I've never seen that in type theory or elsewhere. it seems to make no sense. Please say where you have ever seen that as type theory? — TonesInDeepFreeze

To put it categorically, I'm referring to the definition of the set Nat as the carrier of an Initial Algebra

Hence any variation of the ontological proof must be suspect, since by their nature they seek to demonstrate the existence of something not found in their assumptions. — Banno

Anselm's ontological argument is essentially an inductive definition of god, analogous to the inductive definition of the natural numbers in type theory.

1. Posit an initial imagined god ; g(0).
2. Given any imagined god, specify the existence of an improved 'realer' god; g(s+1) = improve g(s)
3. Define a 'perfect' god in terms of the fixed-point g(inf) = improve g(inf)

Construed this way, the god specified in step 3 isn't a deduction relative to 1 and 2, rather it constitutes the definition of a fixed point for improve with respect to the premises 1 and 2.

As with the creation of the set of natural numbers, such arguments aren't empirically meaningful so they must be imperatives, about how to think about 'god' in the case of religion, and about how to use the sign Nat in the case of mathematics, i.e. as a sign signifying an unspecified number of iterations of step 2.

One important aspect of mindfulness is the experience of thoughts. The philosophy of mindfulness may be important here in being about observing thoughts rather than simply reacting to them. — Jack Cummins

To me that sounds like undirected introspection and a potential recipe for worsening mental illness. Isn't 'mindlessness' a more accurate term for what "mindfulness" is supposed to be? i.e. to avoid paying attention to thoughts by channelling attention elsewhere in order to reduce rumination and introspection?

I don't think it is possible to passively and objectively observe the mind; that seems erroneously suggestive of the myth a passive subject watching a distant cinema screen in the Cartesian Theatre.

To what extent is it possible to step outside of the chain of reactivity? In this thread, I am seeking to start a critical discussion about the nature of mindfulness as a state of awareness. To what extent is the idea helpful as a basis for coping with stress or as a philosophy for finding balance in life? — Jack Cummins

Everything that is a known to definitely reduce stress, e.g gardening, hiking, caring for a pet, painting etc seem to fit the definition of "mindlessness" i gave above, namely relaxed extrospection . If that is what mindfulness is actually supposed to be, then most people already practice it and intuitively know about it.

With 'mindfulness' i see at best a superfluous concept, and at worst a detrimental and mistaken ideology.

As argued in other threads, logical necessity isn't the same thing as empirical necessity or epistemic certainty. Modal logic is in general a fallacy if it's modal operators are interpreted that way, for one cannot know the properties of every possible world until one has literally counted and inspected all of them.

On the other hand, its a mistake to think of ontological arguments as being empirical arguments, for they are really an expression of faith, i.e. of deontic necessity. So the above argument is really a computational way of expressing religious faith.

Personally, I think ontological arguments are interesting when properly considered, and wonder if they have potential application in the secular religion called psychotherapy.

For that matter, as I've already asked, what is your definition of "absolute infinity"? — TonesInDeepFreeze

Absolute infinity refers to a semantic interpretation of a mathematical, logical, or linguistically described entity, relative to which the existence of said entity cannot be independently verified, empirically evaluated, or constructed with respect to a finite amount of data.

Absolute infinity arises when an analytic sentence is mistaken for an empirical proposition. Quine's famous example "All Bachelors are unmarried men" can be held as being true by definition in the mind of a particular speaker, but in doing so it can no longer be regarded as being representative of how a community of speakers might use the words "bachelor" and "unmarried man", given the limitless potential of them using the words non-equivalently.

Analogously, in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension, whereupon it is inconsistently alleged that a finite description of a function can somehow represent a limitless amount of information that is also exact.

The alternative interpretation corresponding to "potential infinity" is to consider the definition of such entities as being vague and verifiable, as opposed to being semantically precise but unverifiable.

Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded". — TonesInDeepFreeze

It is practically equivalent to the common definition of potential infinity as being a "non-terminating" sequence that is never finished but occasionally observed after random time intervals.

Actually my phrasing is a slightly weaker statement, considering that potential infinity is usually used in the context of monotonic sequences, as in infinitesimals or infinitely large numbers. The important thing regarding the common definition of potential infinity is that in order to obtain a value or extension, the process constructing the sequence must be "paused" after a finite amount of time. This constitutes a random stopping event, in the sense that the time of the pause is not defined a priori at the time when dx is declared to be infinitely small or x to be infinitely large.

