But the scary thing is that when it comes to Trump supporters and Trump himself, the facts don't matter. — Wayfarer

Don't underestimate the disdain for government in the USA. This goes way back to the government's anti-communist practises, Kent State shootings, and beyond. With the war of independence, the civil war, etc., it's in the blood of the American, as an essential ingredient of the "melting pot".

This disdain for government has been well documented, highly popularized, and romanticized by the media in the sixties and seventies, with memes like "flower power", and the Hollywood image of "The Wild West". It has assaulted us with demonstrations like Oklahoma City 1995, and it now insults us with the Trump presidency. One important fact which we ought not overlook, American citizens who hate the rule of the democratic government, are still allowed to vote. Therefore the American government has no inherent defence against anarchy. Self-destruction of the government is completely acceptable, so we ought not be afraid of it.

We lack a philosophical basis for that outlook. The world's elite have already decided that Earth as we know it is doomed... — Wayfarer

There is a type of selfishness involved here. It's very difficult for some of us to understand this form of selfishness, and to do so requires that we dismiss the idea that each one of us is a part of a larger whole. The idea that each one of us is a part of a larger whole is the illusion which makes selfishness incomprehensible. Selfishness is the reality though. Therefore we must dispel this idea that each one of us is a part of a larger whole, as an idea which is detrimental to our endeavour to understand the reality of selfishness. We may replace this idea with the idea that each one of us ought to desire to be a part of a larger whole. Then selfishness, though it is apprehended as very real, is understood as immoral..

It is not enough to point it as bad, with mathematics you must demonstrate an alternative system with superior utility, something that is better. Once a system with superior utility is at hand, the exchange would be immediate, you won't need any will power. People will readily exchange older cars for new more efficient ones if they can afford to. Its a pragmatic argument. — Zuhair

There is a problem with pragmatics though, which I explained in the prior post. Utility is judge in relation to the end, as the means to that end. And the particular end, in its particular nature, is particularized, itself conformed, by the means. This leaves the means and the end in a necessary relationship of cause and effect. This particular end can only be produced by this particular means. There is really no such thing as demonstrating a better way, because a different way will bring about a different end. Therefore to judge the utility of mathematical systems it is required to judge the goals, or ends of the mathematicians.

This means that we must judge what the mathematician is trying to do with the system, rather than what the mathematician is actually doing with the system, and relate what is attempted to what is actually produced. Now, "the good" exists in potential, as the thing striven for, the desired end. When we look directly at "the good", it looses its nature as something particular, and is apprehended as a generality. For example, you might have a craving for a particular type of junk food, but when you look at that desired thing as "the good", you see that many different types of food will fulfil your hunger. As "the potential" to fulfill the need, there is always numerous options as to the particular thing to fulfill the need. So this is the first step to overcoming the deficiencies of pragmatism. We need to de-particularize the nature of 'the good" (as the end, what is sought), apprehending its true nature as something general. In understanding "the good" in this way,(i.e. the truth about "the good"), we see that pragmaticism has no bearing, it has no grounding, because utility can only be judged in relation to a particular end, and there is no such thing as the one and only end which will fulfill the need. Therefore pragmatism may be dismissed as insufficient because it provides no "system" for relating one particular end to another.

This is the way to judge a habit. We apprehend, and list all effects brought about by that action, these are "the goods", the ends produced by the action. And, we also apprehend and list the true goods, what we understand as the truth concerning the goodness or badness of these effects. The habit can be judged as bad, if there are bad effects of the action. As I stated earlier, every time that "the infinite", "infinity", or the mitigated "infinitesimal", occurs in a mathematical application, this can be judged as a bad effect of the mathematical habit.

I just want to give an example of a sentence that is highly related to the finite mathematics, that can find a solution in a system that speaks of infinite objects that mathematicians seems to agree upon. That of Fermat's last theorem! This can be solved in ZFC. It's not yet know if it can be solved in PA. However the theorem is clearly about arithmetic, and its formulated in the language of PA, so it is not essentially about any infinite object. But a theory speaking about infinite objects (i.e. ZFC) can prove it. Now I'm not claiming here that ZFC had contributed to the argument of the proof of that theory, certainly not. But seeing that it is provable in ZFC and yet not known to be provable in PA yet, speaks a lot of that issue. — Zuhair

I must say, I don't really know what constitutes a mathematical "proof", so this example is lost on me. However, I would say that any proof which utilizes "infinite", or "infinity", is not a sound proof. Infinity, by its very nature is unresolved, so assuming it as a premise of a mathematical proof, for the purpose of resolving an unresolved issue, only creates an illusion of resolving the issue by premising that the unresolved (infinity) has already been resolved. In other words, the unresolved is inherent within the premise, so the conclusion doesn't really provide a resolution.

