I'll take Sextus Empiricus and Pyrrhonism, but it's doubtful whether skepticism can be classed as a form of wisdom. Perhaps it is better classified as a way toward wisdom.

But this would cast a shadow of doubt on all the methods, techniques, or practises, as to whether they are actually wisdom, or ways toward wisdom. If wisdom is what is produced from the practice, therefore something other than the practise itself, it would be an end to which the practise is a means. Then we need to be able to judge the various practises themselves as to efficacy for obtaining that end. This requires that we have a determination as to what wisdom itself is, as something separate from the practise, which is observed to be the result of the practise. Otherwise we can list all sorts of practises with no criteria as to how they are related to "wisdom".

Actually I think carbon dioxide was first, and more natural to the planet. It took many years of plant forms producing O2 through photosynthesis before there was sufficient free oxygen for the ozone layer, and higher life forms.

So I don't think your depiction of a cyclical dependence is very accurate.

Anyone mention Augustine of Hippo yet? He's probably one of the wisest human beings to have ever lived. Augustine's writings will never grow old, and like Plato's, they are very relevant today.

There's something about modernity that is inimical to the traditional idea of wisdom. — Wayfarer

The modern attitude, generally, is that any old knowledge, especially if it has a theological base, has been thoroughly supplanted by modern scientific advancements. So anything old is seen as outdated and incorrect, having been replaced by the new knowledge.

However, wisdom is composed not only of knowledge, but also of experience. And experience is produced from the temporal extension of being. So a large part of wisdom is understanding the principles which have stood the test of time.

A contradiction is the conjunction of a statement and its negation. — TonesInDeepFreeze

Like for example, when someone says "an empty set has no elements", and also says "the elements of the empty set A are the same elements as the elements of the empty set B". The former says the empty set has no elements, the latter states its negation "the elements of the empty set...".

Or, even if one were to say that an empty set has no elements, absolute and not contingent, and then states the conditional "if X is an element of the empty set...", as if the set is only contingently empty, that would also constitute a statement and its negation.

Don't you agree?

A philosophy crank is more like it. You have zero familiarity with the 20th century literature on the philosophy of set theory. You haven't read Maddy, Quine, or Putnam. You have no interest in learning anything about the philosophy of set theory. When I mentioned to you the other day that Skolem was skeptical of set theory as a foundation for math, you expressed no curiosity and just ignored the remark. Why didn't you ask what his grounds were? After all he was one of the major set theorists of the early 20th century. — fishfry

If some of this is relevant to the points I've made, then provide some quotes or references. Otherwise what's the point in mentioning something which is not relevant?

Mine is the perfectly standard interpretation, comprehensible to everyone who spends a little effort to understand it. Two sets are the same if and only if they have the same elements. Formally, if a thing is in one set if and only if it's in the other; which (as we will shortly see) includes the case where both sets are empty. — fishfry

You haven't addressed the point. To have "the same elements" requires a judgement of elements. Having no elements is not an instance of having elements, and there are no elements to be judged. An empty set has no elements therefore two empty sets do not have the same elements, because they both have no elements. Therefore two empty sets are not the same.

Are you saying that two things can be "the same" but not equal? Are you sure whatever you're on is legal in your jurisdiction? — fishfry

We've been through the law of identity before, and you still show no desire to understand it.. No two things are the same, according to that law. If it's "the same", then there is only one thing. That's what "the same" refers to according to the law of identity, one and the same thing. The law of identity dictates that we use "same" to refer to only one thing, so it is impossible that two distinct things are the same. However, two distinct things may be equal. Therefore "equal" is not synonymous with "the same".

In other words "If 2 + 2 = 5 then I am the Pope" is a true material implication. Do you understand that? Do you agree? Do you have a disagreement perhaps? — fishfry

Sorry fishfry, but you'll need to do a better job explaining than this. Your truth table does not show me how you draw this conclusion.

Now I claim that for all XX, it is the case that X∈A⟺X∈BX∈A⟺X∈B. That is read as, "X is an element of A if and only if X is an element of B.

In sentential logic we break this down into two propositions: (1) If X is an element of A then X is an element of B; and (2) If X is an element of B then X is an element of A."

Now for (1). If X is a pink flying elephant, then it's a person on the moon. Is that true? Well yes. There are no pink flying elephants and there are no people on the moon. So this is line 4 of the truth table, the F/F case, which evaluates to True. So (1) is true. — fishfry

You have already determined that there are no elements in both sets A and B. This is predetermined, they are empty sets. So your starting point, "(1) If X is an element of A then X is an element of B" is not relevant, there are not elements. That's like saying if C and D are both green, when you've already determined that they are not green. It's an irrelevant premise, and your entire appeal to material implication is unacceptable. We already know that there are no elements of both A and B, so that premise concerning the elements of A is not applicable. The two premises "A is an empty set", and "if X is an element of A" are fundamentally contradictory.

I'll give you credit for at least addressing the point now. It was a nice try, but your attempt is a failure.

I just gave you a formal proof to the contrary. An object is in A if and only if it's in B. If A and B happen to be empty, that is a true statement. — fishfry

As explained above, it's actually an irrelevant, and inapplicable statement. And it can only be applied under contradiction. When a set has been determined as empty, then to talk about objects within that set is contradiction. So an attempt to apply this statement to empty sets is contradictory. Look at what you're saying 1)There are not any objects in set A. 2) An object is in A if... See the contradiction? When you've already designated A as having no objects, how does it make sense to you to start talking about the condition under which there is an object in A? Do you agree that there is contradiction here?

As you can see, your attempt at a formal proof is a failure due to contradicting premises.

You interpret your own ignorance as deception by others. Pretty funny. — fishfry

The thing with this type of deception, is that you can either recognize it as deception, and reject it, or you can join it, and become one of the deceivers. This is why mathematics is similar to religion (Tones will disagree), the authors have good intentions, but once falsity is allowed into the premises, deception is required to maintain respect for the premises amongst the masses. When the deception has been pointed out to you, as I have, then you can either reject it and work toward dismantling the system which propagates it, or you can support it with further deception. You it appears, are choosing to be one of the deceivers.

For purposes of this discussion, we take the two predicates as absolute and not contingent. You're just raising this red herring to sow confusion. The only one confused here is you. — fishfry

This thoroughly supports my argument. If the empty sets are necessarily empty, absolute and not contingent, then to talk about the conditions under which there are elements in those sets is very clearly contradiction.

That's what infinity in mathematics gives us, ungainly lacunas.

So, we're, in our "ordinary" lives, stuck with rules that are neither justified to our satisfaction nor universal in scope. — TheMadFool

Maybe consider that those ordinary concepts are not composed of rules at all. It's possible that when we see non-ordinary concepts like mathematics as composed of rules, through a faulty extrapolation we wrongly conclude that ordinary concepts are also composed of rules.

What if, in keeping with Wittgenstein's ludological analogy, rules are more about making the "game" more fun, more interesting and less about justification? In other words, rules don't need to be justified in that they have to make sense, instead they have to ensure the "game" is enjoyable, exciting, and pleasurable but also "painful" enough to, ironically, make the "playing" the "game" a serious affair. — TheMadFool

Living is like that, enjoyable, exciting, pleasurable, and painful. Living is not a game though, nor is it composed of rules. And I don't think rules are necessarily about making life more fun, they are about obtaining ends, goals. Although this is one way of making life more fun..

