Don’t forget dark energy. You haven’t budgeted for that. — apokrisis

I see! Much obliged.

I should have written 'the map = territory fallacy" by which I mean idealists tendency for confusing – conflating – epistemology (i.e. what I/we know) & ontology (i.e. what there is), that is, there is not anything more than what I/we can 'experience'.

it is just that there are maps all the way down. There is no territory.
Always the Hegelian. That's the fallacy / incoherence of idealism I mean. — 180 Proof

I feel there is a kind of conundrum here. You are right, my expression leads to problems. If there are maps all the way down there is no way to tell whether one map is more accurate than another. ontology collapses into aesthetics. It is impossible to intelligibly uphold that view.

On the other hand assuming there is a territory requires a leap of faith and the assumption of an archimedic point which ultimately leads to some sort of foundationalism. Every foundation leads to problems because there is no way it reveals itself. Claims lead to counter claims. Ontology collapses into metaphysics. Maybe I should read Levinas...

The Lorentz strange attractor caused excitement as a model for that reason. — apokrisis

Most of those systems iterate a single complex function. My approach has been infinite compositions of differing functions, producing imagery like the one I use for my icon on TPF.

It seems to me what we have discussed here regarding math and chaos is that along one line of thought math describes chaos and along another line of thought it creates chaos.

You have mentioned symmetry breaking several times in posts. I know practically nothing of it, but it seems to somewhat parallel the fundamental notion of chaos theory, sensitive dependence on initial conditions. How do you perceive it? Does it resonate in metaphysics?

On the other hand assuming there is a territory requires a leap of faith and the assumption of an archimedic point which ultimately leads to some sort of foundationalism. Every foundation leads to problems because there is no way it reveals itself. — Tobias

Pragmatism simply says we take that leap of faith - form a hypothesis - and test it. Our opinions of what is foundational then emerge from that process of engagement.

Furthermore, we know that this process of reasoned enquiry is forming our subjective self as much as it is forming our opinion of what constitutes the outside world.

This is the feature, not the bug, of map-making. Contra idealism, we - as subjects - don't exist beyond the pragmatic modelling relation we form with the world. I am me in terms of the habitual view I come to take of the world with which I interact. My semiotic Unwelt is a running model of "me" in the "world".

So the subjective is entangled in the objective when it comes to reasoned inquiry. And that is a good thing. It is how I as an ego, with will, purpose and creativity, exist along with the worldview I am productively constructing.

Epistemology somehow got hung up on Cartesian doubt. It divided folk into naive realists and mystically-inclined idealists.

Peircean pragmatism is based on a sounder psychology. We engage with reality on the basis of revisable belief. And our own subjectivity is a product of that constructive engagement. Our choices about what are the "right" ways to frame reality emerge from a debate starting at that point.

Most of those systems iterate a single complex function. — jgill

Natural growth processes are crudely modelled by iterative functions in that the functions build on their own history of accidents. Some arbitrary set of initial values is plugged into the equation and some larger pattern may emerge. It is all completely exact and formal - apart from the fact that some human has plucked the starting conditions out of the air and then run an eye over the results and found the output "exciting" for some reason. That part is completely informal - outside it being woven into a system of scientific modelling.

As with cellular automata, the mathematician sees a pattern emerge from the algorithm and finds it striking because it is a pretty pattern. Maybe even a suggestive pattern. Possibly even what looks like a pair of butterfly wings that might seem to stand as a good model of bistability in a natural system.

It all gets very exciting - a la Wolfram - because it seems to say that (heuristically tuned by some fiddling about to find the lucky equations), maths is showing how simplicity produces "lifelike" natural complexity.

But it is then so easy to skip over the many steps needed to start using these sparkling new toys as actual scientific models. I had close experience with this when I was involved in the debates over how to apply "chaos" models to neuroscience in the 1990s. It was disappointment with the ratio of hype to insight that pushed me onwards to hierarchy theory, biosemiosis and the larger story of dissipative systems.

In short, the problem with deterministic chaos and other "exact" algorithmic approaches is that the formality gets abused by the informality of their interpretation. Pretty patterns get cherry picked. Worse still, the fact that complexity appears to emerge "magically" in supervenient fashion becomes weaponised by reductionist metaphysics. It is used to confirm old atomistic prejudices about how the world "really works".

