've been meaning to return to this for a while now, but just haven't had time. — Esse Quam Videri

I'm glad you're back.

The argument about measurement that you provided in your reply is interesting, and I can see how it is relevant to question of whether (or in what sense) a countably infinite set can be said to "exist". — Esse Quam Videri

The issue of platonism is more about the existence of any bijection in general, and the question of whether a measurement exists without requiring someone to measure it. It's a form of naive realism, which in our conventions and educational habits, we tend to take for granted. We look at an object, a tree, a mountain, etc., and we assume that it has a corresponding measurement, without requiring that someone measures it. Then, when someone goes to measure it, the correct measurement is assumed to be the one which presumably corresponds with the supposed independent measurement. This type of realism requires platonism because there must be independent numbers and measuring standards which exist independently from any mind, in order that the object has a measurement before being measured.

But if we understand that numbering conceptions, and measuring conceptions are products of the human mind, then it's impossible that an independent object could have a measurement prior to the measurement being made by a human being. This rules out the possibility that the natural numbers could have a measurement, or be countable because we know that human beings could not count them all.

The argument about measurement that you provided in your reply is interesting, and I can see how it is relevant to question of whether (or in what sense) a countably infinite set can be said to "exist". But the word "exists" can have different meanings depending on the context. Within the context of ZFC set theory, to say that a countably infinite set "exists" doesn't imply that it exists in some Platonic heaven. That's not to say that you couldn't interpret it in a Platonic way, just that nothing in ZFC itself forces this interpretation. — Esse Quam Videri

So we could say, that numbers "exist" in a way other than platonic realism, but we must consider what would be meant by this. We need to ask, what is the criteria for existence. Consider the difference between the following two statements. 1. "the set of natural numbers between 0 and 5". 2. "{1,2,3,4.}". We might say at first glance that they both say the very same thing, and they both necessitate the existence of those four integers, but this would not be correct. That is because the first is a formula, and the existence of the named integers requires that the formula be carried out correctly. So we need to respect this difference, the existence of the integers in the first example is conditional, or contingent, on "correctness", and in the existence is necessary. And when we say 1, 2, 3, ..., or use the successor function, the existence of those numbers is conditional, contingent on correctness. Now we have the problem mentioned above, the natural numbers cannot have a measurement, because the procedure cannot be carried out to the end, "correctly".

The inclination might be to deny that distinction between necessary existence and conditional existence, which I provided. But we cannot do this because we need to account for the reality of human error. A formula does not necessitate the existence of numbers because error may arise in a number of different ways. The formula might be carried out incorrectly, or it might be a mistaken formula in the first place. So we might add, "the designated numbers exist if the the formula is properly formulated, and if it is carried out correctly. But that makes it conditional.

So to say that "a countably infinite set exists" is just to say "ZFC ⊢ ∃x CountablyInfinite(x)". The actual derivation follows very simply from the axiom of infinity in combination with the definition of "countably infinite". — Esse Quam Videri

So you are talking about a conditional existence. The supposed existence of the natural numbers, is dependent on the correct procedure. The issue with the definition of "countably infinite" is that the procedure cannot be carried out. The formula states something which is impossible to correctly finish, therefore the numbers cannot exist.

Furthermore, platonism doesn't solve the problem because the infinite is defined as being impossible, so the numbers cannot even exist in a platonic realm. That would be like saying that the full extension of pi exists in a platonic realm, when this has been demonstrated to be impossible.

I presently suspect that the structure of the uncertainty principle, that concerns non-commutative measurements, is a logical principle derivable from Zeno's arguments, without needing to appeal to Physics. — sime

I agree. Many people conclude that calculus solved Zeno's paradoxes. I've argued elsewhere, that all calculus provided was a workaround, which was sufficient for a while, until the problem reappeared with the Fourier transform.

I'm sorry. I should have said "separates", not "divides". — Ludwig V

I don't think this makes any difference.

Can you think of a form of measurement that is not counting - apart from guessing or "judging"? — Ludwig V

Sure, I believe measuring is fundamentally a form of ordering. So most comparisons which are intended to produce an order are instances of measuring. Get a bunch of people, compare their heights, and order them accordingly. That's a form of measuring.

