So, without observers, nothing is related to anything whatever. That is the thrust of the OP — Wayfarer

It seems to me that the OP is about how the mind explains the world.

A sub-topic is about does the mind explain the world using mathematical necessity or causal contingency.
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As per above, this question can be asked not only of patterns, but of phenomena generally..........................There are patterns that appear in inorganic nature, in crystals, snowflakes, larva formations etc. — Wayfarer

I agree with the introduction to Pinter's book Mind and Cosmic Order

The book’s argument begins with the British empiricists who raised our awareness of the fact that we have no direct contact with physical reality, but it is the mind that constructs the form and features of objects. It is shown that modern cognitive science brings this insight a step further by suggesting that shape and structure are not internal to objects, but arise in the observer.

I read this as saying that patterns exist in the mind, not the world.

↪Philosophim What a cromulent response! :smile:

This just pushes the question back a level -- why is it effective? — J

Ha ha, glad you got the reference! It is accurate because it is not contradicted by reality, and that's it. If I can hold a word, phrase, or set of built up claims that are not contradicted by reality, then I have 'accuracy'. Meaning that accuracy is not about capturing 'what really is'. Its about creating a representation of the world that isn't contradicted by 'what really is'. If we use deduction, and the steps that lead to knowledge I noted in the paper, its the most reasonable way to think about the world, and the closest we can ever get to capturing 'what really is'.

But we can have multiple terms and viewpoints of the world that are effective. For example, we can use math in base 2,3,4 or any really. They're all 'accurate'. This is where the other two points I made come into play. The most likely reason we use base ten is because we have ten fingers, which are great tools to start counting with. And that's really it. We use base two in logic gates, because there are only two states for a gate "on or off". We use hexadecimal for memory, as it turns out to make more sense when you combine several binary gates together for functionality.

Wouldn't it make more sense for you (if I've understood your thinking here) to abandon any talk of accuracy or truth? — J

No, accuracy is still extremely important, and should be the ideal goal of knowledge. Accuracy doesn't mean necessarily that you've discovered what is true, but it does mean that you're not in contradiction with what is true. Truth of course being, 'What is'.

Religion is a good example of holding a set of terms that may be both accurate and useful depending on culture and a lack of other terminology. Lets say you don't have an objective morality, but common people who till fields all day need a guide. You know that government isn't well liked by people, after all, they take your money with taxes. People like to have a greater purpose then just living. We need people to treat each other right to build a society. The idea of a "God" that can answer complicated questions that these types of people do not have the education, or real need to consider, can be incredibly useful.

Why should I not steal from my neighbor? Because God, your creator who loves you/will kill you has deemed it so. Its part of a greater plan then yourself. When you die, you'll go to heaven/hell based on your performance here. So be happy that tilling the fields is fulfilling Gods plan, don't lust after your neighbors hot body when you're already married, raise your kids well, and treat people right.

"God" in this case is not necessarily contradicted by reality. "Why can't I see God?" "God is all around us, but you can't see or hear him." "How do I know there's an afterlife." "Have faith, don't you feel it inside of you?"

Of course, increased education like neuroscience and epistemology start to reveal that "God" is simply a plausible invention of the mind, not a real applicable reality. But for a common person tilling fields, "God" may actually give them greater purpose, and answers the basic life and moral questions that one might think of while plowing a field at 2 in the afternoon. Satisfaction with their lot in life generates good work, and a life that feels fulfilled. It can create a common ideology the village can gather around that isn't government. It can give a sense of freedom and personal worth.

Suppose bleen is "green and "first observed" during or before 2004," or "blue and 'first observed' after 2004." Could you go walk around where you live and determine what was grue or bleen? — Count Timothy von Icarus

If you understood the essential properties and context of what grue and bleen is, then yes. Notice I say context, because the context of what grue is to a scientist who studies light waves and a local and colloquial understanding of grue can be different. The important part is to first establish what the definition is in context. Only then can you go about applying that definition to confirm or deny its match.

Suppose there is a famous green landmark in your town and it got flattened by a tornado in 2006. It was rebuilt with largely with materials salvaged from the original, but has a substantial amount of new material. Is it bleen or grue? — Count Timothy von Icarus

Same response.

What if only small parts of it were replaced each year since 2004? — Count Timothy von Icarus

Seems a bit Theseus here, so why don't we go to the source?

