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Logic & First Principles, 14: Are beauty, truth, knowledge, goodness and justice merely matters of subjective opinions? (Preliminary thoughts.)

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We live in a Kant-haunted age, where the “ugly gulch” between our inner world of appearances and judgements and the world of things in themselves is often seen as unbridgeable. Of course, there are many other streams of thought that lead to widespread relativism and subjectivism, but the ugly gulch concept is in some ways emblematic. Such trends influence many commonly encountered views, most notably our tendency to hold that being a matter of taste, beauty lies solely in the eye of the beholder.

And yet, we find the world-famous bust of Nefertiti:

The famous bust of Nefertiti, found in Thutmose’s workshop (notice, how subtle smiles will play a role in portraits of beautiful women)

Compare, 3400 years later; notice the symmetry and focal power of key features for Guinean model, Sira Kante :


Sira Kante

And then, ponder the highly formal architecture of the Taj Mahal:

The Taj Mahal

ADDED: To help drive home the point, here is a collage of current architectural eyesores:

Current Eyesores

Added, Mar 23 — Vernal Equinox: The oddly shaped building on London’s skyline is called “Walkie-Talkie” and due to its curved surface creates a heating hazard at the height of summer on a nearby street — yet another aspect of sound design that was overlooked (this one, ethical):

Louvre as seen from inside the Pei pyramid

Since it has come up I add the Louvre’s recent addition of a Pyramid (which apparently echoes a similar temporary monument placed there c. 1839 to honour the dead in an 1830 uprising). Notice, below, how symmetric it is in the context of the museum; where triangular elements are a longstanding part of the design as may be seen from the structure below the central dome and above many windows. Observe the balance between overall framework and detailed elements that relieve the boredom of large, flat blank walls. Historically, also, as Notre Dame’s South Rose Window so aptly illustrates, windows and light have been part of the design and function of French architecture. Notice, how it fits the symmetry and is not overwhelmingly large, though of course those who objected that it is not simply aligned with the classical design of the building have a point:

Yet again, the similarly strongly patterned South Rose Window at Notre Dame (with its obvious focal point, as well as how the many portraits give delightful detail and variety amidst the symmetry) :

Notre Dame, South Rose Window

Compare, patterning, variety and focus with subtle asymmetry in part of “Seahorse Valley” for the Mandelbrot set:

Seahorse Valley zoom, Mandelbrot set

I add, let us pause to see the power of spirals as a pattern, tying in the Fibonacci sequence and thus also the Golden Ratio, Phi, 1.618 . . . (where concentric circles as in the Rose Window, have much of the same almost hypnotic effect and where we see spirals in the seahorse valley also):

Here, let us observe a least squares fit logarithmic spiral superposed on a cut Nautilus shell:

Let us also note, Da Vinci’s Vitruvian Man, as an illustration of patterns and proportions, noting the impact of the dynamic effect of the many S- and J-curve sculptural forms of the curved shapes in the human figure:

Note, a collage of “typical” human figure proportions:

Contrast the striking abstract forms (echoing and evoking human or animal figures), asymmetric patterning, colour balances, contrasts and fractal-rich cloudy details in the Eagle Nebula:

The Eagle Nebula

Also, the fractal patterning and highlighted focus shown by a partially sunlit Grand Canyon:

Grand Canyon

And then, with refreshed eyes, ponder Mona Lisa, noticing how da Vinci’s composition draws together all the above elements:

Mona Lisa — the most famous portrait
A modern reconstruction of what Mona Lisa may have looked like on completion

Let me also add, in a deliberately reduced scale, a reconstruction of what the portrait may have originally looked like. Over 400 years have passed, varnish has aged and yellowed, poplar wood has responded to its environment, some pigments have lost their colour, there have apparently been over-zealous reconstructions. Of course, the modern painter is not in Da Vinci’s class.

However, such a reconstruction helps us see the story the painting subtly weaves.