As Frank points out,
It really comes down to which view best accommodates what we do with math.
— frank
And Meta's view undermines most of mathematics, despite what we do with it. — Banno

You mention "what we do with math", but are neglecting something very important, "what we can't do with math". This is the limitations, like the uncertainty principle mention above. We do a lot with math, sure, but there is a lot more we would be able to do if we work out some of the bugs. Then there's the even worse problem of the many things that people believe we do with mathematics, which we really don't. Some people think that calculus has solved Zeno's paradoxes. It has not. Some people think that mathematics allows us to determine the velocity of an object at a single instant in time. It doesn't. Some people think that mathematics has provided a way to make infinite numbers countable. It has not. That's what I'm talking about. To have the attitude that math is perfect, ideal, therefore it is wrong to subject it to philosophical skepticism is the real problem.

I guess Meta is a math skeptic. — frank

I like to apply a healthy dose of skepticism to any so-called knowledge. Nothing escapes the skeptic's doubt.

(The finitude of an object's exact position in position space, becomes infinite when described in momentum space, and vice versa. Zeno's paradox is dissolved by giving up the assumption that either position space or momentum space is primal) — sime

i don't consider this to be a solution, because the result is the uncertainty principle. What you indicate is two distinct concepts of space which are incompatible, "position space", and "momentum space".

It depends, as I explained earlier, how you define "countable". I don't say that it's just all just a matter of definitions, but it's probably a good idea to get those agreed so that we can be sure we are talking about the real issues. As it is, we don't agree and so we never get to identify and discuss the real issues. — Ludwig V

We went through the common definition of "countable" provided by jgill, and the contradiction remained. So I really don't know what type of definition of "countable" you might be thinking of.

I'm not sure what you mean by "serves as a medium". — Ludwig V

"Medium is commonly defined as "something in a middle position". If something is between two things, it is distinct from each of the two as in the middle.

But the point of a succession is that every step (apart, perhaps, from 0) has a predecessor and a successor. That is what it means to say that n is between n-1 and n+1. It is not wrong to say that 2 unites 1 and 3 and it is not wrong to say that 2 divides 1 and 3. But it is wrong not to say both. — Ludwig V

Yes, every step has a successor, but the succession is described as a continuous process. No individual step can serve as a division between the prior step and the posterior, as each is continuity, not a division. To say that one step is a division would produce two distinct successions, one prior one posterior. then the one which served as the divisor would have no place in either of the two successions.

So I dont't understand what you are saying here, especially what you mean by "2 divides 1 and 3". One divided by two produces a half, and three divided two produces one and a half. But it doesn't make sense to say that two acts as a division between one and three in the way that you propose.

This just turns on your definition of what it is to count something.
Using a ruler to measure a (limited) distance means counting the units. Obviously, we need enough numbers to count any distance we measure. So having an infinite number of numbers is not a bug, but a feature. It guarantees that we can measure (or count) anything we want to measure or count.
I maintain that if you can start to count some things, they are countable. You maintain that things are countable only if you can finish counting them., It's a rather trivial disagreement about definitions. But I do wonder how it is possible to start counting if I can only start if I can finish. — Ludwig V

So it looks like you disagree with my premise that counting is a form of measurement. Since you claim that starting to count something is sufficient to claim that it is countable, then if we maintain consistency for other forms of measurement, puling out the tape measure would be sufficient to claim that the item is measurable. Since this is obviously not true, it seems you are claiming that counting is not a form of measurement at all. How would you define "countable"?

So your argument is that 2 is not between 1 and three. — Banno

That's exactly right. To say that 2 is between 1 and 3 is to say that it serves as a medium. However, in the true conception and use of numbers, 1, 2, 3, is conceived as a unified, continuous idea. This unity is what allows for the simple succession representation which you like to bring up. No number is between any other number, they are conceived as a continuous succession. To say that 2 comes between 1 and 3 is a statement of division, rather than the true representation of 1, 2, 3, as a unity, in the way that the unified numbering system is conceived and applied.

Well, no. You claimed there is a contradiction, repeatedly, but never showed what it was. So go ahead and quote yourself. — Banno

"Infinite" means limitless, boundless. The natural numbers are defined as infinite, endless. limitless. All measurement is base on boundaries. To say a specific parameter is infinite, means that it cannot be measured. Counting is a form of measurement. Therefore the natural numbers cannot be counted. To propose that they are countable, is contradictory, because to count them requires a boundary which is lacking, by definition.

