikely. The only way we could think different is if we imagine an infinity of infinite throws. Then our propensity switches to thinking it almost sure that the 1 in infinity outcome would be among all the combinations that happen. Now, it couldn't not ... for all practical purposes.

To sum up, the multiverse scenario I was addressing only permits a one-shot propensity view. It was about the likelihoods within a single infinite Big Bang space.

Then a simple combinatrics view - the one that counts only entropy or degrees of freedom - would give you a naive number for how many times some exact combination of atoms and thoughts could appear in just such an infinite space.

But I argued that simple combinatrics is simply going to be wrong. A realistic calculation of the odds has to include correlations and the emergent constraints on combinations that will result.

If that propensity calculation could be done correctly, my gut feeling is that the probability of that propensity would shrink towards zero, or almost surely not, even with the infinity of a multiverse to play with. The magic of infinity would lose its power to conjure up every possibility.

The big "if" is doing that particular calculation. But I think Scott Aaronson provides some of the right conceptual tools here - https://www.scottaaronson.com/papers/npcomplete.pdf

You are just ignoring the fact that your scenario demands an infinite creation of multiverses. That is different from figuring the odds of repeated configurations within just the one multiverse with “a fair die”.

There is nothing to motivate your assumption that the one multiverse would be so atypical. A multiverse with a die that produced your selective outcome could not be believed to be random. There would be no proper basis for such a presumption. It would be mad not to believe the die was loaded.

So nice try, but no dice.

Oh, and to shoehorn in a point of politics, it might be argued, on the basis of the above, that philosophies of rugged individualism are thus philosophies of ecological infantalism, or else ecological sickness. — StreetlightX

If you are interested in the best account of this, try Stan Salthe’s story on the immature-mature-senescent arc of living systems. And it would account for social systems as well.

But your hope to tie rugged individualism to sick or infantile ecology is lefty nonsense.

A senescent ecology is just one so well adapted to a particular life that it becomes brittle, lacking in degrees of freedom to recover from perturbations.

An immature one by contrast can exuberantly spend degrees of freedom to recover from knockbacks, yet is rather wasteful in being not yet well adapted to some particular life.

You can cash that out in sociological and political terms. But not so crudely as you seem to want to suggest.

Jeez, you were serious?

The odds of landing on the face marked Earth might be 1/∞. The odds of landing on the face marked Earth an infinite number of times in a row is another matter. It could only be 1/∞ in relation to an infinite ensemble of multiverse creations. So in a multiverse of multiverses, you would almost surely get your one multiverse in which every planet wound up being replica Earth, faithful down to us speaking in Korean about flower arranging, or whatever other modal possibility we could imagine.

I agree that presuming infinity entails any madness you care to suggest. But first, you would have to motivate this new tale of yours about multiverses of multiverses where a random process can then turn out its most unlikely possible result with certainty.

It might not. It might land on one face an infinite number of times. It might have been Mars all the way down. — Michael

That would be helluva loaded dice. Get you banned from the cosmic casino for sure. It just wouldn’t fit the description of being random.

And if every planet in the universe could have been replica Mars, then we would be Mars. Venus and Saturn would be Mars. So at least we can rule your hypothesis out observationally.

Each of your regions would contain about 10^120 degrees of freedom. That would be the entropy content of a Hubble radius. So, naively, we would be talking about the chances of one configuration of that magnitude being exactly repeated.

The odds against it are vast, yet finite. So granting an infinite array of such volumes, the configuration would repeat just by accident. Indeed it would occur an unlimited number of times.

Of course, a replica earth with a replica you and me would seem to require even more information to specify it. But even if we toss in another million orders of magnitude to allow for a more negentropic story - one that includes all the information discarded in the course of some evolutionary history - infinity will still ensure that all possible arrangements of finite regions with finite contents must repeat their configurations. It is just a consequence of the atomistic description of the set up.

I’m not a fan of this ergodic analysis. As I’ve indicated, I think it fails to take into account the negentropy of information discarded through interactions. It presumes every degree of freedom is independent. Yet in the real world, particles have dependencies. They interact. And that leads to non linearities that evolve in exponential time rather than polynomial time. In short, the complexity of an actual Hubble volume would grow at a rate that the simple statistical picture cannot even follow.

But if we do treat a Hubble volume as if it is just filled by an ideal gas, then ergodic simplicity applies. The multiverse argument carries if the ontology is atomistic.