A wealthy young lady sits in a three-quarters pose . . . already a subtle asymmetry, in an ornate armchair, on an elevated balcony overlooking a civilisation-tamed landscape; she represents the upper class of the community that has tamed the land. Notice, how a serpentine, S-curved road just below her right shoulder ties her to the landscape and how a ridge line at the base of her neck acts as a secondary horizon and lead in. Also, the main horizon line (at viewer’s eye-level) is a little below her eyes; it is relieved by more ridges. She wears bright red, softened with dark green and translucent layers. Her reddish brown hair is similarly veiled. As a slight double-chin and well-fed hands show, she is not an exemplar of the extreme thinness equals beauty school of thought. The right hand is brought over to the left and superposed, covering her midriff — one almost suspects, she may be an expectant mother. Her eyes (note the restored highlights) look to her left . . . a subtle asymmetry that communicates lifelike movement so verisimilitude, as if she is smiling subtly with the painter or the viewer — this is not a smirk or sneer. And of course the presence of an invited narrative adds to the aesthetic power of the composition.

These classics (old and new alike) serve to show how stable a settled judgement of beauty can be. Which raises a question: what is beauty? Like unto that: are there principles of aesthetic judgement that give a rational framework, setting up objective knowledge of beauty? And, how do beauty, goodness, justice and truth align?

These are notoriously hard questions, probing aesthetics and ethics, the two main branches of axiology, the philosophical study of the valuable.

Where, yes, beauty is recognised to be valuable, even as ethics is clearly tied to moral value and goodness and truth are also valuable, worthy to be prized. It is unsurprising that the Taj Mahal was built as a mausoleum by a King to honour his beautiful, deeply loved wife (who had died in childbirth).

AmHD is a good place to start: beauty is “[a] quality or combination of qualities that gives pleasure to the mind or senses and is often associated with properties such as harmony of form or color, proportion, authenticity, and originality. “

Wikipedia first suggests that beauty is:

a property or characteristic of an animal, idea, object, person or place that provides a perceptual experience of pleasure or satisfaction. Beauty is studied as part of aesthetics, culture, social psychology, philosophy and sociology. An “ideal beauty” is an entity which is admired, or possesses features widely attributed to beauty in a particular culture, for perfection. Ugliness is the opposite of beauty.

The experience of “beauty” often involves an interpretation of some entity as being in balance and harmony with nature, which may lead to feelings of attraction and emotional well-being. Because this can be a subjective experience, it is often said that “beauty is in the eye of the beholder.” However, given the empirical observations of things that are considered beautiful often aligning with the aforementioned nature and health thereof, beauty has been stated to have levels of objectivity as well

It then continues (unsurprisingly) that ” [t]here is also evidence that perceptions of beauty are determined by natural selection; that things, aspects of people and landscapes considered beautiful are typically found in situations likely to give enhanced survival of the perceiving human’s genes.” Thus we find the concepts of unconscious programming and perception driven by blind evolutionary forces. The shadow of the ugly gulch lurks just beneath the surface.

Can these differences be resolved?

At one level, at least since Plato’s dialogue Hippias Major, it has been well known that beauty is notoriously hard to define or specify in terms of readily agreed principles. There definitely is subjectivity, but is there also objectivity? If one says no, why then are there classics?

Further, if no, then why could we lay out a cumulative pattern across time, art-form, nature and theme above that then appears exquisitely fused together in a portrait that just happens to be the most famous, classic portrait in the world?

If so, what are such and can they constitute a coherent framework that could justify the claim to objective knowledge of aesthetic value?

Hard questions, hard as there are no easy, simple readily agreed answers. And yet, the process of addressing a hard puzzle where our intuitions tell us something but it seems to be forever just beyond our grasp, is itself highly instructive. For, we know in part.

Dewitt H. Parker, in opening his 1920 textbook, Principles of Aesthetics, aptly captures the paradox:

Although some feeling for beauty is perhaps universal among men, the
same cannot be said of the understanding of beauty. The average man,
who may exercise considerable taste in personal adornment, in the
decoration of the home, or in the choice of poetry and painting, is
at a loss when called upon to tell what art is or to explain why he
calls one thing “beautiful” and another “ugly.” Even the artist and
the connoisseur, skilled to produce or accurate in judgment, are often
wanting in clear and consistent ideas about their own works or
appreciations. Here, as elsewhere, we meet the contrast between feeling
and doing, on the one hand, and knowing, on the other.

