What about art semi-automatically generated by a neural network that in effect produces novel images with high artistic potential by interpolating the patterns that exist within large databases of artistic, natural and cultural images? Or that transfers the statistical qualities of an artist's style onto an arbitrary image to produce a novel 'painting' in that artist's style?

Who is the artist here, and who owns the results?

In computer programs infinity is used as a free-variable that might later become bound to a random finite number. So it is a logical concept rather than a concrete number.

Can you show that idea with practical example? — Zelebg

Dark Energy hypotheses in physics are currently the most fashionable example as to why falsification isn't used in practice. Rather than considering the Hubble data of the speed of receding galaxies as refuting General Relativity, Physicists instead 'fix' GR as being true by proposing new and (individually) untestable auxiliary hypotheses so that GR still 'works in combination. In fact, to my understanding Dark energy isn't even at the stage of being a well-defined 'hypothesis'.

Auxiliary information also includes the trivial and taken-for-granted assumptions that your instrumentation is in 'full working order', that the laws of physics haven't changed since you began the experiment, that you aren't hallucinating, etc. etc. In short, no hypothesis is ever tested in isolation, and the auxiliary assumptions upon which the credibility of experiments rests aren't even exhaustively stateable, let alone formally stateable. Hence the reason why falsification isn't a good model of science or epistemic judgements in general. It's rooted in the archaic notion of logical Atomism - the idea that language has legible denotational semantics where the truth of a proposition stands or falls in isolation of the truth of every other proposition. But this is only true in toy-world scenarios described in an artificial language.

It's a complex, ill-posed and frankly outdated assertion. Firstly, an observation O can only materially entail the contradiction of a hypothesis H in a closed finite world. For in an open-world, the meaning of the material implication O => ~H isn't empirically reducible to observations, and is instead an auxiliary hypothesis, A, which isn't itself entailed by some other observation on pain of infinite regress. So in an open world we have A => ( O => ~H) , and hence O => (~A OR ~H)

- We know that we cannot enumerate all halting Turing machines, so for every supposedly complete list of halting Turing machines I can find another halting Turing machine that is not in the list.
Does that mean that the set of halting Turing machines in uncountable? No! It only means that there is no way to enumerate that list!. — Mephist

Yet saying that there is no way to enumerate the total computable functions is somewhat ambiguous, for as previously mentioned we can use brute force to simulate every TM on every input and enumerate on-the-fly the algorithms that have so far halted on all their inputs. Furthermore, the classical logician with realist intuitions will go further to argue that there is a definite matter-of-fact as to the set of computable functions and will therefore believe in the independent existence of a 'finished' enumeration, interpreting the limitations of finite constructive arguments to produce such an enumeration as being epistemic limitations rather than metaphysical limitations. Instead they will simply appeal to the Axiom of Choice to claim the independent existence of completed enumerations of the computable total functions.

As we said earlier, this full enumeration cannot be used if the diagonal function d(x) is to be both computable and total; otherwise if the realist diagonalizes the hypothetical full enumeration of computable total functions, then d(x) cannot be computable, for the enumeration ensures that d(x) is total. That d isn't computable is obvious, since it involves running nearly every Turing Machine for an infinite amount of time and then diagonalizing, meaning that it's godel number g is infinitely long and that it assumes an infinite value when evaluated at d(g)=d(g)+1

We know that Cantor's Theorem concerning the cardinality of the power-set of integers isn't a constructive proof, for we cannot enumerate and diagonalise only the Turing machines representing the recursively enumerable sets due to the Halting Problem, so we must enumerate the larger set of TMs.

And yet, the entire set of countable TMs can be diagonalised to prove that the set of countable TMs are "uncountable", by dynamically enumerating the halting TMs so as to ensure the termination of the diagonal TM for each of it's inputs; but in fact all that my proof of "uncountability" amounts to with respect to Turing Machines, is the construction of an enumeration of Turing Machines in such a way that the diagonal Turing machine cannot be part of the enumeration. This is analogous to enumerating the odd numbers and then diagonalising them to construct an even number.

To spell out the difference, in the case of Cantor's Theorem the constructed enumeration of the sets of natural numbers is considered to be prior to the construction of the diagonal set, but in the case of my method, the enumeration of TMs was constructed via the construction of the diagonal function. In other words, selectively constructing a non-exhaustive infinite enumeration via a diagonal procedure isn't a proof that a bijection with the natural numbers doesn't exist under a different enumeration. And in the case of Turing Machines we know that such a bijection does exist.