I will remind you, that Pythagoras demonstrated the irrational nature of the square. The relation between two perpendicular sides of a square produces the infinite, which as I argued above is bad. This makes the square a truly impossible, or irrational figure. And, all "powers" are fundamentally derived from the square. Therefore any exponentiation is fundamentally unsound in relation to a spatial representation..

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.

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".

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

I don't believe in time. — prothero

Do you believe that there is a difference between past and future? If you belief in a real difference between past and future, how can you make that compatible with the non-belief in time?

Can't find time in experience, just the relative ordering of events as seen from a specific point of view. — prothero

How can you not find time in experience? Isn't this so-called "specific point of view", which underlies experience obvious evidence of time? Would you expect to be able to bring events from your past, and put them into the future, so that you might avoid the bad experiences, or take possible events from the future and put them into the past, so that you might ensure good experiences? The "specific point of view" which you refer to is a brute fact of reality, and clear evidence that time is, as well..

*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.

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.

Wittgenstein also demonstrates that not only is there certainty which is subjective, but there is objective certainty, which is akin to knowing. Objective certainty is backed up with facts, evidence, or good reasons. — Sam26

I don't think this is true. Wittgenstein seeks objective certainty in On Certainty, but is incapable of finding it. He clearly does not ever demonstrate that there is such a thing. He defines it, but cannot demonstrate that the identified thing, objective certainty, actually exists. That's why Wittgenstein is known as a skeptic.

Objective certainty is backed up with facts, evidence, or good reasons. — Sam26

This is incorrect, subjective certainty, "inner conviction", is backed up by facts, evidence, and good reasons. But facts, evidence, and good reasons are insufficient for objective certainty, which is to exclude the possibility of mistake. This is because mistake is a consequence of actions, which by their nature occur in particular circumstances. The facts, evidence, and good reasons, must be judged for applicability in the particular circumstances, and the possibility of mistake is inherent within that judgement.

Take it to The Lounge. Maybe you'll find some sympathy. Or you might just bore everyone to sleep.

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.

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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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.

Banno seemed to think that a system could exist without balance, so I thought it might be a good idea to clear up this misunderstanding. You chose to defend the erroneous principle. Now Banno nor you, appear interested in clearing up the misunderstanding.

Google 'system definition ' — ovdtogt

Banno already took this approach, and I produced the one off Wikipedia

From Wikipedia on "system": A system is a group of interacting or interrelated entities that form a unified whole. — Metaphysician Undercover

The interaction between entities spoken of here can be described in terms of force, and the force of these interactions must be in some way balanced in order that we can speak of the group of interacting entities as a unified whole. The boundary of the system is a description of the limits of the unity, which is the parameter of balance, equilibrium. An interaction of entities outside this description, have not the described unity because they have not got the prescribed equilibrium of that system.

Systems are human constructs maintained and serviced by humans and can not be used as an analogy to natural processes. — ovdtogt

That's a bold statement. I think "systems theory" does exactly that, uses systems as an analogy to natural processes. Do you believe that systems theory has no premise to support what it is doing?

I am very much a believer in using balance as a metaphor to understand 'reality' but fail to see why defining a 'system' is in any way enlightening. — ovdtogt

Let me get this straight. You come here directing talk about open and closed systems in my direction, and now you say that you have no interest in understanding what a system is. How precious is your naïve mind?

Systems that are not in a state of equilibrium are a commonplace — Banno

That a system does not have a particular form of equilibrium does not mean that it doesn't have another form of equilibrium. If a system is not in some form of equilibrium then it is not a system. But there may be numerous types of equilibrium which a particular system doesn't partake in. Something completely without any form of equilibrium whatsoever, might be a transitional condition, intermediate between systems, accounting for the corruption and generation of systems, but this is not a system itself.

Unstable systems exist. But are much less likely to last than stable systems. — DanielP

A system must "last" for a period of time or else it does not exist. If it lasts for a minute, a second, or an hour, it has the equilibrium required to qualify as that specified system, for that amount of time. "Last" is a relative term, so we might be able to measure degrees of instability in relation to temporal extension. However, as I explained already, there is a fundamental balance implied by the concept of "system" and to the extent that the balance will not last forever, we say that the system (balance) is unstable.

