Because we can prove what the result would be, we do not have to actually carry out the pairing of every rational number with a natural number. Proof is a further refinement of prediction, beyond even calculation. Of course it's impossible to count the elements of an infinite set as you would the elements of a finite set. But for the results we're interested in here, you do not need to. That is the point. We already know what the result would be if it were in fact possible. — Srap Tasmaner

I demonstrated already, it's not a proper "proof" because it relies on a false premise. This produces your incoherent, unsound conclusion "we can prove what the result would be". The incoherency of proving that the result of an impossible task is anything other than incompletion, is obvious.

I can put it another way: what you cannot calculate, you must deduce. — Srap Tasmaner

Deduction from false premises produces absurdities. That's what Zeno is famous for having demonstrated.

We don't need much ontology. Quantification will suffice. — Banno

Maybe not much, but some. Claiming that numbers "exist" is ontology. If you avoid the ontology, then what are you quantifying?

How do you know that the natural numbers go on for ever? — Ludwig V

Mathematical ideals are produced by definition. People decided that this would be really good, and so the system was designed and maintained that way.

So they are countable in the sense that some of them can be counted and we cannot find any numbers in the sequence that cannot be counted. — Ludwig V

But the issue is whether an infinite quantity is countable. Any finite quantity is, in principle countable. But, since "infinite" is defined as endless, any supposed infinite quantity is not countable.

This problem is strictly confined to Platonism, which treats a number as an object which can be counted. So it inheres within the principal axioms of set theory which premises mathematical objects. Numbering is the means by which we measure a quantity of distinct, individual things. When we assume that a number is a distinct individual thing (Platonism), then we might be inclined to measure the quantity of numbers (cardinality).

The problem which jumps out, is that now we are trying to measure the measurement system with itself. And the ontological issue is that it is fundamentally false to represent a number as a distinct individual thing which can be counted. In reality "a number" is a concept which has its meaning in relation to other numbers (ordinality). Therefore we cannot isolate "a number" to be a distinct object, it would lose its meaning and no longer be able to serve its purpose as the concept it was meant to be. Therefore Platonism, which treats ideas as distinct objects which can be counted is ontologically unacceptable.

Ah, so this is about actual and potential infinities. My problem with that is that I don't see how the idea of a possible abstract object can work. — Ludwig V

I'll give you a brief description why abstract "ideas" are classed as potential by Aristotle. This forms the basis of his claimed refutation of Platonism, and provides the primary premise for his so-called cosmological argument which demonstrates that anything eternal must be actual.

The Pythagorean Platonists, as distinct from Aristotelian Platonists, insisted that ideas, specifically mathematical ideas, had actual existence as eternal objects, eternal truths in themselves. Aristotle premised that it is the geometer's mind which gives actual existence to the ideas. The actual existence of the idea is within the mind. Therefore if we premise that the idea had existence prior to being "discovered" by the mind (as the assumed eternal), we must conclude that it existed potentially. (The cosmological argument then goes on to show that anything eternal must be actual.) So the refutation of Pythagorean Platonism sets up a distinction between an idea within a mind, and the supposed independent idea. Within the mind it is actual, and however it exists in the medium outside of minds, is as potential. So when we assume numbers to be independent objects rather than thoughts within a mind, then they exist potentially, not as actual objects.

The philosophical parameters for the debate what it means for a mathematical (abstract) object to exist are well enough defined, so that's the debate we are really involved in. — Ludwig V

This is probably the heart of the issue. "Exist" is a term which is properly defined by ontology rather than mathematics. Therefore the discipline of ontology is the one which ought to determine whether numbers exist. Notice, @Banno makes some seemingly random claims about the existence of numbers. Since the distinction between what exists and what does not exist forms the basis for our judgements of true and false, we can't simply make an arbitrary, or completely subjective stipulation, or axiom, which defines "exist". That would imply total disregard for truth.

A grasp of what the problem actually is, rather than misrepresenting what it arises from, might be helpful. — Mww

I think that the problem is that I stated a specific problem. Then Paine produced a quote from Kant, which appeared like it sort of addressed the problem I raised, but really addressed a slightly different problem. Therefore we are actually conflating two different problems. So the assumption of "the problem" is somewhat misleading because I raised one problem, and the quote from Kant addressed a different problem, and i treated it as if it was supposed to address the problem I raised. The problem I raised wasn't ever really addressed.

There's a category error that involves thinking that because we can't start at one and write down every subsequent natural number, they don't exist. — Banno

There's an ontology which presumes that numbers exist, it's called Platonism. It's been demonstrated to be a very problematic ontology, and many philosophers claim that it was successfully refuted by Aristotle, as inconsistent with reality.

It is also well-known that those issues do not arise in the same way at the macro scale. — Srap Tasmaner

That's the problem with this type of issue. The supposed universal principles work extremely well in the midrange of the physical domain. Since the midrange is our worldly presence, and that is the vast majority of applications, we tend to get the impression that the principles are infallible, and "true". However, application at the extremes evidently produces problems. Therefore we must take the skeptic's eye to address the real possibility of faults within the supposed ideals.

Logic and mathematics are mental tools or technologies, habits of mind, that we have developed for dealing with things at the macro scale. — Srap Tasmaner

What you call "the macro scale" is really the midrange, the realm of human dealings. Other than the micro scale and the macro scale, we need a third category which might be called the cosmological scale.

This is unsurprising since our mental lives consist, to a quite considerable degree, of making predictions. Logic and mathematics enable us to figure out ahead of time whether the bridge we're building can support six trucks at once or only four. — Srap Tasmaner

It is true, that this midrange scale, what you call the macro scale envelopes pretty much the entirety of our day to day lives. However, as philosophers with the desire to know, we want to extend our principles far beyond the extent of the macro scale. And this is where the issue of incorrectly representing infinity may become a problem.

For example, let's say that the macro scale is in the range of 45-55 in a scale of 0-100. So we might hypothesize and speculate about that part of reality beyond our mundane 45-55 range. If the application of mathematics, to the physical hypotheses leads to infinity in both directions at what is really only 35 and 65, then we have a problem because we place the majority of reality beyond infinity. And, if we close infinity by making it countable, then there is no way for us to know that there is even anything beyond 35 and 65. It appears from our physical hypotheses that we have reached infinity, therefore the extreme boundaries. And, if the mathematics has closed infinity, in the way that it does, then by that principle we actually have reached infinity. Therefore, by that faulty closure of infinity, 35 and 65 are conclude as the true ends of the universe, the true limits to reality, when reality actually extends much further on each side.

Which leads, at last, to my point, such as it is: there is something perverse, right out of the gate, about the insistence on "actually carrying it out". It misses an important point about the value of logic and mathematics, that we can check first, using our minds, before committing to an action, and we can calculate instead of risking a perhaps quite expensive or dangerous "experiment". ("If there is no handrail, people are more likely to fall and be injured or killed" -- and therefore handrail, without waiting for someone to fall.) — Srap Tasmaner

I don't see how this is relevant. The issue is not properly with "actually carrying it out", the problem is with the assumption that it is possible to carry it out. The defining feature of "infinite" renders it impossible to carry it out. So when we say that it is possible to carry out something which is defined as impossible to carry out, this is a problem regardless of "actually carrying it out".

This denigrates the status of "impossible". Now, "impossible" is a very important concept because it is the most reliable source of "necessity". When something is determined to be impossible, this produces a necessity which is much stronger and more reliable than the necessity of inducive reason. So the necessity of what is impossible forms the foundation for the most rigorous logic. For example, the law of noncontradiction, it is impossible for the same thing, at the same time, to both have and have not, a specified property. this impossibility is a very strong necessity. In mathematics, the impossible, and therefore the guiding necessity, is that we could have a count which could include all the natural numbers. if we stipulate that this is actually possible, then we lose that foundational necessity.

The natural numbers turn out to go on forever, and we can prove this without somehow conclusively failing to write them all down. — Srap Tasmaner

So this exposes the problem. We know that the natural numbers go on for ever. Therefore it is impossible to count them, or that there is a bijection of them. They could not have all come into existence therefore it is impossible that there is a bijection of them. This impossibility is a very useful necessity in mathematics. So if we stipulate axiomatically, that it is possible to count them, or have a bijection, then we compromise that very useful necessity, by rendering the impossible as possible.

To see the demonstration that the rational numbers are equinumerous with the natural numbers and complain that it is not conclusive because no one can "actually do them all" is worse than obtuse, it is an affront to human thought. — Srap Tasmaner

This is a misrepresentation of what I am arguing. My claim is that it is definitively impossible to count the numbers. Therefore to represent this as possible is a contradiction. This has nothing to do with whether a human being, computer, or even some sort of god, could "actually do them all". The system is designed so that they cannot be counted. Nothing can do them all, and this is definitional as a fundamental axiom. So, whether or not anything can actually do them all is irrelevant because we are talking about a definition. Therefore, to introduce another axiom which states that it is possible to do them all, is contradictory.

Allow me to apologize if my previous replies came off as an attempt to ridicule you. That was not my intention. — Esse Quam Videri

No, my apology too, I didn't intend to imply that you have done this, in particular. But I might mention@Banno, and a few other members in the past.

I see that what I've said so far has not convinced you. That's understandable. That said, I'm not sure I have the ability to express my critique any more clearly than I already have. I say that not in an attempt to blame you for misunderstanding me, but more as an acknowledgement of my own limitations in that regard. I still stand by my arguments, but I'm not sure how to productively move the discussion forward from here. Thanks. — Esse Quam Videri

As I said, I've learned a lot in my past discussions, so I'll offer you a perspective which you may be able to make sense of. Let's suppose that bijections simply exist without needing to be carried out as a procedural thing. This might be what's intended with the term "function".

For example, imagine that there is forty chairs in a room somewhere. There is simply an existing bijection between the chairs and the integers, so that the count is already made without having to be counted. It's just a brute fact that there is forty chairs there, without anyone counting them. This is a form of realism known as Platonic realism. The numbers simply exist, and have those relations, which we would put them into through our methods, but it is not required that we put them into those relations for the relations to exist.

I discussed this before, with @Banno I believe, in a discussion about the nature of measurement. The example was a jar of marbles. Our common intuition is that when there is a jar of marbles, or something like this, there is a measurement, a count, associated with it, the number of marbles which are in the jar. "Truth", or the correct count, would be to produce a count which corresponds with this already existing relation. You can see that this is completely different from a procedural "correct count". The procedural correctness is produced by performing the procedure correctly according to the rules, and the answer then is the correct answer without any necessary assumption of an independent measuring system (Platonic Ideals) already related, "truth".

