Nobody can agree upon what Kant really meant, even when Kant was still alive and responding to criticism. That said,

If Kant is interpreted to be an identity phenomenalist, meaning that he considered the concept of noumena to ultimately be ontologically reducible to "appearances" when appearances are taken in the holistic sense of the entirety of one's experiences, then he would, like other empirically minded philosophers such as Berkeley , Hume and Wittgenstein, have regarded the metaphysical Hard problem as a misconceived pseudo-problem that results from mistakenly reifying the concept of "mental representations" as being a literal bridge between two qualitatively different worlds. But this would say nothing of Kant's views regarding the semantically 'hard problem' of translating noumena into appearances.

In Kantian terminology, the natural sciences do not make a distinction between noumena and appearances; for any physical entity describable in any SI units can be treated as either a hidden variable or as an observation term at the discretion of the scientist in relation to his experimental context. This doesn't imply that the sciences are committed to one world (whether phenomenal or physical) or both; it only implies the practical usefulness of ignoring the semantic relationship between theory and phenomena, which has been the case so far for the majority of scientific purposes that fall outside of epistemology.

If Kant was astute, he would in my opinion have regarded his phenomena/noumena distinction as being a practical distinction made for the purposes of epistemology, as opposed to a metaphysical distinction, for obvious reasons pertaining to the creation of philosophical pseudo-problems.

Asking for a scientific explanation of consciousness, is like asking an artist to paint a canvas into existence.

Scientific explanations are grounded in empirical evidence, so it is nonsensical to demand of science an explanatory account of what empirical evidence is, which is what asking for a scientific explanation of consciousness amounts to.

The Hard "Problem" does exist, but only in the sense of a semantic issue.

The Hard problem should not be regarded as a deficiency or bug of the natural sciences, but as a positive feature of the natural sciences; the semantics of the natural sciences should be understood as being deliberately restricted to the a-perspectival Lockean primary qualities of objects and events (for example, as demonstrated by the naturalised concept of optical redness) so as to leave the correlated experiential or 'private' concepts undefined (e.g phenomenal redness). This semantic incompleteness of the natural sciences means that the definitions of natural kinds can be used and communicated in an observer-independent and situation-independent fashion, analogously to how computer source-code is distributed and used in a machine independent fashion.

If instead the semantics of scientific concepts were perspectival and grounded in the phenomenology and cognition of first-person experience, for example in the way in which each of us informally uses our common natural language, then inter-communication of the structure of scientific discoveries would be impossible, because everyone's concepts would refer only to the Lockean secondary qualities constituting their personal private experiences, which would lead to the appearance of inconsistent communication and the serious problem of inter-translation. In which case, we would have substituted the "hard problem" of consciousness" that is associated with the semantics of realism , for a hard problem of inter-personal communication that can be associated with solipsism and idealism.

The issues discussed in this thread primarily concern the necessity of complex valued integers and rationals in relation to entangled quantum states, their interactions and the Born rule.

The issues you raise concerning the existence, usefulness and intelligibility of the continuum of reals as part of the foundations of QM is valid albeit tangential to that discussion. Furthermore, the issues you raise are avoided in quantum computer science that is grounded in alternative mathematical foundations for QM that are constructive, computable and usually finite, such as Categorical Quantum Mechanics that is the underlying foundation for the ZX calculus. Those theories retain the essential underlying logical properties of complex Hilbert Spaces that are necessary for formalising quantum computing applications, including the conjugate transpose operator and unitary and self-adjoint operators, but without retaining the continuum of reals and the non-constructive propositions of complex Hilbert spaces.

In PI Wittgenstein opined that a central phenomenological distinction between a voluntary action versus an involuntary action, is that in the latter case an action is accompanied with a feeling of surprise, whereas in the former case feelings of surprise are absent.

Elsewhere he made it clear that he didn't believe in an absolute theoretical distinction of the concepts. So he evidently didn't hold much regard for the 'pseudo-problem' of free-will. Certainty, the meanings and use-cases of those conceptual distinctions in say, behavioural psychology, are radically different from their application in logic and mathematics, phenomenology, criminal law, physics, etc.

E.g consider the fact that in Physics the causal order doesn't have to be taken as being the same as the temporal order, and in the causal analysis of a given system the "first cause" is defined arbitrarily according to it's use value; a presentist can consistently interpret their present actions as being the first-cause of their subsequent observations, including those observations that they interpret as memories.

