Then it would be impossible to create a reasonable hierarchy like you were talking about, if the meaning of tight and loose could vary. — Metaphysician Undercover

You can! The hierarchy would be more of a mold, a frame, that suites a generality purposes. Of course in the particular application the hierarchies would differ.

By the way do relativity theory speak about rules about the mathematical objects used to write its laws with? Aren't those mathematical objects a part of the theory? I don't think relativity theory assumes that numbers for example have a mass, or that they move with a speed less than light, etc.. Those mathematical objects are fixed, eternal, unchangeable. It's the physical objects that the rules of relativity theory applies to. I don't think that the mathematical objects and rules that it uses has anything to do with relativity theory. Imagine that number 1 for example will rut with time? That's crazy! Isn't it.

I agree that mathematical objects are ideal. And when using them in applications one must be cautious about hidden mathematical assumptions that might causing blurring or even faulty theories.

We use possibly fictional objects to display the mathematical rules with, because this is the most evident way in which it can be presented. Most of these rules, as well as the objects manipulated are non-spatio-temporal. But I think we can have pseudo-spatio-temporal objects representing mathematical worlds, thus in some sense approximating the real world. But I think also that nothing of the rule physical world law about physical objects would be applicable to these realms either.

In nutshell I think that set theory involving "collections" "elements of collections" "sets" "members of sets" etc.. all of these can be well understood in terms of Mereo-topology in a fairly easy manner. The hierarchy of sets is I think very essential to understand higher kinds of mathematics. For who's to say even those can possibly find some application in the real world?!

y the way do relativity theory speak about rules about the mathematical objects used to write its laws with? Aren't those mathematical objects a part of the theory? I don't think relativity theory assumes that numbers for example have a mass, or that they move with a speed less than light, etc.. Those mathematical objects are fixed, eternal, unchangeable. It's the physical objects that the rules of relativity theory applies to. I don't think that the mathematical objects and rules that it uses has anything to do with relativity theory. Imagine that number 1 for example will rut with time? That's crazy! Isn't it. — Zuhair

Objects existing in relationship to each other are objects existing in relationship to each other. If physics uses contradictory premises concerning objects existing in relationship to each other (the premises of relativity, and the axioms of mathematics being contradictory) then there is a problem.

It is irrational for you to claim that the "fixed, eternal, unchangeable" objects of mathematics are not subject to the laws of physics, unless you were to produce principles to support a dualist ontology. In that case, we'd have two distinct types of objects, and we'd have to start all over with our discussion of what constitutes an "object", starting with two distinct "objects". If we do not adhere to true principles, derived from the real existence of objects, we might as well allow that the construction of mathematical objects (being imaginary) does not need to adhere to any principles at all. What's the point in even assuming parts, and loose or tight relations at all, when it would be much easier to have eternal objects which have no parts whatsoever? Then the collections of such objects (sets) are not objects at all, but imaginary collections.

It seems to me like you want some half ass sort of compromised system for the existence of "mathematical objects" where you adhere to the principles of physical objects (loose and tight connections) to an extent, but when the principles of physical objects contradict the principles of eternal objects, which you desire to assume for the sake of simplicity, you are ready to throw these principles out the window in order to cling to the false facility of Platonism..

We use possibly fictional objects to display the mathematical rules with, because this is the most evident way in which it can be presented. Most of these rules, as well as the objects manipulated are non-spatio-temporal. But I think we can have pseudo-spatio-temporal objects representing mathematical worlds, thus in some sense approximating the real world. But I think also that nothing of the rule physical world law about physical objects would be applicable to these realms either. — Zuhair

The problem is, that mathematicians are manufacturing, creating, objects. These objects might be completely fictional, imaginary, and not intended to represent the real world at all. Or, in application, these objects might be intended to produce a representation of the real world. We need to decide which is the case. Are we using mathematics to model the real world, or are we using mathematics to create fictional, imaginary worlds? What good is the wishy washy position of saying that these objects are "in some sense approximating the real world"? Then those who want to use mathematics to model the real world will be dissatisfied, and those who want a fantastic, purely fictional mathematics will also be dissatisfied.

Yes, I think there is an intermediate position. Mathematics is producing rule following obedient fictional objects and scenarios. However, those happen to have applications in the real world. I suspect that the matter is not accidental. There is seemingly some common grounds between imagination and the real world. Some rules about arithmetic works fine when applied to real objects, and it really succeeded in increasing our understanding of the real world around us. On the other hand obviously there are rules that are not applicable to the real world like having infinitely many numbers, etc... I think logically obedient rule following imaginative scenarios do have some common grounds with reality.

These objects might be completely fictional, imaginary, and not intended to represent the real world at all. — Metaphysician Undercover

This is TRUE of many mathematical disciplines. For example a lot of set theory stuff is so imaginary that it might not even find any application at all. However, no one can really tell. Even imaginary numbers turned to have applications, even non-Euclidean Geometry turned to have applications. The problem is that we don't know really what our reality adheres to, or even what discourse about obviously imaginary objects could be useful in applications about the real world. There must be some shared realm for those applications to exist. The problem is that if we take Quines-Putnam indispensability argument, then even those non-spatio-temporal features of mathematical object might need to be accepted as part of reality, even though not a physical concrete kind of reality, but some kind of reality there!? The mathematician usually do not bother with these philosophical ground. All of what he cares for is the analytic consequences of his assumption, which for clarity and simplicity they are usually stipulated outside of the confines of space or time or both, or within the confines of some imaginary world that has its own space and time characteristics, as well as its own part-whole relationship with respect to eternity issues in it. Most mathematicians work primarily in a Platonic world! Philosophy comes later!

Yes, I think there is an intermediate position. Mathematics is producing rule following obedient fictional objects and scenarios. However, those happen to have applications in the real world. I suspect that the matter is not accidental. There is seemingly some common grounds between imagination and the real world. Some rules about arithmetic works fine when applied to real objects, and it really succeeded in increasing our understanding of the real world around us. On the other hand obviously there are rules that are not applicable to the real world like having infinitely many numbers, etc... I think logically obedient rule following imaginative scenarios do have some common grounds with reality. — Zuhair

I agree that correspondence with the real world is not accidental, and these principles are adopted for usefulness. But I don't think that any non-useful rules would be accepted. The reason why there is infinitely many numbers is so that we can count anything. It doesn't matter what the world might consist of, we will still, in principle be able to count it because we have infinitely many numbers.

I think we ought to consider a difference between corresponding with the real world, and being useful in the world. The two are clearly not the same. Mathematical rules I believe, are produced to be useful. This means that they do not necessarily correspond with reality, nor do they even have "some common grounds" with reality, they simply interact with reality by means of us using them. Perhaps this process, the activity of interaction, may be called a common ground, but we have to be careful to recognize that although it "grounds" the mathematical rules, it doesn't ground the real world. So for example some people say that the laws of physics describe the foundation for existence in the universe, but this is not really the case. The rules of physics are how we apprehend existence, in the universe, but we may be missing a whole lot, and therefore the rules of physics don't really describe the foundation for existence in the universe.

