I answered this in my most recent post to you. Given two ordinals, it's always the case that one is an element of the other or vice versa. — fishfry

OK, this makes more sense than what you told me in the other post, that one "precedes" the other. You are explaining that one is a part of the other, and the one that is the part is the lesser..

This is NOT true of sets in general, but it IS true for ordinals, and that's what makes the construction work. — fishfry

I assume that an ordinal is a type of set then. It consists of identifiable elements, or parts, some ordinals being subsets of others. My question now is, why would people refer to it as a "number"? Say for instance that "4" is used to signify an ordinal. What it signifies is a collection of elements, some lesser than others. By what principle is this group of elements united to be held as an object, a number? Do you know what I mean? A set has a definition, and it is by the defining terms that the sameness of the things in the set are classed together as "one", and this constitutes the unity of the set. In the case of the "ordinals", as a set, what defines the set, describing the sameness of the elements, allowing them to be classed together as a set?

No, as I'm pointing out to you. It's true that every ordinal is cardinally equivalent to itself, but that tells us nothing. You're trying to make a point based on obfuscating the distinction between cardinal numbers, on the one hand, and cardinal equivalence, on the other. — fishfry

The issue, which you are not acknowledging is that "cardinal" has a completely different meaning, with ontologically significant ramifications, in your use of "cardinally equivalent" and "cardinal number".

Let me explain with reference to your (I hope this is acceptable use of "your") hand/glove analogy. Let's take the hand and the glove as separate objects. Do you agree that there is an amount, or quantity, of fingers which each has, regardless of whether they have been counted? The claim that there is a quantity which each has, is attested by, or justified by, the fact that they are what you call "cardinally equivalent". So "cardinal" here, in the sense of "cardinally equivalent" refers to a quantity or amount which has not necessarily been determined. Suppose now, we determine the amount of fingers that the hand has, by applying a count. and we now have a "cardinal number" which represents the amount of fingers on each, the glove and the hand. In this sense "cardinal" refers to the amount, or quantity which has been determined by the process of counting.

Other way 'round. A cardinal number is defined as a particular ordinal, namely the least ordinal (in the sense of set membership) cardinally equivalent to a given set. — fishfry

Do you agree with this characterization then? An ordinal is a type of set, and a cardinal is a type of ordinal. Logical priority is given to "set". So do you agree that a cardinal number is not an object, but a collection of objects, as a set? Or, do you have a defining principle whereby the collection itself can be named as an object, allowing that these sets can be understood as objects, called numbers?

Right. I can live with that. I know I have the same number of fingers as my glove, but I don't know how many fingers that is. — fishfry

But this is an inaccurate representation. What you are saying, in the case of "cardinal numbers", is not "that. I know I have the same number of fingers as my glove, but I don't know how many fingers that is", but that there is no "number" which corresponds with the amount of fingers in my glove, until it has been counted and judged. You can say, I know I have the same "amount" of fingers as my glove, but you cannot use "number" here, because you are insisting that the number which represents how many fingers there are, is only create by the count.

Cardinal equivalence is a relation between two sets. It's not something a set can have by itself. — fishfry

But you already said a set can be cardinally equivalent with itself. "If nothing else, every ordinal is cardinally equivalent to itself, so the point is made."

I see where you're going with this. Given a set, it has a cardinal number, which -- after we know what this means -- is its "cardinality." You want to claim that the set's cardinality is an inherent property. But no, actually it's a defined attribute. First we define a class of objects called the cardinal numbers; then every set is cardinally equivalent to exactly one of them. But before we defined what cardinal numbers were, we couldn't say that a set has a cardinal number. I suppose this is a subtle point, one I'll have to think about. — fishfry

Yes this exemplifies the ontological problem I referred to. Let's say "cardinality" is a definable attribute. Can we say that there is a corresponding amount, or quantity, which the thing (set) has, regardless of whether its cardinality has been determined? What can we call this, the quantity of elements which a thing (set) has, regardless of whether that quantity has been judged as a number, if not its "cardinality"?

But when you got up that morning, before you came to my party, you weren't a room 3 person or whatever. The assignment is made after you show up, according to a scheme I made up. Your room-ness is not an inherent part of you. — fishfry

I see this as a very dangerously insecure, and uncertain approach, epistemically. See, your "scheme" is completely arbitrary. You may decide whatever property you please, as the principle for classification, and the "correctness" of your classification is a product simply of your judgement. In other words, however you group the people, is automatically the correct grouping.. The only reason why I am not a 3 person prior to going to the party is that your classification system has not been determined yet. If your system has been determined, then my position is already determined by my relationship to that system without the need for your judgement. It is your judgement which must be forced, by the principles of the system, to ensure a true classification. My correct positioning cannot be consequent on your judgement, because if you make a mistake and place me in the wrong room, according to your system, you need to be able to acknowledge this. and this is not the case if my positioning is solely dependent on your judgement.

If you go the other way, as you are doing, then the position is determined by your subjective judgement alone, not by the true relation between the system of principles and the object to be judged. So if you make a mistake, and put me in the wrong room, because your measurement was wrong, I have no means to argue against you, because it is your judgement which puts me in group 3, not the relation between your system and me.

Incorrect: We should not use 'least' if we don't mean quantity.

It is typical of cranks unfamiliar with mathematical practice to think that the special mathematical senses of words most conform to their own sense of the words or even to everyday non-mathematical senses. The formal theories don't even have natural language words in them. Rather, they are purely symbolic. Natural language words are used conversationally and in writing so that we can more easily communicate and see concepts in our mind's eye. The words themselves are often suggestive of our intuitions and our conceptual motivations, but proofs in the formal theory cannot appeal to what the words suggest or connote. And for any word such as 'least' if a crank simply could not stomach using that word in the mathematical sense, then, if we were fabulously indulgent of the crank, we could say, "Fine, we'll say 'schmleast' instead. 'schmardinality' instead'. 'ploompty ket' instead of 'empty set' ... It would not affect the mathematics, as the structural relations among the words would remain, and the formal symbolism too. — TonesInDeepFreeze

OK Tones, explain to me then what "least" means in "the mathematical sense", if it is not a quantitative term. It can't be "purely symbolic" in the context we are discussing. For example, when fishfry stated von Neumann's definition of a cardinal as "the least ordinal having that cardinality", through what criteria would you determine "least", if not through reference to quantity?

df: K is a cardinal iff K is an ordinal and there is no ordinal j less than K such that there is a bijection between K and j.

There is no mention of 'cardinal' or 'cardinality' in the definiens. — TonesInDeepFreeze

Here's another example. Look at your use of "less than". How is one ordinal "less than" another, without reference to quantity?

You wouldn't call it "my" theory of relativity, or "my" theory of evolution, just because I happened to invoke those well-established scientific ideas in a conversation. — fishfry

Yes, in the context of the example we are discussing, I would. Unless you were quoting it word for word from another author, or explicitly attributing it to someone else, I would refer to it as your theory. I believe that is to be expected. Far too often, Einstein's theory, and Darwin's theory are misrepresented,. So instead of claiming that you are offering me 'Cantor's theory', it's much better that you acknowledge that you are offering me your own interpretation of 'Cantor's theory', which may have come through numerous secondary sources, unless you are providing me with quotes and references to the actual work.