But this raises doubts about Cantor's original diagonal argument, for I might have been lucky enough with my original enumeration of TMs to produce a diagonal function without requiring any shuffling of the enumeration. Therefore Cantor's original argument isn't proof enough that the power-set of N is literally larger than N.

A proof as to why the real numbers are absolutely countable, in spite of all pretences to the contrary:

1. Enumerate the undecidable set of total functions within the entire set of enumerable Turing Machines of one argument {f1(x),f2(x),f3(x),..}, by running every Turing Machine in parallel on each input x=1,x=2,..., and shuffling their enumeration over time as necessary, so as to ensure that fn is defined when run on input x=n.

2) Define the Turing-Computable total function g(n) =fn(n)+1.

Congratulations, you've "proved" that the countable set of Turing machines is "larger" than the countable set of Turing machines.

Firstly there is the problem of confabulation, especially when subjects are pressed for more detail; are people accurately reporting their earlier experience, or are they improvising novel content over a vague memory? With memories, it's never clear which parts are authentic.

Also, what are the necessary descriptive features of an apparition? Or is an apparition more of a hypothesis about experience?

My dozen or so successful experiments with day-time and wake-induced sleep-paralysis were personally interesting, but i have no idea how to report most of them, and they were bizarre and my memory is vague. None of them involved aliens or apparitions. My favourite experience wasn't even visual, but more of a blind ecstatic trance accompanied with a feeling of moving at lightening speed and a wooshing sound like a jet engine. Ironically, the dullest experience was a high-definition visual OBE that lasted for about a minute, that felt so real and ordinary that I briefly wondered whether I was actually awake and physically in my kitchen, but then realised I must have been asleep because the kitchen clock told the wrong time, plus there was this dizzy sensation from feeling pulled around on that mythical elastic cord that I never actually saw...

An important question is the relationship of the Axiom of Choice (AC) to the Law of Excluded Middle (LEM), for Classical Logic is normally distinguished from Intuitionistic Logic on the basis of the latter axiom rather than the former axiom. Furthermore, intuitionists often claim that AC is constructively acceptable by interpreting AC to refer to the very construction of a function, for intuitionists do not accept the existence of non-constructive functions. This is very confusing, because AC's natural role is to refer to an unspecified function for which we do not possess a constructive description. This situation arises all the time in computing when a program points to an externally provided input data-stream that the programmer cannot further describe.

Now according to the SEP's article on the Axiom of Choice , AC implies LEM in the presence of two further axioms, namely Predicative Comprehension (PC) and Extensionality of Functions (EF). The former says that the image of every predicate applied to individuals is a set, whereas the latter says that every extensionally equivalent pair of sets has the same image under every Set Function.

https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog

The author of the article proves that PC & EF => ( AC => LEM), but he then argues that whereas PC is constructively valid, EF isn't. See the argument, but I don't find his argument about EF persuasive, at least as I understand it.

In my current view, EF is also constructively admissable, implying that the precise difference between classical logic and intuitionistic logic is AC as much as it is LEM, which then cements the view that classical logic describes a game between two players, whereas intuitionisitic logic describes solitaire.

A further motive for my view (and indeed the most natural motive), is that classical logic involves sequents of the form (a AND b) => x OR y, where it isn't known which of x or y is true, in which the negation of one implies the existence of the other. On the other hand intuitionistic logic only involves sequents with a single conclusion, of the form (a AND b) => x. Thus there is indeterminism in the case of classical reasoning, but not in the case of intuitionistic reasoning.

I interpret the Axiom of choice to be a 'prayer to nature' to send me the desired object already made. The axiom refers to my opponents choices that are not modelled in the formalism i am using.

If I have no internal strategy for constructing a basis from my formalism of a vector space, then I am reliant upon nature sending me a basis, which I have no control over. But suppose nature never sends me a basis?

Arguments between constructivists and classical logicians are caused by a fundamental disagreement about the nature of proof. The former equates proofs with fully-determined algorithms under the control of the mathematician, whereas the latter allows proofs to interact with nature in an empirically-contigent and indeterminate fashion.

Unfortunately, classical logicians are usually in denial about what they are actually doing. Instead of admitting that their notion of proof is empirically contigent and not internal, they insist their notion of proof is internally constructive in a transcendental platonic realm.

A non-computable real number r refers to a truly random infinite process, and yet the distinction between a truly random infinite process and a pseudo-random infinite process isn't finitely testable, since any finite prefix of r is computable. Since r cannot be finished, at any given time r can be equally interpreted as referring to an under-determined pseudo-random process. Yet any process we specify ourselves is fully determined. Therefore r can only be interpreted as referring to a process of nature that we are observing but that we ourselves haven't specified and have only incomplete knowledge of and control over.