This is TRUE of many mathematical disciplines. For example a lot of set theory stuff is so imaginary that it might not even find any application at all. However, no one can really tell. Even imaginary numbers turned to have applications, even non-Euclidean Geometry turned to have applications. The problem is that we don't know really what our reality adheres to, or even what discourse about obviously imaginary objects could be useful in applications about the real world. — Zuhair

The problem with usefulness, and pragmatism in general, is that many things can be useful, in many situations. If you need to pound in a nail, you can pick up a rock and hit the nail, instead of using a hammer. So speculators may think up wildly imaginary theories, and people applying the mathematics will pick up what is available, and put it to use. Therefore we need some standards of efficiency, or something like that, by which to judge usefulness. This is complex. We need clearly defined goals, which in itself is difficult because our own goals are often not clear to us. Then we need the means for judging whether the goals are adequately being achieved.

The problem is that if we take Quines-Putnam indispensability argument, then even those non-spatio-temporal features of mathematical object might need to be accepted as part of reality, even though not a physical concrete kind of reality, but some kind of reality there!? The mathematician usually do not bother with these philosophical ground. All of what he cares for is the analytic consequences of his assumption, which for clarity and simplicity they are usually stipulated outside of the confines of space or time or both, or within the confines of some imaginary world that has its own space and time characteristics, as well as its own part-whole relationship with respect to eternity issues in it. Most mathematicians work primarily in a Platonic world! Philosophy comes later! — Zuhair

I agree with this, and I am completely on board with you here. Maybe, as philosophers, we can analyze this separation between the Platonic world of mathematicians, and the real world which we live in. The difficult thing here is to understand how there can be such a separation in the first place. Let's say that the separation was created, it was manufactured, produced by dualist principles. Like the example above, with infinitely many numbers, the goal was to enable us. To measure the world, we need a measurement system which transcends the world, it must be capable of measuring anything possible in the world. As you say, we don't know what's in the world before we measure it, so we must have a system capable of measuring anything. Therefore the measurement system is based in the assumption of infinite possibility, whereas the real world consists of limited possibilities.

Do you agree that this is the basis of that dualist separation between the real world and the Platonic world? The human mind apprehends the world as consisting of numerous possibilities. In order for it to understand each, every, and any possibility, the mind assigns to itself, the capacity to understand infinite possibilities. But that assignment is wrong, because the human mind is restricted by the real world, being a part of the human body, and so its capacity to understand is really restricted. So the human mind has created this dualist premise, and all these dualist principles, in an attempt to give itself the capacity to understand anything, and everything, when in reality it doesn't have that capacity. That Platonism is self-deception. It was far a good cause, but when it runs its course and we see that it is impossible for it to give us what it was designed to give us, we need to get rid of it.

Do you agree that this is the basis of that dualist separation between the real world and the Platonic world? The human mind apprehends the world as consisting of numerous possibilities. In order for it to understand each, every, and any possibility, the mind assigns to itself, the capacity to understand infinite possibilities. But that assignment is wrong, because the human mind is restricted by the real world, being a part of the human body, and so its capacity to understand is really restricted. So the human mind has created this dualist premise, and all these dualist principles, in an attempt to give itself the capacity to understand anything, and everything, when in reality it doesn't have that capacity. That Platonism is self-deception. It was far a good cause, but when it runs its course and we see that it is impossible for it to give us what it was designed to give us, we need to get rid of it. — Metaphysician Undercover

Yes. Actually I find myself in total agreement with what you said in this particular last comment. Of course we are speaking about mathematics judging it according to its possible usefulness. Some mathematicians might reject to look at mathematics from this perspective, seeing it external to what really constitute its cor content.Seeing it like judging chess by how much profit it brings to the chess player? They might insistent that mathematics is the discipline devoted to some imaginary world were there is no impediment to carrying out strict rule following scenarios, so giving itself the maximal freedom in doing so, and enjoy that practice all for itself without caring about whether it would be applicable or not. Something like the position of "Art for Art" and "Art for people". Your views here suite "Mathematics for science", while some mathematicians might insist on "Mathematics for Mathematics". But I agree with you that the importance of mathematics is related to its role in furthering our understanding of the world, and if it didn't have that rule, it wouldn't have had all that fame and seriousness in studying it, it would have been something like chess, a mental game, a kind of sport, or even art.

I agree with the duality policy. The real issue is how to judge when a mathematician is going a stray? I mean as far as possible contribution to knowledge is concerned (i.e. application). I think a real foundation of mathematics must help direct mathematicians towards producing more beneficial mathematical theories. But how to judge this? I think this is a very important question? We need a foundation for applicable mathematics! But I'm almost very sure that a lot of mathematicians, possibly the most, wouldn't care the hell for that, they'll view it as too restrictive, and favor diving deep into the world of logically obedient rule following scenarios, no matter how wildly far their imaginative worlds are from reality. Sometimes I think this is like the dualism of religion and state in secular states. Let the mathematicians dive deep into the imaginary platonic world they like, and let science work with its strict observance to reality moto. The important matter is not to confuse both. We only need to coordinate both at applications!

That Platonism is self-deception. It was far a good cause, but when it runs its course and we see that it is impossible for it to give us what it was designed to give us, we need to get rid of it. — Metaphysician Undercover

The real problem is even if it is false, still the logically obedient strict rule-following themes it negotiates can prove to be extremely useful, even if in part. The real problem is that we'll never know at which stage it will "run out its course"? Possibly one day foundations for 'applicable' mathematics would issue, having clear cut edge between what is beneficial and what is not?! Perhaps by then this platonic dream would vanish! perhaps?! but I don't really know where such a thing would start? or even if it could start really? Until such alternative is found, we'd better keep the current dualist stance.

Your views here suite "Mathematics for science", while some mathematicians might insist on "Mathematics for Mathematics". — Zuhair

I find that hard to imagine, Mathematics for Mathematics. What would this consist of, people studying and producing mathematical principles just for the sake of doing that, and none of them actually doing anything with the mathematics? So people create mathematics, they study to understand mathematics, and they never apply the mathematics. That's a very odd thought. But in the university I went to, Mathematics was in the Arts department. I think it was studied as a useful art though.

I agree with the duality policy. The real issue is how to judge when a mathematician is going a stray? I mean as far as possible contribution to knowledge is concerned (i.e. application). I think a real foundation of mathematics must help direct mathematicians towards producing more beneficial mathematical theories. But how to judge this? I think this is a very important question? We need a foundation for applicable mathematics! But I'm almost very sure that a lot of mathematicians, possibly the most, wouldn't care the hell for that, they'll view it as too restrictive, and favor diving deep into the world of logically obedient rule following scenarios, no matter how wildly far their imaginative worlds are from reality. Sometimes I think this is like the dualism of religion and state in secular states. Let the mathematicians dive deep into the imaginary platonic world they like, and let science work with its strict observance to reality moto. The important matter is not to confuse both. We only need to coordinate both at applications! — Zuhair

Well you're right, mathematics as an art provides a freedom which is appealing to many. One can demonstrate all sorts of very beautiful things just by applying mathematics to mathematics. That is actually the beauty of mathematics, its very nature is incredibly beautiful. But suppose we can separate math for math's sake, creating beauty in mathematics, from applied math, which is math for some other purpose. The pure art of mathematical beauty would just be there to look at and think about, and the artists would have to warn people against trying to apply that math (some mathemagicians could put some real freaky tricks in there which are thought to be incredibly beautiful). The other mathematicians, creating mathematics for a purpose, would have to be disciplined so as to reign in that freedom, and keep things directed toward the proper goal. I think the two have already been confused as eloquence is becoming more and more of an important part of theories.

