The Wittgensteinian intuition regarding the identity sign, roughly put, and defended here by MU, is that it is the meta-logical expression of synonymy, which upon full analysis of the expression concerned, is eliminated to yield substitution operations among 'non equal' logical terms, each denoting distinguishable objects. This point of view has been shown to be as expressive and as consistent as the logical interpretation of identity, even if on occasion standard theorems of mathematics need amending.

Not quite. Not "For all Godel numbers p," but instead, for all x, x being any natural number. Some xs will be ps, most not. Them that aren't won't be (encode) a proof, and them that are also won't be a proof. — tim wood

Sorry, that was actually a typo. Nevertheless this exchange perhaps serves as a useful reminder that any number represents a legitimate theorem relative to the provability-predicate of some Godel-numbering system.

Furthermore, for any provability predicate Prov('X','S') interpreted as saying 'X' encodes a proof of 'S', it isn't actually knowable which numbers represent legitimate proofs, due the possibility that Peano-arthmetic is inconsistent and proves absurdity, together with Godel's second incompleteness theorem which forbids the possibility of PA representing it's own consistency. Neither humans nor God can ever claim knowledge of PA's consistency, for all we can have are proofs of consistency relative to the consistency of other systems, which begs the question.

Of course, it is standard practice to explicitly state consistency as an assumption when we informally interpret PA and discussing incompleteness, so your remark is valid, after a bit of clarification. My political agenda here is actually concerned with how to interpret PA without assuming consistency, in light of Godel's results. For there are arguably many conceptual benefits to be had by dropping the consistency assumption, that demonstrate the fundamentally empirically contigent, vague and indeterminate nature of logic and mathematics that is better understood as being a posteriori in nature.

Incompleteness is the result of unlimited universal quantification in Peano's axiom of induction, that takes us from the non-controversial constructive semantics of quantifier-free Primitive Recursive Arithmetic that represents the predictably terminating algorithms, to the controversial non-constructive interpretations of PA that represents every possible algorithm. Logic should therefore replace the simplistic sign of universal quantification for a richer collection of signs that distinguishes the different use-cases of the original sign, whilst also making explicit the relation of the Axiom of choice to universal quantification. David Hilbert in fact did use a system closer to this, called the epsilon-calculus during his attempts to prove consistency, and it is in fact making a come back.

And the universal quantifier is both part of the proof, and proved separately within the proof. — tim wood

Consider universal quantification over the negative-provability predicate in PA, that is informally interpreted as saying " For every 'S', 'S' doesn't prove 'G' ".

We know for a fact that this has no syntactical expression in PA, since if PA is consistent then the statement isn't decidable (via the Diagonalization lemma) and if PA is inconsistent then PA is unsound and has no interpretation whatsoever. We also know that for any particular 'S' and 'G', PA can decide whether or not 'S' derives 'G'.

Therefore the universal quantifier above cannot have the ordinary meaning of every, which ought to have been the central conclusion of Godel's incompleteness theorem. In stark contrast to the commonly accepted idea that Godels' sentence is self-referential with definite meaning.

So we have two choices. Either we stick to our a priori philosophical concepts and abandon mathematical logic and it's physical embodiment as being an unsuitable language for expressing and justifying philosophical truth. Or we revise our philosophical intuitions to match what PA and its physical embodiment can and cannot express. I'm saying the latter.

I read this as your taking exception to the use of existential (i.e., existential and universal) quantifiers unless it/they "abbreviate an independent proof of the fact concerned...". So, if I say x>3 is true for all x greater than three, this is neither true nor valid, subject to a proof of the "for all"? — tim wood

This is acceptable, because the contradiction it expresses can be represented without universal quantifiers. It is analogous to having a function that by design terminates, and that converts an arbitrary number into a number or another arbitrary number. the conflation of all with arbitrary is perhaps the central source of controversy and misunderstanding in logic.

If "the present" refers to the specific context in which it is used,i.e. it is an indexical, referring to different things on each and every occasion that it is uttered, then to speak of the present as 'changing', is merely to point out that we can remember using the words "the present" differently. The idea of a "changing present" might be eliminated if we instead uttered unique indexicals in place of it on each and every occasion.