Defining potential infinity in terms of a "non-terminating" process is problematic however, given the fact that 'non-termination' isn't a verifiable proposition if interpreted literally, which is a concept belonging to absolute infinity. What is important to the definition of potential infinity is pausing a process to obtain a finite portion of a sequence, whereupon one might as well regard restarting the process as starting a new process. Then consider the fact that any of the finite extensions generated by pausing a process are countable and isomorphic to the integers. These are my considerations when thinking of "potential infinity" in terms of a priori unbounded finite numbers instead of in terms of a "non-terminating" process. However, my definition might cause confusion due the fact it is more general and includes random variables with unbounded values.

Your whole line of argument has sputtered. As well as you still have not addressed that you terribly misrepresented Brouwer — TonesInDeepFreeze

Nope. You need to reread the article.

I interpreted the puzzle to refer to words, hence

"ofouro"

Moreover, you mention internal set theory, but internal set theory is an extension of ZFC, so every theorem of ZFC is a theorem of internal set theory. So, if one rejects any theorems of ZFC then one rejects internal set theory. Moreover (as I understand), non-standard analysis and internal set theory make use of infinite sets, and though (as I have read) there is an intuitive motivation of 'potentiality' in internal set theory, I do not find 'potentially infinite' defined in those papers or articles I have perused. Let alone that you are now combining non-standard analysis with potential infinity without reference to where that is in any mathematical treatment (possibly there is one, but you have not pointed to one). — TonesInDeepFreeze

Overall, good observation.

To my knowledge, it isn't possible to point to a complete formalisation of potentially infinite logic, because it doesn't yet exist. All we have are fragments or incomplete axiomatizations of the concept, that have been invented by different logicians over the years with respect to different systems for purposes other than the current discussion. This is somewhat similar to the proliferation of different programming languages. There isn't any inter-subjective agreement as to how to formulate open-world reasoning.

- The I, S and T axioms that Edward Nelson introduced are useful for formulating what it means to reason with as-of-yet unconstructed elements of an unfinished set, in spite of the fact he proposed the axioms in the context of ZFC as an alternative to model-theoretic non-standard analysis in ZFC. ZFC is of course inadmissible for the purposes of this discussion, and is the reason why his formalisation doesn't tend to be associated with formalizing potential infinity. But the I,S and T axioms, divorced from the problematic axioms of ZFC appear to be a relevant fragment of some formalization of potentially infinite logic.

-Brouwer's notion of choice sequences, i.e. unfinished sequences, serve as the template for potentially infinite sequences but his formulation doesn't to my understanding provide what I,S and T does.

Choice sequences allow the expression of unfinished sequences, e.g

{1,2,3,...}, where the dots "..." are understood to mean "to be continued"

Brouwer introduces continuity axioms that define what it means to prove a universal proposition over a domain that consists of such unfinished sequences. However, his concepts, at least to my understanding, doesn't permit direct talk about numbers that we have presently declared, but cannot currently quantify, e.g. "The height of the tallest human being who will ever live"

- Linear Logic, as opposed to intuitionistic logic, is the logic i would associate with Intuitionism and potential infinity, because it is a resource conscious logic. Again, as you might say, it is "not evidently associated with p.i", especially in view of it's exponential fragment. Squinting at axioms to see their practical significance is still an unfortunate necessity.

What is a potential infinitesimal — jgill

It is the reciprocal of a potentially infinite number, e.g. a random value taken from the codomain of the rational valued function 1/x.


'non-standard analysis' is the correct umbrella term, but it is already befuddled by the various alternatives that fall under it, some of which receive rightful criticism for obscuring matters even further, e.g the hyperreals .

Infinite sets come into play in Calculus 1. What pedagogy would you propose for people to find derivatives without infinite sets? — TonesInDeepFreeze

You mean without absolutely infinite sets, presumably. The overall approach would be to stress that the mathematical notion of a derivative approximates the real-life practice of directly measuring the quotient of two arbitrarily small intervals Dy and Dx with respect to some observed function. This is opposite to the conventional way of thinking, which construes the practice of measuring a slope as a means of approximating an ideal, abstract and causally inert mathematical derivative.

With this in mind, the classical definition of df/dx with respect to the (ε, δ)-definition of a limit, can be practically interpreted by interpreting ε to be a potential infinitesimal, and δ as representing a random position on the x axis given the value of ε , which when applied to the function yield df and dx as potential infinitesimals, i.e. finite rational numbers, whose smallness is a priori unbounded.

Yes. I mean that different implementations of the constant of pi will yield different values. Orthodox convention says that those values are 'truncations' of some ideal value. The problem is, if one asks what that ideal value is, one can only be referred back to the intensional definition of pi, which isn't the same thing as a value. Hence the only conclusion that can be reached, is that an ideal value of pi doesn't exist, and that so-called 'approximations' of pi aren't approximations of anything specific that is external to them.

Hence the value of pi is ambiguous in the same sense that 'one metre' is ambiguous, in appealing to the uncertain contigencies of practical experiments with finite resolution.