The real problem is even if it is false, still the logically obedient strict rule-following themes it negotiates can prove to be extremely useful, even if in part. The real problem is that we'll never know at which stage it will "run out its course"? Possibly one day foundations for 'applicable' mathematics would issue, having clear cut edge between what is beneficial and what is not?! Perhaps by then this platonic dream would vanish! perhaps?! but I don't really know where such a thing would start? or even if it could start really? Until such alternative is found, we'd better keep the current dualist stance. — Zuhair

It may not be that difficult. To begin with, any application in which the infinite is approached, in any way, is an application where the false premise of Platonism is causing a problem. The mathematician can devise all sorts of different ways to deal with the occurrence of the infinite, but these just disguise the problem. The very nature of infinite, and the nature of application (being practise), makes it impossible that the infinite could be encountered in any application. The mathematician might say 'we have to be able to apply the infinite, it's part of mathematics', but really all that the infinite is, is a thing of beauty, a beauty which is negated by any misguided attempt to apply it.

The mathematician might say 'we have to be able to apply the infinite, it's part of mathematics', but really all that the infinite is, is a thing of beauty, a beauty which is negated by any misguided attempt to apply it. — Metaphysician Undercover

I'm not really sure of that? But as part of history of mathematics, mathematics prior to the 19th century were very cautious when speaking about the infinite. They actually almost avoided it, and only used finite kind of numbers and entities (albeit infinitely many). Not only that! Most of hard core mathematics can be encoded in very weak systems of set theory, however those systems need the infinite most of the time, but they are very weak infinite systems like first and second order arithmetic.

Can you describe this "need" for me? If mathematics prior to the 19th century got along fine without speaking about the infinite, where does this need to apply the infinite, in set theory, come from?

Suppose the purpose of the infinite is, as I described. It is assumed so that we can measure anything. No matter how large the magnitude which we've already encountered, we can always measure something larger. That is the principle of the infinite, it's an open ended scale so that we can always go bigger in our comparisons (measurements) Perhaps you don't agree that this is the reason why we assume infinity, but let me start with that assumption anyway.

The problem I apprehend in set theory is that there appears to be a perceived need to measure the infinite itself. Why would we want to measure the infinite? Suppose you're applying mathematics in measurement, and infinity rears its ugly head. (Consider that the infinite is the most beautiful mathematical principle, as a principle devised for unbounded usefulness, but when it occurs in practise it is the most ugly situation). What is indicated, by the appearance of infinity in the practise of applying mathematics, is that we've encountered something which we cannot measure. That's what I think. What I also think is that the proper way of dealing with this situation is to take a very good, analytical look at the thing we are trying to measure, in relation to our principles of measurement, and determine why infinity appears. The principles being applied are not properly related to the infinite to allow the thing to be measured. The purpose of infinity is to enable us to measure anything, so if it appears in the measurement, there must be an incompatibility between the principles applied, and the thing being measured. We must therefore determine this incompatibility, and devise the proper principles suited to measure the object we are measuring.

However, it appears to me, like set theory takes the wrong approach when the infinite rears its ugly head. Instead, set theory proceeds toward an irrational resolution of this problem, by devising a way to measure the infinite. It's irrational because the infinite, as a principle which gives us the capacity to measure anything, can only be effective to this end, if it is separate (transcendent) from the things being measured: it is necessarily an Ideal. If we allow it to be placed in the category of things which we can measure, we negate the transcendent nature of the infinite The principle, the infinite, can no longer give us the capacity to measure anything by transcending everything, because we've allowed the principle to become diluted, by designating it as one of the things which we are trying to measure.

Can you describe this "need" for me? If mathematics prior to the 19th century got along fine without speaking about the infinite, where does this need to apply the infinite, in set theory, come from? — Metaphysician Undercover

This is indeed a very big subject. And what you've said is reputable. Indeed some imminent mathematicians took that stance, and even a much more aggressive stance of even not allowing for existence of large finites (see finitism and ultrafinitism) let alone infinities. One can indeed write whole volumes on that subject. However, I'll present a possible counter-view:

In agreement with your assumption of having the infinite as a capacity for unlimited measurement in applications about finite objects. I'd say that It is not just infinitude of natural numbers that we need for the sake of such unlimited measurement, we need to stipulate useful relations and functions (operators) on them, so relations like "equal", "smaller than", "greater than", and functions like "summation", "multiplication", "exponentiation" etc.. all of these are needed. So we need SENTENCES in a language that uses those functions and relations between natural numbers, those sentences would be the axioms, and can be the theorems deduced in the systems having logical and mathematical inference rules starting from those axioms. So I insist that we need "sentences" in the language of arithmetic, which doesn't include only natural numbers, but also includes relations and functions on those natural numbers. Now due to Godel's incompleteness theorems, there is no effective system that can capture all true sentences of arithmetic! Now notice here that I'm speaking about sentences of arithmetic and their terms only range over natural numbers, i.e. there is no infinite object whatsoever symbolized in that language, so it is totally about finite objects or descriptions about finite objects, so it is the kind of language that we think it can possibly have applications in our world, viewing all objects in our world being finite. Now we come to the role of theories about the infinite, i.e. theories that speak about infinite objects like actual infinite sets for example, which as you said, and I think it is generally agreed that our physical world seems to be incompatible with their existence in it. However, despite this incompatibility those infinite theories can prove some true arithmetic sentences (those that only range over natural numbers using finitely long formulas) to be true that the theories restricted to the finite objects fail to prove! Now we want the infinite to give us the capacity of measurement as you said, but you need the tools for those measures, and the tools for those are not just the existence of infinitely many naturals, but we need sentences about some relations and operations on them, and the main problem is that we don't have a theory restricted to the finite realm that can effectively give us all of those sentences, which are infinite in number by the way. Or even if those useful arithmetical sentences are finite in number, still we don't have a theory about finite objects that can capture all of those finite sentences, or even if we can have, we don't know which theory is that. So theories speaking about infinite objects can indeed prove some of those arithmetical sentences about the finite realm of them, and those can be useful sentences. So that's why we go to the infinite. Actually Hilbert was under the impression that all of infinite mathematics is just a conservative extension of the finitary one, and that all of what we need is what those infinite theories speak about the finite realm of them. However, he turned to be wrong. And this makes it even more necessary to go to theories speaking about infinite objects, since those can be indispensable for that sake, since those are stronger theories (deductively speaking) and so can prove sentences about the finite realm of them, that weaker theories limited to finite objects cannot do. So this would indirectly support the measure-ability view that you've raised! We go to theories about the infinite, in order to have more and more useful SENTENCES about the finite world! And so use these sentences necessary to produce the additional tools for measurements in our world, and so furthering knowledge about our finite world.

If the theories about the infinite realm proves to be indispensable for harvesting some useful arithmetical sentences about the naturals, i.e.that no theory limited to the finite realm can effectively prove them, then this would make the case for mathematical investigation of the infinite!

Not only that, sometimes those infinite theories might not be indispensable for the above sake, but being more powerful they make proofs about such sentences much easier to have, and so one can contemplate many versions of equivalent systems about the finite world, much easier than when working with theories restricted to just finite objects. It is this ease that is also important, since it would have heuristic value to discoveries in the finite realm.