. It's exactly like chess. I'm teaching you the rules. If you don't like the game, my response is for you to take up some other game more to your liking. — fishfry

I'm a philosopher, my game is to analyze and criticize the rules of other games. This is a matter of interpretation. If you do not like that, then why are you participating in a philosophy forum?

As much as you, as a mathematician are trying to teach me some rules of mathematics, I as a philosopher am trying to teach you some rules of interpretation. So the argument goes both ways, you are not progressing very well in developing your capacity for interpreting. But if you do not like the game of interpretation, then just do something else

But I have already explained to you in my previous post, that "a set is characterized by its elements" is merely an English-language approximation to the axiom of extentionality, which actually says, — fishfry

This is why the axiom of extensionality is not a good axiom. It states something about the thing referred to by "set", which is inconsistent with the mathematician's use of "set", as you've demonstrated to me.

That is the axiom that says that two sets are equal if they have exactly the same elements. And by a vacuous argument -- the same kind of argument that students have had trouble with since logic began -- two sets are the same if they each have no elements. — fishfry

We've already been through this problem, a multitude of times. That two things are equal does not mean that they are the same. That's why I concluded before, that it's not the axiom of extensionality which is so bad, but your interpretation of it is not very good. But I now see that the axiom of extensionality is itself bad.

The formal symbology is perfectly clear. And even if it isn't clear to you, you should just accept the point and move on, so that we can discuss more interesting things. — fishfry

In case you haven't noticed, what I am interested in is the interpretation of symbols. And obviously the symbology of the axiom is not perfectly clear. If you can interpret "=" as either equal to, or the same as, then there is ambiguity.

What's true is that given any thing whatsoever, that thing is a pink flying elephant if and only if it's a person on the moon. So the axiom of extensionality is satisfied and the two sets are equal. If you challenged yourself to work through the symbology of the axiom of extentionality this would be perfectly clear to you. — fishfry

Actually, I'm starting to see that this, what you claim in your vacuous argument, is not a product of the axiom of extensionality, but a product of your faulty interpretation. By the axiom of extensionality, a person on the moon is equal to a pink flying elephant, and you interpret this as "the same as". So the axiom is bad, in the first place, for the reasons I explained in the last post, and you make it even worse, with a bad interpretation.

The axiom of specification allows us to use a predicate to form a set. The predicate is not required to have a nonempty extension. — fishfry

You really do not seem to be getting it. If, we can "use a predicate to form a set" as the axiom of specification allows, then it is not true that a set is characterized by its elements. It's characterized by that predication. The two are mutually exclusive, inconsistent and incompatible. Specification allows for a nonempty set, I have no problem with this. But to say that this set is characterized by its elements is blatantly false. It has no elements, and it is characterized as having zero elements, an empty set. So it's not characterized by its elements, it's characterized by the number of elements which it has, none. .

Of course not. It's not a matter of cardinal equivalence. The elements themselves have to be respectively equal. — fishfry

Yes, this is the problem with the axiom of extension, in its portrayal of the empty set. It is saying that if two specified sets each have zero elements, then "the elements themselves" are equal. However, there are no such elements to allow one to judge the equality of them. So there is no judgement that "the elements themselves" are equal, because there are no elements to judge, and so the judgement of cardinal equivalence, that they have the same number of elements, zero, is presented as a judgement of the elements themselves.

You ought to recognize, that to present a judgement of cardinal equivalence, as a judgement of the elements themselves, is an act of misrepresentation, which is an act of deception. I know that you have no concern for truth or falsity in mathematical axioms, but you really ought to have concern for the presence of deception in axioms.

Of course not. It's not a matter of cardinal equivalence. The elements themselves have to be respectively equal. {1,2} and {1,2} are the same set. {1,2} and {3,47} are not. — fishfry

Now, do you agree, that when there are no elements, it makes no sense to say that the elements themselves are respectively equal? What is really being judged as equal is the cardinality. They both have zero elements.

The axiom of extensionality tells us when two sets are the same. — fishfry

No, the axiom of extensionality does not tell us when two sets are the same, that's the faulty interpretation I've pointed out to you numerous times already, and you just cannot learn. It tells us when two sets are equal.

That faulty interpretation is what enables the deception. Equality always indicates a judgement of predication, and in mathematics it's a judgement of equal quantity, which you call cardinal equivalence. When you replace the determination of the cardinality of two empty sets, "equal", with "the same", you transfer a predication of the set, its cardinality, to make a predication of its elements, "the same as each other". I believe that's known as a fallacy of division.

You are right about that. But that's because we are making up examples from real life. Math doesn't have time or contingency in it. 5 is an element of the set of prime numbers today, tomorrow, and forever. The "people on the moon" example was yours, not mine. I could have and in retrospect should have objected to it at the time, because of course it is a temporally contingent proposition. I let it pass. So let me note for the record that there are no temporally contingent propositions in math. — fishfry

Well, "pink flying elephants" was your example, and it's equally contingent. The issue of temporally contingent propositions raises a completely different problem. The only truly necessary empty set is the one specified as "the empty set". As your examples of square circles and married bachelors show, definitions and conceptual structures change over time, so your assertion that mathematics has no temporally contingent propositions is completely untrue. It may be the case that "the empty set" will always refer to the empty set, necessarily, but how we interpret "empty" and "set" is temporally contingent. So temporal contingency cannot be removed from mathematics as you claim. This is the problem of Platonic realism, the idea that mathematics consists of eternal, unchanging truths, when in reality the relations between symbols and meaning evolves.

In the Aristotelian scheme, matter is characterized as potential, and form is actual. And of course both are real aspects of reality, with a qualification though, pure potential as prime matter, (matter without form). is demonstrated by the cosmological argument to be impossible. This point is sometimes debated because Aristotle lays out a lengthy description of what prime matter would be, if it were real, only to demonstrate that it cannot be real. If one does not grasp the cosmological argument it appears like Aristotle supports the reality of prime matter.

This says in effect that if two sets have exactly the same elements, they're the same set. But the way it's written, it also includes the case of a set with no elements at all. If you have two sets such that they have no elements, they're the same set; namely the empty set. — fishfry

You do not seem to be grasping the problem. If a set is characterized by its elements, there is no such thing as an empty set. No elements, no set. Do you understand this? That is the logical conclusion we can draw from " a set is characterized by its elements". If we have no elements, we have no set. If you do not agree with this, explain to me how there could be a set which is characterized by its elements, and it has no elements. It has no character? Isn't that the same as saying it isn't a set?

So we cannot proceed to even talk about an empty set because that's incoherent, unless we dismiss this idea that a set is characterized by its elements. Can we get rid of that idea? Then we could proceed to investigate your interpretation of the axiom of extensionality, which allows you to say "If you have two sets such that they have no elements, they're the same set; namely the empty set", because "empty set" would be a coherent concept. Until we get rid of that premise though, that a set is characterized by its elements there is no such thing as a set with no elements, because such a set would have no identity whatsoever, and we could not even call it a set.

