Is a trail with a fork ( ---< ) vague or ambiguous?

If one intends to use the trail as a path to a destination in mind, then the trail is ambiguous. If one merely intends to use the trail without a destination in mind, then the trail is vague. And if one doesn't intend to use the trail, then the trail is neither vague nor ambiguous.

Godel's 2nd incompleteness theorem is not that certain systems can't be proven consistent, but rather that if they are consistent then they can't be proved consistent by certain means — TonesInDeepFreeze

Constructively, the implications of the incompleteness theorems are stronger than that. The consistency of certain systems (PA and the like) cannot be constructively proved by any means. All that can potentially exist in the constructive sense with regards to such systems are i) potential constructive proofs of absolute inconsistency via brute-force evaluation of the underlying sequent calculus until the inconsistency is unearthed, and ii) conditional proofs of relative inconsistency, where the inconsistency of one system implies the inconsistency of another. E.g Gentzen proved that the inconsistency of PA implies the inconsistency of PRA + transfinite induction on the ordinals.

Personally, I am tempted to interpret Gentzen's result as denying the meaningfulness of epsilon zero (i.e the first limiting ordinal) as being considered a "well-founded" ordinal. For while it might be shown one day that PA is inconsistent, it can never be shown to be consistent unless one begs the question. By semi-decidability it is meaningful to ask if epsilon zero isn't well-founded, but it isn't meaningful to assume or hypothesise that it is well-founded,

There is so much complexity I find it hard to believe that a little meaningless self-reference of the kinds we are talking about will gum up the cogs in the machinery. — T Clark

Recall that Peano arithmetic might turn out to be inconsistent. In which case, an application of a deductive system based on such arithmetic might result in physically untrue predictions via explosion.

Wittgenstein made the point in Philosophical Remarks (IIRC), that whilst such inconsistencies would lead to physically untrue predictions if applied blindly, there is no reason why the occurrence of such events would discredit uses of the system for which inconsistency plays no role. And since it is impossible to predict the existence of mathematical inconsistency before it arises (due to the the second incompleteness theorem), there is no reason to fret about the possibility a priori. We only need to patch our systems as we go.

Wittgenstein's remarks weren't targeted towards scientists or engineers, but towards philosophers who sought to establish epistemological foundations of mathematics.

Turing on the other hand was worried, due to his interests in artificial intelligence, where such systems might be applied blindly.

Apples and Oranges.

Both Turing and Wittgenstein understood that sufficiently complex formal systems (.e.g Peano arithmetic) cannot be known to be consistent a priori due to the halting problem, and that inconsistency in practice must be patched as and when problems arise in application, similar to legal precedent or a sport.

Wittgenstein argues, using the example of 20th century applications of logic and mathematics, that if logical paradoxes and incompleteness results of higher-order logic have no practical implications, then why should philosophers worry?

Turing's point should be understood in relation to artificial intelligence and automation; whilst it is true that logical paradoxes and inconsistencies aren't relevant to manual applications of mathematical modelling in bridge design, they are potentially relevant with regards to the automation of bridge design in which artificial intelligence reasons in higher-order mathematics.

So they should be regarded as being on the same side, considering the fact that both had no time for platonic superstition, and that both were making different points.

I've only skim read the above article, so correct me of i'm wrong, but i'm of the impression that Dummett (according to Devitt) is equating, or maybe even equivocating, intuitionism with intuitionistic logic. Under the latter interpretation it is indeed the case that the truth of a proposition is synonymous with, and describable in terms of, the sole activities of the mathematician. One can say in this case that the syntax is the semantics. But under the former interpretation the issue is more subtle and already accommodates my (potentially incorrect and off-base) understanding of Devitts position.

According to intuitionistic logic and intuitionistic type theory, the construction of anything, including the natural numbers and arithmetic, is exhaustively described by axioms and rules of inference, in that terms in the logic make no reference to contingent events of the outside world. So for example, the definition of "one" is reducible to an infallible function called "Successor" operating on a directly observable state "O" to yield in every case a directly observable state "SO". ( The infallibility here looks suspiciously platonistic if the logic is interpreted as being literally true, as opposed to interpreted as being an approximate model of something or a set of normative principles).