Of course there is still the objection that such additional sentences about the finite world that theories encountering infinite objects prove, these would prove to be FALSE arithmetical sentences, and so be misleading rather than useful, and would be deemed as waste of time and efforts. That's of course possible, but until such an argument is provable, the window is still open for mathematical investigation of the infinite, under the hope that it might play the above-mentioned role, and so we need to give it the benefit of doubt!

Of course there are other more radical objections to your line of view, like the mathematics for mathematics viewpoint, and like the other direction objection that is our physical world itself being of ACTUAL INFINITE reality and that our current physical theories and observations being erroneous about that aspect, etc... I didn't want to go to those, because I honestly think that the bulk of evidence supports a finite (or at most potentially infinite) outcast of our universe, and that mathematics ought to be useful in understanding that universe, and therefore I approached it from that perspective as given above.

Human experience does not contain anything without end/limit.
Finite provides boundaries that allow measurement.
Mathematics tries to manipulate the imaginary (mental) concepts using methods designed for the finite.
Am still waiting for anyone to measure a one ended stick.

Hi Zuhair, I'm having a little difficulty understanding some parts of your post, but I'll provide my interpretation with some criticism, and you can tell me where I misunderstand.

I'd say that It is not just infinitude of natural numbers that we need for the sake of such unlimited measurement, we need to stipulate useful relations and functions (operators) on them, so relations like "equal", "smaller than", "greater than", and functions like "summation", "multiplication", "exponentiation" etc.. all of these are needed. So we need SENTENCES in a language that uses those functions and relations between natural numbers, those sentences would be the axioms, and can be the theorems deduced in the systems having logical and mathematical inference rules starting from those axioms. So I insist that we need "sentences" in the language of arithmetic, which doesn't include only natural numbers, but also includes relations and functions on those natural numbers. — Zuhair

I take it that you are saying here, that we might need to allow for infinity in any scale of measurement. If that's what you're saying, I don't think it's true. I think that each parameter, "greater than", "heavier than", "denser than", etc., has its own definition. This creates a sort of category, and the descritpiton of the category provides the limits to what is measured within that category, therefore infinity is excluded by these descriptions or definitions. We might say that each of these categories is a specified "quality" and the scale is produced to enable measurement of that quality. The determining features of that quality exclude the possibility of infinity within the scale. So it doesn't make any sense to say "infinitely great", "infinitely dense", "infinitely heavy", "infinitely hot" or any such thing, because these are defined qualities, and to fulfill the criteria of any quality requires that the thing being measured can be related to the thing which forms the scale for measurement of that quality, and this excludes infinity..

So we need SENTENCES in a language that uses those functions and relations between natural numbers, those sentences would be the axioms, and can be the theorems deduced in the systems having logical and mathematical inference rules starting from those axioms. So I insist that we need "sentences" in the language of arithmetic, which doesn't include only natural numbers, but also includes relations and functions on those natural numbers. — Zuhair

The "SENTENCES" act to describe the various qualities, and they may be set up as rules for application of the numbers. So a scale consists of sentences which are rules for the application of numbers. The most fundamental sentences are the axioms which are the most general rules for application. This is where it gets tricky (watch out for mathemagicians). The question is, are there any true general rules (axioms), which are applicable to all mathematical applications, or, is each set of axioms tailored to a particular type of application (measuring a particular quality). In terms of "sentences" then, are there any sentences which may act as rules for all mathematical applications, or is every sentence designed for a particular type of application. I suggest that we allow the possibility of a sentence which allows the use of mathematics for any scale, to measure any quality, as a fundamental axiom, and this would be a sentence describing the infinitude of numbers. Some qualities would require one type of scale, others another type of scale, and the numbers must be infinitely pliable to adapt to all scales. The fundamental axiom therefore, would be an axiom of order, order being required for any form of measurement, also allowing for infinity so long as the infinity is ordered. Allowing for infinite disorder is nonsense.

Now due to Godel's incompleteness theorems, there is no effective system that can capture all true sentences of arithmetic! Now notice here that I'm speaking about sentences of arithmetic and their terms only range over natural numbers, i.e. there is no infinite object whatsoever symbolized in that language, so it is totally about finite objects or descriptions about finite objects, so it is the kind of language that we think it can possibly have applications in our world, viewing all objects in our world being finite. — Zuhair

You ought to be able to see that Godel's approach is backward. A sentence is useful for describing a quality. Once the quality is described, we can take the mathematical principle of infinite applicability and apply it to the described quality. But it doesn't make sense to try and turn things around, making the infinite applicability of mathematics into a quality which can be described by a sentence. So naturally, Godel cannot find that sentence. The infinite applicability of mathematics must be inherent within mathematics itself, and therefore quantitative, and not qualitative. It cannot be described. Even my above description, using "order" does not do justice to "quantity", because it attempts to hand quantity a quality, which is to assign that impossible sentence. So the meaning of "quantity" and "infinite" must remain independent from any descriptive sentence which would assign to these a quality. Such an assignment would be a restriction to the thing which has been designated as unrestricted.

So we cannot approach in that backward manner of attempting to assign a restriction to the unrestricted. We start with the unrestricted, "quantity", and proceed toward measurement by using sentences of restriction which are derived from the thing to be measure. We observe the thing to be measured, and we produce sentences of restriction which are applied to the mathematics, restrictions which are designed to enable measurement of that particular type of thing. So there is no random or arbitrary restrictions placed on the natural numbers, each restriction is placed for a particular reason, dependent on the apprehended quality.

Now we come to the role of theories about the infinite, i.e. theories that speak about infinite objects like actual infinite sets for example, which as you said, and I think it is generally agreed that our physical world seems to be incompatible with their existence in it. However, despite this incompatibility those infinite theories can prove some true arithmetic sentences (those that only range over natural numbers using finitely long formulas) to be true that the theories restricted to the finite objects fail to prove! — Zuhair

According to what I've said above, we cannot come to theories about the infinite in this way. We accept the infinite as a starting point. We make theories concerning the things we observe, and restrict the infinite for application accordingly. To turn around, and face the infinite, with the intent of restricting it for no particular purpose is an irrational move. That's what theories about the infinite do, they restrict it with descriptive sentences. And if this is done for no purpose other than to describe the infinite, it's crippling.

I would think that if we want to proceed toward understanding the infinite, we must approach from a different direction, other than mathematical axioms, which by their nature are composed to restrict the infinite for various purposes. We must therefore approach from the premises which assign to "quantity" its infinite capacity. This means that we must understand "infinite" in terms other than descriptive terms; descriptive terms being applicable to quality only, and used in mathematics for the sake of restricting quantity. Are there sentences which give to "quantity", "infinity", without resorting to description?

Now we want the infinite to give us the capacity of measurement as you said, but you need the tools for those measures, and the tools for those are not just the existence of infinitely many naturals, but we need sentences about some relations and operations on them, and the main problem is that we don't have a theory restricted to the finite realm that can effectively give us all of those sentences, which are infinite in number by the way. Or even if those useful arithmetical sentences are finite in number, still we don't have a theory about finite objects that can capture all of those finite sentences, or even if we can have, we don't know which theory is that. — Zuhair

The infinite gives us the appropriate capacity for measurement simply by assumption. We assume that we have that capacity, and so long as we do not restrict it, it persists, as the fundamental premise. We make the sentences concerning relations and operations according to our observations, and the qualities which we desire to measure. We never need "all of those sentences" we produce them as required, dependent on our observations.