The reason this issue came up, is because of the so-called measurement problem in quantum physics. In quantum physics it has been demonstrated that there cannot be an already existing independent measurement. So measurement is not a case of producing the result which corresponds with the already existing relation, it must be a matter of correctly carrying out the procedure.

Therefore, I argue that this is actually the true nature of "measurement" in all cases, that the correct answer is always a matter of carrying out the prescribed procedure correctly. Consequently I also argue that the Platonic realism which supports the other, intuitive notion of measurement, that there are independent numbers, which are already associated with things, as the true measurement, is misleading. This issue becomes very evident in the notion of infinity.

The formal definition I provided to you (or similar variation) is the one you will find in many of the standard textbooks on Real Analysis, Set Theory and Discrete Mathematics that discuss countably infinite sets. This is why it confuses me when you say that you don't believe that this is the standard formal definition of "countably infinite". — Esse Quam Videri

You don't seem to understand the problem. "Countably" implies a procedure which you continue to deny. When we looked at the definition of "countable" it is defined by "capable", which implies "able to" perform a specified procedure. Then you claimed that mathematicians use a different definition of "capable" which doesn't imply the ability to perform a procedure. That's when I accused you of intentionally trying to obscure the issue, instead of facing the reality of it.

Likewise, and for the same reason, I am also confused by your insistence that the definitional existence of a bijection requires that the bijection be temporally or procedurally executable. Within the global mathematics community it is commonly understood and accepted that procedural execution is not a requirement for definitional existence. This is why you will not find such a requirement listed in the aforementioned textbooks. This is also why I previously stated that adding this requirement would amount to something like an external constructivist critique of the dominant paradigm. — Esse Quam Videri

Well, it appears like "the global mathematics community" is mistaken then. When something is defined in terms of the capability to perform a procedure, and then it's understood that actually being able to perform that procedure is "not a requirement" for fulfilling the criteria of that definition, then this is obviously a mistaken understanding. Don't you agree? And please, live up to your claim of "open to being mistaken on these points".

I am very much open to be mistaken. I have had numerous discussions with mathematicians on this forum, and have learned a lot, altering my perspective on many things. This issue though, as I see it, is so simple, clear, and obvious, that it would require a substantial argument to prove that I am mistaken here. But that substantial argument has not been forthcoming. People simply assert that I am mistaken, and ridicule me for arguing against "the global mathematics community", as a form of appealing to authority, rather than actually addressing the matter with clear principles.

How on earth do you imagine all the natural numbers? — Srap Tasmaner

I can't, neither can you. Get the point?

If you re-read my reply carefully you will see that I did not say that mathematicians do not use the word "capable", but that they use it in a different way. — Esse Quam Videri

I know you said this, but I do not believe you . The concept of "capable" is very straight forward with very little ambiguity. It means having the ability for. So, if you read through to the end of my post, I requested that you provide this special definition of "capable", which you claim mathematicians are using.

"A is countable" means "∃f such that f is a bijection between A and ℕ". That's it. There is nothing procedural in this definition. That was my point. — Esse Quam Videri

You are wrong again Esse. "Countable" is defined as a form of "capable" which is defined as "ability for". Therefore it is very clear that something procedural is referred to by "countable". Producing a bijection is a procedure. That is the point Magnus took up with Banno. You might obscure this fact with reference to 'function", and insist on a separation between "function" and "procedure" or employ a variety of other terms to veil this reality, but all this amounts to is a dishonest attempt to obscure the facts, deception.

Why do you keep insisting on things which you really ought to know are wrong? That is the problem. Instead of acknowledging, 'oh yeah, there are some problems with mathematical principles, and this is one of them', you go off and try to hide the problem. You see, in philosophy we meet these sorts of problems all the time, everywhere, in metaphysics, theology, free will, mathematics, physics, biology, etc.. Philosophers are critical, and look for these issues, that is critical thinking. Those things always pop up, because knowledge evolves, and what was once cutting edge becomes old, a then the problems get exposed. The faster knowledge progresses the more these issue get overlooked, and they multiply.

Now, philosophical criticism seems to be expected in some fields, relative to ancient ideas like metaphysics, theology, etc.. When a philosopher demonstrates problems in an ancient concept of God for example, this does not surprise anyone. However, in my experience on this forum, there are certain fields, mathematics and physics, for example, where criticism is regarded as unacceptable. It's like the dogma takes hold of the people, and is adhered to in such a religious manner, that criticism (heresy) must not be allowed. Those who faithfully uphold these principles seem to be programmed to disallow criticism. When problems are pointed out, they deny that their chosen dogma and ideology could even have such issues, and use whatever means possible to hide those features.

The critical point here is that these issues, which we as philosophers point out (inconsistencies and contradictions), are not unusual in human knowledge. They are common, widespread, extending throughout all the fields of knowledge. They are nothing to be ashamed of. We all make mistakes, and the human species in general is a growing and learning culture. The real problem arises from failure to recognize mistakes as mistakes, when they are exposed and the ensuing denial. That ought to elicit shame.

This is just one example of the way in which, when you change one feature of a language-game (conceptual structure), you often have to change the meaning of other terms within that structure.
So, "countable" in the context of infinity cannot possibly mean the same as "countable" in normal contexts. In the context of infinity, it means that you can start counting the terms and count as many as you like, and there is no term that cannot be included in a count; the requirement that it be possible to complete the count is vacuous, since there is no last term. It's not a problem. — Ludwig V

Let's say that any language game is always evolving. Someone will dream up a new idea, or a new rule, in one's own private mind, and propose it to the others. They start using it, and if the others accept it, it becomes integrated into the game. If the new rule is not consistent with what's already existing then the others ought to notice this, point it out, and rectify the situation. Adopting it for use, would appear to justify it, and if it is inconsistent with some existing rule, that would be a faulty justification. It's analogous to someone offering you a proposal, and instead of thinking about it, to determine if you really agree, you just accept it, and carry on.

Obviously there is a problem in the concept of "countable". I submit that your proposal would not solve the problem. You are suggesting that when it becomes evident that the recently accepted rule is really contradictory to a previously existing rule, and ought not have been accepted in the first place, that we ought to just alter the definition of the offending word in one of the rules. But this is still not acceptable within a logical system because it amounts to equivocation. What this would do is simply obscure the obvious problem, contradiction, with a less obvious problem equivocation. Then all the problems created by what is really a contradiction would be obscured, hidden and more difficult to determine. This would amount to intentional deception, to recognize a problem of contradiction, then try to hide it behind equivocation. That's like taking a shotgun to your problem, blowing it to smithereens, so that you're left with a multitude of little problems instead of one big one.

For example, how about "there is no rational that you cannot place on the number line"? — Ludwig V

How does this make sense to you? To "place on the number line" is a procedural expression, to use Esse's word. We know that it is impossible to make the procedure of placing all the rationals on the number line. Therefore the proper conclusion and procedural statement is exactly opposite to what you propose: "there will always be rationals which you cannot place on the number line".

Although Kant claims "a sufficient reply" in that passage, I don't think he provides that at all. The problem he says arises from an assumed "difference on kind" between the intuition of space as an object, and the intuition of time as an object. Then he says that if we consider that there is no such difference between looking inward, and looking outward, the difficulty may disappear. I assume that the point is that this becomes two different directions, within the same medium, "intuition" in this case. They are relative, "one of them appears outwardly to the other",

However, I believe Kant's conclusion, which follows, proves that the above premise is false.
He says:

...and the only difficulty remaining is that concerning how a community of substances is possible at all, the resolution of which lies entirely outside the field of psychology, and, as the reader can easily judge from what was said in the Analytic about fundamental powers and faculties, this without any doubt also lies outside the field of all human cognition. — "Critique

The issue is that he now refers to "a community of substances", and questions how this is possible. He concludes that resolution of this "lies outside the field of all human cognition". But the only reason why the resolution to this problem lies outside the capacity of human cognition is that he has incorrectly reduced space and time to two dimensions of the same thing. Assuming this one medium, "intuition", which is apprehended by looking inward (temporally), restricts his capacity to determine a multitude of substances, which requires the spatial intuition for separation.

When we look inward, guided by the intuition of time, as Kant did, to reduce space and time to two distinct directions within a single medium (intuition), we do not apprehend the spatial separation required for a plurality of "substances". This is because by looking inward to uncover the intuitions, we are already within the domain of time. And when we turn around and look outward from this perspective, the spatial separation required for a multitude of "substances" cannot be supported if space and time are of the same kind. We are within the domain of time, the intuition of time governs, and we are actually just looking in a different direction in time.

Contrary to Kant's conclusion, that the separation of distinct substances is "outside the field of all human cognition", we ought to simply conclude that Kant's primary premise is incorrect. The intuitions of space and time are not simply a matter of looking two different directions in the same medium. This is easily supported by our understanding of time, which already gives us two opposing directions, past and future. Since these two are properly understood as "opposite", it is impossible to unite them to produce one direction, which space would be opposed to, as described by Kant. Therefore we can conclude that Kant's premise is unsound, and so is his conclusion.

The key word in all this seems to be "all". You might as well bold it each time you use it. — Srap Tasmaner

I don't see anything special about that word. Why do you think I should embolden it?

You disagree, and so far as I can tell only because anyone who tried to do this would never finish. — Srap Tasmaner

That's right, we know, by the defining features of the system, that no one could ever finish this task. It is impossible, by definition.

So, tell me how it is that you claim "it's a known fact that you can line up all the rationals"? Has someone produced this line of all the rationals, to prove this fact? Of course not, because it is also a known fact that this is impossible to do, because no one could ever finish. What's with the contradiction?

what are you referring to with this phrase, "all the positive integers"? I know what I would mean by that phrase; I genuinely do not know what you mean. — Srap Tasmaner

I probably mean the very same thing as you're thinking. Jgill raised the the issue of the meaning of "countable", and provided a reference. The definition from that referred page was: "capable of being put into one-to-one correspondence with the positive integers". So, think of what a "positive integer" is, a whole number greater than zero, and imagine all of them. Now do you know what I mean?

Here's an example to consider Esse. Would you say that someone is "capable" of producing the entire decimal extension of pi? If not, then why would you say that something is "capable" of being put into one-to-one correspondence with all of the positive integers? Or do you equivocate on your meaning of "capable"?

