Firstly, good spot about PA |-/- Prov S --> S

As you point out, that false derivational assumption which I took for granted and which led me to

PA |-- Prov('G') <--> ~Prov('G')

implies that PA is inconsistent due to LOM, something I overlooked as i don't think classically.

In fact, that 'Reductio ad absurdum' constitutes a weaker version of Lob's theorem, specifically applied to G.


But all of that said, from a constructivist position it doesn't seem surprising given that ~Prov already doesn't reflect non-derivability. Lob's theorem is just further ammunition for identifying truth directly with the derivable sentences, and for rejecting the interpretation of "prov" as meaning provability, and consequently for rejecting the semantic philosophical interpretation of incompleteness as implying 'arithmetically true but undecidable propositions'.

Let us suppose that everything you say is true. This still does nothing to address two facts: (1) the set of true formulas is not arithmetically definable, but the set of provable formulas is, whence the two must be distinct; — Nagase

Correct me if i'm wrong, but aren't we both referring to the fact that the negation of PA's provability predicate doesn't actually enumerate what isn't derivable in PA?

(2) truth is not conservative over PA, whence it can't be redundant. I sketched that argument in my first post here precisely so we did not get entangled in fruitless discussions about how we can know that G is true or about the Kirby-Paris theorem. — Nagase

If by truth not being redundant, you are merely referring to the difference between

PA |-\- S (i.e. the event that PA doesn't derive S) versus PA |-- ~Prov('S')

then I think we're in agreement. But in this event, the notion of truth is redundant in the sense that it doesn't transcend the notion of derivability within PA, which is what I took the OP's point to be.

As for G's "truth" value, I can only recognise it as having undefined truth value, both in the sense of it's unknown status as a derivable theorem of PA, and also in the sense of it's decoded arithmetical interpretation pertaining to the undecidable solutions of Diophantine equations. As an aside, i don't recognize the truth of Goodstein's Theorem, due to it's proof relying on non-constructive notions pertaining to transfinite induction.

Obviously, that particular argument assumes the soundness of PA, which you have disputed (this is a minority position, but one that I respect, if only because in the case of Nelson it generated some interesting mathematics). But this is not necessary for the argument to go through: one can start with Q and argue that any recursively axiomatized theory that extends Q will fall into the same problem, namely truth will be arithmetically undefinable and theoremhood will be arithmetically definable. Since no one that I know of doubts the soundness of Q (not even Nelson), the argument should go through. — Nagase

I'm not so much of the view that PA will turn out to be inconsistent within my life time, but rather of the view that talk of the consistency of PA is meaningless in both demanding infinite evidence and also in being logically inexpressible, as demonstrated formally by Godel's second incompleteness theorem.

By the way, if your ii.c) is correct, then PA is inconsistent. In any case, that is not a valid substitution instance of ii.a): ii.a) says merely that (assuming soundness) PA |- S iff PA |- Prov('S'), not that PA |- S <-> Prov('S') (the latter is a reflection principle and is actually not provable in PA). — Nagase

ii a) doesn't express soundness in the sense of begging an external notion of arithmetic truth, rather it expresses the ability of Prov to enumerate what is derivable within PA:

PA |-- S <---> PA |-- Prov('S')

this is a provable equivalence, because Prov(''S') merely enumerates what can be derived in PA and it doesn't lead to false derivations, because it directly encodes PA's axioms and rules of inference.

ii c): By itself ii c doesn't imply inconsistency, because it is specifically stated with reference to the Godel sentence G that diagonalizes the negative provability predicate:

PA |-- ~Prov('G') <--> Prov('G')

Only if PA |-- G is derivable does ii c lead to inconsistency.

ii c) is equivalent to Tarksi's undefineability of truth applied to PA, in which PA's derivability predicate, Prov, is taken to be it's "truth predicate" . It also implies the constructive content of Godel's first incompleteness theorem, namely that ~Prov is a misnomer in not being able to enumerate what PA cannot derive.

That's why i said i agreed with Mad fool, for reasons you hinted at. Truth, as far as mathematical logic is concerned, IS derivability , for mathematical logic cannot be partitioned into a metalanguage talking about an independent object language, for both languages are isomorphic to one another and suffer from the same undecidability.

I'm with the Madfool, paradoxically for reasons hinted at by nagese.