But according to intuitionism objects and numbers can also be lawless, where an object is said to be "lawless" if it's existence and/or value isn't decided by the formal system it is part of, but by something not described by, and external to, the formal system.

For example, the standard model of Heyting-arithmetic, which is the intuitionistic logic equivalent of Peano arithmetic, does not include terms for representing the random outcome from physically tossing a die, whereas a corresponding system in intuitionism can, where such a term is said to be "lawless", meaning that the term refers to a value that isn't internally decided by the logic and whose value only potentially exists. In software engineering, such objects are often called "Promises" and are used by programs to denote external random events of the future, such as unreliable and uncertain server responses, where the possibilities of successful replies, failed replies and no replies must be handled by the program logic.

It should also be remarked, that the founder of intuitionism, Brouwer, considered mentally-created numbers as being lawless in so far as they aren't consciously associated with the outcome of a formula.
So by this philosophy, a "lawful" number is merely a value that is logically interpreted, either consciously or practically, as being of the codomain of a formula.

In summary, it isn't the case that intuitionism equates truth with operational construction as in intuitionistic logic. Intuitionism also generalises the intuitionistic-logic notion of proof to include terms that are merely potentially referring, and that when referring refer to contingent events external to the prior observations, construction practices and prior knowledge of the creating subject. But at the same time, intuitionism does not assume that such potentially referring terms are actually referring until as and when the terms are externally initialised with values. Therefore according to Quine's maxim "to be is to be the value of a bound variable", intuitionism cannot be described as realistic.

I get the initial impression that Devitt is arguing for a very weak form of "realism" that merely denies the reduction of truth to principles of construction stateable by a priori laws. Nevertheless, to my understanding his views seem to be already accommodated by intuitionism, which isn't considered to be a realist philosophy of mathematics.

My questions are therefore:

Every computer program that interacts with the world is describable in the language of intuitionism, in which terms presently either have values or don't have values. What in addition, if anything, does Devitt think needs including?

Is Devitt's very weak realism, as i understand it, really acceptable to common-sense realists? how is it different to negative theology?

One issue that tends to get overlooked with anti-realism, is that it is consistent for a person to assert anti-realism for himself, but to assert realism for other people. This is because the grammar of anti-realism is rooted in perspective, and since people don't share the same perspective, they cannot assert the same definition or understanding of anti-realism.

For example, if Alice and Bob are watching the sunset together and Alice falls asleep, Bob can verify "The sunset looks beautiful, while Alice cannot see the sun", but Alice cannot verify this. Furthermore, if Alice slipped into a coma and died of a drug-overdose, Bob could then assert "The sun still exists, but Alice does not".

In other words, Bob's anti-realism about the sun and it's relationship to himself does not extend to his understanding of the sun in relation to Alice. If Bob is to understand Alice's observations about the sun as being about his sun, then he must understand her remarks in the sense of correspondences between her perceptions of his sun and his experiences of his sun. In effect, he understands Alice as being a brain in the vat that he calls his world.

In my view there's nothing naive about 'naive anti-realism', so it doesn't need moderation. So-called moderate anti-realism is either a generalized form of realism in disguise, else it is a contradiction in terms.

One is either a realist about a proposition P or one isn't. If one is an anti-realist about P, then to assert P is to know P, so an anti-realist cannot say P and ~Know P. This isn't a problem, provided the anti-realist gives an account of "false" assertions so as to eliminate the need for ~Know P, as i'll mention below.

Firstly, suppose a mathematician says "The Goldbach conjecture P might be provable in Peano Arithmetic, but we just don't know". What he means is something like " A well-formed formula named 'P' hasn't so far been obtained as the conclusion of a proof object of PA".

If the above example from classical logic is expressed by saying " P or ~P and ~Know P" then it should be understood that the sign "~" of logical negation indicates that "Know P" isn't a referring term because a proof-object of P doesn't presently exist. Consequently the proposition " P or ~P and ~Know P" can be deflated to "P or ~P" without loss of meaning, where "P or ~P" is trivially true as per the definition of PA. Also notice that "P or ~P" and "Know (P or ~P)" are identical propositions in referring to the axiom known as LOM.