Therefore when a physicist makes the observation x = 0.14 +/- 0.0001, he could be equally described as stating an interval of rational numbers or as stating an interval of real-numbers. If this sounds wrong, "because the real numbers are uncountable, whereas the rational numbers are countable", recall Skolem's Paradox that the set of real-numbers actually possesses a model in which they are countable. The only important thing to know is whether the physicist fixed the result or whether he measured the result, for constructing a certain number is different to measuring an uncertain number - this difference isn't easy to express in either classical or constructive mathematics.

Unless 'Living within a simulation' is defined in such a way as to be falsifiable, the hypothesis is meaningless. And yet it is imaginable that the falsification of any particular simulation hypothesis can be simulated under an alternative simulation hypothesis. Therefore it doesn't make sense to ask the question in a general absolute sense.

As a matter of fact, Wittgenstein plays an important part in my epistemology. Consider where I talk about the many uses of the word know, which is taken from the PI and especially OC. — Sam26

But what about ontics? If the meaning of a proper noun, say "Elvis Presley", is considered to be a family-resemblance of uses across partially over-lapping language-games, then the question as to whether Elvis Presley is dead or alive doesn't have a single definite answer. In Karoake, as far as i'm concerned, if it looks like Elvis and sounds like Elvis, then it's Elvis.

Our understanding of the "soul" of a person has much in common with our attribution of sense to their name, which is retained after the expiration of a referent bearing that name. If an anti-realist concerning other-minds ontologically prioritises sense over reference, he will answer questions concerning immortality very differently to the realist, for these opposing view-points use different and incompatible criteria for personal identity.

Realists also vehemently disagree with one another. One argues that "Elvis" is token-identical with the current state of a corpse buried at Graceland and that evidence for Elvis's reincarnation on earth or in heaven is therefore logically impossible. The other says "Elvis" is type-identical with a class of potential physical objects, and that reincarnation is therefore physically possible but unlikely. The presentist argues "Elvis" is meaningless because nothing bears the name in his present vicinity etc.

When you argue for the possibility of evidence of consciousness surviving the body, what is your understanding of a proper-name?

A number is a definite unit of measurement, a definite quantity. The so-called "irrational number" is deficient in the criteria of definiteness. The idea of an indefinite number is incoherent, irrational and contradictory. How would an indefinite number work, it might be 2, 3; 4, or something around there? — Metaphysician Undercover

In the example of sqrt(2), Alice creates a computer program f(n) for calculating sqrt(2) to n decimal places and sends f(n) to Bob. Bob receives f(n), then he mysteriously decides upon a value for n, then runs f(n) and sends the result to Alice.

When irrationals are identified as being rational-number generating algorithms, irrationals are obviously well-defined. But of course we are rarely interested in algorithms per-se, and we also like to use irrationals as numerals, when for instance we use them to label the length of the diagonal of a square.

Now obviously, our visual impression of a square is vague to a certain extent, as is any physical measurement of a square, and if we repeatedly measured the diagonal of an actual square with very high precision we would obtain a range of (rational valued) measurements. In other words, by labelling the diagonal of a square with, say, sqrt(2), we are using Sqrt(2) not as denoting an algorithm, but as denoting an arbitrary rational or distribution of rationals, that is yet to be determined through a process of (repeated) measurement.

This is another example of why game-theory is a good model of mathematics use.

IMO, the heart of the problem is that the notations of both classical and constructive logic do not explicitly demark the analytic or a priori uses of logic pertaining to activities of computable construction in which every sign is used to refer to a definite entity or process, from logic's a posteriori or empirically-contigent uses in which some or all of it's signs are not used to denote anything specific a priori but whose meaning is empirically contigent upon nature providing some (possibly non-existent) outcome at a future time.

This is why I consider communication games to be the most important interpretation of logic. For it identifies the constructive content of logic with the permissible sequences of actions that can be taken by player A, and the 'non-constructive' content of logic with message-replies that A receives from a player B as a result of A messaging B. The existence of a message-reply from B is not-guaranteed a priori, and A's message to B is only said to be meaningful as and when A receives a reply from B.

A constructive real number refers to an algorithm constructed by A for generating a sequence of integers. In the case of a non-constructive real number, A invokes the "Axiom of Choice" , which is interpreted as A 'outsourcing' the creation of the real number, by sending B a message requesting B to send A an arbitrary sequence of integers. The sign for a non-constructive real number has no specific meaning or referent until A receives a stream of integers from B.

