t was only a matter of time before they impeached Trump for word crimes. It was too difficult for them to find actual crimes, so they reduced themselves to scouring his statements for transgressions of speech, and then lying about them to make them seem worse than they are. — NOS4A2

Sorry, but word crimes are actual crimes, especially when you're the president of the United States of America, because your words actually have power.

Apparently the House, having no respect for the most fundamental of Constitutional principles, namely the separation if powers, refuses to cede the power to the Senate to hold its trial as it sees fit. — Hanover

Why is it not the discretion of the House, to deliver the papers when they see fit? It does not make sense that the Senate can force the House to deliver the papers at any particular time.

Republicans do know where they have to stand. During Nixon's time, they were far more confident where they stood. They could throw away Nixon and be confident that they would have enough popular support in election (even if the Dems got Carter later). Now they aren't so confident about themselves anymore, hence they will defend to the last man Trump, even if they hate the guy privately. — ssu

What does this say, that the Republicans are convinced that they cannot come up with a better candidate than Trump? That's pathetic.

The President has been accused vaguely of "high crimes and misdemeanors" — Hanover

The accusation is not at all vague, it's very clear. The evidence presented is somewhat vague though, because key witnesses have not yet testified.

Impeachment is democracy turning in on itself, where our representatives vote out our representatives. — Hanover

And Mr. Putin has an extremely rejoiceful Christmas!!!

So, it seems that God’s guiding principle when judging the fate of humankind is not the amount of evil but the existence of a minimal quantity of righteousness which one might call it the quanta of righteousness. — Jacob-B

Life itself is essentially good, so to kill it off because the accidental, evil, has become overwhelming, is fundamentally irrational.

All that is required is one kernel of good, at any given time, because the good will take root and flourish, while the evil will die off in the future.

I directed the OP towards a (highly online) reference that explains how mathematicians disarmed his or her objections over a century ago. The date is relevant only because the OP ignored this reference and continued to insist that the methods of analytic geometry are unfounded. The relevant foundations were provided by mathematicians operating around the turn of the 20th century. — quickly

If you think that the objections have been resolved, then you're simply wrong. And pointing to some "highly online" reference (whatever that means) does not make you right. If you would take the time to produce the supposed resolution we could show you how it simply covers up the problem rather than solving it.

(note that when I say that change is more fundamental I’m not saying that there is nothing that temporarily stays the same within the change.) — leo

Isn't this contradictory though? If there is something which stays the same, within change, then how can change be more fundamental? If "being" is inherent within change, then there is no change without being, and change cannot be more fundamental.

1. Change is immediately evident to us, you agreed with that, whereas being is not immediately evident. — leo

This is a good point, but we must ask why is change immediately evident to us. Then we see that change is only evident against a backdrop of "being". Without that back drop, nothing would be evident. This is the issue that you need to consider more carefully, what makes it possible that change is evident to us. Pure, and absolute change, would be the randomness I described, everything different at every moment. And, you can't just dismiss this by saying that's not the type of change I'm talking about, I'm talking about change that has being within it, because then the change you are talking about cannot be more fundamental.

2. If being (absence of change) was most fundamental then there wouldn’t be change by definition, yet there is change. If being changes it is no more being. — leo

This is not true. As I've described, for being to change all that is required is a cause. This is where I'm trying to lead the discussion, toward "causation", but by insisting that change is fundamental you have no need to consider causation. "Change" just is, being the most fundamental, and there is no need for causation. But when being is placed as more fundamental, then we need a cause of change.

3. Change cannot be an illusion because it would be a changing illusion, and thus there would be change. Whereas absence of change can be an illusion, it can be change appearing to be unchanging. — leo

"Illusion" presupposes a being which suffers the illusion, so this argument is not applicable. You seem to be forgetting, that all these terms we are using are applied by us, human beings. So we cannot remove from the picture, the fact that we are discussing the human perspective. That is why, in response to your #1 above, I said we need to consider why change is most evident to us. It appears like you take the "us" for granted, and want to move on towards analyzing what we perceive, as what is most fundamental, but we can't do that because you've already placed "us" as prerequisite, and therefore the most fundamental.

4. An experience is made of parts, for instance there can be simultaneously a feeling and a thought. There may be one part that is seen to be unchanging, but in order to see it as unchanging there is another part that is changing, for instance a thought that is interpreting some part of experience as unchanging. The thought itself is changing, if it wasn’t changing it wouldn’t come to see the other part as unchanging, it would remain stuck on a past thought. So within experience there is always change, the experience as a whole is changing. — leo

This is fine, but again you need to accept that any appeal to "experience" presupposes something which is experiencing. Therefore the thing which is experiencing is more fundamental than the experience itself, as necessary for the experience.

6. Even if we can find regularities within the change, these regularities wouldn’t exist without the change. Even if something remains temporarily the same within the change, that doesn’t make it more fundamental than the change. There can be sameness within change but there can’t be change within sameness. — leo

So this approach is pointless. We cannot proceed to analyze the thing experienced, "change", as if it is more fundamental than the thing which experiences the change. What you say, that one of these, sameness or change is "within" the other is irrelevant at this point, because it cannot be determined until we establish the proper relation between the two.