But algorithmic complexity is merely mechanical complication, not true organic complexity. It is all bottom-up construction and lacks top-down evolving constraints. It exists in a frictionless and sterile world that has no final cause, even in the most basic form of a thermodynamic imperative.

So when I talk about chaos in the natural world sense, I am indeed not thinking it starts and stops in the reductionist trinkets generated by iterative functions. I am clear that these toy systems offer useful tools and arguments. But their shortcomings are just as visible.

You have mentioned symmetry breaking several times in posts. I know practically nothing of it, but it seems to somewhat parallel the fundamental notion of chaos theory, sensitive dependence on initial conditions. How do you perceive it? Does it resonate in metaphysics? — jgill

Symmetry breaking is a huge subject - especially as I've spent the past year really trying to figure out my own view of how it all holds together from a systems science or holist perspective.

But a short answer on this specific question is that spontaneous symmetry breaking has sensitivity because what we are saying is that a system is so poised that absolutely any perturbation would tip its state.

Take the usual examples of a pencil balanced on its point, or Newton's dome with a ball perfectly balanced on the apex of a frictionless hemisphere. The pencil and ball are objects in a state of symmetry, being at rest with no net force acting on them, so they should never move. But then we also know that the slightest fluctuation - a waft of air, the thermal jiggle of their own vibration, even some kind of quantum tunnelling – will be enough to start to tip them. The symmetry will be broken and gravity will start to accelerate them in some "randomly chosen" direction.

So metaphysically, this is quite complex. Some history of constraints has to drive the system to the point that it is in a state of poised perfection. The symmetry has to be created. And that then puts it in a position where it is vulnerable to the least push, that might come from anywhere. The sensitivity is created too. The poised system is both perfectly balanced and perfectly tippable as a result. The situation has been engineered so randomness at the smallest scale - an infinitesimal scale - is still enough to do the necessary.

All this is relevant to the OP - as the Big Bang is explained in terms of spontaneous symmetry breaking. And thus the conventional models have exactly this flaw where the existence of the "perfect balance" - a state of poised nothingness - is just conjured up in hand-waving fashion. And then a "first cause" is also conjured up in the form of "a quantum fluctuation". Some material act - an "environmental push" - tips the balance, as it inevitably must, as even the most infinitesimal and unintentional fluctuation is going to be enough to do the job of "spontaneous" symmetry breaking.

The sensitive dependence on initial conditions is unbounded - and hence becomes helpfully something we don't even need to be talking about when pondering the "cause" of the Big Bang. A fluctuation was surely there at the beginning to tip the inflaton field down its potential well, or whatever. But as an efficient cause, it becomes the most minor and random of events. Any other fluctuation would do just as well as the nudge that set things rolling.

Anyway, again we have the "exact mathematical models" that are indeed used very productively to model the creation of the Universe. One can write the various differential equations that generate some particular inflaton potential to explore. The Higgs, the dilaton, the self-interacting, the massive scalar. The pencil is poised. It must surely tip. We can generate a bunch of theoretical patterns and argue about how closely the observables match the latest CMB data.

But from a metaphysics point of view, there is so much to add about what is going on behind the models - the assumptions that have to be built in as their motivation.

Such as how does nature arrive at a generalised state of critical instability - a pencil balanced on its point? And how does that relate to the fact that nature also needs some unintentional fluctuation - even if it is infinitesimal - to start the game going at some actual point in time.

This is where I bring in the contrary view where fluctuation is unbounded and symmetry states emerge as the constraint of fluctuation to some infinitesimal (Planckian) grain. You start with the absoluteness of an everythingness - chaotic or scalefree fluctuation. And then a state of global order crystallises as a generalised constraint of fluctuation to some single universal scale (the scale scaled by the three Planck constants).

So you wind up with a quantum vacuum - a thermal equilibrium state where everything might be fluctuating, but all the fluctuation is compressed to a minimal effective scale. The vacuum as a whole is decohered to a state of simple looking classical Lorentzian symmetry.

A ground has been created. And that becomes in turn its own next level of order-production in the form of the gauge excitations – the standard model particle content - that are the further "symmetry breakings" of the Cosmos as an expanding~cooling heat sink structure.