Another point is that everyone has trouble with vacuous arguments and empty set arguments. If 2 + 2 = 5 then I am the Pope. Students have a hard time seeing that that's true. The empty set is the set of all purple flying elephants. A set is entirely characterized by its elements; and likewise the empty set is characterized by having no elements. John von Neumann reportedly said, "You don't understand math. You just get used to it." The empty set is just one of those things. You can't use your common sense to wrestle with it, that way lies frustration. — fishfry

You are not grasping the distinction between 'characterized by its elements', and 'characterized by its specification' which I'm trying to get though to you. When you say "the set of all purple flying elephants", this is a specification, and this set is characterized by that specification. There are no elements being named, or described, and referred to as comprising that set, there is only a specification which characterizes the set.

Every set is entirely characterized by its elements. — fishfry

Where do you get this idea from? Clearly your example "the set of all purple flying elephants" is not characterized by its elements. You have made no effort to take elements, and compose a set You have not even found any of those purple flying elephants. In composing your set, you have simply specified "purple flying elephants". Your example set is characterized by a specification, not by any elements. If you do not want to call this "specification", saving that term for some special use, that's fine, but it's clearly false to say that such a set is characterized by its elements.

This is what happens when we proceed deep into the workings of the imagination. We can take a symbol, a name like "purple flying elephants", or any absurdity, or logical incoherency, like "square circles", each of which we assume has no corresponding objects However, we can then claim something imaginary, a corresponding imaginary object, and we can proceed under the assumption that the name actually names something, a purple flying elephant in the imagination. You might then claim that this imaginary thing is an element which characterizes the set. But if you then say that the set is empty, you deny the reality of this imaginary thing, and you are right back at square one, a symbol with nothing corresponding. And so we cannot even call this a symbol any more, because it represents nothing.

The set of pink flying elephants is an empty set. The set of people on the moon is an empty set. And the axiom of extensionality says that these must be exactly the same set. — fishfry

Now you've hit the problem directly head on. To be able to have an empty set, a set must be characterized by it's specification, as I've described, e.g. "pink flying elephants". So. the set of pink flying elephants is one set, characterized by the specification "pink flying elephants", and the set of people on the moon is another set, characterized by the specification "people on the moon". To say that they are exactly the same set, because they have the same number of elements, zero, is nor only inconsistent, but it's also a ridiculous axiom. Would you say that two distinct sets, with two elements, are the exact same set just because they have the same number of elements? I think you'll agree with me that this is nonsense.

And to say that each of them has the very same elements because they don't have any, is clearly a falsity because "pink flying elephants" is a completely different type of element from "people on the moon". If at some point there is people on the moon, then the set is no longer empty. But the two sets have not changed, they are still the set of pink flying elephants, and the set of people on the moon, as specified, only membership has changed. Since the sets themselves have not changed only the elements have, then clearly they were never the same set in the first place.

Of course, you'll claim that a set is characterized by its elements, so it was never "the set of pink flying elephants in the first place, it was the empty set. But this is clearly an inconsistency because "pink flying elephants was specified first, then determined as empty. So that is not how you characterized these sets. You characterized them as "the set of pink flying elephants", and "the set of people on the moon".

If you had specified "the empty set", then obviously the empty set is the same set as the empty set, but "pink flying elephants", and "people on the moon" are clearly not both the same set, just because they both happen to have zero elements. The emptiness of these two sets is contingent, whereas the emptiness of "the empty set" is necessary, so there is a clear logical difference between them.

There is only one empty set, because the axiom of extensionality says that if for every object, it's a person on the moon if and only if it's a pink flying elephant, that the two sets must be the same. — fishfry

I don't know why you can't see this as a ridiculous axiom. You say that a "person on the moon" is a "pink flying elephant". That's ridiculous.

We know from the law of identity that everything is equal to itself. So what is the set of all things that are not equal to themselves? It's the empty set. And by the axiom of extensionality, it's exactly the same as the set of pink flying elephants and the people on the moon. — fishfry

See the consequences of that ridiculous axiom? Now you are saying that a pink flying elephant is a thing which is not equal to a pink flying elephant, and a person on the moon is not equal to a person on the moon. Face the facts, the axiom is nonsensical.

You are confusing the axiom of extension, which tells us when two sets are the same, with the other axioms that give us various ways to build sets or prove that various sets exist. — fishfry

Obviously, the axiom of extension is very bad because it fails to distinguish between necessity and contingency.

When, for example, Metaphysician Undercover repeatedly misunderstands certain notions in mathematics, there is a point at which one concludes that he is simply not participating in the game. One might then either turn away or attempt to follow the path of the eccentric. The question becomes one of what is to be gained in going one way or the other. — Banno

When. after repeated attempts, the rules are apprehended as impossible to understand, due to the appearance of inconsistency and incoherency, the best course for this person is not to participate in that game.

What's trivial is saying that the Vitali set is "specified" because all its elements are real numbers. That's like saying the guests at a particular hotel this weekend are specified because they're all human. It's perfectly true, but it tells you nothing about the guests at the hotel. That's why your point is trivial. — fishfry

It's not trivial, because it's a demonstration of what "specified" means. If you specify that the guests are all human, then clearly that is a specification. If you do not appreciate that specification because it does not provide you with the information you desire, then the specification is faulty in your eyes. But it's false to say that just because you think the specification is faulty, then there is no specification. There is a specification, but it is just not adequate for you. That is simply the nature of specification, it comes in all different degrees of adequacy, depending on what is required for the purpose. But an inadequate specification, for a particular purpose, is in no way a total lack of specification.

A set is entirely characterized by its elements. — fishfry

Do you see then, that if "A set is entirely characterized by its elements", then a so-called empty set is not possible? If there are no elements, under that condition, then there is no set. A set is characterized by its elements. There are no elements. Therefore there is no set. If we adhere to this premise, "the set is entirely characterized by its elements", then when there is no elements there is no set.

Some sets are specified by predicates, such as the set of all natural numbers that are prime. — fishfry

This is logically inconsistent with "a set is entirely characterized by its elements", as I explained in the last post. Either a set is characterized by its elements, or it is characterized by its specified predicates, but to allow both creates the incoherency which I referred to. One allows for an empty set, the other does not.

"By reference?" No. The Vitali set is characterized by its members, but I can't explicitly refer to them because I don't know what they are. It's a little like knowing that there are a billion people in China, even though I don't know them all by name. — fishfry

We've been through this already. You clearly have referred to the members of the Vitali set. You've said that they are all real numbers. Why do you believe that this is not a reference to the members of the set? You can say "all the people in China", and you are clearly referring to the people in China, but to refer to a group does not require that you specify each one individually.

This seems to be where you and I are having our little problem of misunderstanding between us. It involves the difference between referring to a group, and referring to individual. I believe that when you specify a group, "all the guests at the hotel" for example, you make this specification without the need of reference to any particular individuals. You simply reference the group, and there is no necessity to reference any particular individuals. In fact, there might not be any individuals in the group (empty set). You seem to think that to specify a group, requires identifying each individual in that group.

This is the two distinct, and logically inconsistent ways of using "set" which I'm telling you about. We can use "set" to refer to a group of individuals, each one identified, and named as a member of that set (John, Jim, and Jack are the members of this set), or we can use "set" to refer simply to an identified group, "all the people in China". Do you see the logical inconsistency between these two uses, which I am pointing out to you? In the first case, if there are no identified, and named individuals, there is no set. Therefore in this usage there cannot be an empty set. But in the second case, we could name the group something like "all the people on the moon", and this might be an empty set.

