Hopefully in the next few decades, the tide will turn, and philosophers and scientists will come to their senses and accept the fact that we have internal conscious experineces that are different and distinct from neurologica activity. — George Cobau

But the general scientific position would be that the mind part of the equation is broadly some kind informational process. So for a long time, there has been a standard physicalist dualism which treats the mind as software running on hardware.

Now a lot more can be said about how to understand that. But the rival naturalistic metaphysics you have to argue against is the one that divides nature into matter and symbol. Folk feel pretty certain that the brain is some kind of information processing device.

This does make more sense than dividing the world into two disconnected kinds of substance - one that is inert matter, the other which is some kind of perceiving "soul stuff" ... that has no conceivable natural structure or laws, only the usual supernatural kind of existence.

Neuroscience knows for sure the brain does some kind of information processing. The structure of that has been mapped in laborious detail. That is what neuroscience does.

So on what grounds do you challenge a naturalism, a physicalism, which already recognises the further "surprise" of the possibility of information processing as part of what is nature?

Both sides could still be acting in good faith, but the structure of exchange they've fallen into is not what they think it is. — Srap Tasmaner

Putting aside the merely rhetorical tactics that folk use, I'd say there is a largely unrecognised issue of logical structure in play.

And that is that the laws of thought that work for arguing over particulars are different from the dialectical laws for arguing over the metaphysically general.

So the laws of thought require an answer to be right or wrong in the fashion of the law of the excluded middle. Everyone understands that there has to be a winner and a loser, a truth and a falsity, when it comes to a question over some particular individual fact. The LEM says your choices are limited to either/or.

But arriving at generality - which is the usual goal of any metaphysically-tinged debate - should logically result in the and/also outcome that is a dichotomy, some mutually exclusive and jointly exhaustive pair of correct ideas. The dialectical result is two formally complementary truths. When folk end up arguing diametrically opposed positions, that is a good result for the debate - if they can realise that by going at each other in Socratic fashion, they have laid bare the division that can then generally ground all consequent acts of individuation or particularisation.

We are familiar with all the traditional metaphysical dichotomies that dialectical philosophy almost immediately threw up, and which became the basis of modern rational thought.

Particular~general, for a start. Or flux~stasis, discrete~continuous, part~whole, one~many, matter~form, chance~necessity, atom~void, local~global, action~direction, potential~actual, mind~body, signal~noise, freedom~constraint ... the list really does go on and on.

So chances are, where people feel adamant that they are right about something in a general fashion, it is because they have seized on one pole of a polarity.

Is reality fundamentally something continuous or is it fundamentally discrete? A really convincing case can be made for either of those positions. And the LEM seems to rule already that only one particular answer could be right - the other is automatically false. But if the concepts of discrete and continuous are perfectly matched in this complementary fashion, then now we have the happy outcome that is the deeply useful fact of a metaphysical dichotomy.

Somehow both these answers must be true as the ultimate bounds on existence. They become the thesis and antithesis out of which the synthesis - in terms of a world of individuated particulars - can then appear.

So the LEM is a logical structure that fosters the expectation that any decent argument is going to reduce all possibilities to some single correct answer. It is either/or.

But larger than the LEM is the dialectic. That is how you arrive at the space of possibilities, the space of individuated particulars, to which the LEM could apply. And so dialectical reasoning is expansionary. It points you from the symmetry breaking that is a dichotomy towards the holism of a triadic or hierarchical generality. The resolution that is a synthesis. You have two bounding extremes and now all the possible balances that lie in-between.

So people get passionate because in any decent philosophical argument, the most completely opposed alternatives are the ones that are going to feel the rightest. Violent disagreement is the way to get progress - so long as there is then the follow-up of the synthesis that unites those opposites in a generally sensible way.

Impressive!

You lack imagination. Everyone doesn't feel the same way you do. Not just about death, but about all the other difficulties you go on about. Whether or not they are happy, most people get it. Get life. The point. You don't. That says something about you, not about life. — T Clark

Yep. The best bit of advice to the young is that to have a good death, you have to start with a good life. Death-bed interviews stress people's regret at not having more adventures and not taking more time with their close relationships.

Positivity or negativity are habits of mind, so it helps to begin learning the right choice at an early stage. You become what you practice.

'Society' rarely names an 'answer' so much as a problem to itself be explained: what are the conditions which made society respond in this way? And those answers will generally be local, historical, and concrete (even as they can play a part in broader anthropologies). — StreetlightX

Yeah. Social and anthropological science may seek out the deep natural structures that are the organising forces beyond the mere passing contingencies of history. But what do they know, hey? Bloody Platonists!

Apparently.

Thus begins the descent into point-for-point responding. — darthbarracuda

So either the complaint is no response, or too much response.

I get it. Nothing in this life will make you happy. :meh:

One argument I have presented before and here now is that humans are out of balance with nature by their very nature. We're too intelligent, too creative, too self-aware. — darthbarracuda

But if that is so, that is a sociocultural fact. We aren't born that way. We have to learn these things as skills. And so we have the possibility of making some collective choices.

That would be the central question that a naturalistic moral philosophy would be targeted at. If we are responsible for the culture that makes us, what kind of people do we really want to be, and thus what kind of cultural environment should we be producing to ensure that?

I mean, we already do have that kind of conversation. As a general rule - sample any random poster here - folk would support a romantic/individualist ideology as the ethos to promote. But then, is the outcome really functional? Does it result in a reasonable balance? As you say, does it produce people who are "too intelligent, too creative, too self-aware" for the collective good?

I think that objectively, there is something to the civilised/enlightenment mindset that is the core of the modern developed society. And also the individualist/romantic creative edge is part of that balance as well. There is the naturalistic makings of a flourishing psychosocial system in that cultural formula.

But then once we are talking therapeutics, that is why positive psychology gets it right and pessimism so wrong. If you find yourself out of balance personally, positive psychology offers a prescription to match the problem while pessimism is just an excuse to wallow in a state of learnt helplessness.

In crisis, turning towards the civilising, and away from the romantic, is the sensible way to go, just for self-preserving reasons.

Living "in tune" with nature just isn't good enough for us. Metaphorically speaking, nature kicked us out and we're on our own. — darthbarracuda

Here we go. Nature kicked us out. The lament of the lonely child turning angrily on its parent. Society filled our heads with romantic ideals and now the bastard expects us to go out and live them.

But in fact society also says it wants you to live as a mature, civilised, member of the collective. So even worse, you are getting mixed messages!

Well again, this may be a commonplace confusion, but that is why a more sophisticated philosophical or therapeutic frame is so important.

We can understand the dichotomy that a flourishing natural system is based on. It relies on being able to express both poles of its fundamental being - both the competitive and the co-operative, both the private and the collective, both a civilised core and a creative individualistic fringe.