Imagine the universe when it just was a gas of radiation, or even a dust of weakly interacting particles. You could go into deep space right now and sample cubic metres of vacuum. Each will average about a dozen hydrogen atoms. Even within our own visible universe, you would think you would get exact repeats - the same configuration of a cubic metre of vacuum.

So the multiverse argument is very straight forward in itself. The flaw would be in its assumptions. Like the willingness to discount interactions and the fact they may screw up any simple statistical extrapolations with their non-polynomial complexity.

If there are two states and infinitely many universes they could be 0, 1, 1, 1, 1, 1, ... — fishfry

Yeah. But you get to pick these infinite sequences out of an infinite hat. So you would pick that exact sequence an infinite number of times.

If 0 is the "earth" state, there is no other earth. 1 is maybe Mars. So Mars exists infinitely many times but not earth. — fishfry

But now you are changing the rules of your own game. Instead of Earth = 0, not-Earth = 1, you are saying reality only has the two options of Earth or Mars. And for some reason, nothing else will get pulled out of the hat.

Furthermore, 0, 1, 1, 1, 1, 1, ... has the same probability as 1, 0, 0, 0, 0, ... . So those exact sequences are equally improbable. The typical sequence will be more like 1, 0, 0, 1, 1, 0, 1, ... And what interpretation are you assigning to that in terms of physical outcomes?

Maybe there's only one earth even though there are infinitely many copies of Mars. It's perfectly possible. — fishfry

Your maths doesn't give that result as I've argued. You are trying to hardwire in the restriction that Earth or Mars is the binary choice that reality is having to toss a coin on. But this is about a coin with an infinity of faces - one for every possible state of the world. And it gets toss an infinity of times, so lands on every one of those faces an infinite number of times.

The multiverse is pure madness in other words. And maybe that ought to give pause to any Cantorians round about these parts. Maybe the maths version of infinity is not that robust either? Heh, heh.

it has never been a really alien concept that there are undetectable portions of our universe. — noAxioms

No. That was indeed necessary to explain how a Big Bang universe could be so remarkably thermally homogenous. And before that, just to resolve Olbers paradox.

Are those places other universes? Not like they're discreet with boundaries where one stops and the next starts. — noAxioms

I agree they are not other universes. And even if our known Big Bang universe with its light cone structure were spatially infinite, then I still think important constraints on the "modal realist" version of the multiverse will count.

So spatial infinity would seem to guarantee that there should be an infinity of Earths where you and me are having this exact discussion - plus every other even faintly similar or utterly different interactions. We could be discussing hair-do's, speaking in Korean, typing random sequences. And the fact any of those might be the case would mean that all those varieties of cloned Earths would have to be infinite in number themselves. There would be an infinite number of replica planets with us speaking Korean, etc.

There just is no end to the madness once you let actual infinity run riot in your ontology.

MWI suffers this because it can't in fact define what causes a branch universe to form. It tries to confine the branching to stuff like simple spin-up/spin-down entanglements. But that is way too ad hoc as it stands. Every photon emission in history could just as well have landed up being absorbed anywhere in the future eternal visible universe. Try extracting the decohered thermal signal from that.

Anyway, even in a spatially infinite universe, we would presume that it all expands and cools in the same way. And cooling steadily - or in fact, exponentially - removes material possibilities. If every portion of the universe is losing energy density at a shared rate, that means there is only a tiny time window for replica earths to actually form.

So the Universe might have infinite space to play with, but a very finite amount of time. Now the madness of infinity means there will still be an infinite number of worlds where we are saying all this in Korean, etc, but infinitely less than there could have been because of a strict time constraint.

If nothing else, there will be infinitely more space between each of these supposed exact replicas. You would have to travel infinitely further to get that kind of big surprise.

Well perhaps not infinitely, literally. But near enough FAPP. :)

Well, only if the space was smaller than the hubble-sphere — noAxioms

Agreed. But any kind of topology could be screened off in that fashion with a sufficiently small hubble radius.

Anyway, I see now that SophistiCat rather confused things by talking about a toroid rather than spherical geometry. The simplest compact space that would make the point about the Universe being "finitely infinite" would be a sphere.

The story would then not change no matter how big your hubble factor. Or at least not until all that escaping light came back at you from the opposite direction. :)

I play asteroids in a flat 2-torus space, not on the surface of a donut embedded in three-space. — noAxioms

Apologies. I thought your mention of a torus was a typo. Didn't realise it had been introduced into the thread. If we lived in a 3-torus, we would be able to detect that global alignment as you say. We could get places quicker, with less energy, by going in one direction rather than another.