Pure and unapplied logic makes no empirical claims and only expresses linguistic rules for rewriting a term. All logically valid sentences are reducible via rules of term-rewriting to "X=X", where the equality sign designates the term on the left hand side to be an abbreviation of the term on the right-hand side.

If in the course of reducing a logical expression one obtained the opposite, all this would mean is that one's use of terminology isn't consistent. The solution is either a wholesale adjustment of the axioms that defines one's terminology, or to forbid on a case-by-base basis any derivation that leads to contradiction. For example, if Peano arithmetic was discovered to be accidentally inconsistent, a possible solution is to retain it's rewriting axioms for arithmetic, but to forbid any derivation beyond a certain size.

Philosophers have an unfortunate tendency to mistake ordinary uses of equality as denoting a physical relation between things rather than as being a linguistic relation between terms. For example, if the word "Now" is considered to refer to presently moving objects we arrive at the Hegelian contradiction "Now isn't now". But all this means is that our definition of 'now' is inconsistent. The contradiction is removed by replacing each and every use of "now" with a unique and new term, such that we are never tempted into equivocating one "now" with another.

In practice, a "non-computable" process, that is to say a truly random process, is indistinguishable from an unknown pseudo-random process, since we can only ever observe a finite number of observations generated by an unknown process. For the unknown process to be declared as 'truly random' it can never terminate and it must pass an infinite number of tests that refute each and every conceivable Turing computable algorithm that could have generated it.


The practical impossibility of distinguishing an unknown pseudo-random process from a really random process provides us with ammunition for rejecting an absolute ontological distinction between randomness and lawfulness. In which case, the question as to whether nature is computable or not is meaningless.

Instead we need only invoke a game-theoretic distinction between a controlled process referring to a process we create and control ourselves using an algorithm, versus an uncontrolled process that is part of nature that we only query and make speculative computer models of.

The stance that 'knowledge consists of instructions and the ability to follow them' is the epistemological philosophy known as constructivism. In my view this is misguided, because

i) Most of our knowledge and inferences consist of analogies and analogy-making, rather than consisting of recipes and the ability to follow them. Indeed, the Church-Turing thesis is purely the expression of an analogy between mathematical formalism and practical rule-following by humans.

ii) Constructivism cannot be self-justifying without pain of infinite regress; so-called 'constructive' reasoning actually consists of implicit analogical inferences expressed as axioms that lack further constructive justification or explication.

iii) In practice most of our so-called 'constructive' inferences are outsourced to external oracles we call 'calculating devices'. But unless one holds strongly realist beliefs in causality and identifies the logical description of a machine as an expression of a physical hypothesis, the output of a calculator cannot be said to be 'constructed' from it's inputs. Indeed, a central function of logic is to be able to describe the world in an agnostic fashion without committing to speculative physical theories.

Classical Logic together with Model-Theory and the Axiom of Choice accommodates our analogical leaps of faith known as "truth by correspondence" that stem from our non-deterministic interactions with nature better than intuitionistic logic, since the latter is purely the expression of games of solitaire played according to known rules.

Do you understand Turing's answer to the Halting problem? Just as Cantor's diagonal argument shows that not every infinite set of numbers can be put into 1-to-1 correspondence with the Natural numbers, so do the various undecidability results, starting from Church-Turing thesis, show that indeed there are mathematical objects that cannot computed. Not everything can be calculated/computed by a Turing Machine. — ssu

Recall that the proofs of Godel and Cantor correspond to Turing-computable algorithms. Are you meaning to suggest that a deterministic machine can follow a set of rules to prove that there exists a theorem that cannot be deterministically derived by those rules? Or that proves in a finite number of steps the existence of a literally uncountable universe? That's what the common (mis)interpretation of Godel's and Cantor's results amounts to.

It is certainly possible that nature is really random in the sense of falsifying any proposed theory-of-everything. But this is to speculate about nature, rather than to deduce a logical conclusion.

We ought to treat the existence of non-computability and incommeasurability much more seriously than we do. Yet mathematicians push them aside and think somehow that they are 'negative' or something that ought to be avoided.

I personally think that absolutely everything is mathematical or can be described mathematically. Huge part is just non-computable. When we would understand just what is non-computable, we would avoid banging our heads into the wall with assuming that everything would be computable. — ssu

Computability isn't a mathematical assumption, rather computability refers to the very activity of construction by following rules. Since mathematical logic consists only of the constructive activity of rule-following, the idea that mathematical logic can capture the non-constructive notion of "non-computability" is a contradiction in terms. None of Cantor's conclusions are really captured in his (ironically constructive) syntactical expressions. He is merely projecting his theological intuitions onto logical syntax.

