Right, and that axiom must be taken as a “given.” However, I wouldn’t say that giving the realist position the status of axiom renders it “off limits” to the skeptic who is unconvinced of its supposed “self-evidence.” — PessimisticIdealism

Agreed. The relevant issues to my mind would be about what logical consequences follow from it (e.g., is it coherent or does it lead to inconsistency or absurdity?), what it is useful for (e.g., science, everyday communication), and also how alternative definitions fare (e.g., idealism).

I found an interesting statement by Einstein related to this:

"The physical world is real." That is supposed to be the fundamental hypothesis. What does "hypothesis" mean here? For me, a hypothesis is a statement, whose truth must be assumed for the moment, but whose meaning must be raised above all ambiguity. The above statement appears to me, however, to be, in itself, meaningless, as if one said: "The physical world is cock-a-doodle-do." It appears to me that the "real" is an intrinsically empty, meaningless category (pigeon hole), whose monstrous importance lies only in the fact that I can do certain things in it and not certain others. — Einstein, 1918

I'm going to translate your argument in terms of something concrete. Consider an apple on Bob's kitchen table, which I'll abbreviate as 'the apple'.

P1) The realist argues that “the being of the apple is independent of its being known.”
P2) In order to know whether or not "the being of the apple is independent of its being known," one must “know the apple when the apple is not being known.”
P3) In order to compare “the being of the apple as known” to “the being of the apple when it is not being known,” one must “know the apple” and “know the apple when it is not being known.”
P4) One cannot “know the apple when it is not being known” without performing a contradiction.
C) Therefore, the realist position is untenable.

P1 states a definition, i.e., what realism means, or how a realist uses their words. P2 says that to know whether or not that definition is true would entail knowing that the apple exists at those times that it is not known to exist - a contradiction.

So one counterargument is that a definition operates like an axiom in mathematics. An axiom is not something that is itself proven via a rule or process. Instead you either use it or not for pragmatic reasons.

Let me re-emphasize my thought-experiment: Suppose the world changes overnight so that it becomes impossible to model an implication (per se and of course especially for our human minds). It's hard to see why and how, but just bare with me. Wouldn't that mean that MP becomes impossible as well, in contrast to a day before where it was not only possible, but necessary? Doesn't that prove the induction problem for logic as well? — Pippen

If you lived in a world where nothing followed necessarily from anything else, then MP wouldn't apply. Perhaps, for example, a quantum vacuum where particles just pop into and out of existence and you are a Boltzmann brain.

However that wouldn't be a case of MP failing, merely that it has no application or use in that scenario. Whereas the rule p; p ->q; not q is inconsistent and so couldn't apply to any scenario.

Nowadays, Hume's intuition about the sun is considered to be quite right:

The Solar System will remain roughly as we know it today until the hydrogen in the core of the Sun has been entirely converted to helium, which will occur roughly 5 billion years from now. — alcontali

The sun will still be around in five billion years, therefore Hume was right that the sun might not rise tomorrow? :-)

It sounds more or less like how I've always interpreted MWI (which, more on topic, strikes me as very similar to Lewis' notion of actuality being indexical), but the RQM formulation of those ideas seems even more clear and elegant. — Pfhorrest

Yes, RQM is a high-level abstraction that preserves nice features like locality, theory completeness (i.e., no hidden variables or ad hoc changes) and treats all physical systems as fundamentally quantum (including observers). The downside is that it is not explanatory in the sense that MWI is.

FWIW, my take on the relationship between MWI and modal realism is that they can be considered equivalent if we take a "possible world" to be something slightly different from what Lewis takes it to be, which also meshes better with Kripke's semantics about accessibility, which always struck me as really bizarre from a Lewisian perspective (e.g. the notion that something might be necessary from one possible world but contingent from another, when "necessary" should rightly mean "true in all possible worlds"). — Pfhorrest

Fair enough. I agree that accessibility is important, e.g., Alice seeing heads (from a quantum coin toss) in her world means it is impossible for twin Alice to see heads in her world. But it seems to me that interference effects between worlds also need to be factored in which is a function of probability amplitudes, not merely classical possibilities. For example, it might seem possible for a particle travelling through a Mach-Zehnder interferometer (with equal path lengths and no sample) to arrive at Detector 2, but it isn't.

Do you believe in the MW interpretation of I may be so bold? — Wallows

"Believe" is too strong. However MWI and Rovelli's RQM are my preferred interpretations.

what are we to make of claims made in physics as counterfactual definitiveness? Don't they derive from Lewis' work on possible worlds or Kripke? — Wallows

No. Everett developed his interpretation prior to Lewis' and Kripke's work. And the earlier Copenhagen Interpretation rejected counterfactual definiteness from the beginning (as do almost all interpretations - see the comparison table).

The Many Worlds interpretation is additionally factually indefinite in the sense that a quantum coin toss lands on both heads and tails (in their respective branches).

How much intersectionality between Kripke semantics and theories like the Many World Hypothesis lend to each other? — Wallows

None, it seems to me, since they're describing very different things. "Possible worlds" is a tool for modeling abstract hypotheticals or counterfactuals. Whereas Many Worlds is an interpretation of a physical theory (i.e., what actually occurs, according to Many Worlds).

Compare, for example, a classical coin toss with a quantum coin toss. The possible classical coin toss outcomes could be modeled as "possible worlds", yet no Many World's branching occurs. Conversely, the quantum coin toss outcomes would likely count as a single "possible world" (since both branches are actual and potentially interfere or even eventually merge on a Many Worlds view).

That sounds intriguing and intuitively correct, but unfortunately, also difficult to verify, because these nonstandard numbers are infinite cardinalities. So, yes, if there is a proof it will be encoded in one of these infinite cardinalities, which is indeed not a natural number n. — alcontali

Yes.

So I think there's a lot of baggage that comes along with accepting the Gödel sentence in one's logic system. I suggest that the sentence is not truth-apt.

Thanks for your detailed reply - definitely helpful for working through this.

However that Wikipedia link doesn't provide an external reference and a search on "arithmetic unsoundness" doesn't return anything useful. From my own investigation, it seems instead that ~G being true is understood as correct for non-standard models. See the quotes below.

Any model in which the Gödel sentence is false must contain some element which satisfies the property within that model. Such a model must be "nonstandard" – it must contain elements that do not correspond to any standard natural number (Raatikainen 2015, Franzén 2005, p. 135). — Truth of the Gödel sentence - Wikipedia (italics mine)

Namely, if there are independent statements such as GF, F must have both models which satisfy GF and models which rather satisfy ¬GF. As ¬GF is equivalent to ∃xPrfF(x, ⌈GF⌉), the latter models must possess entities which satisfy the formula PrfF(x, ⌈GF⌉). And yet we know (because PrfF(x, y) strongly represents the proof relation) that for any numeral n, F can prove ¬PrfF(n, ⌈GF⌉). Therefore, no natural number n can witness the formula. It follows that any such non-standard model must contain, in addition to natural numbers (denotations of the numerals n), “infinite” non-natural numbers after the natural numbers. — 2.6 Incompleteness and Non-standard Models - SEP (italics mine)

On the other hand, I completely agree that in the model M, the sentence G loses its intuitive meaning. That’s the key to this business.

G is a statement that’s all about natural numbers. It’s supposed to encode the English language statement “I am unprovable”. But what it actually says is a bit more like this:

“There is no natural number n that is the number of a proof of this statement”.

(The idea, remember, is that Gödel figured out a way to assign numbers to proofs.)

However, the model M contains nonstandard natural numbers as well as the usual ones. In the model M, one of these nonstandard numbers is the number of a proof of G. So, G does not hold in the model M.

So, G doesn’t hold in M, but that’s because M has nonstandard numbers. We can loosely say that there’s a nonstandard number which is the number of an “infinitely long proof” of G. — John Baez - professor of mathematics at the University of California, Riverside

Going back to this again.

Yes, there is a problem there. In a nonstandard model it would say that G is provable, while it isn't. — alcontali

Consider a non-standard model where ~G is true.

Since ~G says that G is provable then, if ~G is true, G is provable. Now ~G is true (in that model) therefore G is provable (in that model).

It seems you disagree with this. Which part and why?

Yes, there is a problem there. In a nonstandard model it would say that G is provable, while it isn't. — alcontali

But we're discussing non-standard models where ~G is true. Since ~G says that G is provable then, if ~G is true, G is provable.

However since G isn't derivable from the axioms of the theory, G is not provable. Therefore there can be no models where ~G is true.

Which would seem to render discussion of models with ~G as true as moot.

Yes, I get that provability means true (or false) in all models.

What I don't get is why the negation of G shouldn't be interpreted as saying that G is provable. Since G is saying that G is not provable, then it seems to me that to negate G is just to say that G is provable.

We're obviously interpreting ~G differently, but I don't understand how you're interpreting it, nor what you think I'm specifically getting wrong in the above.

If, in a nonstandard model, G is false, then ~G is true there. — alcontali

Is that the same as saying, "If, in a nonstandard model, G is false, then G is provable there"?

So, per the earlier Wikipedia quote, what does it mean that the Godel sentence (G) is false in some (non-standard) model of Peano arithmetic? Since that implies that G is provable, isn't that an inconsistency?
— Andrew M

No, ~G would still not be provable, because to that effect G needs to be false in ALL models.
Provability of G means: G is true in ALL models
Provability of ~G means: G is false in ALL models
If it is true in some and false in others, that means: it is not provable nor disprovable in the theory. — alcontali

Working through the logic, G is:

• This sentence is not provable

So ~G is:

• The sentence "This sentence is not provable" is not true

Now ~G (as with G) is not provable since it can't be derived from the axioms of the theory. Therefore, according to what you've said, ~G (as with G) must be true in some models of the theory and false in others.

However since ~G negates its inner sentence, ~G unpacks further as:

• The sentence "This sentence is not provable" is provable

Or, more simply:

• G is provable

If that is correct, then it seems that ~G can't be true in any model of the theory, since G isn't provable.

There is no such thing as the consistency of a model. They fit the bill (of the theory) or they don't. — alcontali

OK. So, per the earlier Wikipedia quote, what does it mean that the Godel sentence (G) is false in some (non-standard) model of Peano arithmetic? Since that implies that G is provable, isn't that an inconsistency?

This is a hard question! — alcontali

I see the issue as similar to the Halting problem. To the question of whether the pathological program halts or loops, the answer is that such a program cannot exist. Similarly, to the question of whether the Godel sentence is true or false, the answer, it seems to me, is that without a process for inferring truth (or falsity), such a sentence cannot be truth-apt. It's simply a pathological sentence like the Liar sentence.

Thanks for the explanations.

Gödel's incompleteness theorems also imply the existence of non-standard models of arithmetic. The incompleteness theorems show that a particular sentence G, the Gödel sentence of Peano arithmetic, is not provable nor disprovable in Peano arithmetic. By the completeness theorem, this means that G is false in some model of Peano arithmetic. [Wikipedia] — alcontali

OK. Does that just mean that non-standard models of Peano arithmetic are inconsistent? Or is there more to it than that?

The next sentence from that Wikipedia entry says:

However, G is true in the standard model of arithmetic, and therefore any model in which G is false must be a non-standard model.

Can there be an alternative arithmetic model where the Godel sentence is neither true nor false?

And, if so, can such a model be both consistent and complete?

True(S) will actually work fine. It is True(%S) that does not work. Funnelling sentences through the number-theoretical module of the system in order to determine their truth is not allowed. However, you are still allowed to funnel it through the pure logic module of the system with True(S). — alcontali

OK. What I'm suggesting is that an arithmetic sentence could be defined as only being true (or false) if it is, in principle, provable from axioms (or in the case of a false sentence, that its negation is provable). On that premise, the Godel sentence:

• This sentence is not provable

would be neither true nor false. To expand:

If the Godel sentence were false then it would be provable. But in a consistent system, only true sentences are provable. Therefore it cannot be false.

Alternatively, if the Godel sentence were true then it would not be provable. But that contradicts the above premise that true sentences are provable. Therefore it cannot be true.

Thus, on the initial premise above, the Godel sentence would not express a proposition. In this way, it would parallel the Liar sentence which is also normally considered neither true nor false. And also the Halting problem, which no program can in principle satisfy.

Returning to this...

The formal statement of Tarski's undefinability theorem is, of course, expressed in terms of the diagonal lemma:

That is, there is no L-formula True(n) such that for every L-formula A, True(g(A)) ↔ A holds.

So, there does not exist such number predicate True(%S) because there would always exist exceptions to the proposition that: S ↔ True(%S). That would render the entire theory inconsistent. — alcontali

Yes, in effect the diagonal lemma can generate the Liar sentence, as follows:

• This sentence is not true

Which, in turn, leads to contradiction in the system. However it raises the question as to what it is about the Liar sentence that makes it problematic. As I see it, the deeper reason is that it promises a sentence that can be evaluated as true or false. However instead of delivering on that promise, it merely returns the same sentence which makes the same promise (and therefore entails an endless cycle).

The Truth-Teller ("This sentence is true") is similar in that it also doesn't deliver a sentence that can be evaluated, merely the promise of one. While assuming it to be either true or false doesn't lead to any contradiction, that truth value is arbitrary. Again not particularly useful or informative.

Now consider the Godel sentence which is, in effect:

• This sentence is not provable

Unlike the Liar sentence, assuming the Godel sentence is true doesn't lead to contradiction. Whereas assuming the Godel sentence is false does lead to contradiction.

So the usual conclusion is to accept incompleteness - there is at least one sentence in the system that is true but unprovable. Though an alternative conclusion is that the system is complete but inconsistent.

However there is a third alternative. Specifically, the Godel sentence has the same problem as the Liar and Truth-Teller sentences. It promises to deliver a sentence that can be evaluated (such as "12 > 3" which is provable or conversely "12 < 3" which is not provable (or provably false)). But instead of delivering on that promise, it merely returns the same sentence which makes the same promise (and again entails an endless cycle).

The seemingly natural solution would be to exclude Provable(%S) as a primary predicate (as with True(%S), per Tarski) thus preventing the Godel sentence from being generated via the diagonal lemma.

A belated thanks for your reply! I've spent some time working through the proof of the diagonal lemma and do have a few questions.

However I thought I'd first attempt my own ELI12 explanation of the diagonal lemma. Any critiques welcome.

Consider the following sentence:

• Simon says this sentence

That is an example of a sentence that refers to itself. What Simon purportedly says is the sentence itself and the predicate is Simon says. Notice that the sentence is always true when Simon says it and always false otherwise. If we name the sentence as S, we can express that result as:

• S is true if and only if Simon says S

Now suppose that the language we're considering doesn't support the word this (as is the case for certain formal languages). However it does support variables and substitutions. We can construct the sentence 'Simon says v' where v is a variable and have v be replaced by the entire quoted sentence. The result would be:

• Simon says 'Simon says v'

That substitution procedure is called diagonalization. But we have a problem. If Simon says the full sentence (meaning that it's true that Simon says "Simon says 'Simon says v'"), then what the sentence itself asserts is false (since Simon didn't just say "Simon says v"). The problem is that the inner and outer sentences express different ideas. But there is a solution. We can construct the sentence to instead refer to the result of the substitution procedure, i.e., 'Simon says the diagonalization of v'. Diagonalizing that sentence gives:

• Simon says the diagonalization of 'Simon says the diagonalization of v'

And now we're done! If Simon says the full sentence, then he is saying the diagonalization of the inner sentence. Which is just what the sentence itself asserts. So the full sentence is the fixed point of the predicate Simon says. It is true if and only if Simon says it. Naming the sentence as S₂:

• S₂ is true if and only if Simon says S₂

Generalizing the above for any predicate (F) and sentence (X), we get:

• X ↔ F(X)

Which is the diagonal (fixed point) lemma. (Ignoring Godel numbering to keep it simple.)

From here it is a small step to Tarski's Undefinability Theorem and Godel's Incompleteness Theorems. Just replace the predicate F with IsNotTrue and IsNotProvable respectively and work through the implications.

It is just about the fact that we can define the "isNotProvable" predicate as a number predicate. — alcontali

If no true but unprovable X has been found to satisfy "X ↔ isNotProvable(%X)", then why should we consider it to be a satisifiable definition?

Also, would "X ↔ isNotTrue(%X)" be considered a satisifiable definition? I assume not, but that then raises the question of the criteria for judging that a definition is satisfiable.

Thanks, that was very interesting.

OK, so my understanding is that one sentence that hits (true,true) for isNotProvable is the sentence that asserts that it is itself not provable. How is that expressed as a mathematical sentence?

Also, why is it thought to be true? Is it simply that assuming that it is provable leads to contradiction?

OK, so to go back to the third step in your initial post, K could be isNegative. And so isNegative(%M) is false. Then, per the fourth step, any false sentence will have the property isNegative.

Given the above, it seems that there doesn't have to be a true statement that is not provable. There could instead be a false statement that is provable. So it would be a choice between incompleteness and unsoundness?

I'm also curious about what happens with isFalse. It seems that it will never hit the diagonal. Is that a problem for the diagonal lemma?

Take any property, e.g. isLargeNumber. Say that isLargeNumber is true for numbers above 10^20, and false for numbers that are smaller.

Now, the diagonal lemma says that you can always find a true sentence for which isLargeNumber is true. You can also always find a false sentence for which isLargeNumber is false. — alcontali

So a very long sentence could have a Godel number above 10^20 (say, "0=0 and 1=1 and ... 10^20=10^20"). And that sentence could be true or false.

But how about the property isNegativeNumber? That property will never be true.

OK, to briefly recap:

The sentence "12>3 ; 12-3>3-3 ; 9>0" is a proof of the theorem "12>3". So the sentence "12>3" is both true and provable. It thus hits the diagonal for that sentence and property (in this case, isProvable).

Also, any sentence can be encoded to (and decoded from) a unique natural number (the Godel number).

The diagonal lemma says that it is always possible for any arbitrary property about numbers to hit the diagonal. This means that you can always find a true sentence that has the property but also a false sentence that does not have the property. — alcontali

This is the part I don't understand. How is this proved?

So, counting letters in utf8 is just another numbers game. — alcontali

Cool. No problem with that.

It is also just another numbers game. Say that "12>3" is a simple theorem in arithmetic. Then, the following sequence of sentences is one possible proof:
1) 12>3
2) 12-3>3-3
3) 9>0
In 3) we hit axiom 15 in the equivalent axiomatization: i.e. zero is the minimum element.

So, now we convert the theorem to a number:

utf8("12>3")=49506251
utf8("12>3 ; 12-3>3-3 ; 9>0")=495062513259324950455162514551325932576248

So, now we can say that theorem 49506251 is provable because it is associated to another number. 495062513259324950455162514551325932576248, which is its proof. Therefore, the predicate isProvable(49506251) results in true. — alcontali

Thanks! Very clear.

Still, in an idealized world the "isProvable" predicate can really be implemented. — alcontali

Meaning that ideally every true statement has a corresponding proof and every false statement has no corresponding proof (or perhaps a corresponding disproof)?

And is that the diagonal? (i.e., false/not provable, true/provable)

From there on, it will still not be able to avoid hitting the diagonal lemma: there exists a true sentence which "isNotProvable". — alcontali

OK, how does this part work?

Well, the term "heavy" was probably a bad choice. I couldn't think of a some good predicate, because the literature pretty much never mentions one. It needs to be something calculable true or false about a sentence. The literature typically says something like: "The sentence S has property K". Any idea of what could be a good example for K? — alcontali

I think either something about the form of the sentence (e.g., HasAnEvenNumberOfLetters) or about the derived Godel number as a number (e.g., your earlier IsEven and IsPrime predicates). IsProvable doesn't seem to meet that criteria since it depends on the meaning of the sentence.

Well, since "heavy" is just an arbitrary choice, I wouldn't worry about that. As long as you can calculate the property from a number, it should be ok. For example: "the face is green" should probably work better, because a face can be represented as a number, and figuring out that is green, is just a calculation on its bits and bytes. So, the diagonal lemma says that it should always be possible to construct a face that is green, but also one that is not green. — alcontali

I'm not sure I follow. "heavy" is a predicate in your third step. But "the face is green" is not a predicate. It's fine as a sentence however (with a Godel number).

I guess what I'm having trouble with is step 3 where you're looking for a property that could not possibly apply to any sentence, yet is predicable of it. Perhaps something that is necessarily false? Such as HasZeroLetters (where a valid sentence must have 1 or more letters).

Especially its proof is considered to be incomprehensible. That is a problem because both Gödel's first incompleteness theorem and Tarski's undefinability theorem trivially follow from this lemma. Let's attempt to come up with a more intuitive explanation anyway. — alcontali

I like what you're attempting here. The first two steps seem okay to me.

Third step. Let's now specify a property that could not possibly apply to any sentence

Let's define K(%M) = false. Say that K means "heavy". (It could mean anything, really). There is no number or associated sentence that is "heavy" because K always returns false. We just do not want a heavy sentence in our system. Seriously, we are trying to shoehorn the situation here to prevent anything from being provably heavy. — alcontali

On my view, this is a category mistake. Sentences aren't the sort of things that are or are not heavy, so K(%M) wouldn't be truth-apt.

Look at that! We now have a sentence P that has the property of being heavy, even though K, the heaviness predicate, always returns false! How is that possible? — alcontali

It can happen when non-truth-apt sentences are treated as truth-apt. Similar to manipulating the liar sentence or "the King of France is bald" sentence.

From your later post:

3. utf8( 1=1 or 99=99 or 0=0 ) = 496149321111143257576157573211111432486148 ⇐⇐ BINGO!

Example 3 is true and even. So, it is a solution, i.e. a witness to the lemma. Just for the sake of the argument, let's now try to find an arbitrary false sentence that is odd:

4. utf8( 0=1 ) = 486149 ⇐⇐ BINGO!

Example 4 is false and odd. So, it is also a solution, i.e. a witness to the lemma. — alcontali

Nice example. Makes sense to me.

So, you can also play this game with the "isProvable" predicate, but the really interesting game is with the "isNotProvable" predicate, because as soon as you win the game by finding a true sentence, you will have a true sentence that also "isNotProvable", and that is considered to be an astonishing result in metalogic and metamathematics. — alcontali

So would this show that some true sentences are not provable, or just that those "true" sentences are problematic in some way, similar to the liar sentence or the above "heavy" sentences?

"All things are subject to interpretation. Whichever interpretation prevails at a given time is a function of power and not truth." -- Somebody other than Nietzsche — frank

There could be any number of reasons why a particular interpretation prevails. Epistemology deals with the norms for assessing claims, of which power and authority are only two possible issues. But truth itself is not one of those norms, it is simply what the interpretation asserts, and is being assessed.

For example, Donald says it is raining outside. Should Rudy accept his assertion?

Perhaps Donald is a power figure. He might fire Rudy for disagreeing with him. Or perhaps Donald has a PhD in Meteorology. If anyone knows whether it's raining, surely he does. Or perhaps Donald is reliable and trustworthy, the sort of person who doesn't make rash assertions.

Maybe Rudy can ask around to see what others say, or perhaps launch an investigation. Or, if all else fails, he could look out the window and see for himself. Though there are always edge cases where people can be mistaken about what they thought they saw.

What truth has going for it is that the world stands behind it. Nonetheless, it can also be the last option remaining after people have exhausted all the preferred alternatives.

3. Develop a culture that treats the commons with respect. — Banno

Yes, but how is that culture developed? Elinor Ostrom (a political economist) won a Nobel prize in 2009 for her findings on this:

Contribution: Challenged the conventional wisdom by demonstrating how local property can be successfully managed by local commons without any regulation by central authorities or privatization.
...
Work: It was long unanimously held among economists that natural resources that were collectively used by their users would be over-exploited and destroyed in the long-term. Elinor Ostrom disproved this idea by conducting field studies on how people in small, local communities manage shared natural resources, such as pastures, fishing waters, and forests. She showed that when natural resources are jointly used by their users, in time, rules are established for how these are to be cared for and used in a way that is both economically and ecologically sustainable. — Elinor Ostrom - Nobel Prize

Elinor Ostrom and her colleagues looked at how real-world communities manage communal resources, such as fisheries, land irrigation systems, and farmlands, and they identified a number of factors conducive to successful resource management. One factor is the resource itself; resources with definable boundaries (e.g., land) can be preserved much more easily. A second factor is resource dependence; there must be a perceptible threat of resource depletion, and it must be difficult to find substitutes. The third is the presence of a community; small and stable populations with a thick social network and social norms promoting conservation do better.[47] A final condition is that there be appropriate community-based rules and procedures in place with built-in incentives for responsible use and punishments for overuse. When the commons is taken over by non-locals, those solutions can no longer be used. — Non-governmental solution - Wikipedia

Those "community-based rules and procedures" are an example of internalizing the externalities that Wallows mentioned. Their function is to ensure that those that benefit from the shared resource also bear the costs of their use that would otherwise be borne by others.

And, finally:

A resource arrangement that works in practice can work in theory. — Ostram's law

What is the standard to prove to you mind body dualism? — MiloL

There isn't a standard. Mind-body dualism is the result of a language confusion. As Gilbert Ryle put it, "Descartes left as one of his main philosophical legacies a myth which continues to distort the continental geography of the subject. A myth is, of course, not a fairy story. It is the presentation of facts belonging to one category in the idioms appropriate to another. To explode a myth is accordingly not to deny the facts but to re-allocate them."

It's like watching a game of football played in good spirit and then wondering how that spirit interacts with the teams. Or where it goes when the teams leave the field.

I don't know what sort of problems the Chinese were solving when the encountered negative numbers so that's a dead-end for this discussion. — TheMadFool

It's a fascinating topic. For the details, see Solving a System of Linear Equations Using Ancient Chinese Methods by Mary Flagg. Two quotes from that paper:

"The Nine Chapters is a series of 246 problems and their solutions organized into nine chapters by topic. The topics indicate that the text was meant for addressing the practical needs of government, commerce and engineering."

"[ I] read the Nine Chapters as a boy, and studied it in full detail when I was older. [ I] observed the division between the dual natures of Yin and Yang [the positive and negative aspects] which sum up the fundamentals of mathematics." - Liu Hiu (the third century mathematician)

--

So the Cartesian number line and the Chinese red and black rod system are different ways to conceptualize negative numbers. In the former, a negative number is less than zero. In the latter, positive and negative numbers are duals. "Nothing" (zero) can either be the absence of a number or the secondary consequence of duals canceling, such as with a $2 sale and a $2 purchase.

To illustrate, here's one problem from The Nine Chapters on the Mathematical Art:

Problem 8: Now sell 2 cattle and 5 sheep to buy 13 pigs. Surplus 1000 cash. Sell 3 cattle and 3 pigs to buy 9 sheep. There is exactly enough cash. Sell 6 sheep and 8 pigs, then buy 5 cattle. There is 600 coins deficit. Tell: what is the price of a cow, a sheep and a pig, respectively?

Note that the Chinese considered selling as positive and buying as negative. So the 2 sold cattle would be represented by 2 red rods, the 5 bought cattle would be represented by 5 black rods and "exactly enough cash" would be represented by a blank space.

This problem is on p10 of the linked paper, the answer on p15 and the ancient Chinese array along with the modern matrix representations are on p30.

My question is how can it be that the Chinese knew about negative numbers, defined as numbers less than zero, and didn't know about zero itself? — TheMadFool

According to Wikipedia they did know about zero but just lacked a symbol for it.

Red rods represent positive numbers and black rods represent negative numbers.[7] Ancient Chinese clearly understood negative numbers and zero (leaving a blank space for it), though they had no symbol for the latter. — Counting rods

As an example of how that might work, suppose you have $2 of assets and $5 of debt. In determining your net financial situation, you note that a dollar of debt negates a dollar of assets. So this can be represented with a row of 2 red rods and a row of 5 black rods. Removing 2 red rods and 2 black rods leaves 3 black rods (and a blank space where the red rods were). Thus, in effect, you have $3 of debt and no assets.

But it is a mistake to say that QM definitely shows us particles being in multiple locations. It doesn't. — petrichor

Agreed, the measurements are the data points. Whereas the underlying structure of the world is still a matter of interpretation.

One interesting thing to realize is that nobody has ever even seen a photon in flight! Such things might not even exist except in models. — petrichor

Since photons are the means by which we see things, I suppose we would need to bounce photons off the in-flight photon in order to see it. Unfortunately photons don't interact with each other. And even if they did, a photon would be too small to see. Nonetheless a human retina can respond to a single photon (in a dark room).

Also QM is generally considered to scale up to the whole universe. For example, quantum behavior has been observed in objects visible to the naked eye such as with the piezoelectric tuning fork experiment (with about 10 trillion atoms).

Are you thinking that would simply indicate that the subconscious mind and the conscious mind are working in unison? (This is not a rhetorical question: If so, how is that explainable?) — 3017amen

I suppose I don't really think in terms of "subconscious mind" and "conscious mind", I just see mind as an abstraction over an agent's intelligent activity. If an abstraction creates a philosophical problem, we can just go back to the concrete scenarios (e.g., a driver attentive to the immediate environment versus thinking about something else). The investigation of the processes involved seem to be a matter for science.

Hey Andrew, just curious, would that be more in keeping with an Idealist model? — 3017amen

"Consciousness causes collapse" has a history of being associated with quantum mysticism and is generally dismissed by physicists these days.

The other interpretations are usually considered realist by their proponents. This paper provides a useful classification into so-called intrinsic realist interpretations (including Pilot wave theory and Many Worlds) and participatory realist interpretations (including Copenhagen, RQM and QBism). So that may be of interest.

Accordingly, I was thinking about the conscious and subconscious mind creating two separate realities:

"Sometimes, you are so much into cognitive processes and imagination that your existence shrinks down to only physical presence because you are mentally somewhere else. Missing road turns while driving or adding wrong ingredients while cooking are common examples in this regard."

Does that mean we can perceive two realities at one time viz. our consciousness or conscious states of Being? — 3017amen

I read that quote as noting that sometimes we are highly focused on our immediate environment, whereas at other times we are distracted or thinking about other things. I don't see any out-of-the-ordinary philosophical implications there.

Yep, the great analogy continues. — ZhouBoTong

Glad you enjoyed it! There's a lot of interesting work being done in quantum foundations and learning it is an effective way to give one's philosophical assumptions a workout.

I don't know enough about QM to know if you are right with this analogy, BUT I SURE HOPE YOU ARE :smile: This seems a great analogy that does help even idiots like me to understand. — ZhouBoTong

Thanks! It's even adaptable to your favorite interpretation:

Pilot wave theory: An invisible river guides the boat.

Many Worlds: There is a boat on each fork of the river.

Copenhagen: There is no river until you launch the boat.

RQM: In your reference frame there is a boat on the river.

QBism: You should believe there is a boat on the river.

Consciousness causes collapse: Your mind creates the river. And the boat.

Instrumentalism: We don't talk about the river.

For one possible physical picture, consider a river that forks around an island. You could represent the river symbolically as:

River = left fork + right fork

The river exists in both locations. You could also navigate a boat along the river, i.e., along either the left fork or right fork. As long as one is clear on whether one is talking about the overall state (River) or one of simpler states (e.g., the left fork), then there need be no confusion.

Aside: The actual physical picture outside of measurement is a matter of interpretation. What physicists do agree on is that the superposition provides information for calculating the probability of measuring the particle in one of the locations.

Yes. It seems fairly intuitive in this case, and the only reason i brought it up is that it illustrates another feature of Monty's knowledge that is somewhat forgotten but necessary. - that Monty knows not only where the car is, but also what your choice is and responds selectively to both. — unenlightened

Exactly. :up:

You are just making my point. You choose to include elementary particles into things that you call "objects." — SophistiCat

Yes, but it's motivated by the ordinary definition, not an arbitrary choice. I think the relevant characteristic of objects is that they are concrete things that can be observed or have their properties measured. That's compatible with "a material thing that can be seen and touched".

The question of whether objects can occupy the same space at the same time or not seems to me to be more associated with classical physics rather than something necessarily associated with the term.

What does it mean, precisely, for two things to be in the same quantum state? — petrichor

A quantum state contains all the information about a quantum system. For two photons to be in the same state means there is no information, in principle, that distinguishes them. Which leaves us with cardinality (i.e., 2 photons) but not individual photon identities.

For why this is significant, consider the Hong–Ou–Mandel experiment again. Suppose the photon entering the beam splitter at the top is named A and the photon entering at the bottom is named B. In state 2, photon A ends up at the bottom and photon B ends ups at the top. Whereas in state 3 photon A remains at the top and photon B remains at the bottom. However according to QM, there is no information that would physically distinguish those two states - they are physically equivalent. So their respective amplitudes add which manifests as destructive interference, as observed. Whereas classical physics (in which those states are physically different) is unable to predict what is observed.

Also, when it comes to interference effects, aren't we just adding waves, like in the example of water ripples I gave earlier? And isn't the wave in this case a probability wave? — petrichor

Yes. But it's worth noting we're not observing waves, we're observing particles at specific positions or with specific spins (or whatever we choose to measure).