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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
H, the structure and quantity manifested in the Mobius strip are demonstrably independent of our concepts and abstractions. Cut it around in the middle vs 1/3 way across and dramatically different results obtain, because of its structural and quantitative properties, which can properly be said to be embedded in space and in the body. Further to this, by contrasting an ordinary cylindrical loop we can readily demonstrate one side in a 3-d object vs two, and one edge vs two. We use terms and form concepts about unity and duality but those concepts reflect a real albeit abstract property. Our thoughts are not controlling or inventing the behaviour, they are responding to the behaviour.In the case of the Mobius strip there is documentation of such a strip in Roman times, though there are no records of certain astonishing properties. Cutting around would likely have been harder to do, though spring shears existed and pivoted scissors seem to have been invented by the Romans c 100 AD, they were also known in the far east. KFkairosfocus
March 23, 2019
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ba, I was talking about everyday experiences, such as the Mobius strip that kf has now mentioned 27 times. I understand that there are other issues when you think about this at the quantum level.hazel
March 23, 2019
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as to:
Of course what happens in the world is independent of what we think about it. Who said otherwise?
Hazel, did you even read the reference that you yourself gave me the other day?
More Than One Reality Exists (in Quantum Physics) By Mindy Weisberger - March 20, 2019 Excerpt: “measurement results,, must be understood relative to the observer who performed the measurement”. https://www.livescience.com/65029-dueling-reality-photons.html Yet Hazel, regardless of how he (Weinberg) and other atheists may prefer the world to behave, and as the present experiment that you yourself referenced further verifies, quantum mechanics itself could care less how atheists prefer the world to behave. As your referenced article itself stated, “measurement results,, must be understood relative to the observer who performed the measurement”. https://uncommondescent.com/intelligent-design/viruses-devolve/#comment-674639
bornagain77
March 23, 2019
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Of course what happens in the world is independent of what we think about it. Who said otherwise? It is the abstractions we make about what happens that I am talking about. To quote myself above avove
It seems clear to me that even though individual objects exist in the physical world, the abstract ideas of quantity, addition, and equality arise in the minds of beings with rational, abstract cognitive abilities such as we have. If the world were just monkeys, 2 + 3 = 5 would not exist as an abstract fact, which is of course different than saying the pebbles wouldn’t be sitting on the rock.
hazel
March 23, 2019
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H, I actually think it is clear that the difference is about the distinct identity, nature and being of entities which we interact with and form concepts about. The mobius strip is a key test, as its behaviour when cut around is radically different if you cut at the middle (1/2 way across) or 1/3 way across. This is strictly independent of our thoughts, concepts, opinions, beliefs, agreement or disagreement, and it directly demonstrates that key aspects of structure and quantity are embedded in the world independent of our thoughts. KFkairosfocus
March 23, 2019
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KF,
both are true and are so regardless of who is or is not willing to agree.
Indeed.daveS
March 23, 2019
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kf writes,
And BTW, knowledge is yet another abstract entity
Yes, another abstract entity in the minds of human beings. I'm pretty sure you don't get the point I've made in 337 and repeated in 355. I don't mean "agree with it", I mean "get it." I don't deny the abstract entities exist. It is where and how they exist that our differences lie.hazel
March 23, 2019
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DS, both are true and are so regardless of who is or is not willing to agree. Marks' point is a case of known unknowns, the Heisenberg-Einstein uncertainty issues are other known unknowns. A big problem is Rumsfeld's unknown unknowns. KFkairosfocus
March 23, 2019
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F/N: I updated the OP to include a picture on Walkie-Talkie's street scorcher problem. KFkairosfocus
March 23, 2019