On the other hand, whereas we ordinarily speak of the "the present itself as changing", as if "the present" was a rigid designator, for some reason we tend to merely think that our knowledge and remembrances of an immutable past has changed, which indicates that we tend think of "the past" as partly an indexical in relation to our present state of knowledge and remembrance, and partly a rigid-designator referring to an immutable and transcendental temporal object.

Now the main point of contention here, as i see it, is whether or not the concept of the past deflates to our interaction with "present" appearances, including memories. If it does, then we can eliminate "the past" in the sense of an immutable entity that transcends phenomena, and as with 'the 'changing present', we would merely be grammatically wrong to speak of the "the past" as changing.

↪simeI have to refer you to the proof itself. It relies on recursion and ω-consistency. I quote this: "Every ω-consistent system is obviously also consistent. However, as will be shown later, the converse does not hold." (The Undecidable, Ed. Martin Davis, 1965, p. 24).

PA may be inconsistent, but I take Godel's qualifications on his system P, of which he says, "P is essentially the system which one obtains by building the logic of PM around Peano's axioms.." (10), & ff., as sufficient to regard his claims as rigorous. As he observes later (p. 36), "In particular, the consistency of P is unprovable in P, assuming P is consistent (in the contrary case, of course, every statement is provable).

should not be informally interpreted as saying "n is not a proof of S for n < ω".
.... and this should not feel in the least bit troubling or surprising , for there is no formal justification to support the heuristic and informal interpretation of universal-quantifiers as denoting each and every member of an infinite domain , unless that is to say, the universal quantifier in question was constructed using the axiom of induction.
— sime

Agreed. Informal remarks about his proof are often not quite right. But I think it's pretty clear that the axiom of induction or something like is a main piece of his proof. — tim wood

Peano's axiom of induction isn't an axiom of logic, and plays no part in standard proofs of Godel's incompleteness theorem. It can however be shown to be responsible for causing incompleteness, in the presence of the other axioms of peano arithmetic.

The reason this matters, is because in logic the use of an existential quantifier should not be informally interpreted as bearing witness to a fact, unless the quantifier is used to abbreviate an independent proof of the fact concerned that does not beg the use of this quantifier in a circular fashion.

For otherwise we might just as well say that "a universal quantifier has proved a universal truth, because the universal quantifier says so".

The universal quantifier in "Godel's sentence" G which supposedly says "For all Godel numbers p, p does not encode a proof of Godel sentence G", isn't an abbreviation of an independent proof of G's non-provability. Therefore this quantifier should not be given this common (mis)interpretation.

Certainly there are similarities between the 'seven deadly sins' and symptoms of mental illness. Was the early christian concept of 'sin' more pragmatic than the modern christian concept?

Because a large percentage of NDEs and pre-death or death-bed visions are interactions with the deceased. Therefore, one can conclude based not only on this, but given all the other points that have been made, that we are much more than simply this body (the brain, etc). — Sam26

I don't follow. All that exists are verbal reports and their proximal neurological correlates that are roughly clustered, as is expected by the shared structure of human brains that learn and interact within a shared culture.

These observations tend to reinforce the scientific usefulness of the Cartesian perspective, that mental states should be considered local, discrete and internal to each and every human brain.

Certainly there are drawbacks and oversights of the standard Cartesian view, but I don't seem them as being intimately connected to the phenomena of NDEs.

There is much to be said regarding interactionist conceptions of personal-identity that are holistic and non-reducible to particular events or entities, and yet I still fail to see the relevance of NDEs in relation to the question of mortality, even if NDEs violated our existing understanding of causal relations by producing verifiable paranormal results:

For according to standard type-physicalism, NDEs are by definition types of mental events that correspond to types of biological functioning and so NDEs cannot imply anything transcendent of biological functioning, even if paranormal claims were verifiable.

Conversely, if one's conceptualization of personal identity is sufficiently fuzzy , then immortality is assured by definition, even if NDEs fail to produce verifiable paranormal results.

Whatever a person's beliefs are regarding subjective continuity, I cannot see how those beliefs can be challenged or supported by evidence. For any belief concerning the relationship of the physical world to consciousness isn't an empirical hypothesis, but a definition of 'consciousness'.

" {|||||,|||||||} 'equals' {||||||||||||} " isn't visually acceptable to me. The left side looks too short.