Really the same situation applies. What does the culture think is Theseus ship? There might be a society that believes ownership only happens for five years, and after five years the original thing has changed so much that it is something completely knew and it needs to be registered with the state again.

We can have another culture that believes ownership is tied with purchase and then use. So as long as he uses the ship, its his no matter the part replacements. But if he abandons it for a year, its no longer his.

Its all about "What are the definitions in the context?" Once you have that, then you do applied knowledge. The problems often come in thinking that applied knowledge happens, then we wonder what the definitions are. That's an improper use of knowledge. That's when we're discovering what the definitions should be.

In the case of grue and blue, we're not asking what the definitions should be, we already know what they mean. Our decision is then, "should we use grue or blue"? And I mentioned earlier, as long as both are accurate to the point they are not contradicted by reality, its really a personal choice. It may be as simple as, "I like one word more than another" or as complicated as, "Those dirty grueians are a stupid people that I find inferior. Blue is obviously superior and the 'right' way to identify a color."

A theorem (as opposed to an equation that's given a real-world interpretation) isn't described as effective, it's described as true, or at least provable in L. — J

It depends on your definition of 'true'. If you mean 'true and false' as logically valid, that's not the same as 'true means what is real'. 'Real' being, "What exists despite our knowledge".

Do you want to abandon that way of talking? If a correspondence theory of truth demands that we do so, I'd argue that it represents a reductio ad absurdum and should be rejected on that ground. — J

In the case of 'truth as reality', there is no correspondence theory of truth. The end result is, "Knowledge is the best logical process we have to understand reality, but it cannot prove that it is a 100% accurate comprehension of truth. That is beyond human limits"

And it all comes down to the fact that 'reality' does not care what we think. Tomorrow we could discover physics is wrong. Does that mean we didn't know physics? Of course not. Today with all we can reason, physics is what is known. So we go with that until shown otherwise. Maybe physics is a 100% accurate representation of truth. Maybe one day we'll discover that its not. But logically, its the best thing we have right now.

If a correspondence theory of truth demands that we do so, I'd argue that it represents a reductio ad absurdum and should be rejected on that ground. — J

Very harsh. The word "true" has different meanings in different contexts.

As regards the equation ${d = 0.5 * g * t^{2}}$, i) I can say that it is true, meaning that it works, ii) I can say that it is true, meaning that it has the correct structure of an equation, iii) I can say that it is true, meaning that it is important to me, as in a true friend, iv) I can say we don't know whether it is true or not, as we don't know what obtains in the world, v) I can say I don't know if it is true or not, as I don't know what the symbols represent.

The scientist can use meaning i) for "truth" and the philosopher can use meaning iv) for "truth".
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What would be the point of limiting ourselves in this way? — J

Yes, the correspondence theory does not tell us about things like pain, which, I believe, doesn't exist in the world.

In the proposition "it is true that I feel pain", what purpose does the word "true" have, in that the proposition says no more than "I feel pain"

It seems that "truth" only arises in the correspondence between the mind and the world.

If you understood the essential properties and context of what grue and bleen is, then yes

You can tell when things were created down to the year just by looking at them? When you drive through a neighborhood you know the year each house was built just as readily as the the color it is painted?

the case of grue and blue, we're not asking what the definitions should be, we already know what they mean. Our decision is then, "should we use grue or blue"? And I mentioned earlier, as long as both are accurate to the point they are not contradicted by reality, its really a personal choice. It may be as simple as, "I like one word more than another" or as complicated as, "Those dirty grueians are a stupid people that I find inferior. Blue is obviously superior and the 'right' way to identify a color."

This seems like "because difference is not a logical contradiction it is arbitrary."

"Personal preference," isn't uncaused anyhow, people prefer things because how the world is. Human languages distinguish between shape and color for reasons other than arbitrary social convention for instance. One can feel shape but not color.

There is variance in how disparate cultures that developed in relative isolation cut up the world. At the same time, there isn't nearly the variance we should expect if this was entirely arbitrary.

Likewise, it is not purely in virtue of arbitrary social convention that "leap year biology," "a chemistry of odd numbered days," and a physics for each sex are not used as divisions of the sciences. If someone argued that chemistry should be split into chemistry done by people with blue eyes and chemistry done by people with other colored eyes, or argued that we should divide chemistry into pre and post 1990 chemistries, or a chemistry of federally recognized holidays, they would be rebuffed for non-arbitrary reasons. This might be filtered through "personal preference," but personal preference doesn't spring from the aether uncaused and neither do our concepts and languages.