But nihilism/existentialism/pessimism/anti-natalism is just a tradition of romanticist lament. It is trying to tell the whole story based on just its one angle.

Yet the antinatalist argument is that, despite this relationship, procreation is still an act of supreme manipulation. Someone is brought into existence without permission. — darthbarracuda

Now we get into a romantic view of humans as transcendental beings with transcendental rights. We are way off track when it comes to any properly naturalistic analysis.

So sure, use the familiar legalistic jargon. Try to persuade by rhetorical device what can't be sustained by logical argument.

A naturalistic morality does say society has super-organismic reality. So there is a level of being that transcends each of us as individuals. But also that this is a balancing act - a fair trade. We need that society for there to be the "us" - the self-aware us - that could even care about permissions and manipulations.

So we collectively get to write that script - within ecological limits. Or if we can in fact transcend those limits - in techno-optimism fashion - then we even get to rewrite that ecological script.

It is all to play for really. You just have to understand the game. And pessimism really doesn't. As philosophy, it is quite useless as a tool of human forward-planning.

As I told Baden, with respect to anything else, a "mixed bag" would not be acceptable. You would want something better. You'd tell the manager of the restaurant to please send out a better meal thank-you-very-muchly, this one's over-cooked. It's edible, sure, but it tastes like crap. The manager comes back with a bottle of meat sauce instead. Is that acceptable? Would you return to this restaurant? — darthbarracuda

If you made a bad choice in going to this restaurant, would you seek to make a better choice next time? Or would you simply never enter another restaurant in your entire life?

Rational folk would do one thing. Anti-natalists might do the other.

The crucial part of my argument that I do not think you responded to was the necessity of negative value and the contingency of positive value. — darthbarracuda

I can see that you need to make the negative a foundational truth and the positive a passing delusion. That is what your story hinges on. And I've responded to that how many hundreds of times now? :)

Life is terminal struggle, that's what it is. You're given a burden (mortality) and must find a way to carve out a small part of the world just for yourself so you can postpone death for as long as possible. Life may be comfortable now, but a single toothache, migraine, or kidney stone throws it into a wreck. — darthbarracuda

This is what keeps our conversation going. Gems like this. You seem to live in such a different world.

What you imagine: a Rousseau-esque return back to nature's harmony, is a pipe dream. — darthbarracuda

Yes, of course, this is exactly what I said. Or rather, exactly the kind of half-baked position that would be weak enough to leave your own half-baked position feel like some kind of suitable balance. Honours even.

But no. I'm expecting you to do more work here. Come up with a real counter to my real position.

My greater peeve about thinking of society as some sort of Creator, especially in the case of persons, is that our understanding of consciousness is limited (possibly permanently since the investigator of personhood is always a person). — frank

The social constructionist position would be more sophisticated than how you paint it. It is a co-construction story. Society shapes us as "persons". And we, as those persons, build society.

So for society to persist, in the form of some cultural set of habits, it must be able to produce the right kinds of people. And those people would need to generally re-build or re-produce the social culture that formed them, while also adding sufficient variety or spice to keep the evolutionary game going. The next generation can potentially start shaping some new cultural habits.

The collective whole thus has an organismic unity. We can speak of human society as something that actually exists in an evolutionary sense - a new kind of super-organism that adapts and responds to the world it makes.

This kind of feedback loop is already familiar in biology. We do talk about environments or ecosystems as having living or organismic reality. Earth has a uniquely oxygen rich atmosphere because bacterial life found that oxidative reactions offer the best energetic bang for buck. Life made the Earth that way for its own good reason.

And now humans, as socially constructed creatures, have just about remade the Earth again, transforming it into a planet dominated by farming and cities. The bulk of planetary biomass is entrained to the new human ecosystem. We have brought about the anthropocene age. Life is being redefined now in our image.

Of course, that may be a fleeting affair. But it has the same organic logic - the logic that actually needs to be applied to analyse the human condition through a social constructionist lens.

So perhaps the real problem with social constructionism is the extent to which it might encourage the idea that the outcomes of what it is to be "human" are either arbitrary or near completion.

It seems pretty clear that we are in a runaway period of explosive growth - the immature or "weed" phase of the ecological lifecycle. Come back in a thousand years and we will have a better idea of whether our particular social/linguistic adventure was a successful kind of "Creator".

Genes, as the creator of ecosystems, have got a pretty great track record. Words, as the creator of sociocultural systems and thus individuated personhood, are still to prove their value at the planetary scale of organic organisation.

I'm tired of not being taken seriously, having my entire argumentative essay reduced to a single paragraph and then straw-manned, and then mocked for putting forward my honest thoughts on the matter. — darthbarracuda

But do you take my own position seriously - that structurally we would expect nature to produce a mental balance? A mixed bag would be the logical evolutionary story?

So I showed you how weak your argument is by showing it could equally well be used to argue its opposite. It becomes merely a prior evaluative frame that could make one version seem more right than the other, as @baden points out.

You and @schopenhauer1 are choosing to hang your hat on a structuralist argument. I've replied that natural structures are founded on balancing acts. They require a unity of opposites to be anything at all. So evolution must produce a mixed bag of hedonic states. We need to be adapted to our worlds. And so we need to be able to move across a wide range of emotions as appropriate, while generally seeking some kind of peaceful, neutral, mild, equanamity as the central tendency.

The issue of course is that we are social creatures, self-aware through language and social construction. So we have inherited a pretty functional psychobiology. But it was well adapted to our first million years or so of hunter/gatherer social lifestyle. And in a tearing hurry, the space of 8000 years, we have invented a succession of new lifestyles.

So yes, it may be the case that modern life is structurally shit. Folk are reared on romantic notions of their existence. Society has become a giant economic machine, out of control of a community level living. We are too self-aware in a particular way - our heads filled with the idea of being the heroes of our own unique sagas. And society has become a consumerist, planet-destroying, rat race.

There is plenty about the current state of things - the existing human structure of life and thought - that a philosopher could decry. We could do better.

So what I dispute - what I see as actually part of the problem itself - is this half-arsed pessimism you guys promote. Sure, there is structure. And sure, that structure might not be well balanced right at this time in history. But then is the answer to live a life where you have basically given up on making a personal difference? Is the best choice to make yourself more miserable, and try to make others as miserable, rather than focusing on what could be done about the situation?

Why the anti-natalist focus on not having children? Any of us who are parents will agree that it is a choice that should be carefully considered. But it also has the potential to be hugely rewarding and affirming.

Having children is one of life's big risks, big responsibilities, hence big adventures. From a personal point of view, it should be regarded in the same way as all such risk/reward opportunities, like relationships or travel or sport or enterprise or anything that requires taking a chance, plunging into life, seeing what happens.