Science on the other hand, attempts to predict the course of nature using a particular set of rules. Yet there is no reason to believe that the course of nature follows any particular set of rules. Hence we could say that nature might be "non-computable", but this "non-computability" cannot become the object of mathematical study on pain of contradiction.

A topologically structured set that only possesses the notion of "neighbouring" elements does not in itself possess 'holes', regardless of the finiteness of the set.

The notion of a continuum is relative to the notion of a 'hole', which is describable in terms of the absence of a topology-preserving surjection between two topologically structured sets.


"Gaps" only exist within the line of countably computable reals for platonists who think that non-computable numbers exist. Even if they are granted the existence of an uncountable number of non-computable reals to occupy those "gaps", the resulting model of the reals cannot control their cardinality, suggesting to the Platonist further holes, forcing him into constructing hyperreals and so on, without ever being able to fill the gaps in the resulting continuum.

This is another good reason for rejecting the non-constructive parts of mathematical logic.

Like the morning and evening star, water and H2O, temperature and molecular motion, Samuel Clemens and Mark Twain, the empty set and 0, or the charge of every electron in the universe.

Or that damned ship that had all its parts replaced during its voyage. — Marchesk

Well obviously in each case, the terms on the left and right side do not possess the same sense, so are not practically substitutable, so are only substitutable in a theoretical sense that makes potentially falsifiable counterfactual claims. Now standard dogma alleges that Superman is only de dicto different from Clarke Kent, but that de re they are one and the same person.

" In the context of thought, the distinction helps us explain how people can hold seemingly self-contradictory beliefs.[4] Say Lois Lane believes Clark Kent is weaker than Superman. Since Clark Kent is Superman, taken de re, Lois's belief is untenable; the names 'Clark Kent' and 'Superman' pick out an individual in the world, and a person (or super-person) cannot be stronger than himself. Understood de dicto, however, this may be a perfectly reasonable belief, since Lois is not aware that Clark and Superman are one and the same. " - https://en.wikipedia.org/wiki/De_dicto_and_de_re

Yet in "Superman 3" Superman was poisoned and divided himself into Evil Superman and Clarke Kent who then fought a physical fight. I submit this as evidence that Superman is not Clarke Kent de re .

Your intuition is constructively valid, since constructivists identify the real reals with the computable binary sequences, that is to say, the binary sequences that are computable total functions, and since they reject the premise of Cantor's theorem that the power-set of the natural numbers exists (due to most of it's elements being non-computable), they instead regard diagonal arguments as merely showing that a hypothetical enumeration of only the genuine real numbers cannot be formulated.

Unfortunately in Cantors day there wasn't much attention paid to the underlying algorithms used to generate binary sequences, and hence he failed to admit the critical distinctions between

-The Total functions corresponding to the countable set of reals that can actually be constructed by an algorithm that halts to produce a digit for every argument representing a position of the sequence.

-The Provably Total functions corresponding to the subset of constructive reals that are 'a priori' deducible, in the sense that their underlying algorithms can be proved to halt without requiring the algorithms in question to be actually run.

-The Non-Provably Total functions corresponding to the subset of 'a posteriori' deducible constructable reals , whose algorithms in fact halt when run, but whose halting cannot be proved without running the algorithms in question.

-The Partial functions that fail to produce digits for every position of the sequence, and hence fail to represent a legitimate number, a property which is generally unknowable.

Cantor's diagonal argument when applied to the countable set of provably total functions, constructs a 'diagonal' total function, i.e. an additional valid real number, that is no longer provably total, and we also know that the combined set of provably total and non-provably total functions is countable (via enumerating their respective Turing machines).
Yet we cannot apply a diagonal argument to the set of non-provably total functions to produce an additional real number, for their very non-provability forbids us from knowing apriori whether or not our list contains only total functions representing genuine real numbers. Therefore there is no constructively acceptable diagonal method for producing a new real number from an enumeration of all and only the real numbers.

Identicality isn't a description of appearances, it is an adopted convention that grants the inter-substitution of two or more distinguishable things in every situation. As an adopted convention, it doesn't make sense to ask whether two things really are identical.

To me the past is a deducible concept without referencing external realities — Devans99

This expression sounds very anti-realist to my ears. Namely that the past is deducible i.e. in some sense a living construction out of present sense-data and current activities, as opposed to being an inductively inferred and immutable hidden reality that cannot be observed.