It's a bit like saying that the score in a baseball game is tied -- without saying what the score is. Maybe that helps. — fishfry

OK, so let's start with this then. In general we cannot determine that a game is tied without knowing the score. However, if we have some way of determining that the runs are equal, without counting them, and comparing, we might do that. Suppose one team scores first, then the other, and the scoring alternates back and forth, we'd know that every time the second team scores, the score would be tied, without counting any runs. Agree? Is this acceptable to you, as a representation of what you're saying?

If one thing is defined in terms of some other thing, the latter is logically prior. As is the case with cardinal numbers, which are defined as particular ordinal numbers. — fishfry

Here's where the problem is. You already said that there is a cardinality which inheres within ordinals. This means that cardinality is a property of all ordinals, it is an essential, and therefore defining feature of ordinals. So we have a sense of "cardinality" which is logically prior to ordinals, as inherent to all ordinals, and we also have a sense of "cardinal" number which is specific to a particular type of ordinal.

I'd agree that given some ordinal number, it's cardinally equivalent to some other sets. It doesn't "have a cardinality" yet because we haven't defined that. We've only established that a given ordinal is cardinally equivalent to some other set. — fishfry

Don't you see how this is becoming nonsensical? What you are saying is that it has a cardinality, because it is cardinally equivalent to other sets, but since we haven't determined its cardinality, it doesn't have a cardinal number. In essence, you are saying that it both has a cardinality, because it is cardinally equivalent, and it doesn't have a cardinality because it's cardinality hasn't been determined, or assigned a number.

Let's look at the baseball analogy. We know that the score is tied, through the equivalence, so we know that there is a score to the game. We cannot say that because we haven't determined the score there is no score. Likewise, for any object, we cannot say that it has no weight, or no length, or none of any other measurement, just because no one has measured it. What sense does it make to say that it has no cardinal number just because we haven't determined it?

Note per your earlier objection that by "least" I mean the ∈∈ relation, which well-orders any collection of ordinals. If you prefer "precedes everything else" instead of "least," just read it that way. — fishfry

Actually, this explains nothing to me. "Precedes" is a relative term. So you need to qualify it, in relation to something. "Precedes" in what manner?

No. Cardinal equivalence is logically prior to ordinals in the sense that every ordinal is cardinally equivalent to some other sets. At the very least, every ordinal is cardinally equivalent to itself.

When you use the word "cardinality" you are halfway between cardinal numbers and cardinal equivalence, so you confuse the issue. Better to say that cardinal equivalence is logically prior to ordinals; and that (in the modern formulation) ordinals are logically prior to cardinals. — fishfry

Yes, this demonstrates very well the problem I described above. Because the set has a "cardinal equivalence, it also necessarily has a cardinality, and a corresponding mathematical object which you call a cardinal number. Why do you think that you need to determine that object, the cardinal number, before that object exists as the object which it is assumed to be, the cardinal number?

Socrates does not make the proper distinction between a tuning and what is tuned. It is not more or less a tuning, it is more or less in tune. — Fooloso4

It appears to me, like you're totally missing Socrates' argument. There is no such thing as "more or less in tune". Either the waves are in sync or they are not. Either it's in tune or not, this is not a matter of degrees. The point Socrates makes,93d- 94a, is that a group of notes is either in harmony or not, and there is not a matter of degrees here. But a soul has degrees of wickedness and goodness. So that is one reason why the soul is not a harmony. Either the parts are in harmony or not, and there is no matter of degrees in this situation. But, there is a matter of degrees of goodness with the way that the soul rules the body. That is why the soul is not a harmony.

The main point though, is made at 93a, "One must therefore suppose that a harmony does not direct its components, but is directed by them". This point is built upon at 94b: "Further, of all the parts of a man, can you mention any other part that rules him than his soul, especially if it is a wise soul?" He then explains how the soul rules by opposing what the body wants, and if the soul were a harmony of parts such an opposition would not be possible.

Well, does it now appear to do quite the opposite, ruling over all the elements of which one says it is composed, opposing nearly all of them throughout life, directing all their ways, inflicting harsh and painful punishment on them, at times in physical culture and medicine, at other times more gently by threats and exhortations, holding converse with desires and passion and fears as if it were one thing talking to a different one... — 94c-d

The proper analogy to good and bad souls would be good and bad tunings. — Fooloso4

The point is that there is no such thing as good or bad tunings. Being in tune is an objective fact of the wave synchronization, and if it is out of tune, it is simply not in tune, not a matter of a bad tuning, but not in tune at all. But the soul is not like this, it has degrees of goodness and badness.

The problem for moderns, is that 'prior to' must always be interpreted temporally - in terms of temporal sequence. However, I think for the Ancients, 'prior to' means logically, not temporally prior. 'The soul' is eternal, not in the sense of eternal duration, but of being of an order outside of time, of timeless being, of which the individual is an instance. I think that comes through more clearly in neo-Platonism but the idea is there from the outset. — Wayfarer

Yes, I believe this understanding of the two distinct senses of "eternal" is very important in metaphysics. What we have now, in our modern conception of "eternal", is a notion of infinite time, time extended eternally. This is because with materialism and physicalism, the idea of anything outside of time, (which is the classical theological conception of "eternal"), is incomprehensible.

I believe Aristotle's cosmological argument actually demonstrates that the idea of infinite time is what is incomprehensible, and this forces the need for something outside of time ("eternal" in the theological sense). So it's a matter of how one apprehends the boundaries. Is all of reality bounded by time (physicalism), or is time itself bounded?

But you say it's "my" bijective equivalence as if this is some personal theory I'm promoting on this site. On the contrary, it's established math. You reject it. I can't talk you out of that. — fishfry

No, I think you misinterpret this. I say it's "your" bijective equivalence, because you are the one proposing it, not I. So "yours" is in relation to "mine", and anyone else who supports your proposition (even if you characterize it as "established math") is irrelevant. If you wish to support your proposition with an appeal to authority that's your prerogative. In philosophy, the fact that something is "established" is not adequate as justification.

Two sets are bijectively equivalent if there is a bijection between them. In that case we say they have the same cardinality. We can do that without defining a cardinal number. That's the point. The concept of cardinality can be defined even without defining what a cardinal number is. — fishfry

This is what I do not understand. Tell me if this is correct. Through your bijection, you can determine cardinality. But are you saying that you do this without using cardinal numbers? What is cardinality without any cardinal numbers?

What I think is that you misunderstand what "logically prior" means. Here's an example. We define "human being" with reference to "mammal", and we define "mammal" with reference to "animal". Accordingly, "animal" is a condition which is required for "mammal" and is therefore logically prior. Also, "mammal" is logically prior to "animal". You can see that as we move to the broader and broader categories the terms are vaguer and less well defined, as would happen if we define "animal" with "alive", and "alive" with "being". In general, the less well defined is logically prior.

Now let me see if I understand the relation between what is meant by "cardinality" and "cardinal number". Tell me if this is wrong. An ordinal number necessarily has a cardinality, so cardinality is logically prior to ordinal numbers. And to create a cardinal number requires a bijection with ordinals, so ordinals are logically prior to cardinal numbers.