According to my copy of On Certainty, Wittgenstein only used the word "hinge" in 3 places, two of which concern the same remark, and none of which refer to a type of proposition.

"
340. We know, with the same certainty with which we believe any mathematical proposition, how
the letters A and B are pronounced, what the colour of human blood is called, that other human
beings have blood and call it "blood".

341. That is to say, the questions that we raise and our doubts depend on the fact that some
propositions are exempt from doubt, are as it were like hinges on which those turn.

342. That is to say, it belongs to the logic of our scientific investigations that certain things are in
deed not doubted."

343. But it isn't that the situation is like this: We just can't investigate everything, and for that reason
we are forced to rest content with assumption. If I want the door to turn, the hinges must stay put. "


From which it becomes clear that 'hinges' refer to the calculations that determine the meaning of "truth" in a given instance of reasoning.

"
653. If the proposition 12x12=144 is exempt from doubt, then so too must non-mathematical
propositions be.

654. But against this there are plenty of objections. - In the first place there is the fact that "12x12
etc." is a mathematical proposition, and from this one may infer that only mathematical propositions
are in this situation. And if this inference is not justified, then there ought to be a proposition that is
just as certain, and deals with the process of this calculation, but isn't itself mathematical. I am
thinking of such a proposition as: "The multiplication '12x12', when carried out by people who
know how to calculate, will in the great majority of cases give the result '144'." Nobody will contest
this proposition, and naturally it is not a mathematical one. But has it got the certainty of the
mathematical proposition?

655. The mathematical proposition has, as it were officially, been given the stamp of
incontestability. I.e.: "Dispute about other things; this is immovable - it is a hinge on which your
dispute can turn."

656. And one can not say that of the propositions that I am called L.W. Nor of the proposition that
such-and-such people have calculated such-and-such a problem correctly.

"

In other words, "necessarily true propositions" are either in fact only contingently true, else they refer to those which are held true by convention, but whose necessary truth is nevertheless subject to revision whenever the convention changes.

Similar considerations led Quine to publish his rejection of the analytic-synthetic distinction a couple of years later in "The Two Dogmas of Empiricism".

I asked a math PH.D. and they said Pi is an exact number. How can an irrational number be exact if we can't even reach the last digit ever? — TiredThinker

It is exact in the a priori intensional sense of being defined as an equation or algorithm with instantly recognizable form.

It is inexact in the a posterori extensional sense of being a sequence of rational numbers, for the reason you point out; pi as a constant is ambiguous - just ask Matlab.

If i were you, I would skip trying to decipher Wittgenstein's informal, vague and incomplete prose which constitutes the beginning of post-analytic philosophy, and jump straight into reading Quine's Word and Object, which gives a more developed and precise account of the semantic holism that both he and Wittgenstein arrived at. From their similar points of view, the classical distinction between idealism and materialism loses it's intelligibility.

I'm willing to concede that my colleagues and I have produced mathematical contributions that are worthless, but calling classical mathematics "junk logic" and "crudely expressing ideas" is a ridiculous accusation. On the other hand, that may not be what you are saying. It's hard to work through some of your lengthy paragraphs. Probably just me. — jgill

sure, its an overstatement born of frustration with somewhat outdated formal traditions that still remain dominant in the education system.

A groom, hand on heart, vows sincerely to the bride " I will always remain faithful". Later that afternoon, he runs off with the bridesmaid. Did he really contradict his earlier vows, or does a contradiction exist only in the minds of those who misconceive the nature of infinity ?

Perhaps one might say that to view the groom as contradicting his earlier vows amounts to a definition of 'negated absolute infinity' - but this interpretation is unnecessarily problematic in asserting the negation of a statement that isn't a verifiable proposition with verifiable meaning.

'infinity' as a noun does not ordinarily have a mathematical definition, though 'is infinite' does. A mathematical definition is never circular nor a tautology. — TonesInDeepFreeze

In which case, you surely agree that absolute infinity isn't a semantically meaningful assignment to a mathematical entity, for any semantic interpretation of the symbol of infinity as referring to extensional infinity, is question begging.