The conceptual distinction between the inductive inference of causes from effects versus the deductive 'construction' of effects given causes seems to lie at the heart of disagreements between the realist and anti-realist. The latter wants to bring these two concepts much closer together by interpreting causal induction constructively as a generalised form of deduction.

- I have thoughts, these thoughts from a causal chain. The present exists, there are thoughts that I am no longer having, so the past exists. There are thoughts that I will be having so the future exists. I can label each thought with an integer. Assuming a past eternity, then the number of thoughts would be equal to the highest number. But there is no highest number, so a past eternity is impossible? — Devans99

Yes, the realist thinks of "past eternity" metaphorically in geometric terms, as an infinitely long line beginning at, say, zero and ending at positive infinity at the "the present", which obviously cannot be constructed. The anti-realist can reverse this metaphor by labelling the present with zero, and considering the past to be 'created on the fly' as and when evidence of the past becomes available.

And of course, you presumably mean that you have present thoughts which you interpret as being 'past-indicating' and present thoughts you interpret as being 'future-indicating'. Although recall that an appearance per-se does not refer to either the past or future, as exemplified by a randomly generated image that by coincidence looks historical. And recall that we can doubt the veracity of our memories if we judge our present circumstances to contradict them. So we cannot make an immediate identification of appearances, memories, thoughts or numbers with points on a physical-history timeline.

A believe in infinite past time is therefore akin to a belief it is possible to count 'all the numbers'. — Devans99

That only holds if one is a realist about the past who believes that the object of history is an unknowable reality in being unobservable and transcending present and future information.

In contrast, according to anti-realism the very truth of a past-contingent proposition reduces to present information. if the present state of information is ambiguous with respect to two historical possibilities, then according to anti-realism there is no matter of fact as to which historical possibility is true. Moreover, since the anti-realist never wants to claim 'perfect' knowledge of the past, he must insist that the past is infinitely extensible in a literal sense as and when new information becomes available.

For example, suppose Archaeologist A at time T unearths evidence E implying the existence of a fact F at time T', where T' < T. In stark contrast to the realist, the anti-realist considers A and E to constitute part of the very meaning of F, such that the truth of F is a function of T.

There is a great deal of empirical evidence that the speed of light is a universal constant obeyed by everything in the universe; we have been measuring it for 100s of years and we currently know it within a measurement uncertainty of 4 parts per billion.

Saying the statement "nothing travels faster than light" is about the language of physics seems to me to be equivalent to saying the statement is a natural law — Devans99

Sort of, yes.

Consider the statement "All objects have a temperature at or above zero Kelvin". Interpreted from a realist's perspective, the sentence is synthetic and makes potentially falsifiable empirical claims, and in this sense is considered to be a "natural law".

But from the anti-realist's perspective, the statement is analytic and merely states that negative numbers play no role in our physical concept of temperature states; From this perspective, the statement is part of the convention of our physics language and in being fixed by convention isn't considered to be 'up for grabs'.

Yet as Quine pointed out, conventions often undergo dramatic revision every so often in order for a language to improve it's expression of new observations. But as Carnap pointed out, given any state of evidence, there is freedom as to the physics convention used. So there will invariably be disagreement as to whether any statement taken in isolation is false, true or meaningless. So in this sense there is indeed 'equivalence'.

The language of physics is our model of natural laws after all - so I maintain a belief that the natural laws of the universe are time-aware. .This suggests time is more than just a human invention. — Devans99

Unfortunately there isn't a single viable language of physics. Holistically, all viable languages account for the same observed phenomena, but each language suggests different prescriptions as to what are the most informative future experiments to conduct.

Time is a human concept of convenience
— sandman

Yet there is a universal speed limit - the speed of light - and speed = distance / time, so it appears that something / some mechanism within the universe must be 'time-aware' else the speed limit could not be enforced - so time seems not just a human concept - it seems to be part of nature. — Devans99

An anti-realist with respect to time, might say that "nothing can travel faster than the speed of light" is a statement about the grammar of special relativity, rather than being a factual statement about the world.

The reason why special relativity 'concludes' that nothing can travel faster than c relative to any inertial frame of reference, is because otherwise causality would be violated by faster-than light objects moving 'backwards' in time to inform the past.

If we could make empirical sense out of this idea of causality being violated, then special relativity could not rule out the possibility of faster-than-light objects. But we cannot make empirical sense out of the idea of causal violations, since it leads to empirical contradictions. Therefore an anti-realist might argue that "faster than light travel" isn't a false proposition but a meaningless sentence. In which case "nothing travels faster than light" is a statement about the language of physics rather than a negative proposition about the world.