Where I have a problem is with the cardinality which is logically prior to the ordinal numbers. It cannot have numerical existence, because it is prior to ordinal numbers. Can you explain to me what type of existence this cardinality has, which has no numerical existence, yet is a logical constitutive of an ordinal number.

Cardinality is inherent. — fishfry

In other words, you agree that it's incorrect to say that ordinals are logically prior to cardinals. That is, unless you are just trying to hide a vicious circle by saying that a cardinal number is defined by its ordinality, and ordinality is defined by cardinality. But in the case of a vicious circle of two logically codependent things, it is still incorrect to say that one is logically prior to the other. So despite my lack of understanding of your "bijective equivalence", it is still you who is mistaken.

His argument is that Harmony is a universal. What is at issue is the difference between the universal and particular. Harmony itself is prior to any particular thing that is in harmony. — Fooloso4

The argument is not about universals. It is a question of whether the activity required to produce, or create, an organized system of parts (the harmony), is necessarily prior to that organized system of parts. Read 93-95.

So yes, cardinality is already inherent within the ordinals. Each ordinal has a cardinality. I — fishfry

Well then it's incorrect to say that ordinality is logically prior to cardinality. If there is already cardinality inherent within ordinality then the closest you can get is to say that they are logically codependent. But if order is based in quantity, then cardinality is logically prior.

Not at all. Not "more or less," but "prior in the order," if you prefer more accurate verbiage.
You insist on conflating order with quantity, and that's an elementary conceptual error. In an order relation x < y, it means that x precedes y in the order. x is not "smaller than" y in a quantitative sense. I can't do anything about your refusal to recognize the distinction between quantity and order. — fishfry

"Least", lesser, and more, are all quantitative terms. So as long as you are using "least" to define your order, it is actually you who is conflating quantity with order. If you want a distinct order, which is not quantitative, you need something like "before and after", or "first and second". But first and second is a completely different conception from less and more, and would not be described by "least".

If you want to emphasize a difference between quantity and order you need to quit using quantitative words like "least", when you are talking about order. However, I should remind you, that "least" is the term you used for Von Neumann's definition. If Von Neumann used the quantitative word "least", in his definition, then I think it is just a faulty interpretation of yours, which makes you insist on distancing quantity from order. In the reality of mathematical practise, order is defined by cardinality, not by anything like "first and second". So cardinality is held to be logically prior, regardless of what you claim.

The modern definition is the von Neumann cardinal assignment. Von Neumann defined a cardinal as the least ordinal having that cardinality. — fishfry

Isn't this circular? Doesn't "least" already imply cardinality, such that cardinality is already inherent within the ordinals, to allow the designation of a least ordinal? Then the claim that ordinals are logically prior to cardinals would actually be false, because more and less is already assumed within "ordinal".

The soul is that which imparts life to the body in the first place (105c - d). Without the soul there would be no body. — Apollodorus

This is why the immaterial soul is prior to the material body.

Right, but a lyre is not a living thing. It is not capable of self-movement or self-attunement.

Wayfarer makes an important point: — Fooloso4

Wayfarer's point explains why we must conclude that the immaterial soul is prior to the material body.

And when we proceed further down this route, we see that to account for the real order which inheres within inanimate things, we need to assume an immaterial existence (God) , as prior to the material things of the world.

With all his talk of opposite forms Socrates neglects to consider Harmonious /Unharmonious or — Fooloso4

I don't think Socrates neglects this at all. In fact, it is focused on in many dialogues. When the mind succumbs to the desires of the body, and is overwhelmed by these desires, to the point of irrationality, then the mind no longer rules, and the person gets into an unharmonious, or disordered state.

The question is why Socrates neglected this argument? — Fooloso4

I don't see that you have a point. As I already pointed out to you, what is referred to by "the tuning of a lyre" does not exist independently of a particular lyre. The tuning of a lyre is always carried out, and must be carried out on a particular lyre. What is independent of the particular lyre is the principles by which a lyre is tuned, or as I said earlier "how to tune a lyre".

Second, the argument that the soul is a harmony means that the fate of a particular soul is tied to the fate of a particular body. — Fooloso4

But Socrates demonstrates, by the argument we've been discussing, that this idea, "that the soul is a harmony" is false.

The analogy with the lyre is not with a lyre that needs to be tuned but that is tuned, that is, in harmony. — Fooloso4

But a lyre does need to be tuned. It doesn't magically tune itself, and if used, it rapidly goes out of tune. So there is a very clear need to assume that there is something which tunes it. Likewise, there is a very clear need to assume that there is something which causes an organism to be organized. That's the soul.

I do not know the tuning of the lyre, but let's say the strings are tuned in 4ths or 5ths. The standard is independent of any particular lyre, but whether this particular lyre is in tune cannot be independent of the tension of the strings of this lyre, and that tension cannot be achieved when this lyre is destroyed. — Fooloso4

Yes, and the belief that the soul is like a particular lyre being in tune (a harmony), is the belief which Socrates dismisses as faulty. So the fact that this particular instance of being in tune (a harmony) is destroyed when the lyre is destroyed, is irrelevant to what Socrates is arguing, because he argues that the soul is not like a particular instance of being in tune (a harmony).

God is supposed to be a necessary being. Something is necessary if it is true in every possible world. — Banno

I think you're using "necessary" in a way different from how the classical theologians used it. God is said to be necessary in the sense of "required for".

Logic is needed in order to have the discussion, not as a consequence of the discussion. — Banno

Here's an example of that sense. In the same way that you say logic is needed to have a discussion, theologians say that God is needed to have the world which we have.

Yes, you can conceive of a possible world, which does not require God, but that's irrelevant because God is determined to be necessary for the world which we actually have.

See, it's a different sense of "necessary". It is the sense which describes how a contingent thing actually exists. A contingent thing requires the appropriate efficient cause to bring its actual existence from a mere possibility. The efficient cause is said to be necessary for the thing's existence.

The tuning does not tune the lyre or body, the lyre or body is tuned according to the tuning. It must exist in order to be tuned. — Fooloso4

This is strangely worded. If it is true that the act of tuning is what causes the lyre to be tuned, then it contradicts this to say "The tuning does not tune the lyre or body", as you do say. I think we must admit that it is the act of tuning which causes the lyre to be tuned, so we can't accept what you say here, "the tuning does not tune the lyre or body".

But if the argument is accepted then the soul is not immortal. The destruction of the lyre means the destruction of its tuning, and analogously the destruction of the body would mean the destruction of its tuning. How a lyre or body is tuned according to the relationship of its part is not affected, but the tuning of this particular lyre or body certainly is when the lyre or body is destroyed, — Fooloso4

Socrates argues against the position that the soul is like being tuned, ( a harmony in my translation) for the reason I described, the soul is more like the cause of being tuned, which is the act of tuning. When a particular lyre is no longer tuned, the cause of it being tuned, the act of tuning, is no longer tuning that particular lyre, but it is still tuning other instruments..