It is only in a now over-crowded world that having children becomes some kind of collective social issue where we might talk about putting on the brakes. And as we know, the problem for "developed" nations is really the opposite already. They are breeding too little to maintain a healthy demographic balance.

But then - from the truly dispassionate view of a moral philosopher - we could say things will work out even so. Nations rise and fall. The human story will roll on and find some new steady state balance - some kind of story that also lasts a million years. And that may even be a return to hunter/gatherer level existence, 500,000 survivors scratching a living, after the big collapse.

So unless you actually believe in some transcendent/romantic ontology - humans as the chosen beings - then you have to view all this through the lens of naturalism. And nature has its natural structure - one based on a dynamical balancing act.

Life is a mixed bag because that is how nature works. That is my argument. But go on ignoring it by claiming I'm simply the mirror opposite of you - an optimist, a pollyanna, or whatever other glib counter helps to keep your own game going a little longer.

A mixed bag? Generally all right? Which one is it? — darthbarracuda

Both for most people much of the time.

But anyway, here is my absolutely killer argument. :grin:

Pessimists are selective and strategic in their attack on life. What they cannot attack, they criticise optimists for over-valuing. Or they attempt to psychoanalyze optimists as being "elated" or "deluded", because the existence of the optimist is incompatible with their negative narrative and must be "explained away" via some evanescent category. If optimism did not hold at least some element of truth, it would have been demolished from the get-go. Optimism would be definitively shown to be incorrect, not simply asserted to be incorrect.[/quote]

Most people focus on contingent pain. Metaphysical pessimists see the structural aspects. — schopenhauer1

Bullshit. The structural aspect is the dichotomy of the good and the bad, the rough and the smooth, the burdens and the transports.

So the "subtle" response you get is that you are telling a one-sided tale of unrestricted regret. There is only the bad. Even when things are good, that's when things are really, really bad, because now you have to deal with the possibility of that goodness being missing.

You have built yourself a rationale. It may have some kind of truth for you. You may just be very unlucky and stuck in a basically depressed state. But philosophically, you need to deal with the fact that your story lacks the kind of naturalism that understands life to be a mixed bag. And that is generally all right.

Then also the bit you really don't want to hear. If life is not as you would like it, then a large part of that could be because you have constructed this self-reinforcing rationale of pessimism - perhaps as a "coping" mechanism.

Pessimism is bad philosophy. Plain and simple. Fortunately there are other choices on the menu.

For one thing, this metaphor of a frame he uses, when you cash it out and ask a philosopher what frame they are working with, the answer is presumably a set of propositions. Some of them might be very banal, some of them might be empty metaphorical handwaving, some of them might be substantively interesting, and of the latter, it doesn't seem cretin-headed in the least to investigate whether they may be true or not. — jkg20

Yep. The further thing in play is a coherence angle on truth. Theories of truth - in the pragmatic view - arise out of the dynamical relation between coherent theories and their correspondence with acts of measurement.

So this makes structuralism legitimate. We do have an interest in the over-arching and unifying coherence of any putative reference frame. We are concerned about the particular truths of particular claims - the correspondence issue. But we are also concerned with how overall a paradigm hangs together in a generally coherent (but not actually prescriptive) fashion.

A prime business of philosophy is the uncovering of the rational structure that is objectively, ontically, an aspect of the world.

No, because the 'measuring, is done by an actual person, so again becomes an entirely subjective activity leading to total relativism.
— Pseudonym

Yeah, not dealing with this kind of sophistry. Thanks for your interest. — StreetlightX

I see that Pseudonym has struck on the obvious point. Bryant's blog post is nice as far as it goes. It is standard pragmatism/modelling relations/cogsci. But what is missing is the further fact that our "philosophical frames" are responsible not just for constructing the "truth" of the world, they also construct the "truth" of the self that is living that world. The making of ourselves - as the believers, the doers, the intenders - is the other pole of the modelling relation.

So Bryant's presentation, and SX's talk of "sense", is still rather representational - not yet out of the correspondence approach to truth. What also needs emphasis is that the subjective part of the deal is also being fabricated because of the larger thing that is a modelling relation between a self and its world. The self comes into focus as a structure, a set of persisting interpretive habits taking a particular view of sense, through the development of a relationship that "works".

The triadic nature of this modelling relation becomes important. Once we leave behind the simple dyad of the perceiving mind and the perceived world, we have to recognise how a self~world relation forms in terms of a semiotic umwelt. The self becomes a structure of habits able to read the world - the thing-in-itself - as some set of signs or marks.

And as Bryant emphasises, our conscious world is the one where we mostly know what we can ignore. We have a reference frame that dictates the sense we make of things. We can attend to what are the significant signs, the events that matter, because we matchingly have formed a view in which we need pay no attention to everything else. A frame is a filter separating signal from noise.

So this is a general epistemological story. We become a self - some structure of intepretive habits - by learning how to mostly ignore the world. The less "we" are a direct reflection or representation of its buzzing, blooming confusion, the more we are in fact "a point of view". We are autonomous beings to the extent we are able to be partial about the sense we make of the world "as it really is".

There is no real surprise in any of this. As I say, its standard pragmatism. But the way that modelling is also responsible for the construction of some particular kind of individuated selfhood is a missing ingredient in SX's take on Bryant.

This has consequences. Obviously many folk want philosophy to reveal some kind of useful truth. And as a first step, it does quickly reveal the epistemic truth that we are only modellers constructing a view of "a world" - the umwelt world that has "us" in it - rather than somehow minds seeing reality as it "really is".

But the naive reading of that is to think that because all possible experiential views are a pragmatic construction, then any view goes. Philosophy would have as its project the willy-nilly production of alternative viewpoints that one can just "try on" for size.

This is in fact dangerous if we become how we think. If there is no actual "self" at the back of it all - this self is only an outcome of mental structures of interpretance being formed - then picking on bad philosophies will result in bad habits of thinking and a bad selfhood emerging out of that.

We are what we eat they say. So yes, there are an abundance of philosophical frames. But we should take some kind of active approach to picking out the structures we learn and internalise, because that is what we ourselves are going to be formed by. We have to be prepared to say some philosophies, some frames, are better than others on that score.

I think it is an unconsidered meta question. Has philosophy become too unrestricted in the frames it is willing to consider? Has it bred a lot of bad points of view, bad habits of thought? Is philosophy able to arrive at its own "best self"?

I would say that of course philosophy does have some kind of noble history. It's role is as a central civilising influence. It has been progressive as a historical project. It's not some kind of disaster.

But does philosophy realise that is has this general "mission"? There is something to aspire to.

And my point here is that the recognition of that would require a dethroning of the rather romantic view of selfhood that rather infects many people's philosophy. Folk cultivate subjectivity as the right thing to be doing.