"Change" is what we experience, what is immediately evident to us. So this "change" is already "within" us, as that which is experienced within us. The "us" is already more fundamental than the change experienced, so this points us in the direction of looking inside ourselves, to see what constitutes "us", in order to determine what is more fundamental. Therefore to determine whether change is within sameness, or vise versa, we need to look inside ourselves. Instead of looking at the thing observed, we need to look at the observer, because we can only approach the thing observed through the intermediary of the observation, and the observer is presupposed as more fundamental than the observation.

8. As a reason against seeing change as fundamental, you said you think if change was fundamental then everything would change and the world would be complete chaos and randomness. However I disagree, because why would we have to assume that everything would be changing randomly? Smooth change still counts as change. — leo

I see that you don't quite understand this objection. If there is anything which remains unchanged from one moment to the next, this qualifies as sameness. In order to place change as more fundamental, we need to be able to conceive of change complete devoid of sameness, thus demonstrating the priority (fundamentality) of change. Otherwise we have a duality of change and sameness, and no principles whereby the claim that one is more fundamental than the other might be justified.

So, from my perspective, being or sameness is placed as the most fundamental. And, we can conceive of being without change, this is absolute rest. From this point, we need a cause of change. You might argue that the "cause" is itself a change, and therefore the being which is at absolute rest must coexist with this change. However, traditional metaphysics has adopted the principles required to conceive of this cause of change as distinct from change itself, in the same way that a cause is distinct from the effect. From the perspective of "change", cause and effect are inherently tied together as "an event", but from the perspective of "being" the cause and the effect are distinct.

9. You say that “the idea of a thing which never changes comes from the necessity of ending the infinite regress”. As to how that infinite regress arises, you say: “Change is a difference in relations between things. So if a thing changes, the relations between its parts have changed. But if a part can change, then it must be composed of parts, and so on to infinite regress.”. But notice that in this reasoning you assume in the first place that there are unchanging things that exist, which is what you end up concluding. If you don’t assume that unchanging things exist fundamentally, then there is no infinite regress and so no necessity to conclude that there are things which never change. — leo

This would be a valid objection, if you yourself could conceive of change without being, as a starting point. Since you cannot, you have already assumed "that unchanging things exist fundamentally". Therefore my starting premise is acceptable to both of us. Until you can adopt a starting point of absolute change, which is the complete randomness that I described, you cannot dismiss my starting point of "unchanging things that exist fundamentally". Therefore your objection has no place, because you yourself have also assumed that unchanging things exist fundamentally. Once you see that we cannot remove being or sameness in any absolute way, because this is unsupported and irrational, yet we can remove change in an absolute way, then the consequences of this will appear more reasonable to you.

I see why it means: democracy is not perfect, but I'm not sure why you're saying it doesn't work... — VagabondSpectre

The "most free democracy" will provide more freedom than what is good.

I know how that sounds, but the rest of the world has ostensibly been watching Trump get away with apparently criminal acts, and since America is supposed to the best and baddest and most free democracy around, it sends the message that democracy doesn't work. — VagabondSpectre

Think about what "most free democracy" means and you'll see why democracy doesn't work.

A couple of months ago the forum was infested with bad theology. Now it's bad maths. — Banno

In mathematics, instrumentalism reigns supreme. Anyone who rejects instrumentalism, opting for truth as a first principle, is accused of "bad maths". However, the terms of "bad" and "good" receive their meaning from morality. So to resolve the issue of whether it is really the instrumentalist, or the truth seeker who has "bad maths", we would need to apply moral principles.

Do you have any moral principles which show that instrumentalism is better than truth seeking?

I'm saying that your objections are more than a century out of date. — quickly

How is the date of the objection relevant? If it's a reasonable objection then it's a reasonable objection, regardless of the date.

And, as Devans99 indicates, the issues have not been "resolved more than a century ago". They've simply been ignored.

I think everything is unique in some way or other. If two things appear to be identical they would still be different if they are in different locations for example. — believenothing

This is Leibniz' principle, "the identity of indiscernibles". It states that if two objects can be said to have the very same properties, then they are identical. "Identical" means having the same identity, and by Aristotle's law of identity, this means that they are actually one and the same object. Therefore it is incorrect to say that this is "two objects", because it is actually one and the same object being referred to. The law of identity, "a thing is the same as itself" represents the uniqueness of every object. Essentially, it says that an object is unique.

It is conceivable, however, that the wardrobe chest I just got delivered to my house made of compressed wood contains two identical wardrobe chests, both occupying the same space at the same time and in the same respect. — god must be atheist

You can say this, that two distinct things can occupy the exact same place, at the exact same time, but is this really conceivable? There is a difference between what you can say, and what you can conceive. To actually conceive of what you say here, you would need a conception of the spatial temporal existence of an object which would allow for this. I believe that "multiverse" theorists allow that the same object exists in many distinct universes, and multiverse theory is supported by the many-worlds interpretation of quantum mechanics. This allows one to say that the same object is many objects, separated by being in many universes. However, if the multiple wardrobes you refer to, are actually in separate universes, it would not really be correct to say that they occupy the same space at the same time, because the particular space and time being referred to is a property of the particular universe..