The tuning of a lyre exists apart from any particular lyre. — Fooloso4

Well, I don't think this is really true. There are principles to be followed in tuning the instrument, but the tuning itself is dependent on hearing the particular notes and judging the relation between them as the desired ones. So the tuning does not exist apart from the instrument, as it is dependent on the instrument making those tones so that they may be judged.

It is this relationship of frequencies that is used to tune a particular lyre. — Fooloso4

See, it is necessary to have those tones, in order to have tones with that the relationship between them. Just having the principle does not constitute "the tuning of a lyre" To state the principle, or relationships between frequency, or lengths of similar strings, in mathematical terms, or however you state it, does not give you "the tuning of a lyre". It gives you 'how to tune a lyre'.

Analogously, the Tuning of the body exists apart from any particular body, it is the relationship of bodily parts, but the tuning of any particular body suffers the same fate as the tuning of any particular lyre. — Fooloso4

The argument against the soul as a harmony, is not intended to say anything about the existence of the soul after death. That's why Socrates goes to the other dialectical argument (argument from the meaning of words) afterwards. The harmony argument shows that 'how to tune a lyre', the principle concerning the relationship between tones, is prior to 'the tuning of a lyre'. So the soul is prior to the body, by having that principle of how to create harmony within the parts of the body. So at 95 c-d he explains how the proof that the soul is prior to the man, does not prove that the soul is immortal. It may be the case that entering the body of a man is the decline of the soul, that this is the beginning of the end.

I find the most compelling and important argument in The Phaedo is the argument against the soul as a harmony. As a harmony, continues to be the populist view today as emergence; life is something which emerges from properly aligned material parts. But Socrates' argument actually demonstrates that the soul must be prior to the body, being the cause of alignment of the parts, rather than the harmony which is the result of such alignment. This is important because it provides us with the basis for understanding the nature of free will, and other fundamental ontological principles.

But pi is not a particular real number? How can I have a conversation with you? — fishfry

You must know by now, that I do not accept "real number" as a valid concept. Your insistence that I must accept real numbers as a premise for discussion with you, is simply an act of begging the question.

What the capacity of free will demonstrates to us, is that there is no continuity of existence from past to future. This means that any existing object must be recreated at each moment of passing time. In theology this principle is understood as God being required to maintain existence, It is why Newton proposed his first law of motion as supported by the Will of God.

But Pi obviously is a principle, the ratio of the circumference of a circle to it's diameter. So, clearly it's you who is wrong, to say that pi is "the definition of a particular number". Again, you really amaze me with the nonsense you come up with sometimes fishfry.

Pi is the ratio of the circumference of a circle to it's diameter. Why is that not a "principle of mathematics"?

It's clearly wrong that the ancients knew pi as a real number.

Pi is a particular real number, known to the ancients. Hardly a principle. — fishfry

This is clearly wrong. The ancients did not have real numbers, so they could not have known pi as a real number. They knew pi as the ratio of a circle's circumference to it's diameter. Further, they discovered that this ratio is irrational. You really amaze me with the nonsense you come up with sometimes fishfry.

Are you saying that because humans are physical and sets are a product of the human mind, that sets are therefore physical? — fishfry

No, I was saying that human actions are limited by the physical world, and mathematical thinking is a human action therefore it is limited by the physical world.

Well then your point is trivial and pointless. Everything is physical if we can imagine it. The Baby Jesus, the Flying Spaghetti Monster, the three-headed hydra, all physical because the mind is physical. Whatever man. Pointless to conversate further then if you hide behind such a nihilistic and unproductive point. — fishfry

Obviously, I wasn't saying "everything is physical". Metaphysically, I believe in the immaterial, or non-physical. But human thoughts, as properties of physical beings, do not obtain this status.

Can we please stop now? — fishfry

Sure, you seem to have run out of intelligent things to say.

How do I find out where my likes are?

Is this where we can come to get liked? Maybe the incels would benefit from something like this. A zero on your forehead might look pretty good to some people.

Some aspects of mathematics is so obviously fictional that it is UNREASONABLE that math should be so effective in the physical sciences. — fishfry

The vast majority of mathematical principles, like pi, and the Pythagorean theorem which we discussed, are not fictional, and that is why mathematics is so effective. So there is nothing UNREASONABLE in the effectiveness of mathematics. You can argue from ridiculous premises, as to what constitutes "true", and assume that the Pythagorean theorem is not true, as you did, but then it's just the person making that argument, who is being unreasonable.

If you drop a set near the earth, it doesn't fall down. Sets have no gravitational or inertial mass. They have no electric charge. They have no temperature, velocity, momentum, or orientation. In what sense are sets bound by the real world? — fishfry

The word "set" is a physical thing, which signifies something. And it only has meaning to a human being in the world, with the senses to perceive it. Therefore sets, as what is signified by that word, are bound by the real world.

except that -- stretching a point -- mathematical objects are products of the human mind and the human mind is bound by the laws of nature. So perhaps ultimately there's a physical reason why we think the thoughts we do. I'd agree with that possibility, if that's the point you're making. — fishfry

Now you're catching on. Consider though, that the physical forces of the real world are not the "reason why" we think what we do, as we have freedom of choice, to think whatever we want, within the boundaries of our physical capacities. The physical forces are the boundaries. So we really are bounded by the real world, in our thinking. We do not apprehend the boundaries as boundaries though, because we cannot get beyond them to the other side, to see them as boundaries, they are just where thinking can't go. Therefore it appears to us, like we are free to think whatever we want, because our thinking doesn't go where it can't go.

You will not understand the boundaries unless you accept that they are there, and are real, and inquire as to the nature of them. The boundaries appear to me, as the activity where thinking slips away andis replaced by other mental activities such as dreaming, and can no longer be called "thinking". We have a similar, but artificial boundary we can call the boundary between rational thinking and irrational thinking, reasonable and unreasonable. This is not the boundary of thinking, but it serves as a model of how this type of boundary is vague and not well defined.

Further, we have a boundary between conscious and subconscious, and this is closer to being a real representation of the boundary of thinking. The subconscious activity of the mind is not called "thinking", So dreaming is not thinking. You can see that the descriptions of these two types of boundaries are somewhat similar, being modeled on some form of coherency. A form of coherency marks the difference between reasonable and unreasonable, and a different type of coherency marks the difference between waking mental activity (thinking), and dreaming (mental activity outside the bounds of thinking). In the latter division, you ought to see clearly that it is the physical condition of the body (being asleep), which provides the boundary that we model with a description as to what qualifies as thinking, and what does not. However, there is a coherency or lack of it, within the mental activity, which corresponds with the physical boundary which we model as the difference between awake and asleep. In the case of reasonable/unreasonable, we have cases of physical illness, and intoxication, which demonstrate that the boundary, which is a boundary of coherency, has a corresponding physical condition.

So here's the point. We have mental activity which is thinking, and mental activity which cannot be classified as thinking. Therefore there must be a boundary to thinking which separates it from that other activity. The difference is described as a difference in coherency But, since there are real world physical differences which correspond with the described boundaries of coherency, I propose that it is the real world physical boundaries which impose upon our mental activity, the inclination to create corresponding mental boundaries of coherency. So for instance, the law of non-contradiction is a boundary of coherency. To violate the law is to put oneself outside a boundary of coherency. But the law of non-contraction is a statement of what we believe to be impossible, in the real physical world. So it is the acknowledgement of this physical impossibility, as being impossible, which substantiates the assumed mental boundary of coherent/incoherent.