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Bradley shows that one cannot claim that metaphysical knowledge is impossible without contradicting oneself. Robert Marks states that we can know that things exist which are unknowable. Luckily most of us agree with these two statements.daveS
March 23, 2019
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H, we cannot know completely and perfectly, but as F H Bradley highlighted, the very act of denying any knowledge of actual reality is by implication such a knowledge claim. It refutes itself. I would argue that knowledge of certain key structures and quantities is knowledge of aspects of reality. And BTW, knowledge is yet another abstract entity. KFkairosfocus
March 23, 2019
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F/N: Walkie Talkie melting car parts and shop front stuff, scorching carpets etc: https://www.bbc.com/news/uk-england-london-23930675 (see egg frying demo: https://www.youtube.com/watch?v=XBPq6B_soAw ) and no, I don't buy the excuse this was not foreseen. The seasonal motion of the sun in the sky (part of the seasons due to the tilt between our plane of orbit and axis of rotation which precesses) has been well known for thousands of years, as is the concentrating effect of a curved reflecting surface. Vitruvius, in discussing what an Architect should know, speaks to that sort of astronomical knowledge, c what 30 BC? This red flags the quality of regulatory review on the building. KFkairosfocus
March 23, 2019
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I thought that might be what you referring to. I accept that external reality exists, and that our perceptions of it, at the macroscopic level that they exist, are in general accurate. I understand the argument that we can't really know the thing-in-itself, but I don't find that a useful idea to attach myself to, even though, technically, it is true.hazel
March 23, 2019
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H, the gap between the inner phenomenal world (appearances) and the noumenal (things in themselves), the phrase is by no means original to me. I think it is from Kantians, not Kant himself. KF PS: F H Bradley -- a noted British philosopher -- on the issue:
We may agree, perhaps, to understand by metaphysics an attempt to know reality as against mere appearance, or the study of first principles or ultimate truths, or again the effort to comprehend the universe, not simply piecemeal or by fragments, but somehow as a whole [--> i.e. the focus of Metaphysics is critical studies of worldviews] . . . . The man who is ready to prove that metaphysical knowledge is wholly impossible . . . himself has, perhaps unknowingly, entered the arena . . . To say the reality is such that our knowledge cannot reach it, is a claim to know reality ; to urge that our knowledge is of a kind which must fail to transcend appearance, itself implies that transcendence. For, if we had no idea of a beyond, we should assuredly not know how to talk about failure or success. And the test, by which we distinguish them, must obviously be some acquaintance with the nature of the goal. Nay, the would-be sceptic, who presses on us the contradictions of our thoughts, himself asserts dogmatically. For these contradictions might be ultimate and absolute truth, if the nature of the reality were not known to be otherwise . . . [such] objections . . . are themselves, however unwillingly, metaphysical views, and . . . a little acquaintance with the subject commonly serves to dispel [them]. [Appearance and Reality, 2nd Edn, 1897 (1916 printing), pp. 1 - 2; INTRODUCTION. At Web Archive.]
kairosfocus
March 23, 2019
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kf writes, "We cannot get away from abstracta." I agree, and said this at 337:
It seems clear to me that even though individual objects exist in the physical world, the abstract ideas of quantity, addition, and equality arise in the minds of beings with rational, abstract cognitive abilities such as we have. If the world were just monkeys, 2 + 3 = 5 would not exist as an abstract fact, which is of course different than saying the pebbles wouldn’t be sitting on the rock. I think one reason it seems compelling to say 2 + 3 = 5 exists outside of our minds is because we can’t think of what the world is like without using the the concepts already present in our minds. We have to use our minds to describe the world, so how can we describe what it is like when it is not being described by a mind? It seems to us that the abstraction is a necessary part of the world, and in the world, but that is just a projection of our understanding overlaid on the world we observe. We can’t think about the world without using the abstract concepts we have in our minds.