Of course, as we saw above, reflective (and perhaps, aided) observation of case studies can support an inductive process that tries to identify principles and design patterns of effective artistic or natural composition that reliably excite the beauty response. That can be quite suggestive, as we already saw:

  • symmetry,
  • balance,
  • pattern (including rhythms in space and/or time [e.g. percussion, dance]),
  • proportion (including the golden ratio phi, 1.618 etc)
  • unity or harmony (with tension and resolution), highlighting contrast,
    variety and detail,
  • subtle asymmetry,
  • focus or vision or theme,
  • verisimilitude (insight that shows/focusses a credible truth/reality)
  • echoing of familiar forms (including scaled, fractal self-symmetry),
  • skilled combination or composition
  • and more.

We may see this with greater richness by taking a side-light from literature, drama and cinema, by using the premise that art tells a story, drawing us into a fresh vision of the world, ourselves, possibilities:

Already, it is clear that beauty has in it organising principles and that coherence with variety in composition indicates that there is indeed organisation, which brings to bear purpose and thus a way in for reflective, critical discussion. From this, we reach to development of higher quality of works and growing knowledge that guides skill and intuition without stifling creativity or originality. So, credibly, there is artistic — or even, aesthetic — knowledge that turns on rational principles, which may rightly be deemed truths.

Where, as we are rational, responsible, significantly free , morally governed creatures, the ethical must also intersect.

Where also, art has a visionary, instructive function that can strongly shape a culture. So, nobility, purity and virtue are inextricably entangled with the artistic: the perverse, ill-advised, unjust or corrupting (consider here, pornography or the like, or literature, drama and cinema that teach propaganda or the techniques of vice) are issues to be faced.

And, after our initial journey, we are back home, but in a different way. We may — if we choose — begin to see how beauty, truth, knowledge, goodness and justice may all come together, and how beauty in particular is more than merely subjective taste or culturally induced preference or disguised population survival. Where also, art reflecting rational principles, purposes and value points to artist. END

PS: To document the impact of the beauty of ordinary things (we have got de-sensitised) here are people who thanks to filtering glasses are seeing (enough of) colour for the first time:

Similarly, here are people hearing for the first time:

This will be a bit more controversial, but observe these Korean plastic surgery outcomes:

Comments
KF@308, and what happens if I break one of the sticks in half? :) The fact that we can devise a system to accurately model the real world doesn’t mean that this system existed before us. It is an indication that we are good at seeing patterns.Brother Brian
March 20, 2019
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BB, Only if Mathematics is able to accurately address reality. Which means, what we say and conceive is objectively true in material part. Again, I point to an extremely simple case: is it true that 3 + 2 = 5? Or, that ||| + || --> ||||| ? In short, do our mental exercises and studies regarding structure and quantity in material part accurately describe the substantial reality of structure and quantity in the extra-mental world? If we are locked into an inner world of appearances and concepts, we are dealing with mental games and have surrendered the claim to truth regarding reality, much less warranted truth. (And yes, I suspect, here, the long shadow of the kantian ugly gulch; which is self-referentially incoherent.) KFkairosfocus
March 20, 2019
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KF
H, if Mathematics does not in the end accurately and substantially refer to anything outside the human mind then it is just a mental game. KF
I think you are misrepresenting what Hazel is saying. Mathematics is a powerful tool that we use to model the universe around us. To categorize it as a “mental game” is disingenuous at best. But the fact that it is powerful doesn’t mean that it somehow exists outside the human mind. This is a testament to the human mind, not to some ill-defined necessary being.Brother Brian
March 20, 2019
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H, if Mathematics does not in the end accurately and substantially refer to anything outside the human mind then it is just a mental game. KFkairosfocus
March 20, 2019
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I don't think dismissing logical mathematical systems as "just a mental game" is justified.hazel
March 20, 2019
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F/N: Guardian on the Walkie Talkie building: https://www.theguardian.com/artanddesign/architecture-design-blog/2015/sep/02/walkie-talkie-london-wins-carbuncle-cup-worst-building-of-year It has singed shopfronts, melted cars and caused great gusts of wind to sweep pedestrians off their feet. Now the Walkie Talkie tower, the bulbous comedy villain of London’s skyline, has been bestowed with the Carbuncle Cup by Building Design (BD) magazine for the worst building of the year. Responsible for a catalogue of catastrophes, it is hard to imagine a building causing more damage if it tried. It stands at 20 Fenchurch Street, way outside the city’s planned “cluster” of high-rise towers, on a site never intended for a tall building. It looms thuggishly over its low-rise neighbours like a broad-shouldered banker in a cheap pinstriped suit. And it gets fatter as it rises, to make bigger floors at the more lucrative upper levels, forming a literal diagram of greed . . . . The headquarters of the Royal Institute of Town Planners stands two streets away. “It’s a daily reminder,” sighs one employee, “never to let such a planning disaster ever happen again.” Just a reminder, KFkairosfocus
March 20, 2019
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BB, let's start simply. Is it true that || + ||| --> ||||| ? Is or is that not a mathematical truth? Does it need to be pinned in an axiomatic system and derived from theorems to be true -- or more exactly, proved i.e. a result from a game? If your concept of Mathematics cannot stand up to joining two sticks to three more to obtain five, and if that fiveness is just a mental game to you, I would advise a pause to re-think. KF PS: And BTW, Mathematics is not science.kairosfocus
March 20, 2019
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KF
BB, does mathematical truth exist?
I'm not sure what you mean by this. Do mathematical proofs exist? Yes. That is what makes mathematics different than the rest of science. But that still doesn't mean that mathematics exists outside and independent of the human mind.Brother Brian
March 20, 2019
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H & DS, I again point to the mobius strip as a direct empirical demonstration of structure and quantity embedded in the world. I have long since pointed out how the requisites of a distinct possible world lead directly to numbers as necessary but obviously abstract entities antecedent to creatures such as ourselves. Most recently, I have discussed the issue that a possibility space is not nothing, KFkairosfocus
March 20, 2019
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BB, does mathematical truth exist? If so, what do you mean by truth. If not, why not. And, why do we see the uncanny effectiveness of math in the real world. Where, I again point to the mobius strip as a direct, mind-independent manifestation of structure and quantity embedded in real world bodies. The responses to this case and similar ones actually tell us far more on the roots of disagreements than any metaphysical skepticism on your part. For, the case in hand is a direct, empirical, readily replicable demonstration. KFkairosfocus
March 20, 2019
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hazel:
But there is nothing in my own experience that leads to that hypothesis, so I ground my understanding in human beings, human minds, and human symbol systems.
Then your experience is very, very limited and you have clearly never read of the exploits of Srinivasa Ramanujan and how he came about his knowledgeET
March 20, 2019
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Brother Bogart:
One side believes that math exists somewhere outside the human mind and others don’t find the arguments or evidence for this compelling.
Those "others" don't have an argument or any evidence. So who cares if they want to live in denial? Srinivasa Ramanujan is evidence enough that math exists somewhere outside the human mind. Ignorance of Srinivasa Ramanujan is not an argument.ET
March 20, 2019
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hazel:
As I discussed in 258, I don’t think the idea that these exist apart from human beings in some separate Platonic realm is feasible.
