In threads such as this, it is helpful to begin by investigating the 'opposite' notion of unconsciousness, which at first glance appears to be a simpler concept and which people mistakenly take for granted; for they naively presume that they have a sound understanding of the concept of "unconsciousness", a presumption which they naively carry forward when using their understanding of 'unconsciousness' as a baseline when constructing and appraising theories of consciousness.

First of all, one should examine situations in which they claim not to have been conscious of anything. For example, take the claim that one wasn't conscious of anything while one was sleeping last night. If this were indeed the case, then how could one possibly know it? Isn't empiricism, i.e. conscious verification, supposed to be the most authoritative methodology for making epistemological claims, in which case, isn't unconsciousness an inadmissible concept? Or are we supposed to put faith in pure reason here and unquestionably accept the testimony of others?

One is tempted to say that one can only deduce one's state of unconsciousness in retrospect via a rational reconstruction of what happened in the past. This conclusion ought to encourage a focusing of attention on the meaning of retrospection, including the nature and existence of the past itself , a topic which impinges upon debates concerning the nature and existence of time, space and causation.

At the very least, thinking about sleep and unconsciousness illuminates the conceptual inter-dependencies of philosophies of consciousness with philosophies of time and philosophies of causation, in which the debate between realism and idealism is ever present. Sadly, most neuroscientists appear not to grasp the conceptual scope of their investigations and instead derive trite and unenlightened conclusions.

The myth of state secularism is but a special case of the is/ought fallacy, for there isn't an objective basis for ethics, including the ethical matters of law and government policy.

To my faint recollection, Hume never denied the ability to associate observations, and neither did he deny the imperative mood. By my understanding, his remarks are only suggestive of scepticism that causal and logical necessity are objective properties of objects. i.e he would have accepted anti-realistic understandings of logic and causation, especially those that make no commitment to synthetic a priori propositions, such as Hacker's interpretation of Wittgenstein.

So only in the context of infinite experiments we could say something is truly random? — Haglund

If you only accept the existence of potential infinity then you believe that every process, whether real or mathematical, eventually terminates after a finite amount of time. In which case lawfulness and lawlessness are disrobed of their metaphysical status as distinguishable properties attributable to things in themselves.

E.g we might say that the eventually terminating process {1,2,3, ...} begins 'lawfully' for the first three elements and then continues unlawfully for an unspecified amount of time. Which is only to say that it looks initially similar to another well-known sequence, such as {1,2,3,4,5,6,7,8,9,10}, before continuing in either an unspecified fashion or in a fashion that is expected to look unfamiliar to the average person, before eventually halting.


To take a physical example, take the 'law' that every electron has the same mass. Obviously one cannot measure every electron and so the law isn't empirically verifiable. Following the positivism of Karl Popper, we might either regard "every electron has the same mass" as being as a norm of linguistic convention that is held true no matter transpires in the future, or we might regard it as being a semi-decidable empirical proposition that is falsifiable but not verifiable. What positivism took for granted is the view that the semantics of logic and language is absolutely knowable a priori without being contigent upon and epistemically restricted by the available empirical evidence, even as the world referred to by language and logic is accepted as being absolutely unknowable and empirically contingent.

But suppose we grant that the meaning of thought and language is as uncertain and as empirically contingent as the world to which it refers and instead treat logic, language and world on the same epistemological and ontological footing. Then not only will we be unable to empirically verify that every electron has the same mass, but we will also be unable to empirically establish the meaning of the word "every" in the proposition "every electron has the same mass" - for the word "every" can only be given a definite empirical interpretation in cases involving explicit enumeration over a finite number of entities, but in the present case we have no idea as to how many electrons there. Therefore the meaning of "every electron" is indefinite unless electrons are said to have the same mass by definition, in which case we have a 'law' of convention rather than of matters of fact.

In summary, if the rules of logic are treated as being empirically contingent and epistemically bound by the same principles of verification as the theories of the world they are used to formulate, then the empirical meaning of "infinite quantification" can only be interpreted as meaning 'indefinite finite quantification' as opposed to meaning 'greater than finite quantification' In which case, the question "does every electron have the same mass?" is equivalent to asking "does an undefined number of electrons have the same mass?" which falls short of describing or being supportive of the metaphysical ideas of lawfulness/lawlessness which become superfluous and cannot gain empirical footing.

On the other hand, if the rules of logic are understood to embody norms of conduct and linguistic representation, as opposed to embodying empirical contingencies, then the rules of logic do express lawfulness, namely the conduct and ethics of the logician.

It is the simultaneous presence of both types of meaning in science and logic that leads to the normative notion of "law" being misconstrued as an empirical matter of fact.

