Looking at the way I use some parts of my body doesn't necessarily tell me why I have those parts.

How could nipples define us sexually when both men and women have nipples? I don't know why I have nipples, nor do I know why I have fingers or toes.

Why do you think a man has nipples? Why do you think a female has a small penis? Why do you think a man can grow breasts? Why do you think both men and woman have testosterone and estrogen hormones in their body? — ovdtogt

I don't see how that's relevant, you might as well be asking why males and females both have mouths and noses. If you want to start with something real, start with the y chromosome.

It is obvious you lack even the most basic knowledge of biology. Your ignorance is to vast to fathom. — ovdtogt

Show me a human body which is both male and female, and I will argue that it is neither. Let the vastness of my ignorance overwhelm you!

There isn't a third possibility here. — Wittgenstein

Yes there clearly is a third possibility, it's not a case of either Cantor is right, or the op is right, because both Cantor and the op might be wrong.

… like Sq root 2, which was discovered to be irrational by the Ancient Greeks, much to their consternation. — unenlightened

Actually I think it was Pythagoras who first proved that the square root of two is irrational. And I think it really frustrated him because it demonstrates that the geometrical figure, the square, cannot be a real figure, The two perpendicular sides of a square are incommensurable. This is similar to the problem with pi. It appears like there is an incommensurability between one dimension and another which makes two dimensional figures inherently irrational. It's very odd when you think about it because it casts doubt onto our understanding of spatial existence in terms of dimensions. It may well be that spatial existence would be better represented in another way.

Of course the possibility of mistake still exists, and if "resolution" requires removing that possibility, it would remain unresolved. But that's why I retained the option that both are wrong. However, broadening the mind to numerous options, presented from numerous different perspectives increases one's understanding of the subject, thereby lowering the probability of misunderstanding, even when there is no "resolution".. Therefore you are wrong to assert that doing this wouldn't tell you anything you didn't already know. The mistake is to assume that there is a "resolution" when there is not.

Relax, give the man some time.

Actually, there is a third possibility, as I said, and that is that both Cantor and the op are wrong. This raises the probability that the mathematicians are idiots to about 67 percent. The number of mathematicians involved is irrelevant, as herd mentality demonstrates. The number of people carrying out an action has no bearing on whether the action is correct.

What's the probability for each case. — Wittgenstein

50/50.

How odd, you dismiss an argument you don't understand and don't even try to. — Umonsarmon

It's not odd and it's not dogma. It's just straightforward logic: — SophistiCat

Actually, if two proofs prove contradictory things, then there is a problem with one or both of the proofs. To say I understand one, but not the other, and I accept the one that I understand, therefore the other is wrong, as SophistiCat did, is illogical because the acceptance of the one may be based in a failure to see that its unsound, a mistaken understanding. Until you can exclude the possibility of mistake from your understanding, it is illogical to reject demonstrations which would show that your understanding is mistaken. .

Selfishness is real, but so is the tendency to cooperative endeavour. Some individuals are more selfish, some are more cooperative. There is ample evidence for both. So your assertions about one being real and the other an illusion are false, to the extent that they do not reflect empirical facts. — Pantagruel

I didn't say that cooperation isn't real. I said the belief that each one of us is a part of a larger whole is not a true belief. What there is no evidence of, is the idea that cooperating with another person makes the two distinct individuals who cooperate into a unified whole. Therefore what is untrue, and is not reflective of "empirical fact", is the idea that a number of individual human beings could compose a unified whole.

Human character is both altruistic and egotistic simultaneously as the human body is both male and female. — ovdtogt

This is nonsense. An individual human body is not both male and female, it is one or the other. A human body is either male or it is female, it is not both.

Left and right are both part of the same continuum. — ovdtogt

This, as well, is nonsensical. Right and left refer to opposite directions from a given perspective. They are not both part of the same continuum because there is a necessary dividing point between them, which is the given perspective.

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.. .