Likewise, to take a non-mathematical example, consider this weekend's boxing match between Tyson Fury vs Deontay Wilder. To assert today, on Thursday the 7th October, that "Either Fury or Wilder will win, but nobody knows for certain" is merely to say "Fury or Wilder will win", which it should be noticed is a description of todays state of affairs, and not of a hypothetical future state of affairs that doesn't presently exist. Therefore, even if this weekend's fight ends in a draw, today's assertion should be seen to remain true, in so far as it is an accurate description of todays state of affairs.

We don't experience the same token of an "inner experience" many times. A token of an experience can be timestamped. You cannot have the same timestamped token of an experience twice. You clearly do not understand the type/token distinction if you think this. You can only have the same type of experience twice. — Luke

Doesn't your timestamp proposal amount to fitting experience to theory, rather than vice versa?

I used to have similar thoughts when contemplating McTaggart's "Unreality of Time". The idea being to regard the terms of the A -Series (i.e. Past, present, future) as being indexicals into a B series of "time-stamps", in an attempt to deflate the A series into the B series, in order to establish the unreality of change.

So for example, the potentially perplexing A-series observation "now is no longer now" becomes after substitution "08:53 is no longer 08:54" which is immediately seen to be nonsensical. Here a time-stamp is meant to refer to a sensation under the assumption that one's set of sensations are unique and ordered -which amounts to the assumption that the B series is phenemenologically real.

I've since come to realise that the middle Wittgenstein had a somewhat similar but arguably better conception expressed in terms of his cinema analogy. With it he identified the A-series of consciousness with an image shown on a cinema screen, and the B-series with the ordered-frames of the movie encoded in the photographic film that was projected onto the screen.

Unlike my proposal, Wittgenstein's proposal (to my understanding) is much weaker, in making no assumptions as to the phenomenological reality of the B series. If I understand him correctly, all that can be talked about, to use his analogy, are case-specific use-cases in which the image on the screen happens to be relatable to the frames on the film-reel. In other words, there aren't always criteria available by which to say that an experience is unique or different from another experience.

Q. "A cognition-invariant, involuntary resistance to ineluctable facts of the matter."

A. What are "percepts"?
— George Berkeley plays Jeopardy

A definition of realism is a contradiction in terms, for the ultimate purpose of any definition is to reduce theoretical nomenclature to observations and actions. Otherwise the definition is useless and unintelligible.

If a self-described realist claims to be making a metaphysical assertion founded upon reason, and if he accepts that the tribunal of reason are his experiences, then at the very least he is describable as being a methodological solipsist, even if he believes to have obtained conclusions that aren't reducible to personal experiential verification...


How should the claims of the realist be understood ? are realists really asserting metaphysical claims in spite of whatever they say, or are they at best making epistemological claims that every solipsist can agree with?

I expect that a realist might play down the importance of his personal experiences, saying that he accepts as a matter of pragmatism the judgements and reports he receives by trustworthy or authoritative third-parties, assuming the they cohere with his personal understanding of the world. But then the question remains; for isn't the realist still founding his claims upon[ i]his[/i] experiences, even if such experiences are "second-hand", so to speak?

If an exploration robot trained by machine learning, whose engineering was understood, started speaking about the it's discovery of a mind-independent reality, are there any conditions in which we would take it's claims seriously? I think definitely not. Under no circumstances would we interpret its stimulus-responses as being anything more than the result of it's internal state and present environmental interactions.

In what senses, if any, is the following thought empirically meaningful? : "my present experience is different from my previous experience"

Suppose that I (somehow) denote my "previous" sense-data with the label "R", and that I directly denote my present sense-data S with the label "S". Paradoxically, the condition of empirical meaningfulness entails that R, although conceived as being "previous" to the "present" experience S, is nevertheless contained within S, for otherwise "R" would not be an empirically referring term. In actuality, "R" is being used as an indexical referring what is presently being remembered as part of S.

In other words, R and S are analogous to the concept of two adjacent ordinal numbers, say R = { 0 } and S = { { 0 } , 0 } , while their labels "R" and "S" are analogous to the corresponding numerals, i.e. "R" = 1 and "S" = 2.

In mathematics it could be said that every finite ordinal is part of the same first limiting ordinal. Analogously it could be said that all experience, whether past , present or future, is of the same experience.

Therefore the reason one says "my present experience" isn't to assign a quality of "being present" to the experience, but because one is using a subset of his experience as a memory, and is using "present experience" as an indexical referring to the other part.