The assumption of static meanings is a foundational axiom of epistemology. If that axiom is rejected, then there cannot be a substantial and objective notion of epistemic error, beliefs cannot be identified with mental states and people can only be said to make predictions.

Second-order skepticism about the existence of static meaning is antithetical to first-order skepticism about the truth of our theories. The way I look at it, not only do we have Gettier problems, we cannot even be certain that we really have Gettier problems!

In science, and especially data science and machine learning, Occam's razor is often misunderstood to be an a priori principle. This can encourage biased and erroneous inductive inferences, typically in cases of Bayesian model selection or Bayesian averaging with respect to a family of different theories, where the 'prior' confidence assigned to the predictions of a particular theory is taken, without justification, to be inversely proportional to it's 'description length'.

The above principle can only be applied non-controversially when a supplementary argument is given to justify why the theories are described in the way they are, for otherwise the description lengths assigned to each candidate theory is arbitrary. E.g a diagonal straight line is only 'simpler' than a diagonal sine wave when the coefficients of both lines are given in terms the Standard Basis corresponding to the Cartesian axes. But the opposite is true when both lines are described in terms of a Fourier basis.

Well, I suppose that arguing, instead of Occam's razor per se, that one should present a hypothesis or theory in the simplest available manner is better than presenting such information in a convoluted or inflated way. — Manuel

Which goes towards explaining what Occams razor actually is; the principle of Occam's razor is our post-hoc revision of our linguistic conventions in response to our observations, so that our language encodes our most validated theories as efficiently as possible. Occam's razor shouldn't be mistaken for an a priori principle of inference, rather it should be understood to be a prescription for revising our linguistic conventions so that our past-conditioned expectations are easier to communicate and describe.

Definitions don't need to be observer independent. For example, the Cambridge Dictionary defines beauty as "the quality of being pleasing, especially to look at, or someone or something that gives great pleasure, especially when you look at it"

I agree that one only knows that coffee has a strong flavour after drinking it, in that the drinker reacts to the taste of the coffee. But even so, is it still not the case that the coffee has a strong flavour, not that the coffee causes a strong flavour? The drinker of the coffee discovers a property of the coffee. — RussellA

I would say that it depends on perspective, and more generally how the given term is used.

It is certainly the case that one often uses language tautologically, as for example in the case of private perceptual judgements. For example, ordinarily I might judge my socks to be 'white'. In this situation I am using 'whiteness' to mean my experience of my socks - I am not estimating their colour as being the effect of a hidden-variable that is a theoretical term of public discourse, e.g. 'optical whiteness' as referred to by Physics - rather i am defining what "whiteness" is in my judgemental context.

The interesting thing about continuations, is that they seem to accommodate such private analytic judgements. Take the continuation

Whiteness :: For all r, (whiteStimulus -> r) -> r

The intended meaning is that the public meaning of 'whiteness' is the hypothetical set of outcomes that might occur in response to anything acting upon a particular class of stimuli called "whiteStimuli'", in any conceivable fashion.

Then take the function (whiteStimulus -> r) to mean Bob's private interpretation of a 'whiteStimulus'. From Bob's perspective, it is tautologically the case that a 'whiteStimulus' is indeed a 'whiteStimulus'

By inserting the identity function id :: white-stimulus -> white-stimulus into the previous continuation, we get

Whiteness id :: white-stimulus

We can think of the term (Whiteness id) as representing Bob's private understanding or use of the public definition of Whiteness, which as shown, is indeed is of type 'white-stimulus'.

So the public definition of whiteness as a continuation isn't in contradiction with the subjective 'private language' use-cases of whiteness by each speaker of the linguistic community, but accommodates them in the same way that it accommodates the objective physical definition of 'whiteness' in terms of the physical responses of optical estimators,.

However, continuations seem to present the problem of infinite regress; for what exactly is the definition of the type called 'white stimulus' here? presumably in some use-cases, such as in physics it is taken to be another hidden variable that is another continuation.

White-Stimulus :: For all r , ( someType -> r) -> r

Whilst in other use-cases, such as Bob's perceptual judgements, it refers to a 'given' of experience that is decided by tautological judgement.


Continuations obviously aren't the whole story, nor even necessarily part of the story for there are problems, but they seem useful in conveying the open-ended, counterfactual and inferential semantics of terms as well as accommodating the differing perspectival semantics of individual speakers.