On the contrary. Since everything is equal to itself, the empty set is defined as {x:x≠x}{x:x≠x}. I rather thought you'd appreciate that, since you like the law of identity. The empty set is in fact the extension of a particular predicate. — fishfry

I must say, I really do not understand your notation of the empty set. Could you explain?

The empty set is the extension of the predicate x≠xx≠x. Or if you like, it's the extension of the predicate "x is a purple flying elephant." Amounts to the same thing. — fishfry

This doesn't help me.

Since the empty set is the extension of a particular predicate, your point is incoorect. — fishfry

Actually you don't seem to be getting my point. The point is that if a set is characterized by its predicates, then an empty set is possible, so I have no problem with "the empty set is the extension of a particular predicate". Where I have a problem is if you now turn around and say that a set is characterized by its elements, because this would be an inconsistency in your use of "set", as explained above. A set characterized by its elements cannot be an empty set, because if there is no elements there is no set. Do you apprehend the difference between "empty set" and "no set"?

I don't know what you're doing. i don't know what your point is.

...

I can't really follow your logic. — fishfry

Perhaps it's a bit clearer now?

If as you agree, all sets in standard set theory are composed of nothing but other sets; and that therefore every nonempty set whatsoever can be said to have elements that are sets; then isn't the fact that the elements of any set have in common the fact that they are sets, a rather trivial point? — fishfry

I don't see that as a trivial point, because not only is "set" undefined, but also "element" is undefined. So we have a vicious circle which makes it impossible to understand what type of thing a set is supposed to be, and what type of thing an element is supposed to be. What is a set? It's something composed of elements. What is an element? It's a set.

Under this description, a particular set is identified by its elements, not by a specification, definition, or description. Do you see what I mean? Under your description, any particular set cannot be identified by the predicates which are assigned to the elements, because it is not required that there be any assigned predicates.. But there still might be such an identified set. So a set must be identified by reference to its members. This is why, under this description of sets, the empty set is logically incoherent. A proposed empty set has no members, and therefore cannot be identified.

If, on the other hand, a set is identified by it's specification, definition, or description, (which you deny that it is), then there could be a definition, specification, or predication which nothing matches, and therefore an empty set.

Hopefully you can see that the two, identifying a set by its elements, and identifying a set by its predications, are incompatible, because one allows for an empty set, and the other does not. So as much as "set" may have no formal definition, we cannot confuse or conflate these two distinct ways of using "set" without the probability of creating logical incoherency.

By saying that "set" has no definition, we might be saying that there is nothing logically prior to "set", that we cannot place the thing referred to by the word into a category. But if you make a designation like "there is an empty set", then this use places sets into a particular category. And if you say that a set might have no specification, this use places sets into an opposing category. If you use both, you have logical incoherency.

Therefore it is quite clear to me, that the question of whether a set is identified by reference to its elements, or identified by reference to its specification, is a non-trivial matter because we cannot use "set" to refer to both these types of things without logical incoherency.

OED: specify, "to name or mention". Clearly the set you called "V" is not unspecified, and it's you who wants to change the meaning "specify" to suit your (undisclosed) purpose. Sorry fishfry, but you appear to be just making stuff up now, to avoid the issues.

I like the way that Plato introduces the idea of agency in relation to harmony, at 92c, where he has Socrates say: "How will you harmonize this statement with your former one?" Then by the middle of 93, he's right into the need for agency: "Does not the nature of each harmony depend on the way it has been harmonized?"

I understand Pythagorean cosmology to have been very scientifically advanced for the time. I think they promoted the idea that the entire cosmos consisted of waves or vibrations in an ether, and the various existents were harmonies in the vibrations. Anyway, the cosmos was understood to be highly ordered, as consisting of harmonies. I believe Plato has done a very good job arguing that such an ordered system of harmonies requires agency for its creation. The fact that agency was implied, but not accounted for, was a serious flaw in the Pythagorean cosmology So it had to be dismissed, and the Neo-Platonists produced a replacement cosmology which allowed for the reality of agency.

All multiverse theories fail at their core, because they are pure speculation without evidence. — Philosophim

That's what I see as the principal issue. Evidence is derived from "our" universe, and we generally do not allow conflicting evidence as this is contradictory. If there are multiple universes with conflicting evidence, then we need some principles whereby we could distinguish our universe from others, allowing that conflicting evidence could be acceptable. This is not simply a matter of distinguishing one possible world from others, but what distinguishes our world from all the other possible worlds. At present, there are no principles which would allow us to distinguish one universe from the rest, as "our" universe, except that it's the one we have evidence of.

This is the age-old old ontological question, which is yet to be answered, what distinguishes our world from the many logically possible worlds, as the real world. The simple answer is "evidence", But the many different ways in which evidence may be interpreted produces ambiguity in that distinction. If the ambiguity leads us to believe that every logically possible world is just as real as every other, then we lose the standard by which our world is distinguished from other logically possible worlds, as the real world.. In other words, evidence no longer helps us.

We are at an impasse. — Fooloso4

It appears to me, like you refuse to accept that agency is an essential part of harmony, and that Socrates' description of harmony, as something produced by agency, is a much better description than Simmias' which neglects the role of agency.

There is a similar issue with modern physicalism and the physicalist's conception of emergence. Order, and organization, by the conception of emergence, is said to simply emerge from disorder. Of course this is contrary to empirical evidence, as it totally neglects the observed role of agency in the creation of orderly structures. I believe that this type of conception is promoted by atheists who approach this issue with a bias which encourages them to unreasonably reject the requirement of agency.

The real numbers include some numbers that are in VV and many that aren't. In what way does that specify VV? That's like saying I can specify the people registered at a hotel this weekend as the human race. Of course everyone at the hotel is human, but humanity includes many people who are not registered at the hotel. — fishfry

That's right, to specify that they are real numbers is to specify, just like to specify that the guests at the hotel are human beings is to specify. The fact that a specification is vague, incomplete, or imperfect does not negate the fact that it is a specification.

How so? — fishfry

I told you how so. You've specified that the set contains real numbers. You are the one who explained to me, that 'set" is logically prior to "number", and that not all sets have numbers as elements. This means that "set" is the more general term. How can you now deny that to indicate that a particular set consists of some real numbers, is not an act of specifying?

And the people at the hotel are humans. As are all the people not at the hotel. If that's all you mean by specification, that all I have to do is name some arbitrary superset of the set in question, then every set has a specification. If that's what you meant, I'll grant you your point. But it doesn't seem too helpful. It doesn't tell me how to distinguish members of a set from non members. — fishfry

Good, you now accept that every set has a specification. Do you also agree now that this type of specification, which "doesn't tell me how to distinguish members of a set from non members", is simply a bad form of specification?

Anyway, let's go back to the point which raised this issue. You said the following, which i said was contradictory:

First, the elements of a set need not be "the same" in any meaningful way. The only thing they have in common is that they're elements of a given set. — fishfry

Do you now see, and agree, that since a set must be specified in some way, then the elements must be "the same" in some way, according to that specification, therefore it's really not true to say that "the elements of a set need not be "the same" in any meaningful way." So we can get rid of that appearance of contradiction by stating the truth, that the elements of a set must be the same in some meaningful way. To randomly name objects is not to list the members of a set, because a set requires a specification.

What I am trying to get at, is the nature of a "set" You say that there is no definition of "set", but it has meaning given by usage. Now I see inconsistency in your usage, so I want to find out what you really think a set is. Consider the following.