I like your definition of a system, Metaphysician Undercover. And maybe what I was going for was once a systems loses stability or balance, it might lose its status as a system. — DanielP

Thanks Daniel, I'm glad that someone here can understand reason.

No "balance" as it appears to be used in this thread means equal proportions, or in the case of a system, equilibrium. In order that a group of entities may exist as a unified whole, there must be equilibrium in their interactions. Without that equilibrium there is no reason why the group of entities can be called a unified whole.

A balanced system is any system that persists in time to eternity. — ovdtogt

You're not addressing what I am saying. An eternal system is an "ideal" balance, but a state of equilibrium need not last forever to have existence for a period of time. That a state of equilibrium must last forever to be balanced is pure nonsense.

Because we live in an expanding Universe the sum total of everything is decay (i.e increasing in entropy) and therefor nothing is permanently in balance. — ovdtogt

Who said anything about a "permanent balance"?

I know the difference between an open system and a closed system. This makes no difference to whether or not a system, as "a system" has an internal balance. A system is a coherent whole, and without a balance between internal forces, there is no coherency. An open system is still inherently balanced or else it could not be called a system, it would just be objects interacting in a random way, no coherent whole.

Perhaps you are referencing a different understanding of "balance" from me. It is not necessary that a system's parts be motionless for that system to be balanced, only that the motions be balanced.

Check it on google — Banno

From Wikipedia on "system":

A system is a group of interacting or interrelated entities that form a unified whole.

What I am trying to explain to you is that for interrelated entities to form a unified whole there is necessarily a balance in the interactions between these entities. It's very obvious, and common sense really. If there is no balance, there is no unified whole, just interacting entities. You may insist on denying this fact, without any support for your denial, that's your prerogative.

It's quite clear, from his distinction between "subjective certainty" and "objectively certain", that Wittgenstein begins with an assumed separation between "I am certain that...", and "it is certain that...". "Objectively certain" here, would refer to a true transcendent form of certainty, like what we commonly refer to as facts independent from human acknowledgement.

I think, as you imply, he criticizes this sense of "objectively certain", pointing out that it is unjustifiable. What he sets up instead, is a structure which grounds "it is certain that..." in a type of inter-subjectivity supported by language use. The result is a sense of "it is certain that..." which is consistent with justified, but has no requirement for objective truth.

I offered a brief starting point. Time as a concept is derived from our apprehension of a separation, division, between past and future.

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.

Then what do you think a system is? I think a system is a whole, which is composed of parts. And, for the parts to exist as a whole it is necessary that there is some sort of balance. Otherwise you'd have a random collection of objects and not a whole nor a "system". To speak of a system without any balance is contradictory nonsense. For a system to have any temporal extension (therefore existence), there must be balance in the internal forces.

You do believe that there are internal forces within a complex whole don't you?

The mechanisms are mysterious... — Enrique

No, they are not. — SophistiCat

Sophisticat is in complete denial of the reality of human ignorance. 'There's nothing unknown out there, we already know it all.'

OK, welcome to the club. Have you read the thread?

194. With the word "certain" we express complete conviction, the total absence of doubt, and thereby we seek to convince other people. That is subjective certainty. But when is something objectively certain? When a mistake is not possible. But what kind of possibility is that? Mustn't mistake be logically excluded? — On Certainty

Following a rule allows one to be judged by others as correct. It obviously does not provide what is necessary for certainty of the subjective type. And it cannot provide what is necessary for certainty of the objective type because the possibility of mistake can only be excluded if we know that the person is following the specific rules, which are applicable in the particular set of circumstances. To exclude the possibility of mistake requires not only that one follows a rule, but that the rule being followed is the rule which will exclude the possibility of mistake.

Some suggest that we add in Purgatory as a means to solve the binary afterlife debate, leaving our options as Hell, Purgatory, and Heaven. This is still immoral and there is a division among very similar people who will receive eternal damnation in Hell and people who will make it to Purgatory, eventually making it into Heaven. Even if it takes several years to get into Heaven, it is still more satisfactory than eternal damnation in Hell. I believe adding in Purgatory is also an issue for the division between those going to Purgatory and those going to Heaven. Although they will all eventually make it into Heaven, it seems immoral that very similar people will either have to work for their place in Heaven whereas some will receive eternal salvation without the effort of Purgatory. — Bridget Eagles

I don't see why Purgatory doesn't solve the problem. The time spent in Purgatory varies according to the individual. Two similar people will spend a similar time in Purgatory, but not the same time. Therefore the problem, which is the issue of small differences between people (if there even is such a thing to begin with), corresponding to large differences in reward/punishment, is avoided. The reward/punishment scale allows for each individual to receive one's just desert.