Well, they do. All information exists regardless of us. That is just the way it is in this Intelligently Designed universe.ET
March 20, 2019
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Hazel
This difference is probably the background source of most of the disagreements on this subject that go on here.
I don't think there is any "probably" about it. One side believes that math exists somewhere outside the human mind and others don't find the arguments or evidence for this compelling.Brother Brian
March 20, 2019
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hazel,
This difference is probably the background source of most of the disagreements on this subject that go on here.
Amen to that.daveS
March 20, 2019
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Yes, Dave. There are those that think all this math can and does exists in the Mind of God, or some other Being, and thus ground the "Platonic nature" of math there. But there is nothing in my own experience that leads to that hypothesis, so I ground my understanding in human beings, human minds, and human symbol systems. This difference is probably the background source of most of the disagreements on this subject that go on here.hazel
March 20, 2019
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F/N: Guardian on London (the spoiled skyline city) and new year's resolutions for architects: https://www.theguardian.com/artanddesign/architecture-design-blog/2014/dec/31/new-years-resolution-architects-2015-smart-cities-poor-doors And, concerns on skylines: https://www.theguardian.com/uk-news/2016/aug/27/londoners-back-skyscraper-limit-skyline Concerns on skyscrapers: https://www.theguardian.com/artanddesign/2014/apr/29/top-10-worst-london-skyscrapers-quill-odalisk-walkie-talkie KF PS: Money clip from the last:
Sprouting over every corner of the city, most are of an architectural quality that recalls the outskirts of Dubai or Shenzhen. The overall impression is of an unplanned free-for-all, a steroidal frenzy of building tall, with little attention to individual design quality, or the cumulative effect that these scattered hulks might have on the city. The Planning Decisions Unit of the Greater London Authority, the body responsible for greenlighting these schemes, begs to differ. "It is simply not true to say these towers haven't been planned," says director Colin Wilson. "They have been very carefully planned. But we prefer to use a flexible framework, rather than a rigid masterplan. This liberty is what makes London successful." The London Plan, the mayor's rulebook for development across the capital, supports tall buildings where they "create attractive landmarks enhancing London's character". It states that such developments "should be of the highest design quality … attractive to look at and, where appropriate, inspire, excite and delight". [--> as in should reflect sound aesthetics individually and in context] So how are these rules shaping up in reality?
kairosfocus
March 20, 2019
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hazel, Math Guy, KF, From post #288,
Perhaps Hazel’s view of platonism existing in minds is correct, just not human minds.
Is it necessary to posit another "participant" (God in particular) in our mathematical activities? What evidence is there which supports this? As Math Guy stated earlier:
Gödel’s second incompleteness theorem implies an infinite hierarchy of ever more complicated (but true) statements. Since human minds are bounded at some level and an infinite hierarchy is unbounded, there will be theorems too complicated to be comprehensible by human minds.
Yes, but we do what we can, and it turns out to be rewarding as well as practically useful.
The theorem cited actually shows that no single formal system encompasses all of mathematics. So all of Hazel’s logical consequences, known or unknown, comprise only a tiny sliver of mathematics, since those consequences are derived from within a single formal system.
I'm not sure that hazel means that we must choose a single formal system and stick with it forever. We can explore various systems around the hierarchy MG described, within our abilities (which is what mathematicians actually do, as far as I know). One can prove theorems in Peano Arithmetic before lunch, and then switch to some stronger, large cardinal theory after lunch.daveS
March 20, 2019
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MG: I see your:
Johannes Kepler wrote (translated from Latin) “Geometry is one and eternal shining in the mind of God. That share in it accorded to humans is one of the reasons that humanity is the image of God.” My own opinion is that “geometry” can be replaced by much more in the above quote. On the other hand, humans are bound by finite time and space. Our mathematical deductions require formal systems like ZFC with finitely many axioms (or at most recursively many: consider the Axiom of Replacement) and proofs with finitely many steps. However as I noted previously, any such formal system that is capable of forming the set of integers is necessarily incomplete and contains true sentences that are unprovable within the system. God (by definition) is supernatural and not bound by finite processes (or countable processes, or …..) and likely can “see” the validity of such truths just as we can immediately recognize 2+2 =4 without having to count objects.
I think that fits with my remarks just above. What do you think? KFkairosfocus
March 20, 2019
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PS: Let us now also contemplate infinite, trans spatial, trans temporal necessary being Mind. Could not such a Mind know all possibilities and whatever is knowable? Let me add, including any particular decimal digit of sqrt2 or pi etc. Would not that Mind constitute, instantly, a domain -- as opposed to a concrete location that is only contingently present in some space and time etc -- a realm that transtemporally knows all possibility spaces, actualised or not? Would that not then fill the bill on being a Platonic domain? One, that is eternal such that discoveries, calculations or creations/inventions we make at particular times and places are automatically trans-temporally, trans-spatially present to such a Mind . . . we can even see that an extradimensional being could fill this bill just as for one y, any x is available in the plane. Is such a Mind inherently silly, incoherent, impossible of being? (Where, serious candidate necessary beings are either impossible of being or else actual; such actual beings are part of the fabric for any world to exist.)kairosfocus