Modern science can find no such thing that answers to 'the sensation of being in pain'. That's the problem I'm attempting to address. — Isaac

Yes, as was Wittgenstein throughout his entire career.

I would first suggest reading Bertrand Russell's Analysis of Mind. Without reading this book, it isn't easy to fathom the ideas Wittgenstein was criticising and the problems he was attempting to solve, especially with respect to his remarks regarding 'private language'. Wittgenstein's 1930's transcript known as 'Philosophical Remarks' is also necessary reading to understand his intermediate thinking and to put his later remarks into the right context. Both of these works are freely available online.

Notably in PR, Wittgenstein gives what is to my understanding the first 'zombie argument' against behavioural understanding of 'Other Minds', predating David Chalmers zombie arguments by 65 years. But unlike Chalmer, he phrased the zombie problem in terms of the indistinguishability of sense when interacting with a person versus a zombie, as opposed to Chalmer who posed the problem in a more realist fashion in terms of metaphysical substances.

Ultimately, the late Wittgenstein was an anti-realist who didn't think of 'other minds' in the sense of other substances or in terms of property dualism. Rather he emphasised the role that ones own sensations play when one attributes so-called 'other' minds to other people.

Determinism and non-determinism aren't real properties of the universe, for as demonstrated with the Middle-Earth fictional universe, these terms have no descriptive value with respect to any complete data-set. Therefore it makes no sense to speak of the universe as a whole as being either determined or not.

Probability theory hints at what those terms describe; they describe semantic relationships between data-sets or theories. For example, the sequence {1, 2, 3} is "determined" in relation to the sequence {a,b,c} under the assignment a --> 1, b --> 2, c-->3. But it is undetermined with respect to the set of sequences beginning {1,2,...} which include it as a special case.

Consider historical counterfactuals; Must Hitler have invaded Poland? In spite of appearances, the meaning of this question isn't about the literal existence of a possibility available to the German government in the year 1939, rather it concerns the relation of a model of the actual event to a hypothetical set of "similar" circumstances, such as that defined by a historical simulator, where it is the notion of "similarity" that is actually the focus of the question.

In Automata Theory, non-determinism is the existence of two applicable state-transitions from a given state. This isn't an epistemic notion.

The author JR Tolkien determined what happened in the world of Middle-Earth, but he didn't specify what would have happened in the story if alternative courses of action were taken. So is the world of Middle-Earth deterministic , non-deterministic, or neither?

Is even the world of a choose-your-own adventure story, where alternative courses of action can be chosen by the reader, describable with either of these adjectives?

Are we saying the same thing only you maybe more technically correctly? Or something different?
All I'm saying is that if I throw that fair die, I shall get any of 1 through 6, and that I shall expect each to occur about 1/6th of the time. — tim wood

I was responding to the common belief that chance represents ignorance. We know from experience that the odds for a "fair die" landing on any side, if tossed in a typical fashion, are roughly 1/6 in the sense of relative frequencies. This is essentially the definition of what a "fair die" is.

But a priori, we don't even know that. For in the case of an unknown die that isn't necessarily fair, all we know a priori is that the odds for any outcome is between 0 and 1. Nevertheless, there persists a convention which assigns a uniform distribution in the case of an unknown dice. But this is misleading for it conflates knowledge of a fair die with ignorance of an unknown die, and also leads to inferential biases in the case of an unknown die that aren't warranted.

If all you know is that an event has n possible outcomes, there is nothing more than can be said, and chance cannot be quantified.

What does this mean? In words? — Neoconnerd

It is saying that the physical propensity for obtaining the respective possible outcome, is above zero and less than 1. Without additional information, or assumptions, a more precise set cannot be assigned to the outcome.

What is imprecise about assigning 1 - 6 as possible outcomes of the throw of a die? — tim wood

It is imprecise because probability intervals are assigned to outcomes, rather than numbers.

e.g. P ( dice throw = six) = (0,1)

Which only express the fact that throwing a six is possible, but not certain.

If I throw a dice the chance that I throw any of the six numbers is 1/6. The dice rolls determined towards its destined number.

Our lack of knowledge gives rise to chances. If we somehow could know the initial state of the dice and the exact interactions with its environment then the final numbeŕ could be known.

Well, that's the naive argument. In practice, it can't be known in priciple. Which doesn't mean that the process is not determined. It is.