At this point, I think we ought to distinguish between the object itself, and the observed qualities of the object. The restrictive sentences are always produced for the sake of measuring particular qualities. We do not assume the capacity to measure the object itself. This seems like it would be a little nonsensical as all of our observations are of particular qualities. What would we be trying to measure, as the object itself? Therefore the observed finitude of the object is a function of its qualities. Observation of the object's qualities, and the conclusion that qualities are real, a fact of the object, produces the conclusion that the object is itself finite. So to say 'an object is finite' is to say nothing more than 'an object has finite qualities'.

In all of our observing and measuring of qualities, we really do not ever get to what it means to be an object, and this is what it means to be something which "has" qualities. So here, to understand the existence of the object itself, we must turn to something other than mathematical principles. From these other principles we can begin to understand "the object" in a different way, as a fundamental unity (perhaps as various qualities unified). And unity in relation to multiplicity is the fundamental principle of mathematics. So when we assume a unity of a multiplicity of qualities, as an object, we have one thing which is at the same time many things, and potentially an infinite number of things, so long as we maintain the distinction between the thing (one) and the qualities (many).

So theories speaking about infinite objects can indeed prove some of those arithmetical sentences about the finite realm of them, and those can be useful sentences. So that's why we go to the infinite. — Zuhair

According to what I stated above, the idea of "infinite objects" is a misguided one. The object, as the thing, is always one, a simple unity. The multiplicity as one, is what is unified, under the named and identified "infinite objects". And this multiplicity is a property of the identified thing, we might call it the qualities of the named thing, it consists of numerous things. To speak of a multiplicity of objects is to class those together as one unity. Then "the numerous objects" is a quality of that multiplicity which is referred to as one unity. So we have an object which is described as a multiplicity. We allow that the multiplicity which composes that one identified object, may be infinite. But it is incorrect to refer to that proposed infinity as an infinity of objects, because it is really an infinity of parts, the qualities of that mentioned object, which is the named collection. The thing identified as "infinite objects" is really the object itself, so "objects" ought not be used here and it is really nonsense to speak of "infinite objects".

Of course there are other more radical objections to your line of view, like the mathematics for mathematics viewpoint, and like the other direction objection that is our physical world itself being of ACTUAL INFINITE reality and that our current physical theories and observations being erroneous about that aspect, etc... I didn't want to go to those, because I honestly think that the bulk of evidence supports a finite (or at most potentially infinite) outcast of our universe, and that mathematics ought to be useful in understanding that universe, and therefore I approached it from that perspective as given above. — Zuhair

I think it is important to recognize that a sentence about something will describe a quality. As a quality, the thing referred to is finite. We might allow that the object itself, with that quality is infinite in the sense of potentially having an infinity of qualities,, but this is a self-defeating assumption because it assumes an object which is immeasurable, and the purpose of assuming the infinite is to make all things measurable. And that is also why it is irrational to allow that the infinite itself is an object. We allow for the possibility of infinite qualities to account for the unknown qualities which we have not observed. But this assumption of "'the possibility of infinite qualities" is only made because we know that our knowledge will never be complete. It doesn't indicate that we assume that there actually is an infinity of qualities to any object because this would be assuming the object as fundamentally unknowable.

Hi Zuhair, I'm having a little difficulty understanding some parts of your post, but I'll provide my interpretation with some criticism, and you can tell me where I misunderstand. — Metaphysician Undercover

Hi Metaphysician Undercover! I also cannot understand what you've wrote. I think we are departing a part.

My argument was a very simple argument. I was simply speaking about Peano arithmetic "PA". Peano arithmetic only speaks about natural numbers, which you can in some sense imagine them as indices of the quantity of members in finite sets, so we are speaking here about measurements related to FINITE objects, and so the language of PA doesn't encounter any mention of the infinite, although it allows for the potential of having infinitely many naturals, but it doesn't speak of that infinitude of naturals as an object on its own right. The only objects that PA speaks about are naturals which are in some sense measures of finite objects. So generally speaking PA would be the kind of a theory that is expected to have applications about objects in our finite (or potentially infinite) universe. So all sentences written in the language of PA are statements about finite objects, so they all speak about the state of affairs related to finite objects, as we regard them to be potentially applicable!

The problem is that MOST of sentences written in the language of PA are not provable in PA. So we are missing a lot of sentences that might have useful application in our real world, because PA cannot prove them. However those arithmetical sentences can be proven from theories that encounter speech about existence of infinite objects, like set theory for example, so ZFC can prove arithmetical sentences which cannot be proven in PA. Notice that I'm speaking about arithmetical sentences, i.e. sentences about natural numbers, i.e. statements about measurement of the FINITE, so those are statements that can have applications in our real world, and some of those sentences are provable in ZFC while PA cannot prove them!

Not only that there are sentences that PA happen to prove,i.e. theorems of PA, but yet the proofs of those sentences are too long to be first discovered depending on PA alone, that it is much easier to prove in ZFC, because the proofs are much much shorter. In this way ZFC can help us discover theorems of PA itself.

You see what I'm trying to say here. That's why we revert to ZFC, because it is more powerful and more expressive than PA, that we can even prove theorems of PA itself using ZFC in a much easier way! And also using ZFC we can in addition prove sentences written in the very language of PA itself, that PA itself cannot prove. And those sentences can express facts about measurements of finite objects, and so might have applications.

but this is a self-defeating assumption because it assumes an object which is immeasurable, and the purpose of assuming the infinite is to make all things measurable — Metaphysician Undercover

I read all of the above account of yours carefully. It is really nice. I should say that it added a lot to my knowledge. So Thank you for that.

There is a lot of depth to what you are saying. I won't here make my final stance with or against it, since I need myself more time to examine it in a deep sense. For now I'll outline the potential difference between what you've presented and the commonly held views in foundations of mathematics.

What you said runs against standard set theory. For example you adopted the strategy of assuming the infinitude of observations, of possible qualities (largely its a stance made because of our incompleteness rather than being a truthful statement about the real world, as you maintain). However you refused to put the infinite among those qualities, and you emphasize that its not, it is just a pure quantity. The problem is that the common modern understanding in set theory of the infinite runs to the opposite of that. It views a lot of grades of infinitude, and the absolute allowance of all observations is something that NO formula in set theory or logic can describe. We may paraphrase it as saying that there no limit to measurability made inside theories, but this here applies to various degrees of the infinite as well as to finities. So you are not differentiating between the 'absolute capacity' of measurement, which is sometimes ironically called by some set theories as the absolute infinite, [which you call the "infinite" by the way], and the various grades of the infinite, the latter ones are using your terms qualities, and they can be measured in an effective manner, while the former one (the absolute infinite) is what you cannot measure nor can formalize it as an axiom, and using your terms I would describe it as not really a quality, its a pure quantity (using your terms), this absolute infinite is something that no set theorist tries to capture by its axioms or theories, its an unlimited tendency of measurements. But the scope of measurement in your case is limited to the finites, while for the traditional set theorist it doesn't stop their, it can encounter various grades of the infinite, he will answer you that you are confusing $\omega$ [the first infinite] with the absolute infinite, and this is wrong. The scope of allowable measurements with the set theorist is vastly larger than the narrow one you are allowing. That's the difference.