If you understood the essential properties and context of what grue and bleen is, then yes

You can tell when things were created down to the year just by looking at them? When you drive through a neighborhood you know the year each house was built just as readily as the the color it is painted? — Count Timothy von Icarus

I misunderstood the question then. To tell the age that a house was built, you need verifiable records and trust in their accuracy.

This seems like "because difference is not a logical contradiction it is arbitrary." — Count Timothy von Icarus

No, I'm not saying that. It is arbitrary in terms of 'accuracy'. It is not arbitrary in terms of individuals or cultures. As I noted, language can be a matter of preference and power in addition to accuracy. Preference and power are arbitrary in relation to accuracy, if the only thing one cares about is accuracy. But people often care more about things other than accuracy. Would I elevate these differences to the same level as 'intelligent decisions"? No.

Human languages distinguish between shape and color — Count Timothy von Icarus

This would be about accuracy. Saying, "That feels like the color red" is something only a person with synthesia would experience. So we people without synthesia don't view the above sentence as having any accuracy.

If someone argued that chemistry should be split into chemistry done by people with blue eyes and chemistry done by people with other colored eyes, or argued that we should divide chemistry into pre and post 1990 chemistries, or a chemistry of federally recognized holidays, they would be rebuffed for non-arbitrary reasons. — Count Timothy von Icarus

Yes, like the idea that 'slaves shouldn't read'. Such a division is about cultural power over others, not anything accurate to the nature of being a slave.

This might be filtered through "personal preference," but personal preference doesn't spring from the aether uncaused and neither do our concepts and languages. — Count Timothy von Icarus

I never implied that it did. I'm just noting that personal preference can override why one would use a term over another if you had two terms that accurately described the same situation under consideration. And in some cases, personal preference can override accuracy. Of course, the accurate term will be much more likely to accurately reflect the world, but if one is not interested in that, then it is dismissed.

three kids who want to play with the same toys — Leontiskos

Common solutions: we introduce other toys so that everyone gets something (not an option in our example); no one gets it (not allowed in our example); they each get the whole thing because they will play with it together (not helpful for consumables, as in our example, which is why we split them); we divvy up not the toy but the time playing with it, take turns, and we can even measure the duration of those and make them equal-ish.

(My youngest son used to come to me brandishing minor premises like this: "Dad, don't you think Andy has been on the computer a long time?" Andy is his older brother. Just look at the beautiful deviousness of this, the command of rhetoric ― how young they master the dark arts of persuasion.)

I go to the car dealership and I am offered whole cars. They don't let me buy a half car for half the money. — Leontiskos

You know what would be cool? What if you could give them part of the price and someone else, say a bank, gave them the rest, and you got to take the whole car right now. Then you could have an arrangement with the bank where you pay them back over time, a little each month, and the bank could charge interest to compensate them for the risk they're taking. I think that could catch on! I think that could end up being the way almost everyone buys a car.

See also.

I agree with the introduction to Pinter's book Mind and Cosmic Order

The book’s argument begins with the British empiricists who raised our awareness of the fact that we have no direct contact with physical reality, but it is the mind that constructs the form and features of objects. It is shown that modern cognitive science brings this insight a step further by suggesting that shape and structure are not internal to objects, but arise in the observer.

I read this as saying that patterns exist in the mind, not the world. — RussellA

But there's a danger here of 'subjectivising' the whole question of the nature of logic, principles, and mathematical regularities. Pinter is not advocating subjective idealism, although he mentions it. His thesis is that all cognition (and not just human, but any organism's) is organised around gestalts, meaningful wholes, by which we navigate the environment. This applies as much to bees gathering honey as a scientist gathering data. So whilst these gestalts originate in the mind, there is no possibility of understanding or cognizing the world apart from them, as without them we would be presented with featureless chaos. So to say they don't exist 'in the world' is to miss the point. There would be no 'world' without them.

So what I think needs to be questioned is why we think a hard and fast separation can be made between mind and world in the first place. Going back to the original post:

The question raised is whether a distinctively mathematical explanation (DME) for physical facts truly exists – whether “the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws.”