You can say that chance is a subjective feature that we project to, for example, the world of dice. A dice has a 1/6 chance of showing one of the six (if the dice is ideal). — Zweistein

The subjective interpretation of probabilities as representing ignorance which you appear to be assuming, is a logical fallacy in my understanding. Lack of knowledge should give rise to possibilities only. Moreover, it is impossible for anyone to distinguish ignorance from objective uncertainty before the fact. Such distinctions can only be drawn after the fact.

To represent uncertainty in terms of probabilities isn't to assign a particular distribution, but to assign a set of distributions, which is sometimes referred to as assigning "imprecise probabilities". In the case of complete ignorance of a thrown die, one should assign the entire set of distributions with six possible outcomes, which amounts to saying that one only knows that there are six possible outcomes.

Furthermore, having described one's state of ignorance by using a set of distributions, one shouldn't then average over the set, as Bayesians often do, to obtain a precise "ignorance prior", for this fallacious practice amounts to an attempt to extract information from ignorance.

It makes sense to say " I am not my body ", even if it doesn't make sense to say "He is not his body". Only by conflating the subjects of these sentences does a contradiction arise.

Wittgenstein as a phenomenologist. Presumably not of the Heideggerian school? — Banno

A precise answer to that question can be found here.

Wittgenstein was not promulgating coherentism. But I have an interest in reconciling Davidson - Quine's intellectual son - and Wittgenstein, so I'm interested, if confused.

Wittgenstein's so-called "Anthropological Holism"
— sime
Start there. What is it? Who called it that? — Banno

Fodor et al. as described here

For Wittgenstein, even the concept of a machine implementing an algorithm is relative to human customs, rather than what machines themselves are doing per-se. He is a logic anti-realist. Unfortunately, his understanding of rules as being ontologically dependent upon the background context of human customs for their recognition is perpetually misinterpreted as referring to meaning and rules being epistemically dependent on social feedback, something which isn't helped by the terrible wikipedia article on the subject.

I don't know any other word with which to describe Wittgenstein other than a coherentist, given his abandonment of foundationalism and recognizably Quineian stance towards semantics, nothing the 1951 publication date of the Two Dogmas of Empiricism that was two years prior to the posthumous publication of PI in 1953. There have been some recent attempts to divide their points of view, but I personally find them unconvincing.

It might be considered cheating to a phenomenologist, but a useful short-cut to clarifying Wittgenstein's ideas is to read Quine's Word and Object, The Two Dogmas of Empiricism, and Truth by Convention. A scientific analysis of semantics helps to dispel the same myths that Wittgenstein dispelled through phenomenological investigation, including the myth that meaning is founded upon either convention or upon 'stimulus-synonymy' as envisaged by the logical postivists and the earlier Wittgenstein. It is generally believed among the academic community, that the later Wittgenstein's so-called "Anthropological Holism" isn't reducible to either social or personal convention or to 'Tractatarian names'. Yet people insist on jumping to conclusions through quote mining , and then attributing nonsensical views to the author.

When under a local anaesthetic for dental treatment, it is certainly possible to feel ambiguity or uncertainty as to whether or not one is in pain. And self-reinforcing beliefs and observer-effects come into play, where one asks "am I really in pain, or just imagining it?

Recall Wittgenstein's analysis of Moore's proposition later on in the book, namely "It is raining, but I believe it is not raining", which Moore had previously considered to be nonsensical when considered in the present tense. Wittgenstein, if i recall correctly. envisages the possibility of this sentence making sense in a situation in which one finds oneself consciously disbelieving that it is raining whilst observing oneself to be behaving otherwise. Our thoughts can belie our actions.

Both intelligence and stupidity refer to behaviour that is interpreted to be goal-driven. In the case of intelligence, the goal state is considered to be desirable, whereas for stupidity the goal state is considered to be undesirable, thereby making the distinction between the two concepts relative and subjective.

The question is really about the realism of counterfactuals as well as the question of backward causation, and the reality of the temporal order.

For instance, suppose that If Bob lights a fuse, then a bomb will explode. At first glance, this appears to imply a temporal order in which Bob's action as cause precedes the bomb exploding as effect. But this clause can also be restated in reverse by saying that if the bomb is observed not to explode, then Bob couldn't have lit the fuse.