So calculating according to a rule is enough. Even if it is legitimate to ask if the rule itself is reliable, we shouldn't expect by doing so to find another, higher level, transcendent rule. In the end it is in the very following of the rule that one attains correctness and reliability. — jamalrob

This is not conducive to certainty though. One could look at contradictory rules, and correctness could be obtained by following either one. Now all that person would have is uncertainty as to which rule to follow. Calculating according to a rule may be sufficient for correctness but it's not sufficient for certainty.

I just don't understand the insult culture around here. Over the past couple of years I've had to take extended breaks from this forum because someone started piling on personal insults at me over technical matters on which they happened to be flat out wrong. Not because I can't snap back; but because I'm perfectly capable of snapping back, and that's not what I'm here for. I'd suggest to members that whenever they throw an insult in lieu of a fact, perhaps they should consider whether they've got any facts. — fishfry

Fishfry, you appear to be very sensitive. From my experience, if someone points out to you, your misguided way, you take it as an insult. This is philosophy, so you ought to learn to take this as an attack on the ideology which has guided you, rather than an attack on your person.

An unstable system will change - that's what being unstable is. — Banno

A system is necessarily stable. That's what being a system is, a whole with temporal extension, and to exist as a whole requires stability in the relations of the parts. That is a balance. A "system" is an ideal, a model by which we judge the relations between things.

The fact that we can describe the degree to which the balance is not perfect (eternal temporal extension) in terms of instability does not mean that the balance is not there. Balance is implied by the descriptive term "system" and "unstable" refers to the imperfections of that balance.. .

Key take-away is the hold was put on the Ukraine aid because of the “the President's concern... — NOS4A2

Says it all.

Trump has been making deals for half a century so I suspect you have little clue what you’re talking about. — NOS4A2

The Trump deal. Take the money. Let the company go bankrupt. Creditors don't get paid because the money's been taken.

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.

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.

Meaningful disagreement, by my lights, is the sort of disagreement you have with someone while understanding the words they say. So we do not share the same belief. But I understand the statement the belief is about — Moliere

Thank you for clarifying that. So your point is that there is no disagreement concerning the meaning of the words, the disagreement is about something else, some other "belief". So how do you construe the disagreement itself as being meaningful?

You had attributed "meaningful" to "disagreement" in "meaningful disagreement", which I thought was incorrect. Now I see that what you really meant was that there is meaning, and agreement in the understanding of the words, yet disagreement concerning something else, some belief other than the belief in what the words mean.

How do you think that this belief exists as something other than the belief in the meaning of the words? Suppose we pass judgement of true or false, or some such thing, on the meaning of the words which we understand. If we disagree on this judgement, as we often do, how can this disagreement be meaningful?

Now here's the point. Your so-called background of agreement, upon which you apprehend a "meaningful disagreement", is simply the meaning of the words, itself. But this is not the background at all, it is the foreground, the surface, the shallows. The true background is the principles we hold (beliefs) by which we make judgements of true or false. The meaning, and all these agreements and conventions are the foreground, while in the background lie these judgements of true or false, where disagreement is abundant. Disagreement is abundant because such judgements are often based in intuitions, attitudes, feelings, and emotions, rather than rational logic. This is why the background is a background of disagreement, and agreement is conjured up in the foreground, by conscious minds. But the conscious mind is just the tip of the iceberg, and we often cannot even say why we believe some things and not others, because those principles often extend deep into the subconscious. So the background is full of disagreement.

If you allow that disagreement can be meaningful, you open the abyss of meaning without agreement. This is how we defend ourselves against the nonsense of Platonism, by showing that meaning emerges with agreement, and is not the property of some eternal objects. But then the background from which meaning emerges must be something other than agreement.

The point now, is do we simply say as I do, that this "other than agreement" is disagreement, or do we try to argue like Banno and some others, that it is actually some form of agreement? We might be best off to place it in a category distinct from agreement/disagreement, but how would we keep ourselves from getting lost then?

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.