* The notion of 'potentially infinite' is of course central to important alternatives to classical mathematics. However, as far as I know, formalization of the notion is not nearly as simple as the classical formalization of 'infinite'. Therefore, if one is concerned with truly rigorous foundations, when one asserts that the notion of 'potentially infinite' does better than that of 'infinite' one should be prepared to accept the greater complexities and offer a particular formalization without taking it on faith that such formalizations are heuristically desirable, as we keep in mind that ordinary mathematical application to science and engineering uses the simplicity of classical mathematics as one first witnesses in Calculus 1. — TonesInDeepFreeze

The semantic notion of absolute infinity (whatever that is supposed to mean) isn't identifiable with the unbounded quantifiers used in classical mathematics, logic and set theory, due to the existence of non-standard models that satisfy the same axioms and equations without committing to the existence of extensionally infinite objects. Science and engineering continues to work with classical mathematics , as well as classical logic, due to their vagueness, simplicity and brevity as a junk logic for crudely expressing ideas, which usually cannot be fully formulated or solved in those notations due to the inconvenient truths of software implementation and physical reality. Most software engineers don't regard themselves to be mathematicians or logicians, due to historical reasons concerning how mathematics and logic were initially conceived and developed.

* What writings by intuitionists are fairly rendered as describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" of any kind? — TonesInDeepFreeze

SEP's article on intuitionism is a useful introduction for understanding the notion of Brouwer's tensed conception of mathematics, i.e. mathematics with lazy evaluation, that rejects both formalism and platonism, in which unbounded universal quantification is understood to refer to potential infinity, which leads to his formulation of non-classical continuity axioms. In a similar vein, Edward Nelson's Internal Set Theory adds tenses to Set Theory, by distinguishing the elements of a set that have so far been constructed that have definite properties, from those that will potentially be constructed in the future, that have indefinite properties.

Unless infinity is formally identified with a finite piece of syntax, whereupon becoming a circularly defined and empirically meaningless tautology, infinity cannot even be said to exist inside mathematics, let alone outside.

Potential infinity, as the intuitionists keep stressing and as programmers demonstrate practically, is the only concept that is needed, both inside and outside of mathematics, that refers to finite entities of a priori indefinite size.

The myth of absolute infinity is what give the illusion of mathematics as being an a priori true activity that transcends Earthly contingencies.

In the parlance of computer science, criteria constituting what it means to obey a given rule falls under Denotational Semantics and in the case of a function refers to it's tabular definition.

For example, part of the tabular definition of the total function f(x) = 2x can be specified as
{(0,0), (1,2), (2,4)}. In general, we can provide partial definitions of f in terms of partial functions.

A central question that denotational semantics is supposed to answer, is given that we only have the time to write down partial functions, what does it mean to assert that f(x) has a complete tabular definition as a total function?

In contrast, how a rule is followed, which is in this case concerns how f(x) is computed, is addressed by Operational Semantics. For computer science, this refers to the infinite number of possible pathways for computing the value of a function in accordance with it's specification (as described in terms of denotational semantics)

Lastly, axiomatic semantics specifies the imperative implementation of a function as a computer program running on a finite state machine (recalling that the denotation of a function doesn't possess the notion of a state).

For natural languages, an individual's mental interpretation of their public language, which eludes ostensive definition, is analogous to the operational and axiomatic semantic aspects of formal languages which elude denotational definition.


The paradox of logic that Wittgenstein was colloquially referring to, that was initially raised by Lewis Carroll and formally expanded upon by Quine in his attack on the Analytic-Synthetic distinction, involve the fact that there isn't a way to derive the complete denotational semantics of any given function, either by fiat or by appealing to some other form of semantics, due to the essential incompleteness of any type of semantic specification. To put it colloquially, it isn't possible to give an exhaustive account of what it means to obey a given rule, because a tabular definition of the said rule can never be finished, implying that the intended meaning of a rule is publicly under-determined.

To use the example above, how can we nail-down the complete tabular definition of the total function
f(x) = 2x ? At most we can write a finite portion, and then intimate the rest with dots:-

f(x) := {(0,0), (1, 2), (2 ,4 ), ...}

but how can the gesticulated meaning of the dots "..." in this context be interpreted to refer to an implicit yet unambiguous definition? One might try appealing to supposedly finite denotational semantics in the form of recursion :-

f(0) = 0,
f(x) = f(x-1) + 2

But then we need to complete a table specifying how to map the variable x to f(x) for every possible value, which is impossible, so we have gained nothing. (Consider the fact that every computer program implementation of 'f' will overflow at some value for x, that varies in accordance with the operational and axiomatic semantics of the CPU, OS and compiled executable that varies in each and every use case).

Domain Theory is the theory appealed to by computer scientists for completing denotational semantics in such a way as to pretend that the 'private' axiomatic and operational semantics of a function are independent of it's 'public' denotational semantics. The theory fails to acknowledge the essential incompleteness of denotational semantics and merely hides the fact by implicitly defining the total function f(x) = 2x to be the fixed point of a functional F( g, x) , e.g

F(g,x) :: (Int -> Int) -> Int -> Int
F(g,0) = 0
F(g,x) = g(x-1) + 2 If g(x-1) is defined, else
F(g,x) = undefined

Applying F to the totally undefined function called 'bottom' and iterating repeatedly, leads to the increasing sequence of partial denotations

F (bottom, x) = {(0,0), otherwise undefined }
F( F(bottom, x), x) = {(0,0), (1,2), otherwise undefined }
F( F(bottom, x), x) = {(0,0), (1,2), (2,4), otherwise undefined}
...