Now, you are proposing a type of thinking "pure mathematics", which is not at all bounded by the real physical world. How could there possibly be such a thing? As I explained, the mental activity which is called thinking, is already bounded in order that it be separate from the mental activity which is not thinking, and there is a corresponding physical condition, being conscious, or awake, which provides the capacity for thinking. Since this physical condition is required for, as providing the capacity for, thinking, then thinking is necessarily bounded by the real physical world. A person cannot go in thought, beyond the capacity given to that person by one's physical body.

Suppose we allow that thinking might move past the boundaries of coherency, (which I admit we have created), to be not at all bounded by coherency. The problem is that there is a corresponding real world boundary which is responsible for the creation of the boundary of coherence. Do you allow that subconscious mental activity, and dreaming, are thinking? See, it makes no sense to say that thinking can go beyond the bounds imposed upon it by the real physical body which capacitates it.

Now suppose we say that there is a special type of thinking, "pure mathematics", which we give that privilege to. How can we even call this thinking? We create boundaries of coherency to define what "thinking" is, keeping "thinking" within the range of conscious mental activity, but now you want to allow a special type of thinking which is not bound by this rule. In reality, "pure mathematics" is a special type of thinking, so it has stricter binding of coherency than just thinking in general does. And, corresponding with those binds of coherency are features of the real physical world.

In math, violating the "fundamental principles" is how progress is made. — fishfry

I agree with this fully. But the need to violate fundamental principles just means that what was once considered to be a boundary of coherency can no longer be consider such. It does not negate the real physical boundaries, which the boundary of coherency was meant to represent. The boundary of coherency did not properly correspond, with the real world boundary, and therefore it needed to be replaced. The need to replace fundamental principles is evidence of faulty correspondence.

I've just shown that some of the greatest advances in math have been made by blowing up the opinions of the world. What happens is that the opinions of the mathematical world change. Or as Planck said, scientific progress proceeds one funeral at a time. Meaning that the old guard die off and the young Turks readily adopt the radical new ideas. — fishfry

Yes, yes, I think you truly are catching on. The need to change mathematical principles is a feature of poor correspondence. It is not the case that pure mathematics is thinking which is not bounded, because it truly is bounded, as described. But it is thinking which does not adequately understand the real world conditions which are its boundaries. It does not understand its boundaries, that's why it might even think as you do, that it is not bounded. Therefore it often poorly represents these boundaries, and when the boundaries become better understood, the representations need to be replaced. Understanding the true real world boundaries is what produces certainty.

Sure the pieces are made of atoms, but there is no fundamental physical reason why the knight moves that way. — fishfry

Yes there is a physical reason for this. The pieces cannot be floating in air, nor can they randomly disappear and reappear in other places, nor be in two places at once. There are real physical restrictions which had to be respected when the game was created. So a board and pieces, with specific moves which are physically possible, was a convenient format considering those restrictions.

This is the problem with your claim that mathematics is not bound by real world restrictions. You can assert that it is not, and you can create completely imaginary axioms, such as a thing with no inherent order, but when it comes to real world play (use of such mathematics) if these axioms contain physical impossibilities, it's likely to create problems in application. The creator of chess could have made a rule which allowed that the knight be on two squares at once. or that it might hover around the board. But then how could the game be played when the designated moves of the pieces is inconsistent with what is physically possible for those pieces?

I have full respect for the notion that mathematical axioms might be completely imaginary, like works of art, even fictional, but my argument is that such axioms would be inherently problematic when applied in real world play. You seem to think that it doesn't matter if mathematical axioms go beyond what is physically possible, and it's even okay to assume what is physically impossible like "no inherent order". And you support this claim with evidence that mathematics provides great effectiveness in real world applications. But you refuse to consider the real problems in real world applications (though you accept that modern physics has real problems), and you refuse to separate the problems from the effectiveness, to see that effectiveness is provided for by principles which are consistent with physical reality, and problems are provided for by principles which are inconsistent.

That math is inspired by the world and not bound by it? To me this is a banality, not a falsehood. It's true, but so trivial as to be beneath mention to anyone who's studied mathematics or mathematical philosophy. — fishfry

I've explained to you very clearly why it is false to say that mathematics is not bound by the real world. Perhaps if you would drop the idea that it is a banality, you would look seriously at what I have said, and come to the realization that what you have taken for a banal truth, and therefore have never given it any thought, is actually a falsity.

But it's still a formal game. — fishfry

I've gone through this subject of formalism already. No formalism, or "formal game", of human creation can escape from content to be pure Form. You seem to be having a very hard time to grasp this, and this is why you keep on insisting that there's such a thing as "pure abstraction". Content, or in the Aristotelian term "matter" is what is forcing real world restrictions onto any formal system. So when a formal system is created with the intent of giving us as much certainty as possible, we cannot escape the uncertainty produced by the presence of content, which is the real world restriction on certainty, that inheres within any formal system.

That's an interesting point. Yet you can see the difference between representational art, which strives to be "true," and abstract art, which is inspired by but not bound by the real world. Or as they told us when I took a film class once, "Film frees us from the limitations of time and space." A movie is inspired by but not bound by reality. Star Wars isn't real, but the celluloid film stock (or whatever they use these days) is made of atoms. Right? Right. — fishfry

Let's take this analogy then. Will you oblige me please to see it through to the conclusion? Let's say that abstract art is analogous to pure, abstract mathematics, and representational art is analogous to practical math. Would you agree that if someone went to a piece of abstract art, and started talking about what was represented by that art, the person would necessarily be mistaken? Likewise, if someone took a piece of pure abstract mathematics, and tried to put it to practice, this would be a mistake.

Bear in mind, that I am not arguing that what we commonly call pure math, ought not be put to practice, I am arguing that pure math as you characterize it, as pure abstraction, is a false description. In other words, your analogy fails, just like the game analogy failed, the distinction between pure math and practical math is not like the distinction between abstract art and representational art.

I recognize the difference between pure and applied mathematics. And you seem to reject fiction, science fiction, surrealist poetry, modern art, and unicorns. Me I like unicorns. They are inspired by the world but not bound by it. I like infinitary mathematics, for exactly the same reason. Perhaps you should read my recent essay here on the transfinite ordinals. It will give you much fuel for righteous rage. But I didn't invent any of it, Cantor did, and mathematicians have been pursuing the theory ever since then right up to the present moment. Perhaps you could take it up with them. — fishfry

I do not reject fiction, I accept it for what it is, fiction. I do reject your claim that pure mathematics is analogous to fiction. Here is the difference. In fiction, the mind is free to cross all boundaries of all disciplines and fields of education. In pure mathematics, the mathematician is bound by fundamental principles, which are the criteria for "mathematics", and if these boundaries are broken it is not mathematics which the person is doing. And, these boundaries are not dreamt up and imposed by the imagination of the mathematician who is doing the pure mathematics, they are imposed by the real world, (what other people say about what the person is doing), which is external to the pure mathematician's mind. This is why it is false to say that pure mathematics is not bound by the real world. If the person engaged in such abstraction, allows one's mind to wander too far, the creation will not be judged by others (the real world) as "mathematics". Therefore if the person wanders outside the boundaries which the real world places on pure mathematics, the person is no longer doing mathematics.

edit: This again is the issue of content. If the content is not consistent with what is judged as the content of mathematics, then the person working with so-called "pure" abstractions cannot be judged as doing mathematics. Therefore the abstractions cannot be "pure" as there are restrictions of content, as to what qualifies as mathematics.