hazel
March 23, 2019
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kf, what is this "Kantian ugly gulch" of which you speak so often?hazel
March 23, 2019
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PS: Such abstracta then extend very naturally to aesthetics and ethics too. Namely, are there true principles of value/worth, aesthetical or ethical? Where if something gives us pleasure to contemplate, can that be grounded on objective principles? Yes, tracing to symmetry, balance, harmony, variety, etc. -- many of which pivot on the embedding of structure and quantity in the world of objects. Are such contemplations autonomous? No, all aspects of human reality and cognition are morally and rationally governed. Some aesthetics pleasures begin to drift into the oppressive -- the castrata singers of the past being a case in point. We forgo certain aesthetical experiences on moral grounds, for cause . . . castrata being a capital example, the last such having been laid to rest many years ago. Rumours regarding certain pop stars notwithstanding. Likewise, pornography can be aesthetically attractive as the notorious case of D H Lawrence demonstrated. But, again, moral considerations are at stake. Anatomy textbooks contain images that could be used in a pornographic manner -- for prurient purposes, but have a very important and legitimate use. A katana of traditional manufacture is indubitably highly refined and beautiful, but may be used to murder. The state of Japan finds it an important contribution to culture to support as living treasures, the expert artisans who to this day produce such swords. Even so strapped a country as Jamaica finds it a vital contribution to national progress to provide support to writers. By contrast, London has clearly done significant damage to its aesthetic worth by granting permission to erect the Walkie Talkie building and the like -- something that can have measurable impact on tourism and on degradation of environment that supports various social pathologies with even criminal manifestations. Ironically, in other contexts, even such marred entities may be an aesthetic improvement. And more.kairosfocus
March 23, 2019
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DS & H: I note, given a running TM, the options per logic of process are that that state continues [does not halt] or changes [halts]. The former in principle is infinite so cannot be directly observed but can be inferred similar to say the cycling of a long division exercise . . . our first imposed halt on an algorithm. The two alternatives, in context, are mutually exclusive and exhaustive -- they dichotomise possibilities.If no halt, then continue.(If you dispute such, kindly explain why.) The remarks on truth-value realism are a discussion on the nature of propositions: AmHD: "5. Logic a. A statement that affirms or denies something. b. The meaning expressed in such a statement, as opposed to the way it is expressed." Collins: "2. (Philosophy) philosophy a. the content of a sentence that affirms or denies something and is capable of being true or false b. the meaning of such a sentence: I am warm always expresses the same proposition whoever the speaker is." RHK Webster's: " 6. Logic. a statement in which something is affirmed or denied, so that it can therefore be significantly characterized as either true or false." These document the standard understanding: A proposition is an assertion about reality, internal or external, that X is so or else that ~X is so, which may be accurate or inaccurate, beyond whatever we may happen to know at a given point on whatever degree of warrant we may have. In a Kantian ugly gulch haunted world, it is unsurprising that some would seek an alternative (but notice the subtle self-reference -- whether a proposition refers to what is the case or not is itself a proposition.) The key issue is that we must distinguish accuracy of reference from adequacy of warrant. That something refers to what may or may not be the case in reality is one thing, that we have a belief or perception that it is or is not so is a second, that we may be in doubt is a third, that we have warrant sufficient to responsibly accept it as so is a fourth. But the second through the fourth do not alter the first, they pivot on it. Nor does resorting to the Language Game model escape this issue: A language of Mathematicians L-M in which X is true/~true and a different one for Philosophers L_P where such is not the case, becomes obscure and self-referentially incoherent as this discussion is itself a discussion of what is or is not the case. The infinite regress problem opens up. Note, how SEP continues:
many nominalists endorse truth-value realism, at least about more basic branches of mathematics, such as arithmetic. Nominalists of this type are committed to the slightly odd-sounding view that, although the ordinary mathematical statement (1) There are primes numbers between 10 and 20. is true, there are in fact no mathematical objects and thus in particular no numbers. But there is no contradiction here. We must distinguish between the language LM in which mathematicians make their claims and the language LP in which nominalists and other philosophers make theirs. The statement (1) is made in LM. But the nominalist’s assertion that (1) is true but that there are no abstract objects is made in LP. The nominalist’s assertion is thus perfectly coherent provided that (1) is translated non-homophonically from LM into LP. And indeed, when the nominalist claims that the truth-values of sentences of LM are fixed in a way which doesn’t appeal to mathematical objects, it is precisely this sort of non-homophonic translation she has in mind. The view mentioned in the previous note provides an example. This shows that for the claim Existence to have its intended effect, it must be expressed in the language LP used by us philosophers. If the claim was expressed in the language LM used by mathematicians, then nominalists could accept the claim while still denying that there are mathematical objects, contrary to the purpose of the claim.