If you substitute “logical” for “mathematical,” you can see that the question is very much about whether our analytical (for lack of a better term) knowledge imposes itself on the physical world in a way that is genuinely explanatory. Jha et al. think not, and present a strong argument that all purported DMEs are actually rephrased or disguised versions of the causal explanations that ordinarily obtain. — J

So this is really a question about the relationship of logical necessity (DME) and physical causation (contigent causal laws). And behind that question, I think there's a lurking pre-supposition, namely, that logical necessity can only be 'in the mind', while physical causation is a matter of empirical (hence contingent) fact. This all goes back to David Hume ('the godfather of positivism') and his scepticism with respect to causal relations. But, as per Kant's answer to Hume (and, I think, supported by Pinter), even though logical necessity is restricted to the relationship of ideas, the causal regularities of nature are presupposed by reason in order for us to make sense of nature.

I think the sentiment against this relationship goes back to the same basically nominalist and empiricist attitude that animates most analytical philosophy, which is unwilling to admit that ideas - mathematical ideas, logical principles - might possess any kind of intrinsic reality which can't be reduced to 'contingent causal laws'.

So what I think needs to be questioned is why we think a hard and fast separation can be made between mind and world in the first place. — Wayfarer

:100:
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I think the sentiment against this relationship goes back to the same basically nominalist and empiricist attitude that animates most analytical philosophy, which is unwilling to admit that ideas - mathematical ideas, logical principles - might possess any kind of intrinsic reality which can't be reduced to 'contingent causal laws'. — Wayfarer

In our mind we have mathematical ideas such that ${d = 0.5 * g * t^{2}}$ (distance with time of a falling object under gravity), and logical principles such as the Law of Non Contradiction whereby I cannot feel pain and not feel pain at the same time.

If mathematical laws cannot be reduced to contingent causal laws, what caused d to be ${0.5 * g * t^{2}}$ rather than ${3.0 * g * t^{2}}$, for example?

If logical principles cannot be reduced to contingent causal laws, what caused my inability to feel pain and not to feel pain at the same time, for example?

This is good. I would amend it slightly: Jha et al are asking the very question about whether logical principles can be more than contingently causal. They’re not at all “unwilling “ to entertain the idea; it’s the topic of their paper. They do end up arguing against it, true.

This is Jha et al’s argument, more or less. Math only appears to be causal when we state the problem in terms that remove, or demote to “background conditions,” the physical constraints that actually provide the explanation.

Having not read the paper, what do they mean by "causal?" This seems a lot more plausible if they are making the common contemporary move or only considering efficient cause.

It seems to me that the distinction in play here is also between essence and existence. What something is does not explain that it is, but it may very well explain formal cause even from a purely mathematical frame.

Jha et al. argue that various presuppositions such as the principle of mass conservation and the physical integrity of individual objects (this has to be assumed in order to get “whole and unbroken”) — J

Disagree, because the question is clearly independent of the exact physical content. You can draw arbitrary boundaries on the world and apply this question.

Q3: Why must the LNC hold (under the usual constraints) as a principle of thought?
Q4: Why can’t my cat be on my lap and in Paris at the same time? (constraint: I live in Maryland) — J

Again, I think these rules are so abstract that they do not depend on the physical content. The fact your cat can't be simultaneously be in Maryland and Paris is because you have chosen to define "Maryland" and "Paris" in ways that are mutually exclusive and so one is not the other. But there is no need to do this for any physical things, whether in the trivial senses we talk about all the time or in more fundamental ways; for instance, some quantum interpretations will ascribe an ontological realism to the idea that a thing can be in two places at once in the sense suggested by traditional conceptions of quantum superposition.

The fact your cat can't be simultaneously be in Maryland and Paris is because you have chosen to define "Maryland" and "Paris" in ways that are mutually exclusive and so one is not the other.

We haven't chosen this arbitrarily though. The difference between North America and Europe is not something we simply stipulated. If there is good empirical evidence to believe anything at all then surely there is good warrant to believe that when one steps off the Metro at the Silver Springs station one is not likely to find oneself staring at the banks of the Seine.

But there is no need to do this for any physical things, whether in the trivial senses we talk about all the time or in more fundamental ways; for instance, some quantum interpretations will ascribe an ontological realism to the idea that a thing can be in two places at once in the sense suggested by traditional conceptions of quantum superposition.

Well, I suppose ones attitude towards reductionism and smallism will probably guide the extent to which one thinks quantum foundations is particularly relevant here. On the one hand, there seems to be increasing consensus around the idea that there is no hard dividing line between "quantum and classical worlds." On the other, there is strong consensus in physics that the same living cat cannot be simultaneously in College Park and strolling the the Champs-Élysées.