However, my argument above (the one you've answered to) is not that deep. It only says that theories that have capture SOME infinite objects, are vastly stronger (deductively speaking) than theories that only capture the finite, that's why technically speaking, those stronger theories can help even prove some theorems of strictly weaker theories that only speak about the finite, not only that it can prove theorems spoken about in their language that those weaker theories cannot prove, and those sentences are of the kind speaking about infinite objects and relations between them and properties of them, so they are (generally speaking) the kind of sentences expected to have application in our finite world. This mean that theories speaking about infinite (as well as finite) objects can aid in measurements of the actual world via proving those sentences of them that are concerned with the finite realm of them. It supply us (technically speaking) with more and more sentences about finite objects, and so enrich our knowledge base and potential to make descriptions in our finite world. Its a pure technical issue. So they are useful and can make contributions to our finite world, although they are theories that have the capability of speaking of infinite objects (pure unities with infinitely many qualities).

Just being a pure unity with infinitely many qualities, it doesn't mean that it is unknowable, or that it is not measurable! That's the basic message that set theory is telling us. The earlier philosophers who thought that, like Aristotle, were mistaken!!! Cantor clearly showed how to make descriptions of various grades of the infinite, and he successfully did so, without being encountered with any inconsistency so far (it has been more than a century since then!).

Since contemplating SOME of the infinite as objects, or as qualities in your terms, did enrich our parcel with sentences about finite sets and numbers, which can possibly have applications, then it enriches our potential for application, so we are justified in doing such contemplation. Its a technical utility point of view.

"What is the difference between actual infinity and potential infinity?"
All matter can potentially be part of a black hole.

The only objects that PA speaks about are naturals which are in some sense measures of finite objects. So generally speaking PA would be the kind of a theory that is expected to have applications about objects in our finite (or potentially infinite) universe. So all sentences written in the language of PA are statements about finite objects, so they all speak about the state of affairs related to finite objects, as we regard them to be potentially applicable! — Zuhair

Here's the issue I have with this position. Let's assume that numbers are measures applied to finite objects. Prior to application, we need a rule, or rules of application, and these are in the form of sentences. Naturally, the sentences, rules, must be based in some actual understanding of what a finite object is, in order that the application be useful. For example, in geometry the goal is to measure the spatial limitations of objects. The objects are finite because they have spatial limitations, and the goal is to measure these limitations. So we have developed some understanding of these spatial limitations through observations, and produced axioms of geometry from this understanding.

Now let's go deeper, and relate this principle to the axioms for applying numbers. The goal here is clearly to measure objects, but we must be careful in the assumption that the objects are finite. We could probably say that natural numbers are intended to count finite objects, but if we consider rational numbers, which are infinitely divisible, the object loses its finitude in that way. Now we would have two distinct rules applicable to the use of numbers for measuring objects, rules for measuring finite objects and rules for measuring infinite objects. Because of this, we would need further rules to distinguish whether the objects to be measured are finite or infinite. Therefore we need some principles for understanding the infinity of an object. In the example of geometrical axioms, above, we have a clear understanding of the finitude of the objects to be measured, they have spatial limitations.

Now we want to say that these spatial limitations do not provide a complete, or absolute form of finitude, because the object with spatial boundaries, may be infinitely divisible. The rational numbers allow us to make such divisions, but then the irrationals pop up, and we see that our ways of dividing finite objects are somewhat deficient. So I think we can say that there are natural limitations to dividing objects, which makes infinite division of an object irrational, but we really do not understand these limitations.

The dilemma now is how do we restrict our rules for the application of measuring objects, such that objects are necessarily finite, and therefore measurable, when we do not know the real principles whereby an object is restricted to being finite. We have rules of geometry which restrict the spatial extension of an object, but we do not understand, and therefore cannot produce the rules, to restrict the "intension" (idiosyncratic use) of the object. Until we properly understand the intension, we do not have adequate rules to restrict the object's intension, therefore we cannot truthfully say that the object being measured is finite. This is a problem inherent within division, that we have irrational ratios, (a type of contradiction), which becomes very evident in harmonies and wave problems. We do not understand the real spatial constraints which restrict the division of objects, so the rules which we apply are simple modifications to infinite divisibility. Therefore our rules are rules for measuring infinite objects which we pretend, by tweaking the rules with terms like Infinitesimal, are actually rules for measuring the finite object.

The problem is that MOST of sentences written in the language of PA are not provable in PA. So we are missing a lot of sentences that might have useful application in our real world, because PA cannot prove them. However those arithmetical sentences can be proven from theories that encounter speech about existence of infinite objects, like set theory for example, so ZFC can prove arithmetical sentences which cannot be proven in PA. Notice that I'm speaking about arithmetical sentences, i.e. sentences about natural numbers, i.e. statements about measurement of the FINITE, so those are statements that can have applications in our real world, and some of those sentences are provable in ZFC while PA cannot prove them! — Zuhair

Now, with set theory you jump to the assumption of "infinite objects". What grounds this assumption? We use numbers to measure objects, and the rules for measuring are based in the natural restrictions of the object. Having restrictions is what makes them measurable, but also what makes them finite. In the last post, I called these restrictions "qualities". The qualities of an object are what we measure.

Let's say that because an object has restrictions, boundaries, making it finite, this allows that there is a multitude of objects, more than one. Now we want to count that multitude and this is different from measuring an object (its restrictions, qualities), it is measuring a quantity. This is completely distinct from the act of measuring objects, it is an act of measuring a quantity. Therefore we can make a conclusion here. As explained above, we need principles, statements, rules based in the understanding of what an object is, prior to making rules about measuring objects, now we need an understanding of what a quantity is, prior to making rules about measuring quantities. Again, we are confronted with the very same problem described above, which is the consequence of our inability to understand the nature of divisibility. The plurality, multiplicity, "quantity", is completely dependent on how we divide things up into individuals. The rules which we have for dividing our environment into individual things are the rules which govern the "quantity" of things.

So you are not differentiating between the 'absolute capacity' of measurement, which is sometimes ironically called by some set theories as the absolute infinite, [which you call the "infinite" by the way], and the various grades of the infinite, the latter ones are using your terms qualities, and they can be measured in an effective manner, while the former one (the absolute infinite) is what you cannot measure nor can formalize it as an axiom, and using your terms I would describe it as not really a quality, its a pure quantity (using your terms), this absolute infinite is something that no set theorist tries to capture by its axioms or theories, its an unlimited tendency of measurements. — Zuhair

I view this distinction between the absolute infinite, and various grades of infinite as unjustified, and actually a category mistake. The absolute, and the relative are categorically distinct. To place "the infinite" in both categories, is actually impossible, and all it does is give two quite distinct meanings to the term, inviting equivocation. For example, it's like the difference between absolute rest and relative rest.in physics. If we accept the principle that all rest is relative then absolute rest is meaningless to us. But if we accept absolute rest as a meaningful proposition, then relative rest can no longer be called "rest" because "rest" is just assigned as a reference point, while the thing designated to be at rest is still in movement compared to absolute rest.

The same is true for absolute infinite and various grades of infinite. If infinity is relative, various grades, then "absolute infinite" is meaningless because if there was an absolute infinite it would mean that the other various infinities are just not true infinities and ought not be called such. So all this does is give "infinite" two very distinct meanings, inviting equivocation.