The illusion of f(x) = 2x as a definite total function with complete denotational extension is generated by appealing to the definition of f as the fixed point f := F( f, x) , which can then used as a definition for the earlier expression

f(x) : = {(0,0), (1,2), (2,4), ...}

At the fixed point f, the above functional F ignores it's function parameter entirely and so the definition of f in this case is more or less identical to the earlier recursive definition of f above. Hence all this definition does is reinterpret the denotational ambiguity of f in terms of the denotational ambiguity of functionals.

The lesson here, is that the meaning of any word or rule isn't definable in closed form, ergo

i) The meaning of mathematics isn't reducible to logical axioms, and neither is the meaning of logic.

ii)The meaning of language is under-determined by, and cannot be grounded in, any explicitly stated convention, whether publically or privately given, as Quine and in high probability Wittgenstein, concluded,.

Simply put it, if you don't need to work, some can choose then not to work. And then you basically slide off "the society", even if the welfare state does provide you housing and free health care. In that case you look for a job only so many times and then say f*k it. And what it creates is apathy. — ssu

If that were true, then why are so many rich people, including pensioners who no longer have to work, highly productive?

Apathy is the product of alienation rather than the product of financial security.

That doesn't tell me how an action can be true or false. I get up, go into the kitchen, and get a glass of water. Is that action true or false? — T Clark

Unless the action is related to context, a truth value isn't assignable.

Certainly in the context of predictive modelling, a truth value is assignable by definition of the context concerned. One can certainly be an anti-realist about truth in such contexts, but this isn't to deny the concept of truth or to identify truth with utility.
.

Pragmatism doesn't say anything about the truth of actions. How can an action be true or false? — T Clark

If engineers develop a model on the basis of past experience, their words and actions assent to some notion of truth.

Responsible naturalists would put it differently. They would say that the time to believe non-natural explanations - idealism, gods, reincarnation, that only consciousness exists, whatever it may be - is when there is good evidence for them. These concepts then become knowledge and presumably, a part of naturalism. There's a Noble Prize as yet unclaimed. — Tom Storm

Idealism isn't an explanation and shouldn't be associated with superstitious beliefs in the supernatural. Rather, Idealism is a subjective interpretation of the concepts defined by naturalism, in terms of the experiences of the observer. In other words, Idealism is a form of phenomenalism, but without necessarily implying the possibility of a phenomenalist theory of meaning.

To view naturalism as being ideologically opposed to idealism is to imply that naturalism isn't an empirically grounded belief system.

We need to distingush two forms of ambiguity

Intensional Ambiguity of Extensions: A given extension, e.g. a sequence [s(1),s(2),s(3),...], corresponds to an infinite number of functions. This is an epistemic form of ambiguity studied by Theoretical Machine Learning and Statistical Learning Theory.

Extensional Ambiguity of Intensions: A given description of a function, say f(x) = 1/x, corresponds to an infinite number of possible extensions, e.g not only [1,1/2,1/3], but also [1,1/2,0,312,9998].
Although we immediately recognise the latter as being false, such pathological interpretations cannot be exhaustively ruled out by any finite description of f(x) . This is a semantic form of ambiguity that Quine and Wittgenstein were concerned with, that machine learning and statistical learning theory typically ignores.

Following Quine in Truth By Convention (1936), it is impossible to exhaustively define a function extensionally in terms of a graph-plot of the function's values, since any graph plot is finite, leaving many semantic holes. Therefore the meaning of a function cannot be explicitly stated by convention, and the same is true for the meaning of logic.

The upshot is that conventions cannot explicitly describe or prescribe how users use mathematics and language in general, which implies that linguistic conventions are largely a post-hoc expression of how people decide to use language in practice, rather than the converse.

Materialism has no meaning for a materialist"! — Alkis Piskas

This is why i don't believe that self-avowed materialists are materialists. Their identity isn't the same as their orientation. Materialists cannot relate their perceptions of objects to their thoughts concerning 'material objects' without pain of contradiction. They are smuggling their own brand of phenomenalism into their private definition of materialism whilst being in denial about it.

if the sequence is for example N, then the correct algorithm is "list all natural numbers". And natural numbers don't come to an end. — ssu

To understand the paradox using your example, you have to distinguish the intensional definition of a function, such as one reproducing the natural numbers

i.e. f(n) = n for all n in Nat

from a potentially infinite list of elements such as

S = { 1, 2 , 3, ... }

that looks like it might be a prefix of Nat.