Godel sentences do not support the colloquial interpretation they receive, that is to say "G is true because G is unprovable, assuming Peano Arithmetic is consistent". This is because

i) Firstly, the hypothesis that PA is consistent is potentially falsifiable, but it can never be verified. Therefore it is irrational for logicians and philosophers to assume or even talk about PA's 'infinitely complete' hypothetical consistency. Instead, they should only talk about PA's consistency within a limited finite scope of derivations.

ii) PA cannot talk about what isn't provable in PA, which is the central conclusion of Godel's first incompleteness theorem, and corresponds to both the Halting Problem and Tarski's undefinability of truth within PA. In general, for any sentence S of PA we have

a) PA |-- S <---> PA |-- Prov('S')

e.g. PA derives S if and only if PA derives Prov('S')

b) PA |-- G <--> ~Prov('G') For any Godel sentence G.

Therefore b substituted into a leads to the conclusion

c) PA |-- ~Prov('G') <--> Prov('G')

In other words,

PA |-- ~Prov('S') does not imply that PA does not derive S .

i.e. Peano arithmetic cannot enumerate the formulas that PA cannot derive (duh).

Therefore G does not say that "G isn't provable"

iii) The arithmetical theorems that PA cannot derive, e.g. Goodstein's Theorem, aren't constructively acceptable in any case, due to them appealing to the Axiom of Choice. These theorems cannot be computationally or otherwise empirically verified in principle without begging the question. Therefore they should not be considered as having a truth value.

In conclusion, an undecidable proposition of PA cannot be said to have a truth value, unless it is added as an additional axiom, in which case it is vacuously true due to now being provable as an axiom.

Unfortunately, Classical Logic and it's accomplice Model Theory, give this illusion that arithmetic is a meta-language for defining the truth of PA or vice versa from 'outside the system'. This isn't the case, due to the very ability of each system to encode each other. They are therefore one and the same theory, for formal purposes and are as equally undecided as each other.

I'd imagine that the topology of 'good' music as it is formally represented, is a disconnected archipelago of irregular islands, each with many holes, quite unlike the densely connected forest of logic theorems.

Presumably, a composer is like a drunkard who awakens on one of these islands with no idea how he got there.

s

Heisenberg's uncertainty principle is in any event an epistemological and NOT an ontological fact. It's a limitation on what we can know (with our current theories) and says nothing about what truly is. — fishfry

That view assumes counterfactual definiteness; the belief that the possibility of stopping a moving arrow to construct a definite position implies that the moving arrow must have a real and precise but unknown position when it isn't stopped or it's position otherwise measured.

Yet this unquestioned assumption of counterfactual definiteness is the reason why Zeno's paradox appears paradoxical. To my understanding, Zeno's arguments are perfectly sound, which means that i have no choice but to reject counterfactual definiteness in order to resolve the paradox, and is the reason why i believe that Zeno ought to have stumbled across the underlying logic of Heisenberg's principle (when it is interpreted ontologically).

Of course, the rejection of counterfactual definiteness is only one means of making sense of quantum entanglement and which is also the view of the Copenhagen interpretation, which means that Heisenberg uncertainty is interpreted as ontological ambiguity/incompatibility, rather than as epistemic uncertainty.

I've come to the conclusion that the uncertainty principle should have been discovered by Zeno of Elea, nearly 500 years BC, since in my opinion the only satisfactory way of resolving Zeno's paradoxes is by recognising the incompatibility of the notions of momentum and position - something which is immediately self-evident in ordinary experience, and obvious after one abandons the dogmatic assumption that counterfactual experimental outcomes exists.

As for special relativity, it's scope was narrowly concerned with the logical consistency of theories such as maxwell's equations that employ temporal indexicals but otherwise lack explicit temporal frames of reference. So I think it's right to point out that special relativity isn't particularly relevant to phenomenological puzzlement and concerns about the nature of time, but at the same time SR cannot be criticised on that ground, for the nature of phenomenological time wasn't the theory's intended purpose and SR leaves the relationship between theoretical space-time and ordinary experience undefined.

Firstly, to satisfy the hedonistic imperative the machine must offer a genuine and sustainable alternative to reality, whereby one isn't constantly reminded that they are only obtaining virtual goods. For the presence of an underlying sense of having a fraudulent experience would mean that the machine isn't by definition an optimal experience machine. So i think it should be taken for granted for purposes of discussion that the machine is optimal in this sense.