Is stoicism really a personal philosophy?

One of the things i find ironic about conservatives, and especially the more radical christian conservatives who preach self-responsibility, self-reliance, self-motivation etc which are values very much aligned with stoicism, is the fact they rely on preaching and perpetual communal gatherings to instil it, a permanent ongoing situation that is actually in direct contradiction to their message of self-motivation, self-reliability and self-responsibility.

I think that the conservative/evangelical mind-set is probably more resilient to life's misfortunes, which explains conservatism's appeal, but their actual practice for instilling and maintaining that mindset is communal and involves the social regulation, coercion and motivation of individuals, rather than individuals regulating and motivating themselves.

I think you would be clearer by referring only to the syntactical notion of derivability, since the diagonal lemma does not refer to truth, and neither does it assume the law of excluded middle. Any derivation of the lemma that did appeal to LOM would not be constructively acceptable, invalidating any consequent formula that appealed to the lemma which would include Godel's and Tarski's theorems, which are in fact constructively accepted.

The diagonal lemma only states that for every well-formed formula f(x) of one free variable, there exists a sentence s, such that the derivation of s implies the derivation of f('s') and vice versa, and that this fact is itself derivable.

Earlier, I was referring to the syntactical fact that when you wrote

∀f∈F:N→{false,true}∃s∈S:s↔f(┌s┐)

'f' isn't assumed to be a provably total predicate function. For example, when the lemma is applied to derive Godel's incompleteness theorem f refers to ~prov('s'), and whilst the disjunction "~prov('s') or prov('s')" might be derivable for any 's' via an appeal to LOM, axiomatic arithmetic cannot consistently decide which part of the disjunction is the case. Therefore any derivation of the diagonal lemma that appeals to such a hypothetical function isn't a permissible derivation of of Peano arithmetic.

I agree with you that we lack a good definition for general intelligence. But as my example of a thing that is clearly as intelligent as us but can't predict all our associations demonstrates, even our intuition doesn't agree with the Turing test as what is intelligent. We need to keep working to understand what intelligence is and as I currently see it, the way the Turing test is used in this work and in things like AI development, it diverts us into a path that is harmful. It is quite obvious that a transistor based general intelligence doesn't need to be able to speak any language in an indistinguishable way from humans and that that would be an inefficient and unnecessarily complex way to program general intelligence - yet people tend to see that as an important goal right now. Harmful, I say! — Qmeri

Whether or not a particular Turing test is appropriate in a given situation is largely a question concerning the breadth of the test. For example, if testing whether a computer 'really' understands Chess, should the test be very narrow and concern only it's ability to produce good chess moves? or should the test be very broad to even include the ability of the computer to produce novel metaphors relating chess to the human condition?

Personally, I don't interpret the spirit of the Turing test as making or implying ontological commitments regarding how AI should be programmed or trained , or as to how intelligence should represent sensory information with language, or even as to what intelligence is or whether it is ultimately reducible to metrics. Neither do I understand the spirit of the Turing test as being prescriptive in telling humans how they ought to judge a participant's actions. Rather I understand Alan Turing as very modestly pointing out the fact that humans tend to recognise intelligence in terms of situationally embedded stimulus-response dispositions.

In other words, the specifics of what goes on inside the 'brain' of a participant is considered to be relevant only to the functional extent that the brain's processes are a causal precondition for generating such situationally embedded behavioural repertoires; the meaning of language and intelligence being undetermined regarding the implementation of stimulus-response mappings.

Indeed, an important criterion of intelligence is the ability to generate unexpected stimulus-responses. Hence any formal and rigid definition of intelligence solely in terms of rules, whether internal in describing computational processes inside the brain, or situationally in terms of stimulus-response mappings, would be to a large extent an oxymoron.

Yes the Turing test is anthropomorphic, but why is that a problem in the absence of an 'objective' alternative?

Not even a logical language can be identified without mirroring. Recall Wittgenstein's example of an alien tribe stamping their feet and grunting in a way that is compatible with the rules of Chess. Only if we recognised their culture a being similar to ours might we assert they were playing Chess.

Recall that in the Turing Test, a human evaluator has to decide purely on the basis of reading or hearing a natural language dialogue between two participants, which of the participants is a machine. If he cannot determine the identities of the participants, the machine is said to have passed the test. Understood narrowly as referring to a particular experimental situation, yes the Turing Test fails to capture the broader notion of intelligence. But understood more broadly as an approach to the identification of intelligence, the Turing test identifies or rather defines intelligence pragmatically and directly in terms of the behavioural propensities that satisfy human intuition. The test therefore avoids metaphysical speculation as to what intelligence is or is not in an absolute sense independent of human intuition.