If the sequence is random, no such function exists. Each outcome (B or R) is not determined by a function. Isn't that the definition of a sequence of random choices? — Haglund

Yes, according to the classical understanding of randomness, or rather we should say unlawfulness - in mathematical logic "randomness" tends to refer to algorithmic randomness that refers to the compressibility of a definable sequence such as Chaitin's Constant , whereas we are referring to the common understanding of "randomness" that indicates that the values of a sequence are being produced in an algorithmically unspecified manner -something that the intuitionists call unlawfulness, which includes sequences generated step-wise by repeatedly tossing a coin, or through the 'free willed' choices of the mathematician. For intuitionists, the distinction between lawfulness and lawlessness is practical rather than metaphysical or ontological - if a sequence is generated by using an algorithm, then it is can be said to be "lawful" in pretty much the same way that a good citizen might be said to be "law abiding" - neither of these examples appeals to the metaphysical notions of logical or causal necessity.

What you describe is the classical way of thinking that upholds the traditional philosophical dichotomy between lawfulness versus lawlessness . The classical ontological distinction between a lawless sequence versus a lawful sequence begs the existence of absolute infinity in order to conclude that a 'completed' infinite extension is possible that can be subsequently tested as to whether it corresponds to a definable function. But if the empirically meaningless notion of absolute infinity is rejected for the empirically meaningful weaker notion of potential infinity in which infinite sequences are understood to refer to unfinished sequences of a priori unknown finite length, then the previous conclusion can no longer be considered as meaningful and consequently there is no longer an ontological distinction between unlawful versus lawless processes; all we we can only speak of are similarities between finite observed portions of two or more processes that haven't thus far finished.

That every choice is based on pure chance? If you assess a finite sequence, BRRBRBRBRRBBRRBRB... (which probably ain't random since I typed it right now) and you find a program leading to this sequence, but can this be done with every sequence? Say that I base my choice on the throwing of a coin. Taking the non-ideal character of the dice into consideration and throwing it randomly (by making random movements). Will there always be a function a pattern, beneath the sequence? Is there non-randomness involved? If the underlying mechanism is deterministic, and we're able in principle, to predict an R or a B, can't we say the initial states of the throws are random? — Haglund

When repeatedly tossing a coin ad infinitum, at any given time one has only generated a finite number of outcomes that is always identical to the prefix of some definable function. A classicist is tempted to speculate " this infinite process if continued ad infinitum might eventually contradict every definable binary total function, and hence be lawless", but such speculation isn't testable as it again begs the transcendental idea of a completed infinity of throws only with respect to which lawfulness and lawlessness become ontologically distinguishable.

How about the genetic code? That determines outcomes, does it not? — Wayfarer

When we say that genotypes 'determine' phenotypes we are implictly referring to a class of situations that we recognize as bringing about this determination via the empirical contingencies of nature that we are unable to fully describe, control or predict. And so we are not appealing to causal or logical necessity when we recognise this determination, and are only appealing to our expectations of nature with respect to this recognisable class of situations. We could have alternatively said that genotypes 'miraculously' produce phenotypes with respect to such situations that bring about the magic of nature.

The ontological distinction between miracles and mechanics begs the principle of sufficient reason, which is but another form of absolute infinity in disguise.

Not sure I understand. Why shouldn't determinism be meaningless in such a universe? I understand that from the outside of such a universe all the events in that universe can be known. If you are part of it, your being in it prohibits knowing all happenings, that's clear. But while in it you can still say there is determinism. Without actually knowing what's determined. — Haglund

To put in another way, I am basically arguing that determinism and non-determinism aren't descriptive of phenomena and therefore shouldn't be considered as being applicable to reality considered in itself. Determinism and non-determinism are descriptive of theories and beliefs concerning the consequences of hypothetical actions, but these concepts are not descriptive of phenomena.

For instance, suppose that if Bob (P)resses a button, then it either results in a (B)lue light or a (R)ed light: P --> B OR R. Then we might say this theory is "non-deterministic". Equivalently, we could drop philosophical nomenclature for logic , and simply state that our hypothesis is a product in a suitable category.

But notice that the above theory is simply stating that if Bob presses a button, then one of two possible outcomes are expected. It isn't describing observations of the actual world:

For suppose that Bob presses the button a potentially infinite number of times, and this results in a potentially infinite sequence of 'random' outcomes, {B, R, B, B, R, R, ...}. The previously observed outcomes can be vaguely summarised as 'possibilities' using the co-product, yet there is no objective test for a random process; for at any time t, the sequence of lights generated so far is always describable by some computable function, and at any time t, any previously assumed computable hypothesis about the generating process of the lights might be falsified. Therefore as far as phenomena are concerned, there is no discernable distinction between a deterministic process and a non-deterministic process.