So metaphysically, this is quite complex. Some history of constraints has to drive the system to the point that it is in a state of poised perfection — apokrisis

You do write the most interesting posts. This is related to catastrophe theory as well. A frightened guard dog teeters on the edge of attack or retreat, like a pencil balanced on its point, or a married couple at the edges of each other's nerves, the slightest provocation and a serious collapse of wave functions.

As with cellular automata, the mathematician sees a pattern emerge from the algorithm and finds it striking because it is a pretty pattern. Maybe even a suggestive pattern. Possibly even what looks like a pair of butterfly wings that might seem to stand as a good model of bistability in a natural system. It all gets very exciting - a la Wolfram — apokrisis

He had such high hopes with what he considered a new science. Like almost everyone who attempted to read his massive book, I gave up after a few hundred pages.

But it is then so easy to skip over the many steps needed to start using these sparkling new toys as actual scientific models — apokrisis

I have always enjoyed mathematics as a completely abstract playground, never having illusions, nor wishes, that my modest research would have applications. A method of accelerating convergence of certain function expansions as continued fractions years ago, that may have had use in computations in QM. And more recently, a surprising application of convergence of infinite compositions in a paper on decision making within groups. But all the rest unicorns in a velvet sky.

I've wondered whether fixed points (attracting, repelling, indifferent) have any metaphysical properties. Stanislaw lem's ergodic theory of history presents a counterpoint to the butterfly effect in Chaos theory: certain social movements are so strong that minor fluctuations have little to no effect on large scale outcomes.

You do write the most interesting posts. This is related to catastrophe theory as well. — jgill

Thanks. Catastrophe theory was both one of my earliest intellectual thrills and disappointments. It seemed to promise so much and yet deliver so little. It had little practical application and just stood as a signpost to the realisation that nonlinearity is more generic than linearity in nature.

I've wondered whether fixed points (attracting, repelling, indifferent) have any metaphysical properties. Stanislaw lem's ergodic theory of history presents a counterpoint to the butterfly effect in Chaos theory: certain social movements are so strong that minor fluctuations have little to no effect on large scale outcomes. — jgill

My systems science approach is predicated on global constraints that produce local stability. So fixed points emerge due to top-down acting constraints on possibility.

The tricky bit is then that the local degrees of freedom thus created have to be of the right kind to rebuild the whole that is creating them. It is a cybernetic loop where the system maintains its structure in a positive feedback fashion.

So fixed points are important as the emergently stable invariances of a physical system. The symmetries that anchor the structure of the self-reconstituting whole.

This is the guts of physical theory. Lorentz symmetry gives you the “fixed point” behaviour of spacetime, and the Standard Model gauge group gives you the invariances which in turn define the “inner space” structure of particle interactions.

My systems science approach is predicated on global constraints that produce local stability. [ ... ] So fixed points are important as the emergently stable invariances of a physical system. The symmetries that anchor the structure of the self-reconstituting whole.

This is the guts of physical theory. — apokrisis

:fire: :up:

Take the usual examples of a pencil balanced on its point, or Newton's dome with a ball perfectly balanced on the apex of a frictionless hemisphere. The pencil and ball are objects in a state of symmetry, being at rest with no net force acting on them, so they should never move. But then we also know that the slightest fluctuation - a waft of air, the thermal jiggle of their own vibration, even some kind of quantum tunnelling – will be enough to start to tip them. The symmetry will be broken and gravity will start to accelerate them in some "randomly chosen" direction. — apokrisis

Such symmetry is pure fiction, a useful principle employed by mathematicians with no corresponding reality in the world. The type of things that you cite, which would break the symmetry, would never allow such a symmetry in the first place.

So metaphysically, this is quite complex. Some history of constraints has to drive the system to the point that it is in a state of poised perfection. The symmetry has to be created. And that then puts it in a position where it is vulnerable to the least push, that might come from anywhere. The sensitivity is created too. The poised system is both perfectly balanced and perfectly tippable as a result. The situation has been engineered so randomness at the smallest scale - an infinitesimal scale - is still enough to do the necessary.