The fact that people have a use for coffee means that the presence of coffee causes things to happen. However, coffee is not defined by what it may cause to happen, coffee is defined by what it is, a dark brown powder with a strong flavour. — RussellA

So in your opinion, 'dark brown' and 'strong' are observer independent properties of coffee that everyone can point at? Recall that the taste and colour of coffee is relative to perspective. Different organisms and processes react differently to coffee. From my perspective, how can i understand your use of "dark brown" and "strong" except as an observable effect of you drinking coffee?

Here's a type-theory inspired suggestion for explaining or dissolving ineffability: Identify the meaning of a word with it's effects in relation to a given stimulus. This idea is a generalisation of "meaning-of-use" known as causal semantics.

E.g take the integer 2, which in Haskell can be written

2 :: Integer

where 2 is by definition the result of 1 + 1

On the other hand, if we identify 2 with it's effects, this means interpreting 2 :: Integer to be equivalent to the following type

2 :: (Integer --> r) -> r, where r is of arbitrary type (not necessarily a Nat).

In other words, here the meaning of 2 is the effect that 2 has on every function of type (Integer -> r) that takes an integer and returns an object of type r, where r is arbitrary and refers to any type. In functional programming, the latter representation of 2 is known as a continuation'.

In Haskell, 2 can be converted to a continuation by writing ($ 2), i.e.

2 :: Integer

whereas

($ 2) :: (Integer --> r) -> r,

Example applications of the latter type include

($ 2) (+3) = 5
($2) print = "2" as the display output of a computer monitor.

i.e r isn't necessarily an abstract type, but can refer to physical events.

in Haskell, the form 2 :: Integer is considered to be fundamental and the meaning of it's continuation is derived from this consideration. But in general there is nothing stopping us from treating the continuation as semantically fundamental. This stance has the benefit of allowing the meaning of a type to be generalised so that it is always incomplete, evolving and contingent upon the affairs of the physical world, e.g. effect r could refer to a physical or psychological response to a symbolic instance of integer 2, such as sense-data created by the mind of a human in response to a 2, or to the operations of a physical machine reading 2 as input.
"
In terms of continuations, the public meaning of "coffee" is of type

Coffee :: (Coffee-stimulus -> r) -> r

where 'coffee-stimulus' is the type of a perspective-relative hidden variable that isn't publicly shared (since only reactions to stimuli are publicly available). So if a person's reaction to a coffee-stimulus is of type (Coffee-stimulus -> r), then the effect of 'coffee' on that person is by definition implicitly included in the public definition of "coffee", in spite of the fact the public definition of coffee does not know about or explicitly include that person reaction.

Edit : I realise the last paragraph is technically problematic. For instance does 'sense-data' refer to r or to 'Coffee-stimulus' ?

Drop the word "objective" if it gets in the way.

Both an observer on the earth and one in orbit around the sun will agree that, for an observer on the earth the earth remains stationary, while for an observer in orbit around the sun it moves. Movement is relative to the frame of reference and can be translated from one frame to another. Basic relativity. — Banno

Relativity indeed lacks the concept of objectivity in being a family of conditional propositions of the form
x --> p(x), where x is a given frame of reference. As conditional propositions they are mutually consistent as you point out, and since the theory of relativity does not assume the existence of any particular frame of reference it isn't descriptive of any particular world.

On the other hand, we like to think that multiple observers exist who occupy one and the same universe in different frames of reference. The problem is, if we accept the reality of different frames of reference, say x' and x'', then relativity implies the unconditional conclusions p(x') and p(x'') that appear to be mutually inconsistent if interpreted as referring to one and the same world, e.g the Earth moving and not moving.

So to restore consistency it seems to me that one must either reject in a solipsistic fashion the existence of other frames of reference, or reject relativity, or accept the conclusions of relativity as referring to different worlds.

But there are other ways to resolve "the conflict". Either the cases are equivalent and can be transformed from one to the other as in the geocentric/heliocentric example, or one account is wrong or insufficient, as in the Herodotus/Thucydides example.

Inventing the paraphernalia of worldmaking is surely overkill. — Banno

So objectively speaking, is the Earth moving or not? Can objectivity be relative?

I avoid the rain by staying inside. Hence, it is not ineluctable; and not real. — Banno

When it is raining outside, you cannot "avoid" that it is raining outside "by staying inside". Btw, your example doesn't concern ontology, Banno, which, in the context of my remarks, isn't relevant. — 180 Proof

This line of discussion leads towards the topic of irrealism; for we can at least claim

A. Each individual has a different conception of reality, that is incommensurable with respect to each and every other persons conception of reality; different individuals aren't using a common basis of understanding when they each refer to 'reality', for their understanding of reality is relative to their unique perspectives.