The main issue I have with an optimal experience machine in the above sense, is due to the fact that many of our most positive experiences come about through overcoming adversity, i.e. unpleasant situations. But doesn't this imply that the machine must also simulate unpleasant experiences? In which case, how do we know that reality isn't already the ultimate experience machine?

Consider a 'roguelike' video game, where the player explores a dungeon that is generated 'on the fly' in response to the player's actions. Here, the history of the game world and the future of the game world are identical, and so one could describe the game-world as having a potentially infinite past, if and only if, the game world has a potentially infinite future.

Presentism sees the actual world in similar respects; all that exists is the present state of information, and so the denotation of a temporal "beginning" is arbitrary, as is the distinction between past and future.
To use computational terminology, presentism understands history as being lazily evaluated.

The reason we don't experience contradictory propositions is precisely because what we experience is information, and if there is no information, then there is nothing to experience - except for the visual experience of the seeing scribbles on a screen or hearing sounds spoken - which is information, but about something else that isn't about what is being written or said. — Harry Hindu

Right. And so the word "contradiction" doesn't mean zero information, for that is nonsensical, but refers to conflicting sources of information, actions, intentions, judgements and so on. A "true" contradiction can be taken to refer to an unresolved conflict that is logically implied.

For example, conflicts of judgement that are present in discrete borderline categorisation problems, as in being in the kitchen and not in the kitchen, are not resolvable by introducing more linguistic precision, for the same borderline problem resurfaces on a finer level of semantic granularity; here the "true" contradiction refers to the fact that the concept of discreteness cannot be reconciled with the existence of borderline cases. It's all well and good hoping that the conflict is potentially resolvable, but there is no reason to believe that all such conflicts are resolvable.

A contradictory statement says nothing at all, and is therefore useless. It is basically asserting something and then walking back that assertion at the same time resulting in a net zero amount of information. It is basically scribbles on a page, or sounds in the air — Harry Hindu

Yes in the sense of contradictory propositions. Nobody of course, experiences contradictory propositions - which goes to show that the general meaning of "contradiction" isn't to refer to propositions but to conflicts, such as the conflict between the definition of a language and it's application, or the rules of a sport and the moral notion of fair-play etc.

Three examples of "true" contradictions come to mind

i) Self-negating universal imperatives, i.e. hypocritical statements such as "Don't live by rules!".

ii) When a semantic distinction is more fined grained than is expressible in the language used, such as when standing in a doorway and thereby "being in the kitchen but not in the kitchen".

iii) When a semantic distinction is vague or uncertain, such as "a heap of sand" that isn't defined in terms of a particular numeric range of sand grains Hence "heaps of sand" exist, but no particular collection of sand grains constitutes a heap.

ii + iii are contradictions that programmers have to deal with, but they also present challenges for self-learning autonomous agents, that like human beings must somehow internalise a truth-language distinction.

I suspect that like humans, AI agents will also behave in a logically inconsistent fashion relative to their self-knowledge.

One should not forget that even logical laws are doubtable whenever they are interpreted extensionally as referring to a collection of real-world objects, as opposed to when they are used intensionally as rules of production.

Most of the time logic is used and appealed to directly, without reference to extensions, because it is used for asserting normative statements. The logical description of an ideal electronic circuit is comparable to the expression "Tidy your room! because i said so!". This ideal use of logic is comparable to ethics and it makes no sense to speak of epistemic doubt here. In contrast, if a circuit description was thought to universally describe the operations of real world electronics, this is obviously a highly doubtable proposition.

Physical laws are the joint expression of normative sentiment and physical description and so they aren't pure propositions in the ideal philosophical sense. The normative part is expressed by the use of universal quantifiers that aren't falsifiable and which say "Every X is a Y". But they don't need to be falsifiable, for their purpose is political,namely to assert scientific and cultural policy in the same way as the electronic circuit design that implicitly asserts "Intel should make chips this way".

Getting back to the original discussion, consider how one determines measurement precision. Isn't it's very definition ultimately in terms of the reproducibility of experimental results? In which case, if repeated experiments fail to reproduce results, then by definition measurement precision is lacking.

