The category theoretic description of the natural numbers does away with the pesky elements of the natural number "set" and externalises the notion of 'element' onto the objects counted.



https://en.wikipedia.org/wiki/Natural_numbers_object

And so the notion of elements isn't needed for the pure purpose of constructing abstract numbers.

Does that help?

..and no basis for calling it true. Reincarnation becomes a form of life that does not make contact with truth or falsehood. It's use - meaning - can only be in its social function. — Banno

Scientifically speaking, I agree, but of course, your argument also applies to the "you only live once" position, so it's a moot point.

But i don't agree that scientific truth and metaphysical truth are synonymous, due to the fact the latter directly concerns the logic of first-person experience, whereas the former is precluded from coming into contact with first-person experience due to the public semantics of scientific discourse , where identity relations are decided by public agreement with respect to propositions stateable in the third-person. Hence it is possible, imo, to accept the public meaningless of the scientific question, whilst accepting the private question to be metaphysically meaningful, and even potentially answerable in some philosophically critical sense.

To give a related example, if i awaken from a coma then I am said to have been "previously unconscious" by definition of the circumstances i am presently in, which includes such things as medical opinions i hear from loved ones around me, a brain-scan i am presented with showing an absence of critical neurological activity, and my self-observed tendency to abstain from memory recalling behavior (amnesia). As with the question of reincarnation, it is logical for me to ask "But was I really unconscious previously", or only in the tautological sense decided by public convention, where my "previous unconsciousness" is ironically decided by present observations that make no actual reference to a non-existence of first-person experience per-se?

Personal identity isn't a topic that science is able to investigate, because identity relations are part of logic and ontology rather than empirically deducible matters of fact, and science is compatible with any set of identity relations, provided they are consistent.

Given any assumed set of identity relations, science only has the power to decide whether or not a given observable process conforms to those relations. For example, if a caterpillar is defined as being identical to the resulting butterfly, then scientific experiments have the potential to confirm whether or not a given caterpillar is identical to a given butterfly. But the result can neither confirm nor deny the reality of the assumed identity relation.

There a multiple cultural and practical factors as to why western culture has converged onto an assumed set of identity relations that makes rebirth not merely physically impossible but logically impossible. I think part of the reason is that scientific theories are initially easier to understand relative to an atomic ontology, such as periodic tables and subatomic particles, than holistic process ontologies. This is also reflected in logic and mathematics, where most students find set theory with elements easier to understand than category theory without elements.

Given that intuitionists who reject the existence of actual infinity also reject the law of excluded middle, they are likely to disagree with the assumption that the universe must either be finite or infinite.

For the intuitionist, truth is synonymous with verification of some sort, meaning that according to this stance there are no unknowable true propositions. This implies that if it is unknowable in principle as to whether the universe is finite or infinite, then there is no transcendental matter-of-fact as to which is the case.

Also, it should be mentioned that the commonest use-case of potential infinity involves an agent querying nature for the value of an unbounded variable and accepting the received response (if any). Therefore it is false to claim that denial of actual infinity entails denial of external reality.

Personally, I am sympathetic with regards to beliefs in rebirth, due to logical reasons connected to the temporal philosophy of presentism that ontologically prioritises the present to the extent of rejecting the literal existence of the past. The implication is that memories aren't so much the recordings of bygone and static states of existence, but are part of the very meaning of what the past presently is.

Essentially by this view, the first-person subject is static and exists only "in a manner of speaking", with the concept of change applying only to presently observed things.

I personally have no problem with someone calling a single grain of sand a heap, even if that isn't my cup of tea. What harm could it do, and who am I to decree otherwise?

There is no a priori linguistic definition of "heap" in terms of any specific number of grains of sand,
— sime

Yes, that is the problem. — bongo fury

Why is linguistic imprecision a problem? "Heap" trades referential precision for flexibility, whilst retaining the necessary semantics for useful, albeit less precise communication.

The only theory of meaning Wittgenstein ever published was in the Tractatus, which was a solipsistic, subjectivist or idealistic doctrine of meaning that constituted conclusions he drew via methodological solipsism , that is to say by phenomenological investigation strictly in the first-person, that discounted the applicability and relevance of third-personal scientific rationalisization.

And the later Wittgenstein, whose solipsistic methodology remained the same as the earlier Wittgenstein and who now directly asserted that philosophy was purely therapeutic and descriptive and wasn't in the business of proposing theories, didn't immediately contradict himself by proposing the frankly ridiculous theory attributed to him that meaning is grounded in inter-subjective agreement or in some publicly obeyed rule-set sent decreed from above by the guardians of meaning in Platonia.

The confusion here, seem to partly stem from the public's lack of understanding of the positivistic epistemological ideas of his time that he was attacking, as well as a general lack of awareness regarding Wittgenstein's so-called "middle period", in which he wrote about his phenomenological inquiries and negative conclusions that there was no hope of obtaining a phenomenological theory of meaning of the sort proposed his earlier self proposed.

But that doesn't mean Witt then concluded "in that case, by appealing to the law of excluded middle realism is true. I propose a new epistemological foundation in which there is only one sort of meaning that is decided by the public, platonia or scientific naturalism in a mind-independent reality". All he concluded is that due to the overwhelming complexity and uncertainty of phenomenological analysis, it is impossible for himself to give an exhaustive and unconditional phenomenal theory accounting for his own use of words.

It is therefore understandable, as to why Wittgenstein was sympathetic towards Heidegger and could personally relate to Being and Time on the one hand, while at the same time insinuating that Being and Time was nonsensical when viewed as a collection of propositions with an inter-subjectively determinable truth-value.

Nonsense doesn't mean "false", it merely refers to an inability to determine the sense of a word when it used in a context from which it did not originate. Wittgenstein's sympathies towards Heidegger demonstrate that he did not believe the most important types of meaning to be inter-subjectively decided. Only inter-subjective meaning is inter-subjectively decided.

We can all agree that we can relate to Being in Time, without pretending to ourselves that we understand each-other's understanding of this work when viewing our agreement from the perspective of a different language-game.

Any finite number of grains of sand does not have this property.
— sime

So... isn't a heap? — bongo fury

There is no a priori linguistic definition of "heap" in terms of any specific number of grains of sand, which is why "heap" must be logically represented as referring to a potentially infinite number of grains of sand.

The role of potential infinity in a logical specification is to act as a placeholder for a number that is to be later decided by external actors or the environment, rather than the logician or programmer.

Agreed. But what is the smallest number of grains that would need considering by speakers as a particular case? Is it 1? — bongo fury

It is you and only you who gets to decide the answer to that question whenever you are next confronted by a growing or diminishing collection of sand grains.

I would be surprised if there isn't a population study that has attempted to quantify the mean number of grains of sand at which speaker of English judge a collection of sand to be a heap. A few hundred grains?? A few thousand?

like imagining a heap of sand that never changes after a grain is removed or added.
— sime

... leading to the conclusion (incompatible with a premise, or there's no puzzle) that a single grain is a heap. Does that happen also with your "infinite" element, so that it can evaluate to 1? — bongo fury

No, that isn't the case. To summarise, a heap of sand can be defined as the list:

Heap := [ Heap, grain]

The list remains constant, regardless of how many grains are added to it or subtracted from it. Any finite number of grains of sand does not have this property. That is precisely what it means to say that a heap of sand has no inductive definition.

Where i differ with the OP, is his belief that it is an approach unconnected to ideas of fuzziness or ambiguity. This isn't the case, because the practical usage of infinity, such as the use of infinite loops in computer programs, is to defer the termination of the program to the environment. Or in the case of heaps of sand, the semantics which concern the precise moment when an actual heap of sand is considered to be mere grains of sand, isn't linguistically specified a priori but is decided by speakers on a case specific basis.

Your proposed solution sounds to be in a logical sense the same as mine, which is to consider the heap to be a co-inductive type.

First consider the recursive definition for the set of inductively constructed lists.

Inductive List := { [ ] AND x : Inductive List }

Evaluating this equation by substituting the left-hand side into the right-hand side is analogous to building every possible collection of sand-grains by starting from nothing and adding a grain at a time. Yet we don't call a collection of sand-grains constructed by this process a heap, because "heaps" aren't defined by induction.

In contrast, the set of co-inductively constructed lists is defined by switching the AND for an OR:

Co-inductive List := { [ ] OR x : Co-inductive List }

The set of co-inductive lists contains the inductive set of lists, but because it isn't obligated to build lists by starting from empty, it also includes an additional list of "infinite" length. By "infinite" we are merely referring to the fact this set contains the definition of list that might never terminate upon iterated evaluation.

The former type of list corresponds to the natural numbers, whereas the latter type corresponds to the conatural numbers, otherwise known as the extended natural numbers that includes a numeral for infinity, which in computing can be used to denote programs will never halt to produce an output.

I think that the grammar corresponding to "a heap of sand" is analogous to the "infinite" element of a coinductive list, which as you say is like imagining a heap of sand that never changes after a grain is removed or added.

What's a "theorem prover"? Computer program? A tutor? — jgill

More generally, it is a dependently typed logical programming language, with clause resolution and other rules of logical inference, together with SAT solvers, methods of analytic tableaux and heuristics for automated or interactive theorem proving.

Lean with Mathlib, to my understanding is a state of the art approach to logic and mathematics programming that embodies the above principles , an approach which together with advances in deep learning theorem proving will undoubtedly revolutionise mathematics education , automation and research. Remarkably, the mathematics library of lean now codes the proofs of a lot of undergrad level mathematics.

https://leanprover.github.io/live/latest/

I don’t regard professional mathematicians as being trustworthy or helpful with regards to the literal truth of their subject, for several reasons.

1. They can be expected to exhibit a political bias towards inflating ontological claims in mathematics , in the same way that professional chess players can be expected to inflate the importance of chess and exhibit bias against other games. Traditionally this is bias is evident in the continued prevalence of classical logic in the justification of mathematical claims and the high percentage of mathematicians who are platonists, a position that is troubling to any engineer or computer scientist who unlike pure mathematicians have actual responsibilities regarding the physical actualization and interpretation of mathematical results.

2. The subject of ontological claims and commitments is the domain of logic rather than of mathematics, and the classical mathematician isn’t logic savvy, unlike today’s generation of mathematics undergraduates who are studying mathematics using theorem provers from the outset.

yes, sorry, i should have clarified that i was referring to number in the extensional sense, i.e. as an obtainable state of a calculation. This is because i understood the OP as questioning the existence of sqrt 2 in this extensional sense.

Naturally, Sqrt 2 exists intensionally in the constructive sense of an algorithm that generates any finite cauchy sequence of computable reals, whose limit x fulfils the axiomatic definition

sqrt 2:= x : x^2 = 1.

Furthermore, it is constructively provable that the limit is irrational in the sense of being separated from any computable rational number. And the sqrt of 2 intensionally exists in this sense, irrespective of whether the limit of this process of calculation is axiomatically accepted as being a number in the extensional sense.

Of course, normally when we assign numbers to physical lengths we aren't resorting to logical construction, nor even necessarily to calculation, which means that our practical use of numbers is logically under-determined. We could for instance, declare the hypotenuse of unit length triangles to be "real" lengths that correspond to extensional numbers relative to some novel logical construction of numbers, in which the unit lengths of the other sides of the triangle only exist intensionally as limits in this number system.

In my opinion, constructive mathematics should permit a plurality of axiomatic systems to represent our use of numbers. That way we don't have to make arbitrary a priori decisions as to which numbers only exist as limits.

Does Collingwood undermine his own arguments, by begging his own ontological assumptions, when he makes a hard distinction between absolute presuppositions and propositions?

Initially, SEP's article on Collingwood says that the difference between presuppositions and propositions isn't one of content, but one of role:

"Whether a statement is a “proposition” or a “presupposition” is determined not by its content but by the role that the statement plays in the logic of question and answer. If its role is to answer a question, then it is a proposition and it has a truth-value. If its role is to give rise to a question, then it is a presupposition and it does not have a truth-value. Some statements can play different roles. They may be both propositional answers to questions and presuppositions which give rise to questions. For example, that an object is for something, that it has a function may be a presupposition which gives rise to the question “what is that thing for?”, but it may also be an answer to a question if the statement has the role of an assertion. "

So far, so good, for no Positivist could disagree with that; whenever an assertion is understood to be meant presuppositionally it isn't used in a truth-apt sense, but as a temporary conditional assertion upon which a subsequent course of a truth-apt epistemological enquiry is founded. This is to understand a presupposition as being a modest and consciously subjective assertion that is comparable to an axiom or a Wittgenstein 'hinge proposition'. But then immediately after that (emphasis added), the article makes a stronger claim on behalf of Collingwood, saying

"Philosophical analysis is concerned with a special kind of presupposition, one which has only one role in the logic of question and answer, namely that of giving rise to questions. Collingwood calls these presuppositions “absolute”. Absolute presuppositions are foundational assumptions that enable certain lines of questioning but are not themselves open to scrutiny."

At first I wondered if SEP was overstating Collingwood's position. For if the distinction of these types of sentences is one of role rather than content, then presumably he was merely pragmatic and did not think that his distinctions had significant implications with respect to epistemological conclusions, but only with respect to the defence of a plurality of epistemological methodologies, none of which have the capacity to contradict the assertions that the other methodologies arrive at. But then the article makes it clear that his position is even stronger:

" Collingwood’s account of absolute presuppositions generates an interesting angle on the question of scepticism concerning induction. Hume had argued that inductive inferences rely on the principle of the uniformity of nature. If it is true that the future resembles the past, then inferences such as “the sun will rise tomorrow” are inductively justified. However, since the principle is neither a proposition about matters of fact nor one about relations of ideas the proposition “nature is uniform” is an illegitimate metaphysical proposition and inductive inferences lack justification. The principle of the uniformity of nature, Collingwood argues, is not a proposition, but an absolute presupposition, one which cannot be denied without undermining empirical science. As it is an absolute presupposition the notion of verifiability does not apply to it because it does its job not in so far as it is true, or even believed to be true, but in so far as it is presupposed. The demand that it should be verified is nonsensical and the question that Hume ask does not therefore arise:

…any question involving the presupposition that an absolute presupposition is a proposition, such as the question “Is it true?” “What evidence is there for it?” “How can it be demonstrated?” “What right have we to presuppose it if it can’t?”, is a nonsense question. (EM 1998: 33) "


If the SEP is portraying his beliefs correctly, then I think Collingwood has jumped the sharked from making reasonable and modest epistemological commentary, to arriving at a dogmatic position regarding presuppositions that appears to beg the very ontological premises which he claimed philosophy isn't about. For you cannot insist that Hume was mistaken to question the uniformity of nature on the basis of it being an absolute presupposition, without adopting the dogmatic ontological standpoint that absolute presuppositions constitute objective existential claims.

Isn't denying the existence of sqrt of 2 on the grounds that it isn't a computable number a bit like denying the existence of a "heap" of sand on the grounds that a "heap" isn't derivable from a granular definition of sand?

The "Sqrt 2" is at the very least, pragmatically useful as a moniker for the hypotenuse of a certain class of visually recognisable triangle, and it should be remembered that we have as much empirical justification for labelling the hypotenuse with sqrt 2 as we have for labelling its other sides with "1".

Zeno's reaction to his paradox is also similar to yours, in his conclusion that the existence of motion is impossible on the grounds that motion cannot be constructed from positional information. But the converse is also true: a position, in the logical sense, cannot be constructed by slowing down motion. Motion and position are irreconcilable concepts pertaining to mutually exclusive starting conditions of a system and mutually exclusive choices of disturbance of the system by an observer thereon after. Each concept can only informally represent the other as a "limit" that can only be approached but never arrived at.

The constructivist isn't forced into believing in the literal existence of hypertasks as the platonist might insists, rather the constructivist only needs to deny the existence of a universal constructive epistemological foundation.

I suggest you google Nelson Goodman and his ideas concerning irrealism that more or less convey the basic structure of community-level solipsistic logic. His argumentation embraces and accommodates, rather than rejects, the inevitable conflicts of speech that arise in community discussions of truth, such as when every member of a community of solipsists declares himself to be the centre of the universe.

The traditional way of thinking is to assume that whenever a community of speakers discuss the universe in an absolute sense, they must be referring to the "same" universe. But this is proposterous according to a Goodmanian irrealist, according to whom each and every speaker cannot transcend their personal frames of reference and so cannot refer to the same universe in an absolute sense, even when they insist otherwise.

Consquently the irrealist understands every assertion, including assertions of absolute truth, as being relative to the speaker and of the form "according to speaker X assertion Y is true".

Not all individualists are solipsists, and not all solipsists are individualists.

And why can't a solipsist be a realist? after all, the thought that the external world has independent existence is just a thought, and solipsists accept the existence of thoughts.

think you are focusing too much on the fact that theoremhood is not strongly representable in PA, with the consequence that you are ignoring the fact that it is weakly representable in PA. Indeed, while theoremhood is not computable, it is computably enumerable. In other words, there is an algorithm which lists all and only theorems of PA. It exploits the fact that, given your favorite proof system, whether or not a sequence of formulas is a proof of a sentence of PA is decidable. Call the algorithm which decides that "Check Proof". Here's an algorithm which lists all the theorems of PA, relative, of course, to some Gödel coding:

Step 1: Check whether n is the Gödel number of a sequence of formulas of PA (starting with 0). If YES, go to the next step. Otherwise, go to the next number (i.e. n+1).

Step 2: Decode the sequence of formulas and use Check Proof to see if it is a proof. If YES, go to the next step. Otherwise, go back to Step 1 using as input n+1.

Step 3: Erase all the formulas in the sequence except the last. Go back to Step 1, using as input n+1.

This (horrible) algorithm lists all the theorems, i.e. if S is a theorem of PA, it will eventually appear in this list. Obviously, this cannot be used to decide whether or not a given formula is a theorem, since, if it is not a theorem, then we will never know it isn't, since the list is endless. But, again, it can be used to list all the theorems. My point is that there is nothing comparable for the truths, i.e. there is no algorithm that lists all the truths. In fact, by Tarski's theorem, there can be no such algorithm. So, again, the two lists (the list of all the theorems, the list of all the truths) are not the same, whence the concepts are different. — Nagase

I'm still confused as to where and how we disagree. I suspect the issue might be mostly terminological, due to your use of modal notions versus my deflationary/constructive terminology. However it is possibly worth recalling that PA must have the following theorem for any provability predicate

PA |-- Prov('G') --> ~G for any Godel sentence G.

In other words, there cannot be an exhaustive and infallible enumeration of theoremhood within PA. - and here we aren't referring to the failure of ~Prov to enumerate the non-theorems - rather we are referring to the inability of Prov to correctly enumerate all and every derivable theorem.

We can of course construct an infallible Prov by defining it to enumerate only the godel-numbers that have been independently determined to be proofs via brute-force checking. In which case Prov along with the godel-numbering system are merely redundant accountancy of what we've already derived, and in which case Prov sacrifices exhaustivity for infallibility.

Alternatively, prov could be an 'a priori' algorithmic 'guess' as to theoremhood , in which case it can be exhaustive , e.g. by guessing "True" to every godel number, but at the cost of infallibility, in guessing both correctly and incorrectly as to theoremhood.

But we're at least both in agreement it seems, that there isn't an algorithm for listing all and only the actual theorems of PA - which is precisely the reason why truth should be constructively identified with actual derivability, as opposed to being identified with whatever is unreliably or incompletely indicated by a 'provability'/'truth' predicate.

So where and how do we diverge?

The upshot of all this is that, in my opinion, constructivists should resist the temptation of reducing truth to provability. Instead, they should follow Dummett and Heyting (on some of their most sober moments, anyway) and declare truth to be a meaningless notion. If truth were reducible to provability, then it would be a constructively respectable notion. But it isn't (because of the above considerations). So the constructivist should reject it. (Unsurprisingly, most constructivists who tried to explicate truth in terms of provability invariably ended up in a conceptual mess---cf. Raatikainen's article "Conceptions of truth in intuitionism" for an analysis that corroborates this point.) — Nagase

Dummett's overall arguments sound roughly similar to my deflationary position of truth in mathematical logic'; software engineers don't say that the operations of a software library has no truth value, rather they define truth practically in terms of software-testing, without appeals to the choice axioms, or the Law of excluded middle.

What i think classical philosophers overlook is that the absolute consistency of PA isn't knowable, or even intelligible. As an absolute notion, undecidability is meaningless.

thanks for the reply, will get back later.

In your view, if flourishing has to be the intention of a moral action, then how should moral intentionality be determined?
— sime

Hi sime, sorry, did not understand the question, can you restate in a different form? Thanks! — Thomas Quine

What is your position regarding moral intention, moral freedom, moral responsibility and moral competency? How are these things definable and measurable?

Presumably you don't consider social utility to be sufficient grounds for defining morality, for otherwise, morality is indistinguishable from luck....

I don't think Löb's theorem supports the constructivist position. That's because truth is generally taken, prima facie to obey the capture and release principles: if T('S'), then S (release), and, if S, then T('S') (capture). But what Löb's theorem shows is that proof does not obey the release principle. So there is at least something suspicious going on here. — Nagase

Constructively, Prov doesn't fail either the capture or release properties of a truth predicate with respect to decided sentences, rather Prov doesn't supply negative truth value when classifying undecided sentences that no axiomatic system can decide without either begging the question, or by performing a potentially infinite proof search equivalent to doing the same in Peano arithmetic.

Perhaps part of the confusion/suspicion comes from overlooking the following symmetry in what Peano arithmetic cannot derive

PA |-/- (For All S: S --> Prov('S')) ( since compilation is potentially infinite)
PA |-/- (For All S: Prov('S') --> S ) ( since decompilation is potentially infinite)

Löb's theorem only deduces the existence of an as-of-yet undecided object on the assumption that the decompilation process of it's respective code terminates. And yet the undecided object will only compile into a code in the first place, if it is decidable. Therefore Löb's theorem does not have constructively relevant implications.

Why would you want, let alone expect, a truth predicate to capture and release the properties of potentially infinite objects whose existence is potentially non-demonstrable?

Moreover, one can show that the addition of a minimally adequate truth-predicate to PA (one that respects the compositional nature of truth) is not conservative over PA. Call this theory CT (for compositional truth). Then CT⊢∀x(Sent(x)→(Prov(x)→T(x)))CT⊢∀x(Sent(x)→(Prov(x)→T(x))), where "T" is the truth predicate. As a corollary, CT proves the consistency of PA. So truth, unlike provability, is not conservative over PA. — Nagase

In other words, introducing new axioms to represent undecided formulas generally permits the derivation of new sentences in a vacuous manner.

Finally, you have yet to reply to my argument regarding the computability properties of the two predicates, namely that one does have an algorithm for listing all the theorems of PA, whereas one does not have an algorithm for listing all the truths of PA. So the two cannot be identical. — Nagase

The set of theorems of PA isn't recursive due to the halting problem, meaning that any proposed test of theoremhood by a "truth predicate" is bound to be either incomplete or to contain an infinite number of mistakes.

Consequently, the "truth" of PA consists of the explicit construction of each and every theorem, doing everything the hard way.

Edit: I rushed this post, so came back and rectified some mistakes.

As an aside, being an engineer you sound well positioned to lead a digital nomadic lifestyle close to the outdoors

In your view, if flourishing has to be the intention of a moral action, then how should moral intentionality be determined?

1. Proof is necessary for truth [Godel assumes and thus proves his incompleteness theorems]
2. The incompleteness theorems proves that proof is unnecessary for truth
3. Proof is unnecessary for truth (from 2)

1 and 3 contradict each other, no? This is a meta-cognitive statement regarding Godel's thought processes. — TheMadFool

Yes, and your confusion is the fault of classical modal interpretations of logic that insist on misinterpreting PA's provability predicate as being meta-logical in character or representing knowledge, which it doesn't unless it is applied to theorems that are derivable without meta-logical reasoning.


The reason for all the persistent confusion that remains nearly 100 years after godels' proofs, is because Godel's theorems were baptised with their official philosophical interpretation as relating logic to a transcendental plane of mathematical reality, before the invention of the Turing Machine that collapsed the meta-logical distinction between logic and arithmetic by grounding them both in shared computational semantics; operations that also explicitly recognise the nondeterminism of provability due to the halting problem.


As an aside, the reason why the halting problem constitutes the quickest and easiest proof of Godel's incompleteness theorem is due the recognition that algorithms must halt to produce an answer.

For the logician, the predicate "A is provable" simply means that A is mechanically derivable by following the roles of the respective axiomatic system i.e.

"A is provable" := |--A

No other informal nomenclature about 'truth' is useful or necessary.

However, most philosophers describe logic using alternative nomenclature and then appeal to modal heuristics that only leads to confusion, misunderstanding and unnecessary arguments, for they define

"A is true" := |-- A

"A is provable" := |--- Prov('A')

Formally of course, Prov is equivalent to a prolog compiler written in the language of PA that translates theorems of PA into programs that when executed reveal the theorem of PA they they encode. Writing Prov as Comp, Lob's theorem no longer seems remotely surprising.

Recall that if we have PA |-- A , then we can also derive

PA |-- Comp('A') -> A

( i.e, a compiled program for a derivable theorem can be disassembled to reveal the theorem. )

Lob's theorem says

PA |-- ( Comp('A') --> A ) -> A

( i.e. if any compiled program is executed, then it's conclusion is the same as what the disassembler returns . )

So far, Lob's theorem isn't remotely surprising.

But in the case of Godel sentence G, if we assume that PA is consistent and hence that G isn't derivable, the following cannot be true by Lob's theorem:

PA |-- ( Comp('G') --> G )

i.e. if an undecidable statement is compiled, then it cannot be disassembled.

Bur recall that G is the statement saying

G <---> ~Comp'G')

This indicates that the only way to disassemble Comp('G') would be to substitute G into itself an infinite number of times. So it shouldn't be surprising that Comp('G') doesn't imply that it can be disassembled back into G (assuming consistency).

I suspect that those who find Lob's theorem surprising do so because they assume that compilation is always invertible.

I once witnessed a similar mistaken belief expressed in a technical report written at MIT discussing algorithms for Bayesian statistics.

They argued that if you only know that your data is Gaussian distributed, but that you don't know it's location and scale parameters, then you should assign a particular 'non-informative prior' over these parameters, say a uniform distribution, and then integrate out these parameters analytically in order to obtain a 'predictive distribution' - which is essentially to accept the wisdom of crowds.

But ignorance about a distribution's parameters cannot imply any particular distribution of outcomes, rather it implies every distribution of outcomes that is consistent with one's state of ignorance.

For example, if we don't know anything about the bias of a coin, then we shouldn't assign a particular distribution to the probability of getting heads, say P(heads)= 0.5 - as is done by naive Bayesian practitioners who appeal to maximum entropy or the principle of indifference, rather we should assign the interval (0,1) i.e.

P(heads) = (0,1) i.e. 0 < P(heads) < 1

As another example, how about "there is a number greater than zero and smaller than every real number". Is it true or false? Does it even make sense to say that it's either true or false? Or is it more correct to say that it's provable if using the hyperreals and that its inverse is provable if not? — Michael

Unless we appeal non-constructively to the Axiom of Choice, we obtain a set of 'hyperreals' that is isomorphic to a subcountable set of natural numbers with their usual ordering relation. So in a superficial sense we can have infinitesimals in peano arithmetic by defining {1,1/2,1/3..} to be greater than {0,0,0...} even though we have only a subcountable number of such 'hyperrreals'.

Does empirical statistical evidence count as rational? This intelligent "fluke" seems to be built-in to the mathematical foundation of Nature. :smile: — Gnomon

Your very article points out the biased nature of people's predictions.

"However," adds Berdahl, "there is a great deal of evidence that people have strong biases in estimation and decision tasks."

Hence the crowd might be wise, but it has nothing inherently to do with probability or the laws of nature, per se. All that the law of averages can say is that ignorant opinions cannot completely drown out expertise.

Nope, the "wisdom of crowds" is false. There is no rational reason to believe that averaging naive guesses increases accuracy, unless by fluke the average of these naive guesses is the right answer.

Firstly, good spot about PA |-/- Prov S --> S

As you point out, that false derivational assumption which I took for granted and which led me to

PA |-- Prov('G') <--> ~Prov('G')

implies that PA is inconsistent due to LOM, something I overlooked as i don't think classically.

In fact, that 'Reductio ad absurdum' constitutes a weaker version of Lob's theorem, specifically applied to G.


But all of that said, from a constructivist position it doesn't seem surprising given that ~Prov already doesn't reflect non-derivability. Lob's theorem is just further ammunition for identifying truth directly with the derivable sentences, and for rejecting the interpretation of "prov" as meaning provability, and consequently for rejecting the semantic philosophical interpretation of incompleteness as implying 'arithmetically true but undecidable propositions'.

Let us suppose that everything you say is true. This still does nothing to address two facts: (1) the set of true formulas is not arithmetically definable, but the set of provable formulas is, whence the two must be distinct; — Nagase

Correct me if i'm wrong, but aren't we both referring to the fact that the negation of PA's provability predicate doesn't actually enumerate what isn't derivable in PA?

(2) truth is not conservative over PA, whence it can't be redundant. I sketched that argument in my first post here precisely so we did not get entangled in fruitless discussions about how we can know that G is true or about the Kirby-Paris theorem. — Nagase

If by truth not being redundant, you are merely referring to the difference between

PA |-\- S (i.e. the event that PA doesn't derive S) versus PA |-- ~Prov('S')

then I think we're in agreement. But in this event, the notion of truth is redundant in the sense that it doesn't transcend the notion of derivability within PA, which is what I took the OP's point to be.

As for G's "truth" value, I can only recognise it as having undefined truth value, both in the sense of it's unknown status as a derivable theorem of PA, and also in the sense of it's decoded arithmetical interpretation pertaining to the undecidable solutions of Diophantine equations. As an aside, i don't recognize the truth of Goodstein's Theorem, due to it's proof relying on non-constructive notions pertaining to transfinite induction.

Obviously, that particular argument assumes the soundness of PA, which you have disputed (this is a minority position, but one that I respect, if only because in the case of Nelson it generated some interesting mathematics). But this is not necessary for the argument to go through: one can start with Q and argue that any recursively axiomatized theory that extends Q will fall into the same problem, namely truth will be arithmetically undefinable and theoremhood will be arithmetically definable. Since no one that I know of doubts the soundness of Q (not even Nelson), the argument should go through. — Nagase

I'm not so much of the view that PA will turn out to be inconsistent within my life time, but rather of the view that talk of the consistency of PA is meaningless in both demanding infinite evidence and also in being logically inexpressible, as demonstrated formally by Godel's second incompleteness theorem.

By the way, if your ii.c) is correct, then PA is inconsistent. In any case, that is not a valid substitution instance of ii.a): ii.a) says merely that (assuming soundness) PA |- S iff PA |- Prov('S'), not that PA |- S <-> Prov('S') (the latter is a reflection principle and is actually not provable in PA). — Nagase

ii a) doesn't express soundness in the sense of begging an external notion of arithmetic truth, rather it expresses the ability of Prov to enumerate what is derivable within PA:

PA |-- S <---> PA |-- Prov('S')

this is a provable equivalence, because Prov(''S') merely enumerates what can be derived in PA and it doesn't lead to false derivations, because it directly encodes PA's axioms and rules of inference.

ii c): By itself ii c doesn't imply inconsistency, because it is specifically stated with reference to the Godel sentence G that diagonalizes the negative provability predicate:

PA |-- ~Prov('G') <--> Prov('G')

Only if PA |-- G is derivable does ii c lead to inconsistency.

ii c) is equivalent to Tarksi's undefineability of truth applied to PA, in which PA's derivability predicate, Prov, is taken to be it's "truth predicate" . It also implies the constructive content of Godel's first incompleteness theorem, namely that ~Prov is a misnomer in not being able to enumerate what PA cannot derive.

That's why i said i agreed with Mad fool, for reasons you hinted at. Truth, as far as mathematical logic is concerned, IS derivability , for mathematical logic cannot be partitioned into a metalanguage talking about an independent object language, for both languages are isomorphic to one another and suffer from the same undecidability.

I'm with the Madfool, paradoxically for reasons hinted at by nagese.

Godel sentences do not support the colloquial interpretation they receive, that is to say "G is true because G is unprovable, assuming Peano Arithmetic is consistent". This is because

i) Firstly, the hypothesis that PA is consistent is potentially falsifiable, but it can never be verified. Therefore it is irrational for logicians and philosophers to assume or even talk about PA's 'infinitely complete' hypothetical consistency. Instead, they should only talk about PA's consistency within a limited finite scope of derivations.

ii) PA cannot talk about what isn't provable in PA, which is the central conclusion of Godel's first incompleteness theorem, and corresponds to both the Halting Problem and Tarski's undefinability of truth within PA. In general, for any sentence S of PA we have

a) PA |-- S <---> PA |-- Prov('S')

e.g. PA derives S if and only if PA derives Prov('S')

b) PA |-- G <--> ~Prov('G') For any Godel sentence G.

Therefore b substituted into a leads to the conclusion

c) PA |-- ~Prov('G') <--> Prov('G')

In other words,

PA |-- ~Prov('S') does not imply that PA does not derive S .

i.e. Peano arithmetic cannot enumerate the formulas that PA cannot derive (duh).

Therefore G does not say that "G isn't provable"

iii) The arithmetical theorems that PA cannot derive, e.g. Goodstein's Theorem, aren't constructively acceptable in any case, due to them appealing to the Axiom of Choice. These theorems cannot be computationally or otherwise empirically verified in principle without begging the question. Therefore they should not be considered as having a truth value.

In conclusion, an undecidable proposition of PA cannot be said to have a truth value, unless it is added as an additional axiom, in which case it is vacuously true due to now being provable as an axiom.

Unfortunately, Classical Logic and it's accomplice Model Theory, give this illusion that arithmetic is a meta-language for defining the truth of PA or vice versa from 'outside the system'. This isn't the case, due to the very ability of each system to encode each other. They are therefore one and the same theory, for formal purposes and are as equally undecided as each other.

I'd imagine that the topology of 'good' music as it is formally represented, is a disconnected archipelago of irregular islands, each with many holes, quite unlike the densely connected forest of logic theorems.

Presumably, a composer is like a drunkard who awakens on one of these islands with no idea how he got there.

s

Heisenberg's uncertainty principle is in any event an epistemological and NOT an ontological fact. It's a limitation on what we can know (with our current theories) and says nothing about what truly is. — fishfry

That view assumes counterfactual definiteness; the belief that the possibility of stopping a moving arrow to construct a definite position implies that the moving arrow must have a real and precise but unknown position when it isn't stopped or it's position otherwise measured.

Yet this unquestioned assumption of counterfactual definiteness is the reason why Zeno's paradox appears paradoxical. To my understanding, Zeno's arguments are perfectly sound, which means that i have no choice but to reject counterfactual definiteness in order to resolve the paradox, and is the reason why i believe that Zeno ought to have stumbled across the underlying logic of Heisenberg's principle (when it is interpreted ontologically).

Of course, the rejection of counterfactual definiteness is only one means of making sense of quantum entanglement and which is also the view of the Copenhagen interpretation, which means that Heisenberg uncertainty is interpreted as ontological ambiguity/incompatibility, rather than as epistemic uncertainty.

I've come to the conclusion that the uncertainty principle should have been discovered by Zeno of Elea, nearly 500 years BC, since in my opinion the only satisfactory way of resolving Zeno's paradoxes is by recognising the incompatibility of the notions of momentum and position - something which is immediately self-evident in ordinary experience, and obvious after one abandons the dogmatic assumption that counterfactual experimental outcomes exists.

As for special relativity, it's scope was narrowly concerned with the logical consistency of theories such as maxwell's equations that employ temporal indexicals but otherwise lack explicit temporal frames of reference. So I think it's right to point out that special relativity isn't particularly relevant to phenomenological puzzlement and concerns about the nature of time, but at the same time SR cannot be criticised on that ground, for the nature of phenomenological time wasn't the theory's intended purpose and SR leaves the relationship between theoretical space-time and ordinary experience undefined.

Firstly, to satisfy the hedonistic imperative the machine must offer a genuine and sustainable alternative to reality, whereby one isn't constantly reminded that they are only obtaining virtual goods. For the presence of an underlying sense of having a fraudulent experience would mean that the machine isn't by definition an optimal experience machine. So i think it should be taken for granted for purposes of discussion that the machine is optimal in this sense.

The main issue I have with an optimal experience machine in the above sense, is due to the fact that many of our most positive experiences come about through overcoming adversity, i.e. unpleasant situations. But doesn't this imply that the machine must also simulate unpleasant experiences? In which case, how do we know that reality isn't already the ultimate experience machine?

Consider a 'roguelike' video game, where the player explores a dungeon that is generated 'on the fly' in response to the player's actions. Here, the history of the game world and the future of the game world are identical, and so one could describe the game-world as having a potentially infinite past, if and only if, the game world has a potentially infinite future.

Presentism sees the actual world in similar respects; all that exists is the present state of information, and so the denotation of a temporal "beginning" is arbitrary, as is the distinction between past and future.
To use computational terminology, presentism understands history as being lazily evaluated.

The reason we don't experience contradictory propositions is precisely because what we experience is information, and if there is no information, then there is nothing to experience - except for the visual experience of the seeing scribbles on a screen or hearing sounds spoken - which is information, but about something else that isn't about what is being written or said. — Harry Hindu

Right. And so the word "contradiction" doesn't mean zero information, for that is nonsensical, but refers to conflicting sources of information, actions, intentions, judgements and so on. A "true" contradiction can be taken to refer to an unresolved conflict that is logically implied.

For example, conflicts of judgement that are present in discrete borderline categorisation problems, as in being in the kitchen and not in the kitchen, are not resolvable by introducing more linguistic precision, for the same borderline problem resurfaces on a finer level of semantic granularity; here the "true" contradiction refers to the fact that the concept of discreteness cannot be reconciled with the existence of borderline cases. It's all well and good hoping that the conflict is potentially resolvable, but there is no reason to believe that all such conflicts are resolvable.

A contradictory statement says nothing at all, and is therefore useless. It is basically asserting something and then walking back that assertion at the same time resulting in a net zero amount of information. It is basically scribbles on a page, or sounds in the air — Harry Hindu

Yes in the sense of contradictory propositions. Nobody of course, experiences contradictory propositions - which goes to show that the general meaning of "contradiction" isn't to refer to propositions but to conflicts, such as the conflict between the definition of a language and it's application, or the rules of a sport and the moral notion of fair-play etc.

Three examples of "true" contradictions come to mind

i) Self-negating universal imperatives, i.e. hypocritical statements such as "Don't live by rules!".

ii) When a semantic distinction is more fined grained than is expressible in the language used, such as when standing in a doorway and thereby "being in the kitchen but not in the kitchen".

iii) When a semantic distinction is vague or uncertain, such as "a heap of sand" that isn't defined in terms of a particular numeric range of sand grains Hence "heaps of sand" exist, but no particular collection of sand grains constitutes a heap.

ii + iii are contradictions that programmers have to deal with, but they also present challenges for self-learning autonomous agents, that like human beings must somehow internalise a truth-language distinction.

I suspect that like humans, AI agents will also behave in a logically inconsistent fashion relative to their self-knowledge.