But if A is true, then how does one avoid the conclusion of irrealism?

B. Each individual has a different reality; there isn't a shared reality that individuals are occupying and describing.

On the other hand, each of us will probably insist that we possess a concept of 'shared reality', if only because we communicate to each other and to ourselves in a common language whose semantics aren't publicly defined in relation to the perspectival judgements of a particular individual at a particular moment in time and space.

But isn't even this supposedly aperpsectival concept of 'shared reality' relative to perspective, and thus not a defence against irrealism?

The pre-theoretical notion of reality , e.g Johnson's pain when kicking a rock, should be distinguished from the ideology of realism that often accompanies, but is not implied by, the use of a naturalized ontology such as in the natural sciences. The ideology of realism interprets the inter-subjective usefulness of a naturalized ontology as evidence that reality transcends and grounds the subject and his perspective, which the idealist and anti-realist reject as being incoherent.

Universal definitions that apply to an infinite number of cases are not extensionally exhaustive descriptions of anything, in spite of appearances to the contrary. A universal definition that quantifies over an infinite domain is an intensional and prescriptive definition, i.e. a speech act, that is given in relation to an indeterminate number of future observations, as in "Put all dirty socks onto this pile".

Therefore any proposed universal definitions of "reality", "truth", "existence", "equality" etc can only be prescriptive rules of language for standardising the public expression of individual judgements that are made on a case by case basis. Such universal definitions don't describe their future applications before the respective future judgements are made, and the outcomes of said judgements aren't dictated by the a priori universal definitions - only the expression of such judgements can be said to be determined a priori by the universal definitions.

Definition of 'extensionally meaningful? — TonesInDeepFreeze

The extensional meaning of a set are the items it refers to, in contrast to the definition of the set in terms of a formula, that is to say it's intentional meaning. Countably infinite sets cannot be given an extensional definition for obvious reasons, which is why finitists object to the reality of such objects, even if conceding that such 'sets' have instrumental use for generating numbers.

That's not a definition of anything, let alone the set of natural numbers. — TonesInDeepFreeze

Its just short-hand for the inductive definition of the Naturals in terms of an F-algebra with respect to an arbitrary category in which 1 --> N represents '0' and N --> N represents the successor function that corresponds to the Dedekind infiniteness property.

'N is Dedekind infinite' means that there is a 1-1 correspondence between N and a proper subset of N. There's no need to drag isomorphism into it. — TonesInDeepFreeze

That's a fair enough remark, given that only the right arrow is involved.

'

The function {<j j+1> | j in N} is provably a 1-1 correspondence between N and a proper subset of N, so it proves that N is Dedekind infinite (and notice, contrary to your incorrect claim, choice is not involved). But that proof is not a definition of anything, let alone of the set of natural numbers. — TonesInDeepFreeze

I never said that Choice was involved in the definition of dedekind infinity, i said that the presence of Choice causes all infinite sets in ZF to become Dedekind infinite by default, which is a major failing of ZFC in ruling out the only sort of "infinite" sets that have any pretence of physical realisability in the sense of extension.

There's no consideration of intensionality in the illustration. — TonesInDeepFreeze

You do recognise that Dedekind-infinite sets aren't extensionally meaningful, right?

So if one writes down an inductive definition of the natural numbers

1 + N <--> N

where <--> is defined to be an isomorphism, then to say N is "Dedekind-Infinite" means nothing more than to restate that definition.

Inappropriate extensional analogies for understanding dedekind-infinite sets , such as unimaginable and unobservable completed infinite sets of hotel rooms are going to appear paradoxical .

As I recall, it's not a perpetually growing hotel. Rather, it' a hotel with denumerably many rooms and rooms and denumerably many guests, one to each room. — TonesInDeepFreeze

That's right. But we have to distinguish between the extensional concept of a number of hotel rooms that can be built, visited, observed, realized etc, versus the intensional concept of a countably infinite set of rooms. The latter refers not to a hotel, but to a piece of syntax representing an inductive definition of the natural numbers.

The paradox is due to conflating intension with extension. Keystone is right to raise objection.

The hotel is not finite. It has infinitely many rooms. — TonesInDeepFreeze

A perpetually growing hotel that always has a finite number of rooms is still an infinite set, because there isn't a bijection between any finite set and the number of rooms in the hotel. But such a hotel isn't describable in ZF if the axiom of choice is assumed, because it forces Dedekind-infiniteness upon every infinite set.