But is taking some ontic notion of subjectivity to its "logical" extreme an actually healthy way to frame matters? Is that really the umwelt you would choose to dwell in as your personal view of "the world that has you in it"?

I set out the challenge.

:ok:

An edge marks the boundary of a region, a point marks the boundary of a line segment. A region is two or three dimensional, a line is one dimensional. Why are you intent on producing ambiguity? — Metaphysician Undercover

For pity's sake. Can't you see you are just saying what I said?

A line is a 1D edge to a 2D plane. A point is a 0D bound to a 1D line. So you are simply choosing to pretend to be confused by the fact that we use terms that speak to the specifics of some act of constraint.

Yes, a line is an edge to a plane. And a point is only an "edge" to a line. But if you can't see that in the context of my account that the similarity of the nature of the constraint, the form of the symmetry breaking, is exactly the same, then I've no idea how to talk about interesting ideas with you.

Again, that's not true. Geez, what are they teaching in school these days, that kids like you get so mixed up? — Metaphysician Undercover

You're taking the piss now? Or maybe you are 90+. Seems possible.

Two lines may cross at any random angle, and represent two distinct dimensions. — Metaphysician Undercover

And those two distinct dimensions would be distinct because ....?

[Clue: it rhymes with "morthogonal".]

Theoretically, we could assume an infinite number of rays around a point, and assign to each ray a dimension, such that there would be an infinite number of dimensions. That classical "dimensions" are produced by right angles, and are therefore orthogonal is completely arbitrary. — Metaphysician Undercover

I don't know what they taught you at high school Granddad but you are just imagining any number of rays in a spherical co-ordinate space - a description that is dual or dichotomous to the usual Cartesian one. https://en.wikipedia.org/wiki/Spherical_coordinate_system

If you did go to big school any time in the last century or two, you would have learnt that higher dimensional geometry doesn't work like that. You could indeed have an infinity of spatial dimensions, but they would all have to be orthogonal to each other as that is the critical thing making them a distinct dimension of the one connected space.

Why do 3D knots come undone in 4D? Trying working that one out.

I agree with you here up until you said "existence". — schopenhauer1

But the context of that is my own earlier posts in this thread.

So in terms of metaphysics, the question becomes what is the most universal goal? And one obviously sensible answer is the limitation of instability. If any kind of world is going to exist - given the primal nature of chaotic action - then it has to develop the kind of regularity that gives self-perpetuating stability. — apokrisis

So I was specific that existence = persistence in the face of instability or chaos. I am talking about Peircean process metaphysics.

However, you seem to make the illegal move to apply it to any and every subject in a totalizing fashion. — schopenhauer1

Illegal? It could be warranted or unwarranted - the evidence can decide the case. But it is illegal to hypothesise?

Besides killing any other angles of inquiry (which would be taking advantage of the open-endedness of philosophy I was talking about) you are quick to dismiss all else to constrain your framework, thus limiting possibilities of other frameworks. — schopenhauer1

Hey, do you see me trying to rule out hypotheticals by resort to rhetoric like SX? I welcome your hypotheticals. I just introduce them to the facts of reality. Nature has already chosen what is true. :)

But more important than this, you apply such methods/language-games to problems such as the Mind-Body problem. This is where your theory is in deep water and breaks down. Where math is all modeling, you try to overmine the modeling language-game (constraints/symmetry breaking, etc.) to experience itself, and then when people accuse you of never penetrating beyond the models- you defensively go back to the Romantic vs. Enlightenment rhetoric to hand-wave the rebuttal. Your argument becomes a circularity back unto the modeling. — schopenhauer1

Blah, blah, blah.

You are back to front. Peircean metaphysics begins in phenomenology. And it is not surprising that I arrived at Peircean metaphysics via a dissatisfaction with the prevalent reductionism and epiphenomenalism in mind science (and philosophy of mind).

So Peirce (like Rosen and others) made the deep structural connection that can connect epistemology and ontology.

Mind is a modelling relation. Epistemic fact.

Our model of mind is then going to be a model of this as a suitably general ontic fact. Modelling - or semiosis - is how minds arise in a natural fashion.

Then completing this philosophical trajectory, even matter may be explainable as a pansemiotic fact. Matter exists as an (attentuated or effete) form of the same essential modelling relation ... in some intellectually useful sense.

And guess what. As I keep saying. Physics has gone that way. Everything that exists is the product of informational or holographic constraints on entropic degrees of freedom.

Keep up with science and it is pan-semiotic.

Now, I agree with you very much about your ideas as they relate to math. I have no problem with that move. Its the totalizing of its application to all areas that this becomes questionable. — schopenhauer1

Questions are fine. This is the bleeding edge of metaphysical speculation. You ought to be questioning.

I'm just reminding that I've already replied many times on the same questions. And the criticisms are not penetrating.

I would also note that the reason why Peirce (and all the others I would cite) are getting it right is because they are structuralists, they are thinking in terms of fundamental mathematical basics.

Vagueness, dichotomies, hierarchies - these are all mathematical-strength concepts. They capture the architecture (the architectonics!) of Nature because they begin from first logical principles. They are what symmetry/symmetry-breaking looks like when described in general mathematics.

Now - because they are logical/holistic arguments - they are not the kind of maths you do a lot of calculation with. They are the meta-models rather than domain specific models. But there are then plenty of those kinds of models too now - all the stuff arising out of condensed matter physics, non-linear dynamics, whatever.

So maths itself is doing a better job of describing the structure of nature as it actually is.

As I said, it started out with geometry - existence in space, with time and energy left out of the equation. That was what made the Platonism objectionable - what is real about the form of a triangle?

But replace the bloodless triangle with some real life dynamical flow - like a fractally branching river - and suddenly you really are starting to talk about Nature in a way that has fundamental unifying scope. Suddenly you can see why Nature has to express fractal order so as to be able to exist - or rather, persist as a now regulated and equilibrated source of instability.

Mathematicians seem to think about creativity in mathematics this way; a certain 'accuracy of ideas' which doesn't immediately reduce to the accuracy of a proof. — fdrake

So inference to the best explanation - the principle of least action in practice. We jiggle the bits about until it all snaps into place with a holistic best fit.

That is, we start with a broken symmetry - some "problematic" that is a collection of disjointed parts. And then we probe for the symmetry, the global coherence, that must have originally connected them.

Proof follows because that is the formal (re)construction. It is the creation of the bottom-up deductive path that connects us securely to the top-down glimpse of the Platonic reality.

So first comes the abductive leap that allows us to see the fragments in an inductively retrospective light. We see the smashed glass across the floor, the cat innocently licking its fur on the bench. In a flash we see how the symmetry of the vase got broken.