Of course collections would have different meaning across all applications, but they will have consistent meaning within the same application. Like how number 1 can have different meaning across applications — Zuhair

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

I find your idea that an object cannot have parts unless its subject to temporal separability as un-supported. Especially under imaginary grounds. — Zuhair

Of course we can imagine things which would violate that proposed law, but the whole point is to exclude from our principles, things which are physically impossible. If we allow mathematical principles to include imaginary things which are physically impossible, and we apply mathematics within physics which employs inductive conclusions that exclude such things as impossible, then we will be employing contradictory premises in the very same application, as I described.

Relativity theory denies the possibility of eternal unchanging relations between parts (absolute rest). But if mathematical principles allow for eternal unchanging relations, then we have contradictory premises. To resolve this problem we cannot change our description of the physical world without loosing accuracy. So we must change these mathematical principles to provide consistency. That it would be difficult to make such changes, or that the existing principles are supported by simplicity, is no excuse.

I'd say even if that platonic realm is FALSE (i.e.doesn't exist), still, the logical-mathematical rules displayed in them are not necessarily false. And they can hold of some real scenarios, and so can possibly find applications, and that what really matters! — Zuhair

Sure, such mathematical rules would be applicable, and in the vast majority of cases they would give us very accurate conclusions. That is because the vast majority of cases don't deal with things like eternal objects, and infinity is never approached. And, when eternal objects are approached (fundamental particles for example), we can be fully aware of the faults within the principles, and take the conclusions with a grain of salt. However, as these faulty principles become more accepted, and work their way deeper and deeper into the hierarchical structure of the mathematical axioms, their application becomes more commonplace. At this time, they are subsumed by other principles, and we could loose track of when they are actually being applied, and not notice the mistakes which they produce.

They are not names of relations. Naming of relations is a different subject, and I've never attempted to speak about it in any of my prior comments. I've been always speaking about naming collections, and so speaking about naming objects, and not relations. — Zuhair

The point is that your description of the distinction between "loose" and "tight" does not provide us with an indication as to what these terms really mean. The reality is that the parts of a collection are either loose or tight depending on what type of relation they have with each other. Therefore there are all sorts of different types of sets, which may be named dependent on the relations between the parts. We cannot just classify loose and tight sets, just like we cannot just class soft and hard physical objects, there are all sorts of different type of objects which we name, like 'cars' and "houses". The naming of a set, as it is supposed to be the naming of an object ought not be any different. The name ought to indicate to us what type of an object that particular set is, by indicating something about the relations of its parts, or its use, or something like that. W#hen someone says "house", or "car", it brings to mind a specified type of object, the same ought to be the case when sets are named.

When a set say set x names some collection C, then we call each "element" of C (i.e. each singular part of C) as a "member" of x. In some sense membership would copy element-hood but transfer it to an object external to the collection, that is to the name of the collection. But you need not confuse "membership" as a name for "element-hood", No! That is not the case. Membership is not a name, it is a relation, so it is not an object. — Zuhair

So this is a basic type of set. The set "C" has "members", "elements", that is what is specified of this type of set. Notice that there is no specification as to the relationship between members, the relations might be either loose or tight. Would you agree that the so-called "empty set", and the set with only one member, are different types of sets from the set which has members. We could have a set "A", which is empty, and a set "B" which has one member, and then the names A, B, C, would each specify a different type of set, therefore the name would be useful. We could then proceed to look at the different types of relations between members and have all sorts of subcategories of type C, but A and B would be distinguished as completely different.

Now through membership relation and sets (i.e., names of collections), one can easily define a hierarchy of sets. And that build-up proves to be an extremely useful tool in our understanding of many mathematical entities. And the witness to that is SET THEORY. In particular ZFC set theory (Zermelo-Frankel set theory with Choice), which proves to be very powerful in understanding mathematical entities and rules, through the iterative buildup of a hierarchy of sets. — Zuhair

Unless your hierarchy is built on real principles of distinguishing real difference between types of sets, that hierarchy will be arbitrary and misleading. Therefore the hierarchy is not built on the names themselves, but the type of set which the name identifies. We cannot make up relations between named sets, without first having a complete grasp of the relations between the members of the sets themselves, which gives us an understanding of the type of set, because such relations would be completely imaginary. Understanding the relations between the members, gives an understanding of the type of set, and understanding the type of set will allow us to proceed toward establishing relations between the types of sets. ZFC proceeds without a proper understanding of the types of sets, to produce imaginary relations between imaginary types.