In certain contexts, we might treat the expressions 5+7 and 12 identically, as for example when discussing what our shared formal convention says concerning our customs of numeric substitution. Yet in mathematics applications, these expressions cannot be treated the same, as simply demonstrated by children who must sum with their fingers.

It is wrong to insist that our mathematical convention grounds the meaning and truth of mathematics in application. For instance, it is incorrect to claim that "the conventions of logic a priori determine that 5+7=12, whereas the physical calculation merely confirms it".

Consider, for example the addition of two summands that are so large that their summation cannot be precisely determined in any individual physical experiment, let alone by hand. Here there isn't a clear distinction between the truth of the summation according to convention, versus the confirmation of the summation bu physical demonstration.

And in visual psychology, it should not be regarded as an error if a test subject reports that he saw 5+7 as 13. It simply means that visual phenomena are not a good model of ordinary arithmetic and vice versa.

Sure it is: 1 is not a proof, 2 is not a proof,..., n+1 is not a proof,... The limitation is against going on into the transfinite. I'm not looking at the moment - subject to correction - but I think the relevant qualification is call ω-consistency (omega-consistency) — tim wood

It is precisely because Peano arithmetic might be ω-inconsistent that Godel's constructed sentence should not be informally interpreted as saying "n is not a proof of S for n < ω".

.... and this should not feel in the least bit troubling or surprising , for there is no formal justification to support the heuristic and informal interpretation of universal-quantifiers as denoting each and every member of an infinite domain , unless that is to say, the universal quantifier in question was constructed using the axiom of induction.

"If I take this anesthetic, then I will lose consciousness."

When said by a third party, this is an empirically verifiable and empirically contingent proposition - for the loss of consciousness in a third party has a behavioral definition.

But what about in my own case when I take the drug? Here, I do not have a behavioral definition for loss of my own consciousness. All i can have in this case, is a wakeful sense of amnesia in relation to memories I have of previously ingesting the anesthetic.

Here, to refer to my sense of amnesia as being equivalent to an earlier loss of my own consciousness would be a tautology.

In another example, there is the canonical witness for Gödel's first incompleteness theorem:


S = "S is not provable in T"


It is a witness for the theorem that says that there exist expressions in the language of first-order logic that are not provable from any possible choice of axioms (phrased in the same language).

S happens to be formally not provable from T. Well, that is what S says. So, in a sense, S is even logically true. — alcontali

Certainly, if S is such that S implies ~S, then an axiomatic derivation of S entails inconsistency. And Godel exhibited an arithmetical proposition of this form. But here, S should not be formally interpreted as a referring to its lack of provability. For this heuristic interpretation of S is terribly misleading, because the universal quantifier in S that supposedly ranges over the godel numbers of every proof, isn't obtained by mathematical induction. Therefore it's interpretation as referring to every proof isn't justified.

The senses in which 1+1=2 is said to be an analytic sentence or tautology, are not the same as the senses in which 1+1=2 is said to be an empirically contingent proposition.

What estimating feels like.

Well, I suppose if one can't coherently suggest that something might be wrong or mistaken. I'm not sure what that could be. You say 'logically speaking', but even the laws of logic can't be assumed against the commonsense argument. — Paralogism

Formality makes no difference here. The soundness of an argument refers to external validation.

What are permissible criteria for either defining or testing a mistake in the reasoning process?

Logically speaking, mustn't such criteria be considered as being at least partially-independent of the reasoning process, as for example when we consult a calculator?

So if you acknowledge the possibility that your reasoning could be mistaken, then haven't you already begged the existence of at least a partially independent world ?

It seems to me that no matter how bad or stupid a politician or idea is, the BBC can be trusted to normalize and promote it in the name of journalistic objectivity.

Assume I've given all the relevant information, for the sake of discussion. — bert1

sure, - of course my point is to say that it is impossible to given an exhaustive account of all the potentially relevant information that necessitates an action; the consequence being that free-will and determinism cannot be absolute polarities but are terms used for the relative comparison of two or more specific situations.

In the scenario, I am very hungry and want one of the cakes. I have a choice whether to eat a cake or not (according to street). The deliberation involves feelings of hunger and desire (nothing else). I eat one of the cakes. Not eating one of the cakes in this circumstance would involve other factors which I have not given (i.e. madness, cream allergy, diabetes, obesity, hallucination etc). My choice to eat one of the cakes is highly determined.