March 20, 2019
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H, Let us not conflate an abstract, logic model world with necessary entities that are framework to any possible world. Say, at some particular place and time as actors in our world, we create some game or model, which is thus in effect a formal system with its logic of being and of process spanning a space of possibilities . . . in what we could call a realm of ideas, perhaps even a transfinite space -- e.g. a game with a token that moves on an endless tape following a program (resemblance to a Turing Machine is intended). That model spans an abstract space of possibilities we cannot exhaust already but we know that space well enough to accurately refer to it, to learn and express truths about it, maybe even profound truths that lead to a discipline we now call Computer Science. But that also means that several distinct individuals can work with and hold in mind truths regarding that space. The space is abstract, it has some sort of individual- or group- independent reality, it intersects with the real world in ways that constrain logic of physical beings, and it is significantly independent of any particular individual's thoughts or those of the collective community working with it. After all, individuals and the community may err about it but then would be subject to the correction of more accurate reasoning. We thus see abstract logic model worlds with possibility spaces framed by the logic of being, which have sufficient independence of individual thinking or even intersubjective consensus to be truly objective. Objectivity does not entail or require concreteness nor is it dependent on individual or collective opinions. So far as I can see, to be objective we need a framework that warrants claims as credibly true and so reliable, as well as a tolerably effective means of detecting and improving on our errors. None of that requires that the domains so contemplated only comprise concrete, material entities. Or even that we have arrived at comprehensively absolute truth as a body, i.e. while we know some self evident and some other necessary plumb line truths, there is no need to assume or pretend that our system as a whole or for the most part is free of errors. Hence, the concept: improve on. We also see here a good reason to set aside nominalism (above and beyond its incoherence), as here we see abstract possibility spaces independent of our fallibility and shaped by the logic of being. It seems to me that we need to stretch our concept of being to include abstract entities, quantities and structures and to recognise the power of logic especially when coupled to being and possible worlds speak. Pardon, I now add a clip from Davies and Walker on phase or configuration or state space, an important concept in Statistical Thermodynamics and related fields . . . which BTW also brings out the significance of fine tuning and of particular start-points:
In physics, particularly in statistical mechanics, we base many of our calculations on the assumption of metric transitivity, which asserts that a system’s trajectory will eventually [--> given "enough time and search resources"] explore the entirety of its state space – thus everything that is phys-ically possible will eventually happen. It should then be trivially true that one could choose an arbitrary “final state” (e.g., a living organism) and “explain” it by evolving the system backwards in time choosing an appropriate state at some ’start’ time t_0 (fine-tuning the initial state). In the case of a chaotic system the initial state must be specified to arbitrarily high precision. But this account amounts to no more than saying that the world is as it is because it was as it was, and our current narrative therefore scarcely constitutes an explanation in the true scientific sense. We are left in a bit of a conundrum with respect to the problem of specifying the initial conditions necessary to explain our world. A key point is that if we require specialness in our initial state (such that we observe the current state of the world and not any other state) metric transitivity cannot hold true, as it blurs any dependency on initial conditions – that is, it makes little sense for us to single out any particular state as special by calling it the ’initial’ state. If we instead relax the assumption of metric transitivity (which seems more realistic for many real world physical systems – including life), then our phase space will consist of isolated pocket regions and it is not necessarily possible to get to any other physically possible state (see e.g. Fig. 1 for a cellular automata example).
[--> or, there may not be "enough" time and/or resources for the relevant exploration, i.e. we see the 500 - 1,000 bit complexity threshold at work vs 10^57 - 10^80 atoms with fast rxn rates at about 10^-13 to 10^-15 s leading to inability to explore more than a vanishingly small fraction on the gamut of Sol system or observed cosmos . . . the only actually, credibly observed cosmos]
Thus the initial state must be tuned to be in the region of phase space in which we find ourselves [--> notice, fine tuning], and there are regions of the configuration space our physical universe would be excluded from accessing, even if those states may be equally consistent and permissible under the microscopic laws of physics (starting from a different initial state). Thus according to the standard picture, we require special initial conditions to explain the complexity of the world, but also have a sense that we should not be on a particularly special trajectory to get here (or anywhere else) as it would be a sign of fine–tuning of the initial conditions. [ --> notice, the "loading"] Stated most simply, a potential problem with the way we currently formulate physics is that you can’t necessarily get everywhere from anywhere (see Walker [31] for discussion). ["The “Hard Problem” of Life," June 23, 2016, a discussion by Sara Imari Walker and Paul C.W. Davies at Arxiv.]