So we have at most a concept of epistemic uncertainty at play. But I don't see anything in the above that refers to the actual world; .

If the universe is assumed to be causally closed and contain a finitely bounded amount of information, then both determinancy and indeterminacy can be rejected as meaningless concepts on the grounds that neither concept can say anything normative or descriptive about a universe that is considered to be a complete dataset.

Asking whether or not a finitely bounded universe is deterministic or not is like asking whether J R Tolkien's world of Middle Earth is deterministic or not. The question only makes sense relative to some conception of the transcendental, relative to which the world in question can be regarded as incomplete. In the case of the complete works of Middle Earth, the applicable transcendental concept would be the author J R Tolkien , whom when considered from an external perspective transcedendental to Middle Earth can be said to have determined the events of Middle Earth. But when Tolkien and his books are considered together as a complete joint system, the question is again meaningless.

This is more or less the same observation that Bertrand Russell made when he commented to the effect that the concept of cause and effect adds nothing to the joint description of the motions of the stars and planets.

Causality is really a means of talking about experimental interventions, in which the actions of an experimenter, i.e the 'causes', are considered to be 'transcendental interventions' with respect to the experiment he is performing. Causality is therefore a "metalogical" concept rather than a logical concept, when we consider a logical system to be finite number of axioms with finite proof lengths that are self-contained.

Science therefore doesn't need causality per se, but only the concepts of internal versus external reasoning relative to the theories in question, plus a notion of material implication as provided by relevance logic.

The modern conception of logic (as represented by categorical quantum linear logic) is interactive and game theoretic (e.g quantum, linear), where the role of logic isn't to determine or even to predict the outcomes of experiments (which amounts to superstition and fortune-telling), but merely to define protocols of scientific investigation and to document the outcomes of such investigations.

In game-theoretic fashion, the material implication A --> B is weakly interpreted to refer to some process of interaction between an observer and his environment, a process that in general is vaguely understood and unreliable. For example, 'A' might stand for a message that Alice sends to Bob, and 'B' might stand for a response that Alice expects to receive from Bob in return. Understood this way, logical implication represents an expected or intented dialogue between interacting entities, rather than representing epistemic certainty with respect to a supernaturally infallible process. The role of the modern logician is thus akin to the role of a tennis umpire, who adjudicates and documents the conduct of interacting actors, whilst remaining agnostic with respect to the outcome of the game.

I suggest reading about Linear Logic, that greatly clarifies and narrows the distinction between logical, causal and modal necessity, even if causality isn't directly discussed in the majority of articles on the topic.

Modern philosophical confusions about the relationship of logic and causality are largely due to the fallacy of 'material implication' - a classically valid mathematical rule of inference adopted by Frege and Russell that is inadmissible for casual reasoning.

According to 'material implication', the hypothesis, rule or law A --> B is equivalent to a data-set of the form NOT A OR B:

A --> B <---> (NOT A OR B) ,

where (NOT A OR B) refers to elements of the set {( A = FALSE, B = FALSE), (A = FALSE, B = TRUE), (A = TRUE, B = TRUE))

Common sense should instantly recognise this rule as unreasonable, made worse by the the fact that (NOT A OR B) OR (NOT B OR A) is a tautology, which if material implication is accepted implies that
A --> B OR B --> A is true, i.e. that for any event types A or B, either A must cause B OR B must cause A.

In Classical logic, material implication is true as a result accepting the law of excluded middle. On the other hand, intuitionistic logic that rejects LOM thereby rejects material implication, whereby only the inference (NOT A OR B) --> (A --> B) is intuitionistically valid.

But if causal theories are supposed to summarise and describe our experimental interventions in the course of nature, then even the intuitionistically valid latter inference rule is inadmissible , considering the fact that even if (NOT A OR B) is observed, this doesn't necessarily imply that manipulating events of type A influences events of type B, since A type events might not be relevant to B type events. This leads us to so-called 'Relevance Logics', which includes linear logic, in which A --> B is interpreted to mean 'One A-type resource transforms into one B-type resource'.

In my opinion,

Metaphysical Solipsism : True by definition.
Methodological Solipsism : Unavoidable.
Psychological Solipsism : Dangerous and unhealthy, avoid at any cost.