However I really don't mind at all which cake I eat. I make a choice and eat the jam doughnut. The question is, is this decision determined or not? I don't think it is. I think it is a free arbitrary choice. Is this even possible do you think? It's logically possible. Is it metaphysically possible? Physically possible? Psychologically possible? Or is there always a determinant? — bert1

At least according to the compatibilist logic of Hume, a 'free willed' action must at least be superficially describable as being 'caused' by one's will; the charge of determinism being avoided, by understanding the concept of necessity as referring to states of psychological compulsion as opposed to causal relations per se that merely comprise of an inferential attitude in relation to a previously observed conjunction of events.

Another source of compatibilism, as i'm trying to point out, lies in the indeterminacy of the very meaning of determinism, as contradictory as that might sound.

Regarding vagueness, indulge me with this idealised scenario, which I grant might be impossible to actually exist. Just as the non-existence of perfect circles does not stop us calculating using assumptions of perfect circles when designing machines, I want to contrast the concepts of free and determined choice by using an idealised scenario. — bert1

Isn't the concept of the perfect circle also relative to the situation? 3.14159 being a 'more' perfect description than 3.141 regarding the circumference of the circle on my monitor?

Analogously, is there a notion of perfect determination that applies in every case?

From my perspective, by linguistic convention I should at least say that "you have a choice as to what you eat" with respect to the imprecise situation you put forward.

But is what I am saying a claim about yourself, or is what I am saying a mere figure of speech that linguistic convention dictates to be a permissible description of the imprecise observable situation that you put forward?

Supposing you now reveal that you have an allergy for cream. Then i might now say "it appears that you don't have that much of a choice relative to my previous understanding, given your newly admitted allergy for cream"

The question is, does there exist an absolutely precise and exhaustively describable circumstance that you can describe, or that I can observe, under which I am at least permitted to say without fear of controversy, that you have absolutely no choice but to take one of the presented options?

And whatever I am permitted to say here, would this now be a claim about yourself, or again would it be merely a figure of speech in relation to the imprecisely defined concepts 'exhaustive' and 'absolute'?

Compare your question to other borderline questions that are in relation to vague concepts:

Is this adolescent an adult?
Are these grains of sand a heap?

Let's say I have a choice between having a chocolate eclair and a jam doughnut and I don't mind which. However I somehow manage to choose one. Is my choice determined or free? — bert1

Ordinarily, freedom refers to an availability of alternative courses of action, in relation to a partial specification of the influences bearing upon a decision making process.

Conversely, determination refers to a causal or logical relation within a partially specified context.

These concepts are therefore compatible, in virtue of them being under-determined. Their use in any given situation is analogous to describing a glass of water as being half-full and half-empty.

So basically you are questioning the meaning , ontological status and normative value of counterfactual propositions?

Stated this way, we can hopefully avoid at least some of the circularity and vagueness concerning the meaning of free-will (or determinism).

When asking counterfactual questions like "What if Hitler had won the war?", there are philosophers who regard such questions as referring not to our past, but to either the potential outcomes of potential new experiments here or elsewhere, or to the histories of other potentially existing Earth-like planets whose circumstances are sufficiently identical to ours to be considered suitably analogous to answer such questions.

To me, this interpretation of counterfactuals seems to avoid the main epistemological concern of 'world-intervention' skepticism - that the past could merely be a story in which we play our part. For even if the past is merely a story, counterfactual inferences under the potential outcomes interpretation presumably remain viable, assuming that one isn't also skeptical of induction. For our story might be a-causal, but who is to say that it cannot follow a pattern?

Formally speaking, actual infinity merely refers to the axiom-of-infinity. But this isn't a good answer, because this does not account for the controversial motives for introducing the axiom.

The underlying dilemma is the result of different interpretations of the informal sign "..." used to denote partially elicited sets, and how these different interpretations lead to different conclusions concerning the very meaning of a set, including what sort of sets are admissible in mathematics.

Ordinarily, in statements such as {0,1,2,3,...}, the sign "..." is used to state that the "set" refers to a rule (as in this case, the rule of adding one and starting from zero), as opposed to an actually completed and existent body of entities. This is synonymous with potential infinity, that appeals to one's temporal intuitions regarding a process whose state is incremented over time.