I agree that in order to experience change there has to be something that stays the same in relation to what is changing, for instance a thought, because if everything was changing randomly including our thoughts we wouldn’t even have a static thought that would tell us we are experiencing randomness, we wouldn’t have any memory and so on. But even though some things temporarily don’t change in relation to some other change, we don’t have to assume that it is necessary that some thing never changes. We have no evidence that something can never change forever, while we have evidence of change. — leo

Right, I agree with this. But I'd say it's more like this, something unchanging is necessary for experience itself, it's fundamental to experience. And, as you conclude, it is not necessary for this unchanging thing to be never changing. This is why causation becomes paramount. How is it that a thing can be the same for a while, then not be the same. A cause of change is necessary.

The idea of a thing which never changes comes from the necessity of ending the infinite regress. So both ideas, that there is something unchanging, and that there is something which never changes, are produced by logical necessity. To ground experience, and give it reality, we need to assert an underlying consistency, sameness,and also to give reality to the thing which we experience, the sensible world, we assume the existence of an underlying "matter" the fundamental element which never changes.

We are certain of change but not of being. — leo

This is why it appears like we are more certain of change than of being. Change is fundamentally evident to us, while the idea of being is produced by logical necessity. Change is the premise, and that there is something "the same" is the conclusion produced by the fact that not everything changes. Logic proceeds from the more certain to the less certain. But as you'll see below, we can turn around and face those premises, as potentially uncertain themselves, and look for the most certain of all premises.

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What we interpret as not changing might be simply change that is not perceived, for instance something might seem unchanging and yet by looking more closely we see change. Also, if one part of experience is not changing while another part is changing, the whole experience is changing as a whole, so again change appears as more fundamental than being. For these reasons I think it will be more fruitful to see change as fundamental rather than being. — leo

I don't think it is possible that everything is changing. This would mean that from one moment to the next, absolutely everything changes. Then there would be no consistency whatsoever, and the entire world would be complete chaos and randomness. It would be completely impossible for us to understand the world at all, because we could make no principles about how things would be from one moment to the next, because such a principle is based in assuming that something stays the same from one moment to the next.

That is why, despite the fact that we experience things as changing, the most certain of all premises is the premise that something stays the same. This is basically the principle which Plato impressed on us. We must get beyond the illusory world which the senses are handing us, to look at the reality of intelligible principles. All the premises concerning change, which we derive from our sensations of the world, have fundamental uncertainties inherent within. So we must look beyond sensation, toward what makes sensation possible in the first place, to derive the most certain of all premises, from which to build any structure of knowledge.

Would you agree with the idea that fundamentally what we call a “relation” is a thought, an experience? The idea that “change occurs” follows from experiences that are seen to change. Where does the idea of a “relation” come from? Doesn’t it come from seeing that some part of experience is correlated with some other, that the two parts change not independently from one another, but together in some way? — leo

This is exactly why we must place sameness, or being, as the most fundamental principle. If we do not, we cannot get a true perspective of what a "relation" is. You have brought "relation" into the experience, as if it is something which is part of the experience, when in reality we see a "relation" as a part of the thing experienced. Consider the map and the territory analogy. A relation is part of the territory, and we map it using principles. So within the experience, there are principles not relations, and we use the principles to map the relations which are outside the experience as part of the world being sensed.

Now consider principles themselves. We could say that there are relations between principles, but that would imply the principle is an independent thing existing by itself, relative to other principles. However, principles don't really exist like that, they are inherently connected to one another, supporting each other and dependent on each other, so it is somewhat incorrect to portray them as independent objects existing in relations to each other.

Now, see that you and I come to agreement about the nature of our experience. As you say "the two parts change not independently from one another, but together in some way". This is because the "two parts", which I portrayed as "principles" above, do not exist separately from one another, as independent things. There is dependency. So let's say that "parts" do not exist as independent objects, and they do not exist in relations with each other. Let's say that there is a "whole", and the part exists as a part of the whole, and being part of a whole is something other than a relation, it's some sort of dependency.

Where I’m going with this is that you were saying that a thing without parts cannot change on its own, but if you agree that a relation is fundamentally an experience, a thought, a thing, then again why would that experience or thought or thing change on its own? If you say that this relation is made of parts, and that this is why the relation can change, then we’re back to asking why do the parts of that relation change in the first place? — leo

So this is a very good question, and I'll show you how I can resolve it. A relation is now something outside of the thing. There is not "relations" within the thing, but dependency between parts. Within a thing there are parts, but the parts don't exist by relations. The parts are like principles which exist more like in a hierarchy of dependency. Now the question is what causes a thing to change, so we must look to the structure of this hierarchy of principles to understand this.

What is implied here is that we pay attention to Aristotle's distinction between the two ways of depicting change, locomotion (change of place), and internal change. Now we are focused on internal change, and the question is how does a thing change. To understand this we need to understand how a hierarchy of principles exists and changes. Changing a fundamental principle will have a huge effect, while changing a fringe principle will have a minor effect. But the question is what causes a principle to change in the first place.