All this is relevant to the OP - as the Big Bang is explained in terms of spontaneous symmetry breaking. And thus the conventional models have exactly this flaw where the existence of the "perfect balance" - a state of poised nothingness - is just conjured up in hand-waving fashion. And then a "first cause" is also conjured up in the form of "a quantum fluctuation". Some material act - an "environmental push" - tips the balance, as it inevitably must, as even the most infinitesimal and unintentional fluctuation is going to be enough to do the job of "spontaneous" symmetry breaking. — apokrisis

So this entire proposal of "spontaneous symmetry breaking" for the creation of the universe is pure "hand-waving" nonsense in the first place. The proposed "state of poised perfection" is an impossible ideal, which could have no corresponding reality, and the entire proposal is a non-starter. This is just an attempt to validate platonic realism, by placing a mathematical ideal, "symmetry", as prior to the physical universe.

My systems science approach is predicated on global constraints that produce local stability. So fixed points emerge due to top-down acting constraints on possibility. — apokrisis

In a social setting suppose a very famous and compelling person, say, "William", attracts followers. "Jack" wants to be close to William, but must push aside others surrounding his target. He finally gets very close and stays there, inching forward. William, who remains himself in all of this, is like an attractive fixed point. Now, suppose William, always comfortable with himself, begins to change, but still attracts many if they come at him from a certain direction, but if they come towards him from his bad side he repels them dramatically. Now William has become an indifferent fixed point. Finally, suppose William becomes sullen and angry to all save himself, so that his closest friends flee from him. He has become a repelling fixed point.

However, suppose William is a leader, like a Czar born into his position. He is psychotic and teeters on the edge at all times. If he is provoked in the slightest way, he will explode, taking down friends and enemies alike, his symmetry having been broken.

My systems science approach is predicated on global constraints that produce local stability. So fixed points emerge due to top-down acting constraints on possibility.

The tricky bit is then that the local degrees of freedom thus created have to be of the right kind to rebuild the whole that is creating them. It is a cybernetic loop where the system maintains its structure in a positive feedback fashion.

So fixed points are important as the emergently stable invariances of a physical system. The symmetries that anchor the structure of the self-reconstituting whole. — apokrisis

This is a mathematically formal approach that seeks to find data points of a model of a physical path?
Or does the model apply to a swirling (nominally static) 'hurricane' at the center of a great storm moving over the landscape as dictated by low air pressure sheared by troughs transferring heat from the tropics to moderate zones?
To me, the difference is that the first is a platonic model, the second is a Heraclitean process.

So fixed points are important as the emergently stable invariances of a physical system. The symmetries that anchor the structure of the self-reconstituting whole — apokrisis

An example would help. Intriguing.

For example, the fixed point behaviour that anchors renormalisation in quantum field theory.

RE: The Ultimate Question of Metaphysics
SUBTOPIC: QFT Normalization?
※→ apokrisis, et al,

Hummm ... normalization in quantum field theory (QFT)?

I am not sure I understand your application of this idea of normalization relative to “metaphysics.” When we speak of “normalization” • and • “renormalization” in contemporary times (relative to QFT), it is my understanding that we are referring to Wavefunction Properties, Normalization, and Expectation Values. Vector space of potentially infinite dimensions is implied. But QFT and (say for instance) consciousness are NOT inexplicably linked. QM cannot make thoughts and dreams a “reality” absent other external influences. And in the simplest way of looking at Metaphysics, you are studying “reality.” We are looking at a “Hilbert Space.” But it is very difficult to apply since we do not have a system of wavefunctions that describe a Metaphysical system While QM and the basic systems of Geometries (Euclidean, hyperbolic and elliptical) can help in the detection and experimentation of questions pertaining to “reality” (Metaphysics), the solution to any question of “reality” is not ensnared by the various systems of symmetry and constraints implied by the QFT.


Most Respectfully,
R
Tuesday, July 26, 2022

the fixed point behaviour that anchors renormalisation in quantum field theory — apokrisis

Beyond my pay grade. :smile:

My mammalian brain still asks where did it come from ? And from this I just realise that this question will never really have a satisfactory answer.

Thoughts ? — Deus

I also don't think there's a satisfactory answer. Assuming the brain has evolved to solve problems (or maybe just the single problem of replication), it's tempting to understand looking for causes in terms of looking for levers and buttons. I want to exploit what I perceive as causal relationships. From this POV, it's tempting to say that 'real' why questions link determinate items to determinate items, all within the causal nexus of reality understood as a web of such items. To ask why there is a causal nexus in the first place is to ask for something within that nexus to explain it, which doesn't make sense to me.