This is why "open sets" are used to justify unmeasurable spaces, resulting in an incoherent concept of continuity. Incoherent "continuity" is the result of the false opinion that there exists numbers "between" numbers, instead of representing the numbers as a unified concept.

Look, to say that something is infinitely heavy or light means its weight cannot be measured. To say that it is infinitely long or short means that its length cannot be measured. To say that it's infinitely hot or cold means that its temperature cannot be measured. To say that it is an infinitely large or small quantity means that its quantity cannot be measured.

Why do you think the proposition that the natural numbers is countable does not contradict the proposition that the natural numbers are infinite, in the way I explained?

But in addition to the usual thngs nominalism rejects, Meta rejects the notion that numbers as values of variables. while nominalists say numbers aren’t abstract objects, they undersntad that they can still be quantified over. Meta says that numbers aren’t things at all — they’re modifiers like “pink”. That blocks: — Banno

Correction. There is no difference between a number and a numeral, the number is the symbol. What the symbol represents is an abstract value. It's a category mistake to say that a value is an "object" unless we define "object" in the sense of a goal.

Regardless of what you assert, to say that the value represented is an object called "a number" is platonism. Calling it an imaginary, or fictional object, doesn't fulfill the ontological criteria of "object". Therefore we'd have to treat it as an idea because treating it as an object would be a false premise. We cannot truthfully treat a fictitious object as an object, because it is an idea and the existence of ideas is categorically distinct from the existence of objects.

I think you're discounting the importance of community. If it's not stretching your spine out of shape, you can go along with the rest of the phil of math and write it as platonism. It's a little nod to the deep bonds that hold us together over the millennia as our brothers and sisters try to take freakin' Greenland and what not. — frank

This issue is more complicated though. The Neo-Platonists took Plato's name and claimed to continue Plato's school, but their ontology is consistent with what you call platonist. Aristotle's school claimed to be the true Platonists but the Neo-Platonists took the name. So you have to take on the Neo-Platonists, and tell them that they should call themselves Neo-platonists, as not true Platonists. But this problem has been around for millennia, and they do not like being accused of misrepresenting Plato, they like to claim the true continuation of Plato's teaching.

We can make it simpler for you: How many whole numbers are there between one and three? — Banno

i say it's a loaded question, like "have you stopped beating your wife?". If we give up on the idea that there are numbers in between numbers, we get rid of an infinity of problems from infinitely trying to put more numbers between numbers. This supposition that you have, that there are numbers between numbers is very problematic.

Set the supposed contradiction out. — Banno

I did it all ready in this thread, numerous times. If you're truly interested go back and reread my posts. But I'm tired of it. And I know you, you'll just deny anyway so what's the point?

“Countable” is defined as “there exists a bijection with ℕ (or a subset of ℕ).” I bolded it for you — Banno

Right, begging the question. "There exists a bijection with N" is explicitly saying "N is countable". Are you kidding me in pretending that you don't see this?

I would ask one favor though. Stop capitalizing the P in Platonism. The phil of math view of platonism. Plato pitted opposing ideas against each other, so for instance, in Parmenides, he outlines a lethal argument against the Forms. That's why they use a little p: platonism. — frank

I agree. but my spell check doesn't like little p platonism. And, I count the distinction as unimportant because there really would be no such thing as big P Platonism if we maintained that distinction. Plato pitted ideas against each other so there's no real ontological position which could qualify as big P Platonism. So they end up being the same meaning anyway.

Looks like Trump has a Nobel Prize... — NOS4A2

I'm kind of surprised that he didn't just make his own.

You got it bro!
We measure the object not the measurement tool. The standard metre cannot be measured. Numbers are the tool, not the thing to be measured.

I can't really follow anything you are saying.

Now many integers are there between zero and five? — Banno

Again, "integer" is a faulty concept, because it assumes that "a number" is a countable object. That's exactly the problem I explained to you. We ought not treat an idea as an individual object. Providing more examples of the same problem will not prove that the problem does not exist. The problem of Platonism is everywhere in western society, even outside of mathematics, so the examples of it are endless.