However, my argument above (the one you've answered to) is not that deep. It only says that theories that have capture SOME infinite objects, are vastly stronger (deductively speaking) than theories that only capture the finite, that's why technically speaking, those stronger theories can help even prove some theorems of strictly weaker theories that only speak about the finite, not only that it can prove theorems spoken about in their language that those weaker theories cannot prove, and those sentences are of the kind speaking about infinite objects and relations between them and properties of them, so they are (generally speaking) the kind of sentences expected to have application in our finite world. This mean that theories speaking about infinite (as well as finite) objects can aid in measurements of the actual world via proving those sentences of them that are concerned with the finite realm of them. It supply us (technically speaking) with more and more sentences about finite objects, and so enrich our knowledge base and potential to make descriptions in our finite world. Its a pure technical issue. So they are useful and can make contributions to our finite world, although they are theories that have the capability of speaking of infinite objects (pure unities with infinitely many qualities). — Zuhair

This could only be true if we could develop an understanding, and therefore applicable rules concerning what it means to be an infinite object. To me, as I've outlined above, the idea of an infinite object only arises because of our inability to understand the nature of divisibility. We do not understand the true constraints and restrictions on divisibility in our world, so we posit infinite divisibility. This creates the infinite divisibility of an object, and the idea of an infinite quantity of objects, both. An infinite quantity of objects is derived from the infinite divisibility of the universe. Therefore I believe that "infinite object", or "infinite objects" represents a misunderstanding of the natural boundaries which objects have. Unless "infinite object", or "infinite objects" can be given some real meaning, as referring to something real in the world, supported by real principles, it's simply nonsensical talk..

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Therefore I believe that "infinite object", or "infinite objects" represents a misunderstanding of the natural boundaries which objects have. Unless "infinite object", or "infinite objects" can be given some real meaning, as referring to something real in the world, supported by real principles, it's simply nonsensical talk.. — Metaphysician Undercover

It doesn't matter whether its makes sense in relation to the real physical world or not! The point is that it can serve as a strong reducer of proof lengths of theories that only speak about objects related to our real world, thereby it can aid our analytic deductive reasoning about our real world, it makes it technically easier to analytically follow up truthful assumptions about our own world, because it "shorten" proofs. That rule of making it easier to follow up the analytic consequences of truthful assumptions about our real world, is a very useful tool, as as far as this property is concerned then it makes full sense to introduce them as Tools to help understand our reality in a much faster and easier way than without them.

If it doesn't make sense in relation to the real world, then it cannot be a true premise. Therefore the proofs which are derived are unsound. This is the matter of "eloquence". Eloquent proofs are not necessarily sound proofs. You are persuaded by the ease of the proofs, not by the soundness of the proofs.

If it doesn't make sense in relation to the real world, then it cannot be a true premise. Therefore the proofs which are derived are unsound — Metaphysician Undercover

Unfortunately you are not following what I'm saying. I'm speaking of two roles that a stronger theory (deductively speaking) can play. The FIRST role is in proving sentences in a language about finite objects that are NOT provable from some known finite theories having SOUND principles. Here you can make your objection above. But your objection above doesn't extend to the SECOND role. The second role is that stronger theories do facilitate provability within SOUND theories themselves!!! I need you to concentrate here, suppose you have a sound theory, by that I mean a theory whose axioms are coined in relation to the reality, i.e. they are realistic rules, and so they are sound, i.e. they meet reality, now supposing that this theory is consistent, then provability of a consistent theory whose axioms are sound, is always sound, i.e. its theorems always conform to reality, since provability (in consistent systems) preserves soundness of axioms. Now I'm speaking about those kinds of theories which we think that they'll have many applications in the real world. Now even for those theories, the stronger theories which are not reality based (like those speaking about the infinite), still do possess much expressive analytic power that can enable them to prove theorems INSIDE the realistic theories in a much shorter manner, here with this situation the proofs are sound of course, because they are only shortening proofs of a sound theory. This technical assisting role is very important. You and others in philosophy might underestimate it, because this second role is in principle dispensable! But there is a great difference between "in principle" and "in practice", I'd agree that they are in principle dispensable, but in practice they are not, because we are humans, so theorems of sound axiom systems that are provable from very long proofs will not be discovered by the human mind, while the assisting stronger systems would enable discovering those theorems because they can prove them in shorter steps, and then afterwards we can go back to the original sound theory and find the long proof of those theorems. So the stronger imaginary (potentially unrealistic) theories can play the role of a technical guide of the weaker more realistic ones. Truly the FINAL acceptance of the theorems would be by establishing their provability in the sound systems by finding those very long proofs, but finding these long proofs can be assisted by provability in the stronger theories, even if the stronger ones are not realistic. The reason is because the stronger ones have more expressive analytic power. Its a pure technical matter.

I don't understand why you call these theories, which are not based in sound premises "stronger theories". They are clearly weaker.

You and others in philosophy might underestimate it, because this second role is in principle dispensable! But there is a great difference between "in principle" and "in practice", I'd agree that they are in principle dispensable, but in practice they are not, because we are humans, so theorems of sound axiom systems that are provable from very long proofs will not be discovered by the human mind, while the assisting stronger systems would enable discovering those theorems because they can prove them in shorter steps, and then afterwards we can go back to the original sound theory and find the long proof of those theorems. — Zuhair

The problem though, is that the so-called stronger theory is unsound, and therefore the conclusions produced are unsound. It may be that some of the conclusions will later be proven to be true, but some might later be proven to be false. So there is really no point in using the so-called stronger theories, because they cannot give us any certainty in the conclusions.

You refer to such a theory as a "technical guide", and say that they are aimed at practise. So lets say that they are like hypotheses. We apply them in the attempt to prove whether they are true or false. So we must be willing to reject them when they are proven to be false.

I don't understand why you call these theories, which are not based in sound premises "stronger theories". They are clearly weaker. — Metaphysician Undercover

*stronger* is a logical term. Theory A is stronger than theory B if and only if every statement provable in B is provable in A, but not every statement provable in A is provable in B.

You refer to such a theory as a "technical guide", and say that they are aimed at practice. So lets say that they are like hypotheses. We apply them in the attempt to prove whether they are true or false. So we must be willing to reject them when they are proven to be false. — Metaphysician Undercover

Yes! Of course. If the stronger theory proves to mislead us about provability in the weaker theory most of the time, then we reject it. On the other side if most of the times it proves to be helpful, i.e. assist us in proving theorems of the weaker theory, then we would say that it is playing a conservative extension role over the weaker theory most of the time, then we adopt it as a guide only.

*stronger* is a logical term. Theory A is stronger than theory B if and only if every statement provable in B is provable in A, but not every statement provable in A is provable in B. — Zuhair

You take theory B, and add some unsound premises and voila, theory A. Everything in B is provable in A, and even more is provable in A due to the additional (unsound) premises. I don't see how that's helpful. Sure, it gets you to the conclusion you want quicker, but that conclusion is unsound, fabricated by adding the premises require to produce the desired conclusion.