Of course for the development of set theory, all of our units are un-breakable over time, and they don't change their tight connections with time, so they are remotely different from natural objects which rut over time or combine with other objects to build bigger units, etc... Here in the platonic mathematical imaginary world, all individuals (units) have non-changeable tight connections over time. So they are as you said "eternal". Then we can freely form collections of them using the descriptive tool, and with the help of the naming relation, we can speak of a hierarchy of them, which helps us encode almost all of mathematical entities in it. Thus serving as a FOUNDATION for MATHEMATICs. — Zuhair

As I said, the Platonic eternal, unchangeable object must be considered as imaginary, impossible, and must be excluded as not real, until its reality may be demonstrated. Therefore to base a set theory in this assumption is to start from a false, unjustified premise.

I'm speaking within the confines of a mathematical realm, some platonic realm in which time doesn't cause any change to connection relations. So what is actually separate is always separate, so separable is separate, and so temporal x spatial connection is immaterial in this realm. We only have spatial connection and separation. That said we need to revert again to loose versus tight connections. — Zuhair

But this is unreal, impossible, therefore I deny an such "speaking" as irrelevant, because you are speaking illogically, of the impossible as if it were possible.

My account entails that the existence of connections between parts of an entity is what qualifies that entity to be an object. So having loose connection is fairly enough for that quest. You don't need tight connections between parts of an entity to qualify it for being an object! NO! loose connection can do the job, so an entity in which loose connections between its singular parts exist, is perfectly qualified of being an "object. However, you need tight connections to form units (singulars) but units are just special kinds of "objects", so an entity that has tight connections over its parts and it itself doesn't have that kind of connection to external objects, that would qualify it to be a unit object. But objects need not be units. They can be totalities of loosely connected units, or what I call as "collections". So as such collections qualifies for being "objects". I hope this resolves the confusion. — Zuhair

But this is total nonsense. In the "platonic realm", all objects are eternal, so the distinction of loose and tight means nothing because one cannot be more susceptible to corruption than the other. And "loose and "tight" cannot refer to spatial relations because eternal objects are non-spatial. Therefore your specified relations are completely illogical, there is nothing to justify "tight" or "loose".

Ok, I agree it would be eternal since its not actually breakable. But why it can have no parts? Any object is itself a part of itself. — Zuhair

Again, this is illogical nonsense which has been accepted into set theory, the idea that a thing can be a part of itself. The adoption of that idea into set theory is representative of the breakdown in logic of set theory. "Part" means 'some but not all of', and "whole" means 'all of'. If we redefine "part" such that 'all of' may be 'part of', then we loose the essence of the distinction between part and whole, rendering a corrupted ontology.

The eternal thing cannot change, that's what defines its being as eternal. But if a thing is composed of parts, then the parts must exist in relations to each other. Sense observations demonstrate to us that all relations are changing (relativity). Therefore it is impossible that an eternal thing is composed of parts.

Even if we have an object that is eternally not breakable, still it can have many parts connected by tight connection in a manner that renders it a unit, it doesn't mean it doesn't have proper parts, it only means its no breakable to them, but it can have them always as parts of it. — Zuhair

As I've shown, your distinction of "tight" and "loose" amounts to meaningless nonsense. If the relations between an object's parts are so tight that these relations cannot change with an eternity of passing time, the claim that there are parts in relation to each other, rather than one changeless entity without parts, is unjustified. If the relationship between two so-called "parts" of an object cannot change, then the claim that they are "parts" is not justified. That is one object, a whole without parts.

In real life having eternal objects is itself faulty. — Zuhair

Exactly, therefore we must establish proper principles to distinguish between imaginary eternal objects, and real objects. An eternal object, by nature of what it means to be eternal, cannot consist of parts. This presents a problem to set theory. A set, by its nature consists of parts. Therefore those who work with "set theory" must resist the contradictory notion of treating sets as if they are eternal objects.

Now if we work in an imaginary space in which time has no effect, i.e. doesn't change connection relations of objects to each other, still it is imaginable for those objects to have parts, so having parts is not a function of temporal separability as you hold. Not only that, still without time we can fathom of having objects that are composed of units that are loosely connected to each other. So we can have collections having many elements. — Zuhair

Now, don't you see this as illogical? There is no such thing as a space in which time has no effect. that's like assuming absolute rest, which relativity theory has ruled out as impossible. So if we adopt this premise as a basis for "set theory", we are working from a premise which inherently contradicts relativity theory. If one were to apply this "set theory" in physics, all sorts of confused conclusions would follow from that application of contradictory premises.