Presumably, if a bomb is defused in a state of ignorance as to Bob's earlier actions, one can at least conclude that defusing the bomb didn't alter the earlier fact as to whether or not Bob previously lit the fuse. Or does it? For there isn't a way of testing the counterfactual as to what would have happened earlier in the past had the bomb been allowed to detonate. Furthermore, if the bomb is very big then any potential evidence as to Bob's earlier actions might get destroyed by it's detonation. According to a presentist interpretation of history, there aren't any facts about the past that transcend the state of the present. This logically implies that in a finite universe where history cannot be preserved indefinitely, if a sufficiently big bomb explodes, then it must explode for no reason.

The dead, however, present as being utterly devoid of consciousness. This goes beyond reflex action or stimulatory response...one can tell when the brain has died, and cellular metabolism has ceased. From this state, this "death", there appears in my experience to be no regaining of any level of consciousness whatsoever. — Michael Zwingli

Yes, that might be true from your perspective when it comes to you appraising the mental state of other people. But can you speak meaningfully and authoritatively about past cases of your own unconsciousness? what empirical criteria are you using in this case? Is it really possible for you to infer necessary conditions for the existence of your own mind via analogical reasoning from your understanding of the necessary conditions for other minds?

Rationalists have a tendency to overlook and equivocate the vastly different empirical criteria used for defining unconsciousness of the first-person, as opposed to when defining the unconsciousness of another human being.

In the latter case, a human being is publicly defined as 'being unconscious' according to behavioural criteria, such as bodily reflexes failing to respond to stimulation. But in the former case, when we speak of personal unconsciousness of ourselves, we aren't using a public behavioural definition and we only speak of personal unconsciousness in the past tense. Ironically a person makes this past-self judgement according to observations they presently make.

For example, a person wakes up in the morning from a deep-sleep and concludes that they didn't exist during the previous night , due to not remembering anything over that period. But how does the person determine that they don't remember anything about the previous night? They ask themselves what they remember and they observe that "nothing" comes to mind, but "nothing" here means that whatever is presently sensed or comes to mind is considered to be irrelevant to the question. Yet this purported 'evidence' for past-personal unconsciousness in effect constitutes present empirical criteria for defining what is meant by past-personal unconsciousness, which might suggests to an empiricist that the concepts of past-personal-unconsciousness and present personal consciousness aren't ontological opposites.

This article provides relevant context regarding the history and evolution of Wittgenstein's later thought. The SEP's article on private language is also recommended reading.

In my opinion, it only makes sense to discuss Wittgenstein's remarks when they situated in the context of the traditions of both analytic philosophy and phenomenology.

In my opinion, yes, on my understanding that it is linguistic convention which ultimately decides whether a sentence is true or false (for on any physical understanding of verbal behaviour, every assertion can be understood to be a true representation of it's physical causes, and therefore true).

Every person has their own linguistic convention and associated agenda, and conventions come into either conflict or cooperation for political reasons. Discussions and debates which on the surface look like passive disputes over the objective nature of shared truth, are ultimately analysable in terms of the resolution of socio-political objectives. But i don't see this as a nihilistic conclusion and more like an alternative view of philosophy.

Recall that the later Wittgenstein was negatively reappraising the phenomenalist doctrine of logical atomism that he, Russell and Mach previously assumed, where names are interpreted as stand-ins for sense data as suggested by the picture theory of meaning. But this solipsistic doctrine implies a fixed and idiosyncratic relationship between word meanings and the perceptions or mental states of a solitary speaker of the language, which in turn implies that the public meaning of such names in a community of speakers is synonymous to the use of indexicals like this, him, here, it, with names only communicating raw presence among speakers without a shareable accompanying description of what is perceived. Such a conception of names essentially ignores context, and in particular the (overwhelmingly complicated) inferential semantics that constitutes the descriptive function of names, both in the case of public communication and in the case of private introspection. As Wittgenstein illustrates, the actual usage of words , both in public and in private, ultimately defies any purported private or public definitions, except in the most trivial and useless cases. As with scientific and mathematics terminology, the definition of a natural language is forever playing catch-up after the facts.

The actual referents of a belief are it's immediate physical causes; so whenever a speaker asserts a so-called "false" belief, any alleged epistemic error exists solely in the minds of the listeners, due to their misidentification of the causes of the speaker's assertion.