The elements of a set need have no relation to one another nor belong to any articulable category or class of thought, OTHER THAN being gathered into a set. — fishfry

Since we now see that a set must have a specification, do you see how the above quote is inconsistent with that principle? Since a set must have a specification, a set is itself an "articulable category or class of thought". And, it is not the "being gathered into a set" which constitutes the relations they have with one another, it is the specification itself, which constitutes the relations. So if you specify a set containing the number five, the tuna sandwich you had for lunch, and the Mormon tabernacle choir, this specification constitutes relations between these things. That's what putting them into a set does, it constructs such relations.

Now here's the difficult part. Do you agree that there are two distinct types of sets, one type in which the specification is based in real, observed similarities, a set which is based on description, and another type of set which is based in imaginary specifications, a set produced as a creative act? Do you acknowledge that these two types of sets are fundamentally different?

There is no need for outside agency. This view is much closer to our scientific understanding of physiology and homeostasis. — Fooloso4

You might place the agency within, as immanent, but the main point is the lack of agency in Simmias' argument. And, when agency is accounted for the agent must be prior to the body, because the body only exists as an organization of parts. Therefore a separate soul, prior to the body is a necessary conclusion.

It is not a correction, it is a different concept of the soul. It is a soul that is completely separate from the body. — Fooloso4

It is a correction, a move toward a more realistic conception of the soul. It's more realistic because agency is a very real part of life (look at Aristotle's potencies of the soul, self-nourishment, self-movement, sensation, intellection), and therefore must be accounted for. And when it is accounted for, the agent which causes the parts to be ordered is necessarily prior to the ordered parts, which is the body. Therefore it is necessary to conclude the existence of a soul which was prior to, and independent from the body.

The argument is as follows: soul is an attunement, vice is lack of attunement, and so the soul cannot be bad and still be a soul because it would no longer be an attunement. What is missing from the argument is that being in or out of tune is a matter of degree. Vice is not the absence of tuning but bad tuning. — Fooloso4

We went through this already, bad tuning cannot be called tuning. If I go to an instrument and start adjusting it to put it out of tune, I am not tuning the instrument. One can change the tuning, by altering adjustments, but if you move toward being out of tune, this cannot be called "tuning".

You continually refuse to recognize that tuning is an act, so you refer to "the tuning", as a static state, But if you would recognize the true nature of tuning, as an act which cause the instrument to be in tune, you would see that if you change the instrument in the wrong direction it cannot be called "tuning".

This is why at 92, the soul as a harmony (static thing), is contrasted with learning (an activity) as recollection The two are incompatible because one is described as a static thing while the other is an activity. What Socrates demonstrates is that "the soul" is better described as an activity "tuning", which causes the harmony, rather than the static thing which you all "the tuning". But since "the body" is understood as a thing, this produces the necessary separation between soul and body.

You previously denied that something can be more or less in tune, but, as any musician or car mechanic can tell you, that is simply not true. — Fooloso4

The point is that the activity, which will affect "the tuning", which we call "tuning" when we respect the "ing" suffix, will alter the instrument in one way or the other, and if it is the other, it cannot be called "tuning". You continually deny the reality that "tuning" properly refers to an activity, insisting that it means "in tune".

The problem with 94c is that there is such a thing as singing out of tune, internal conflict, acting contrary to your own interests, and so on. — Fooloso4

Right, this is acting in a way which is contrary to the direction of the soul, and the reason why the soul needs to inflict harsh punishment to break bad habits, as described. It is not a problem to Socrates' argument, but the first step to you acknowledging the difference between a static state, and an activity. You think there is a problem, but it only appears as a problem because you haven't moved toward recognizing "the soul" as an activity, and breaking away from that static state you call "the tuning". That's why the soul is a "form" for Aristotle, and forms are actualities.

In the Republic passions and desires are in the soul. It is a matter of one part of the soul ruling over the other parts of the soul. Why does Socrates give two very different accounts of the soul? Does the soul have parts or not? Are desires and anger in the soul or in the body? Why would he reject attunement in the Phaedo and make it central to the soul in the Republic? — Fooloso4

I do not see that this is a "different account". The soul, as an activity which rules over all the parts of the body must be present to all parts. So passions and desires, as emotions, are movements of the soul, and there is no inconsistency.

. In addition to those above there is the problem of the identity of Socrates himself. — Fooloso4

I don't see any problems above, except your failure to recognize the distinction between an activity and a state. I agree that "identity" is an issue when we assign personality to an activity, but that's why Aristotle formulated the law of identity, in an explicit way, to resolve this problem. Aristotle's law of identity allows that a thing which is changing may maintain its identity as the same thing, despite changing.

So ∼∼ partitions the real numbers into a collection of pairwise disjoint subsets, called equivalence classes, such that every real number is in exactly one subset. By the axiom of choice there exists a set, generally called VV in honor of Giuseppe Vitali, who discovered it, such that VV contains exactly one member, or representative, of each equivalence class. — fishfry

You are specifying "the real numbers". How is this not a specification?

You're wrong. I just demonstrated a specific example, one that is not only famous in theoretical mathematics, but that is also important in every field that depends on infinitary probability theory such as statistics, actuarial science, and data science.

I know you have an intuition. Your intuition is wrong. One of the things studying math does, is refine your intuitions. — fishfry

Actually, you're wrong, your set is clearly a specified set.

You can tell me NOTHING about the elements of VV. Given a particular real number like 1/2 or pi, you can't tell me whether that number is in VV or not. The ONLY thing you know for sure is that if 1/2 is in VV, then no other rational number can be in VV. Other than that, you know nothing about the elements of VV, nor do those elements have anything at all in common, other than their membership in VV. — fishfry

This is not true, you have already said something else about the set, the elements are real numbers.

Still Metaphysician Undercover must also agree that when he says that @jgill and I have infinite regress wrong, he's incorrect about that too. If both interpretations are the same, everyone's right. — fishfry

I'll agree with Tones, the two ways are just different ways of looking at the same thing. That's why I said the Wikipedia article is consistent with the SEP. I do believe there are metaphysical consequences though, which result from the different ways, or perhaps they are not consequences, but the metaphysical cause of the difference in ways. The principal consequence, or cause (whichever it may be), is the way that we view the ontological status of contingency.

he is not talking about some invisible act. The tuning of what is tuned is not the act of tuning, but rather the result. — Fooloso4

At 86 is how Simmias describes what you translate as "tuning". At 94 is where Socrates corrects Simmias,.with a more true description of "tuning", as an action consisting of the ordering or directing of the parts .

This is the Socratic method, he allows participants to offer their own representations of what is referred to by a term; "beauty" in The Symposium; "just" in The Republic; etc., and he demonstrates how each one is deficient. Then he moves toward a more true representation.

You are refusing to accept Socrates' correction, that the true representation of "tuning" must include the act which directs the parts, causing them to be in tune. So you're still insisting that Simmias' representation is the true description of "tuning", despite the deficiency demonstrated by Socrates, and the obvious absence of agency, which is an essential aspect of "tuning".
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There is in this theory no outside agent or principle acting: — Fooloso4

Yes, that's the whole point, in that theory, the one offered by Simmias, there is no outside agency. This description, offered by Simmias, requires no agency for "a tuning" to come into being. But Socrates demonstrates that Simmias' position is untenable, as has been thoroughly explained by Apollodorus. Then, Socrates offers a more realistic description of "tuning", a description which includes the agency which is obviously involved in any instance of tuning.