So it’s an output tool, like a painting on canvas or printed pages. But what is the vision in feeding AI words and lines from existing poems? There is no vision except to create what is now redefined as a poem, as art. There is no poet, only the programmer. The vision then becomes that of the reader, as in reader-response literary theory. There is no vision of the artist because the construction of the poem is random and the meaning accidental. — Brett

Modern creative algorithms are accessible to anyone, and have many tuneable parameters that the artist himself can control in accordance with his artistic vision, to influence the style, subject etc . But perhaps this is tangential to the discussion.

Is the creative process a self-directed and inwardly driven process determined by the artistic foresight of the visionary artist? or is it a blind and partly external environment-driven process, whereupon external stimuli bring about artful stimulus-responses in a person said to have artistic temperament?

I suspect that Cartesian minded internalists who believe in the former might have a harder time reconciling AI and the arts compared to externalists who view the artist as a shambolic director of
haphazard external processes.

sime

Art algorithms ... accelerate the pace of art revolutions.
— sime

Can you give evidence for that? — Brett

Art has moved in tandem with the accelerating technological trajectory that began with the invention of electronics in the Victorian era. Without semi-autonomous content creation tools, the present creation of vast and open virtual worlds wouldn't be possible. Perhaps we could call the potential algorithmic output of content-creation tools "meta art" in comprising a distribution over art objects, but it is still 'art' in the traditional sense in being ultimately shaped by the vision of it's users who program it, feed data to it, and tweak it's responses. And these tools also complement and fuel the need for traditional artists who produce hand-crafted content, as for example in virtual world content creation where there are neither enough algorithms nor artists to produce the infinite amount of diverse content required.

Artistic inspiration invariably refers to an external source of information and stimulation for the artist from where they get their ideas. A poetry generating algorithm is just another source of artistic inspiration, whose output a poet will tweak to his own satisfaction. This is little different to the use of 'story dice' as random source of inspiration.

In my opinion, AI art does not represent a paradigm shift in terms of the meaning of art, rather it reveals a significant part of the preexistent process called artistic inspiration and democratises it. Art algorithms are really about increasing the economy of scale of art production and they accelerate the pace of art revolutions.

If Presentism does not entail the reality of passage (i.e. the A-Theory), then are you arguing for the position of Presentism + B-Theory, i.e. that only present objects exist and that temporal passage is not real? I have never heard of this before. This is like the converse of the Moving Spotlight theory (Eternalism + A-Theory). I can only note this is at odds with the definition of Presentism given in most places, including the SEP article on Presentism:

presentism can be understood as the following conjunction:

(PC) (i) Only present things exist,
& — Luke

Yes, in my view the logic of presentism, or at least what I call presentism, leads to a reduction of the so-called "A series" into a perspectival interpretation of the B series, that is fully coherent with the best scientific theories, including special relativity for all empirical and practical purposes. The reason why I call a perspectival interpretation of the B series "presentism", is due to the fact that tenses are treated as indexicals, where an indexical can be considered to be an act of pointing to something, where the "something" is empirically undefined up until something actual and specific is pointed at.

In fact, i'd consider presentism to be the temporal logic of perspectivalism. From wiki

Perspectivism (also perspectivalism; German: Perspektivismus) is the view that perception, experience, and reason change according to the viewer's relative perspective and interpretation. It rejects both the idea of "one unchanging and essential world accessible to neutral representation by a disembodied subject."

I also consider the entire work of Wittgenstein to be a commentary of the logic of presentism, starting from the Tractatus that reduced every proposition including tensed propositions to observable empirical relationships among present atomic elements. See Hintikka for more on Wittgenstein's implicit philosophy of Time. The philosophical investigations is also useful for explaining the conflicting intutitions between presentists and growing block enthusiasts; presentists point out that "past" and "future" are actually used as empirically undefined indexicals, as opposed to growing block enthusiasts who think of the meaning of "past" and "future" in terms of mental imagery they assign to those notions, mental imagery which they overlook is part of the very present.

I notice that Wiki has also mentioned a similar criticism of the "spotlight/growing block" theory from the perspective of indexical tenses. In fact its the only criticism wiki documents against the growing block theory:

https://en.wikipedia.org/wiki/Growing_block_universe


Can a growing block enthusiast explain to me what explanatory value their theory of time adds to a present observation of a physically growing block? What is the nature of the "meta-time" that this growing block of time must have in order to grow time? and what about the meta-meta-time needed for that and so on?

But yesterday was a different day to today, just as tomorrow will be.
. — Luke

According to the indexical theory, to say "tomorrow will become yesterday" is to merely to express an intention to redefine the meaning of "yesterday", "today" and "tomorrow" so as to give the illusion of temporal passage. In other words, the indexical theory is an anti-realist stance. It's relevant to the issues you raise, because the use of tenses as indexicals is routinely overlooked in philosophical discussions.

You say that a denial of passage need not involve a denial of the past and future, but if "the state of the river is also our notion of "the present", then isn't this a denial of past and future? This seems to imply that we have no 'notion' of past or future states by which to judge that the present has changed. — Luke

All that is being denied is a notion of "temporal passage" that is distinct from the passage, of say, of a speeding train relative to the readings on a stop-watch. In other words, that the notion of temporal passage reduces to relations among appearances, which can also include whatever experiential content one has temporarily assigned to the notions of "past" and "future".

If understood indexically, the past is always the past and the future is always the future, for yesterday is always yesterday, and tomorrow is always tomorrow....

I tend to agree that a true presentist who rejects the existence of the past and future would be unable to judge which time is present. However, in reality, I think we are all able to ascertain this and can talk meaningfully about temporal passage. But this is not the focus of this discussion — Luke

For similar reasons I disagree that a denial of passage of time involves the denial of past and future, since "past" and "future" can similarly be interpreted as indexicals.

We can say that the state of the river has changed relative to the state of a photograph. But if the state of the river is our notion of "the present", then we can no longer say that the river has changed relative to the present.

I believe that McTaggart was making a similar deflationary argument when he concluded the unreality of the A series.

I disagree that presentism entails the reality of passage, because presentism might interpret the word "now" as being an indexical that cannot refer to the same set of affairs twice. If that is the case, then temporal passage cannot be referred to.

Recall Heraclitus, who said that it is impossible to step twice into the 'same' river twice. Here he is implying that the meaning of "same river" refers to a constant, say a static memory, relative to which the state of the actual river can be said to have changed. In contrast, if it is denied that the meaning of "the present" is fixed, then the river cannot be said to have changed relative to the present.

If the phrase "the present" is always substituted for the current international atomic time, then the sentence "the present has changed" is no longer grammatically permissible.

To many enthusiasts, panpsychism isn't so much an explanatory theory of consciousness, but an Occam's Razor style argument that non-living systems should be considered to have identical metaphysical properties as living systems, on the basis that there is no falsifiable justification for considering their metaphysical properties to be different.

From this perspective, pan-psychism is in a logical sense very close to if not indistinguishable from eliminative-materialism, the difference being that panpsychism doesn't consider subjects who claim to possess consciousness as being factually false, but as being necessarily and vacuously true in virtue of consciousness being a universal and hence tautological property. From this perpsective, the main difference between panspsychism and eliminative materialism is optimism.

When it comes to general epistemological questions of the form 'Can science study X'? the answer depends on the extent to which X is considered to constitute the very meaning of scientific practice. In the event that X is considered to ground the meaning or truth conditions of scientific practice, science can only be said to study X if science is considered to be it's own meta-science. But that assumption in turn raises worries and doubts as to the consistency, meaningfulness and reliability of the consequently circular scientific epistemology.

Consider similar questions: Can and to what extent can science study causality? or the existence of space, time and phenomena? or the reliability of epistemology suitably naturalised? can it even be meaningfully asserted that science can study the cosmos?

I don't think someone could be very functional having their self identity undermined as we see in cases of amnesia and dementia. It is useful to keep track of who you are and exhibit a consistent personality. — Andrew4Handel

In other words, the "self" is a useful idea with practical utility. But does that warrant the promotion of the "self" to the status of ontological primacy?

Most everything you believe has come from the testimony of others, if you doubted most of it you would be reduced to silence. Professors, books, language, science was given to you by others, you probably had little to do with creating the information yourself. — Sam26

I'm specifically referring to the trustworthiness and reliability of the verbal reports of experimental subjects in psychological experiments where they are tasked with giving self-reports, possibly including explanations for their own behaviour. A testimony of a subject taken at face value can be terribly misleading when it comes to understanding the actual underlying proximal and distal causes of the subject's verbal behaviour, for there is no reliable mapping between a person's use of sentences and their psychological state, and people don't possess introspective access to the causes of their own behaviour.