In other cases such as "my shopping list is {Chicken,wine,orange juice,...} ", the dots might denote either

i) an abbreviation for a particular, finitely describable list that is already existent, but only partially described on paper

or

ii) An indication that a list is abstract and only partially specified, that the reader is invited to actualize for himself via substituting his own items, or rule of extension.

or

iii) a mystical sign, referring to "actual infinity" in a sense that is empirically meaningless, physically useless and logically a mere piece of syntax, but which nevertheless has psychological value in causing giddy vertigo-like sensations in true-believers when they contemplate the unfathomable.

Unfortunately, because "..." is informal notation with at least three completely distinct operational uses in addition to having private psychological uses, people continue to conflate all of these uses of the dots, causing widespread bewilderment, philosophical speculation and moral panic up to the present day.

The definition:

n>1: n! = n * (n-1)!
n=1: n! = 1

is indeed somehow circular, but that is the essence of recursion. It works absolutely fine. Wittgenstein does not seem to handle that. — alcontali

Wittgenstein is rather attacking the heuristic semantic notion of "self reference" in relation to the iterative evaluation of a sequence of expressions via recursive substitution. Unless the iteration eventually halts, the resulting sequence isn't even sentence, never mind a proposition. Yet if the iteration is halted, each resulting sub-expression has non-equivalent arguments.

No, I think it is clearly not consistent. But the distinction between past and future is obviously "the present" — Metaphysician Undercover

From SEP
"
McTaggart distinguished two ways of ordering events or positions in time. First, they might be ordered by the relation of earlier than. This ordering gives us a series, which McTaggart calls the B-series. A second ordering is imposed by designating some moment within the B-series as the present moment. This second ordering gives us a series that McTaggart calls the A-series. According to McTaggart, in order for time to be real both series must exist,although McTaggart holds that, in some sense, the A-series is more fundamental than the B-series."

Yet aren't "the past", "the future", "the present" etc, indexicals that refer to different things on each occasion?

Supposing that each of us always carried a mobile phone and that we agreed to eliminate "the present", "now", " currently" etc. from public discourse by replacing each of their uses with the exact current reading of the International Atomic Time supplemented with the Gregorian calendar. Likewise, we respectively do the same for "the past" and "the future" by replacing their use with time-intervals that are before or after the exact current TAI time.

Doesn't this elimination of temporal indexicals also eliminate all talk of change, and therefore reduce MacTaggart's A series to his B series?

I would be interested in knowing more about Ayer's rejection of memory as a means of distinguishing between past and future. Could you elaborate, or cite a reference?

It seems to me that experience (which happens in the present) is more than capable of distinguishing between before and after (e.g., cause and effect), and designating the measurable change: time (per Aristotle). — Galuchat

I'm not particularly knowledgeable about Ayer's particular ontological views regarding the relationship between memory, phenomena and time, and I am certain that Ayer, like all of us, had no problem acknowledging the practical role that memory serves as (unreliable) testimony to the truth of past-contingent propositions- i'm only referring to his general acknowledgement that the doctrines of logical positivism and verificationism failed -see for instance his interview with Bryan Magee. We still do not possess a theory spelling out what we mean by meaning, evidence and truth, especially in relation to past-contingent propositions for which there cannot exist direct observation or immediate testimony:


Is it logically consistent to be an empiricist who accepts a hard ontological distinction between past and future?

Is the semantic distinction between the past and future somehow reducible to appearances or to relations between appearances, or to potential appearances as a function of potential experiments?

How should physics and computer science categorize "future-directed" behavior in humans and other agents?

How can this be reconciled with the causal theory of reference which identifies the meaning of an utterance with it's causes?

the philosophically enlightening thing about the craft of software engineering is it's cut-throat pragmatism that makes explicit what mathematics and logic are really about, such that the disease of philosophical speculation cannot take hold. There aren't any software engineers debating whether an infinite loop really runs forever, unlike a significant proportion of set-theorists, who as a result of refusing to get their hands dirty in practical application, end up associating mathematical infinity with the religious idea of eternity.

It is certainly true, that from a pure meaning-as-use perspective the distinction between the past and the future is much harder to distinguish than it is from an axiomatic meaning-as-reference perspective (which effectively insists upon an a-priori and axiomatic past-future distinction).

We also anticipate both the future (e.g is this oasis I see a mirage?), as well as the past (e.g. will my current archaeological dig verify the massacre that allegedly took place here in 1942?). Of course in hindsight, yesteryear's predictions that supposedly refer to today are now seen retrospectively as mere instances of retro-futurism that in actuality only ever referred to what occurred when yesteryears so-called "prediction" was made ( how can yesterday's predictions even be wrong?)

We cannot definition-ally distinguish past-contingent propositions from future-contingent propositions on the basis of experiential content, unless we are prepared to bite the bullet and call a certain appearance "the past", such as the contents of a memory or photograph. But once we reject this as a mistake, as did Ayer, we realize we are then unable to provide an experiential distinction between past and future, even while we continue to insist on it.

There is of course, a big difference between an eaten Hamburger and a Hamburger sitting in front of us; if an object is called 'destroyed', then there does not exist a direct and local reference to the object that we can point at. There is instead a potentially infinite and interlinked fabric of facts called "the evidence of the destroyed object" together with our investigatory sense of anticipation. Hence an empiricist might be able to equate the past with our current sense of inferential expectation together with today's appearances taken holistically as an inseparably entangled whole. But this of course is too vague to constitute an empirical "theory" of any description.

Nevertheless, at least we can still speak of our expectations as being fulfilled, as for instance when walking up a hill to inspect the view, or when digging in the earth for relics. We can also partially order our historical knowledge in such a way as to minimize the statistical dependence of the occurrence of so-called "earlier" events on the occurrence of so-called "later" events. Perhaps it is possible to go neo-Kantian and argue that today's perceptual judgments necessitate an axiomatic past-future distinction in order to speak of "types" of objects and events. I don't know about this though.

In my view, a metaphysical assertion is meta-cognitive speech-act whose intention is to influence perception, behavior and values, via a wholesale change of view. A Metaphysical debate is the result of differing views or values being simultaneously incompatible, typically with one party being unable to grasp the sense or value of the other party's viewpoint.

I see metaphysical assertions as value-apt, but not truth-apt in a representational sense. After all, to a certain extent what is at stake is the method of representation, which in turn is decided according to what it is considered to be worth representing.

In my opinion, neither type-theory nor category theory are satisfactory in addressing the ambiguity of standard logic regarding infinitary notions.

Take as an example "My local postbox contains some letters". This proposition isn't a full specification of a set, and hence is not representable as a specific algorithm and hence cannot be constructed, and therefore it must be stated axiomatically as a particular set which exists but without having any internal details. Some people might already interpret set theory, type theory or category theory in this way but this isn't thanks to those theories themselves.

- The Axiom of Choice is wrongheaded in two respects:

i) It isn't a logical rule permitting the universal existence of non-constructable sets. Rather, it is used as a non-logical proposition to refer to a specific set in a domain of inquiry whose number of elements is potentially infinite, that is to say, isn't specified in logical description.

ii) The axiom has nothing to do with choice. On the contrary, it is usually used as a reference to sets for which a choice-function isn't specified.

Whereas classical logicians often mistake AOC for a logical rule, Constructive logicians often mistake the Axiom of Choice for a tautology; for they conflate the existence of a choice-function with the existence of a set; yes it is true that 'constructed set', 'computable set' and 'choice function' are synonyms - but this misses the point as to how the axiom of choice is used, namely in order to represent 'externally provided sets' that provide elements on demand, but whose mechanism and quantity of contents are unspecified.

To nevertheless insist that every physical set must be constituted by a computationally describable mechanism, should be regarded as a physical thesis, rather than being a logical theorem. For logic is a purely descriptive language without physical implications. Physicists might use set theory as a means to document their epistemological certainty and uncertainty regarding the internal operation of physically important sets, but this epistemological interpretation of set-theory isn't part of set theory per-se.

The correct semantics of logic and mathematics is game semantics describing the interaction of entities created and controlled by the logician on the one hand, and the entities of his external environment that he is aware of, but for which he has only partial control or specification of. This semantics serves to reinforce the notion that logic and mathematics are languages of empirical description that documents the interactions of the logician with his external world.

In my opinion,

Metaphysics refers to the conventions of language-games that seem to lack a definite or well-understood collective purpose, or when used disparagingly, to a convention that is believed to be unhelpful with respect to some assumed purpose of the language game.

In my view, I think we first have to distinguish two possible interpretations of the assertion 'induction is habit', namely an empirical interpretation versus a grammatical interpretation.

The empirical interpretation is to view the idea of induction as being distinct from the idea of repetition, whereby 'induction is habit' is viewed as an a posteriori empirically contingent assertion correlating two distinct ideas, say, the internal mental state of expectation and its association to external observations of repetition. As you point out, this interpretation appears to be self-undermining, since according to Hume no empirically contingent proposition can have universal justification, which in this case can lead to semantic skepticism concerning the very meaning of induction.

The grammatical interpretation is that induction is defined directly in terms of observed repetition. In which case, 'induction is habit' is a deflationist assertion, i.e an analytic a priori definition of induction without empirical implications, as opposed to being an empirically contingent assertion. In which case semantic skepticism regarding the meaning of induction might be considered to have been circumvented, assuming one is happy to accept the notion of repetition as being an empirical notion that can ground the rational idea of induction without presuming it's existence. Alternatively, if one rejects the idea that repetition is an empirical notion, one might instead be willing to accept the converse, namely that idea that induction is a directly observable mental state (at least in terms of an inward experience of anticipation) and that it is this 'experience of induction' that serves to ground the idea of repetition within the faculty of reason .

Yet however complicated the phenomenology of induction is, presumably we can at least draw the conclusion that induction doesn't have a purely empirical justification nor a purely rational justification.

Unfortunately neither the empirical nor the grammatical interpretation fits with either our counterfactual intuitions concerning induction; For if a cause of type 'A' is understood to entail an effect of type 'B' when situated within a background context 'C', then an absence of 'A' implies the absence of 'B' whenever 'C' remains the same as in the former case. And whilst there are many cases in which we directly observe frequencies of conjoined absence, we mostly appeal to counterfactual reasoning rather than to observation to infer the consequences of absent causes (e.g. what doesn't happen when a nuclear bomb fails to detonate).

At the very least, we cannot argue for causation without appealing to circularity or to counterfactual situations embedded within vague background contexts, for which we cannot explicitly provide a constructive argument, and in which our observations and logic are inseparably interwoven.

Therefore it is coherent to accept mathematical induction as a principle of construction, and yet reject it's interpretation as a soothsayer of theoremhood. — sime

Sorry, that's confusing and misleading, by theoremhood i was referring to truth in the respective model of the axioms.

What i mean is that mathematical induction 'proves' a universally quantified formula in a completely vacuous sense in that the 'conclusion' of the induction, namely the universally quantified formula (x) P(x), is merely an abbreviation of the very axioms i and ii.

On the other hand, for universally quantified formulas over infinite domains that do not possess a proof by mathematical induction, there is no reason to support their informal interpretation as enumerating the truth or construction of the predicates they quantify over, by the very fact that they do not correspond to a rule of substitution.

For in what sense can a formal system speak of universal quantification over all of the natural numbers?
— sime

The standard way is assuming the principle of induction: if a proposition is true for n=0 and from the fact that is true for n you can prove that is true even for n+1, then you assume that is true for all natural numbers. (https://www.encyclopediaofmath.org/index.php/Induction_axiom)
It cannot be verified in a finite number of steps, so it's assumed as an axiom.
But assuming it to be false is not contradictory: you can assume the existence of numbers (inaccessible cardinals) that can never be reached by adding +1 an indefinite number of times (https://en.wikipedia.org/wiki/Inaccessible_cardinal). — Mephist

So then question is, in what sense does the principle of induction speak of universal quantification over all of the natural numbers?

Take any predicate P with a single argument, then take as axioms

i. P(0)
ii. (x) ( P(x) => P(x+1)) ,

where ii. denotes universal quantification over x, where x is a bound variable.

Then we say, "By informal convention, this predicate is now said to be 'true' for 'all' x."

But what exactly has our demonstration achieved in this extremely small number of steps?

Do we really want to insist that our use of these axioms is equivalent to a literal step-by-step substitution of every permissible number into P?

Do we even want to say that we are assuming the existence of this inexhaustible substitution?

All that i and ii define in a literal sense is a rule that permits the substitution of any number for x.