The tuning is not the act of tuning, it is the ratio of frequencies according to which something is tuned. — Fooloso4

Do you not grasp the "ing" suffix on "tuning"? The ratio of frequencies, according to which something is tuned is the principle, or rules, applied in the act of tuning. But these principles do not magically apply themselves to the instrument, an agent is required. The agent is "the cause" in common usage. That is what you are consistently leaving out, the requirement of an agent, and this is what Socrates says is traditionally called "the soul", the thing which directs the individual elements, the agent.

The cause of the lyre being in tune is not the activity of tightened and slackens the strings. If I give you a lyre you cannot tune it unless you know the tuning, unless you know the ratio of frequencies. It is in accord with those ratios that the lyre is in tune. The cause of the lyre being in tune is Harmony. — Fooloso4

This is utter nonsense, and you should know better than to say such a thing Fooloso4. Clearly, "the cause" in common usage of this term, is the activity which results in the instrument being tuned, which is the tightening of the strings. Yes, knowing the principles (ratios), is a necessary condition for the agent which acts as the cause, but the ratios do not constitute the cause of the instrument being tuned, as "cause" is used in common language.

If we refer to Aristotelian terminology, and his effort to disambiguate the use of "cause", we'd see that the ratios would constitute the "formal cause". However, there is still a need for an "efficient cause", as the source of activity. Efficient cause is "cause" as we generally use it. We do not, in our common language use, refer to principles like ratios as causes. Would you see a circle drawn on a paper, and say that pi is the cause of existence of that circle? Or if you saw a right angle would you say that the Pythagorean theorem is the cause of existence of that right angle? Normally, we would say that the person who produced the figure, as the agent, is the cause of the figure's existence, and the principles are static tools which the person employs

Whether the body requires something else acting on it is never discussed. — Fooloso4

Yes, the requirement of something else acting on it is discussed, throughout 94, and I provided the quotes. The body requires something which rules over the parts, and this is the soul. Ruling over, directing the elements, and inflicting punishment on them, clearly constitutes "acting on".

I looked at the SEP article. That is utterly bizarre. An infinite regress goes backward without a beginning. Going forward without end like the Peano axioms is not an infinite regress. — fishfry

I agree. It's nonsense. Regress means going backward. I am more than familiar with these notions, as I investigate dynamical processes going forward as well as those going backward. — jgill

Another example of the division between mathematics and philosophy. But the Wikipedia entry is consistent with the SEP.. You two just seem to twist around the concept, to portray infinite regress as a process that has an end, but without a start, when in reality the infinite regress is a logical process with a start, without an end.

Perhaps it is the idea of "forward" and "backward" which is confusing you. There is no forward and backward in logic, only one direction of procedure because to go backward may result in affirming the consequent which is illogical.

But MOST sets can't possibly have specifications, because there are more sets than specifications, a point I've made several times and that you prefer not to engage with. There are uncountably many sets and only countably many specifications. There simply aren't enough specifications to specify all the sets that there are. Most sets are simply collections of elements unrelated by any articulable property other than being collected into that set. — fishfry

This is what I've argued is incoherent, the assumption of an unspecified set, and you've done nothing to justify your claim that such a thing is coherent. I will not ask you to show me an unspecified set, because that would require that you specify it, making such a thing impossible for you. So I'll ask you in another way.

We agree that a set is an imaginary thing. But I think that to imagine something requires it do be specified in some way. That's the point I made with the distinction between the symbol, and the imaginary thing represented or 'specified' by the symbol. The symbol, or in the most basic form, an image, is a necessary requirement for an imaginary thing. Even within one's own mind, there is an image or symbol which is required as a representation of any imaginary thing. The thing imagined is known to be something other than the symbol which represents it. So, how do you propose that an imaginary thing (like a set), can exist without having a symbol which represents it, thereby specifying it in some way? Even to say "there are sets which are unspecified" is to specify them as the sets which are unspecified. Then what would support the designation of unspecified "sets" in plural? if all such sets are specified as "the unspecified", what distinguishes one from another as distinct sets? Haven't you actually just designated one set as "the unspecified sets"?

The tuning is not the thing that is tuned. The tuning is the octave, 4th, and 5th, the ratios according to which the strings of a lyre are tuned. Analogously, the tuning of the parts of the body too is in accord with the proper ratios. Again, the tuning should not be confused with the body that is tuned. — Fooloso4

I already explained how this interpretation is faulty. "The tuning" is the act which tunes. It is not visible in the tuned instrument because it is prior to it, in time. But the act of tuning is logically implied by the existence of a tuned instrument. This is clearly what Socrates is talking about, because he describes how the soul is active in directing the parts. You continually ignore Socrates' reference to the activity of the soul, which is the way toward understanding that the soul is necessarily prior to the body. Appolodorus gets it:

Harmonia here does not mean a harmony in the sense of melodious sound, but the state of the lyre, brought about by a combination of things, that enables it to produce a certain sound: — Apollodorus

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But now it seems that there might be an alternative. Rather than an incomplete yet consistent account of mathematics and language, we might construct an inconsistent yet complete account... — Banno

"We might construct..." In other words, if we allow that anything goes, then we are able to do anything, so we might also be capable of doing everything. But of course, that's just the imagination running wild. That's why philosophy is considered to be a discipline. But pure mathematics, who knows what that is?

This is Stanford Encyclopedia of Philosophy on infinite regress. "An infinite regress is a series of appropriately related elements with a first member but no last member, where each element leads to or generates the next in some sense."

Notice that there is a starting point, and this is why infinite regress is a logical problem, there is generally an assumption which requires something else for justification, and this requires something else etc.. Numbers in themselves, do not constitute an infinite regress because a number itself does not require a next number for justification. We may justify with the prior number, and finally the concept of "one", "unity", which is grounded in something other than number. So infinite regress in numbers is axiom dependent.

Peano’s axioms for arithmetic, e.g., yield an infinite regress. We are told that zero is a natural number, that every natural number has a natural number as a successor, that zero is not the successor of any natural number, and that if x and y are natural numbers with the same successor, then x = y. This yields an infinite regress. Zero has a successor. It cannot be zero, since zero is not any natural number’s successor, so it must be a new natural number: one. One must have a successor. It cannot be zero, as before, nor can it be one itself, since then zero and one would have the same successor and hence be identical, and we have already said they must be distinct. So there must be a new natural number that is the successor of one: two. Two must have a successor: three. And so on … And this infinite regress entails that there are infinitely many things of a certain kind: natural numbers. But few have found this worrying. After all, there is no independent reason to think that the domain of natural numbers is finite—quite the opposite. — Stanford Encyclopedia of Philosophy

Notice the statement that "few have found this worrying". This is because, as fishfry demonstrates, "pure mathematicians" are wont to create axioms with total disregard for such logical problems which are entailed by those axioms. In other words, there are many issues which philosophers see as logical problems, but mathematicians ignore as irrelevant to mathematics. As pure mathematicians proceed in this way, the logical problems accumulate. This has created the divide between mathematics and philosophy which fishfry and I touched on in the other thread, in reference to the Hilbert-Frege disagreement.