See the regress beginning to gather speed? The very obvious solution is to acknowledge that we may ponder abstracta, which manifest themselves in part by being embodied in reality. As we can directly observe with the peculiar properties of the Mobius strip. We do an operation and see the paper model convincingly has ONE side and ONE edge, by contrast with an ordinary untwisted cylindrical loop of paper. This requires reference to what an operation is, what a loop is, what an edge is, what a side is, even what orientability is, etc. All of these are abstract, we cannot evade abstracta, universals, propositions, numbers, structures, information, energy, time, frequency, states of affairs and much more. Socrates is a name, an abstract entity. A certain bearer [abstract[ of this name is [abstract] a man [abstract]. Men [abstract] are [abstract] mortal [abstract]. Socrates is [abstract] mortal. We cannot get away from abstracta, and the nominalist's denial of same begins the self-referential spiral. The issue is indeed ontological, it pivots on the logic of being, what does it mean for a concrete [an abstract category] entity to exist, and to exist as an example or case of a class? What does it mean for such a universal to exist or be real, given its abstract nature? Ditto for the existence of a proposition. Ditto for a number, starting with one, two, zero. So far as I can see, we can make a simple observation: for some x to be, it is AS an instance of some distinct kind of thing, X-class, X_C. That is, there are in-common core properties that place x in X_C, and to hold distinct identity x, there is some unique aspect of x, A. We are back to law of distinct identity. Where, that x is concrete typically means that it is tangible, directly observable. In observing it, however, we are instantly forced to refer to and frame it in terms of abstracta. x is distinct from the rest of actualised or potential reality ~x, by virtue of some characteristic A, but it already belongs to classes, an actual or potential entity in reality for instance. We therefore must acknowledge that in-common aspect of being, which means that classes are real, exist in a different sense, intangible and abstract. First, obviously in our minds -- another intangible but directly experienced reality that is pivotal in grounding warrant for ever so much more. But, did our particular minds [dubiously . . . ] impose and project X_C unto x? No, down that road lies utter confusion, we recognised that x is or may be and that it therefore has a property of actual or potential existence along with many other things such as ourselves. So, too, we recognised X_C implicitly, the relevant class of x. Let x for the moment be say Earth in the solar system. Absent our planet, we would not exist, absent the Sol system, it could not harbour life including ours. The categories and members are antecedent to us, we recognise the realities we do not invent them. So, abstracta must have existence in some sense, a sense that must include cases antecedent to our existence. Classes hold reality by being natural or invented categories that reflect the in-common aspects of being. A proposition involving classes has existence not just as an utterance or thought but as a thought that asserts something to be true to reality, and may succeed or fail. A number has reality by expressing a particular sort of quantitative property of the world or wider reality and possible worlds within structures that tie such together: N, Z, Q, R, C. Such are aspects of reality, which is antecedent to our existence and independent of our success or failure in attempts to attain knowledge. Prime numbers have properties in common that we happen to find interesting. Between 10 and 20, 11, 13, 17, 19 are prime, 15 is odd but not prime. They are numbers with additional, restrictive properties. Where each is distinct, 11 is not 13 etc. We discover those properties, we did not invent them. Indeed, likely our remote ancestors saw this by noting how pebbles or sheep or fruit could not be evenly shared as intact units among a smaller number of persons, reflecting how concrete circumstances exhibit the in-common properties. (This then, doubtless led to fractions thus rationals, then onward irrationals etc as recognised real classes.) So, wisdom is to swallow the unpalatable but effectively undeniable reality of abstracta. Contemplated by minds, yet antecedent to our own. Real, as connected to any x holding distinct identity. And insofar as one may desire a world of forms that eternally recognises such, a logical candidate -- again peculiarly repugnant to many -- would be a mind at world-root. ut that is an onward debate. KFkairosfocus
March 23, 2019
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Dave quotes the SEP as saying,
For there are various accounts of how mathematical statements can come to possess unique and objective truth-values which do not posit a realm of mathematical objects.
I wonder if there is a nice layman's description someplace of what some of those accounts are?hazel
March 22, 2019
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Thanks Dave - that's very interesting, and I appreciate your finding it. I have already been burned once, so to speak, by expressing an attraction to a philosophical label, but I like the phrase and description, for the most part, of truth-value realism, which is a term I have never heard before.hazel
March 22, 2019
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hazel, Thanks, you are correct. It is actually a separate question after all. I should be talking about truth-value realism apparently:
Truth-value realism is the view that every well-formed mathematical statement has a unique and objective truth-value that is independent of whether it can be known by us and whether it follows logically from our current mathematical theories. The view also holds that most mathematical statements that are deemed to be true are in fact true. So truth-value realism is clearly a metaphysical view. But unlike platonism it is not an ontological view. For although truth-value realism claims that mathematical statements have unique and objective truth-values, it is not committed to the distinctively platonist idea that these truth-values are to be explained in terms of an ontology of mathematical objects. Mathematical platonism clearly motivates truth-value realism by providing an account of how mathematical statements get their truth-values. But the former view does not entail the latter unless further premises are added. For even if there are mathematical objects, referential and quantificational indeterminacy may deprive mathematical statements of a unique and objective truth-value. Conversely, truth-value realism does not by itself entail Existence and thus implies neither object realism nor platonism. For there are various accounts of how mathematical statements can come to possess unique and objective truth-values which do not posit a realm of mathematical objects.