By 'potentially infinite' I mean that S is finite but of unknown size and whose elements are only partially defined a priori (in the above case, only the first three numbers). it's remaining elements are denoted by the dots "..." and are a priori unknown and decided when S is instantiated.

Obviously, until S is fully instantiated it cannot be decided as to whether S is a prefix of N or some other function. The paradox concerns the fact that one cannot know in advance what the prefixes of N are.

One can try to define the prefixes of Nat intensionally as a function, namely

P(n) = {0,1,....n} for all n

But in order to know what all of the prefixes are as implied by this definition, we would need to define what it means to treat n as a free-variable that can be substituted for any natural number. But if this definition of a free-variable is also intensionally defined, we will have gotten nowhere. So the only way to decide what the prefixes are, is to resort to writing them out extensionally for some random number of terms, which we can represent as the potentially infinite set

{ {0}, {0,1}, {0,1,2} , ...}

This set will be instantiated with a random number of prefixes, relative to which it will be decided, in a spur-of-the-moment bespoke fashion, as to whether or not S is a prefix of Nat.

If there genuinely is a pattern with 2,4,8,... then that pattern will describe the number chain or series to infinitum or otherwise it's a wrong pattern or the series of numbers is basically without a pattern, patternless. Here to talk about rules it would be better to talk about algorithms in the general sense. And either you have an algorithm that correctly tells you how the series 2,4,8,... goes or either you have the wrong algorithm or the series is non-algorithmic. — ssu

yes, in the case of a potentially infinite sequence of numbers, it is meaningless to consider any particular function, let alone algorithm, as being descriptive of the sequence unless and until the sequence comes to an end. Until then, one cannot even decide whether or not the sequence is computable. Nevertheless it is meaningful in the meantime to speak of falsified hypotheses in relation to the sequence.

However, the problem goes further than that, because on Kripke's interpretation, the skepticism is calling into question the very meaning of "algorithm", and hence the distinction between algorithmic versus non-algorithmic processes, which computing and constructive mathematics take for granted. Such philosophies treat the definition of an algorithm to be isomorphic with the input-output pairs generated by it's execution, which presupposes the existence of ideal calculators. But as we know practically, physical implementations of algorithms have finite capacity and finite reliability, making the intensional definitions of functions misleading with respect to their implemented behaviour. Kripke is asking how it can be decided that a sequence corresponds to a given total function, given the irreparable inability to define what the 'correct' outputs of the function are for most of it's inputs.

Just contemplate upon the fact that the feelings and mental-imagery that you associate with the future or the past, actually correspond to the present. It isn't possible to think beyond the present in a literal sense.

To clarify Wittgenstein's "epistemology", background historical context is required with regards to "Wittgenstein's Verification Principle" in 1930 that the Vienna Circle enthusiastically adopted for several years, but that Wittgenstein came to reject shortly after he proposed it. According to the verification principle, the sense of a proposition is its means of verification.

Wittgenstein explained the verification principle to Schlick:

"If I say, for example , 'Up there on the cupboard there is a book', how do i set about verifying it? Is it sufficient if I glance at it, or if I look at it from different sides, or if I take it into my hands, touch it, open it, turn over its leaves, and so forth? There are two conceptions here. One of them says that however I set about it, I shall never be able to verify the proposition completely. A proposition always keeps a back-door open, as it were. Whatever we do, we are never sure that we are not mistaken.
The other conception, the one I want to hold, says, 'No, if I can never verify the sense of a proposition completely, then I cannot have meant anything by the proposition either. Then the proposition signifies nothing whatsoever.'
In order to determine the sense of a proposition, I should have to know a very specific procedure for when to count the proposition as verified. "

But, in line with a common tendency displayed in this forum, the logical positivists mistook his idea of verification for a dogmatic theory of meaning. Indeed Wittgenstein himself was briefly seduced by this principle in 1930 before abandoning it. Wittgenstein told the Moral Science Club in Cambridge:

"I used at one time to say that, in order to get clear how a sentence is used, it was a good idea to ask oneself the question: 'How would one try to verify such an assertion?' But that's just one way among others of getting clear about the use of a word or sentence. For example, another question which it is often very useful to ask oneself is: 'How is this word learned?' 'How would one set about teaching a child to use this word?' But some people have turned this suggestion about asking for the verification into a dogma - as if I'd been advancing a theory about meaning. "

Hopefully everyone here will see that abandoning the identity 'meaning is verification' as well as it's weaker cousin 'meaning is dependent upon some process of verification' implies abandoning every theory of meaning , including meaning is proof or derivation, meaning is convention, meaning is contingent upon social verification, meaning is falsification etc. For each of these cases involves appealing to case-specific criteria of meaning that aren't universally employed across all language-games, and are mostly of relevance to formal language-games in which meaning is directly defined identified with verification. Verification criteria aren't generally present in other use-cases of propositions. For example, many everyday uses of "2+2 = 4" don't involve any checking for equality beyond one's immediate first impression.

