It's metaphysical solipsism not the "how can we know" one, because we really can't. WE can't even know if we exist, like I said.

Berkley can argue against and unobserved and unimagined tree all he wants it doesn't make it any less real. It's also why idealism died out I guess and why we follow science. The "if I don't see it it didn't happen or isn't real" is one of the easiest things to disprove.

Everything else you said is irrelevant to the topic. — Darkneos

You appear to have false preconceptions regarding Berkeley's position. I'd recommend studying the SEP article before continuing discussion.

First, it is necessary to distinguish the main types of solipsism and discuss their interrelations.

1) Metaphysical (M)
2) Epistemological (E)
3) Psychological (P)

Initially, it seems that your post concerns E solipsism, in asking "How can one know whether or not one's mind is all that exists?". Thus buried in this question are a concept and a presupposition, namely that one is using a closed a-priori definition as to what one means by one's mind, relative to which one is asking whether there exists a type of evidence, that if observable to ones mind, settles the question as to whether one's mind is all that exists.

M solipsism on other other hand, isn't a presupposition, but a refusal to grant intelligibility to the idea that there exists anything outside of one's mind. This entails that one's mind isn't meant as a closed and static concept that is a priori definable, but as an open and adaptive concept that is rationally and empirically exhaustive of one's concepts and potential experiences to the point of closing off the domain of philosophical and epistemological inquiry.

Philosophies that are sympathetic towards M solipsism are phenomenalism and empiricism. We might recall Berkeley, who rejected the conceivability of an unobserved and unimagined tree, Wittgenstein who questioned the intelligibility of the distinction of idealism and realism, and Charles Sanders Pierce who considered the external world to be congealed mind. These philosophers weren't speculating that mind is an exhaustive substance, as when an E solipsist and his naive-realist opponent considers mentality to be an object for propositional analysis. Rather, those philosophers treated mentality as a meta-linguistic activity that is the very basis of any act of rational and empirical enquiry.

As such, it doesn't make sense to argue for or against M-solipsism, as an M-solipsist will always interpret the arguments of any purported opponent or critic M-solipsistically. Indeed an M-solipsist might even identify as a realist for all epistemological purposes.

P solipsism is a ruminative psychological condition experienced by amateur philosophers and isolated individuals such as astronauts, who mistake their narrow a priori self-concept for the world. Anyone who self-identifies as an M-solipsist runs the risk of experiencing this condition as a result of misunderstanding the meaning of M-solipsism.

In my opinion, the very meaning of a religion refers to it's psychological, economic and political causes and it's intended psychological, economic and political effects. This includes both theism and atheism.

For example, part of the meaning of modern atheism are the unsustainable life-styles we associate with consumer-capitalism, life-styles that Baby Boomers in particular often justify on the basis of their metaphysical belief that "you only live once" . Atheism both drives, and is driven by, consumer capitalism, e.g. retailers preaching to us that we must live this 'one' life to the fullest.

If my opinion is correct, then the rise of sustainable environmentalism throughout the world will be correlated with a rejection of today's widespread atheistic beliefs for metaphysical belief systems that give moral incentive for individuals to live sustainably.

One of the oversights of common-sense atheists is that they reject the existence of the transcendental on the basis of a lack of evidence, and yet they tend not to consider the semantic possibility that the very meaning of transcendental concepts refers to the world. For isn't the psychology and behaviour of a Christian preacher fully accounted for by the physical causes of his behaviour? In which case, what so-called 'claims' asserted by the preacher should the atheist be sceptical about?

If "Steel" is accepted as denoting a purely physical concept, then by definition "steel" cannot be semantically reduced to any individual's private thoughts, experiences or understanding of "steel".

To account for this, if a speaker says "I am thinking about steel", interpret them as saying

"I am thinking about p-steel" - where 'p-steel' is understood to an indexicial. This implies

- "p-steel" has direct and immediate referential content for that particular speaker, and for that particular speaker only. This referential content includes both the speaker's perceptions of their external world and their subjectivity. From the perspective of an external onlooker who tries to understand the speaker, this referential content can be identified with the immediate situational causes of the speaker's utterance of "steel" (and hence nothing to do with any mythical 'public' understanding of "Steel").

- P-steel has no public referential content, except in the sense previously considered.

- "Steel" has no a priori referential content; Every empirical identification of "steel" is an instance of "p-steel" with respect to a particular observer.

Your broad question falls under Meta-Metaphysics.

Here's a good book on the subject.

Anomalous Monism is only concerned with third-personal causal analysis of propositional attitudes, and so it isn't really relevant to the "hard problem". Rather, AM concerns the "soft problem" of inter-translating the public ontologies of scientific psychology and the physical sciences.


"Davidson restricts the class of mental events with which Anomalous Monism is concerned to that of the propositional attitudes—states and events with psychological verbs such as ‘believes’, ‘desires’, ‘intends’ and others that subtend ‘that-’ clauses, which relate subjects to propositional contents such as ‘it is raining outside’. Anomalous Monism thus does not address the status of mental events such as pains, tickles and the like—‘conscious’ or sentient mental events. It is concerned exclusively with sapient mental events—thoughts with propositional content that appear to lack any distinctive ‘feel’."

https://plato.stanford.edu/entries/anomalous-monism/

The "conscious events" that AM doesn't address are those that correspond with our private use of language as indexicals, as in the cry " owww toothache!!" - an occasion that constitutes a bespoke use of language, that in spite of appearances isn't justified by, nor needs to be justified by, a priori established linguistic conventions regarding the public meaning of "toothache"in the referential or functional sense of a noun or verb.