Basically it seems to me that you can’t escape the fact that a thing without parts can change, that it can become something different than it was, which again leads to the idea that change is more fundamental than being. It seems to me that it is a circular reasoning to say that “a thing with parts can change because the relation between the parts can change”, because in saying that you’re essentially saying that the relation can change on its own, or that the parts of the relation which are themselves not made of parts are changing on their own. Do you see what I mean? — leo

I see what you're saying, but the answer is that there is no such thing as a thing without parts. A thing only exists as such a hierarchy of parts, and without that there is no thing. So you've taken an impossibility "a thing without parts", and asserted that this thing can change. But without parts, there is nothing there, no thing.

The issue is the "circular reasoning", which appears from the hierarchical thinking. The hierarchy of parts implies a top position, or base position depending on your perspective, so let's just call it #1 position. Also, there is no hierarchy unless there is something which follows #1. So "hierarchy" implies more than one, yet #1 implies priority. The circularity is avoided by assigning priority to #1. But there is no hierarchy unless there is more than one, and if there is more than one, how does a specific part acquire the position of #1. Therefore we must look to something other than the parts to assign #1 to, and again we meet with causation. There is a balance between the parts in the hierarchy, which allows them to exist as a unity, and the balance is caused. This cause is what we can assign #1 to. So #1 exists not as a part of the hierarchy, but as the cause of it.

To answer the question we can look to the nature of "cause". "Cause" is a temporal concept, and the cause is always in the past. The past cannot be changed. Therefore the #1, being the thing without parts, and the cause of parts existing in a hierarchical balance, necessarily cannot change or be changed, being in the past. The cause does not exist as a relation to the thing, because it is within the thing, like part of the thing yet still not the same as a part of the thing, because of the priority we must assign to it.

f we see change as more fundamental than being then a thing is simply an absence of change in relation to some other change. And then we don’t have to explain how a thing changes, change is what’s fundamental, a partial absence of change is what has to be explained, and we can explain it simply by seeing it as two opposite changes that cancel one another (or as several changes in equilibrium). Would you agree with that? — leo

As explained above, any premise based in change is less reliable, more uncertain than a premise based in being, or sameness. So your suggested approach cannot give us the required degree of certainty.

It'll be history, and there is always people who care about history

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I'm not sure why you think it'll be any different in the Senate. — Hanover

I forgot to mention the most important thing. There's sure to be at least one republican Senator who want's Trump's position. That's why Trump wants the trial to blow by as fast as possible, and the Democrats want to drag it out a bit, let the wannabes stoke the fire..

I'm not sure why you think it'll be any different in the Senate. — Hanover

I see at least three major differences. There's no need for any Democrats in the Senate to defect because conviction is highly unlikely. Also the population represented is different as Senators represent an entire state. And, I think there are a number of Republican senators who have expressed dislike of Trump in the past..

We know Trump is extremely unlikely to be convicted by the Senate. The issue is which Republican Senators will be inclined to vote against Trump to ensure personal re-election, and what kind of division this will create within the party. And if they do not vote against Trump they face the prospect of being replaced by a Democrat.

The only question now is whether this maneuvering will more energize the left or the right in the upcoming election. It's doubtful it will change a single vote from one side to the next, but it might cause more people to go to the polls. — Hanover

There is probably a lot more to this matter than what you make of it here. The Democrats may have layered the strategy. The Senate has a complex election system, with representation by state. It is likely that some Senators will have a tough decision to make. Some Republican Senators will face the prospect of not getting re-elected if they side with Trump. There may be a shake up of the Senate, or there may be division in the Republican party. Either way, the Democrats come out ahead.

I see where you’re getting at, but do you agree that without the experience of change we wouldn’t even come up with the concept of “existence”? Without the experience of change there wouldn’t be thoughts, there would only be a single thought, or a single color, a single experience that never changes, and we couldn’t even think about that experience. So it seems to me that “change” is more fundamental than a “thing”. There can be change that is so random that no specific thing can be identified within this change, and we can’t identify a thing without change.

In that view then and to avoid confusion, maybe we should talk of change instead of existence? — leo

Yes, it doesn't even really make sense to speak of the possibility of experience without change, as change is so fundamental. However, we shouldn't dismiss its dichotomous partner, "being", if we define "being" as remaining the same, through time, continuity, consistency. In experience, we tend to notice things which stay the same for some period of time. In fact, it appear necessary that something stay as it is for some period in order for us to even notice it. Imagine if at every moment, everything nlittle part of existence changed in some completely random fashion. So if we look at the ancient dichotomy of being and becoming (change) it would be difficult to say which is more fundamental to our experience. To notice one seems to require that we notice the other. To get to the bottom of this, we can divide the two in analysis, and see what conditions underlie each of them.

But what are relations, if not things themselves? It seems you are assuming two fundamental distinct entities: things and relations. You are also assuming that a thing without parts cannot change on its own. Why would a relation without parts be able to change on its own, and not a thing without parts? It seems to me that if you assume a thing without parts cannot change you’re running into the same problem concerning a relation without parts. — leo

I don't understand what you could be talking about here. A "relation" requires two things, therefore the relation necessarily has parts. It doesn't make sense to speak of a relation without parts. I definitely was not assuming a relation without parts.