Of course in a mathematical realm in which time is not operable, like "most" of mathematical contexts, then all unit objects in that realm are true units. — Zuhair

However, as I've shown, it is illogical to speak of "relations" in a realm where time is not operable. Relations must be justified. If the relation is spatial, it cannot be free from time. If you propose a type of relation which is not spatial, such a proposition needs to be substantiated, justified.

The problem is that this would add additional features to the picture, namely temporarily, which is not all that desirable in a mathematical realm. For the purpose of defining sets, we can simply hold the dichotomy of loose and tight connection as primitive concepts without relation to time. Our aim is largely descriptive. Since set theory serves as a foundation for mathematics then the particularities of what decides the "units" of a certain mathematical discipline is stuff related to the particularities of that discipline itself, so in Geometry units would be "points", in arithmetic units would be "numbers", in set theory units would be "sets", etc.... Here we are only concerned in introduced a general descriptive framework that can be applicable to diverse mathematical disciplines, and possibility even non-mathematical spheres of knowledge as well! For example the idea of having a "true unit" in time, might be useful in understanding the ontology of time and space? — Zuhair

The glaring problem here is that mathematics is fundamentally a tool which is used to understand the relations between objects, from the most fundamental, "order". In the mathematical realm, change to relations is as you say, not "desirable". However, change to relations is reality. So if mathematics assumes changeless relations, because this is desirable, but changeless relations are not real, then there will be a problem in the application of mathematics.

So we need to look directly at this fundamental idea of changeless relations, and determine whether we can make sense of it. I suggest that there is a type of relation, which is demonstrated by the existence of time itself, which may be changeless. This is "order". We can assume an "order" which is given by time itself, a changeless order which is the passing of time. However, we need to make this consistent with relativity theory.

I haven't got any questions for Mr. Pigliucci at this time, but I have some suggestions which may be useful to help facilitate the success of this project. They are as follows.

1. Allow Mr. Pigliucci to choose which questions (members) to address, directly from this thread.
2. Start with one topic, and create a parallel thread to allow other members to discuss the discussion. The parallel thread will help Mr. Pigliucci to get acquainted with the way that other members relate to his ideas, as the parallel discussions develop.
3. Leave this thread open to receive new topic suggestions at any time, and allow Mr. Pigliucci to choose new topics (members) to engage with, at will.

I would like to see a log running interaction, with numerous members joining the discussion with Mr. Pigliucci. Thank you very much for the efforts of all involved in this project.

Well I do agree that having a common description imply some material connection, but that connection is not the connection that imply inseparability. You can call these connections "loose" connections, as opposed to "tight" connection which is what causes continuity (inseparability), so if object K has tight connection to object L then they are in continuity, i.e. they are not separate, ie. they are in contact; while if object K has loose connection to object L then they are separate. — Zuhair

We agree to an extent, but you seem to have some ambiguity. Material connection is what makes parts into an object, and when its an object, you say that the parts cannot exist as separate objects. Yet now you say that material connection does not imply "inseparability". I would agree with this if we clarify by distinguishing between actually separate, and separable.

I find your distinction between loose and tight connections to be very fuzzy, and I think we can replace it with the more substantial distinction between temporal continuity and spatial continuity. Suppose for example, that the apple exists as a collection of molecules. There is a tight connection between the molecules, and an even tighter connection between the atoms, and this spatial continuity indicates that these parts are not actually separate. Yet the molecules, and atoms are in principle separable, and this is a function of the temporal continuity of the object. The parts are not separate spatially, yet they are separable temporally.

So we need to distinguish temporal continuity from spatial continuity. If the parts are inseparable in the sense of temporal continuity, then the object is eternal, and it would appear contradictory to even talk about the object as being composed of parts. Such an object is the fundamental, or base "unit". It can have no parts because that would imply that the unit is separable in time. The spatial extension of such an object is dubious.

From experience, it appears like spatial extension implies parts which are separable. It's difficult if not impossible to conceive of a object that occupies space, which is not divisible into parts. If you think about this, you will probably be able to conclude that the actual unity, which is an object with spatial continuity, is really an illusory "unity" due to temporal separability. The spatial continuity which we observe really hides the underlying temporal separability, The parts of the spatial object which are connected through a spatial continuity, may appear to have a loose or tight spatial connection, but the true strength of that continuity will only be understood through an understanding of the temporal separability. Since the temporal aspect is what hands the parts of the object separability, the nature of time is better understood through the terms of discontinuity. Therefore we end up with spatial continuity which is an illusion, and temporal discontinuity which describes the reality of the object in terms of separability. The reality of the eternal object (true temporal continuity) is dubious, as synonymous with the object that has no spatial extension.