If I cry "owww toothache!!" , although the noun "toothache" has (many) public definitions that a dentist might use to assess the physical state of my mouth, my cry of "toothache!" bears no semantic relation to the dental definition of toothache, for I am privately using "toothache" as an indexical, rather than publicly using it in the dental sense of a noun. So regardless of whether or not I 'actually' have "toothache" in the sense of a dysfunctional dental property, my cry of "toothache!!" still stands as a fact, even if outsiders are puzzled as to what it could relate to from their perspective.

Although indexicals are excluded as objects of Davidson's analysis, given that indexicals a) serve to ground public definitions in the minds of each and every individual and b) that people use the nouns and verbs of their public language as indexicals in an unpredictable bespoke fashion, indexicals contribute to the indeterminancy of translation and reference that Davidson appeals to in the context of the propositional attitudes he analyses.

There's a useful paper i'd recommend reading with regards to Wittgenstein's relation to Dennett's views:

Consciousness demystified: A Wittgensteinian critique of Dennett's project

S4 Modal logic (which lacks logical quantifiers) is best thought of as a weakening of first-order logic:

Instead of having the particular comonad known as 'universal quantification' and the particular monad known as' existential quantification' which already give first order logic a canonical and a priori definition of "necessity" and "possibility", S4 has a weakly defined arbitrary comonad called "necessity" and an arbitrary monad called "possibility", making it weaker than first order logic.

But gven that modern type theories permit arbitrary definitions of monadic structure in addition to explicitly possessing quantifiers whose use is optional, what justifies philosophers continuing the study of modal logic with it's antiquated and impoverished syntax and weaker modes of justification that ironically encourage misleading over-interpretation by philosophers?

In my understanding, what gives Modal logic continued relevance is the usefulness of Kripke Semantics, i.e. the intuitive and useful concept of an accessibility graph of possible worlds together with propositions that pick out subsets of those worlds, a semantics which the syntax of Modal logic succinctly describes.

But if the semantics of modal logic are merely regarded as the predetermined outcome of 'real' modal operators of an underlying modal logic, as seems to be indicated when philosophers attempt to justify their abstract modal reasoning with respect to an assumed definition of the modal operators, then i think Modal logic is either obsolete, and misleading.

Ironically, I think where Kripke Semantics shines is when it is used descriptively in a data-driven fashion to chart one's present knowledge of possible worlds, without appealing to the necessary implications of a dubious modal operator. For modern logic handles reasoning from dubious assumptions in a much clearer, richer and flexible fashion.

Intentionality is a concept I use when I refer to other people's perspectives, whereas phenomenality is a concept i use exclusively with respect to my experiences.

It makes no sense for me to interpret science as analyzing a first-person subject, therefore it makes no sense for me to interpret science as saying anything either for or against phenomenality.

So you have never been unconscious? I know I have and that you do not have any grounds to doubt my subjective account of having been unconscious. — 180 Proof

'Unconsciousness' is a deceptively named concept, given that its conditions of assertibility are identical to the empirical concept of amnesia.

E.g, " I know I was unconscious last night" ,means something like "When contemplating what happened last night, I associate my experiences with the present, as opposed to the previous night."

The hard problem can be paraphrased by the following Wittgensteinian semantic problem

"How are my perceptual and cognitive judgements that i express using my mother tongue, correlated with the public conventions that define my language"?

Once these two concepts are distinguished, the hard problem ought to evaporate, regardless of whether the two concepts can be put into correspondence. For there isn't a meaningful public answer as to whether or not Mary 'learns' new information about the concept of colour when leaving her black and white world; for none of Mary's perceptual judgements bear any analytic relation to public physical theories about colour .

Of course, Mary is likely to decide to associate her perceptual judgements with said physical theories as part of a private-dialect we might call "Mary's personal physical colour theory"

Naturalised neurological theories are semantically deficient for tacking the hard problem, due to the fact their theoretical concepts are only publicly defined up to third person predication, which restricts their applicability to the description of psychological predicates in relation to the mythical third-person subject.

For example, my perceptual judgement that this apple in front of me is "green" isn't part of any public neurological theory of colour perception. Rather, my perceptual judgements constitute my personal semantic foundation for interpreting public neurological theories of colour perception.

A scientist who fails to acknowledge that a-perspectivalized naturalised science has a 'hard problem' conflates their private interpretations of science with the public theories of science. These aren't the same thing. For instance, Einstein's understanding of General Relativity isn't part of the theory of General relativity; The theory of relativity isn't defined in terms of Einstein's thoughts and observations and the theory doesn't even define observation terms. So Einstein would not be at liberty to use the public definition of Relativity to explain the existence of his frame of reference. Rather, he is at liberty to apply the public definition of relativity to his frame of reference as he sees fit.

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.