(from the SEP entry on platonism in the philosophy of mathematics)daveS
March 22, 2019
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kf writes,
There is a single objectively true answer to the question of whether it halts or not (according to platonists anyway).
To me, too. I don't see what this has to do with Platonism. As I have explained, once a logical system is started, the logical consequences are determined.hazel
March 22, 2019
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KF, I think I mostly agree, particularly with this:
Once it exists — that is a part specification of a contingent state of affairs, and it is fed an input it will halt or else it will not halt...
There is a single objectively true answer to the question of whether it halts or not (according to platonists anyway).daveS
March 22, 2019
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DS, a Turing machine is a highly contingent entity so by definition it will not exist in every possible world, and it is dependent on antecedent causal factors. Once it exists -- that is a part specification of a contingent state of affairs, and it is fed an input it will halt or else it will not halt, but that is often undecidable a priori. That behaviour is the result of its having been set up and having particular rules and structures of operation, perhaps as a model on some computer as we do not usually actually set up a physical instantiation or model, which given the infinite tape is in principle impossible.Likewise, we can see that a set of activities proceeds and for all we know seems to be set to go on endlessly but that is something we cannot observe: a stepwise set of causally connected finite stages spanning an actual infinity is a supertask; we can observe a halt but not a no-halt, though we may properly infer that. That said, once set up and functional, it seems the case that its onward behaviour on an input is independent of what we may think assuming we do not interfere further. The Mobius strip is a clear case of this. KFkairosfocus
March 22, 2019
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KF, You're welcome. (Edit: Interesting, I've never seen pi in hex before). Going back to some points raised earlier about Möbius strips, formal systems, hazel's take on platonism, here's a related question. It sounds trivial, but there is a point.
Suppose we have an arbitrary Turing machine and a particular input for that Turing machine (i.e., a paper strip containing a finite string of symbols from its alphabet). Is it true that the Turing machine either will halt or not halt given this input, and that this behavior is independent of our minds? And further, that this behavior is the same in every possible world?
daveS
March 22, 2019
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DS, thanks, I had only seen some reference and vaguely remembered it. Four suitably framed binary digits in succession would give a hexadecimal digit but that would not readily translate into decimal digits. KF PS: Pi hex: 3.243F6 A8885 A308D 31319 8A2E0 37073 44A4 . . .kairosfocus
March 22, 2019
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KF,
Aren’t there procedures that give arbitrary position digits of pi, or are those in binary form, which does not simply translate to decimal digit form under place value notation.
In case this is not a rhetorical question, yes:
The Bailey–Borwein–Plouffe formula (BBP formula) is a spigot algorithm for computing the nth binary digit of the mathematical constant π using base-16 representation. The formula can directly calculate the value of any given digit of π without calculating the preceding digits.
daveS
March 22, 2019
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KF,
Despite a lot of speculation to the contrary, despite an expectation from evolutionary materialistic scientism (which is self-refuting) computation is simply not a process of rational inference and wider cognition.