Now, let us set one such game as the [study of the] logic of structure and quantity and call it Mathematics. Which just happens to be pivotal to the sciences and to daily affairs in general. Where, we know per Godel, that our contemplations can never span the possibilities on any formal set of start-points (axioms) and we cannot build such a limited set that we can guarantee as coherent. Mathematics is inherently transfinite and irreducibly complex, no finite mind can comprehensively span it. Indeed, we are forced to take it on trust that our more comprehensive thoughts and systems are coherent, i.e. to walk by faith and not by sight, though such can be reasonable and responsible. Where, for me, things like the Euler identity tell me that major domains are perfectly locked together in infinitely precise fit. Where also, there are many independently intelligible and known "core mathematical facts" that constrain axiomatisations. So, just to build our civilisation, to run it and to do science, we are forced to work with an objective abstract realm of structure and quantity that has ability to correct our errors. Reality and being are not constrained to materiality and concreteness. Thus also, let us posit another game, one that creates or contemplates entities that manifest beauty. That is, phenomena, that as contemplated by suitably cognitively capable beings will commonly excite appreciation, a sense of high worth or value tied to form, wonder, pleasure etc; responses which are in material part intelligible, being due to manifest form, proportions, symmetry, patterns, harmony, balance, subtle asymmetry, focus, verisimilitude, dynamism, etc. and so too due to their overall structure and/or composition and creativity. Or, as AmHD put it: "[a] quality or combination of qualities that gives pleasure to the mind or senses and is often associated with properties such as harmony of form or color, proportion, authenticity, and originality." Where, several of these aspects are direct manifestations of structure and quantity, so are intelligible and amenable to derivations and calculations -- as say Vitruvius and Da Vinci showed. Where, yes, mathematical objects can and do fit this conception of beauty, e.g. famously, 0 = 1 + e^i*pi. Or even: i^i = e^-pi/2 = 0.20787956 . . . Let us call this new game, Aesthetics. Why should we suppose that there is no more to this logically governed game than essentially arbitrary individual or intersubjective tastes and opinions, any more than with Mathematics or Physics etc? I can see no good reason to so confine Aesthetics. Aesthetics is thus both a significantly objective study and that study addresses an equally objective substance. Beauty is more than the eye and peculiar tastes of the beholder. KFkairosfocus
March 20, 2019
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Perhaps Hazel's view of platonism existing in minds is correct, just not human minds. Johannes Kepler wrote (translated from Latin) "Geometry is one and eternal shining in the mind of God. That share in it accorded to humans is one of the reasons that humanity is the image of God." My own opinion is that "geometry" can be replaced by much more in the above quote. On the other hand, humans are bound by finite time and space. Our mathematical deductions require formal systems like ZFC with finitely many axioms (or at most recursively many: consider the Axiom of Replacement) and proofs with finitely many steps. However as I noted previously, any such formal system that is capable of forming the set of integers is necessarily incomplete and contains true sentences that are unprovable within the system. God (by definition) is supernatural and not bound by finite processes (or countable processes, or .....) and likely can "see" the validity of such truths just as we can immediately recognize 2+2 =4 without having to count objects.math guy
March 19, 2019
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So, MG, you do believe that "The Game of Life has always existed, and so has every other similar game, including ones on hexagonal grids, ones in which most or all starting generations die quickly, etc."? Even if I made up some kind of system right now that had some logical rules and produced results, you would say that that system had eternally existed in a Platonic realm? I assume that is what you think. I have explained that I find that idea very unreasonable, so I think just trying to make it as clear as I can how I see it, rather than that way, is all I can do in this discussion.hazel
March 19, 2019
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This was written before I saw MG's 284, but I'll post it anyway: I like the way kf phrased things above: I would generalize and say any logical mathematical system has a "possibility space [that] is a map of realities that could be instantiated." Although, as stated above, these are "realities" in that they are elements in the possibility space, not that they exist somehow outside that system. Consider two examples: What is the area of a triangle with sides of 917, 455, and 620 units, and what is the area of a triangle with sides of 3, 4, and 5. Now I am certain (and let us assume this is true) that the first question has never been asked before, and I am certain the second one has been asked. I’m sure we all agree that for each there is unique right answer. Each answer is in the possibility space of all questions about numerical quantities associated with triangles, the vast, vast majority of which have never been asked. They each exist as a determined, unambiguous logical consequence of our system of geometry. So ontologically, they have the same status: each answer is a logical consequence in the possibility space of all questions about numerical quantities associated with triangles. The fact that the second is known (A = 6 u^2) and the first is not is a fact about our knowledge within the system, but not a fact about the elements of the system itself. It is an epistemological fact, not an ontological fact. But wait! I know how to answer the first question! Heron’s formula gives me 126,513.18 u^2, to eight significant digits. Now