This statement of yours is neither a theorem, nor a definition nor a logical consequence of anything from within the formal system. This is a philosophical assertion grounded in a procedural interpretation of "capable" that is foreign to the mathematics. All you are saying here is that the impossibility follows from your definition of "capable", and that you think your definition is the right definition. This is an external critique. At no point have you derived a contradiction from within the system. Therefore, nothing you have said so far justifies the claim that the system is inconsistent. — Esse Quam Videri

Esse, please read what is written. I took the definition from a mathematics site, provided by a mathematician, jgill. The definition was "capable of being put into one-to-one correspondence with the positive integers". Please, for the sake of an honest discussion, recognize the word "capable" in that definition. And please recognize that your diatribe about my use of the concept "capable" is completely wrong, and out of place.

"Capable" is not a concept foreign to mathematics. Mathematicians employ the concept of "capable" with the concept of "countable", and surprise, there it is in that definition. You have no argument unless you define "countable" in a way other than capable of being counted. Are you prepared to argue that "countable" means something other than capable of being counted for a mathematician.

Or, are you proposing that mathematicians have their own special definition of "capable", designed so as to avoid this contradiction. Are you proposing that they have a meaning of "capable" which applies to things which are impossible, allowing that mathematicians are "capable" of doing something which they understand to be impossible? If so, then let's see this definition of "capable" which allows them to be capable of doing what they know is impossible to do.

I'm just wondering if you think somewhere in the rest of the paragraph (following the bolded sentence) you have provided an argument in its support. Is this the post you will have in mind when someone asks and you claim to have demonstrated that "Nothing is capable of being put into one-to-one correspondence with all of the positive integers"? Because it's just an assertion of incredulity followed by a lot of chitchat. (I think you have in your mind somewhere an issue of conceptual priority, but it's not an actual argument.) — Srap Tasmaner

Sorry Srap, it seems you haven't been following the discussion. I suggest you start at the beginning.

This does not negate our knowing it by other means. Kant is only talking about reason, rational thought. We are acquainted with the noumenon through our presence in the world. — Punshhh

That is debatable, and it is really the issue I raised already about the difference between Plato and Kant. Plato allows the human mind direct access to the intelligible, without sense mediation. What you call "our presence in the world" is most likely equivalent with Kant's intuition of time. Time he described as the internal intuition, space the external.

The internal a priori intuition is present to us as "time". The external a priori intuition is present to us as space. The two combined form the conditions of sensibility, providing for the appearance of phenomena. Now to support what you claim, we'd have to be able to separate our knowledge of the internal intuition from our knowledge of the external intuition. This would allow us pure unmediated access to the internal aspect of the human subject, as an "in itself" object, without any influence from the external intuition of space, and the consequent phenomenal appearances.

Kant does not take this route though, as his categories all follow from the combined space and time intuitions. Therefore he proceeds from those two a priori intuitions into the empirical realm and the a posteriori. He does not look toward a further analysis of the a priori and does not adequately separate those two intuitions. I would say that perhaps he leaves this route open, as a possibility though. And, I believe that this is generally the way of phenomenology. A person might look at oneself, a human subject, as purely noumenal, but only by looking exclusively at the temporal intuition, and filtering out any influence from the external (spatial) intuition, if this is possible.

Exactly. "Countable" means something very specific within the formalism. The critique provided amounts to a rejection of that notion, not a derivation of contradiction from within the system. — Esse Quam Videri

That's right, "countable" means something very specific. But as I've demonstrated, the meaning of it, as defined, contradicts the meaning of 'the natural numbers extend endlessly'. That's where the problem lies. The natural numbers have been in use for a long time, with a very specific formulation allowing for infinite, or endless, extension. Then, "countable" was introduced as a term with a definition which contradicts the infinite extension of the natural numbers.

Please see my reply to jgill below.

It all depends on how one defines "countable" — jgill

As usual, I agree with you jgill. Here's the definition you provided: "capable of being put into one-to-one correspondence with the positive integers".

Nothing is capable of being put into one-to-one correspondence with all of the positive integers. We might say that the system was designed this way, to be unlimited in its capacity to measure quantitative value, 'to count'. That's why the system was formulated to extend infinitely. The positive integers derive their extraordinary usefulness from being extendable indefinitely, to be capable of counting any possible quantity. Notice, infinite possibility covers anything possible. To allow that the integers themselves may be counted. or to designate that something may be put into one-to-one correspondence with them all, is to say that there is a capacity which extends beyond them, i.e. that capacity to count them. This is to limit their usefulness as unable to measure that specific capacity. To limit the usefulness of the integers is counterproductive to the various disciplines which use mathematics.

All you’ve claimed so far is that mathematicians are working with a notion of infinity that you don’t accept, and you’ve given some philosophical reasons for rejecting it. — Esse Quam Videri

That's not true. The definition of infinity I use is the one used in mathematics, to describe the natural numbers as unbounded, unlimited, without end. I do not reject this definition of "infinity".

The problem is that this is a philosophical objection, not a mathematical one, and as such it doesn’t justify the claim that the mathematical notion of infinity is contradictory. The mathematical definition is perfectly sound relative to the formal system in which it is embedded. — Esse Quam Videri

Again this is not true. The philosophical objection is based in a fundamental logical principle, the law of noncontradiction. I demonstrated that mathematicians employ contradiction when they claim that the natural numbers are countably infinite, or a countable infinity. By the mathematicians' own definition of infinite, or infinity, it is contradictory to say that an infinity can be counted because "infinite" means that we cannot have such a count, it could never be acquired.

By analogy: suppose we’re playing a game of Chess and, on your turn, you legally move your queen from d1 to a4. Suppose I respond to your move by saying: “that move doesn’t make sense because in real life kings are more powerful than queens and so only kings should be able to move like that”. That may be a fine external critique of the rules of Chess, but I haven’t thereby shown your move to be illegal. Given the established rules, it was a perfectly valid move. — Esse Quam Videri

This is not analogous. I clearly show how the move of the mathematicians is 'illegal' (to use your word) within standard rules of logic, because it is contradictory. The natural numbers are defined as infinite, meaning limitless, endless, impossible to count them all. Then they say the very opposite, that the natural numbers are countable. Clearly, "countable infinity" is a contradictory concept where the first term contradicts the second. These are not my definitions which I have made up for this purpose. This contradiction is within the way that mathematicians themselves define the terms.

Likewise, your objection to the mathematical notion of infinity is a meta-level objection. It doesn’t undermine the internal coherence of mathematics as it is standardly practiced. At most, it shows that the standard mathematical notion of infinity conflicts with your own metaphysical views. — Esse Quam Videri

Again, this is wrong. The incoherence is internal to mathematics. The notion of "infinity" used by mathematicians themselves, is contradicted by the predication they make, when they propose a "countable" infinity. Here's an example much better than your chess proposal because the chess proposal fails to capture the situation.

Lets say we have a concept called "unintelligible" (analogous in this example to infinite). Then, we notice that there are different sorts of unintelligible things, that things are unintelligible in a number of different ways Different sorts of infinities). So, instead of studying the reason for, and the difference between, the different ways that unintelligibility appears to us, we simply name one of the forms of unintelligibility the "intelligible unintelligibility" ( analogous to countably infinite). Then we proceed to compare the other forms unintelligibility to this, under the illusion (falsity by contradiction) that we have made this type of unintelligibility intelligible by naming it so.

That is what the concept of "countably infinite" does. It creates the illusion (falsity by contradiction), that this type of infinity is actually countable. It's far better to use a concept like "transfinite", and state that the transfinite are a special type of infinite, but maintain they are not countable. This would exclude the possibility of an infinite set, or a transfinite set as this is the mistaken venture. It is the attempt to contain the boundless, limitless (infinite) into a set which is defined as an object, that requires the employment of contradiction. Putting limits to the limitless is contradictory.

f you wanted mathematicians to take this challenge seriously as mathematics, it would require proposing an alternative formal framework built around your accepted notion of infinity and showing that it does at least as much mathematical work as the existing one. As things stand, no such reason has been given for abandoning the standard definition. — Esse Quam Videri

The standard definition of "infinite" is not a problem whatsoever. so there is no need to abandon it. The proposal of "countably infinite" is a problem.

I clearly explained why it is not necessary, and actually inappropriate for me to propose an alternative framework. If mathematicians do not understand that they have incorporated contradiction within their framework, and so they are not inclined to rectify this, then I will just keep pressing this point. Maybe they never will.

Excellent use of the chess analogy. — Banno

The analogy is not similar. I have shown that the internal rules of the game (mathematics) are contradictory. Unless noncontradiction is not a rule in the game (mathematics), then the analogy fails. Are you and Esse Quam prepared to take that stance, to insist that the rule of noncontradiction is not a rule in the mathematician's game? if so, you might be able to make the analogy work.

I don't agree. Measurement is not comparison. Measurement is finding the numeric value of the measured objects or movements. — Corvus

How would you determine the numeric value of anything without comparison to a scale? That's what the instrument does, it applies the scale to the item and makes a comparison. Think of the tape measure example, a thermometer, a clock, any sort of instrument of measure.

Yes, I know, but the thing’s identity as itself, the first law of rational thought, is not what the transcendental idea “in-itself” is about. — Mww

Why would you say this? I think it clearly is. Aristotle placed the identity of a thing, in itself. The supposed independent thing is affirmed to have an identity as the thing which it is, independent of anything we might say about the thing. What Kant shows is that this proposed "identity", as a thing, is actually unjustified. The "thing", or "object", is what appears to us as phenomenon, but this appearance is the result of the a priori intuitions of space and time. Therefore, we cannot assume as Aristotle did, that the proposed "thing" has any identity as a thing, independent from what is produced by those intuitions.

This effectively deconstructed the foundation of how we relate to the supposed independent. No longer can we utilize the Aristotelian system of material objects each with a unique form, identity, even that assumption is unjustifiable. We cannot even assume that the independent consists of things. Hegel goes even further to discredit the law of identity. But this completely undermines the notion of "truth". By Kant, we really can't have any knowledge about the independent, so truth by correspondence becomes irrelevant. Further, without independent objects with identity, the law of noncontradiction and the law of excluded middle are left as inapplicable.

But there’s no change in the “in-itself”, so any measure in units of time, are impossible. — Mww

You cannot make that conclusion. Kant leaves us incapable of making any judgements of truth or falsity concerning "the in itself". If we make a primary assumption of change, like process philosophy does, then the "in itself" is nothing but activity. We might start with that assumption, but then we'd be left with the question of why do the intuitions of space and time make the "in itself" appear to consist of persistent objects. That is the issue which Whitehead ran into. Ultimately, I think a form of dualism is required, to account for the appearance of both persistence and change.