The T schema doesn't dictate

1) The type of truth object (sentences vs propositions)
2) The nature of the equivalence relation (analytic necessity vs material necessity vs modal necessity)
3) Whether the schema is used prescriptively to exhaustively define the meaning of "truth" e.g as in deflationary truth, or whether the schema is used to non-exhaustively describe truth but not explain the truth predicate, as in inflationary truth.

Extensionally, Hilbert's Hotel refers to the trivial possibility of indefinitely expanding a finite hotel in such a fashion that guests are reassigned to new rooms as new guests are added. Unfortunately, ZFC cannot distinguish between a hotel that isn't finite purely because it is growing without bound, from a mythical hotel with a countably infinite subset, which as you point out, is an extensionally meaningless assertion, and is partly the fault of the axiom of choice that ZFC assumes.

That "A Hilbert Hotel has a countably infinite subject" refers to a sentence of ZFC, and not an actual hotel.

Material implication in classical and intuitionistic logic is a static relationship that holds between sets , as in "Smoking events might cause Cancer events", where the condition always exists ,even after the consequent is arrived at, due to the fact it is talking about timeless sets rather than time contingent states of processes.

(Smoking "might" cause cancer is due to the fact ~A OR B => A --> B , which doesn't have a conjunction of events in the premise)

For resource-sensitive logical implication that is truly material in denoting conditional changes of state over time, see linear logic for expressing "If I am in the state of smoking then I might arrive at a state of cancer". It has the same form as the above rule, but the premise can only be used one when arriving at a conclusion.

The 'might' here can also be avoided by defining only one axiom of implication in which smoking is the premise. Otherwise the resulting logic expresses multiple and mutually exclusive possible outcomes of smoking, i.e possible worlds are built into the syntax.

For a programming language with native linear types, see Idris.

Can you explain this to me from a computer programming perspective? In your comparison, is the data the output of the function? A function can return a function, but it can also return another object type, like a string. In the latter case, there is a type distinction between between the function and its output, but I don't see how this is unnecessarily rigid. I suspect I'm missing your point. — keystone

I'm basically warning against logicism, the ideology that there is a single correct logical definition of a mathematical object. Thinking in this way leads to unnecessary rejection of infinite mathematical objects, for such objects aren't necessarily infinite in a different basis of description. e.g the length of a diagonal line doesn't have infinite decimals relative to a basis aligned with the diagonal.

Also, the algorithm for approximating sqrt(2) to any desired level of accuracy can itself be used to denote sqrt(2) without being executed.

Why can't we just say that pi is not a number? Instead, it is an algorithm (e.g. pick your favorite infinite series for pi) used to generate a number. This algorithm is potentially infinite in that we can never complete it, but we can certainly interrupt it to generate a rational number. If you interrupt it, maybe you'll get 3.14. Actual infinity only comes into play if you claim that the algorithm can be completed, in which case it would generate a real number - a number with actually infinite digits. This is what I would like to challenge. — keystone

A limitation of that conceptualisation, is that it asserts what might be considered an unnecessarily rigid ontological distinction between functions (intension) and data (extension), which is surely a matter of perspective, i.e the language one uses. Also, recall incommensurability; the length of diagonal lines in relation to a square grid have a length proportional to sqrt(2). The decimal places of sqrt(2) are only "infinite" relative to the grid coordinates.

Any computable total function N --> N can be regarded as a number, whose value is equal to the potentially infinite sequence of outputs it encodes. e.g '3' can be identified with the constant function f(n) = 3, whilst pi can be identified with the computable function whose values if executed are the potentially infinite sequence g(0) = 3, g(1) = 3.1, g(2) = 3.14 ... These numbers can be compared positionwise, with arithmetical operations defined accordingly. However, there are only a subcountable number of such functions, meaning that any set that contains some of these functions either doesn't contain all of them,or contains errors i.e partial functions that fail to halt on certain inputs, to recall the halting problem.

That said, it could be argued that the concept of exact and correct computation, whereby a computer program or function specification is translated by man or machine to a precise and correct result of execution, is an ideal platonistic notion that is incompatible with the austere epistemic and metaphysical conservatism of finitism. In which case one wants a purely extensional treatment of mathematics that doesn't appeal to any notion of computation, in which case see Brouwers intuitionism for a calculus built around choice sequences that appeal only to the existence of resources for memorising data generated by a creating subject.