How does a person experience dementia? presumably he finds himself learning about things that he has good reason to believe he has previously forgotten. But how does he classify an experience as being of something forgotten? Whatever the experiential criteria, perhaps an avid learner should consider dementia to be the ultimate learning experience.

I recall an interview in which Sam Harris (atheist, author, neuroscientist) claims that to believe nonexistence (I mean death) is unthinkable is, as he put it, "...for a lack of trying..." He explains: there are people in Paris, his choice of city, who don't know you exist; in other words, you don't exist as far as Parisians are concerned. That, according to Sam Harrris, is to give you a glimpse of what nonexistence is! — TheMadFool

There are people in Paris observing the Eiffel tower, who are not observing the computer monitor you are observing; in other words, your computer monitor doesn't exist as far as Parisians are concerned.

And so presumably according to Sam Harris, he has given you a glimpse as to what the non-existence of your monitor is.

What I mean is that one can look up the entry for the axiom of choice on nLab, without encountering a rant against ZFC. So I think you're the one adding that part, and not your fellow constructivists / category theorists / programmers or whatever direction you're coming from.

Indeed, nLab expresses choice as "every surjection splits," which they note means "every surjection has a right inverse," in set theory. This formulation is easily shown to be equivalent to the traditional statement of the axiom of choice. There is no distance between the category-theoretic and set-theoretic views of choice. — fishfry

Category theory is a useful meta-language for understanding precisely where the mathematical foundation proposed in the early 20th century goes wrong, as well as being helpful for relating alternative theories. CT is itself philosophically neutral in the sense that it only assumes the presence of identity arrows in a category, but places no other constraints on either the presence or absence of arrows, provided the laws of arrow composition and association are obeyed. Therefore disputes between intuitionists, formalists and platonists carry over into the language.

Like a mathematics department, nlab as an encyclopedia is obviously going to disseminate mathematics in a politically neutral fashion. Or perhaps i should have said, unlike a mathematics department. But political neutrality doesn't amount to reasonableness regarding which mathematics should be prioritised.

You lost me there. Absent AC there is a set that is infinite (not bijective with any natural number) yet Dedekind-finite (no proper subset is bijective with the entire set). I don't know what you mean here. — fishfry

I'm talking about stream objects in computer science, or equivalently the isomorphism
S <----> 1 x S in the Set category, where S is a stream object, 1 is the terminal object, i.e the singular set, and x is the Cartesian product. Unfolding the definition:

S <----> 1 x S <----> 1 x (1 x S) <----> 1 x (1 x (1 x S)) ....

Since each of the arrows is invertible, S clearly has, by definition, a surjection onto any finite set, which is what i meant by Kuratowski infiniteness.

On the other hand, recall that in category theory every element of a set S is an arrow of the form
1 --> S. However, these arrows haven't been specified in my above definition of S, and therefore the number of elements of S is currently zero, i.e. S is the empty set, which is another way of saying that S is a completely undefined set until the first observation is made.

Every time an observation is made, an arrow of the form 1--> S is introduced into the above category, and we can denote the current state of the stream by shifting from left to right in the above diagram. But at every moment of time, the arrow S --> S that is implicit in the the product S ---> 1 x S is a bijection, meaning that is S is forever dedekind-finite.

S cannot exist as an internal set of ZF because it isn't a well-founded set, although it can exist in the sense of an "external set" that is to say, inside ZF as part of a non-standard interpretation of an internal well-founded set. And it it cannot exist in any capacity inside ZFC.

All of which is tantamount to saying that ZF has only partial relevance to modern mathematics in terms of being an axiomatization of well-foundedness, whilst ZFC is completely and utterly useless, failing to axiomatize the most rudimentary notions of finite sets as used in the modern world.