Rather what the OP specifically referenced, which is the infinite numbers between infinitely minute numbers. — LuckyR

I think this matter still has relevance. It is the issue of division. In reality, everything that we attempt to divide can only be divided according to its nature. Nature dictates the way something can be divided. We cut things up very evenly using instruments of measure, but eventually we get to molecules and then atoms, and we are greatly restricted in our capacity to divide "evenly". However, some things like space and time, we might not find the natural restrictions, and so we would be inclined to apply principles of infinite divisibility. Since the mathematical principle of divisibility (infinite) does not correspond with the real divisibility of the substance (space and time), the uncertainty principle is produced.

I would agree with you if the object of this discussion were 'real' infinity as a 'real-world phenomenon'.
I find this 'real' infinity uncomprehensable, and so any speculation about it's properties, seems, well, at the very least, dubious. — Zebeden

Infinity is a "real-world" phenomenon. We have examples of it as the infinite decimal extension of pi, and of the square root of two. The circle, and the square are extremely useful real world applications, yet the principles which validate their use lead us into these real-world infinities.

We might dismiss the problem by saying there is no such thing as a true square, or a true circle, in the real world, and dismiss these conceptions as ideals without real world validation, but that doesn't resolve anything. It just produces a division between conceptions and the real world, where we allow ourselves to employ false premises for the sake of usefulness, and we lose the epistemic value of "truth". Truth is no longer a requirement for knowledge, and we allow that we are not guided toward the truth.

Instead, we ought to look at these issues, where the ideal does not correspond with the real world, as demonstrations which show where our ideals have been compromised by selecting usefulness over truth. They display where our understanding of reality faulters, as reality is fundamentally different from how we represent it. If you just say "I don't care about the true nature of reality, if the principles serve the purpose that's good enough for me" this is a violation of the philosophical mindset which seeks truth. And if we're always happy with the way things are working now, knowledge never advances.

Still, I would argue that if the 'orthodox' view of mathematical infinity solves more problems than it creates, then so be it. — Zebeden

This is not a good standard because the comparison cannot be made. The problems which are solved can be pointed to and numbered. The problems created are associated with the unknown and cannot be counted, nor can the extent or size of the problems be determined. The resolved problems are finite, the created may be infinite and uncountable. So, for example, we created CFCs, and that resolved a whole lot of different problems which we could point to. However, at that time we didn't know what was going on with the ozone, and we couldn't compare the created problems. This is the issue then, the problems created are hidden within the unknown, and only when they start to fester do we take them seriously, and seek out their depth and roots. The example I use above, which displays the problem of unruly use of infinity is the uncertainty principle. We don't know what is hiding beneath that name.

This is why the discussion keeps looping. If you want to move the discussion forward you need to either (1) derive (not assert) an actual contradiction within the accepted mathematical framework (per ↪Banno) or (2) reject the standard framework and present a coherent alternative (e.g. intuitionism, finitism, non-classical logic, etc.). — Esse Quam Videri

1. The actual contradiction is blatant, and I've stated it.
2. Rejection of the framework because it is contradictory and false, is the task of philosophers. Presenting a coherent alternative is the task of mathematicians. Therefore you are wrong to suggest that the one who refutes the framework is obliged to present another.

At this point there is nothing of substance left to discuss. — Esse Quam Videri

The problem is clear. The mathematicians in this forum refuse to accept the refutation, though it is very sound. Because of this, they refuse to get on with the task of producing a coherent alternative. For the philosophers, "there is nothing of substance left to discuss", because the refutation is clear, and the mathematicians remain in denial. Until the mathematicians accept the refutation, and start again at the foundation, the philosophers will have nothing to offer, and there will be nothing of substance to discuss.

Both of you have raised worries about the “doability” of bijection for infinite collections, which suggests a rejection of the identification of existence with formal definability and consistency. That’s a substantive philosophical position. But if that’s the objection, then it isn’t a matter of showing that the usual definitions lead to contradictions (they don’t), but of rejecting the underlying framework. — Esse Quam Videri

I wouldn't characterize this as "worries". It doesn't worry me at all. I just reject falsity for what it is, and since this matter has little if any influence on my daily life it doesn't worry me.

However I think you should reconsider what you say about contradiction. If "infinite" is defined as without limit, then it is clearly contradictory to say that the bijection could be done. It is also contradictory to say that it is doable. Further, it is also contradictory to say that the natural numbers are "countably infinite". Obviously, "without limit" means cannot be counted, so countable contradicts this.

Framed that way, the disagreement would look less like an accusation about the failure of proof and more like a clash of foundational commitments, which is where I suspect the disagreement really belongs. — Esse Quam Videri

I suggest we call a spade a spade. A falsity is a falsity. A conclusion derived from a false premise is unsound. An unsound argument does not constitute "a proof".

I suppose you could argue that mathematicians produce their own rules, and are not subject to the terms of logic. But what would be the point in giving mathematics such an exemption, to proceed in an illogical way. It seems like it would only defeat the purpose of the pursuit of knowledge, to allow for an illogical form of logic.

Magnus's objections are framed as an internal problem with a proof, when they should be framed as external problems with the process being used. — Banno

This is the reason for the distinction between "true" and "valid". Validity is concerned with the internal process. Truth is concerned with the external relations of the premises. "Proof" requires both, and this is known as soundness.

If Magnus rejects the very idea of infinite totalities... — Banno

Clearly, when "infinite" is defined in the usual way, and the way that we understand the natural numbers to be, "infinite totalities" is contradictory.

In no way does this perspective make it impossible to talk about infinite succession. It only applies standard principles of logic to such talk, denying by the law of noncontradiction things like "infinite totality", and denying as false, premises such as "countably infinite". If application of these standard principles of logic expose some of current mathematics as unsound, then that is a problem, which the mathematicians ought to deal with. They ought to accept this, and not whine about having to throw away a whole lot of work.

So constructivism will not help Magnus here. He must resort to finitism - the view that why for any number we can construct its successor, we can't thereby construct the infinite sequence N

. — Banno

I don't understand this objection. As mentioned above, there is no need to reject the idea of infinite sequence, nor is there a need for finitism. The problem is with the idea that an infinite sequence could be completed. That is talk which is unacceptable.

So, we can talk about tasks which will never be completed, and there is nothing wrong with this talk, it makes sense. We can even define a specific task as being impossible to complete, and this makes perfect sense. We can define counting all the natural numbers as such a task which will never be completed, and there is no problem with talking about this. The problem is when we take a task which we have defined as being impossible to complete, such as counting all the natural numbers. and then start talking about it as if it is possible to complete.

To say that the empirical world “arises also from the cognitive faculties of the subject” is correct if it is understood transcendentally rather than causally. The subject does not produce empirical objects, but it provides the necessary conditions under which anything can appear as an object in a unified world.

Kant is not dividing labor between the subject (general concepts) and Nature (particular things). Instead, he is saying that Nature itself is Nature as appearance, which exists only in relation to the subject’s forms of intuition and categories. To invoke “Nature herself” as the source of particular empirical things is to speak as if we had access to Nature as it is in itself. From Kant’s point of view, that is precisely the illusion his critical philosophy is meant to dispel. — Joshs

You have requested a distinction between a "transcendental" understanding, and a "causal" understanding. Can you explain this difference better, for me? "Nature herself" you say, is not the source of empirical things. So nature is not causal in this respect. And, you describe "the conditions" for empirical appearance, as the a priori intuitions. What could be the cause of those empirical appearances then? As empirical appearances they ought to be understandable, and this implies that we ought to be able to speak of causation. If the human mind itself is not taken to be the cause, then they end up as causeless eternal objects, like Platonic objects.

So, yes, the “in-itself” idea can only refer to itself, but from which occurs a problem for the other cognitive faculties, for a reference to itself contains no relations, hence would be worthless as a principle. — Mww

The relation between a thing and itself is what Aristotle called "identity". The law of identity states that a thing is the same as itself. (Philosophers have argued that it is worthless as a principle.) But it is relevant to the thread because it is known as a temporal relation, constituting the temporal extension of a thing. The thing at one moment is allowed to continue being the same thing at the next moment, as it was, even though accidental properties are changing. So identity, the relation which a thing has to itself, is the defining feature of primary substance.

But I will call out the language of “intelligible objects.” I think this is where a deep metaphysical confusion enters. Expressions like “objects of thought” or “intelligible objects” (pace Augustine) quietly import the grammar of perception into a domain where it no longer belongs. They encourage us to imagine that understanding is a kind of inner seeing of a special type of thing. I'm of the firm view that the expression 'object' in 'intelligible object' is metaphorical. (And then, the denial that there are such 'objects' is the mother of all nominalism. But that is for another thread.)

But to 'grasp a form' is not to encounter an object at all. It is an intellectual act — a way of discerning meaning, structure, or necessity — not the perception of something standing over against a subject. Once we start reifying intelligibility into “things,” we generate exactly the kind of pseudo-problems that Kant was trying to dissolve. — Wayfarer

I agree, there is something incorrect about the language of "intelligible objects". But this is the language which comes from Plato, derived from the Pythagoreans who believed that the cosmos was composed of mathematical objects. This perspective is maintained today by mathematicians who employ the concept of "mathematical objects" as essential to set theory. A philosopher may apprehend the fact that mathematical objects are not objects at all, and claim that this must be a metaphorical use. But make no mistake, the principles of modern mathematics state that they are objects, and require that they are objects, for their logical proofs. So in application "intelligible objects" is not a metaphor, but something stipulated by axiom.

Notice in Plato's divided line, those who use the so-called intelligible objects, mathematicians, and physicists for example, have a knowledge at a lower level than the philosophers who seek to understand the true nature of these so-called intelligible objects. I believe that Aristotle made the first definitive step in separating the intelligible "forms", from the conception of "objects". This he did with the law of identity, which applies to material objects, but not to the intelligible. Intelligibility is fundamentally based in similarity (which is a type of difference) rather than the sameness stipulated by the law of identity. So in a sense, it is the sameness (remaining the same as time passes), that we assign to the material object which makes it identifiable as "an object". This is to have temporal extension, to persist as the same thing. But this also makes it unintelligible, because intelligibility is based in similarity which is a sort of difference. Consequently the material object as "the same as itself" is distinguished from the intelligible, which at each instance of occurrence is similar but recognizably different.

Not really, but ignoring the infinite level of irrelevance of the topic is a pretty important omission. — LuckyR

Why do you say the topic is irrelevant"? The concept of infinite is commonly used in mathematics, so there must be at least some relevance.