It is sound! because it is the pure conclusion of the original sound theory. I'm just using theory A as a facilitator and then I'm checking it again in the pure theory B, so there is no harm. For example you have a certain argument for a proof in theory B (the reliable theory), but there are some missing steps, you translate the argument to theory A, sometime theory A manage to fill the gaps, then you go back to theory B itself and check if that filling is correct. There is no problem with that approach as long as we are not depending solely on what theory A is doing. To facilitate matters is an important practical tool. Its no magic, we know why it works, because it increase the expressive power the logical deductive system, it supply you with more tools to solve analytic problems, many times the difficulty in proving a theorem (in the original reliable theory) is not due to you being in the wrong direction, no, its due to poor analytic tools, those are supplied by the more powerful theory that extends A (even if the additional premises of A are unreliable), and that's why it manage to fill the gaps, but we always go back to the original theory and check the complete proof which is solely this time coming from theory B (the reliable theory). Theory B is the final arbiter. We start from B and we end by B; theory A is just an intermediate step quickening provability in B, just a facilitator. That should be right!

I'll put this as a separate reply. Because it speaks about role 1 (see prior comments). Now role 1 is the real issue. Because role 2 is not fundamental, but it is a practical tool, that is to build technically powerful conservative extensions over theories in order to shorten proofs thereby easing provability in the original reliable systems, that was role 2 and there is no problem with it, it is just a technical facilitator. But I'll present here role 1, which is what really matters! With role 1 we take a reliable theory B and add to it axioms about *infinite* objects, under the assumption that those axioms are SOUND. The real take here is that we don't really know if our universe admits existence of a parallel realm to it that admits infinite objects, such that this parallel realm is indispensable for understanding our own universe with. Obviously the parallel realm here is meant to be the Platonic mathematical realm. So lets put this as a HYPOTHESIS. And then add axioms about such a realm that admit infinite objects, and so strengthen the original reliable theory with those additional axioms, to give this idea the benefit of doubt of being true.
Now how to decide if this is true or not. We simply try it and check. Here the stronger system would not be just a facilitator as it was the case in role 2, no, here it would prove additional theorems about the reliable sector of our world, sentences that the original reliable system cannot prove. Now we go and check if those additional proved sentences have applications, if so, then this would mean that the added theory is increasing our knowledge about our real world, and thus the axioms are true! This is challenging! If it fails and proves misleading, then we REJECT the extended system from being a part of useful mathematics, and only keep it as a piece of beautiful analytic school of art (Mathematics for Mathematics).

Lets see how this fair in practice. Now we know that every *finite* fragment of ZFC is arithmetically sound!!! That is: it doesn't prove false arithmetical statements! Now true arithmetical statements are always considered as "possibly applicable" because they are statements about finite objects, they are the kind of statements you've desired as giving unlimited possibility of measuring!!! Now in applications if we are going to apply a fragment of ZFC then this would be of course FINITE, i.e. well take finitely many theorems of ZFC and work within their deductive closure, now ALL arithmetical sentences proved in such a fragment are guaranteed to be TRUE, and so potentially applicable!!! That's why ZFC practically speaking works as a foundational system for useful mathematics.

This is challenging! If it fails and proves misleading, then we REJECT the extended system from being a part of useful mathematics, and only keep it as a piece of beautiful analytic school of art (Mathematics for Mathematics). — Zuhair

The problem is that these axioms are saying things about a Platonic realm of infinite objects, so it's not easy to determine whether they are misleading or not. This is why we need metaphysics, to make that determination. An axiom can be very useful yet still misleading. That's the problem with "use" as a principle, and pragmaticism in general. The success which is derived from use is itself misleading. When we have success we are uninspired to look for a better way. The tool might be the most primitive, awkward tool, but if it brings us success in what we are doing, then we are not inclined to look for a better one, That success misleads us because it hides the fact that we really need a better tool, by making it appear like we have the tool we need

The tool might be the most primitive, awkward tool, but if it brings us success in what we are doing, then we are not inclined to look for a better one, That success misleads us because it hides the fact that we really need a better tool, by making it appear like we have the tool we need — Metaphysician Undercover

Yes, I would agree with that of course. But it need not be so really. The issue here is that if we have something useful, even if it is not the best, we ought not to reject it, we initially accept it, on pragmatic basis, but at the same time we look for a better system. But until this *better* system is in our hands, we keep working with what we have, despite is metaphysical (or other) weaknesses. I think this is a correct policy.

I agree in principle, but I think resolving that type of issue is far more complicated in practise. The way that we do something, including the way that we think, is essentially a habit, so we need to look at this as a matter of breaking a habit. The first issue, is as I said in the last post, if "the way" is observed as bringing success, it will not be seen as something which needs to be changed, and the motivation to find a "better way" will not exist. So the first step is to point out all the problems with "the way" existing, to inspire innovation and invention of a new, better way.

So the second issue is the way that we break a habit, and this is a complex issue. First, it takes very strong will power, and having someone demonstrate a better way (a different course of action toward the same end), I believe is insufficient. And this has to do with the ends involved, "the way" being a means to an end. The end is suited to, conformed to, or particularized by the means, over time through repetition, just as much or more than the means is designed, or conformed to meet the end in the first place. In other words, what we want in general, our goals, is shaped by our practises. This is how the habit sets in, we get accustomed to, and grow to like, and be comfortable with, the luxury that a particular action brings us. The required luxury (goal) will conform to the practise because no particular luxury is necessary. This allows the forming of a relationship of necessity between that particular luxury and the behaviour, the relation of cause and effect. To break the habit requires that we apprehend this particular luxury, which the action brings us, as not good.

Therefore we cannot assume that there is a "better way" of achieving any particular goal which is already being achieved by the existing way, because the causal relation of necessity between the action and that particular goal has already been established. The goal and the way exist in a causal relationship of necessity. We need to dismiss that particular goal as an incorrect goal, by replacing it with a "better goal". So it's not a matter of demonstrating a better way, it's a matter of demonstrating a better goal. Having a different goal will necessitate finding a different way.

There might be numerous distinct strategies for this. One would be to cease the practise cold turkey, leaving a hole where the luxury provided by it once existed, then finding another goal to fill that hole with. Another might be to find that other goal first, then replace the old practise with the new practise as the one necessary for the new goal. However, this latter method is sketchy because goals are not well defined, and they tend to shape themselves to the practise, as is evident in "the habit".

I find myself in full agreement with what you said! However, I do think that imperfections would sooner or later show themselves, no matter how much useful they are. And at that point the habit will break. Sometimes its accidental! Some genius come and discover a system which immediately show superior utility, thus overshadowing the existing system and thereby exposing all of its hidden imperfections by comparison. The switch would be immediate! However, if we want to plan matters, then I'd agree, it won't be so easy, without demonstrating superior utility of the new system, it would be a very hard task to shed the older well accustomed one. On the other hand, there is always the quest of extending theories in mathematics and finding better alternatives, this is an ongoing practice really, even finding ones that suit better metaphysical principles, even those are in development, I know that no one would buy them just for that, but if they conform to reality more than others, they'll be convincing first to philosophers, then gradually over time they would find better utility. So I don't think the matter is so grim.

The only thing "potentially infinite" about an object is that it's infinity of parts can potentially be pointed out.

However, I do think that imperfections would sooner or later show themselves, no matter how much useful they are. And at that point the habit will break. — Zuhair

It still requires will power. It may take a long time to convince people that a particular habit is bad, but once it is recognized as bad, without the will power to stop they will continue to do it.