I don't even trust personal testimonies when it comes to deciding the veracity of the humdrum theories of behavioural psychology, let alone for deciding the veracity of pseudo-scientific mystical hypotheses.

That said, i have sympathy with the sentiments expressed by beliefs in "life after death"; not in the sense of it constituting an empirically contingent and testable scientific hypothesis, but because the opposite notion of 'eternal oblivion' is equally nonsensical.

These are very interesting remarks. Sadly, my knowledge of dynamic logics is sorely lacking at this point in time, but I think dynamic logics at best can only have partial applications; for there are many cases where we need to use a static logic. And it is in these scenarios that the Liar Sentence arises. — Alvin Capello

Certainly the semantic contradiction arises when the meaning of the liar sentence is analysed statically, but there is nothing that necessitates this adoption of tenseless logic in either the construction or analysis of liar sentences.

Indeed the construction of all proofs is a dynamic process over time. In the case of the liar sentence, a typical verbal explanation of the paradox involves alternatively saying "I am telling the truth about my lying, therefore I am lying about my lying, therefore i am telling the truth about my lying...etc". What is static in the construction of this paradox? Isn't the insistence that the liar sentence must be understood statically, the source of the contradiction?

Surely some seeming contradictions can be resolved, but I don’t think this is true of all of them. For instance, I don’t think the Liar Sentence and other similar semantic paradoxes have any consistent solutions, so these are radically contradictory objects on my view. — Alvin Capello

Of course, liar paradoxes are only contradictions if their truth is considered to be atemporal; otherwise these contradiction are avoidable using a tensed logic in which every sentence of a proof is temporally indexed according to the moment of it's creation, wherein the only distinction between premises and conclusions is that the latter is constructed after the former.

In such a tensed logic, liar paradoxes of the form P(t) => ~P(t+1) are consistent and only the simultaneous derivation P(t) and ~P(t) is inconsistent.

There's a difference between using words to denote objects or relationships between objects in the world, and the objects and relationships between objects in themselves that those words represent. The Law of Non Contradiction is thought to be violated only because it can be shown that a contradiction in terms of the relationships between the symbols (i.e. words) that point the objects, can be true. This results for the false equivalence that the symbols that represent objects and the objects themselves are the same, or rather, have the same logical form which they do not. In an actualized sense, nothing can ever exist and not exist at the same and in the same respect. However, in a state of potentiality, the actualized possibility of x and -x exist at the same time and in the same respect, according to my philosophy anyways. — TheGreatArcanum

I agree that contradictions are properties of sentences rather than of matters-of-fact, for I cannot understand what could be meant by contradictory matters of fact. I would also say the same about truth, for I cannot fathom a false matter-of-fact. The principle of non-contradiction is certainly critical to the practice of science, but I see neither justification nor practice of non-contradiction when it comes to philosophy.

Its not about logical atomistic consistency. Wittgenstein should not be in your list sime. — Gregory

I wasn't specifically thinking of logical atomism, i was referring to his consciously self-refuting Tractatus, as well the latter Wittgenstein's philosophical investigations, that isn't logically consistent. For example, his apparent reliance on the imagination to refute the idea of private language. This isn't a criticism, it's just a general feature of philosophical arguments. For many other examples see Graham Priest's "Beyond the Limits of Thought".

PNC is either rejected or violated in the works of many philosophers, e.g. Heraclitus, Kant, Hegel, Wittgenstein... There isn't much evidence to support the logical consistency of philosophy, especially in epistemology. If philosophy is considered to be primarily a normative activity, this doesn't matter. The loss of PNC isn't a great blow, it just means that philosophers are unstable hypocrites with alternating beliefs.

If we are constantly changing our opinions as to the facts of the past on the basis of new information, then why should we believe that the past is real and immutable?

Suppose that in 2030 society obtains decisive historical evidence concerning the identity of Jack The Ripper in 1888, whereby historians thereafter claim that the riddle regarding Jack the Ripper's identity was solved in 2030. Why should we believe that the actual facts regarding the identity of Jack the Ripper in 1888 existed before 2030? What does this assertion add to our calendar-indexed observations?