Perhaps you misunderstood the point I was making. If two distinct things are shown to be in a relation to one another, then by virtue of that relation, we have indicated that those two things are parts of a larger thing. If the "relation" is valid then a larger unity is indicated.

However, things and relations are fundamentally distinct. Relations are what we predicate of things, whereas the things themselves are the subject of predication. So a relation is what a thing is said to have, but it does not make the thing itself. Likewise, a thing has parts, but the parts do not make the thing itself, because the parts must exist in specific relations. These are the analyzed principles of the two above mentioned aspects of experience, parts and relations.

Experience, as a thing, the subject of consideration, has two features, parts and relations between the parts. For the sake of understanding, we say that the parts remain the same, as time passes, and all that changes is the relations between the parts. This is Aristotle's matter and form. The matter remains the same while the form changes. The problem is that we always learn to divide the parts further, then it appears like the part is made of parts with changing relations. To end the infinite regress, some will posit a "prime matter", the fundamental part, not composed of parts, therefore not itself changing, as the basis for all existence. Reality would consist of fundamental parts existing in different relations. The problem is that Aristotle demonstrated this prime matter as illogical,

And if we proceed to assume relations as fundamental, then it doesn't make sense to speak of relations without parts. Therefore we are really missing something in our analysis. What has come up, in much metaphysics is that what is missing here is "the cause". If parts exist in relations to each other, there must be a cause of this. It is our failure to address this feature, that leads to the unending analysis of parts and relations, seeking to find the bottom, the most fundamental, when we are actually neglecting the most fundamental thing, which is the cause of this unity between parts and relations, the cause of parts existing in relations. So to avoid the dead end analysis of parts and relations, we need to turn our attention toward "the cause".

Maybe if we start from the concept of change instead of starting from the concepts of things and relations, we won’t run into these problems. Change occurs, and within that change things can be identified, in that they are parts of the change that temporarily do not change in relation to the rest. What do you think of this? — leo

Yes, if we start with "change", we will see that change requires a cause, and so we are on the right track here.

I don’t see where there is the infinite regress when we say that a part can change, why would we have to assume that a fundamental part does not change? — leo

Change is a difference in relations between things. So if a thing changes, the relations between its parts have changed. There is no other way that a thing could change, that is change, a change in relations. But if a part can change, then it must be composed of parts, and so on to infinite regress. To avoid the infinite regress we assume a fundamental part, what the ancient Greeks called atoms, and in modern physics is fundamental particles. Aristotle demonstrated that this is illogical, as "prime matter".

es we can describe that change. Let’s say you have the experience of ‘white’ (you’re close to a white wall and you’re only seeing white), you might say this is a thing that doesn’t change, but no there is still change, your thoughts are changing, you only see the white as not changing because your thoughts are changing and allowing you to think that. And there the change can be seen as made of parts, one part is the thoughts that you are having and the other part is the sensation of ‘white’ that is not changing in relation to your thoughts, but they form one whole, you can’t see the ‘white’ as not changing without having changing thoughts at the same time. Do you see where I’m getting at? — leo

Sorry, I made a typo, I meant to say we cannot do this without assuming parts, instead of saying "with" assuming parts. My mistake. I meant to say that we cannot explain "change" without assuming parts in relation to each other. Change requires parts.

You've described a potential infinity, but not an actual infinity. To understand an actual infinity we need to understand the actual existence of the elements represented by mathematical language.

Is kind of why I asked about a working definition. It seems to me all those matters depend on context and what is required in and for the context. Kind of a shame you-all didn't. A good topic deserves good grounding. — tim wood

The nature of reality is not an issue here. The nature of an object is. "Reality" is the more general concept, so there is more to reality than just objects. What we are interested in here, is objects.

Here is another example: consider the "potential infinity" defined by the Fibonacci sequence. You can generate every Fibonacci number using a recursive function defining the sequence. In other words, the recursive function defines the first, second, third, and so forth, Fibonacci number. However, you can always consider the collection of elements generated in this way by saying: "suppose that nn is a number in the Fibonacci sequence." What you are talking about, in the latter case, is an infinite set of objects - there is no limit to the number of objects that satisfy this condition, although there are restrictions on the kinds of objects that satisfy the condition. — quickly

Now the issue, which we discussed already in the thread, is whether or not a written numeral necessarily represents an object. In actual usage, the numeral might be used to represent an object, or it might not. If it doesn't represent an object, then any supposed count is not a valid count.

Your example seems to create ambiguity between the symbol, and the thing represented by the symbol. So you would have to clarify whether there is actually existing numbers, existing as objects to be counted, otherwise the claim of "an infinite set of objects" is false. As proof, it doesn't suffice to say that it is possible that a numeral represents an object And actual usage of symbols demonstrates that it is possible that the symbol represents an object, but also possible that it does not. To present the symbol as if you are using it to represent an object, when you really are not, is deception.

No, we didn't really discuss the nature of reality. We discussed the difference between trying to make true descriptions of objects, and creating imaginary figures. Both of these, I would say, are part of reality. The issue I think is whether the imaginary figures qualify to be called objects. So the question would be to define "object", and this is why I turn to the law of identity. An object has a unique identity.

Are you talking about the hypotenuse of a right triangle? — John Gill

Yes.