Then proof is the mopping up operation. All the parts get gummed together to show that the vase did exist as we imagined. We have sound reason to blame the cat as we are certain the vase wasn't just stuck in a cupboard while we weren't looking and meanwhile some random collection of glass just materialised on our floor.

I know apo mentioned using 360 degrees being contingent, but again, the "discovered" aspect I refer to are the concepts behind them. — schopenhauer1

That was MU in fact. I agree with you that the fundamental structures of mathematical thought are the inevitable rules of form or constraint that are there (so "somewhere" a bit Platonic) to be discovered.

So where does the arbitrary or the contingent come in? Two ways.

First - the Peircean point - the deep structures or constraining rules must be themselves emergent from a ground of arbitrariness. They are precisely the states of organisation that will emerge from chaotic possibility itself. Order must evolve to regulate instability. It might be a very minimal order, a very permissive order. Yet still it will be a generalised state of order.

Even statistics depends on bounded action. Randomness can have macro properties like a mean or a variance because there is some kind of global constraint bounding a system of independent variables. You get a temperature or a pressure only when your gas is confined in a flask. And any workable notion of randomness or probability depends on a duality of free local action coupled to definite boundary constraints. Otherwise there just wouldn't be any "statistics" - any macro properties to speak of.

So our very notion of the arbitrary or the contingent only makes metaphysical sense in the context of its "other" - the necessity, the regulation, to be found in some set of bounding constraints. You can't even have the one without the other. Hence there is the Platonic structure to be discovered as the necessary spine of existence. That can't not be the case ... if you do in fact believe in the matching "other" of the accidental or contingent. Each secures the reality of the other in complementary fashion. Hence why SX's orientation, as expressed in the OP, is so off-base from the start.

Then second, in discovering the deep and necessary structure of existence, we humans can gain our own local possibilities of control. We can insert ourselves as agents into the cosmic equation. If we construct mathematical models that encode the basic rules of the game - the game that is symmetry and symmetry breaking - then we can start to use them in our own "arbitrary and contingent" fashion.

We can do things that Nature doesn't seem to be thinking about or caring about. And folk of course find that a big philosophical deal. Suddenly our values and desires start to affect the metaphysical story. We are the source of meanings. We are the source of inventions. We are the source of fictions and fables and beliefs.

All this can lead to a PoMo-style rejection of metaphysical structure and meaning. Dichotomies and hierarchies become dirty words. Humans can transcend nature to become ... well, now it seems philosophy has solid grounds for its Romantic revolt against all "merely" physicalist constraints or necessities.

But this gets the metaphysics wrong. Constraints only constrain. So there is room enough in our cosmos for the very mild and attenuated constraints - the generalised thermodynamic tendencies - and then the much more complex and particular kinds of natural organisation that are reflected in evolved life and mind.

There is no basic conflict here, no real battle that the Romanticist must fight. We don't need to be like SX and fetishise the arbitrary and the plural, anathemise the Platonic and the unitary.

To get back to the 360 circle, I would note how the choice of 360 wasn't so accidental. It seemed important that we find a numbering system that made division into simple fractions easy, while also offering enough divisions to capture the differences that were of (Babylonian/astronomical) interest to us.

Again, some deeper symmetry was the reason for 360 being a choice that lasted. It was a number that offered the kind of symmetry breaking we found most convenient. Given an infinity of numbers we could have chosen, picking on 360 was not an arbitrary act from the point of view of a human having to do the calculations on a regular basis.

Neither can a point have an edge ... A point marks the limit to a line segment. — Metaphysician Undercover

So a point can be the edge to a line? Make up your mind.

It is contradictory to say that a point is a line segment which can't be cut any shorter, because a point and a line segment are fundamentally different. — Metaphysician Undercover

So if we cut away all the line to one side, it is bounded by a point on that edge. And if we then cut away all the rest of the line to the other side, what then? Is the point bounded by a point or is there just the point?

Isn't the fundamental difference that the point is the natural unit of which lines are composed? It has a locally emergent symmetry which marks the place where no more can be cut away. A point is all featureless edge?

A point has zero dimensions, while a line signifies a dimension. — Metaphysician Undercover

But doesn't the point have a location? It exists as a limit on dimension. A line represents a space of points. A point represents that space without a line.

What this indicates is that our spatial concepts, in terms of dimensions, are incorrect. The concept of dimensions of space produce an unintelligibility and therefore must be incorrect. — Metaphysician Undercover

Seems radical. But sure, throw away the concept if you can tell us about something better to replace it.

Or maybe it is your logic that lets you down. My logic expects metaphysical truth to be rooted in the fruitfulness of dichotomies. Yours is instead deeply troubled by discovering dialectical "contradiction" at the heart of things.

I think that the application of the theory of general relativity has proven this to be false, the shortest distance to connect two points is not actually a straight line. — Metaphysician Undercover

You got it exactly backwards. I was saying GR indeed shows that "straightness" is relative. It cashes out as the shortest path, the least action principle, once your geometric intuitions include some proper notion of action or energy density along with the spacetime dimensionality.

That is one of the issues with classical geometry. It all rests on spatial intuitions. It doesn't have a natural way of including energy, thus actual time and change, in the picture. It is a story of the directions but not the actions.

So GR is geometry with energy density included. It is more physically realistic as maths. And now the least action principle comes to the fore when we are thinking about "straight lines". We have generalised Euclidean geometry so that there is just universal curvature. We have removed a major constraint - the one picked out by the parallel lines axiom. And now straightness becomes defined more clearly as the shortest possible (energetic) distance between two points.

GR changed the simple definition of what counts as straight. That was the bleeding point.

I answer this question by saying that the entire conceptual structure which models space in terms of distinct dimensions is inadequate and therefore incorrect. — Metaphysician Undercover

Seems coherent - in being dichotomistic - to me. Dimensions are defined by being orthogonal to each other. Mutually exclusive and jointly exhaustive. The x axis is oriented so that no part of it - apart from a point, an origin - intersects with the world of the y axis.

You can head for infinity in the x-axis without moving an infinitesimal amount in the y-axis. And that is your strong definition of a spatial dimension. You have two disconnected or asymmetric directions that exist for motion~rest.

Remember how that distinction played out in Newtonian mechanics. Spacetime became defined by its energy conserving transformations. Masses could spin inertially on the spot or move inertially in a straight line. These were local symmetries that couldn't be broken down as they were the terminus to the very possibility of a more global symmetry breaking.

Then along comes GR, and even QM, to add the missing ingredients of time and energy back into this mathematical picture. The classical view was certainly correct, but still carried extra constraints that proved themselves to be local and particular rather than cosmically general.