4. {{ }, 0}. This set is a valid set. — Agent Smith

Typically $0 = \{\}$, so the set above might not be valid. That depends on whether zero is assigned to the empty set and whether it's OK to repeat elements when specifying sets (it is essentially harmless, if confusing.)

:up:

0 = { }

True, { } is the union (additive) identity (just like 0) for sets. However,, I question the validity of the claim 0 = { }. Perhaps you could explain but do keep it simple, I no mathematician.

Perhaps you could explain but do keep it simple, I no mathematician. — Agent Smith

It's just that the two most popular constructions of the counting numbers from set theory use the empty set as zero. The best is maybe this one (Zermelo ordinals) : https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

There's a somewhat spectacular journey from the natural numbers all the way to functions defined on the real numbers, and everything is built from/on the 'nothingness' of the empty set. There's nothing 'inside' (typical versions of) set theory at all. Real analysis (and everything else) is just ripples in the nothingness.

:ok:

I recall, vaguely, that it all begins with $\phi$.

Ex nihilo nihil fit Creatio ex nihilo

How can nothing be something?! — Greeks

:up:

I recall, vaguely, that it all begins with ϕ. — Agent Smith

Yes, it's all built of $\emptyset$. I also like thinking of it as the bubbleverse. The empty set is an empty bubble. All the other bubbles are just bubbles containing bubbles. From this point of view, the bracket notation is best. I think you can get away (theoretically if not practically) with the finite alphabet of { and } and , (the brackets and a comma, but maybe just a single bracket-- not sure). So 0 = {}, and 1 = {{}} and 2 = {{},{{}}} , and so on... It's even better to draw circles containing circles, but that's not practical here.

0 = n({ })
1 = n({{ }})
.
.
.

That's how I understand it.

Too...

Barring infinity, what's the solution set to the equation x = x + 1?

Not {0} but rather { }.

What's nothing?

Is 0 = { }?

Cipher (0) isn't a set, it is a pattern in sets?

What sayest thou?

What sayest thou? — Agent Smith

Typically, the concept of 0 is just identified with the empty set. So you don't need your "n" as a kind of function.

It should be mentioned that Benacerraf uses the variety of ways of constructing the counting numbers from sets as an argument against platonism and for structuralism.

I sympathize with the intuition that the counting numbers are the deepest and truest thing in math (maybe with Euclidean geometry also.)

Barring infinity, what's the solution set to the equation x = x + 1? — Agent Smith

You basically need to situate that equation within a number system. Consider that $x^2 + 1 = 0$ has a solution in the complex number system but not in the real number system. There are also strange number systems like finite fields. Or "+" could represent the operation of a group. From this POV, infinity is only a solution if this infinity is carefully defined in a systemic context.

The broader point is that serious mathematics tames intuition, beats it into a kind of Chess with strict rules.

Cool!

Mathematics, though proven as a tool for uncovering truths, is itself not/only partially about truths (truths to be understood in the conventional sense).

Mathematics, though proven as a tool for uncovering truths, is itself not/only partially about truths (truths to be understood in the conventional sense). — Agent Smith

I agree with the point I think you are making. We can think of math as an excellent syntax for expressing truths about our world. One of its best features is how quickly we make inferences within this syntax. It's also lean and efficient. Take a messy scatterplot and transform it into two parameters and a measure of fit. That's some juicy, concentrated info. Like bears eating salmon brains.

:up:

I recall, vaguely, that it all begins with ϕϕ.

Ex nihilo nihil fit Creatio ex nihilo

How can nothing be something?!
— Greeks — Agent Smith

Think of it as pure potential. Zero, or empty set. is nothing, but it is a type of nothing, or nothing of a specific type of thing. If we proceed to say that the specified type is every type, so that it is nothing of any type of thing, then "every type" is a type. And if types are things, (Platonism), then nothing is something.

Zero, or empty set. is nothing, but it is a type of nothing, or nothing of a specific type of thing. If we proceed to say that the specified type is every type, so that it is nothing of any type of thing, then "every type" is a type. And if types are things, (Platonism), then nothing is something. — Metaphysician Undercover

I like to make sure statements like this are enshrined on the forum. :cool: