didn't think he proposed a solution. Rather, it was an example to show that it is impossible to complete a supertask. — Michael

Yes, in other words rejecting iii), namely the idea that one can finish counting an infinite sequence.

Let's first remember the fact that the limit of a sequence isn't defined to be a value in the sequence.

Re : The Cauchy Limit of a Sequence

"When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the limit of all the others"

A converging sequence might eventually settle on value equal to its limit, but even then the two concepts are not the same. So it doesn't matter whether we are talking about Thompson's Lamp, or merely a constant sequence of 1s. In either case, a limit, if it exists, doesn't refer to any position on the sequence, rather it refers to a winning strategy in a type of two-player game that is played upon the "board" of the converging infinite sequence concerned.

So it make no literal sense to consider the value of an unfinishable sequence at a point of infinity, so the meaning of a "point at infinity" with respect to such a sequence can at best be interpreted to mean an arbitrary position on the sequence that isn't within a computable finite distance from the first position. In the newspeak of Non Standard Analysis, such a position can be denoted by a non-standard hyper-natural number, meaning an ordinary natural number, but which due to finite limitations of time and space cannot be located on the standard natural number line.

As for the OP, its triad of premises are inconsistent. For only two of the three following premises can be true of a sequence

i) The length of the sequence is infinite.
ii) The sequence is countable
iii) The sequence is exhaustible

For example, Thompson's proposed solution to his Lamp paradox is to accept (i) and (ii) but to reject (iii). Whereas solutions to Zeno's Paradox tend to start by accepting (iii) but reject the assumption that motion can be analysed in terms of a countably dense linear order of positions, either by denying (i) (namely the assumption that the sequence of positions is infinite, which amounts to a denial of motion) or by denying (ii) (namely the assumption that motion can be used to count positions, for example because the motion and position of an arrow aren't simultaneously compatible attributes).

Sure. That is indeed a different take. I'm taking what I like to think of as a traditional scientific approach, otherwise known as a reductionist materialist approach. Like anyone in this field, I'm driven by a particular set of beliefs that is driven by little more than intuition - my intuition is that reductive scientific methods can explain consciousness - and so a big motivation -- in fact one of the key drivers for me - is that I want to attempt to push the boundaries of what can be explained through that medium. So I explicitly avoid trying to explain phenomenology based on phenomenology. — Malcolm Lett

Consider the fact that traditional science doesn't permit scientific explanations to be represented or communicated in terms of indexicals, because indexicals do not convey public semantic content.

Wittgenstein made the following remark in the Philosophical Investigations

410. "I" is not the name of a person, nor "here" of a place, and
"this" is not a name. But they are connected with names. Names are
explained by means of them. It is also true that it is characteristic of
physics not to use these words.

So if we forbid ourselves from reducing the meaning of a scientific explanation to our private use of indexicals that have no publically shareable semantic content , and if it is also assumed that phenomenological explanations must essentially rely upon the use of indexicals, then there is no logical possibility for a scientific explanation to make contact with phenomenology.

The interesting thing about science education, is that as students we are initially introduced to the meaning of scientific concepts via ostensive demonstrations, e.g when the chemistry teacher teaches oxidation by means of heating a testtube with a Bunsen Burner, saying "this here is oxidation". And yet a public interpretation of theoretical chemistry cannot employ indexicals for the sake of the theory being objective, with the paradoxical consequence that the ostensive demonstrations by which each of us were taught the subject, cannot be part of the public meaning of theoretical chemistry.

So if scientific explanations are to make contact with phenomenology, it would seem that one must interpret the entire enterprise of science in a solipsistic fashion as being semantically reducible to one's personal experiences... In which case, what is the point of a scientific explanation of consciousness in the first place?

Let S denote the set of stairs, let N denote the standard natural numbers and let N* denote the nonstandard numbers. We can model the cardinality of S, which is equivalent to the height of the top of the staircase, by using a non-standard natural number h* from N*. Lets assume

i) There does not exist an injection N --> S
ii) There exists a surjection I ---> S, where I is a subset of N.

Condition i) represents the hypothesis that we do not know how many stairs there are, or equivalently that we cannot know the height of the top stair due to assuming that we will never reach the bottom of the staircase.

Condition ii) represents the physically plausible situation that although we cannot count the stairs, there cannot be more stairs than some finite but unboundedly large subset of the natural numbers.

In other words, we are assuming that S is subcountable.

Let s(n) denote the n'th stair that is visited when descending. Using this order of descent on S, we have a total function S --> N* describing the height of each stair as a non-standard natural number, namely

s (0) => h*
s(1) => h* - 1
s(2) => h* - 2
..

which when written directly in terms of the indices denoting the order-of-descent is a function f

f : N --> N* :=
f (n) = h* - n*


This function describes an infinite descent in N*, and is paradoxical because

1) Every nonstandard natural number e* that is in N* corresponds to some standard number e in N, and vice-versa.

2) We have defined an infinitely descending chain of non-standard natural numbers in N*.

The paradox is resolved due to the fact that the order-of-descent we are using when descending the "infintie staircase" from the top has no recursively definable relationship in terms of the order of ascension when climbing the staircase from the bottom; although Peano's axioms rule out the existence of non-wellfounded subsets for recursively enumerable subsets of the natural numbers, our subset isn't recursively enumerable in terms of those axioms, and is therefore an external subset that cannot be talked about by Peano's axioms.

Sorry Fishfry.

On further reflection the infinite sided die shouldn't need a choice axiom in its construction (e.g a sphere can be painted by working clockwise and outwards from a chosen pole - since there is an algorithm choice isn't needed). But then what of the idea of rolling said die an actually infinite number of times? That surely is equivalent to choice, assuming the rolls are random.

The natural numbers are well ordered in their usual order. — fishfry

Yes, that is true, by Peano's inductive construction of the natural numbers. And a well-order is usually assumed for an infinite sided die, in spite of its construction lacking an inductive specification (for which side should be assigned what number?) - So the assumption of a well-ordered infinite sided die that lacks an inductive definition is the same as a countably infinite set of objects equipped with the axiom of countable choice.

For what it's worth, the fact that we can't put a uniform probability measure on the natural numbers doesn't mean they have to be "all the same number." They're all different numbers. And I can't understand the idea you're getting at. — fishfry

I took the idea to mean that the faces of an infinite die isn't a well-ordered set, unless the Axiom of Countable Choice is assumed. If this axiom isn't assumed, then the sides of the die can only be ordered in terms of their order of appearance in a sequence of die rolls, which implies that unrolled sides are indistinguishable.

I'm quite fond of this potential infinity solution and believe it may be the correct direction to pursue.

However, the die in the paradox possesses an actually infinite number of sides (the set of sides is Dedekind-infinite). What more needs to be said to argue that such a die cannot exist? — keystone

The first problem is one logical inconsistency. In Kolmogorov's treatment, the axioms exclude the proposition; if one introduced such a die as a new axiom, the system wouldn't be consistent. Whereas in my above (very rough) proposal, A Dedekind infinite set is measured directly in terms of its definition rather than in terms of it's cardinality,but which in turn implies that it has lower probability than its subsets, violating additivity, (Here I am assuming that we want to use standard rules for mapping distributions from one set to another. I'm not actually sure if there might be some other workaround than banning Dedekind-infiniteness).

The second problem is one of motive. Is the motive good enough? Consider what it means to say that the Natural Numbers are Dedekind infinite. In type theory, it refers to an object N with an arrow
1 + N --> N that has an inverse ( here 1 denotes zero, and + indicates disjoint union, and the arrow is the successor function). A standard computational reading of this arrow is that it conveys the fact that one can count upwards from zero to an arbitrary finite number of one's choosing and then count downwards to return to zero. In a nonstandard reading one is also allowed to count from an arbitrary position that cannot be reached from zero. But in either case, the arrow doesn't have the extensional significance that set theorists like to assume. That is to say, the arrow doesn't imply that "every member of the natural numbers exists prior to it being counted" , rather the arrow is used to construct as many members as one desires. In summary, we can say that Dedekind-infiniteness is a type of rule that can be used to generate Dedekind-finite extensions of any size that can be freely extended as and when one desires, by applying the rule once more.

In the case of an infinitely sided die, if the die can only be rolled a finite number of times, then its trajectory of outcomes is equivalent to the trajectory of some Dedekind-finite die that by definition is guaranteed to possess an arbitrary but finite number of unrolled sides after the final roll of the die. Is rolling the die a Dedekind-infinite number of times extensionally meaningful? Not according to the functional interpretation of Dedekind-infiniteness, which deems the previous analysis sufficient for the philosophical analysis of the fall of man paradox.

There are two types of infinitely sided dice; those whose set of sides is actually infinite - meaning Dedekind-infinite as in the set of dice sides possessing a countably infinite proper subset, versus those dice whose set of sides is potentially infinite - meaning Dedekind-finite but without having an a priori finite upper-bound on their number of sides.

You need a non-standard probability theory to express the idea of an infinitely sided fair die, but the idea only makes sense for Dedekind-finite dice.

Rather than assigning a real number to a set of outcomes to denote the probability of the set, which doesn't work in the case of infinitely sided dice, we can in the case of a fair and infinite die eliminate the distinction between sets of outcomes and their probabilities, because in this case the probability of a set of outcomes is equivalent to the set itself.

So let P(N) = N

- We simply drop the normalisation factor since it is a constant, and let N directly denote both the set of natural numbers and the probability of choosing a natural number (i.e the value of probability one) .

Let N/a denote the set of natural numbers that is the complement of the subset of natural numbers a. Then

P(N/a) = N/a

P(a) = a

P(a OR N/a) = N/a + a = N

- If N/a is cofinite, meaning that it's complement is finite, then its interpretation as a probability value is larger than the probability value for any finite set of naturals, but is nevertheless smaller than N.

- if a is finite but non-empty, then it's interpretation as a probability value is smaller than the size of any cofinite set and any infinite set, but is nevertheless larger than zero.

- If both N and a are infinite, then their corresponding probability values are of intermediate magnitudes between the two previous cases.

I think this is pretty much all that is needed for a basic non-standard probability treatment of a fair infnitely sided dice, and the resulting measure is both countably additive and normalizes to 'one'.

Lastly, the assumption of Dedekind finiteness is important, because we don't want to derive
P(N) < P(N).

So the intuition stated in the OP, that switching is always the best decision, is represented in terms of the magnitudes of cofinite sets of natural numbers as always being larger than the magnitudes for finite sets, but without succumbing to the false conclusion that the probability of getting a better result is 1, which is a 'bug' of classical probability theory caused by it's insistence upon using standard models of arithmetic.

Huh? Why? Inverted qualia arguments are specifically about different S experiencing different things. The degree of difference is what seems to defeat certain theories. — AmadeusD

Fregean ideas are necessarily perspectival, whereas the public meaning of Fregean sense is a-perspectival. So if by "qualia" you mean to refer to your first-person perspective constituted by your Fregean ideas, then what criteria of comparison to you propose to use to relate your lived and actual qualia with what you abstractly conceive and hypothesize to be my experiences? How could scientific analysis which is deliberately restricted to propositions stated only up to the third person, be of any help here?

On the other hand if by "qualia" you are in some sense referring to both of our experiences, then I presume you are no longer referring either to your actual first-personal experiences or to mine, but to some abstract concept. Which is the starting point of any behavioural, functional or computational third-personal analysis of "shared sensations" "sensation similarity" and so on, whether in type or in token.

Also I don't think language is at all relevant and is in fact a red herring. Presumably deaf, illiterate mutes who aren't blind can see colours. — Michael

Semantics is relevant, due to the aforementioned ambiguity as to what is being referred to when speaking of qualia. In these sorts of discussions, it is often implicitly assumed by participants that "qualia" is meant in some Fregean sense. Which does indeed permit the sort of abstract functional and behavioural analysis that you propose in terms of type-token distinctions and similarity criteria, but which also forgets the reason why "qualia" were included in philosophical parlance in first-place - as a means of bridging the subjective private understanding and use of language in the first person, with the use of physical concepts that speak only in terms of abstract definitions stated in the third-person.

With respect to physicalism, the question is whether or not this difference in colour perception requires differences in biology, and with respect to naive realism, the question is whether or not one of them is seeing the "correct" colour (in the sense that that colour is a mind-independent property of the object). — Michael

Yes, I am in agreement there, although I would say that the question you mention is with regards to the physical analysis of perception, rather than a philosophical analysis of perception which less constrained than physical analysis, since the latter analysis is free to define concepts in relation to Fregean Ideas, which isn't possible in an aperspectival physical analysis.

IMO, when eliminative materialists speak of "consciousness not existing", I interpret them to mean (whether they agree with my interpretation or not), that physical analysis is by definition restricted to the analysis of cognition and perception in terms of Frege's notions of sense and reference which constitute the meaning of "objectivity", but which does not include the meaning of "subjectivity" that refers to the unshareable Fregean ideas that modern philosophers often refer to as "qualia".

First of all, does it make sense to speak of shared sensations?

If the answer to that question is deemed to be negative, then inverted-qualia arguments cannot get off the ground, but in which case aren't necessary for refuting physicalism, for a negative answer to the former question would imply that the set of "Alice's sensations" is both disjoint from, and unrelated to, the set of "Bob's sensations", however they might be labelled. But then again physicalism cannot also get off the ground, since physical concepts are "shareable" by definition.

Recall Frege's semantic distinctions of sense (referring to a term's public usage), reference (referring to what if anything a term signifies) and ideas (referring to a term's private aesthetic meaning that varies from person to person). Then ask if a sensation is shareable in any of those above semantic aspects.

Obviously, Fregean ideas aren't shared by definition, so if by "sensations" we mean the Fregean ideas that each of us subjectively intuits about word meaning, then we can refer to our previous analysis and conclude that that the concept of inverted-qualia is nonsensical.

But if by "shared sensations" we are referring only to Fregean sense (which the word itself might suggest), then we are only referring to the shared public usage of the term "sensation". in which case "inverted qualia" could mean something like when two subjects react equally and oppositely to the same stimulus - a meaning which actually amounts to a physical definition of "inverted qualia", even if the concept says nothing about any underlying Fregean senses that will typically be assumed to exist regardless of the physical concept's silence on the matter. In fact, the very meaning of a "physical concept" might be interpreted to mean a concept that is by definition invariant to the Fregean ideas that individuals privately associate with the concept, in contrast to aesthetic concepts whose definitions are allowed to vary among language users in line with their unique Fregean ideas that they each associate with their shared terminology.

Lastly, the common-sense of naive realism and the classical psychology of perception might lead us to consider "sensations" as lacking any Fregean referents. Indeed, we tend to speak of an object as "looking red to an individual" but not as being red per-se. However, the later Wittgenstein remarked that different types of sensation vary as to the degree that the subject of the sensation attributes the sensation to himself versus the object of his perception. In Wittgenstein's example of a green stinging nettle, he points out that we will tend to refer to the nettle as possessing "green leaf patches" but not as possessing "painful leaf patches", and seemed to imply that the degree to which a sensation-type is attributed to the object of perception was determined by the Fregean sense of the sensation type.

According to Externalism, knowledge is merely true belief, in which the truth-maker (reality) is external to whatever justifications one might offer in the defense of their beliefs. So externalism avoids the Gettier problem of false justifications that produce true beliefs, because it doesn't consider beliefs per se to be truth-apt. Or alternatively, if it is assumed that truth is internally related to beliefs, then externalism denies the existence of beliefs. Either way, externalism eliminates the normative dimension of epistemology, an elimination which many philosophers find problematic, and which is a common characteristic of naturalised epistemology.

"Now this is eternal life" : Was Saint John a presentist?

According to this (possibly unreliable) answer as to the meaning of "eternal life" :-

"
....
It is a mistake, however, to view eternal life as simply an unending progression of years. A common New Testament word for “eternal” is aiónios, which carries the idea of quality as well as quantity. In fact, eternal life is not really associated with “years” at all, as it is independent of time. Eternal life can function outside of and beyond time, as well as within time.

For this reason, eternal life can be thought of as something that Christians experience now. Believers don’t have to “wait” for eternal life, because it’s not something that starts when they die. Rather, eternal life begins the moment a person exercises faith in Christ. It is our current possession. John 3:36 says, “Whoever believes in the Son has eternal life.” Note that the believer “has” (present tense) this life (the verb is present tense in the Greek, too). We find similar present-tense constructions in John 5:24 and John 6:47. The focus of eternal life is not on our future, but on our current standing in Christ. "

In which case the so-called "after-life" of his Christianity is a misnomer, in that it's conditions of verification aren't considered to transcend the present.

Experimental synthetic languages such as Ithkuil and Lojban were designed to improve upon the semantic deficiencies and limitations of natural languages, in the knowledge that many of those deficiencies being responsible for the creation of philosophical pseudo-problems. However, a a learner will inevitably rely upon their mother tongue as a meta-language when learning those synthetic languages, so it is hard to see what the payoffs are in learning such languages in the short to medium term, especially considering the fact that one can reason and communicate poorly in any language.

Also, the more powerful a language is in it's ability to express and disambiguate information, the harder it is to master the language due to the increased complexity of it's semantics. There were no speakers of Ithkuil for that reason, given that it might take hours for a human to compute a sentence. Hence the invention of New Ithkuil

I think 'Kripke's Wittgenstein' was tainted by Kripke's semantic foundationalism. Kripke was correct IMO to conclude that Wittgenstein's concept of rule-following involves appealing to semantic criteria that are independent of the psychological facts of the rule-grasper (roughly speaking), due to the trivial implication that "to grasp" something entails a distinction of the grasper and the grasped thing. However, Kripke was wrong to conclude that the external criteria referred to assertibility conditions laid out by social convention. I think Kripke's incorrect conclusion about the later Wittgenstein's views was due to the fact that Kripke, unlike the later Wittgenstein, could not accept the non-existence of a universal and shared semantic foundation.
For Wittgenstein, any assertibility criteria can be used for defining the meaning of 'grasping' a rule, and not necessarily the same criteria on each and every occasion that the rule is said to be 'used'. And a speaker is in his rights to provide his own assertibility criteria for decoding what he says, even if his listeners insist on using different assertibility criteria when trying to understanding the speaker's words.

I'm not sure I follow. Can you reword this? — Tom Storm

I'm basically pointing to the ancient debates regarding the question as to what grounds personal identity. Does the ground consist of essential criteria, or not? And is the ground context-independent or not? The ghosts of folklore suggest to me, that humans ordinarily do not appeal to essential criteria when identifying a person.

In interviewing people who have experienced ghosts, what I find interesting is how often hauntings come with sound effects and beings present as fully dressed, often in period clothing. I get the theory behind a spirit appearing in some form, as an entity, but in clothing seems a stretch to me. Why would clothes also survive death? And sometimes there are ghost trains, cars and horses and dogs with their drivers or masters. What makes animals or machines come along for the undead journey? — Tom Storm

Isn't our very concept of a person made entirely out of the clothes of contextual accident?

Robots that make "perceptual errors" are only epistemically wrong in the sense of behaving in a fashion that their owners find undesirable. So if humans are robots, then humans don't really make epistemic errors when they fall victim to optical illusions.

I understand everyone else's experiences in accordance with the logic the indirect realism that is in relation to my world that i grasp directly.

In summary, it's direct realism for me, but indirect realism for everyone else.

The old man's position isn't surprising to read, given his defunct beliefs that the structure of syntax reflects universal aspects of neurological processing. I now think of Chomsky as proposing largely unhelpful tautologies for cognitive science and linguistics as those subjects were conceived under his influence in the latter half of the previous century that tended to downplay the external and behavioural causes of thought for ideological reasons (american individualism?), rather than him presenting useful and relevant scientific theories of language and cognition for this age.

I mean, what exactly are "thoughts"? where is the supposed interface between perception, thought and communication?

First of all, a CTC doesn't come in iterations, so if there's a loop, it's like a portal that's open for a while. One can go through (back a day say), and do it again in a day, but not a third time. That's not a contradiction since there's no iteration, only one loop with two different people going through, possibly holding hands. Secondly, no person needs to experience the trip. The loop is likely not something a living being can survive, but getting information through is enough. If at the past end of the loop, data is received concerning news of tomorrow (such as a sports score), that is evidence that it worked, without anybody having to experience it first hand. The sports score constitutes an empirically observable consequence. — noAxioms

But if iteration isn't permitted, then is sending information backwards proof of a loop? Isn't the hypothetical possibility of a temporal contradiction the very motive for the loop interpretation? For if contradictions are ruled out a priori, then what justifies the use of a loop topology?

E.g suppose that it is possible to send sports results backwards in time. If this action "changed" history, then many people (including myself) would interpret this as merely referring to the action producing significantly non-local effects in our present, so that we can preserve the meaning of the word "history" as referring to immutability. On the other hand, if the action cannot "change" history, then what is the proof that anything has actually been sent backwards?

To return to the presentist reasoning I sketched earlier, It is logically consistent to believe that the past of our world is generated 'on the fly', as in a roguelike video game that generates the content of the game world as an effect of the adventurers present actions. In such worlds it might appear that information is sent backwards. E.g the adventurer is in an unknown dungeon with a closed door. Only after he opens the door does the game decide what lies beyond the door. Every adventurer's action has a predictable "forwards" effect e.g pushing a closed door causes it to open, and an unpredictable "backwards" effect, referring to what the action reveals about the world. But no information is actually sent backward, unless the adventurer is allowed to choose the revealed information, say as a consequence of using a magic spell . So the adventurer's ordinary present actions enable the course of history but without controlling the course of history (which in general is decided by a Dungeon Master of which one of his responsibilities is the logical consistency of the game world).

Notably, players don't typically interpret "history change" as time travel, e.g when an adventurer uses a magic spell to re-roll the state of the dungeon around him, but merely as magic affecting the global state of the present. Amusingly, a philosophical dispute once arose between players of the single-player roguelike game Nethack. In that dungeon crawler there exists the "Potion of Amnesia", which if drunk by an adventurer causes the game to delete it's record of the adventurer's knowledge of the game world,whilst leaving the actual game world in tact, meaning that the player must rely on their personal memories when their adventurer navigates and relearns the content of old locations. But isn't that cheating? Shouldn't a true potion of amnesia change the world itself? Players are divided.

Anyone can look at the past, which isn't any sort of retrocausality. I mean, that's exactly what hte archaologists do. It's looking forward or causing some effect backwards that's the trick. Most of the plausible scenarios I have in mind require cooperation at both ends. No travel to a time that isn't expecting you, but rather a portal deliberately held open at both ends to let information or more through. So in that scenario, there's no 'changing' of the earlier time since the travel back to that point was always there. That's the nature of a CTC. SEP had some examples of this, but I find them implausible. — noAxioms

I think that the concept of non-local causal cooperation that you allude to is interesting and useful, but i think CTCs are empirically inconsistent and theoretically unnecessary. For any proposed loop, if you could experience going around it more than once, then the proposed loop would be falsified (since the second iteration would be distinguishable from the first). But if you cannot experience going around the loop, then how you do know the loop exists to begin with? A theory containing a CTC cannot have empirically observable consequences on pain of contradiction. So a CTC can at most be an uninterpretable expression of mathematical convenience rather than a representation of a physically verifiable entity.

As for archeology, how do you know that the practice isn't retrocausal? Consider that the effect of digging into the ground can be expected to produce both predictable consequences that we might call "forwards directed" e.g the dig producing a hole next to a mound of earth, as well as unpredictable consequences that we might call "backwards directed", e.g the dig revealing of a Roman hoard of treasure. For why should the hoard of treasure be assumed to exist before it was discovered in the hole? Why shouldn't the archeologist take some credit for the hoard's physical existence?

Sure, before the dig was commissioned archeologists might have discovered other archeological evidence the day before, that implied that the hoard would be found where the hole would later be dug. But that merely moves the goal posts; why should the hoard of treasure that was unearthed be assumed to exist prior to the establishment of the archeological evidence that they determined day before?

So in short, i think the concept of non-local causal cooperation (Synchronicity?) is a causally permissible concept that aligns with experience, but I cannot say the same about closed time-loops.

According to the philosophy of intuitionism, a sequence that is said to be "without an end", is only taken to mean a sequence that is without a defined end. This is similar to computer programming, where an infinite loop that is declared in a computer program is only interpreted to imply that the program is to be stopped by the external user rather than internally by the program logic.

So in intuitionism (and computer programming), the difference between a finite sequence and an infinite sequence is taken to be epistemic rather than ontological. From the point of view of the producer of the sequence who gets to control it's eventual termination, the sequence could be said to be "finite", whereas from the consumer's point of view who has no knowledge and control of the sequence's termination, the same sequence could be said to be "infinite", or better, "potentially infinite". Or even better, the word "infinity" can be deprecated and replaced by finer-grained terminology that precisely conveys the information that one has at one's disposal in a given situation, without committing to the idea that the information one has is complete.

Amateur (and even some professional) philosophers demonstrate a profound gullibility, in their face-value interpretation of mathematical symbolism. To believe that infinity means "never ending" in an absolute sense just because an upper bound is omitted from a definition, is like believing that a blank cheque cannot bounce.

Two types are extensionally equal if they reduce to the same set of values when the abbreviations used in their respective definitions are expanded out. Nevertheless they aren't intensionally equal unless their definitions are the same before their expansions. Generally speaking, type theory distinguishes intensional equality, also referred to as definitional equality, from extensional equality, due to the fact that the extensional notion of equality is undecidable unless types are restricted to decidable sets. Whilst intensional equality always implies extensional equality, the converse is only true for "extensional" type theories which are those type-theories that define intensional equality explicitly in terms of the fully-expanded extensional equality. But this implies that type-checking in those theories is generally undecidable and very expensive to compute in comparison to intentional type-theories that don't bother to consider extensional equality when type checking. For this reason, general-purpose theorem proving languages tend to be intensional, meaning that two terms or types are only considered to be equal for the purposes of substituting one for the other in a given context, only after the programmer has both constructed a proof-term that they are equal, and has also granted explicit permission to substitute one for the other in that context on the basis of that proof. So the practical difference between extensional and intensional type theories is the degree of automation that they permit during the process of type checking, i.e the burden of proof that they put onto the programmer.

The above Chat GPT examples concern logical behaviourism rather than physicalism. For example, a solipsist who doesn't believe in the literal existence of "other" minds, is on the one hand likely to identify as a logical behaviourist when it comes to interpreting so-called "other minds", and on the other hand as an idealist when it comes to his interpretation of the physical world as being inseparable from his "own" mind.

If the past is assumed to be potentially infinite as opposed to either finite or actually infinite, then it isn't necessarily the case that every cause must either be initial or a successor. Instead, the position of any cause in the order might be determined on-the-fly, and only when it is necessary to preserve causal consistency.

For why should the universe decide before our measurements and observations, what is and is not an initial cause? That question might look contradictory, but only if it is assumed that the universe consists of an absolute order of events whose existence transcends our observations and measurements of it.

But if that assumption is dropped, whereby observations and measurements are deemed to be constitutive factors of the thing being observed and measured (as for example as in subjective idealism and in interpretations of QM that fall under anti-realism), then the question as to the ordering and positions of unobserved events denoted by hidden variables, doesn't have to have an absolute and definite ontological answer one way or the other.

There is a difference between the concept of changing the past versus the concept of affecting the past. Changing the past is that which only a time-lord could do on the basis of his transcendental privilege, as illustrated by an unrestricted variant of the time-travel monopoly game, whose rules permit the players actions to diverge from any fixed-point of the game, which is essentially Classical Monopoly, that incidentally isn't identified as expressing time travel for that very reason (for fictional time-travel is a bunch of inconsistent narrative requirements).

But the ability to merely affect, or rather to construct the past is a weaker condition that only stipulates that the past is a creation made out of available information, including information that is the consequence of players present and future actions, but it doesn't assume that the revealed information is mutable.

Of course, a mere mortal cannot know as to whether the information at his disposal "comes from the future or the past", in order to rule out the possibility of him using that information to cause causal contradictions, and this epistemic restriction isn't modeled in the suggested time-lords game I previously suggested, which grants players transcendental knowledge of the future/past distinction and merely forbids them from acting upon it.

The only means of eliminating temporal omniscience from the game, is to restrict the game to a single iteration of the board loop. But to distinguish the resulting game from a case of single-iteration classic monopoly requires a different approach to the rules and constraints. For example, by granting a random event such as "Bank error in your favor, collect £200" the capacity to impose constraints onto the future actions of the players. Say, by that event triggering the potential introduction of a "cause" card into a deck of cards that some player is guaranteed to draw in the future (but without his prior knowledge).

I'm disappointed that nobody has invented time-travel Monopoly. As in the ordinary case, a game consists of players traversing a spatial loop by rolling dice to land on squares , then making decisions and interacting with other players, and repeating this process until they reach or pass Go. Call this an iteration of the loop.

For a trivial implementation of the time-travel variant, say that whenever a player passes Go, he is forced to land on Go and must wait for the other players to finish the loop. After all players finish the loop, their individual wealth is reset to that of the beginning of a new game (£200?). During the next iteration of the board, the players must perform exactly the same actions as they did in the previous iteration and must re-use the dice-values they previously rolled.

Obviously that example is completely useless and boring, but it expresses the desirable property of an iteration of a Monopoly time-loop, namely that the iteration of the loop is stable, in the sense that the next iteration must proceed in exactly the same way as the last.

To put this condition more generally, a stable iteration must be a fixed-point of some functional F of the players combined observations and decisions. In the case where we always restart the players from Go and reset their wealth to the beginning of game and force them to play as they did in the last loop , then every possible iteration of the board is a fixed-point, since there is no new information leaking from one iteration to the next. This corresponds to setting the functional F to the identity functional.

So the challenge of Board-game game design here, is to design a functional F that allows partial leakage of information from one iteration to the next, such that players have freedom to make decisions that spans several iterations of the loop, albeit with that freedom decreasing from one iteration to the next as the game converges towards a fixed-point, such that the result of the game expresses time-travel, or rather a time-loop, although the iterated gameplay beforehand does not.

It can be a fun exercise to consider how one might change the rules of the board game Monopoly, in order to turn the spatial loop into a time loop. What should the state of the board be for the first player to pass Go? If he lands on a square and decides to buy a house, then which players does this effect and how should the state of the game be backtracked and updated?

If you think about what you're saying, then you also agree with me. If something appears or happens that has no prior reason for its existence, its a first cause. Notice the title says 'a' not 'the' first cause. There is no reason preventing our universe from having multiple first causes in the past, the present, or the future. A first cause has no reason why it should or should not happen. It simply does. — Philosophim

But that's stretching the meaning of "first" to the point of vacuity, for the concept of "first" is only meaningful in relation to a recognizable order with a distinguished bottom element. In the absence of a well-defined order, the concept makes little sense, especially considering that a rejection of the causal order doesn't entail that postulated "first" causes can't have explanations in terms of other causes, but only that such explanations are incomplete, vague, relative, ever changing, etc.

5. Infinitely prior, and infinitely looped causality, all have one final question of causality that needs answering. "Why would it be that there exists an infinite prior or infinitely looped causality in existence? These two terms will be combined into one, "Infinite causality.

6. If there exists an X which explains the reason why any infinite causality exists, then its not truly infinite causality, as it is something outside of the infinite causality chain. That X then becomes another Y with the same 3 plausibilities of prior causality. Therefore, the existence of a prior causality is actually an Alpha, or first cause. — Philosophim

You need to clearly distinguish spatio-temporal causality from your murkier concept of meta-causality.

In a similar fashion, Stephen Hawking once proposed a causally closed cosmological model of the universe , in which the universe was hypothesized to be finite but without a spatio-temporal boundary. Nevertheless, he famously asked "what breathes fire into the equations?". But this philosophical question as it stands cannot be translated into the spatio-temporal language of physics. Furthermore, there isn't a consensus that Hawkings philosophical question is even meaningful, let alone how it should be solved or dissolved if it is.

Another possibility you are overlooking, is the possibility that the very existence of the past and its historical content might not transcend the ever-changing state of the present. In which case, the past is open and indeterminate like the future and there isn't a universal causal order.

This has nothing to do with theological assertions jgill. Forget God. It floors me that I cannot get through to other atheists on this. Truly their fear of this being theological terrifies them to the point of being unable to think about it. I am an atheist. I wrote this. This is about base matter. Its very simple. Don't let fear prevent you from understanding it. — Philosophim

What makes you think that you can conceive of a first cause?

I can for example, conceive of, and indeed witness, a pencil line that has a beginning, and I can also start counting up from zero. But these so-called "first events" that occur in ordinary experience are only conceivable to me because i am able to witness or conceive of other events in time and space that occur earlier ...

In my experience of fellow atheists, they often harbor a peculiarly theological belief in "nothingness", in that they seem to reify the notion as a sort of anti-substance that they envisage as existing before and after substance, out of which they construct myths such as universe as having an objective "beginning",or of personal experience as having a subjective end. (They will deny these charges of course, in the usual spirit of "true believers"). But if we reject this ontological interpretation of nothingness as being nonsensical, then how else are we supposed to conceive of absolutely first (and last) events?

None of this shit is "tangible". "Infinite" is not tangible. That's the issue, because it's not tangible, mathematicians are free to create all sorts of axioms which do not relate to anything physical. But when the mathematics gets applied there is a very real issue of the intangible aspects of reality. And if the axioms which deal with the intangible in mathematics do not properly represent the real intangible, the product is "the unintelligible". — Metaphysician Undercover

To be clearer, I meant that an infinitesimal is "tangible" if it can be finitely described as a total computable function,which implies that the tangeable infinitesimals correspond to an undecidable countable subset of the natural numbers.

But note that by definition, an infinitesimal only has to satisfy the condition that whenever it is multiplied by a number of arbitrary large size, the product is always less than some finite constant. This condition can be satisfied purely by mapping the natural numbers onto a data-structure other than a line. So there exists semantics for infinitesimals (and their reciprocals) that does not imply the existence of infinite time, space or information (which is the unfortunate result of misinterpreting such numbers as literally denoting limitless extensions)

This is what happens when we approach the issue of "the first cause". The calculus turns the first cause into a limit on tangible causation, rather than treating the first cause as an actual cause. But if there is an actual intangible first cause then the mathematical representation renders that first cause as unintelligible, being outside the limit of causation, according to the conventions for applying the mathematics. — Metaphysician Undercover

Similarly, the information implied by a limit is relative to one's method of counting. E.g if we define a number n to be greater than every natural number (which we have the right to do), then infinite extension isn't implied if we choose to start counting within a finite distance from n.

The classical theory of real numbers interprets 1.000... and 0.999... as referring to the same equivalence class of different Cauchy sequences. So it isn't necessarily true that the system of real numbers conflates the sequences 0.999.... 1.00..., for the truth of that hypothesis is decided by assumptions concerning the existence and construction of Cauchy sequences prior to their identification as real numbers. For example, a computational interpretation will identify cauchy sequences with total computable functions, whose Cauchy limits might not necessarily be decidable, and even if they can be proved to exist, their limits might not be decidably different or indifferent. On the other hand, intuitionism interprets the meaning of 0.9999.... extensionally as referring to an unfinished sequence of data, in which case the very notion of a sequence, cauchy or otherwise, as having a definite limit is denied as absurd, meaning that not only is 0.9999 distinguished from 1.000..., it is also distinguished from any other instance of 0.999....

Perhaps we ought to say that the Real numbers cannot be interpreted as directly referring to Cauchy sequences, unlike in the case of the Hyperreals, on pain of the Cauchy sequence interpretation being in conflict with the Archemedian property of the reals that it's axiomatization imposes by fiat, but which the Hyperreals sacrifices for the sake of an illusion of creating "more" numbers.

Also, lets be wary of non-constructive interpretations of Hyperreals, for otherwise one ends up having infinitesimals by fiat that do not denote anything tangible. If we stick to constructive principles, then contrary to popular belief there cannot be more hyperreals than natural numbers, let alone of real numbers, meaning that hyperreals are just reorderings of the naturals , but whose operations aren't necessarily recursive.