Well, no. It is defined as f(n)=n−1 and then shown to be a bijection. — Banno

It is not "shown to be a bijection". It is stipulated to be a bijection. And, it is actually impossible to make that bijection. So what it actually is, is the affirmation of an unjustifiable, impossible, action (bijection). When what is stipulated as done or even doable, as a premise, is actually impossible, this justifies the judgement that it is a false premise.

Yep. that's what a proof does. — Banno

A sound proof requires true premises. A so-called "proof" derived from a false premise, is not a proof at all. Therefore your so-called "proof" is ineptly named because it doesn't fulfil the criteria. It has been refuted. I think you actually know this already, but you tend to deny the obvious brute facts, when they are contrary to what you like to believe in.

Noumenon means literally 'object of nous' (Greek term for 'intellect'). In Platonist philosophy, the noumenon is the intelligible form of a particular. Kant rejects the Platonist view, and treats the noumenon primarily as a limiting concept — the idea of an object considered apart from sensible intuition — not as something we can positively know. And it’s worth remembering that Kant’s early inaugural dissertation already engages directly with the Platonic sensible/intelligible distinction. — Wayfarer

I think the main difference between Plato and Kant, is that Kant denies the human intellect direct access to the noumenon as intelligible object. He describes all of our understanding of any supposed noumenon as derived through the medium of sensation, and those a priori intuitions of space and time.

Plato, on the other hand thought that the human intellect might have direct, unmediated access to the intelligible objects, to apprehend and understand them directly as noumena. This is elucidated by the cave allegory, where the philosopher is able to get beyond the realm of sensations, and grasp with the mind's eye the intelligible objects directly. At this point, instead of the medium of sensation imposed by Kant, Plato proposed "the good" as that which illuminates intelligible objects, so that the philosopher may apprehend them directly.

Notice the difference, instead of sensation and the a priori intuitions coming between the intellect and the noumenon, Plato has the intelligible objects being illuminated by the good, so that the intellect may grasp them directly. This is the highest part of the divided line analogy.

I am not asking for anything. I am just stating that any act of reading measurements is involved with some sort of measuring tools. You cannot read size, weight or time with no instruments or measuring tools. The measuring instruments or tools become the part of reading measurements. You cannot separate them. — Corvus

Actually, measurement in its basic form, is simply comparison. So no "instrument" is required for basic measurements. If Jim is short, and Tom is judged as taller, that is a form of measurement. The tools, standard scales, and instruments, just allow for more precision and complexity, for what is fundamentally just comparison.

To take photos of the speeding cars, it uses camera vision, not the radars. Radars are used for mostly flying objects in the sky and aeronautical or military applications, not for the speed traffic detection.

Why and how does your ignorance on the technology proves that I am wrong? — Corvus

We're talking about measurement, not taking pictures of the measured thing. The radar instrument, with the integrated computer analysis is what measures the speed. The camera does not, it takes a picture of the speeding car, to be sent to the owner. That's why it's called "photo radar", the radar machine measures, and the photo machine pictures what was measured.

This is a good question. Measurement of time is always on change. That is, the changes of movement of objects. It is not physical length. It is measurement of the duration on the start and end of movement the measured objects.

Think of the measurement for a day. It is the duration of the earth rotating once to the starting measurement geographical point. It takes 24 hours. Think of the length of a year. It is the set point where the earth rotates around the sun fully, and returns to the set point, which the duration of the movement is 365 days.

Think of your age. If you are X years old now, it must have counted from the day and year you were born until this day. For this measurement, you don't need any instruments, because it doesn't require the strict accuracy of the reading / counting. However, strictly speaking, we could say that your brain is the instrument for the reading. — Corvus

Giving examples of different lengths of duration doesn't tell me what you think duration is. Your claim was that there is "no physical existence" of that which is measured, "perceived duration". So I asked you, if duration is measured, and it has no physical existence, then what is it? It must be something real, if we can measure it.

The question is easily answered. Duration is the passing of time, which happens at the present. The passing of time is not a physical thing, it is nonphysical, immaterial. So duration is the measured extension of a very real immaterial, nonphysical thing, which we know as "time".

Hmmm…..the in-itself is purely conceptual, as a mere notion of the understanding, thus not real, so of the two choices, and in conjunction with conceptions being merely representations, I’m forced to go with imaginary. But every conception is representation of a thought, so while to conceive/imagine/think is always mind-dependent, we can further imagine such mind-dependent in-itself conceptions as representing a real mind-independent thing, by qualifying the conditions the conception is supposed to satisfy. This is what he meant by the thought of something being not at all contradictory. — Mww

So, does this mean that the mind itself is mind independent? For example, everything that is thought, is mind dependent. But the thinker, being the mind itself is mind independent.

If you could think of some measuring instrument, you will change your mind I am sure. — Corvus

I gave you a couple of examples of measuring instruments, in my examples. I used a tape measure, keeping things nice and simple so as to avoid unnecessary complications. And in the case of measuring time I used a clock. What more are you asking for?

Think of the speed detection machine for detecting cars driving over the speed limit on the road.

The machine monitors the road via the camera vision, and reads the speed of every passing cars. When it detects cars driving over the set speed limit in the machine, it will take photo of the car's number plate, and sends it to the traffic control authorities, from which they will issue a fine and warning letter with the offense points to the speeding driver. — Corvus

I wouldn't use a "speed detection machine" as an example, because I really don't know exactly how it works. I do however know that it works by radar, not "camera vision". So you are just continuing to demonstrate how wrong you are.

Time doesn't have physical existence itself. It is measurement of perceived duration. — Corvus

Then what does "duration" as the thing measured, refer to, if not a length of time? And if it does refer to a length of time, how can there be a "length" of something which has no physical existence?

The OP is correct, yet incomplete. — LuckyR

You wouldn't expect completion from a thread titled "Infinity" would you?

You can use the entire set of natural numbers as your measuring stick, or its power set if that that's not enough, or the power set of the power set, and so on. — SophistiCat

The problem though, is that you really cannot use the entire set of natural numbers as your measuring stick. No one can do this, because by definition, no one can get all those numbers into one's grasp, to use them that way. This renders that statement as false.

Counting infinite sets works the same way, except that you have to set aside certain other assumptions that hold for finite sets but not for infinite sets. — SophistiCat

The "other assumptions" which one must "set aside" are the assumptions that truth is required of a premise, to produce a sound conclusion. Once we dismiss the necessity of truth, then we might assume the premise that the entire set of natural numbers could be utilized in the prescribed way.

Why is it so difficult to see it? — Corvus

You said, an instrument reads the numeric value of an object. There is a few fundamental errors with this statement, which render it incoherent.. Here's some:

1. The person using the instrument reads the number from the instrument.
2. The instrument does not read anything from the object.
3. As I already explained, it is not "the value" of the object itself which is determined by the measurement, but the value of a specific measurement parameter, which we might call a property of the object.
4, The number must be determined relative to a scale. Usually the instrument does this, places the number within a scale. The designated scale, is the property of the property. So in the phrase "5 metres of length", the property of the object is "length", and the property of that property is 5 metres.

For example, if a tape measure is the instrument, one might put it beside an object, according to the criteria of the parameter, width, height, etc. (3). Then the person reads the number from the instrument (1). The instrument does not read anything (2). And, the person must interpret the number relative to a scale, imperial system, metric system, whatever (4). The tape measure might say on it "inches", "centimeters", or something like that.

These same principles apply to the measurement of time:
1. The person measuring reads a number from the clock.
2.The clock does not read anything from the object (time) itself.
3. It is not time itself (the object) which is measured, but a specific parameter which is commonly called "duration".
4. The number read, (4:02 for example) must be determined relative to a scale, atomic scale, solar scale, or something like that.

I can't think instead of you, Banno. If you can't do it, that's fine. But don't make it look like it's the other person's problem. — Magnus Anderson

Classic Banno!

You should get on well with Meta. — Banno

Well, Magnus was very quick to pick up on your nasty habit of straw manning the other person's claims to make it appear like your own errors are the errors of the other person. I wonder why both of us come to the same very peculiar conclusion.

Earth’s magnetic field and gravitational field are in the same space. But the particles associated with those fields are not in each other’s spaces. — Mww

Now the issue I pointed to is that we generally restrict the boundaries of "the object" according to visual information, and that's why we conclude that two objects cannot be in the same space. We cannot see two distinct objects at the very same place. In reality, if we include the parts of the object which we cannot see, numerous objects exist at the same place and at the same time. So for example, the gravity of the moon exists in the same space as the gravity of the earth. And, we really ought to include the object's gravity as part of the object. If we did that, then we'd have to admit that the moon exists in space that the earth also exists in, at the same time.

Furthermore, when distinct identifiable physical objects exist in the same place, like a solution of water and salt, we tend to see the two visually as one object. Then one might be inclined to rationalize how they are really just one object, instead of admitting that two things exist in the same space. So, this idea that two things cannot exist in the same space at the same time, is really just an example of how we are mislead by overconfidence in our sense of vision, toward the unreasonable acceptance of a faulty principle.

But I see your point. It was Feynman in a CalTech lecture, who said fields could be considered things, insofar as they do occupy space. But you know ol’ Richard….he’s somewhat cryptic, if not facetious. — Mww

Feynman was actually very good at explaining complicated physics. I read one paper where he explained how the electricity in a copper wire, which common language says travels as electrons within the wire, actually travels through the field around the wire. This is how an induction motor works.

The first statement says that space and time are relevant to or operative in some domain, which doesn't rule out that they are also relevant to or operative in other domains. The second says they are relevant to and operative in only one domain. If you cannot see the difference in meaning between the two statements then I don't know what else to say. — Janus

Janus, both statements say what space and time "are". "Space and time" is the subject and the statements are definitive as to what space and time are. The subject is not "some domain" which "space and time are relevant to or operative in". What's the point in intentionally switching the subject in your interpretation of one as compared to the other?

That would be a very unusual interpretation of Kant, to say that when he states that space and time are a priori intuitions, he is talking about a domain of a priori intuitions, within which space and time play a role. And, although space and time each play a role within this domain, they are also active in some other domains. Your proposal that space and time cross over from one "domain" to another, is nothing but a category mistake.

I was trying to make you understand what measurement means. — Corvus

All I can say, is that what "measurement" means to you is nothing like what it means to me. And since what you said looks nonsensical to me, I can tell you with a high degree of confidence, that you will never be able to make me understand what measurement means

Why can’t two things occupy the same field without occupying the same space?

If the sun’s light is a field projected from itself, how can it occupy the same field as that which receives it? — Mww

I don't think you quite understand what I meant. What I was talking about is distinct fields in the same place. So the earth has an electromagnetic field which shares the same space as the sun's electro magnetic field. And, the proton and electron, for example, consist of fields which overlap. Therefore the proton and the electron share the same space, but are spoken of, as distinct things.