I used to lucid dream every night as a teenager (whatever that means, see my remark above), but I came to the conclusion that lucid dreaming as a deliberate and willed ideological practice for achieving peak experiences, as advertised in new age,pop psychology, and alt therapy books, is a counterproductive road paved with delusions and misconceptions that leads nowhere, much like the rest of the self-help industry.

Lucidity also comes at the cost of creativity; the more lucid I am, the less interesting and surprising is the dream environment, dream characters lose their autonomy and stop speaking for themselves and I stop hearing novel music. Everything creative and interesting that happens seems to stem from a state of uncontrolled and dissociated non-lucidity in which the self and it's agenda aren't present. For purposes of creativity for it's own sake, i suspect that the ideal amount of lucidity is just enough to start the dreaming process off in a vaguely desired direction and to recall what happened afterwords.

Getting back to the question as to what lucidity is, there are obviously several semi-independent dimensions to the concept, e.g volition, control, vividness and recall, all of which present to some extent in ordinary dreams, and which come at the cost of other dream qualities e.g 'surprisingness' and 'subjectedness' ; isn't it better to ditch the general concept of lucidity for these separate concepts?

In debates between semantic internalists vs externalists it isn't clear that matters of fact are being debated. Both sides of the debate seem only to be cheerleading different linguistic conventions that emphasize different semantics for different purposes. To think otherwise is to grant linguists powers of omniscient authority.

In the first person, when ones uses a name to refer to a present acquaintance, the distinction between sense and reference disappears. The distinction only comes into play when utterances are interpreted as referring to 'non-present' entities. But then it must be asked what is the meaning and usefulness of interpreting such words as designating what is absent? Doesn't designation amount to postponing an extensional interpretation of a name until a satisfactory object is recognised as passing into view?

People tend to forget that ordinary usage of the concept of 'nothingness' refers not to an absence of information, but to irrelevancy of considered information.

e.g when a patient awakens and claims to remember 'nothing' about being in a coma, his claim refers not to his past coma but to the fact he considers his present information to have no relevancy to the question.

From a neurological perspective, it doesn't make sense to interpret memories, or their absence, as referring to an extensional past that lives outside of the present.

In practice, "Artificial intelligence" is merely state-of-the-art software engineering in service of human beings done in accordance with the ideals of human rationality; it is the design and implementation of systems whose validation criteria are socially determined in accordance with cultural requirements, e.g a recommender system must suggest a 'good' movie, a chatbot must argue 'persuasively', a chess engine must respond with a 'brilliant' move, a mars rover must avoid 'dying'....

These sorts of applications aren't differences in 'kind' from early programming applications; they only differ in terms of their degree of environmental feedback and their corresponding hardware requirements. In both cases, software is invented to satisfy human needs and often to reinforce human prejudices.

As for general intelligence, no such thing can exist in either man or machine; to pass a 'general' Turing Test is to pass a highly specialised "human traits" examination that comes at the cost of being unable to perform any single task efficiently, whilst also ruling out the ability to execute of other potentially useful behaviours that humans don't recognise as being rational. (Also, no two humans have the same concept of rationality because they live non-identical lives).

The concept of "consciousness" cannot be divorced from the concept of rationality, because empathy is invoked when judging the rationality of another agent's actions. We put ourselves in the agent's shoes, then fool ourselves into thinking that we were experiencing their consciousness rather than ours.

I don't think it helps to introduce "meh" as a truth value for undecided arithmetical propositions, because that would distort the existent meaning of arithmetical truth values for both the constructive and classical senses of arithmetic.

In the constructive case, the truth value of an arithmetic proposition is considered a 'Win' or 'True' if there exists a proof of the proposition, and is considered a 'Loss' or 'False' if there is a proof of it's refutation. But introducing a truth value for the status of undecided arithmetic formulas is tantamount to calling a failure to prove or refute them a 'Draw', which distorts the concept of mathematical truth by muddying the distinction between a mathematician's abilities and his subject matter.

IMO, in constructive logic it is better to resist assigning a truth value to undecided propositions so that truth values always refer to what has been proved, rather than to what hasn't been proved. Draws should only be considered a third truth value in cases where there is a constructive definition of drawn games such as in Chess, unlike arithmetic that doesn't possess a natural concept of a draw

As for the classical case, the Law of Excluded Middle suffices to denote the truth value of undecided propositions; unlike in the constructive case, the classical meaning of A OR B doesn't entail either a proof of A or a proof of B, therefore A OR ~A interpreted as meaning TRUE OR FALSE suffices as the truth 'value' for undecided propositions of classical arithmetic.