I think we've covered much ground in this thread so it would be difficult to summarize. The pivotal issue seems to be the reality of Platonic objects. So we had an extensive discussion concerning what various mathematical symbols are representative of, whether they represent objects, if so, what kind of objects, and particularly the identity of the objects. The law of identity was prominent. . It appears like axioms which treat an infinite collection as an actual object, require the reality of Platonic objects. However, the point I argued is that mathematical objects do not have an identity which is consistent with the law of identity. .

OK, so your interpretation is (as I understand it) that that a line segment is not composed of infinite points, but is composed of sub-lengths. I am in agreement. I would point out that the length of a sub-length cannot be zero else all line segments would have the same size. — Devans99

The issue here is that there is an incommensurability between distinct spatial dimensions. Pythagoras demonstrated that the ratio between two perpendicular sides of a square is irrational. The same type of irrationality arises from other two dimensional figures, like the circle, with the irrational pi.

This incommensurability is extremely evident in the relationship between the non-dimensional point, and the one dimensional line, according to TheMadFool's explanation.

Whenever we add another dimension to our spatial representations we add a new layer of complexity to this fundamental incommensurability, such that by the time we get to a four dimensional space-time the irrationality involved is extremely complex. What is indicated by this fact, is that our representation of spatial existence, in the form of distinct dimensions, is fundamentally flawed.

People always say that there are things that science can't explain, and it is such a shit, desperate excuse that it might be to blame for the loss of some brain cells in certain people. Why do they simply not realise that science is an ever-expanding subject, that we may just have not discovered an explanation to said happening that 'science can't explain'. — Athen Goh

I think that what happened is that some people came to understand that there are things beyond the limits of understanding through science, because they are beyond the capacity for empirical observation of the human being. These people still had the desire to understand these things, to take their minds where science couldn't go, past the dead ends and road blocks which ancient science came up against, toward understanding the vast reality of what lies hidden beyond the limitations which are the premises of science.

However, there was still many people who had the attitude such as what is expressed above, by you. So the people with the desire to expand their understanding of reality beyond the limitations inherent within the principles of science, had to create the notion of God and religion, to gather the resources necessary for that crusade.

What is it? I can't open it.

Say, ∑n∈Nf(n)∑n∈Nf(n) is not a process like going shopping and returning home, it's a mathematical expression.
Convergence and divergence has concise technical definitions using the likes of ∀∀ and ∃∃.
I challenge you find and understand them. ;) At this point you might be in a position to launch critique.
By the way, you should know that this stuff has practical applications used every day by engineers, physicists and others. — jorndoe

A process is a process. If your intent is to create ambiguity in the definition of "process", such that it is possible to have a completed process, which by definition has no end, then be my guest. Do not expect me to follow along with such contradiction though.

And if you back up, justify, such contradiction with the report that it has practical applications, I would reply that such applications are nothing more than sophistry, deception.

Honestly I think its both cases. Some structures were actually contemplated due to their own beauty in a platonic world, while others raised secondary to observations and need for application as you depicted. I in some sense do agree you that we'll have infinite possibilities if we were to contemplate just purely, but there are definitely some scenarios that are more attractive platonically speaking than others. — Zuhair

I wonder if this is even true. Can you bring an example of an imaginary structure, created neither for the purpose of copying something in the world, nor for the purpose of resolving a specific type of problem. I suppose that it would be very difficult to distinguish whether the structure was created purely for beauty, or for utility. And, if you were to go and create one right now, saying you created it purely for beauty, I would argue that you did it for the purpose of your argument. So we might leave this point as unresolved, or even unresolvable. However, we might still argue our opinions, in an attempt to get the other into our own metaphysical camp.

Example of "mathematics prior to observation" is that the orbit of planets which suits more of an ellipse. Ellipses where there on board since ancient Greek, and their study didn't arise from contemplating planet orbits as you think. No they actually were studies on our earthly structures which are simply about inclined sections of cones. Then Kepler picked what is already available and matched it with observations about planets movements. — Zuhair

OK, I think you're right here, but this does not exclude the possibility that the ellipse was created for another purpose. So it doesn't really force the conclusion that the model was produced prior to having an application. We just might not be a ware of the application it was first designed for.

Other examples include Riemannian n-dimensional geometry, this was contemplated before relativity theory and other recent theories of physics which use many dimensions. Also non-Euclidean geometry was long contemplated by Al-Tusi and also by various mathematicians long before relativity theory called for their use, and they did arise from the pure study of geometry in the platonic realm, mainly becuase of the non-proof of parallelity postulate. Pure Platonic contemplation is not random, and so it pursue interesting alternative structures, and also can pose general mathematics investigating wide array of those structures. — Zuhair

Again, we cannot really resolve the question this way, because there would always be a reason for speculating about non-Euclidian geometry. You say that it is because the non-proof of the parallel postulate, but as I think I indicated earlier, Pythagoras was dissatisfied with the irrational nature of the square, and we also have the irrationality of pi. These are all good reasons to speculate about non-Euclidian geometry, and it would be difficult to prove that utility is not at the base of this dissatiisfaction.