I'm perfectly happy to stipulate so for purposes of discussion. After all, there are no infinite sets in physics, at least at the present time. So, what of it? The knight doesn't "really" move that way. Everybody knows that knights rescue damsels in distress, a decidedly sexist notion in our modern viewpoint. Therefore chess is misleading and unrepresentative nonsense. Nevertheless, millions of people enjoy playing the game. And millions more enjoy NOT playing the game. What I don't understand is standing on a soapbox railing against the game. If math is nonsense, do something else. Nobody's forcing you to do math, unless you're in school. And then your complaints are not really about math itself, but rather about math pedagogy. And I agree with you on that. When I'm in charge, a lot of state math curriculum boards are going straight to Gitmo. — fishfry

So are you agreeing that mathematical infinity has neither philosophical nor scientific relevance and that everyone knows this, or am i right to stand on a soap box and point out the idiocies and misunderstandings that ZFC seems to encourage?

I might point out, though, that assuming the negation of the axiom of choice has consequences every bit as counterintuitive as assuming choice. Without choice you have a vector space that has no basis. An infinite set that changes cardinality if you remove a single element. An infinite set that's Dedekind-finite. You lose the Hahn-Banach theorem, of vital interest in functional analysis, which is the mathematical framework for quantum mechanics. The axiom of choice is even involved in political science via the Arrow impossibility theorem. — fishfry

Obviously, a denial of AC doesn't amount to an assertion of ~AC, given that things are generally undecidable, but i see no counter-intuitive examples in what you present. In fact, many examples you raise should be constructively intuitive if we recall that construction can proceed either bottom-up from the assumptions of elements into equivalence classes, or vice versa, so an inability to locate a basis in a vector space using top-down construction seems reasonable.

As for the sciences, AC is meaningless and inapplicable when it comes to the propositional content. At best, AC serves a crude notation for referring to undefined sets of unbounded size, but ZFC is a terribly crude means of doing this, because it only recognises completely defined sets and completely undefined sets without any shade of grey in the middle as is required to represent potential infinity.

QM has also been reinterpreted in toposes and monoidal categories in which all non-constructive physics propositions have been removed, which demonstrates that non-constructive analysis is dying and going to be rapidly replaced by constructive analysis, to the consternation of inappropriately trained mathematicians who resent not knowing constructive analysis.

Besides, if you have a nation made up of states, can't you always choose a legislature? A legislature is a representative from each state. If there were infinitely many states, couldn't each state still choose a representative? The US Senate is formed by two applications of the axiom of choice. The House of Representatives is a choice set on the 435 Congressional districts. The axiom of choice is perfectly true intuitively. If you deny the axiom of choice, you are asserting that there's a political entity subdivided into states such that it's impossible to form a legislature. How would you justify that? It's patently false. If nothing else, each state could choose a representative by lot. — fishfry

Obviously, the axiom of choice isn't used in the finite case. In the infinite case, the sets of states needs to be declared as being Kuratowski infinite in order to say that the elements of the set are never completely defined, and so a forteriori the size of the set cannot be defined in terms of it's finite subsets.

Secondly, the set should be declared as Dedekind finite, in order to say that the set is an observable collection of elements and not a function (because only functions can be dedekind-infinite).

So, yes, you can choose as many representatives as you wish without implying a nonsensical completed collection of legislatures that are a proper subset of themselves, but formalisation of these sets isn't possible in ZFC, because AC and it's weaker cousin, the axiom of countable choice, forces equivalence of Kuratowski finiteness and Dedekind finiteness.

It upsets some people (Frega, Meta) that mathematical axioms don't necessarily "mean" anything or "refer" to anything. — fishfry

And it should do, for classical set theory and real analysis are misleading and unrepresentative nonsense, unless cut down to the computationally meaningful content. Students who are taught those subjects aren't normally given the proviso that every result appealing to the axiom of choice is nonsensical, question-begging and of use only to pure mathematicians and historians.

What seems clear to me is that if we do live out other lives, in order to do this, there has to be a source that maintains the continuity of the self, otherwise it's difficult to make sense of this idea. — Sam26

Why not just deny the possibility of eternal oblivion by denying the existence of a continuous self, even within a single lifetime? That way you circumvent the need for evidence of reincarnation, and avoid all of the scepticism that the begging of evidence entails.

In my understanding, he was pointing out that our inferences can neither be understood nor be given justification on a definitional basis on pain of infinite regress, a point first raised by Lewis Carroll , whom Wittgenstein was evidently inspired by and indebted to.

The impossibility of infinite analysis, coupled with the problem of theory under-determination, means that the inter-translation of rules and phenomena and even rules and rules is under-determined in both directions. Hence Tractatarian approaches to philosophy are profoundly misguided.