I'm 99% sure you're correct. :PdaveS
March 22, 2019
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MG, 338: >> I was not insinuating that any particular UD characters were a-mats, rather that their position entails yet another logical absurdity. From [H,] 337: “So where does the abstraction 2 + 3 = 5 reside? In the world, or in our minds?” You have asked me (and other readers) an opinion question.>> 1: indeed. A necessary distinction. >> My opinion is both: abstraction is embedded into the fabric of the universe itself, as KF has demonstrated with his Möbius band example.>> 2: Thank you for this recognition of a striking manifest phenomenon that shows something that does not seem to have fully registered with many people in our civilisation. 3: In the context of this and linked exchanges at UD, a key point has been the definition of Mathematics I put on the table: the [study of the] logic of structure and quantity. 4: This points to both a discipline and to a reality-embedded intelligible substance that is there to be explored and teased out in terms of its rational principles, elements and organisation. >> However it takes a certain amount of intelligence (and attention) to observe such abstract relationships.>> 5: Yes, absent certain cognitive capabilities that manifest in abstract language, reasoning and ability to recognise the ability of truth -- accurate description of reality -- to span the gap between our inner world and the outer one of things in themselves, Mathematics is impossible as a disciplined study carried out by individuals in community. 6: I point to the central importance and sign of abstract, concept-rich language, including its extension into Mathematical exposition and computing science. >>As I have remarked @ 247, (some) humans have known since at least 1931 that there exist levels of abstraction too deep to be discovered by human minds. Whether those abstractions are also embedded into our universe or not (i.e. form a branch of applied math) cannot be determined by us.>> 7: A humbling, significant recognition. Let's pull back up 247, as comments get buried under the onward exchange:
MG, 247: >>This appears to be an issue for abstract objects that require a mind to live in because the truth or falsity of the conjecture (or septillionth decimal digit of pi) appear to be independent of any mind. In fact, 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. Where do these theorems reside?>>
8: Objective truth rests in credible, reliable, warranted accuracy of description of reality. 9: It is subjects who can recognise, warrant and state truth, and such subjects in our case are inherently, inescapably limited. 10: Where Mathematics is irreducibly complex and cannot be spanned by a coherent, known to be coherent, constructed axiomatic system. 11: So, there are structural and quantitative realities we cannot warrant on our schemes of thought, but perhaps we may state some of them as conjectures that turn out to be undecidable on particular axiomatic schemes such as ZFC. Which is itself an adjusted system ultimately deriving from the problems of naive set theory. KF PS: Aren't there procedures that give arbitrary position digits of pi, or are those in binary form, which does not simply translate to decimal digit form under place value notation. I confess to using tables of pi as random number tables, and to finding telephone numbers there etc. (My argument for sufficient randomness is want of correlation between two determined systems, the value of pi and the place value decimal system of counting on powers of 10.)kairosfocus
March 22, 2019
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H:
where does the abstraction 2 + 3 = 5 reside? In the world, or in our minds? It seems clear to me that even though individual objects exist in the physical world, the abstract ideas of quantity, addition, and equality arise in the minds of beings with rational, abstract cognitive abilities such as we have. If the world were just monkeys, 2 + 3 = 5 would not exist as an abstract fact, which is of course different than saying the pebbles wouldn’t be sitting on the rock.
I note, again, that we must distinguish the study of the logic of structure and quantity from the structure of structure and quantity that are evidently embedded in the world as part of a pattern of intelligible rational principles it manifests; often through the logic of being. I have pointed to the behaviour of a Mobius strip when cut around 1/3 way in from its edge vs in its middle. Those are embedded structural, quantitative properties in action, utterly independent of and antecedent to our cognition and conception. Likewise, if by some chance two pebbles fall on a rock shelf and later three more, by force of the logic of structure and quantity, there will now be five pebbles there, regardless of whether we have come along to contemplate it. These sorts of events exhibit the logic of being. At another level, such phenomena are pivotal to the ability of our cosmos to support the sort of life we enjoy. Again, antecedent to our existence. Indeed, the ideas of number, structure, wider quantities etc are mental phenomena. The disciplined, logical study of same is also mental. Just as ability to recognise and infer principles of aesthetics is a mental process. Likewise, the underlying inextricably entangled ethical process of moral government of our cognition by known duties to truth, right reason, prudence, fairness etc is inevitably mental as well. Moreover, I find the three are inextricably intertwined. I suggest, yet again, that while monkeys or birds etc evidently do not have the cognitive ability to develop mathematics, aesthetics and ethics as disciplines, there is significant evidence that such considerations are embedded in the world of life and in the prior ordering of an observed cosmos fine tuned in a multitude of ways that support such life. For the first, I point to D/RNA, the involved code and complex algorithms with associated molecular nanotech execution machinery found in metabolic automata with integrated von Neumann kinematic self replication facilities. For the latter I point to our discoveries of just how fine tuned our cosmos' physics is on key structures and quantities -- which are obviously embedded in the world -- that enable C-chemistry, aqueous medium, terrestrial planet in stellar and galactic habitable zone life. In short, there is significant but often overlooked evidence that points to a mathematically framed design of the world of life and the cosmos that hosts it. Such points onward to cognitive function and capacity to carry out such contemplations, i.e. to a cosmos-enabling, mathematically sophisticated designer as the locus of evident design. KFkairosfocus
March 22, 2019
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