His rejection of the principle of verification also demolishes common misconceptions referred to as "Private language arguments" that argue for a rejection of private meaning on the basis of an absence of independent or external verification criteria. Wittgenstein gave an example of private meaning without verification criteria as early as may 1930 in Philosophical Remarks:

"How do I know that the colour of this paper, which I call 'white', is the same as the one I saw here yesterday? By recognizing it again; and recognizing it again is my only source of knowledge here. In that case, 'That it is the same' means that I recognize it again"

As for the status of philosophical skepticism. Wittgenstein did not believe that "hinge-propositions", had prescriptive value. His criticisms of Moore weren't criticisms about the truth of Moore's assertion that he has hands (which depending on one's interpretation of Moore's intended meaning could either be viewed as true, false, both or neither) but were criticisms pointing out the distinction between the use of propositions as auxilliary hypotheses, versus the use of propositions under evaluation.

au contraire, Wittgenstein is saying is that all observations (word usage) are compatible with any conceivable law. — Agent Smith

Not quite. Wittgenstein only criticised logical conceptions of meaning, especially in relation to the view that the meaning of a proposition is static and a priori decidable . He didn't criticise individuals for their idiosyncratic interpretations of rules and language, which generally don't invoke theoretical interpretations of meaning.

There is a world of difference between speculating that an event E must logically follow from the a priori definition of a law L, versus recognising for oneself post-hoc, that E follows from L.

For example, often when you judge for yourself that two colours are the same, (which you usually do without any external guidance), your recognition wasn't contingent upon you invoking a priori definitions of the colours involved and calculating a truth value.

This is the reason why Wittgenstein wasn't a verificationist - meaning doesn't normally involve processes of verification - ergo Wittgenstein wasn't against the idea of private meaning.

Consider the propositions

"The sequence x1, x2, ... is determined by the function f(x)"

"The function f(x) is determined by the sequence x1, x2, ... "

There are two permissible interpretations of the word 'determined' in the above propositions:

A) As an imperative when for instance normatively insisting that " f(x) means the sequence x1,x2,..

B) As a descriptive hypothesis when for instance alleging that a given sequence x1,x2,... obeys f(x)

Classically, the sign 'x1,x2...' is interpreted as an abbreviation for a particular sequence of infinite extension for which there isn't time to write the whole sequence down. Under this identification, interpretations of the form A leads to the identification of f(x) as also denoting a particular domain and image of infinite extension. Thinking of functions extensionally in this way leads to scepticism whenever it is asked if some unbounded sequence S obeys a given f(x), given that only a finite prefix x1,x2,..xn of S can be observed. In conclusion, hypothesis of the sort (B) aren't verifiable when thinking this way.

On the hand, in the Russian school of constructive mathematics, functions aren't directly interpreted as representing entities of infinite extension, but as being finitely describable computable maps whose domain is unbounded. But this can lead to the same impression of such functions as having actually infinite and precise extensions if one thinks of such functions as denoting ideal and physically infallible computation. Thus the same platonistic skepticism about rule-following arises as in the classical interpretation.

The alternative to adopt Brouwer's philosophy of Intuitionism, in which ' x1,x2,... ' is interpreted as referring to partially defined finite sequence of unstated finite length, rather than as referring to an exactly defined sequence of actually infinite length. In other words, x1,x2,... is interpreted as referring to a potentially infinite sequence whose length is unbounded a priori, but whose length is eventually finitely bounded a posteriori at some unknown future date. Likewise, the domains and images of functions are also interpreted as being potentially infinite rather than as being actually infinite. Relative to this philosophy, rule-following scepticism is avoided due to the fact that the meaning of potentially infinite sequences and functions are both understood to be semantically under-determined a priori.

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All of you must be talking nonsense. It's easy to demonstrate that nothing can change in space. How can the same thing exist in two different positions? It's also easy to demonstrate that nothing can change in time. How can the same thing exist in two different points in time? It must be two very similar things, but they are not the same, since they exist at different times. There is no such thing as change. — pfirefry

Nevertheless, one can still create the idea of 'change' by using an indexical such as 'this' to refer to two or more referents, as when recognising that the colour of an object has changed - something that is objectively nonsense for the reasons you point out, and yet subjectively meaningful.

Perhaps one can say that the mind is change, implying that philosophical theories concerning personal identity over time are unnecessary and redundant.

"the primary value of truth and knowledge is for use in decision making to help identify, plan, and implement needed human action." — T Clark

But that is likely to be accepted as true by many non-pragmatists.