But this is not enough. You need representatives, or actually NAMEs, you can also call them tokens, or labels, those would be singular objects (units) that we arbitrarily assign to each collection, but provided that the assignment works along unique lines, I mean each collection is assigned only one name, and each name only names one collection. So although the choice of which object would name a collection is arbitrary, but once done naming of other collections cannot use that name, so the naming function is not totally arbitrary. Of course this is not Ontologically innocent, it involves adding unrelated material into the picture!

But why names? why should we assign an external object that is singular to act as a name to a collection that may have multiple elements, so why represent a multiplicity by a singular object? With external naming, there is no clear intimacy between the name and what is named, the assignment is arbitrary for that particular aspect. And this is what actually happens with naming generally, its artificial, for example the names used in language are all arbitrary, there is no special connection between the string of letters "horse" and the animal group it is used to represent. So that's the question: why we should bring an external object that doesn't bear a necessary relationship to a collection and make it act as a name, actually a "representative" for that collection? — Zuhair

OK, I'm with you here. I apprehend a name as an object, which is a representative of something else, as described. And I wonder why there is a need to bring names into the picture. The type of object which a name is, I think is a tool, and this tool has the purpose of understanding. It's the tool we use for understanding.

The answer is to develop a hierarchical account about collections! This cannot be done in an efficient manner without the use of singular names. The idea is that through this artificially made unique naming process, we can define a new relation, called "membership", that act to copy the relation of element-hood in collections but raises this relation to the name of the collection, and since names are singular objects so they can be elements of collections (while collections when they are non-singular objects cannot be elements of collections, so we can't have a hierarchy of collections in collections using directly the "element-hood" relation!!!), so all elements of a collection wold be "members" of the name of that collection. The "name" of a collection, is what we call as "set" in set theory. So for example the set of objects k,l, denoted by {k,l}, is actually the name given to the collection whose only elements are k,l. so k,l would be "members" of that set, i.e. they bear the membership relation to the NAME of that collection, which is the set itself. Through this copying process of elements to members, one can speak of a hierarchy of sets that are members of sets and so on.... And so indirectly speak of collections of collection of...This would give the powerful mainframe needed to interpret almost all of mathematics. — Zuhair

This I find to be very confused and I am unable to follow. I agree that we may use names to develop a hierarchical account, that is part of understanding, but where I find confusion is in your failure to recognize the distinction between naming an object, and naming a relation. These are two distinct uses of a name, like the difference between noun and verb.

So if we name "membership", what that name actually refers to is the relationship implied by "membership". Now we must guard against deception. We can name membership when no reasonable relationship has been identified. Therefore there is no point to naming "membership", unless to deceive. The relationship ought to be named directly, without the medium "membership". Furthermore, what follows from this, is that this "name", which is the name of the collection, but actually represents membership in the collection, which in itself represents a relation, is an even further layer of representation. So we have three levels of representation now, the name represents the collection, the collection represents membership, and membership represents a relationship. Plato warned us against such multi levels of representation, calling them "narrative". Any hierarchy produced in this manner would be extremely unreliable, as we ought to refer directly to "the good" to produce a hierarchy. Names are tools used for understanding, so the good here is understanding. Multi levels of representation are conducive to confusion rather than understanding.

Now you might be suspicious, and actually object, to such a buildup. Since its pivotal rule is built up through an intermediary that involves some arbitrariness, which is the choice of a name per particular collection of course. So its like building a big building that involves multiple big junks of tightly connected material put on top each other using light joining material, so the the whole buildup is bound to fail! — Zuhair

Exactly! A name is an object with extremely unreliable temporal continuity so we ought not construct structures using names. And, since it is required to produce a multi level representation to make the name into a building material, this makes the structure even more unreliable.

So we needs NAMES, to do the intermediary role in developing a hierarchy of sets of sets of..,etc.. It is the simplest way to do it! And this proves to be very powerful logically speaking, that almost all of mathematics can be encoded in it. — Zuhair

But there is no need for a hierarchy of sets, that is a faulty premise. What is needed is to understand the relationships between things, that is the good which we might put the tools (names) to use toward. The hierarch is produced naturally by understanding the nature of continuity, and the strength of relationships. So we begin with some fundamental determinations concerning the nature of continuity; for example what is prescribed above concerning spatial and temporal existence, and we proceed to name relationships according to their strength, producing a hierarchy of relations. There is no need for a hierarchy of sets.