Nowhere does this definition imply that induction is a test for the truth of every substitution.

Therefore it is coherent to accept mathematical induction as a principle of construction, and yet reject it's interpretation as a soothsayer of theoremhood.

The main teething problem of data-science is the current absence of integrated causal modeling (let alone stakeholder legible causal modelling) which is a situation changing rapidly due to the ongoing invention of problem-domain-specific software simulations, along side the development of probabilistic programming libraries for quantifying simulation accuracy.

In this regard I expect that outside of computer-vision and speech recognition, where deep-learning as a strong inductive bias, the black-box neural network approaches in other problems domains will begin to take a back seat as transparent models of network logic come to the forefront. For you cannot easily inject human-readable logic clauses into a distributed neural network with positive and negative activations.

But our use of real numbers (at least for the most part) is in integrals and derivatives, right?
So the "dx" infinitesimals in integrals and derivatives should be the morally correct model? This is how real numbers are intuitively used since the XVII century. — Mephist

First of all, does our imprecise use of real numbers warrant a precise account?

What does it even mean to say that a real number denotes a precise quantity?

For in what sense can a formal system speak of universal quantification over all of the natural numbers?

Recall that in a computer program, an infinite loop is merely intended to be an imprecise specification of the number of actual iterations the program will later perform; it is only the a priori specified number of iterations that isn't upper-bounded in the logic of a program, that is to say before the program is executed.

Therefore if the syntax and semantics of a formal system were to mimic a programmer's use of infinity, it wouldn't conflate the 'set' of natural numbers with an a priori precise set, but would instead refer to this 'set' as an a priori and imprecise specification of an a posteriori finite construct whose future construction is generally unpredictable and isn't guaranteed to exist for a given number of iterations.

A universal quantifier over N would not longer be interpreted as representing a literally infinite loop over elements of N, but as representing a termininating loop over an arbitrary and unspecified finite number of elements, where the loop's eventual termination is guaranteed, even if only through external intervention that is outside of the constructive definition of the program or formal system.

By interpreting 'infinite' quantification as referring to an arbitrary finite number of elements, the law of double negation is then naturally rejected; for an arbitrarily terminated loop cannot conclusively prove anything in general.

This interpretation also has the philosophical advantage that semi-decidable functions are no longer informally misinterpreted as talking about their 'inability to halt', and likewise for formal systems supposedly 'talking about' their inability to prove certain statements.

So in short, an imprecise and pragmatic model of the real numbers is what is important, corresponding to how numerical programmers implement them in practice, with particular attention paid to numerical underflow.

Arguments for or against anti-natalism depend upon one's ontological beliefs, particularly with respect to personal identity, personal continuity and other-minds. Different positions on these topics will lead to radically different conclusions regarding anti-natalism.

the morally correct model of real numbers is the model most resembling our use of real numbers, i.e. the model that treats the real numbers as being subcountable, wherein the absence of a bijection between the natural numbers and the real numbers is interpreted to imply the following profundity:

There is an absence of a bijection between the natural numbers and the real numbers.

How is it possible for these dream characters that populate a dream world to have, seemingly, an intent of their own? Why would they? I mean, given that a dream world is practically tantamount to living in a solipsistic world, you would assume that you are the only person with a 'mind' within a dream. Yet, even in dreams where I am aware that I am dreaming (lucid dreams), I still find it hard to shake off the preprogrammed belief that the friend I meet, the father or mother I talk with, or the siblings I interact with in a dream have a mind of their own... — Wallows

In my lucid dreaming experiences, a dream character tends to lose their autonomy as soon as I demand them to act autonomously. For example, if I ask them a question in the hope of receiving a novel and autonomous answer they turn into a puppet and say nothing. This effect is presumably related to artist's block.

So if we replaced your brain with sawdust, you'd still be conscious? — Marchesk

For a phenomenologically minded empiricist, the very meaning of a scientific hypothesis is the sense in which experience is said to corroborate or refute the said hypothesis and this sense cannot be transcendent of experience. Therefore this empiricist is likely to reject your question as meaningless and inapplicable in the first-person.

Nevertheless, neuro-phenomenology can potentially have sense in the first-person, in terms of an association between sensory experiences and brain-probing-experiences, as for example in an experiment in which the subject records his experiences when probing his own brain.