If we are unsure that being in Rome, New York differs from Rome, Italy, I think we have left empiricism and the natural sciences behind a long time ago.

I'd add that collapsing these sorts of differences really makes Odysseus's taking 10 years to make it to Penelope inexcusable. :rofl:

This is Jha et al’s argument, more or less. Math only appears to be causal when we state the problem in terms that remove, or demote to “background conditions,” the physical constraints that actually provide the explanation. — J

This might be a possibility.

Is 1+1=2 a mathematical necessity in the world?

It initially seems that nothing in the world has caused 1+1 to be 2, supporting Lange's DME that mathematical necessity is stronger than contingent causation.

Where P = 1+1. The antecedent, the prior situation in the world.
Where R = 1+1=2. The explanans, the necessary mathematical law.
Where Q = 2. The consequent, the subsequent situation in the world.

Let P be one apple alongside apple, in other words 1+1
Let Q be a pair of apples, in other words 2.

But for there to be a pair of apples, this presupposes a prior relation between the individual apples.

In other words, the existence of a pair of apples has been caused by a prior relation between the individual apples.

Therefore, the mathematical law that 1+1=2 is a consequence of a prior relation between 1 and 1.

The prior relation between parts is the cause of the necessity of the mathematical law that 1+1=2.

(However, a subsequent question is, do relations ontologically exist in the world?)

We haven't chosen this arbitrarily though. — Count Timothy von Icarus

No, but in general things are not mutually exclusive like that in our language use. The same physical scenarios can have overlapping descriptions which is not mutually exclusive.

Well, I suppose ones attitude towards reductionism and smallism will probably guide the extent to which one thinks quantum foundations is particularly relevant here. On the one hand, there seems to be increasing consensus around the idea that there is no hard dividing line between "quantum and classical worlds." On the other, there is strong consensus in physics that the same living cat cannot be simultaneously in College Park and strolling the the Champs-Élysées.

If we are unsure that being in Rome, New York differs from Rome, Italy, I think we have left empiricism and the natural sciences behind a long time ago.
40m — Count Timothy von Icarus

I don't think it matters whether it is actually the case or not; what matters is whether someone cpuld plausibly hold this kind of conception of the world in a cohetent way.

But for there to be a pair of apples, this presupposes a prior relation between the individual apples.

Quite right. This seems to me like a manifestation of the classic problem of the "One and the Many." That is, how can being be in one sense "one," i.e. everything interacts with everything else, there are no truly isolated systems, etc. yet at the same time "many" in that we experience very many types of things, and things like appels that are "one" in being the same sort of organic unity, while being "many" in being numerically distinct.

This question was sort of the question of early physics and metaphysics (Aristotle helpfully opens with what might be the first literature review in history on this topic in both the Physics and Metaphysics), and I'd argue that it still sits at the center of much philosophical debate today.

Can we speak of unifying generating principles (causes) producing (but also unifying) a multitude of diverse species? Maybe. For my own part, I think current work in interdisciplinary fields like complexity study, information theory, cybernetics, etc. do offer a sort of vindication of this way of resolving the problem (dominant for a good span of philosophical history) at least.

I wonder if we might think of Hume and co. as a symptom of Taylor's "buffered self," to some degree. Over here is a thing in the world and I (or "my brain") creates a "representation" of it. And in turn I might use language to represent that representation, but I remain separate from the thing represented in a strong sense.

There is a sort of divorce between the mind and the world implicit here.

By contrast, we might consider a dry river bed, which contains information about rain that has fallen nearby over the years. I don't know if it makes sense to think of a dry riverbed as a "representation" of water, but nonetheless it is a sign of water (or a collection of signs). If we examine these signs closely enough we can actually discover a great deal about rain or even vegetation patterns from a very long time ago. And, although the rain and the riverbed are different things, they don't seem to be "divorced" in our usual way of thinking of them. Indeed the rain is what causes the riverbed to exist at all.

Jha et al are asking the very question about whether logical principles can be more than contingently causal. They’re not at all “unwilling “ to entertain the idea; it’s the topic of their paper. They do end up arguing against it, true. — J

I'm doing a bit more reading on this, but meanwhile I found Marc Lange's academic homepage which is definitely worth the visit:

He is the author of the blockbuster bestseller Laws and Lawmakers: Science, Metaphysics, and the Laws of Nature (Oxford, 2009), which is soon to be a major motion picture. More recently, he has written the iridescently lyrical “Because Without Cause: Non-Causal Explanations in Science and Mathematics” (Oxford, 2016). Earlier he wrote the gripping page-turner An Introduction to the Philosophy of Physics: Locality, Fields, Energy, and Mass (Blackwell, 2002) and the steamy, heart-rending Natural Laws in Scientific Practice (Oxford, 2000).

:lol:

Any word yet on casting for L&L? I heard Tom Cruise was in the running to play Science. Or was it Metaphysics . . .

I have input a Q&A into ChatGPT which you can peruse if you're interested

1) So, the neo-logos philosophies might say something like, "If nature has patterns, and our language has patterns, and we are derived from nature, it may be the case that our language is a necessary outcome of a more foundational logic". Thus, the logic would not be transcendental, but (for lack of a better term) "immanent" in nature, not some outside observing entity that is detached from it.
2) Kant never explains why our minds would compose such a world, but evolution does.
3) Wouldn't evolution put a connection between the efficacy of the mind and the world?
4) But what is the world outside of an observer?
5) Otherwise it's just "I have believe" without an explanation, which though is valid in terms of asserting an idea, is not necessarily valid as an fully informed reason for why you think that way. — schopenhauer1

1) 2) 3) I agree in the sense that I believe in Enactivism, where life has evolved over 3 billion years in a dynamic interaction with its environment, meaning that life is a part of the world, not outside it.

2) Yes, Kant argues in the CPR that we have transcendental knowledge of a world the other side of our senses, but never explains how this could work.

4) We are an intrinsic part of the world as the tree is an intrinsic part of the world, but it doesn't follow that because we are an intrinsic part of the world we must necessarily have knowledge about the world that we are an intrinsic part of.

The problem remains that our only knowledge about any world the other side of our senses (sight, sound, taste, touch, smell) arrives through these very same senses.

5) For myself, as an Indirect Realist, I believe that there is a world the other side of my senses that has caused what I experience in my senses, and can justify my belief in such a world.

The problem remains, we may reason about what has caused the experiences in our senses, and arrive at beliefs about any world the other side of our senses, but how can we transcend our senses in order to know what has caused these experiences in our senses?

I know the colour red from my sense of sight. I may have a reasoned belief as to its cause, but how can I ever know its cause?

Even Enactivism (that life has evolved as an intrinsic part of the world) provides no mechanism whereby we can know what exists the other side of our senses.

That is, how can being be in one sense "one," i.e. everything interacts with everything else, there are no truly isolated systems — Count Timothy von Icarus

Yes, Aristotle considered the Platonic argument that he called "The One Over The Many", whose premiss might be expressed as "Whenever two or more things can be properly said to be F, it is by virtue of some thing, F-ness, that they are properly called F"

In the world are observed numerous atoms (philosophically speaking).

For the observer, atom A and atom B are part of one apple and atom C is part of a different apple.

For me, the problem about objects existing in a world independent of any observer, is, what mechanism is there in such a world independent of any observer that relates atom A to atom B but doesn't relate atom A to atom C?

If a correspondence theory of truth demands that we do so, I'd argue that it represents a reductio ad absurdum and should be rejected on that ground.
— J

Very harsh. The word "true" has different meanings in different contexts — RussellA

So, to review the set-up here: The question is whether a standard correspondence theory has to invoke a match between statements in a language and facts about the world. If so, this would seem to rule out using such a theory to describe strictly logical or mathematical facts as true. Yes, you could say that, to demonstrate such facts, we don't need a correspondence theory and therefore we can call them true using a different meaning of the word. But isn't that a stretch? The appeal of a correspondence theory is that it seems to give a common-sense reply to the question of what it means for a statement to be true: We compare it to what is the case, and if it fits, bingo. Obviously it can't be that simple, but the concept is still powerful.

So, Euclid's proof about prime numbers claims to state a truth. When we examine it, we see that while it isn't exactly a "truth about the world," or something that is made true by facts found in the world, it nonetheless appears to express a match of language with something. What is the something? For me, none of the usual-suspect answers involves giving up using "true" to mean "agrees with what is the case." So we shouldn't limit a robust correspondence theory to "facts about the world" -- there are plenty of other facts that we want to call true in the same way. I can't defend that here, but I just wanted to give you a sense of where my objection is coming from. And yes, maybe calling this version of the theory a reductio is a bit harsh, but at the very least it requires a strong argument against its implausibility.