Am I right in suspecting that what you are actually protesting about is the artificial distinction between theory and practice that classical philosophy has been prone to insinuating?

Of course, not only philosophers but mathematicians, scientists and engineers are prone to thinking dogmatically in holding certain propositions, models or techniques to be infallible, lending to occasional calamities such as financial crises. One of the modern culprits of dogmatism is statistical and probabilistic modelling and deep learning for making implicit the assumptions of their respective models. The joke called Bayesian epistemology, which can encourage the delusional practice of smuggling assumptions into a model in the name of not making any assumptions, further adds fuel to the fire.

Here is a description of William James' definition of truth from an article I found on his book "Pragmatism.

Beliefs are considered to be true if and only if they are useful and can be practically applied. At one point in his works, James states, “. . . the ultimate test for us of what a truth means is the conduct it dictates or inspires.” — T Clark

Note that James appealed to such arguments when justifying the beliefs and practice of religion, and Richard Rorty has given pragmatic arguments for increasing the cultural prioritisation of the humanities (including Continental philosophy), relative to the natural sciences, by arguing that different communities in different subjects get to decide their own criteria of truth.

Pragmatism can encourage the identification of truth with what is expedient to believe, in line with post-modern cultural relativism, which I'm pretty sure you don't agree with. Something far from being an ally of the enlightenment values embodied by modern engineering.

Change vs Difference:

In terms of McTaggart's B series, every temporal referent, e.g. date, is different - by definition of "referent" . This is used in calendar logic, but also applies to the above colour difference picture in which one instantaneously recognizes more than one colour referent. In both calendar logic and the above picture, what is called "difference" is recognized without any concept of temporal passage coming into play. ( a stimulus-response to an image shouldn't be interpreted as being an inference, because the future is irrelevant as far as the stimulus-response is concerned)

"Change" refers to two referents being associated with the same indexical. For example, if one thinks that "now" becomes "now", then one creates the confusion of temporal passage. But if instead one thinks of the first act of "now" as referring to '09:09 on 26th Jan' and the second act of "now" as referring to '09:10, 26th Jan' then no change can be acknowledged.

I think it's clear from what I've written that I don't agree. — T Clark

I'm not sure that we do disagree. You presumably agree that modelling assumptions , which are ultimately causal or logical, aren't empirically verifiable, and that on the other hand, unless modelling assumptions are made, to speak of learning from data is meaningless.

It isn't clear to me how to philosophically distinguish epistemological pragmatism from a supposed anti-thesis. I am under the impression that epistemological pragmatism is being defined here in terms of the practicality of the problem pursued, rather than in terms of the method of inquiry which at every step hangs upon intuition regarding non-verifiable assumptions of causality.

"Life after life" (anti-anxiety placebo) is nonsense like e.g. north of the North Pole. — 180 Proof

Does tomorrow come after today, or is today always today?

One cannot justify the usefulness of a model of data without first making ontological commitments. The concept of usefulness only comes after committing to some notion of truth, that cannot be pragmatically determined on pain of circularity.

I think out of sheer intellectual curiosity it can be interesting to try to determine what happens after death. Does it really serve any practical purpose? Maybe not. — Paul Michael

On the contrary it very much does, considering the fact that all moral and ethical conclusions are relative to the premise of death that one adopts. In my opinion, society's beliefs regarding death are very much decided according to the behavioural advantages that result from holding those beliefs, which under capitalism tends to favour beliefs that motivate someone to work and spend intensely, as if they only lived once.

On the other hand, if the general public believed in reincarnation, and hence that there is no escape from the physical suffering perpetuated by unfair economic outcomes and environmental destruction, then I cannot see why they would continue to accept the current system of capitalism.

Cultural atheism under capitalism is more a less a sect of Protestant Christianity rather than being it's antithesis. The myth of the afterlife has only slightly changed, with ethereal promises of a heavenly paradise being substituted for an equally ethereal promise of perpetual nothingness - which most boomers are banking on for their post-humus escape from the mess they created on Earth.

The other conclusions beg the question. They assume that an entity or substance exists within the biology but is not the biology, and second, that this entity or substance can somehow persist beyond the biology itself. It seems to me one should be proven before contemplating the other. — NOS4A2

The presentist/idealist alternative doesn't speculatively assume a soul substance, rather it simply treats first-person experience as ontologically fundamental and unchanging. Of course, it's conclusions beg it's own ontology, but this is unavoidable whatever stance one takes.

The question one needs to ask, is given that different ontological assumptions about life lead to radically different conclusions about death that are in large part tautological, why choose a single ontology as being correct? Why not accept all of them and accept their respective conclusions relative to their respective ontology?