Pure Platonic contemplation is not random, and so it pursue interesting alternative structures, and also can pose general mathematics investigating wide array of those structures. — Zuhair

So let's assume as we would agree, that pure Platonic contemplation is not random. How could we assign anything other than utility (what Plato calls "the good") as the thing which delivers us from randomness? If we were to contemplate pure beauty, completely devoid of utility, wouldn't this be randomness itself?

So in real practice both lines are occurring, the pure investigation of those entities in the platonic laboratory and on the other hand the on-demand construction of mathematical entities to match needed application. We can say that mathematics can work to enrich our knowledge about the world by detecting behaviors in the later that we known in the platonic world (in approximate manner), and also the other direction is also true, that observation in our real world as the source and the motive to contemplate certain platonic structures, so our world enriches mathematics also. It is a bilateral movement. And I think this bilaterality is important. And it should be observed if we are to have mathematics help enrich our knowledge about our world. — Zuhair

I'm not convinced that there is such a thing as pure investigation in the platonic laboratory. I think this would require that we totally remove ourselves from the necessities of life, and the constraints of the physical world, and this is impossible. That is why Plato himself settled on "the good", as that which makes the intelligible objects intelligible. The good inheres within the essence of the intelligible object therefore, as what gives it the characteristic of being intelligible. If we remove this good, we are overwhelmed by randomness. And randomness might itself be the most beautiful thing there is, such that we would be overcome by the beauty of pure randomness, but such beauty would be inherently unintelligible because of the nature of randonmness, and therefore impossible to be the source of any type of structures.

I know, it's because philosophers can't agree on anything when it comes to time.

That's simply an indication that we can do logic without knowing how the logic works. To know how logic works is a completely different issue. This question is Socrates' claim to fame. The artists and skilled craftsmen would claim to "know", because they had a technique which produced the desired results. This attitude extended into all fields, science, mathematics, even ethics and sophistry. Socrates demonstrated that these people who know how to do something do not know how it is that their activity brings about the desired end. Therefore their own claim to "knowing-how" is not grounded in anything, the activity is just a habit, and so is not real knowledge at all..

Not so for Planck Time. You'll need a real, live physicist to discuss this properly. It used to be that this limit was variable according to some physical features. — John Gill

Actually, what I described is exactly the case with Planck time. The limit (Planck time) is the product of the theories being used. This is from Wikipedia: "The Planck time is the unique combination of the gravitational constant G, the special-relativistic constant c, and the quantum constant ħ, to produce a constant with dimension of time.".

You might say that these theories represent something real in the universe, but they only do so to the extent of our understanding. Any misunderstanding creates a limit to the mathematics which is not representative of a real limit in the physical universe. When this is the case, application of the mathematics to observations will produce an abnormal occurrence of infinities, (as we see in quantum physics) as the things being observed go beyond the limits created by the lack of understanding expressed by the theor.

I don't see where you differ with me. Mathematics can also speak of patterns that had not been yet observed! Because it tackle all possible structures in an unlimited manner. That's what I meant when I said *before-hand*, if we had good mathematics about ellipses, parabola, hyperbola, etc.., even before we observed the movements of planets, that knowledge would make it easier for the astronomer to discover the pattern of movement of those planets, because as I said many times humans don't see what they don't look for. — Zuhair

I don't see how this notion of "before-hand" can be realistic. Before-hand, there are infinite possibilities for spatial shapes. So it could not be practical to produce all these possible models prior to observations, then after observing, attempt to fit a model to the observation. What is really the case, in practise is that we see something, observe it and take notes, then we create a model to represent it. So we work from the purest form of mathematics, simple numbers to represent observed occurrences of events, with the most primitive spatial representation of those events, toward creating a more complex spatial form, or pattern, which fits to those occurrences.

I agree that it is necessary to keep our minds open to "all possible structures" but to approach a problem with all possible structures already apprehended, and developed on paper, is not practical because unrestricted possibility approaches infinity. Therefore we take the information presented to us by the particular problem, and create structures as possible solutions, according to what is required, striving to keep our minds open to many possibilities because once we accept one we tend to close our minds to others. And this is not good, because we never actually obtain "the ideal."

f you have the descriptive arsenal before-hand, you'll predict easily the behavior of matters with fewer observations because it would look familiar to what you have experienced in say the platonic world about those orbits. — Zuhair

This is not true, because the "descriptive arsenal" would have to contain all of the countless possibilities. Then, you would have to compare the observations with each of those countless possibilities to determine which description is the best. This is highly impractical, and not representative of the way that we actually proceed. in reality we create the "platonic world" to represent what we have observed.

Notice that we didn't coin the mathematical structures describing orbits (ellipses, hyperbola,etc..) after the observation had been made, we actually imagined it form more trivial observations on our planet, then we freely contemplated more variety of structures in the platonic imaginary world, this free contemplation is what made us arrive at those orbit mathematical structures way before any application was discovered. — Zuhair

I don't think that this is true either. Kepler noted that planetary orbits were not eternal perfect circles as postulated by Aristotle. This knowledge was produced by inconsistent positioning. Kepler approached this problem with numerous possible curves, and found the elliptical orbit to be most suitable. But I don't see any indication that there are any elliptical orbits available for observation on our planet, from which Kepler could have copied the design, and no indication that the design was created for anything other than the purpose of modeling planetary orbits..