So we shouldn't limit a robust correspondence theory to "facts about the world" — J

The question is, where is this "world", inside or outside the mind.

The relationship between the Correspondence Theory and Truth is an enormous subject, as indicated by the SEP articles on Truth and The Correspondence Theory. The Merriam Webster Dictionary has a range of definitions of "truth", and the word "truth" may be used both literally and as a figure of speech.

Even if we adopt the schema "snow is white" is true IFF snow is white, which relates something in language to something in the world, the next question is where is this world. The answer depends on individual opinion, whether one is a metaphysical Realist or Metaphysical Idealist, never mind all the sub-divisions of these two positions.

For example, are there infinitely many prime numbers.

Initially, we can use the Schema and say that the proposition "there are infinitely many prime numbers" is true IFF there are infinitely many prime numbers.

However, Euclid has proved that there are.

Therefore, we can use the Correspondence Theory and say that the proposition "there are infinitely many prime numbers" is true because there are infinitely many prime numbers.

The proposition "there are infinitely many prime numbers" exists in language.

The next question is, where do these infinitely many prime numbers exist.

In particular, where does the number 23 exist? Some say only inside the mind and some say both inside and outside the mind.

However, regardless of where the prime numbers exist, the Correspondence Theory is still valid.

Therefore, we can use the Correspondence Theory and say that the proposition "there are infinitely many prime numbers" is true because there are infinitely many prime numbers. — RussellA

Good, I think we're on the same page. A correspondence theory ought to work independently of the ontological status of various "worlds"; that was why I questioned limiting it to "facts about the world," which from your example I took to refer to the more-or-less-physical world that would exist without us. But now I see that your view is more inclusive, so that's fine.

Kant argues in the CPR that we have transcendental knowledge of a world the other side of our senses, but never explains how this could work. — RussellA

We can ‘see’ things through deductive inference that are not empirically knowable. There’s a sense in which even arithmetic is transcendental in that it reveals aspects of nature which sense could not otherwise discern.

how can we transcend our senses in order to know what has caused these experiences in our senses? — RussellA

In pre-modern philosophy this was generally associated with transcending the self, although that is made more explicit in Eastern philosophies such as Vedanta and Buddhism. In those philosophies the over-valuation of sense-data is deprecated as the root cause of delusion although plainly this has an ethical dimension which is generally absent in Anglo philosophy and the overwhelmingly sensory orientation of modern culture.

IIRC, it's Metaphysics 7, about midway through the chapter to the end where he really gets into essence and identity. Unfortunately, it isn't super straight forward because he is actually building up a case which makes it look like he is contradicting himself, but really he is pursuing a reductio argument that takes a long time to pay off.

For me, the problem about objects existing in a world independent of any observer, is, what mechanism is there in such a world independent of any observer that relates atom A to atom B but doesn't relate atom A to atom C?

Well, a question here is what it means to be "independent from observers." In a certain sense, everything we think of is, in at least some sense, not independent of observers. We have thought of it, therefore it is not independent of our thought. It is in this very broad sense that Parmenides contends that "the same is for thinking as for being."

But what about some sort of ontological dependence relationship? I would say the weight of virtually all empirical evidence is that an apple being an apple doesn't depend on us specifically for its existence. When we leave a room, the apples don't vanish. We can tell they continue to exist because they are subject to corruption. being eaten by mice, etc. while we are gone. We don't crash our cars because once we can no longer see our tires they vanish. In an important sense, since everything is casually connected, we are never really "independent" of things we are not currently experiencing anyhow.

But in the modern context, we might assume that the intelligible whatness of things, their quiddity, is a "mental construct," such that even their position in space and time, along with all their describable qualities is "in our heads." I think modern phenomenology has done a good job recovering the ancient and medieval solution to this problem. Robert Sokolowski's The Phenomenology of the Human Person is a particularly good example. That "everything is received in the manner of the receiver," need not force us to conclude that the world is unknowable precisely because knowledge of things "in themselves" should never have been elevated over knowledge of "things as they interact with anything and everything else." "Act follows on being." If something doesn't interact with anything else, it might as well not be.

I also like Eric Perl's Thinking Being (which IIRC Wayfarer gave good reviews too) which gets at the premodern attitude on this issue. Far from being "naive realists" there was simply a refusal to accept intelligibility as the result of a sui generis act of the mind.