The soul, according to his argument, brings life to the body. — Fooloso4

I don't think this is quite what he is saying. In fact, this is the problematic perspective which Plato believed needs to be clarified. Think about what you're saying, that there is a body, and the soul brings life into it. This is not right. The body does not come into existence without life in it, as if life is then brought into the body. That is the problematic perspective further analyzed to a great extent in the Timaeus. To say that there is a body first, and then life is put into it is not consistent with our observations of living things. The living body comes into existence with life already in it. So it's not a matter of the soul putting life into an already existing body. This is why Plato posited a passive receptacle, "matter". The form is put into matter, which is the passive potential for a body, and then there is a body. But matter is not by itself a body, as Aristotle expounds, it is simply potency which does not exist as a body, because it requires a form to have actual existence.

So we are lead toward the conclusion that life creates the very body which it exists within. And this is why Aristotle defined soul as the first actuality of a body having life potentially in it, to emphasize that the soul is the very first actuality of such a body. The body doesn't first exist, and then receive a soul, the soul is the first actuality of that body. For him, the soul couldn't exist without a body, so he assigned "soul" to the very first actuality of such a body, as a sort of form, which provides for the actual existence of that body. For Plato and the Neo-Platonists, it is necessary that the soul is prior to the body to account for the reason why the body is the type of body which it is. Therefore the soul doesn't only provide the general "actuality" of the living body, but also the more specific type.

[His response to Simmias' argument is that you can't have it both ways. You can't have both the soul existing before the body and the soul being a harmony of the parts of the body.] — Fooloso4

He demonstrates that the soul cannot be a harmony, but allows that the body might still be a harmony created by the soul.

Right. In this case the Form would be Harmony. Just as a beautiful body is beautiful by the Beautiful, the harmonious body is harmonious by the Harmonious. — Fooloso4

I think you are being taken for a ride. There is no "Form of Harmony". — Apollodorus

Right, I think Fooloso4 is reaching for straws here, going outside the argument. and I don't see the point.

There is no “Form of Harmony” in Plato for the simple reason that what we call “harmonious” in Modern English, is “rightly-ordered” or “just” (depending on the context) in Plato. So, the corresponding Form would be Justice, not “Harmony” which does not exist.

In Plato, the proper functioning of a whole, be it a city or a human, is not harmony but justice or righteousness (dikaiosyne). Dikaiosyne is the state of the whole in which each part fulfills its function: — Apollodorus

I think that's right. In The Republic, justice is described as a type of order, in which each person minds one's own business and does one's own part, fulfills one's own function without hindering others from fulfilling their functions.

The question of whether there is an Idea of Justice is similar to the question of whether there is an Idea of Good. These questions cast doubt on the theory of participation. It can be argued that Plato rejects the theory of participation in the Timaeus, when he introduces "matter" as the medium between the Form and the material object.

But we're not talking "fact," if by that you mean the real world. The subject was set theory, which is an artificial formal theory. Set theory is not any part of any physical theory. I pointed out to you that in set theory, everything is a set, including the elements of sets. You responded by saying you hadn't realized that. I thought we were therefore making progress: You acknowledged learning something you hadn't known before. And now you want to revert back to "fact," as if set theory has an ontological burden. It does not. — fishfry

I'm not reverting back. Just because I understand better what I didn't understand as well before, doesn't mean that I am now bound to accept the principles which I now better understand.

Focus. You said that the fact that in set theory everything is a set, leads to infinite regress. I pointed out that the negative integers are an example of an unproblematic negative regress; and that the axiom of foundation rules out infinite regresses of set membership. — fishfry

I suggest you look into the concept of infinite regress. The negative numbers are not an example of infinite regress.

Yes, that didn't last long. But you were more than agreeable the other day. You actually achieved some insight. You realized that a set has no definition, and that its meaning is derived from the axioms. You realized that the members of sets are also sets. — fishfry

No, you said "set" has no definition, as a general term, and I went along with that. But I spent a long time explaining to you how a set must have some sort of definition to exist as a set. You seem to be ignoring what I wrote. Since you haven't seriously addressed the points I made, and you claim not to be interested, I won't continue.

So "2" cannot refer to two distinct but same things? — Luke

Of course not, that's contradictory. According to the terms of the law of identity, two distinct things are not the same thing, so "two distinct but same things" is contradictory if we adhere to the definition of "same" provided by the law of identity.

You cannot have 2 apples or 2 iPhones, etc? — Luke

Those are similar but different things, therefore not the same.

The categories we use are either discovered or man-made. If they are discovered, then how can we be "wrong in an earlier judgement" about them; why are there borderline cases in classification; and why does nothing guarantee their perpetuity as categories? — Luke

I still don't see your point, or the relevance.

First, there is no need for something to order the parts. If you assume that the parts together need to be ordered, then each part would also need to be ordered because each part of the body has an order. — Fooloso4

Right, each part needs to be ordered, towards one end, purpose, function, or whatever you want to call it. Each particular has a specific role within that one unity.

How do you proceed toward the conclusion that there is no need for something which orders the parts toward that unity? Do you think that the parts just happen to meet up, and decide amongst themselves, to join together in a unity? The evidence we have, and there is much of it with the existence of artificial things, and things created by other living beings, is that in these situations where parts are ordered together toward making one united thing, there is something which orders the parts.

There is no evidence of any parts just meeting up, and deciding amongst themselves to create an organized, structure, though there are instances, such as the existence of life itself, where the thing which is doing the ordering is not immediately evident. So your claim that "there is no need for something to order the parts" is not supported by any empirical evidence, while "there is a need for something to order the parts" is supported by empirical evidence and solid inductive reasoning.

Second, in accord with Socrates' notion of Forms something is beautiful because of Beauty itself. Something is just because of the Just itself. Something is harmonious because of Harmony itself. Beauty itself is prior to some thing that is beautiful. The Just itself is prior to some thing being just. Harmony itself is prior to some thing being harmonious. In each case there is an arrangement of parts.

The question is, why did Socrates avoid his standard argument for Forms? It is an important question, one that we should not avoid. — Fooloso4

I don't see the point here. What you are referring to is the theory of participation, which I believe comes from the Pythagoreans. There is a problem with this theory which Plato exposed, and Aristotle attacked with the so-called cosmological argument. The problem is with the active/passive relation. When beautiful things are portrayed as partaking in the Idea of Beauty, then the thing which partakes is active, and the Idea is passive. Then we have the problem that the Idea is needed to be prior to the particular thing which partakes, to account for the multitudes of thing being generated which partake. But there is no principle of activity within the Idea, which could cause participation, because the Idea is portrayed as passively being partaken of.

So Aristotle associates "form" with "actual". And, by the cosmological argument, he determines that there must be a Form which is prior to any particular material thing, as cause of its existence, being the unique and particular thing which it is. This type of Form is associated with final cause.

So, we have the Form which is prior to the particular thing, and responsible for its existence, but we cannot represent this relationship between the particular, and the Form, with the Pythagorean theory of participation, because "participation" does not provide the required source of activity (cause). The source, or cause of activity must come from the Idea, or Form, rather than from the particular thing, which by the theory of participation is said to be doing the partaking. .

I'm missing your point also. What's your gripe about the innocuous Riemann sphere? :chin: — jgill

To make infinite numbers into a circle is to make a vicious circle. It is to say that the beginning is the same as the end. And this is what allows for the faulty view of time which fishfry described.

"2" can also refer to two distinct but same things, such as "things" of the same type or category. — Luke

This is a different sense of "same", not consistent with the law of identity.