"Space and time are the pure forms of intution"―not dogmatic.
"Space and time are nothing but the pure forms of intution"―dogmatic. — Janus

You're talking nonsense just like Corvus is. I see no substantial difference between the two phrases. Why does one appear dogmatic, and the other not dogmatic to you? Are you that sensitive to the qualification of "nothing but"?

For the purpose of logical procedure, when a word such as "space" or "time", is defined in a specific way, then we must accept that the meaning for that word is "nothing but" the prescribed definition. To allow that the word might have a meaning other than the prescribed definition is to invite equivocation, which is a fallacy. You might call this "being dogmatic", but it's really just the process of maintaining validity in logic. If you prefer to throw validity out the window, and equivocate by providing a definition other than the one prescribed, because you feel that logical process is too dogmatic, that is your prerogative. We can all be illogical if we want to.

I'll be quick on the quantum answer as I don't want to distract from your real point. The reason we measure as a wave vs an point is again a limitation of measurements. Lets go back to the waves of the ocean for example. We have no way of measuring each molecule in the wave, and even if we did, we would need a measurement system that didn't change the trajectory of the wave itself. I agree, its not all 'lumbering instruments', sometimes its just the limitation of specificity in measurement. Even then, such specificity is often impractical and unneeded. Fluid dynamics does not require us to measure the force of each atom. — Philosophim

I don't think your analogy works. Particles like photons and electrons, in their relation to electromagnetic waves, are not analogous to water molecules in their relation to ocean waves. The waves in water are composed of molecules, and the movement of these particles comprises the visible wave activity. The case of electromagnetic waves is completely different. The observed wave activity is not comprised of underlying particles. And although the energy is known to be transmitted as wave activity, the transmitted energy can only be measured as particles. This is not an issue of limited specificity, it is an issue having no understanding of the relationship between the material particle which is measured, and the immaterial wave which cannot actually be measured.

You say I think Kant is dogmatic, and I do because Kant, having said we can say nothing about the in itself, inconsistently and illegitimately denies that the in itself is temporal, spatial or differentiated in any way, which is the same as to say it is either nothing at all or amorphous. He would be right to say that we cannot be sure as to what the spatiotemporal status of the in itself else, and that by very definition. — Janus

I think you misunderstand Kant. Since space and time are a priori intuitions, these two are proposed as the conditions for sense appearances which are internal to the human being. Therefore the proposed "in itself" cannot have any "spatiotemporal status". You talk as if it would be consistent with Kant to assign to the in itself, a spatiotemporal status which we cannot understand. That is an incorrect interpretation, because by Kant's principles we cannot describe things such that the in itself can even be said to have a spatiotemporal status. The spatial temporal status is a creation of thiose intuitions.

This is not inconsistent, or illegitimate at all. His claim is that space and time are conditionings which the human body imposes, therefore it would be wrong to think that the in itself would be composed of them. Take the map/territory analogy for example. The intuitions of space and time are part of the map. Even though the symbols on the map are intended to represent some aspect of the territory, it would be wrong to assume that you could go out and find those very symbols existing in the territory. In the same way, Kant implies that it would be wrong to think that there is space and time in the in itself.

So I get that it can rightly be said that the in itself cannot be known to be spatial, temporal or differentiated in the ways that we understand from our experience inasmuch as we have defined it as being beyond experience, but it does stretch credibility to think that something which is either utterly amorphous or else nothing at all could give rise to the world of phenomena. Kant posits it simply on the logical grounds that if there are appearances then there must be something which appears. — Janus

This does not "stretch credibility". Living beings are known to be creative beings. It ought not appear to you as "incredible" that they have created these a priori intuitions of space and time, as useful in their living ventures. Further, it ought not seem unreasonable to you, that the symbols used by a living being may not be in any way similar to the thing symbolized. Does the word "symbol" to you, appear to be in any way similar to what the symbol means or refers to. Likewise, the sense representations produced through the means of the intuitions of space and time, may not be in any way similar to the in itself.

You misunderstood my point. I never said or implied, just 2 folks agreeing on something is objective. My idea of objectivity means - widely or officially accepted by scientific tradition or customs in the world. — Corvus

The issue of the difference between true and justified remains. That a principle is "officially accepted by scientific tradition or customs in the world" implies that it is justified, but it might still be false. If "objective knowledge" requires justification and truth, then "officially accepted by scientific tradition or customs in the world", is insufficient for "objectivity" because the condition of truth is not there. So your proposed definition of "objective" cannot be accepted.

Measurement is not idea. It is reading of the objects in number. Numeric value read by the instruments i.e in case of time or duration, it would be stop watch or clock. The instruments are set for the universal reading methods in numeric value, which is objective knowledge on the objects. — Corvus

I'm sorry Corvus, but this line, ("It is reading of the objects in number") makes no sense to me at all. How could a person read an object, unless it was written language like a book. Are you suggesting that you, or an instrument, could look at an object and see numerals printed on it, and interpreting these numerals forms a measurement? That's craziness.

Your confusion seems to be coming from the fact that you misunderstands the ideas of "measurement". Please read the proper definition from my previous post. It is not property of property. Measurement is always in numeric value of the objects read by the instruments. — Corvus

Yikes! You seem to believe in that craziness.

True enough, but my response would be….my experiences are not on so small a scale. I remember reading…a million years ago it seems….if the nucleus of a hydrogen atom was the size of a basketball, and it was placing on the 50yd-line of a standard American football field, its electron’s orbit would be outside the stadium. Point being, there’s plenty of room for particles to share without bumping into each other. And even if the science at this scale says something different, it remains a fact I can’t seem to get two candles to fit in the same holder without FUBARing both of ‘em. — Mww

I believe that this idea, this common instinct or intuition, that two things cannot occupy the same space at the same time, significantly misleads us in our understanding of the world.

The problem is that we place far too much emphasis on what is seen. so when a multitude of things exist at the same place, we see them as one, and think that there is only one thing there. But if we understand a thing as consisting of fields (for example) then we understand that there is always an overlapping of multiple fields, existing at the same place.

The light from the sun for example, is a field, and the earth is within this field. So the sun and the earth exist in the very same place, as a single object, the solar system, just like the hydrogen atom nucleus and its electron exist at the same place, as the atom. We cannot separate the two, to say that they occupy different places, because it is essential to the atom's existence, as a thing, an object, that their fields overlap each other, interacting with each other, to have a multitude of things existing together at the same place, with the appearance that they are one united thing.

This way of looking at things becomes very clear in a hierarchical model. From one direction to another, some levels may be entirely subsumed within another to to exist completely within that one, whereas some just overlap like Venn diagrams. This is the way we understand the existence of conceptions, logically, like a sort of set theory sometimes. One concept may be entirely within another, and this produces logical priority.

You can list them in a sequence, 1/1,1/2, 1/3, 2/3, 1/4, and so on, and so you can count them - line them up one-to-one with the integers. — Banno

That's funny. Why do you think that you can line them all up? That seems like an extraordinarily irrational idea to me. You don't honestly believe it, do you?

Do you think anyone can write out all the decimal places to pi? If not, why would you think anyone can line up infinite numbers?

That’s actually on point. It’s very close to Bergson’s argument about clock time: what gets measured is not concrete duration itself, but an abstracted, spatialized parameter extracted for practical and mathematical purposes. Precision applies to the abstraction — not to the lived or concrete whole. But then, we substitute the abstract measurement for the lived sense of time. — Wayfarer

I believe this is exactly how the false, determinist representation of time is created. We create abstract "units" of time based on certain activities. The traditional activities are the motions of the earth relative to the sun, day, year. Now we use vibrations of atoms. All the units are totally abstract though, and mark length of "duration". Then we take these abstract units of duration, and assume that they are "the lived sense of time".

The problem is that as measured units of duration is not at all how we experience time. We experience time at the present, as a continuous position relative to a determined past and a future full of possibility. So duration is just an abstract tool we come up with, which is very useful for many purposes, but our true experience of time is not as duration at all.

No two things can be in one space, but any one thing can be in two times. — Mww

I like your post Mww, but this line stands out to me, as erroneous. Many things seem to share the same space, and that becomes problematic for physics. Consider a solution for example. Then we might assume that each molecule has its own distinct space. But within the molecule there are atoms which share electrons, so the distinct atoms overlap each other in spatial position. And separate electrons share spatial position in a shell. Things get even more difficult with fundamental particles which seem to be all over the place, all the time. So as much as it seems like two things cannot be in the same space at the same time, they really are. And that raises some very interesting questions about the nature of space and substance.

Where it does appear to be controversial is insofar as it calls into question the instinctive sense that the universe simply exists “just so,” wholly independent of — and prior to — any possible apprehension of it. But again, that is a philosophical observation, not an argument against science. It is an argument against drawing philosophical conclusions from naturalistic premises. — Wayfarer

The problem I see with "the universe simply exists 'just so,'” is that the nature of time and free will indicates that there is possibility for real change, at each passing moment. This means that the true "just so" of every moment is decided (selected from possibilities) at that moment. Then we need to conclude that the universe is actually recreated at each moment of passing time, to allow for the reality of deciding the "just so" at each moment.

That is why the passing of time is a very mysterious and misunderstood feature of the universe. And, because so much of the universe appears to be determined from one moment to the next, i.e. mass obeying the law of inertia, the simplistic cop-out, is to opt for determinism, and deny the real possibility for change at each passing moment. This leaves a very simple, linear representation of time, but it is one that is not consistent with human experience.

Isn't the measurement objective? — Corvus

Try this explanation.

To claim that the measurement of time is objective requires that we have an object to refer to, to meet the criteria for "true", by correspondence. "True" is a common condition for "objective" knowledge. Without this object, which would be pointed to as the one with the property of "time" which is being measured, there is no possibility of truth by correspondence. Then all we are left with is agreement amongst subjects, and this only means that the measurement has been justified. But there is no way to determine truth without the required object. Therefore such a measurement cannot be "objective".

This is a consequence of special and general relativity. Since the measurement of time is made to be reference frame dependent, there is no single object which the passage of time is a property of. Therefore it is impossible that the measurement of time could be objective.

Measurement is agreed way of setting and counting the figures of objects, be it size, weight or time. — Corvus

You ignored the point I made. "Size", "weight", etc., are not "the object", those terms refer to a specific feature, a property of the supposed object, and strictly speaking it is that specific property which is measured, not the object.