The concept of infinite infinities is already part of mathematics today. Therefore, in your dubious distinction between mathematics and “imaginary fictions”, your placement of infinite infinities on the side of "imaginary fictions" makes no sense; infinite infinities is already on the side of mathematics. Your attempted stipulations to the contrary are pointless. — Luke

My argument is that such things are wrongly called mathematics, due to faulty conventions which allow imaginary fictions, cleverly disguised to appear as mathematical principles, to seep into mathematics, taking the place of mathematical principles. And obviously, it's not a stipulation but an argument, as I've spent months arguing through examples.

How can you argue with the truth of things that are not claimed to be true? Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY says that. — fishfry

I don't see why this is a problem for you. You hand me a proposition, and I refuse to accept it, claiming that it is false. You say, 'but I am not claiming that it is true'. So I move to demonstrate to you why I believe it to be false. You still insist that you are not claiming it to be true, and further, that the truth or falsity of it is irrelevant to you. Well, the truth or falsity of it is not irrelevant to me, and that's why I argue it's falsity hoping that you would reply with a demonstration of its truth to back up your support of it.. If the truth or falsity of it is really irrelevant to you, then why does it bother you that I argue its falsity? And why do you claim that I cannot argue the falsity of something which has not been claimed to be true? Whether or not you claim something to be true, in no way dictates whether or not I can argue its falsity.

It's entirely analogous. Chess is a formal game, there's no "reason" why the knight moves as it does other than the pragmatics of what's been proven by experience to make for an interesting game. And there are equally valid variations of the game in common use as well. — fishfry

Either you are not getting the point, or you are simply in denial. Playing chess, is a real world activity, as is any activity. Your effort to describe an activity, like the game of chess, or pure math, as independent from the real world, as if it exists in it's own separate bubble which is not part of the world, is simply a misrepresentation.

Now, you admit that there actually is a pragmatic reason why the knight moves as it does, and this is to make an interesting game. And of course playing a game is a real world activity. so there is a real world reason for that rule. Now if you could hold true to your analogy, and admit the same thing about pure mathematics, then we'd have a starting point, of common agreement. However, the reason for mathematical principles beings as they are, such as our example of the Pythagorean theorem, is not to make an interesting game. It is for the sake of some other real world activity. Do you agree?

Of course math is inspired by the world. It's just not bound by it. A point I've made to you a dozen times by now. — fishfry

You keep on insisting on such falsities, and I have to repeatedly point out to you that they are falsities. But you seem to have no respect for truth or falsity, as if truth and falsity doesn't matter to you. Mathematics has been created by human beings, with physical bodies, physical brains, living in the world. It has no means to escape the restrictions imposed upon it by the physical conditions of the physical body. Therefore it very truly is bound by the world. Your idea that mathematics can somehow escape the limitations imposed upon it by the world, to retreat into some imaginary world of infinite infinities, is not a case of actually escaping the bounds of the world at all, it's just imaginary. We all know that imagination cannot give us any real escape from the bounds of the world. Imagining that mathematics is not bound by the world does not make it so. Such a freedom from the bounds of the world is just an illusion. Mathematics is truly bound by the world. And when the imagination strays beyond these boundaries, it produces imaginary fictions, not mathematics. But you do not even recognize a difference between imaginary fictions, and mathematics.

News to me. — fishfry

The reason why I can truthfully say that our discussion has never been about how math works, is that you have never given me any indication as to how it works. You keep insisting that mathematical principles are the product of some sort of imaginary pure abstraction, completely separated from the real world, like eternal Platonic Forms, then you give no indication as to how such products of pure fiction become useful in the world, i.e., how math works.

Wow! I am really impressed to realize Omar Khayyam (1048-1131) had the perspicacity to realize his efforts at Non-Euclidean geometry involved notions of space-time. Thanks, MU. I would not have guessed. :chin: — jgill

As I said, one can produce any sort of geometry depending on the particular purpose. My reference to space-time was in reply to fishfry's talk of a specific incidence, the use of non-Euclidian geometry in modern physics

And I'm insane to question the metaphysics of a person who rebukes the skeptic with 'it's science therefore your demonstrations of deficiency are irrational unless your a scientist' It doesn't take a scientist to understand metaphysics..

I'm not arguing my point of view is right, I'm not even arguing a point of view. I'm telling you how modern math works. It's like this, if you don't mind a Galilean dialog. — fishfry

This is not true. You've been making arguments about "pure math", and "pure abstractions". So it is you who is making a division between the application of mathematics, "how modern math works", and pure mathematics, and you've been arguing that pure mathematics deals with pure abstractions. You've argued philosophical speculation concerning the derivation of mathematical axioms through some claimed process of pure abstraction, totally removed from any real world concerns, rather than the need for mathematics to work. So your chess game analogy is way off the mark, because what we've been discussing here, is the creation of the rules for the game, not the play of the game. And, in creating the rules we must rely on some criteria.

Nowhere do I dispute the obvious, that this is "how modern math works". That is not our discussion at all. What I dispute is the truth or validity of some fundamental principles (axioms) which mathematicians work with. This is why the game analogy fails, because applying mathematics in the real world, is by that very description, a real world enterprise, it is not playing a game which is totally unrelated to the world. So the same principle which makes playing the game something separate from a real world adventure, also makes it different from mathematics, therefore not analogous in that way

I have a proposal, a way to make your analogy more relevant. Let's assume that playing a game is a real world thing, *as it truly is something we do in the world, just like scientists, engineers and architects do real world things with mathematics. Then let's say that there are people who work on the rules of the game, creating the game and adjusting the rules whenever problems become evident, like too many stalemates or something like that. Do you agree that "pure mathematicians" are analogous to these people, fixing the rules? Clearly, the people fixing the rules are not in a bubble, completely isolated from the people involved in the real world play. Of course not, they are working on problems involved with the real world play, just like the "pure mathematicians" are working on problems involved with the application of math in science and engineering, etc.. Plato described this well, speaking about how tools are designed. A tool is actually a much better analogy for math than a game. The crafts people who use the tool must have input into the design of the tool because they know what is needed from the tool.

In conclusion, your claim that "pure mathematicians" are completely removed from the real world use of mathematics is not consistent with the game analogy nor the tool analogy. Those who create the rules of a game obviously have the real world play of the game in mind when creating the rules, so they have a purpose and those who design tools obviously have the real world use of the tool in mind when designing it, and the tool has a purpose. So if mathematics is analogous, then the pure mathematicians have the real world use of mathematics in mind when creating axioms, such that the axioms have a purpose.