The "angle" is something totally arbitrary, inserted into spatial conceptions as an attempt to alleviate the described problem of an incompatibility between linear dimensions. — Metaphysician Undercover

It is not arbitrary. What got inserted was the very notion of a dichotomy or asymmetry. Dimensions are distinct due to their orthogonality.

Ask yourself why pi = 180 degrees. Hint: a circular rotation that flips you back to a flat line having transversed its orthogonal "other".

These things aren't arbitrary at all. They are logically founded. They are Platonic strength.

That is why SX is so off the mark in his OP. Maths is "unreasonably effective" because - where it is based on the metaphysics of symmetry and symmetry-breaking - it is finding ways to model the necessary structure of existence. For anything to be, it would have to have being in this self-justifying form.

... and your insults are so lame as well.

Quit complaining and put up a counter argument if you have something to say.

Insults and yet no argument. Curious that.

It is not that the two determinations are teleologically programmed in advance, but rather that from the moment when they occurred (as a contingency), their emergence retrospectively unifies all prior attempts, through the construction of the universal. — Artmachines

PoMo is going to get such a shock when it catches up to 1980s work on universality in dynamical systems. :)

In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system.

https://en.wikipedia.org/wiki/Universality_(dynamical_systems)

The curved line is not a limit to the straight line, the two are categorically different. — Metaphysician Undercover

Hmm. What then of the points that make the circle. Are they not the smallest possible straight edges?

A point is the limit to a line - the zero-D terminus that has greater local symmetry than the 1D line which is having its own symmetry broken by being cut ever shorter, and eventually, infinitely short. A point is simply a line that can't be cut any shorter.

Then for a line to be either straight or curved is itself a question embedded in the 2D of a plane at a minimum. So curvature, or its lack, is determined by the symmetry breaking of a more global (2D) context. A line becomes "straight" as now the locally symmetric terminus of all possible linear wigglings.

Straightness is defined in terms of the least action principle. A straight line is the shortest distance to connect two points. You may be familiar with that story from physics.

So now what can we rightfully say about the points that make up the circumference of your circle?

They are minimal length lines. But are they straight or are they curved? Or would you say the issue is logically vague - the PNC does not apply? No wiggling means no case to answer on that score.

Anyway, it is clear that the straightest line is simply the shortest path in regard to some embedding context. And even you would agree that a circle is composed of points. So the standard view - that a circle is the limit case, an infinite sided regular polygon - holds.

In any case, the importance of symmetry to modelling nature seems to be something about which we do not have a choice - symmetry is at work in the General Theory - so there at least I agree with you. — jkg20

Great.

No, no, this definition of "unit" must be rejected as circular, or an infinite regress, and therefore not a definition at all. — Metaphysician Undercover

That's funny, given a circle is the most fundamentally symmetric type of unit. It stands as the limit to an infinite regress in terms of the number of sides to a regular polygon.

How do we account for the usefulness of pure mathematics in describing and predicting reality? That's a different question, but I'm certainly not convinced that the answer to it requires either mathematical realism or physical realism. — jkg20

Sure. If you read my posts, you will see I am a pragmatist. You’re talking about bread and butter epistemic issues.

But again, do you want to claim that the connection is arbitrary? Do you have reason to believe that nature plays by different structural rules despite the evidence to the contrary?

Yeah. I should have said something catchier like "intra-systemic imposition that responds to no genuine, worldly problematic".

The standard model has its problems and its alternatives/adaptations, and the existence of "gravitons" is contentious — jkg20

... or gravitons aren't even part of the Standard Model yet.

So if you mean by "convinced" "convinced that the Standard Model describes reality as it is in itself independently of our means of modelling it", then no I am not convinced. — jkg20

I'm not asking you to deny that the Standard Model is "only a model". The clue on that score is probably in the name.

The issue here is SX calling particle physics use of symmetry breaking "arbitrary".

So do you think group theory is arbitrary? Are its results contingent in some fashion you can explain?

And is particle physics success in using symmetry maths to account for particle relations arbitrary? When a model fits like a glove, why would we have reason to think that was also merely contingent?

Of course, it could be a lucky accident. No one can deny Descartes his demon.

Not convinced by the Standard Model, hey? You think it might be just a big coincidence?

Sounds legit.

Did you actually contest any of the content of my posts. Must have missed it somehow. :yawn:

Oh please. If mathematical physics tells us that existence is the result of broken symmetry, then who are you to disagree? Get over yourself - your Copernican belief in the worldly problematics that revolve around your Being. Good lord.

You mean that game where the goal, the meta-rule, was to show that no particular rule could hold firm?

Yet, that’d work as prescribed. It would serve the purpose of anti-metaphysicians. ;)

Or put otherwise: there is no 'ultimate symmetry', the breaking of which explains individuation — StreetlightX

In case I left you confused - it does happen - I hope it is clear that symmetry-breaking is what connects a triadic system of symmetries. So the "ultimate symmetry" would be a three cornered structure, if you like.

You have the symmetry of vagueness - a state where (material) contingency and (formal) necessity are differences not making any difference. As Peirce noted, the PNC does not apply.

You have the symmetry of generality - a state where globally there is the continuity that has formally absorbed all possible differences so that they don't make a difference. As Peirce noted, the LEM does not apply.

Then you have that final symmetry of atomistic particularity. Eventually, even constraint no longer makes a difference. Locally, things arrive at the ultimate simplicity of a geometric point, a mathematical identity element, a quantum particle, an informational bit, a semiotic mark, or a fundamental entropic degree of freedom.

Or as the laws of thought would have it, the emergent entity to which the principle of indiscernibles does finally successfully apply.

So symmetry is something to be understood in a formally general fashion - hence why maths is the domain that winds up speaking about it.

But in a holistic metaphysics with a triadic structure, we are talking about three kinds of symmetry-producing limitations. If we pursue symmetry-breaking back to its source, we find it in three types of bounds - Peirce's triad of firstness, secondness and thirdness. Each is a "level" of symmetry - a terminus to a dichotomous "other".

So I think your whole approach mistakes description for prescription, effect for cause: once you suck the life out of problems-in-duration and make the move into a higher dimension where everything can be seen from the perceptive of placing them into neatly-parsed boxes (accidents or necessities? generalities or particulars?), then and only then does development seem to proceed on that basis; but the leap into that dimension is illegitimate: it's simply retroactive ratiocination, the work of philosophical morticians. — StreetlightX

Is there any real effort at thought behind these ad homs?

Sure, there is pragmatism in the weak Jamesian sense of utilitarianism - whatever is good for "someone's" contingent purposes.

But the existence of the ad hoc itself highlights the "other" which is Peircean pragmatism - the metaphysically general kind. Instead of the someone, we are now talking about the generality that is "anyone".