If it is not objective, then everyone will have different way of measurement on days, hours, minutes, distance, size, weight etc, which will make Science and daily life chaotic? — Corvus

That is the definition of "objective" which I tried to steer you away from, so that you could understand the point I wanted to make. If you just want to claim that this definition of "objective" (based in an agreement between subjects) is the only meaningful definition of that term, then I can't make the point, and discussion is useless.

But let me ask you one question. If we define "objective" in the way that you propose, how would you differentiate between "justified" and "true"?

It is. If you read the OP as saying it isn’t, then you’re not reading it right. — Wayfarer

The point I made is that if we adhere to a strict definition of "objective", meaning of the object, then measurement is not objective. This is because measurement assigns a value to a specified property, it does not say anything about the object itself. Assigning the property to the object says something about the object, but assigning a value to the property says something about the property.

The problem with the loose definition of "objective" (agreement amongst subjects) which Corvus is proposing, is that it blurs the distinction between justified and true. If we maintain that objective knowledge requires both, justified and true, and "true" requires correspondence with the object, then simple agreement amongst subjects does not meet the criteria for "objective knowledge".

Isn't the measurement objective? The feel, knowing and perception of time is subjective, but any measurements are objective i.e. by watch or clock, isn't it? Your 1 hour must be same as my 1 hour, and for the folks in the down under, and the folks in the whole world. — Corvus

Well, "objective" has many meanings. Here, you imply that if two people agree, then it is "objective". That would imply a meaning of "objective" which is based in intersubjectivity. So, when I said the measurement is "subjective", this is not inconsistent, or contrary to your use of "objective" here.

Look at it this way. Let's say that ideas and concepts are property of the subject. These things are dependent on the minds of subjects, therefore in a sense, "subjective". Also, we assume physical objects, like the cups I mentioned earlier, which are supposed to be independent. When we talk about these things, their properties etc., we are talking about the objects, hence what is said may be "objective", in the sense of 'of the object'.

Measurement is a very difficult concept because we take ideas and concepts, which are subjective, in the sense described above, completely universal and removed from the objects, and attempt to apply them to objects. The measurement is never objective, because it is always entirely conceptual, property of the subject. Nor is the measurement something we say about the object itself, because measurement is applied to a specific parameter (property) of the object. Notice, a property is said to be "of the object", objective in the sense of something we say about the object. But the measurement is not something we say about the object itself, it is something we say about the specific property. So measurement is twice removed from the object. It is not a property of the object, but a property of the property. It is an idea applied to an idea, therefore subjective.

The observer knows there is activity independent from the observer”. He does indeed. — Wayfarer

So the passage of time itself is independent. Right? Therefore before and after are also independent.

What is subjective (dependent on an observer) is the measurement of the passage of time. Therefore any specific unit, or period of time is subjective (dependent on an observer). Examples of these are a specific minute, a specific hour, today, yesterday, 2021, 1940, etc.

The problem is that most people do not distinguish between the measurement of the passing of time, and the passing of time itself. Then the measurement, which is subjective, is taken to be "time". And so most do not distinguish between physical change (the common means of measuring time), and the thing measured, the non-physical passage of time.

If you take a ruler and measure a blade of grass at one foot long, one foot long is the measurement, it is not the thing measured, being the blade of grass. Likewise, if we measure that it has been 24 hours since this time yesterday, 24 hours is the measurement. It is not the thing measured, which is the passage of time itself. The passage of time is that mysterious immaterial aspect of the independent world, which we do not understand.

Nothing like that is required. What appears mysterious is not some hidden feature of the world, but the fact that the conditions which make the world intelligible are not themselves part of what appears, but are provided by the observer. That is exactly what “transcendental” means: essential to experience, but not visible within it. — Wayfarer

That is exactly what I am disagreeing with. That feature of the world, which we know and measure as the passing of time, is a real, independent, and very mysterious feature of the world. We know that the passage of time is independent from observers from the evidence derived from studies like geology and geomorphology. We know there is activity independent from the observer, and any activity requires the passage of time. Therefore we can conclude deductively that this mysterious aspect of reality, which we know as the passing of time, is independent from the observer.

The passage of time turns out to be "transcendental" in a much more significant and absolute way. Not only does time transcend all experience, but it also transcends all physical existence. This is why modern cosmological theories break down at the so-called "Big Bang". They have not been able to separate the immaterial, nonphysical passage of time from the physical existence of the universe. The former is necessary for, and demonstrably prior to, the latter.

It's a measured reality - and that is a world of difference. 'One second' is a unit of time. As are hours, minutes, days, months and years. But (to put it crudely) does time pass for the clock itself? I say not. Each 'tick' of a clock, each movement of the second hand, is a discrete event. It is the mind that synthesises these discrete events into periods and units of time. That's the point you're missing. — Wayfarer

Yes, time does pass for the clock itself. The passage of time is what is measured by the clock. Seconds, minutes, hours, etc., are the units of measurement. In his "Physics", Aristotle explained very clearly how "time" has two distinct meanings. One sense references a number, which is a measurement, but the other sense references what is measured. The latter is what we know as the passage of time, the former is the units of measurement, seconds, minutes, etc..

For example, if I say there is ten cups on the counter, "ten cups" is a measurement, and the direct referent is an idea a concept consisting of the value being judged, ten, and the idea of what qualifies as a cup. Also, we assume something independent, the room with a counter and objects on the counter which are being counted. This independent existence is what we claim is being measured. We can apply the map/territory analogy here. One sense references the map, the other the territory.

The case of "time" is very intriguing because the thing measured, the passage of time, which is the territory itself, is across the boundary of physical/non-physical, as something nonphysical. This presents us with a very difficult problem, how do we measure something which has no physical existence. The common ontological solution is to simply deny the reality of the passage of time, say it is an illusion or something like that, and reduce "time" to one sense, a measurement. But this is clearly not the correct answer because it is contrary to our experience, which I explained in my last post. It is false to claim that the passage of time is not real.

So when we accept as true, that the passage of time is something real, something immaterial which we attempt to measure, the problem with time becomes very clear. We have not developed the means to measure something immaterial, therefore our measurements of time are very primitive and inaccurate. Furthermore, since we tend to deny the reality of the immaterial we start to deny that there is anything there to be measured in the first place (time is an illusion), then we start to accept anything which works for our desired purposes as an acceptable measurement, and we do not even attempt to try to figure out the true nature of time, and how to measure it correctly.

But the point is, the observer is watching, measuring, deciding on the units of measurement. — Wayfarer

This is the key point here. Since there is not empirically discernible distinctions between one unit of time and another, then "the units" of time are inherently subjective. Unlike the cups on the counter, there is no empirically observable separation between one unit and another, like the spatial separation of the cups, therefore we are forced to produce our own principles to distinguish a unit. What we do is take an observable activity which is repetitive (rotation of the earth, vibration of an atom), and observed to be reliable in comparison with other reliable repetitive activities, and establish this as a standard unit. There is always discrepancies though (starting with leap years) so adjustments need to be made, and relativity theory throws another problem at us.

Ultimately, the passage of time ought to be considered as an immaterial activity, which all material activities may be compared with (measured by). However, this presents us with the problem of determining exactly what this immaterial activity is, so that we might figure out a way to measure it. We actually already have a good idea about what it is, it is a wave activity, the vibration of the cosmos. However, failure to identify the medium within which the waves propagate leaves that activity as completely insubstantial, and unintelligible to us. So the need is to determine the medium of the wave activity, and the means by which waves are propagated. This will guide us toward a more "objective" way of measuring the passage of time.

I looks like we both have an uneasiness with possible world semantics. I think your unease is more with the metaphysics, while mine is with the application. The PI sections you had mentions, 253 to 256 are typically associated with Wittgenstein's argument around private language. Should this extend to possible world semantics? At first glance, I would say "no". Possible worlds are not suppose to be a private language. In PI, a private language is about language only a single individual understands that refers to purely private inner experiences. — Richard B

The issue is the relationship between one possible world and another, and the discontinuity implied by that relationship. If someone assumes that there is an object with the same identity in multiple possible worlds, then this supposed object is nothing but an object of a private language, unless the continuity between distinct possible worlds can be established. So it's not that possible worlds are supposed to be a private language, but that the matter of transworld identity creates a private language problem. The matter is the problem of identifying objects which exist in distinct mental images, as "the same object". That is the private language problem. The assumption of "the same" is unjustified, rendering the proposed "object" being an unity of multiple instantiations, as unintelligible.

Think about the chair in Wittgenstein's example. Suppose I insist that the chair here today is the same chair that was here yesterday. This is analogous to saying that the thing named "Nixon" in one possible world is the same as the thing name "Nixon" in another possible world. So you may ask me to justify my claim that the chair is the same object. Since it is a publicly accessible object, I could say look at it, doesn't it look exactly the same. It's identical. That still might not appease you, because you could say that every room in the building has identical chairs, how do I know that they weren't switched. Then, I could say that someone watched over it in the interim, or refer to security video, and the continuity required to justify my claim could be justified.

In the case of "Nixon", justification cannot be done in the same way because the possible worlds are inner objects, imaginary. This means that the continuity of the proposed object referred to, between distinct possible worlds, must also be imaginary, fictitious, or stipulated. And if the thing referred to as "Nixon", is supposed to be the same thing in multiple possible worlds, this stipulation needs to be justified. What makes it "the same thing"? Obviously it's not identical because the different worlds give it different properties. And the supposed continuity from one possible world to the next is not a temporal continuity, so reference to observation, or surveillance is not relevant. Nevertheless, we need justification. Without justification it is simply a private object which is absolutely unintelligible.

In Wittgenstein's other example, where "S" signifies a private thing referred to as "a sensation", the thing referred to is unintelligible to others, until justification of the use of "S" is provided. It turns out that the use of "S" coincides with an observable rise in blood pressure, and this forms the justification. That could be the "essential property" of the sensation referred to by "S". Now justification of continuity between distinct worlds could be done through an essential or necessary property. We could do the same with the thing referred to by "Nixon". We could say that there are essential, or necessary properties, which identify the thing called "Nixon", in every possible world, and this would suffice for transworld identity. However, notice how this arbitrarily, or subjectively, limits the extent of possibility. The number of possible worlds is thereby limited. This amounts to saying that it is impossible for there to be a possible world where the thing called "Nixon" does not have the named essential properties. Therefore the limiting of the possibilities in this way, itself need to be justified. And the choice of these limiting factors becomes very subjective dependent on purpose. And if we want to allow all possibilities, infinite possibility, we must deny any essential properties, and we are back at an unjustifiable, and completely unintelligible object of a private language.