Modern math is what it is, and nothing you say changes that, nor am I defending it, only reporting on it. — fishfry

The problem is that you have been "reporting" falsely. You consistently claimed, over and over again, that "pure mathematicians" work in a realm of pure abstraction, completely separated, and removed from the real world application of mathematics, and real world problems. That is the substance of our disagreement in this thread. My observations of things like the Hilbert-Frege discussion show me very clearly that this is a real world problem, a problem of application, not abstraction, which Hilbert was working on. And, the fact that Hilbert's principles were accepted and are now applied, demonstrates further evidence that Hilbert delivered a resolution to a problem of application, not a principle of pure abstraction.

It was forced on math by the discovery of non-Euclidean geometry. Once mathematicians discovered the existence of multiple internally consistent but mutually inconsistent geometries, what else could they do but give up on truth and focus on consistency?

I'm curious to hear your response to this point. What were they supposed to do with non-Euclidean geometry? Especially when 70 years later it turned out to be of vital importance in physics? — fishfry

I can't say that I see what I'm supposed to comment on. The geometry used is the one developed to suit the application, it's produced for a purpose. With the conflation of time and space, into the concept of an active changing space-time, Euclidean geometry which give principles for a static unchanging space, is inadequate. Hence the need for non-Euclidean geometry in modern physics.

It's not good or bad, it is simple inevitable. What should math do? Abolish Eucidean or non-Euclidean geometry? On what basis? — fishfry

This is why there is a need for solid ontological principles, an understanding of the real nature of time, the real nature of space. Only through such an understanding will the proper geometry be developed.
This is why it makes no sense to place the "pure mathematician" in a completely separate realm of "pure abstraction". The "pure mathematician" could dream up all sorts of different geometries, and have none of them any good for any real purpose, if the "pure mathematician" had absolutely no respect for the real nature of space.

As evidence I give you "The unreasonable effectiveness of math etc." — fishfry

Sorry fishfry, but this is evidence for my side of the argument. "The unreasonable effectiveness of math" is clear evidence that the mathematicians who dream up the axioms really do take notice, and have respect for real world problems. That's obviously why math is so effective. If the mathematicians were working in some realm of pure abstraction, with total disregard for any real world issues, then it would be unreasonable to think that they would produce principles which are extremely effective in the real world. Which do you think is the case, that mathematics just happens to be extremely effective in the real world, or that the mathematicians who have created the axioms have been trying to make it extremely effective?

You've given me not the slightest evidence that you have any idea how math works. And a lot of evidence to the contrary. — fishfry

Our discussion, throughout this thread has never been about "how math works". We have been discussing fundamental axioms, and not the application of mathematics at all. You are now changing the subject, and trying to claim that all you've been talking about is "how math works", but clearly what you've been talking about has been pure math, and pure abstraction, not application.

Since you're refusing to address the issue, I'll take the time to characterize you, as you did me.

You present us with bad metaphysics, and call it "science", so that when a metaphysician demonstrates the flaws in your metaphysics, you can dismiss it all by saying that the metaphysician has no understanding of that field.

Seems you're incapable of sticking to the topic. Why? Is it because you actually know how completely ridiculous, and completely unscientific, your description of inflation between spatial points is, so you're eager to change the subject?

Until you can empirically demonstrate any real points in space, with inflation between them, I'll continue to assume an infinite number of points between any two points in space, therefore inflation necessarily at an infinite speed.

The best you can do is say "normal rules for inertial frames (including universal speed limits) don't apply". What hurts like a kick in the balls, is that you insist this is science. Oh yeah, science is the discipline where rules don't apply! Have you absolutely no respect for true science? Calling this crap science is the worst disrespect for science that I've ever seen. Call it what it is, will you please, "metaphysics". You will not though, because you know it would be rejected by educated metaphysicians, as terrible metaphysics.

I'm curious where you came across this. I've only seen it once, from Eva Brann, but don't recall if she cited any supporting evidence. — Fooloso4

I've read Plato, Aristotle, and other ancient Greek philosophy. Plato had "the good", and descriptions of how the mind, reason, must prevail over the material body, and Aristotle developed final cause as 'that for the sake of which', the end, but there was not yet a concept of "free will" in the modern sense. In the modern sense, "free will" is the source of activity in an intentional act.

I think the "Passion" of Christ refers in the first place to the suffering of Christ from late Latin passio "suffering, experience of pain". Though, I guess you can use it in the sense of "strong will" if you want to. — Apollodorus

I believe that the Latin passio means equally "enduring" as it means "suffering". So we cannot focus only on the suffering aspect, but we must also consider the enduring aspect. The Passion of Jesus was not a simple moment of suffering, but an extended period which he choose to endure. This is what will power is all about, to endure suffering for the sake of a higher good.

Yeah, I'm using the word an insane amount of times. But in this case, I just meant that you're quite mad. — Kenosha Kid

Sad preacher nailed upon the coloured door of time
Insane teacher be there reminded of the rhyme
There'll be no mutant enemy we shall certify
Political ends, as sad remains, will die
Reach out as forward tastes begin to enter you
— Yes, And You and I

Modelling, hypothesis, observation: so far, so scientific, not to mention that inflationary cosmology comes from scientific research groups, not philosophical ones, and the founders of the theory have won prizes for breakthroughs in science, not metaphysics. — Kenosha Kid

Those are not observations. They are interpretations made through the application of dubious theories. I know that you don't agree that the theories are dubious, but there is no reason to believe that general relativity is applicable toward understanding inflationary theory. Anomalies such as "dark energy" and "dark matter" demonstrate the inapplicability of the these theories which are applied in the interpretation of the observations.

So on that level, calling it metaphysics not science is insane — Kenosha Kid

I don't know why you think that calling cosmology "metaphysics", which is the conventional norm as I demonstrated with Wikipedia quote, is insanity. But I think that calling an hypothesis "science", when the hypothesis is not at all consistent with observations, as the need to assume mystical, magical entities like "dark matter" and "dark energy" demonstrates, is worse than insanity, it's intellectual dishonesty. Calling your metaphysics "science" and trying to back up that claim with faulty interpretations of observations, is nothing other than deception.

You are insane. — Kenosha Kid

Right, we know how you use that word "insane": " the 'inflationary period', while brief, was insanely rapid". When you don't understand something you designate it as "insane". You do not understand me therefore I am insane.

Since you have absolutely no idea as to any of the specifics concerning this "insanely rapid" expansion, it makes no sense for you to call this "science". Clearly, from your description, there might be an infinite number of spatial points, each with an infinite number of spatial points between them, with each spatial point receding from every other, at some absolutely arbitrarily designated speed. The designation of a speed is really irrelevant because if there's an infinity of these points to even the smallest parcel of space, then the speed of inflation is necessarily infinite.

Cosmology – a central branch of metaphysics, that studies the origin, fundamental structure, nature, and dynamics of the universe. — Wikipedia: Outline of Metaphysics

Inflationary theory, as you describe it, Kenosha Kid, is not science, but extremely bad metaphysics.