The very definition of 'alien' is in terms of the respective entity's tendency or capacity to mirror and predict our stimulus-responses for it's own survival. The Turing 'Test' is a misnomer; for the test constitutes a natural definition of intelligence. If we cannot interpret an entity's stimulus-responses as acting in accordance with the basic organising principles of human culture, then as as far as we are concerned, the entity isn't worthy of consideration. So to a large extent, the ability of aliens to speak 'our language' is presupposed in our definitional criteria.

Married and bachelor are two seperate and opposing qualities. — Harry Hindu

As far as i'm concerned, I'm a married-bachelor until the ink of the registrars signature is dry.

Could you continue on to an explanation of what counts as an alternating truth value? Is that what makes it self-negating? If it's true, it is false, etc... — creativesoul

Self-negation, or perhaps to state more accurately, the potential for self-negation, is a common property of negative universal propositions of meta-linguistics, metaphysics and epistemology that declare limits on sense, cognition or knowledge. For example, "All sentences have indeterminate meaning" , "All things are empty of intrinsic existence and nature" , and "Every belief is fallible" are all potentially self-negating propositions. Common coping strategies in the face of such potential contradictions are either to impose an artificial and rigid hierarchy of reference like Bertrand Russell did to avoid Russell's Paradox, or to quit philosophy and declare it to be nonsense as the Early Wittgenstein did, or to accept 'true contradictions' as Hegel did. Accepting alternating truth value is another coping mechanism that understands a person's concept of truth in terms of their present state and rejects the dogma of a static truth concept.

Often self-negation occurs when a conclusion negates it's own arguments, as when Wittgenstein declared that the propositions of the Tractatus are meaningless, after they had served as a 'ladder' to understanding. The later Wittgenstein's "private language arguments" have similarly been interpreted as self-negating "ladder" arguments, and similar remarks have been said about Kant's Critique of Pure Reason. Pure reason certainly can lead to contradictions, yet we don't simultaneously entertain both sides of such philosophical contradictions, rather we use logic to hop from one conclusion to it's opposite and then usually quit philosophising.

I also have a sneaking suspicion that alternating belief states might become a practical problem of artificial intelligence. After all, the human brain is a dynamical system and there is no compelling reason to assume that belief states converge to an equilibrium.

In Godel's incompleteness theorem f is taken to be ~Prov('s'), the negative of the 'provability predicate'.
It is easy to show via the construction of Prov('s') that s -> Prov('s') when s is a theorem. However it cannot be shown, assuming consistency, that s -> ~Prov('s') when s isn't a theorem. Prov is only a computable partial function, therefore a proof of the lemma by way of contradiction cannot be obtained.

Nothing is alternating, though. — frank

That depends on your notion of truth. Classically, you're right; for truth is not traditionally considered to be the property of a sentence or of it's construction, but of a timeless matter of fact referred to by the sentence that is existentially independent of, and external to, the sentence. From that perspective, the notion of 'alternating truth' i have sketched should be interpreted as referring to 'alternating belief' in the truth of a sentence, where a sentence is said to be 'true' merely if one accepts it and 'false' otherwise.

The liar paradox has multiple interpretations and resolutions. For instance, if 'this sentence' is interpreted syntactically as being recursive self-quotation, we end up with an infinitely deep nest of quotes ' ' ' ... ' ' ' that isn't even a sentence, let alone a well-formed sentence.

In my opinion, the Godel sentence used in the proof of the incompleteness theorem is best understood in an analogous fashion, since the proof is purely syntactical. Its semantic interpretation as a sentence asserting it's own lack of provability is a heuristic argument that isn't formally acceptable, because the Godel number supposedly referred to by the Godel sentence is infinitely long when the sentence is recursively unpacked by substituting the sentence into itself.

On the other hand, if 'this sentence is false' is interpreted semantically as being a pair of sentences, each sentence belonging to a different language whose meaning is the negation of the sentence in the other language, then we get the traditional semantic understanding of the sentence as a contradiction.

However, since the liar paradox is a paradox of natural language that is it's own meta-language, as opposed to being a paradox of formal language, my preferred resolution is to consider the liar paradox as being a meaningful sentence (since we can understanding the paradox), that isn't a contradiction, rather it is a self-negating sentence with alternating truth value. This interpretation best describes our use of the paradox. i.e. "It is true - hence it is false - hence it is true... etc"