If you take a look at the SEP's description of combinatorialism, the issue might become more clear to you. Here, the object of the private language is called the "simple". The simple has transworld identity as such, having no essential properties, allowing for infinite possibilities. However, because it has no inherent properties it is completely unintelligible, as an object of a private language. Being a private language doesn't mean that we cannot use the word, it just means that what the word refers to cannot be known. So we can all talk about "the simple", just like we can all talk about "the beetle", but this talk doesn't make the private object referred to, intelligible.

So you'll notice in the SEP, that the different philosophers who use this system, of employing "the simple", have different ideas of what "the simple" refers to. That is because it is essentially an object of a private language, in a case where justification of the use of the term varies according to an individual's preference, or purpose. In general, "the simple" is supposed to be an object whose identity, and existence, cannot be justified, as simply a requirement for unlimited possibility. This leaves justification as completely subjective, because justification is to apply limits, boundaries. Consequently the philosopher is allowed to apply limits, and justify what "the simple" refers to, according to the purpose at hand. But the object of the private language is inherently unbounded and therefore unintelligible, then the philosopher applies limits as desired. However, the application of boundaries. limits, is contrary to the basic assumption, and need within the system. The need is for an unlimited object of the private language, and the corresponding assumption. so we are left with self-contradiction when we try to make the object of the private language, referred to here as "the simple", into something which could be understood.

Reading the SEP, you'll see that the simple has a different meaning for Russell as it does for Wittgenstein. Also Quine and Cresswell suggest a different interpretation. Armstrong argues something different. Ultimately, the ontology of "the simple" is a matter of debate. This is because it is employed as the object of a private language, whose existence cannot be justified. That is the purpose of "the simple", to allow for transworld relations which cannot be justified. And, as the object of a private language use of the term is completely unjustifiable. Attempts at justification are self defeating

He does not say this in the quote I mentioned from N&N. What he says is "Don't ask: how can I identify this table in another possible world, except by its properties? I have the table in my hands, I can point to it, and when I ask whether it might have been in another room, I am talking, by definition, about it." — Richard B

Look at what he is saying Richard B. The table being spoken about is referred to as the one "I can point to", "the table in my hands". That is very clearly how "the table" is defined here, in this context, by Kripke. It is impossible that this table, the one indicated by the definition, as the one "I can point to", "the table in my hands", could be in another room, or else it would not be the defined table.

Therefore his question is answered very easily. He cannot identify this table in another possible world. That is because he has defined the table spoken about as the table here, and now, in this world, the table I can point to, the table in my hands. By defining the proposed object in this way, he denies the possibility that it could be in another possible world.

Physics can describe relations between states using a time parameter, but that parameter by itself does not amount to temporal succession. A mathematical ordering does not yet give us a meaningful before and after. The fact that most fundamental physical equations are time-symmetric illustrates the point: the time parameter in physics functions is an index of relations between states, not an account of temporal succession or passage. Direction, duration, and the sense of "before" and "after" enter only at the level of interpretation, description, and experience. Hence the philosophical problem of "time's arrow", which is understood to be absent from the equations of physics. — Wayfarer

I believe there is something very important hidden within this passage. The "time-symmetric" character of physical equations is a feature of determinism. If everything which occurs is determined, then backward and forward necessarily produce the very same order, only reversed.

It is our sense, our intuition, which tells us that the future is somewhat undetermined, making us realize that we need to chose. This producers the fundamental difference between before and after in our understanding. I can make choices to influence events in front of me, in the future, and even produce the events I want, but the past is fixed and those events cannot now be chosen in that way.

So the matter of "time's arrow" is very real to us. We could assume determinism, fatalism, or whatever, and claim that time's arrow is not a real issue. Nevertheless, in our daily lives we all accept that we cannot alter what has occurred, and we all make choices in relation to the future. And every time we make choices we belie the determinist claim, which the successes of physics inclines us toward.

Now, the underlying importance is related to the way that we understand reality when we reject the determinist approach to time, and accept that the real difference between past and future is demonstrated by the reality of choice. This perspective makes the entirety of physical existence contingent. And what I mean by "contingent" here is dependent on a cause which is selected therefore not necessary.

This is what produces the difficulty for the common notion of physicalism. In denial of physicalism, we do not necessarily insist that consciousness is prior to physical events. All we need to do is to demonstrate how selection is necessarily prior to physical existence. Then if consciousness is demonstrated as posterior to physical existence, we need to identify a type of selection which is non-conscious. Proposals like random chance, symmetry-breaking, or quantum fluctuations, can be shown to be incoherent, and the product of misunderstanding, rather than the required "selection".

"Let's call something a rigid designator if in every possible world it designates the same object, a nonrigid or accidental designator if that is not the case. Of course we don't require that the objects exist in all possible worlds" — Richard B

Notice this condition, it's a rigid designator "if" it designates the same object in every possible worlds. The issue is that possible worlds are abstractions, so there are no objects in any possible worlds. This makes "rigid designator" useless right from the start, unless we go to some form of concretism. But concretism disallows such transworld identity anyway, for other very clear reasons. So "rigid designator" is completely useless.

Here is the way that Wittgenstein elucidated this issue of identity in "Philosophical Investigations", starting from 253.

253. When I say "another person can't have my pains", what is the criteria of identity? Consider: "This chair is not the one you saw here yesterday, but is exactly the same as it". But if it makes sense to say "my pain is the same as his", then it makes sense to say that we both have the same pain.

254 The substitution of "identical" for "the same" is an "expedient in philosophy". The question for the .philosopher is to give an account of the temptation to use a particular phrase. What, for example mathematicians are inclined to say about "mathematical facts", is not philosophy, but "something for philosophical treatment".

255. "The philosopher's treatment of a question is like the treatment of an illness."

256. Now, how can I give an account of my inner experiences? How can I use words to refer to my inner sensations? I could use natural expressions, but suppose there were none, and I had to "simply associate names with sensations".

Richard B, please reflect on what has been exposed here. There is a problem of "identity" when referring to inner things such as sensations. We cannot employ the same criteria of "identity" which we use when referring to physical objects like the chair. This is because in the case of physical objects we distinguish between "identical" and "the same", but we cannot make that distinction in reference to inner things like sensations. So, with the physical object, there is a chair here today, which is "identical" to the chair which was here yesterday, but it is not "the same" chair, if someone switched two identical chairs. In the case of inner things, if I feel a pain today which is "identical" to the one yesterday, I call it "the same" pain, and the two terms "identical" and "the same" are used interchangeably.

In this way, as described by 253, it makes sense to say "pain" may refer to the same thing for me, as it does for you, therefore you and I feel the same pain, and refer to the same thing when we say "pain". However, this only works if we have the criteria required for justification.

So this matter of criteria is the key point for transworld identity, and specifically the notion of "rigid designator". To signify the object which the rigid designator refers to requires some criteria. This would produce the need for necessary (essential) properties, thereby compromising the usefulness of the possible worlds semantics. We need to allow that the designated object has nothing essential, to cover all the possible worlds. But this denies the possibility of criteria. Then, whatever it is, the supposed object, which is designated by the rigid designator, is completely unintelligible in the way described by Wittgenstein's "private language". It is a private, inner thing, with no criteria for identification, hence no way of knowing the thing being referred to.

So, after laying out this platform, Wittgenstein proceed to talk about the problem of producing such criteria of identification, which I call justification of the use of the name.

Kripke's example, I like it because it seems rather apropos for everyday conversations we have about everyday objects. — Richard B

Kripke's example is naive, and not at all a fair, or an accurate representation. He says:

"I have the table in my hands, I can point to it, and when I ask whether it might have been in another room, I am talking, by definition, about it. I don't have to identify it after seeing it through a telescope."

What he says here is demonstrably wrong and deceptive. The spoken about object is actually defined as "the table in my hands". Therefore the question about "whether it might have been in another room" must be answered with "No". It is impossible that "the table in my hands" could be in another room because then it would not be ""the table in my hands". The point is that if we adhere to what he says "I am talking by definition, about it", where "it" clearly refers to the table in his hands, then it is impossible that this object is in another room. If it was in another room, it would not be the designated object "the table in my hands".

So Kripke just makes a deceptive use of language, to produce the appearance that the table in his hands could also be in another room. Clearly though, if we adhere strictly to the example, this claim is false. The object defined as ""the table in my hands" could not possibly be in another room, because that would not be consistent with the definition.

And "rigid designator" turns out to be a nonsensical, unintelligible proposal, for the reasons demonstrated by the private language argument. If the designated object is identified by physical existence "the table in my hands", then it cannot be in other possible worlds. And if we remove the physical existence, then it's a private object with no criteria for identification. Any criteria for identity (essential properties) denies certain possible worlds as not possible, arbitrarily compromising the use of "possible worlds".

This is the issue which Wittgenstein elucidated with the so-called private language argument. First, he lays out the common conditions for "identity" with reference to the chair. The chair in this position today, looks like it's the same chair as the one here yesterday. But, if someone switched it out overnight, then it is not the same chair with the same identity. It is implied that an object, "the same" object, has temporal extension between distinct acts of observation. The identity as "the same object" is therefore supported by an observable (public) temporal continuity.

Then, he plays a little mental trick on us. He goes on to refer to a sensation, which he signifies with "S". Each time that this supposed "same" sensation occurs, he marks an "S". Now the "S" is proposed as referring to "the same" sensation, but it's a trick because there is no continuity between one instance to another. In reality, "S" refers to distinct occurrences of similar sensations.

The problem is that the assumed continuity of the supposed object, referred to by "S", which is necessary for concluding that it is a single object, is private, within the mind of the one who senses it. It is not verifiable by public observation. Therefore the supposed object identified by "S" with its required temporal continuity is a private object, making this language which employs "S" to refer to a single identified sensation, is a private language. The thing referred to by "S" is an imaginary thing, and as such, it doesn't have identity in the common way that the chair has identity.

So this is the issue with rigid designation. The supposed continuity of the object, between one possible world and another, which establishes rigid designation, is completely imaginary, private, like the continuity between one instance of "S" and another. This supposed identity, as rigid designation, cannot be supported by empirical observation. Therefore it is nothing but a private language. "Nixon" refers to the same person in a multiplicity of possible worlds, just like "S" refers to the same sensation. That is, by assumption of a private object. There is no observable temporal extension of the object, and the extension between possible worlds is completely imaginary, unverifiable through (public) empirical observation, therefore the supposed "identity" is private. This constitutes a private language, as we assume an object whose existential extension is completely unintelligible. Now "S", or "Nixon" in the example, refers to a completely unintelligible object, making that private language incoherent.