The spirited part of the tripartite soul in the Republic, for example, is not spiritual in the sense I think you are using the term. — Fooloso4

All perfectly sound, but note that your definition of it is given in a specific context, or domain of discourse, rather than an attempt to define the term 'spirit' in a general sense. — Wayfarer

Right, because Fooloso4 (sorry, I said Apollodorus) had mentioned Plato's tripartite soul as context, in the passage which your reply was in response to, indicated above. The op concerns Plato's cave allegory, and the idea of the tripartite soul as presented in The Republic is very important to grasping the reality of the intelligible realm As the medium between body and mind, this third aspect, passion or spirit, is the means by which dualism escapes the common charge of an interaction problem. The interaction problem is a strawman which monist materialists hold up in a feeble attempt to ridicule dualists.

Further, this is why "the good" becomes so important. The good is what is desired or wanted, and is what actually moves the will, if the will is not allowed to be free from pragmatic influences. ("Will" did not even exist as a philosophical concept at Plato's time, though the underlying principles were being exposed by Plato, becoming a commonly used philosophical term at a later time, around Augustine's exposition on free will).

The problem of the free will, is that while the will moves us to act, it may be bound to material desires and bodily habits, or we may free it from following such habits, allowing it to follow the reasoning of the intellect leading us toward truth and understanding. We may even allow the will enough freedom to contemplate the highest intellectual principles. But intellectual principles, immaterial objects themselves, must be judged for truth or falsity, by the reasoning mind, according to the skepticism of the Socratic method, or Platonic dialectics, and this judgement is itself an act of will. Therefore It is of paramount importance that the soul allow the will complete freedom from the influence of the material body (including even the brain activity which would be composed of acquired habits of thinking), in making those judgements.

We must escape the trap of what is known as "rationalizing". This is why pragmaticism cannot be accepted as providing first principles, because it has been oriented toward 'what works for giving us material luxury' rather than a true honest understanding of "good" which exposes this deficiency.

The reason I don't like 'spiritual' is because of its many different uses, and also the different and sometimes conficting meanings of 'spirit' — Wayfarer

This is why I prefer "passion" when speaking of Plato's tripartite soul. it is much more specific, and I believe completely consistent with what Plato had in mind, strong ambition and enthusiasm, which under the right guidance of a reasonable good, is a prerequisite for achieving the end. But without the guidance of reason, passion becomes contemptuous anger, or unruly lust Also, it is the same word used to describe the "Passion" of Christ, referring to the very strong will of Jesus, to proceed and continue in his course of action intended to deliver us from a corrupted spirituality.

It is what the education in music is supposed to moderate. — Fooloso4

That's right, this is why music becomes a very important aspect of Plato's republic. Nothing stirs the emotions (passion) like music does, and it is understood by Plato that different types of music might lead passion in different directions. So it is proposed that music be used to help direct passion.

I also am ambivalent in respect of the word 'spiritual'. The terms I'm familiar with are psyche, nous, and logos. — Wayfarer

I think what Apollodorus was referring to is the position of "spirit", sometimes translated as "passion" (which I prefer), in Plato's tripartite soul. Passion, or spirit, takes the position of intermediary between body and mind or reason. In the well-ordered, healthy soul, passion allies with reason to exercise control over the body, But in the unhealthy soul, passion is swayed by bodily functions, resulting in an unreasonable mind.

This is important in Plato's comparison between the composition of an individual person, and the composition of the State. His State has three parts just like the human being, being designed after the tripartite being. There is the ruling class, the guardians, and the artisans or trades people (working class I suppose). The guardians are the middle, highly bred like dogs, high-spirited warriors to defend the interest of the State, having great honour, loyalty, and allegiance to the rulers. But in time, as the State starts to degenerate (the cause of degeneration being something to do with numbers), the guardians come to see honour, bravery, courage, ambition, as the highest thing in itself, falling away from the noble and good principles of reason. which are actually higher. Then the guardians are no longer guided by the noble principles of the intellect, they have no allegiance to the rule of reason, and so they are swayed by money, and the material goods of the lower part of the State. This is the unhealthy State.

You can see how the three parts of the human soul, reason, passion or spirit, and body, are comparable to the three parts of the State, rulers guardians, and artisans.

When I was in my early teens, no one at school spoke of “Platonism”. It was always individual authors like Plato, Aristotle, Plotinus. So, when I first read Plato’s dialogues like Timaeus, Symposium, Republic, I was unaware of the existence of a system called “Platonism”. — Apollodorus

We didn't get any education in philosophy in high school, so I wasn't exposed to Plato or Platonism until university.

If we insist that there were major changes, for example, from Plato to Plotinus, we should be able to show what those changes are and to what extent (if at all) they are inconsistent with (a) the text of the dialogues and (b) with how Plato was understood in the interim. — Apollodorus

What I find, is that in Plato's dialogues, Socrates produces unanswered questions. So if Plotinus made some progress toward answering some of those questions, that would constitute a change between Plato and Plotinus.

And the focus of that way of life, at least within the Academy, was the positive construction of a theoretical framework on the foundation of UP. — Apollodorus

I had to do a Google search to find out what Ur-Platonism is:

;

Here I briefly sketch a hypothetical reconstruction of what I shall call ‘Ur-Platonism’ (UP). This is the general philosophical position that arises from the conjunction of the negations of the philosophical positions explicitly rejected in the dialogues, that is, the philosophical positions on offer in the history of philosophy accessible to Plato himself. — Platonism Versus Naturalism, Lloyd P. Gerson, University of Toronto

I really do not see how a "general philosophical position that arises from the conjunction of the negations
of the philosophical positions explicitly rejected in the dialogues", can be called "a theoretical framework". I think these two are miles apart. A position of skepticism, which rejects philosophical positions, cannot be said to provide a theoretical framework. So any supposed theoretical framework would have to come from some principles other than those found in Plato.

We might say that Aristotle build a theoretical framework on UP, but we wouldn't call Aristotelian metaphysics Platonism, it's Aristotelianism.

I'm very familiar with it. But cosmology is metaphysics, not science.

Well, much more than 37bn. Part of inflation theory is that the universe must be much, much larger than the observable universe. However, no magic necessary, just counting. 2c for two adjacent points. Next add a third. You have points A, B and C in a row. A is receding from B at almost the speed of light. B is receding from C at roughly the same speed. How fast is A receding from C? — Kenosha Kid

Don't forget that we might assume an infinite number of points between any two spatially separated points, then we really do have an insanely fast separation. More like a terrible hypothesis though. Anyway, this stuff is not even science at all, so it shouldn't be presented as an example of science, or, @counterpunch.an example of scientific failure.

There's no empirical evidence which indicates that the separation between two supposed points, due to expansion, is limited to c. We don't even know how to find the points which are supposed to be separating from each other. So we might assume an infinite number of such points within any volume of space, and then we have nonsense, Scientists don't even have a vague idea as to how light moves through space which is not supposed to be expanding, so how are they ever going to make theories about how light moves in a space which is composed of points moving rapidly away from each other in all directions? Maybe if they weren't so quick to reject ether theories, the expanding points could be the vibrating particles of a substance.

These issues were not completely resolved in Plato's times and had to be worked out later. — Apollodorus

Actually, if you look really closely, you'll see that the issues haven't been resolved yet. I think you have something to work on Apollodorus. Get back to your studies!