My first impression of your original post, is that you are implying ignorance as to whether you occupy your actual world versus a possible world occupied by someone else. In which case there is a contradiction.

But if by definition you take p, Kp and Bp to correspond to your actual world, then no contradiction arises with respect to the discrepancies with a possible world you talk about.

"I believe it is raining and it is not raining" is logically consistent and possibly true, but not something we would ever assert. — Michael

Not according to many people's grammar of "belief" including mine, although you appear to have company with a certain group of subjective Bayesians, who when designing an experiment insist on talking about their mental states rather than the experiment itself, much to the bemusement of any non-Bayesians present who merely wish to discuss reality.

Personally, if I am prepared to say "I believe X", then i am also prepared to assert "X" and "X is true". So according to my prescriptive usage, Moore's sentence is inconsistent. Only in the past or future tense would i invoke belief concepts.

If you hold the karmic banking system as a strong belief, how does that fit in with crisis management or counselling?
When you are dealing with someone with an acute mental health problem and who cares nothing for karma? — Amity

My impression of Indian culture before it underwent westernisation, is that it's belief in reincarnation encouraged slower and more sustainable lifestyles, but that it's belief in karmic justice encouraged social neglect of the downtrodden.

Question: To what extent do the metaphysical beliefs of a culture become determined by the practical necessities of it's society? Clearly they must be correlated to a certain extent, but do they converge in the long run?

For example, if modern society is to survive then it needs to adopt environmentally sustainable lifestyles together with long-term ecological investments that will benefit future generations more than today's. Does this necessity imply that society's environmentally unsustainable belief that "You only live once" will mutate towards a belief in reincarnation that encourages people to work for tomorrows generations rather than today's ?

If karma has to be taken seriously, then it is to sensible to identify Karma with causality and then recall the practical impossibility of knowing causal relations with any certainty.

After 911 Tony Blair and George Bush decided to divert the trolley in a similar scenario.

Do you have specific examples of why it is morally problematic to respect gender self-identity? — Michael

For example, situational factors that provoke someone to seek gender reassignment surgery, whom having undergone the operation decide they want to revert back after the situational factors are removed.

Isn't a person's self identity largely thrust upon them by society? e.g, couldn't a boy perceive himself to be a girl as a result of bullying that caused him to believe that he couldn't compete as a man and seek support from the opposite sex?

For society to automatically respect self-identification seems morally problematic, because it would mean for society to automatically reinforce the social treatment a person receives, however dysfunctional and situational.

A generally problematic consequence of making a hard distinction between conscious observation and measurement, is that it throws the empirical significance of measurement into doubt; science is supposed to validate theory against observation through measurements, but how is that validation possible if measurements aren't at least partially identified with the conscious observations themselves?

The general reason why science resists taking measurements for conscious observations, is because measurements are taken as referring to the obtaining of observable values, whereby the set of measurements is taken to include both potential observations and actual observations. This is because science is by design not a private language, but a public language for facilitating inter-subjective communication among individuals whose actual experiences are in contradiction with one another.

The conundrum for the realist is, if 'potential observations' are to be of necessary importance to empirically accountable theories, as opposed to being unverifiable dogma for facilitating 'ornamental coping' among the communicating public, then what could potential observations amount to other than actual observations of some sort or other?

Roger Penrose once criticised the many universe interpretation, saying it fails to address the central mystery of Quantum Mechanics which is why can't we directly observe the quantum superposition of live cat and dead cat"?. On the other hand, if potential observations are taken to be semantically equivalent to actual observations of some sort, then one does indeed observe "Live cat + dead cat" - for example by interpreting "live cat + dead cat" to refer to the conditions of state preparation of the respective quantum superposition. This aspect of semantics is of course not what Penrose had in mind.

Until another big theory change comes along, QM is most naturally interpreted as irrealist theory that describes a process of interaction between a particular individual and his world, as opposed to being a realist theory defining a set of propositions that are held true by all observers simultaneously. For verification minded logical positivists, this isn't a defect of the theory since they interpret all theories in this way.

When interpreted in irrealist or idealist fashion, it is logical to associate consciousness with wave-functions in the same way as with any other proposition whether classical or quantum - but it isn't logical to[ think of consciousness in terms of wave-function collapse- for this move prohibits the deflation of "conscious observation " to "observation", since according to verificationism quantum superpositions are consciously observable. It also goes without saying that consciousness cannot be considered a causal event.