So you can wave the flag for the ad hoc story. It is part of my larger story already. It is precisely the kind of contingency that I am generalising away as the differences that don't make a difference when the intent is to reveal the basic structural mechanism at the heart of existence.

Or put otherwise: there is no 'ultimate symmetry', the breaking of which explains individuation; it only seems that way after-the-fact, once you've illegitimately abstracted the concept from the conditions which gave rise to it; Symmetry is always-already broken in some way: there are generalities and particulars, and even stratified hierarchies of such divisions - all this can be granted - but they develop from the 'bottom-up', even if, once so developed, the higher levels attain a consistency of their own (e.g. category theory as a 'response' to problems in algebraic topology). Explanation occurs in medias res, and not sub specie aeternitatis. — StreetlightX

I've explained this to MU above. And I already pointed out that the Peircean view is ultimately triadic - an orthogonal pair of dichotomies - in that it combines a diachronic view with the synchronic. It is a tale of two symmetry-breakings - the developmental one that goes from vague potential to crisply ordered, and the scale hierarchy one which is the developed crisp outcome, the equilibrium state where the local constructing is in stable balance with the global constraining.

So there is room enough in the triadic view to house all your metaphysical concerns. :)

There is bottom-up atomistic construction, for sure. There is actualised contingency or degrees of freedom, for sure. All the regular stuff that makes reality safe for reductionist thinkers can be found in a world that has developed enough to have attained a stable local~global structure.

Once the world is being held in place by the constraints of its own history, it does look securely classical to the typical human observer, sitting right at the Copernican centre of the story.

And you can join all those who blithely takes this as Humanity's right - to see themselves as the spiritual centre around which the Universe then dumbly and mechanically revolves. It is our own personal desires and values that matter - which should inform material reality. We impose ourselves on nature in some metaphysically rightful fashion as the world itself is just some bunch of ad hoc events, lacking any formal or final causes. What completes existence is us, at the centre, doing our thing of expressing ourselves in some kind of glorious free pluralistic fashion.

Talk about reinventing theology.

Meanwhile, science and maths are just getting on with the job of revealing the deep structure of existence. And the reason why they are working can only be understood once you see how they are simply an expression of a Aristotelian/Peircean holistic structuralism.

Folk are confused because the first fruits of maths and science were the presentation of a classical world - the world of the atomistic, mechanical, local, deterministic, etc. That exploded like a bomb in people's thoughts.

But now it is clear how the classical realm is emergent. It is what you get only as existence develops its generalised habits of constraints and forms a clear local~global structure as constraint eventually produces the grainy local limit where atomistic construction actually starts to be a thing.

So the new project is holism. And mathematical physics is deeply engaged with that. It seems to know what it is doing at least.

Does "one" signify an indivisible unit, or does it signify a divisible unit? Numbers like 2, 3, 4, represent divisible units, 2 representing a unity which is divisible into two distinct units. But 1 when understood in this way must be indivisible. If we allow that 1 is divisible, we undermine the meaning of unity. But we need to allow that one is both a unity and is divisible, so we allow two incompatible, contradictory concepts to coexist within one, being signified within one symbol. — Metaphysician Undercover

Remember that in maths, a unit is defined by the identity element - a local symmetry that can't be broken by whatever operation broke the global symmetry. So 1x1=1. Or A-0=A. The fundamental unit is whatever emerges as the local limit on symmetry breaking. The act of quantification results in a quantity where the action no longer makes a difference. Things finally stop changing. You arrive at a fundamental grain so far as that symmetry-breaking is concerned. Now nature just spins on the spot, quantified in good atomistic fashion.

This is indeed the tale of fundamental particle physics. So maths and physics are talking about the same universal mechanism. Reality exists because there was a symmetry to be broken. And then the breaking of a symmetry eventually also hits some local limit. A new state of symmetry is discovered where the individuating, the differencing, no longer results in a difference. You wind up with a smallest Planckian grain of action.

So geometry begins with the fundamental thing of a zero-d point. Dimensionality cannot be constrained any more rigorously than a dot, a minimal dimensional mark. Having found the stable atom, the concrete unit, the construction of dimensional geometry can begin.

Instead of a holistic metaphysics of constraint - the story of how a unit or identity element could naturally exist at the end of a trail of symmetry breaking - we can flip to the more familiar reductionist task of (re)constructing the world from the bottom-up. We have our unit. We can then start framing the universal laws that then do arithmetic with that unit, building a reality up step by concrete step in accordance with a material/effective cause notion of how the world "really is".

So in the mathematical realm where 1 is the identity element - the unit that is unchanged by the kind of change that more generally prevails - it is both part of that world and separate from it. It has that incompatibility which you point out. And that is because it is a re-emerging symmetry.

Globally, a symmetry got broke by the very notion of a division algebra. Division, as an operation, could fracture the unity of the global unity that is our generalised idea of a continuous wholeness - some undifferentiated potential. But then divisibility itself gets halted by reaching a local limit. Eventually it winds up spinning on the spot, changing nothing. A second limiting state of symmetry emerges ... when our original notion of unity as a continuous wholeness finally meets its dichotomous "other" in the form of an utterly broken discreteness.

So it is the usual metaphysical deal. A dichotomy that finds its fullest resolved expression in the form of a local~global hierarchy of constraint. To bound a world takes opposing poles of being. And this is what both physics and maths have worked out - even if the holism that underpins the successes of reductionism is not itself generally appreciated.

It does seem weird. Even science and maths don't really understand why they work so well - why they get at the basic structure of existence. Everyone thinks it is because of their reductionism. And that is certainly what works in a "pragmatic" everyday sense - when the mechanical and atomistic view is good enough to serve our very concrete human purposes.

But that is why Peircean pragmatism, the original metaphysical kind, is important. Existence is a story of how constraints can tame flux or instability, eventually resulting in the irreducible grain - the fundamental units or atomistic actions - from which a resulting counter-action of mechanical constructability can start.

Bottom-up material/efficient causation can be a thing once top-down constraint has forced everything towards a local limit and a symmetry has emerged there which can be the foundation for more semiotically complex constructions.

It ends up treating the pragmatics as mere accidents on the way to some eternal Platonic story which was there from the beginning — StreetlightX

Not really. If we are talking about a pan-semiotic metaphysics now, the goal is to divide reality into its necessities and its accidents. So pragmatism is about some finality being in play and shaping events in a contextual fashion. Things are individuated by constraints as a matter of top-down necessity. But constraints themselves are open or permissive. They only limit to the degree it matters. Beyond that, the accidental or spontaneous can be the rule. Constraints are only concerned by the differences that would make a difference.

So in terms of metaphysics, the question becomes what is the most universal goal? And one obviously sensible answer is the limitation of instability. If any kind of world is going to exist - given the primal nature of chaotic action - then it has to develop the kind of regularity that gives self-perpetuating stability.