Mathematics on the other hand takes a bijection between two sets A and
B to mean there is a rule f such that each element of A is paired with exactly one element of B, and each element of B is paired with exactly one element of A. — Banno

This does not address the point. A rule can contradict another rule within the same system. Saying that there is a rule which allows a specific bijection doesn't necessarily mean that there is not another rule which disallows such.

The bijection is not assumed, it is demonstrated. — Banno

That's false, it's not a demonstration, at best, it's begging the question. You have no definition of "countable", so your conclusion, "Therefore N is countable" does not follow. It would follow, if you provide a definition of "countable" which begs the question. The proper conclusion is N is countable if N is countable. Then a definition of "countable" could be provided which contradicts the infinite nature of the natural numbers, making "N" "countable". Voila, begging the question with contradiction. I think Magnus already explained this to you, so you're just continuing to demonstrate your dishonest denial. I don't see any point to further discussion, you'll only continue to refuse to look at what is shown to you, and rehash the same faulty arguments.

So you do know that the series is infinite without completing the count of them all. — Ludwig V

Of course, why would say that? it's defined as infinite. That's the whole point. It is infinite and infinite is defined as boundless, endless, therefore not possible to count. So any axiom which states that it is countable contradicts this.

And yet, Frodo Baggins exists - in the way that fictional characters exist. They can even be counted. Similarly, numbers exist - in their way. — Ludwig V

Yes, a finite number of fictional thigs is countable. But infinite is defined as endless, therefore it is impossible to count an infinite number of fictional things.

I'm not quite sure that I understand you. I think that it is not necessary for the infinite number of numbers to exist in my mind. — Ludwig V

I'll explain again. If numbers are assumed to be independent Platonic objects, we can assume that bijections simply exist, without needing to be produced by a human being. However, the infinite bijection is a matter of contradiction, even if bijections simply exist. Therefore it ought to be rejected as incoherent. If, on the other hand, numbers are assumed to be fictional objects, created by human minds, the same contradiction still remains. It is the idea that numbers are infinite, yet countable as objects, which is incoherent. So it does not matter how you validate the existence of numbers as objects which can be counted, the incoherency cannot be avoided.

All I need to have in my mind is S(n) = n+1. — Ludwig V

I don't understand why, so many people on this thread seem to think that if they can make symbols which represent something incoherent, this somehow makes it coherent. When we speak contradictions, that's what we do, use symbols to represent something incoherent. Why would you think that writing it somehow makes it coherent? I can say RNR stands for the thig which is both red and not red at the same time, but how does symbolizing it make it coherent? How does "S(n) = n+1" make you believe that an uncountable number of objects is countable? I truly cannot understand this.

It turns out that the disagreement turns on a metaphysical disagreement. Tackling that needs a different approach. — Ludwig V

It's not a metaphysical problem directly. As a matter of contradiction between basic axioms, it is an epistemic problem within the mathematical system (set theory). As explained above, it doesn't matter which metaphysics you use to validate the existence of numbers as countable objects, the problem remains.

I do propose that it could be resolved with a metaphysical solution though. The solution is to reject the ontology which supports the idea that numbers are countable things, along with the mathematics which follows (set theory). An idea is not a thing which can be counted, and that is a basic flaw in the ontology which supports set theory.

Notice, it's not a metaphysical problem in itself. We can assume that numbers and all sorts of ideas are objects, and maintain that ontology. The problem is epistemic. We think that since numbers are objects then they ought to be countable. that's what produces this problem. To resolve the problem we might say that numbers are a type of object which is for some reason not countable, but that creates a problem with the concept of "object". Therefore it's better, and actually provides a better foundation for understanding what concepts are, if we deny that numbers are objects.

If you think Meta has convincingly shown that numbers do not exist, then I suppose that's an end to this discussion. And to mathematics.

But I hope you see the incoherence of his position. — Banno

The point is that a number is not a thing which can be counted, it is something in the mind, mental. I think you understand the difference between physical, sensible things which can be counted, and mental thoughts which are not individual things that might be counted. You did read Wittgenstein's Philosophical Investigations didn't you? Did you learn anything from it?

There is a very significant error in the idea that a measuring system could measure itself.