Ok. But yours is the first mention of epistemology in the thread. Are you suggesting the mysticism isn't rational? — Pantagruel

I'm saying that empiricism and rationalism are sufficiently broad churches so as to accommodate anything that might be called 'mystical'. There is no room for 'mysticism' in philosophy as a distinct third form of epistemological inquiry.

A mystic is just another person who theorises in response to sense data towards the same epistemic ends as a non-mystic. Even if we grant the mystic extrasensory perception and super-powers of reason, his process of inquiry isn't categorically different from the ordinary philosopher.

In epistemology there isn't room for another source of knowledge besides empirical observation and rational thought, for those concepts are considered exhaustive by definition. So to relate mysticism to epistemology requires translating the methods, premises and conclusions of mysticism into the standard epistemological concepts people are already familiar with.

In Euclidean geometry, there is no such thing as a length magnitude of -2. Negation only indicates the direction of the magnitude in relation to a coordinate system. Hence it isn't surprising that by convention constants aren't signed.

To paraphrase and restate what I said earlier, the evolution

Whole Numbers -> Naturals -> Integers -> Rationals-> Reals -> Complex Numbers

accommodates increasingly general uses of arithmetic, which in my opinion and following Wittgenstein's general philosophy, is best understood in terms of games of increasing generality .

The starting intuition that makes the Whole Numbers so compelling initially, coincides with the picture theory of meaning and the reference theory of meaning: Whole numbers are used to denote the process of counting, whereby a number is assigned to a particular object without consideration as to how the object relates to other objects or how the object is used; relative to this semantics, the concepts of 'zero' objects and 'negative' objects make no sense. Also, recall that the whole numbers and integers have the same cardinality. So in the context of counting, they are equivalent.

The Naturals mostly cling to this early intuition, but introduce a 'zero object' to accommodate the concept of balance, say when using weighing scales, and also to denote the situation that exists prior to counting anything.

The previous introduction of zero motivates the construction of Integers with additive inverses, which leads to rejecting the earlier intuition outright; instead of using whole numbers to refer to entities, they are used to represent interactions between two entities, whereby an equation can express the net result of their interactions. So the shift from Nats to Ints marks the shift from denotational semantics to inferential semantics; but this is strictly in the context of exactly two interacting parties, which is denoted by the fact that the negation operator exactly reverses the direction of a given interaction in respecting the law of double negation, e.g -(-1) = 1.

In a three player game, say between Alice, Bob, and Carol, then from the perspective of Carol an interaction has 2 dimensions, namely a vertical dimension whose positive and negative values respectively denote Carol giving to and receiving from Alice, as well as a horizontal dimension representing Carol giving to/receiving from Bob. Thus Carol has 4 combinations of directions to consider, which implies that negation for three player games must respect a law of quadruple negation, motivating the construction of complex natural numbers.

The rationals generalise the integers by providing denotational semantics for divided objects, e.g a cake eaten by two agents, and the reals generalise the concept of divided objects to the concept of processes of dividing, albeit in a flawed way. The complex numbers over the field of reals accommodate everything previous.

A private language can exist; however the private linguist, him/herself, may not understand it. There could be n number of reasons why this is the case, my favorite one being the circularity of the verifying process for meaning: The private linguist can only ask him/herself what a private word means but to ask this question means I'm unsure of the meaning; in essence I must know what I don't know, an impossibility, — Agent Smith

That ignores the fact that

1) People tend to say "I understand" when they mean "I recognize that" - not to mention the fact that people regularly change their mind as to whether they previously understood.

2) Conventions amount to a finite description or prescription of language use, and therefore cannot pin-down the meaning of "understanding".

For example, in the case of Modus Ponens

"For all x, x and x -->y implies y"

is not equivalent to giving a complete table of uses, and does not pin down any particular table of uses. At most it pins down the sense of Modus Ponens by appealing to innate cognitive judgements of the learner, but it cannot pin down the references and use-cases of Modus Ponens, since the meaning of "for all" is left under-determined.

Compare this to the social definition "All Bachelors are unmarried men" - the public certainty do not apply "Bachelor" and "Unmarried man" synonymously, because their cognitive judgements vary - the definition of "bachelor" amounts to a mythology or prescription of word use.

3) Cognitive judgements not only make no recourse to social guidance , but they cannot make recourse to social guidance, on pain of begging the question as to how one is being guided.
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