we know the area, and moments ago we didn’t. I am the first person to ever know the answer to this question, and now I have shared it with you. Did its ontological status change? Is it any more “real” now, or “real” in a different way, than it was just because we have figured it out and expressed it within the written system that we use to instantiate these logical results? Which returns us to the question of in what way is formal mathematics real? Where does its reality reside? Or is that really a poor question, one which implies characteristics of math that are not true, or at least unknown. My answer to that question, to repeat, is that mathematical elements exist as logical possibilities within the symbol systems we use to instantiate our abstract concepts: they are potential concepts that can exist in human minds when those minds use the logical tools they haveto draw out the logical conclusions of those system. Some of them have been articulated and many have not (as my example is meant to show), but they all have the same status as a potential concept with our minds. They aren’t “residing” someplace else, waiting for us to access them, they are “residing” “inside us”, waiting for us to find them within the logical systems we have developed. All interesting stuff to think about!hazel
March 19, 2019
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KF @ 275 "there is a difference between advancing stepwise on a potentially transfinite and actualising same stepwise. The latter is a supertask that cannot be achieved. The former simply indicates onward progress without finite limit. " This appears to be Aristotle's distinction between a potential and an actual infinity. The "supertask" of attaining the actual infinity cannot be achieved by humans in finite space and time. But to a Being not bound by such circumstances......?math guy
March 19, 2019
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H @ 258 The first definition of mathematical platonism is standard and what I agree with personally (the second is silly IMHO). H @ 281 "That is, the things we haven’t figured out yet and the things we have figured out have the same fundamental ontological status: they exist as possible elements of the logical system." Although the above is referring specifically to cellular automata, I would opine the sentence is equally applicable to any consistent formal system, subject to the caveat that "figured out" means logically proved/derived, in which case the qualifier "possible" should be omitted. KF adds "If you want to call that a platonic world, fine.” I would call it a sub-world within the platonic realm because of my post @ 247 : "Gödel’s second incompleteness theorem implies an infinite hierarchy of ever more complicated (but true) statements. Since human minds are bounded at some level and an infinite hierarchy is unbounded, there will be theorems too complicated to be comprehensible by human minds." The theorem cited actually shows that no single formal system encompasses all of mathematics. So all of Hazel's logical consequences, known or unknown, comprise only a tiny sliver of mathematics, since those consequences are derived from within a single formal system.math guy
March 19, 2019
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Thanks, MG. I know you've explained your day job: FWIW: I'm enjoying my retirement. I will be interested any time you c an respond, although I hope when you do you back up a bit and look at relevant posts since the last time you were here.hazel
March 19, 2019
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As I have written previously, I have a job (some of which entails proving theorems!) that prevents regular posts to this forum. So my periodic absence is not necessarily a sign of throwing in the towel, so to speak.math guy
March 19, 2019
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kf writes,
"DS , Near as I understand, Conway’s game of life creates an abstract space of possible states based on how it can be played. much like chess or draughts etc. Are those possibilities actually possible? I would think so; we thus have a case of an abstract, logic model world we can conceive of and represent. So, the possibility space is a map of realities that could be instantiated. The possibilities are obviously abstract but are real possibilities and we may speak truthfully (or make errors regarding) them.
Much of that is correct, except it’s not actually a game like chess: once you create the original configuration, the rules create each succeeding generation in an iterative fashion. However, the question, which is one we return to, is what is the nature of those abstract possibilities (which is a better way of asking the question than “where do those abstractions exist?)? The answer I have offering is that the abstractions are elements of the logical system, and that the logical possibilities are as much elements of the system as the actual states that are instantiated at various times. A state that is instantiated (gen 3 of an original state of 9 cells with the middle cell missing, for instance, which is a common result when people explore the game) is not more or less an abstract element of the system than the 1000th generation of some complex gen 0 that no one has ever investigated. That is, the things we haven’t figured out yet and the things we have figured out have the same fundamental ontological status: they exist as possible elements of the logical system. kf adds, “If you want to call that a platonic world, fine.” No, I don’t think that is fine, because I don’t think what I am describing and what a “platonic world” implies are the same thing. The systems we are talking about are part of human understanding, which requires some instantiation in a symbol system, written or verbal, that includes various beginning axioms, definitions, etc. As I discussed in 258, I don’t think the idea that these exist apart from human beings in some separate Platonic realm is feasible.hazel
March 19, 2019
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KF <blockquote<This sort of thinking is part of the in-progress decay of our civilisation. KF You must be fun at parties. :)Brother Brian
March 19, 2019
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