Thus the "Platonic universals" would be an evolutionary story. The need for stability would be selected for just because stability is definitional of what we mean by existing. It would be the first necessity. And the eternality of that form would be an emergent and immanent fact. It would be what gets expressed by the end in the long run.

And then, some kind of stable world having established, this could be the ground for the development of further, more particular, states of constraint. Other more localised purposes, expressed by more complex forms, could arise - stabilising "sub-worlds".

So it would be pragmatics all the way. Constraints to limit dynamics and produce particular states of entification could keep developing in open-ended fashion. In the most universal view, they would be accidental - differences that don't make a difference to the most universal view. But locally, in being further hierarchical levels of development, they would encode the forms that are necessary to that further level of organised and individuated being.

That is the metaphysical picture. Maths then becomes the science that explores the principles of form. It seeks out the structures, the rules, that stand for the necessities - with the accidental part of reality becoming whatever measured values we might insert into some general rule.

Vagueness is for me the ultimate transcendental illusion: it takes a perfectly valid move - the step from particular to general, always motivated by a particular problem (B&C's 'decision points') - and then illegitimately extrapolates that step into what one might call an 'unmotivated generality'. — StreetlightX

I wasn't talking about vagueness in reference to mathematical notions of symmetry. I was talking about the continuity of generality quite specifically.

In Peircean logic and metaphysics, the particular and the general co-arise from the vague. So vagueness is firstness, then particularity would be secondness, and generality is thirdness.

Sure, that makes vagueness a kind of ultimate symmetry - it gets broken by that definite division into the general vs the particular. But here, for the moment, the focus is on the ontic structuralism story - the general or universal constraint that "being intelligible" has on shaping the particular or the local. That is the territory that maths is exploring by abstracting to discover the most generalised forms that limitations can intelligibly take. The question being asked is what is the most primal kind of individuation.

And given your interest in individuation and contextuality, it is odd that you don't see this immediately. The step from the particular to the general is about discovering what context of constraint is causing the particular to be what it is in the first place. From there, one can start to remove those constraints - in a most general fashion - to get down to what is most primal about individuation itself.

It is in generalising away the constraints that individuate that we arrive back at vagueness as a formless and unindividuated ground. But that vagueness is a material potential. And maths focuses on the issue of the forms that could organise that. That is why maths does not talk about energy, just organisation. That is why maths ends up talking in "Platonic fashion" about the finalities that want to be expressed.

For there to be (persistent) individuation, there needs to be (embodied) constraints. So reality has to be actually organised as a material system. But maths targets just the constraints, understood in terms of logical generalities. It imagines them having their own abstracted existence - as a ghostly organising hand. And it is healthy to do this as it is the way that it can focus on what is essential vs what is accidental in our accounts of nature. Pragmatism relies on being able to know the difference - the differences that make a difference vs the differences that don't.

So basically I can agree with you up right up until the point where you invoke unmotivated generality as a Platonic bow to tie the whole developmental story together. It's this very last step that shifts a perfectly rigorous and valid methodology into a procrustean metaphysics that tries to retroactively fit concrete developments into a pre-ordained story. It's just a theological-Platonic hangover/residue that needs to be rejected. — StreetlightX

But this is just you forcing things into a framework you feel you can quickly reject. It's not the story I tell.

Remember the evolutionary principle at the heart of this. For anything to exist in persistent and individuated fashion, it must mean that some primal state of constraint managed to work. Everything else then follows naturally from the fact that instability could develop limitation.

nor about the TOPIC AT HAND, which is to say that the goods of life do not make up for the continuous burdens of life. — schopenhauer1

So what makes that the correct framing of the situation rather than life being continuously stimulating apart from the occasional interruptions?

Oh the burden of deciding what kind of fun to have today. It is truly unbearable!

LOL. You guys.

I agree with the pragmatic angle, but the conclusion seems more Platonic. Each of these particular turns in the history of mathematical thought were "forced" by the need to move from the particular to the general. A constraint that was breaking a symmetry needed eventually to be unbroken so as to move up to the next level of abstraction or generality.

Irrational numbers - following the Aristotelian argument supplied in the paper - can be seen as having to relax the constraint that a number is either odd or even. The argument shows that an irrational number is not definitely one or the other. So a higher degree of symmetry is obtained by removing this constraint as an "obvious necessity".

The same goes for complex numbers - removing the constraint that number lines be one dimensional. And non-Euclidean geometry - removing the constraint that worlds be flat thus parallel lines apply.

Well, for B&C, the important point to note is that nothing in the math itself forced this choice, rather than the other. — StreetlightX

No. The tension here was between the numerical and geometric view - the geometric one representing the presumed continuity of physical nature, the numerical one representing the desire to talk about that in a system of discrete signs.

So you have the continuity of symmetry and the discreteness or individuation that results from symmetry-breaking. And then the metaphysical tension regarding whether to understand acts of individuation as actual discrete ruptures - stand alone existences - or merely contextualised developments, breakings that have only relatively definite existence.

The disturbing thing was that geometry and algebra did seem to be two commensurate or complementary ways of talking about the same world. But eventually the cracks got revealed.

Numbers work as zero-dimensional and non-geometric signs when their physical dimensionality is suppressed or constrained. But to maintain a complementarity between numbers and geometry as maths seeks to advance, gradually the constraints on that dimensionality have to be relaxed in a systematic fashion to keep the two worlds connected as we move deeper towards maths capable of higher symmetry.

So in summary, it does all start in pragmatic acts of measurement - the semiotic trick of giving names to things. We break the continuity of physical experience by imposing a system of discrete marks upon it.

This is the useful trick. The semiotic ability to construct constraints that break the symmetry of the experienced world. And the Ancient Greeks were dazzled by this new Platonic reality that rational geometry opened up. A fundamental connection between discrete number and continuous dimension appeared to be forged.

And gradually the very nature of that trick - even for the physical world itself, the pan-semiotic Cosmos - got revealed. The whole damn world exists as a constraint on dimensionality, a grand tale of global symmetry-breaking and localised individuation.

So maths only ever had one route by which to recover its physical origins. At every turn, it needed to work out what further localising constraint had been added to the deal that could be successfully generalised away. Mathematical advance had to see that a symmetry had been broken, which could then be unbroken to reveal the higher, more abstracted, less particularised, realm that lay beyond.

Go up a level and you could still give names to things. Topology abandoned geometry's definite measurements of length, but still deals in individuated abstracta. Semiotics could still work as the trick.

But the route was always Platonically predestined and necessary. If existence takes definite shape due to constraints, due to symmetry-breakings, then the only way to understand that is by following the path backwards that abstracts away those constraints, unbreaks those symmetries, to reveal how the how show works.