And I think that's one of the most important jobs of mathematics, to supply such descriptive arsenal that objects in our world can possibly follow. I'd say perhaps, the particle physics objects move along some paths that we don't have the descriptive arsenal necessary to match them with, that's why we remain in ignorance about them. — Zuhair

I agree that we are very close to complete agreement on these issues, that's why I have pointed out the specific places of disagreement with "not true", hopefully to help you see that my perspective is better suited. Though you might bring me around to your perspective instead.

So we're back to this question of art (beauty, aesthetic), or utility. Do mathematicians create all sorts of shapes, forms, and structures simply because they are beautiful, and have them lying around for possible use, or do they create them to serve as solutions to particular problems. You seem to choose the former, that mathematicians create a whole arsenal of beautiful shapes, simply because they are beautiful, then physicists and cosmologists might choose from this collection of designs, those which are suitable to them. I think that mathematicians create their forms with purpose, as potential solutions to particular problems.

I did not ask, "How many numerals are there?" This is immensely important. I asked a question about a human being, namely, "How many numerals did you learn to write down?"

This is the category difference which makes Banno's claim of "done" false.

Consider also that proofs are finite objects. — softwhere

Continuing with the Wittgensteinian perspective, the finitude of the proof would be dependent on the definitions of the terms. The definitions create the boundaries of meaning, required for the proof. If there is any vagueness, or undefined terms in the proof, then the proof cannot be considered as a finite object. Therefore it is very unlikely that we actually have any truly finite proofs, because definitions are produced with words, which themselves need to be defined, etc., ad infinitum. Vagueness cannot be removed to the extent required for the production of a finite object.

This makes mathematics a prototypically 'normal' discourse, and perhaps explains the mixed feelings that metaphysicians have toward it. As I see it, the old dream of metaphysics is to do 'spiritual math' about matters of ultimate concern. Proofs of god, etc. But non-mathematical language seems caught up in time to a much greater degree. 'History is a nightmare from which I'm trying to awake.' — softwhere

Yes, we can class mathematics as "normal discourse", but to characterize "normal discourse", as working with finite objects of meaning, is what Wittgenstein demonstrates as wrong. This is why we must work to purge the axioms of mathematics from the scourge of Platonism, To consider proofs as finite objects is a false premise.

A sum is the total. "Diverges" signifies a direction. It would be a category mistake to say "diverges to infinity" is a sum.

But "the sum diverges to infinity" is not "it can't be done"! — Banno

Well, it's not a sum. And to say that the sum "diverges to infinity" says I can't give you that sum. So if you're not saying "it can't be done" then why can't you give me the sum?

Take the other example - 1-½+⅓-¼...

It converges to two.

You disagree? — Banno

I'd need a definition of "converges" before I'd agree to what you're saying, but if you mean comes closer and closer to, as you continue on the unfinished process signified by "...", then I'd probably agree. But this implies that it's not ever done.

The sum! Wasn't that the task, to sum the sequence?

Face it Banno; you're trying to avoid doing the task by giving some answer which amounts to "it can't be done".

Your answer is not a sum. The task has not been done.

Regardless of whether or not they understand the process, unless they "keep going like this" they cannot be said to have done the task. Understanding the task to be done, and doing it are two distinct things.

And, if someone thinks that understanding the task constitutes doing the task, as you apparently do, then that person actually misunderstands.

OK, suppose you tell someone "keep going like this". At what point have they completed (are done) with that command? When they reach infinity?

Here it means "keep going like this..." — Banno

In other words you're never done.

As I said, ellipsis means unfinished. So using the ellipsis and claiming "it's done" is a false claim.

I think that without having descriptive account on "orbits" like those of Ellipses, Parabolas, and hyperbolas that mathematics beforehand supplied us with, it could have been very difficult to observe how the planets moves, and it would be very difficult to predict their movements. Possibly similar things might apply with the uncertainty principle. I don't know really. — Zuhair

This is a very interesting subject which you bring up here, but my opinion is somewhat opposite to what you say. I think that mathematics allows us to make many very accurate predictions based on statistics and probabilities, without having any accurate description of the mechanisms involved. So for example, Thales apparently predicted a solar eclipse in 585 BC. I think it's common that we observe things, take note of the patterns of specific occurrences, thereby becoming capable of predicting those occurrences, without understanding at all, the motions which lead to those occurrences.

So the ancient people observed the motions of the sun, moon, planets, and stars, and described these motions relative to their point of observation, and could make predictions based on those descriptions. But the motions they described were completely different from the motions we describe of the very same bodies, today. And we say that they were wrong. However, we still insist that motion is relative so we don't even really have the right to say that they were wrong.

It all breaks down as limits are approached: — John Gill

That's ironic, the numbers approach infinity (limitless), as the condition approach the limit. What this indicates is that the limit is created by the principles which govern how the numbers are applied. The limit is created by the numbers approaching infinity, and the principles of application dictate when the numbers will approach infinity.

Sure, but what makes the examples in the OP interesting is the active roll played by the third sex. — Banno

Do you think that the infertile female bees, the workers, do not play an active role?