No, I don't think that Meta has shown that numbers don't exist. I'm inclined to think that he doesn't believe that, either. He has been explicit that he rejects what he calls Platonism, but I don't think it follows that he thinks that numbers do not exist. I'm not sure he even rejects the idea that there are an infinite number of them - since he realizes that we can't complete a count of the natural numbers. I do think that we can't get to the bottom of what he thinks without taking on board the metaphysical theory that he has articulated. — Ludwig V

The point is that "numbers" do not exist as individual countable things. This is a misrepresentation of what a number is, and the problem becomes evident when we allow the infinite capacity of numbering, and then try to count those numbers. So it doesn't matter if you represent the number as an independent Platonic object, or an object of human construct, either way is faulty. A supposed individual number is really an idea, which is dependent on other ideas for its meaning, and cannot be accurately represented as an individual object.

Your view is called finitism. It's from Aristotle. — frank

No, that's not quite right. I reject the assumption of any "mathematical objects" finite or infinite, as Platonism, and unacceptable. It just happens that the absurdity of assuming such objects becomes very evident with supposed infinite objects. In other words, the reality of "infinite", and "infinity" serves very well to demonstrate the falsity of Platonism.

A value can ‘have being’ within a formal system, a constructive framework, or a model, without existing independently as Plato would claim. — Banno

In case you think I did not address this claim, 'having being within a system' is fiction. It can be said that Frodo Baggins has being within a system, but this type of being is well known as "fictional".

Being within a mathematical system, is fictional being. Therefore it is a false premise rendering any supposed "proof" as unsound.

As I said, denial!

When numbers are assumed to be mathematical objects, these objects simply exist independently of any human mind. The supposed object is not in my mind, nor your mind, because it would be in many different places at the same time. Under the assumption of Platonism, a bijection does not need to be carried out, it can be represented, because it is assumed to already exist independently of any minds, so we just need to reference it. Likewise, the natural numbers can be represented with "{1, 2, 3, ...}" but only if they exist independently of any minds.

Do you understand the difference between the representation of a set of objects, and a formula for the procedure which you called "assigning a value"? Could you read 1, 2, 3, ... as a formula for assigning a succession of values?

1 is a number, and every number has a successor. That's enough to show that the natural numbers exist. — Banno

What you stated her is blatant Platonism.

If however, "1, 2, 3..." signifies to you, a formula for the process of "assigning a value" in a specified sequence, rather than representing an infinity of numbers, this is not Platonism. Can you apprehend the difference? I think you can.

And I think, that's why you switched form "to be is to be the value of a bound variable", to, "assigning a value". right after you stated "Platonism is indeed unacceptable". You know the mathematical principles you argue are thoroughly Platonist, and you feel ashamed of this. So you tried to cover this up. Why the dishonesty? Accept mathematics for what it is, and get on with it. Shameful deception and attempts to disguise your ontology get us nowhere.

Every time that you say 1, 2, 3... represents an infinity of numbers, that is blatant Platonism. It is absolutely necessary that the referenced infinity of numbers must have independent existence because it is absolutely impossible that they could exist within any minds.

No, Meta. Quantification or assigning a value does not require Platonic commitment. A value can ‘have being’ within a formal system, a constructive framework, or a model, without existing independently as Plato would claim. — Banno

You were claiming that numbers "exist", and how to be, is to be a value. Now you've totally changed the subject to "assigning a value".

Formally, set theory is just a system of rules. — Banno

Sure, and those rules are axioms about "mathematical objects". When you were in grade school, were you taught that "1", "2", and "3" are numerals, which represent numbers? Notice, "2" is not a symbol with meaning like the word "notice" is. It's a symbol which represents an object known as a number. In case you haven't been formally educated in metaphysics, that's known as Platonism.

Guess it's back to ignoring your posts. — Banno

And I'll opt to believe that you willfully deny the truth, rather than simply misunderstand.

Platonism is indeed unacceptable, but quantification is not platonic. — Banno

Do you recognize that set theory is based in Platonism?

If, "to be is to be the value of a bound variable", then you are obviously talking Platonism. Anytime a value has being, that's Platonism. I'm amazed that you do not understand this, or deny it, or whatever.

To be is to be the value of a bound variable. ω and ∞ are cases in point. In maths, Quine's rule fits: existence is not discovered by metaphysical intuition but incurred by theory choice. Quantification, ∃(x)f(x), sets out what we can and can't discuss. — Banno

As I said, Platonism, which is an unacceptable ontology.

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.