The tasks which you suggest like gardening and caring for a pet aren't what I would call mindlessness, although the only one of them which I ever do is painting. I would say that painting is a form of mindfulness because it is about active attention, especially in relation to the experience of the senses. My biggest example of mindlessness would be going out and getting drunk. I have done it a few times to cope with stress and it involves blotting things out, especially emotional distress. — Jack Cummins

My problem with the term "mindfulness" , is that the term might be interpreted as selectively paying attention to, and thereby inadvertently feeding, preconceived cartesian notions of self/ego. To me the term intuitively implies self-monitoring, self-judgement and self-obsession, which can only feed self-consciousness, introspection and anxiety, and ultimately behavioural avoidance of anxiety provoking situations.

Of course, advocates might say "no, mindfulness is about passive observation and acceptance of the mind". But is the passive observation of the mind a valid concept? Doesn't the very act of paying attention to a thought create it? And how can one even choose to observe passively, given the fact that the very intention to be mindful is agenda-driven?

On the surface at least, mindfulness therapies, especially how they are marketed in consumerist contexts, seem to me like a denial of , or excuse to avoid, the socio-political reality that ultimately determines thought and behaviour.

Also, if a person denies the existence of the cartesian self, then what does 'mindfulness' amount to for that person with that understanding? For that person, doesn't the concept of "mindfulness" become broadened to the point of not excluding any mental or physical activity?

There is no "pausing" and "restarting", only a reference to subsequences leading toward a limit definition. — jgill

If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity?

I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown. Saying "Look at the syntax" doesn't answer the question. Watching how the syntax is used in demonstrative application reinforces the fact that "non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc.

The creation of numbers is a tensed process involving a past, a present (i.e. a pause), and only a potential future.

Simply:

Where do you find "eventually finitely bounded" in Brouwer, or even any secondary source, on potential infinity? Please cite a specific passage. — TonesInDeepFreeze

It is a logically equivalent interpretation of Brouwer's unfinishable choice sequences generated by a creating subject, and I have already presented my arguments in enough detail as to why it is better to think of potential infinity in that way.
.

You said that your claims about the notion of potential infinity are supported by the article about intuitionism. Now you're jumping to non-standard analysis. You would do better to learn one thing at a time. You alrady have too many serious misunderstandings of classical mathematics and of intuitionism that you need to fix before flitting off fro them. — TonesInDeepFreeze

.


Intuitionism is partially aligned with constructively acceptable versions of non-standard analysis. If you want an more authoritative but easy-read sketch, Read Martin Lof's "The Mathematics of Infinity" to see the influence Choice sequences have had on non standard extensions of type-theory (which still cannot fully characterise potential infinity due to relying exclusively on inductive, i.e. well-founded types.

You say, "in mathematics". The ordinary mathematical literature does not use 'absolute infinity' in the sense you do. As I've pointed out to you several times, 'absolute infinity' was a notion that Cantor had but is virtually unused since axiomatic set theory. And Cantor does not mean what you mean. So "in mathematics" should be written by you instead as "In my own personal view of mathematics, and using my own terminology, not related to standard terminology". Otherwise you set up a confusion between the known sense of 'absolute infinity' and your own personal sense of it. — TonesInDeepFreeze

Classical mathematics and Set theory conflate the notions of absolute with potential infinity, hence only the term "infinity" is required there. Not so in computer science, where a rigorous concept of potential infinity becomes needed, and where ZFC is discarded as junk.

Cantor does mean what i mean in so far that his position is embodied by the axioms of Zermelo set theory:

1) The Law of Excluded Middle is not only invalid, but false with respect to intuitionism.

2) The axiom of regularity (added in ZFC) prevents the formulation of unfinishable sets required for potential infinity.

For example, in {1,2,3 ... } where "..." refers to lazy evaluation, there is nothing wrong with substituting {1,2,3 ...} indefinitely for "..."

3) Choice Axioms obscure the distinction between intension and extension, whereupon no honest mathematician knows what is being asserted beyond fiat syntax when confronted with an unbounded quantifier.

3) The Axiom of Extensionality : According to absolute infinity, two functions with the same domain that agree on 'every' point in the domain must be the same function. Not so according to potential infinity, since it cannot be determined that two functions are the same given a potentially infinite amount of data .

We also have Markov's Principle: according to absolute infinity, an infinite binary process S must contain a 1 if it is contradictory that S is constantly zero, and hence MP is accepted. Not so according to potential infinity, due to the fact that 1 might never be realised. This principle is especially relevant with respect to Proof theory, since any proof by refutation must eventually terminate at some point, before knowing for certain whether an unrefuted statement is refutable. So unless we are a platonist who accepts absolute infinity, Markov's principle isn't admissible.

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.