The deeper problem concerns the fact that in Cartesian co-ordinates the notion of vertical and horizontal Euclidean lengths is incommensurate with the notion of diagonal Euclidean lengths; hence the reason why a hypotenuse that is diagonal to the Cartesian axis is assigned an irrational number such as Sqrt(2), which of course isn't a quantity but a non-terminating algorithm for generating a Cauchy sequence.

As 180proof mentioned, If atheism is identified with the absence of belief, then it avoids the 'truth-by-correspondence' problem concerning beliefs that have non-existent referents. However, this is arguably not the case for theism, that your argument can be turned around to defend, by the following argument:

Premise 1 : All beliefs have referents.
Premise 2: Theism is a belief.

Conclusion: The referent of Theism exists, and therefore theism is true.

Personally, I find this argument acceptable, because the idea that a non-existent object can cause belief-behaviour is scientifically unacceptable, leading me to the conclusion that all beliefs are vacuously true in the epistemological sense of truth-by-correspondence of language to something. Hence any substantial notion of truth cannot be in terms of "truth by correspondence" of language to reality, but in terms of ethics and cultural convention.

A mistake of atheists is to assume that the object of theistic beliefs is universal, for there are many potential physical causes of religious behaviour and speech.

Well obviously the reason why people enjoy playing Chess is because its outcomes are uncertain due to players bounded rationality and tendencies to make mistakes, assuming that the skill difference between opponents is roughly even. This is especially the case for the variant Chess960 that in being randomly initialised diminishes the role of opening-theory. The Chess community are well aware that the rules of Chess have to evolve if Chess is to remain an interesting non-predetermined spectacle. Perhaps the game will continue to fragment into more and more alternatives. Personally, I think there are more interesting board-games to professionalise.

Irrespective of dualism, it isn't clear in any case what is meant by physical interaction, due to conflicting opinions as to the metaphysics and existence of causality. If one goes so far as to deny the literal existence of counterfactuals then interaction isn't even a substantive concept. Therefore ontological dualism and more generally, ontological pluralism, don't necessarily imply interaction problems, but only that different descriptions of the world cannot be inter-translated.

All working physicists informally appeal to "directness" whenever they make an inference, even though Physics possess no theory of directness. For otherwise a physicist could not claim to learn anything from an experiment, nor for that matter could he find the sentences of physics intelligible. So although directness/indirectness aren't themselves defined in terms of physical criteria, the converse is true.

Exactly the same issue applies to language in general, for we are taught the meaning of words either through ostensive definition, or by verbal definitions that implicitly appeal to earlier ostensive definitions for their intelligibility. And yet we have no linguistic criteria for translating verbal definitions into ostensive definitions and vice-versa, for languages are only publicly defined up until verbal criteria.
.
In line with language in general, the semantics of Physics is both under-determined and redundant; one Physicist's "natural" object is another Physicist's "metaphysical garbage", because they might each understand physics using different semantic foundations that are rooted in different ostensive definitions.

Everything is 'true' is a position I independently arrived at, without knowing that this epistemological position already existed under the banner of Trivialism.

Essentially, trivialism (at least as I am using the term) says that every belief is seen to be true once the object of the belief is identified with its immediate causes. Trivialism is a corollary of semantic deflationism and presentism, which denies that a prediction can actually refer to a future event by virtue of the future not existing in a literal sense in being a mere indexical.

For example, suppose that Alice becomes convinced that she will win the lottery and buys a ticket. According to the causal theory of reference, her belief that she will win the lottery is nothing other than a report referring to her immediate situation. If in fact she doesn't win the lottery, then according to trivialism she is only said to be "wrong" by reinterpreting the object of her belief to refer to the results of the lottery via a post-hoc revision of linguistic convention.

The collective use of language constitutes an inconsistent convention, for everybody uses the first-person pronoun to refer to a different subject. This is the central oversight in debates over idealism and realism that entirely ignore who is making an ontological commitment, such as the existence of colour.

Ordinarily, if I assert "I am seeing a red apple" the meaning of the sentence cannot be decomposed into two independent assertions, namely one of a subject and another of an object, as is in situation where I assert that someone else seeing a red apple. As far as I'm concerned, red, i.e. my red, exists independently of other people's perceptions of my red, and they cannot possibly know this fact, for whenever they talk about red they are referring to their red. And the situation isn't improved by talking only about "objective" optical properties.

Therefore consider the irrealist alternative; namely that ontological disagreements are partly the result of our collectively inconsistent use of language.