Unless a clear, non-debatable physical example arises the things uncaused may be the empty set. — jgill

A resource-conscious set-theory that only expresses transformations between existent sets, could in principle be developed by introducing "negative" sets, such that the empty set denotes the union of equal sets of opposite polarity, whereby the resulting set-theory operates in an analogous fashion to the string-diagrams of particle physics in which energy is purely transformative without being created or destroyed.

But for some reason the traditions of logic and set theory have remained entrenched in structures such as toposes that forbid an initial object from having incoming arrows from other objects, i.e. their initial objects are strictly initial, which to a layman leads to the unnecessary impression of logical origination.

Not sure what you're saying here but definitely a fan of Kripkenstein. — Apustimelogist

"According to functionalists, mental states are identified by what they do rather than by what they are made of" - IEP

Yet functionalism leaves the very nature of "doing" unspecified, so it is hard to think of what functionalism rules in versus out. The concept of functions/doing is part descriptive and part normative, and related to metaphysical presuppositions about the nature of time, causality and counterfactuals. For instance, can "doing" purely consist of synchronised motions like the contents of a movie? or is agency, causation and the notion of counterfactuals involved?

Kripke came to mind for similar reasons, in his astonishment to learn that the meaning of mathematical functions is intensional, i.e implicit and normative, as opposed extensional i.e explicit and descriptive.

(Data might be interpreted as expressing a function, but cannot ground the meaning of the function or make the meaning of the function explicit, since the latter's meaning is inexhaustible from the perspective of the user of the function who understands it normatively, while being open to interpretation from the perspective of an observer of the purported function who understands it descriptively)

For me, meaning is functional. If our behavior is functionally explained by brains entirely then meaning is as well. — Apustimelogist

If functions are regarded to be nothing more than tools, then it would seem that the intensional meaning of functions is entirely dependent upon the intentional state of the investigator who applies them.

It seems to me that the identification of meaning and function per-se doesn't distinguish function realism from function anti-realism and idealism. (Kripkean skepticism comes to mind again)

The philosophy of mind (which in spite of appearances isn't a particular subject but concerns the whole of the subject of philosophy) is part of science, in so far that the purpose of science is considered to be explanatory in the sense intended to satisfy the existential questions of a particular human being.

The techniques of science and even it's formalized theories can be considered instrumental, but if the purpose of science isn't considered to be instrumentally pragmatic but explanatory in the above sense, then there exists a semantic or explanatory gap between the tools of science and it's supposed goals, which must be filled somehow, leading us back to philosophy and it's patchwork of vague and apparently inconsistent pre-theories

So if we reject the idea that science and philosophy have distinct goals, then what you have described under the heading of the philosophy of mind, is the pitiful state of science as a whole. Also your summary of AI is interesting, because it reflects society's recent obsession with Machine Learning that has up until recently, ignored the normative discipline of symbolic reasoning, which must be addressed if AI is to scale to more difficult problems in a fashion that is reliable and understandable, but that direction opens the can of worms known as the Philosophy of Language, which is at the heart of Philosophy of Mind...

As i see it, the mind-body problem is but one example of the semantic under-determination of scientific theories, and one's tolerance for semantic under-determination depends on what ones goals are.

I think you are giving idealism a realist interpretation, by interpreting " the mind" as a speculated theoretical object or posit, with your infinite-regress arguments resembling those used to attack indirect realism. Ironically, Berkeley's arguments against representationalist materialism were that he found it to be incoherent for reasons which are very similar to yours.

There is no "mind" posited in Berkeley's arguments for subjective idealism in the literal sense you assume, but only ideas referring to the thoughts and observations of the individual.

Nevertheless, Berkeley apparently remained uncommitted to the solipsism which many consider subjective idealism to imply, for although his arguments for idealism were based only on ideas, he was apparently open-minded with regards to the truth of the rationalist doctrines of causality and the external world. Like Malebranche and Hume, Berkeley didn't consider causality to be reducible to observations, for he understood observations in themselves to be inert, like the video frames of a movie. So if causality and externality were to exist, he argued that they must exist in some other mind that exists apart from one's ideas, namely in the mind of god, which ironically leads back to realism.

(I consider Berkeley to have shown that realism is ultimately a theological notion - the speculated existence of external reality in physicalism doesn't seem any less theological to me than Berkeley's mind of god)

This is not an acceptable explanation of causation. An assignment of causation does not exclude the possibility of other things having the same effect. So in the example above, saying that heat causes water to boil does not exclude the possibility that something else as well, such as a drop in pressure, could also cause water to boil. That A is judged to cause B does not exclude the possibility that something else might also cause B as well. — Metaphysician Undercover

You've misunderstood me. Yes, there can be multiple causes for an effect, but when testing for the existence of a causal relation in a series of repeated trials that check that consequents of type B allows follow after antecedents of type A, then it must be assumed as a working hypothesis that there are no other possible causes of B other than events of type A. For otherwise a successful test might only indicate correlation between As and Bs.

You presumably agree that each video frame of a movie isn't the cause of the next video frame in the movie. So even if video frames of type A are seen to always occur before video frames of type B, such that they are in perfect correlation, then you would not want to identify that relation as causation. No?

Which is the reason why counterfactuals come into play. For causation isn't supposed to merely refer to perfect correlation. At least, that isn't how the concept of causation is used by the sciences, in which causes refer to conditional propositions in which the output of the conditional is a function of the input.