But all categories/classifications are equally as fictitious and man-made as the sets and orders you reject. — Luke

When they are based in empirical observation they are not equally fictitious. Remember, fishfry speaks of pure abstraction, and claims that a set might be absolutely random..

Scientists justified both the inclusion and exclusion of Pluto as a planet at different times. Like Pluto, many individual "things" are borderline cases in their classification. Moreover, nothing guarantees the perpetuity of any category/set, or of what defines ("justifies") the inclusion of its members. — Luke

That a person later decides to have been wrong in an earlier judgement, is not relevant.

Furthermore, if you base your mathematics on empiricism rather than on "abstraction" or "fiction", then you must also reject fractions, since a half cannot be exactly measured in reality. — Luke

I do reject fractions, I believe that the principles employed are extremely faulty, allowing that a unit might be divided in any way that one wants. This faultiness I believe, is responsible for the Fourier uncertainty In reality, how a unit can be divided is dependent on the type of unit.

If there are "no real boundaries between things", then acknowledging that "anything observed might be divisible an infinite number of times" is not to "give up on the realism", but to adhere to it. — Luke

That's the case if there are "no real boundaries between things". But I am arguing that empirical evidence demonstrates that there are real boundaries.

t is what he argues against. He does this by changing the terms of the argument. His argument is based on a pre-existing soul, something that is not part of Simmias' argument. — Fooloso4

No, Socrates argument is not based on a pre-existing soul, as I explained. First he demonstrates the faults of Simmias' position. Then he demonstrates that if there is such a thing as the soul, it must be pre-existing, as that which orders the parts to create the harmony. So the argument supports the notion of the pre-existing soul, with reference to the directing and ordering of the parts. Therefore the argument is based in the idea that a harmony requires the directing and ordering of parts, to cause the existence of the harmony, and concludes that what is commonly called "the soul" is what directs and orders the parts.

The conclusion is a pre-existing soul. It does not matter that the conclusion (a pre-existing soul) is presented first, as the thing to be proven. This does not make the argument based in the presumption of a pre-existing soul. What matters is the logical procedure. We can proceed from the premise of "harmony" to a need for something which directs and orders the parts, to the conclusion that the thing which directs and orders the parts (commonly called the soul) pre-exists the harmony. A pre-existing soul is not the base of the argument, but the conclusion.

How can either the number 2 or the numeral "2" represent or mean anything in use if no two things are identical in spatiotemporal reality? Isn't the law of identity the basis of your mathematics? — Luke

Obviously, "2" refers to two distinct and different things. If there was only one thing we'd have to use "1".

No not at all. First, what's wrong with infinite regress? After all the integers go backwards endlessly: ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ... You can go back as far as you like. I'm fond of using this example in these endlessly tedious online convos about eternal regress in philosophy. Cosmological arguments and so forth. Why can't time be modeled like that? It goes back forever, it goes forward forever, and we're sitting here at the point 2021 in the Gregorian coordinate system. — fishfry

Again, this is the difference between fiction and fact. We can imagine infinite regress, and imagine time extending forever backward, but it isn't consistent with the empirical evidence. That's the problem with infinite regress, it's logically possible, but proven through inductive (empirical) principles (Aristotle's cosmological argument for example) to be impossible.

jgill was referring to the Riemann sphere, a way of viewing the complex numbers as a sphere. It's based on the simple idea of stereographic projection, a map making technique that allows you to project the points of a sphere onto a plane. There is nothing mystical or logically questionable about this. You should read the links I gave and then frankly you should retract your remark that the Riemann sphere is a "vicious circle." You're just making things up. Damn I feel awful saying that, now that you've said something nice about me. — fishfry

I beg to differ. Didn't we go through this already in the Gabriel's horn thread. It seems like you haven't learned much about the way that I view these issues. You write very well, but your thinking hasn't obtained to that level. Another example of the difference between form and content.

Jeez Louise man. I say: "The only thing they have in common is that they're elements of a given set." And then you say I "ought to recognize ..." that very thing. — fishfry

Are you denying the contradiction in what you wrote? If they are members of the same set, then there is a meaningful similarity between them. Being members of the same set constitutes a meaningful similarity. You said "the elements of a set need not be 'the same' in any meaningful way. The only thing they have in common is that they're elements of a given set." Can't you see the contradiction? If they are said to be members of the same set, then they are the same in some meaningful way. It is contradictory to say that they are members of the same set, and also say that they are not the same in any meaningful way.

Another example of this same sort of contradiction is when people refer to a difference which doesn't make a difference. If you apprehend it as a difference, and speak about it as a difference, then clearly it has made a difference to you. Likewise, if you see two things as elements of the same set, then clearly you have apprehended that they are the same in some meaningful way. To apprehend them as members of the same set, yet deny that they are the same in a meaningful way, is nothing but self-deception. Your supposed set is not a set at all. You are just saying that there is such a set, when there really is no such set. You are just naming elements and saying "those are elements of the same set" when there is no such set, just some named elements. Without defining, or at least naming the set, which they are members of, there is no such set. And, naming the set which they are elements of is a designation of meaningful sameness.

Here is a feature of imaginary things which you ought to learn to recognize. I discussed it briefly with Luke in the other thread. An imaginary thing (and I think you'll agree with me that sets are imaginary things, or "pure abstraction" in your terms) requires a representation, or symbol , to be acknowledged. And, for an imaginary thing, to exist requires being acknowledged. However, the symbol, or representation, is not the imaginary thing. The imaginary thing is something other than the symbols which represent it. So the imaginary thing necessarily has two distinct aspects, the representation, and the thing itself, the former is called form, the latter, content. And this is necessary of all imaginary things.

The important point is that you cannot claim to remove one of these, from the imaginary thing, because both are necessary. So a purely formal system, or pure content of thought, are both impossibilities. And when you say "these things are elements of the same set", you have in a sense named that set, as the set which these things are elements of, thereby creating a meaningful similarity between them. The point being that a meaningful similarity is something which might be created, solely by the mind and that is how the imagination works in the process of creating fictions. But when something is a creation, it must be treated as a creation.

A very disingenuous point. The elements of a set need have no relation to one another nor belong to any articulable category or class of thought, OTHER THAN being gathered into a set. — fishfry

Again, incoherency fishfry. Can't you see that? There is necessarily a reason why you place them in the same set, and this 'reason why' is something other than actually being in the same set. You are not acknowledging that "being gathered into a set" requires a cause, and that cause is something other than being in the same set. So the relation that the things have to one another by being in the same set is not the same as the relation they have to one another by being caused to be in the same set. And things which are in the same set necessarily have relations to each other which are other than being in the same set, because they have relations through the cause, which caused them to be in the same set.

A set is an articulable category, or class of thought! If a set is not a class of thought, then what the heck is it, jeez louise? And don't tell me it might be anything because it is not defined, because even "anything" is a class of thought.

Ok, you are now agreeing with me on an issue over which you've strenuously disagreed in the past. You have insisted that "set" has an inherent meaning, that a set must have an inherent order, etc. I have told you many times that in set theory, "set" has no definition. Its meaning is inferred from the way it behaves under the axioms. — fishfry

It appears like you didn't read what I said. That a word is not defined does not mean that it has no meaning. As I said, it may derive meaning from its use. If the word is used, then it has meaning. So if "set" derives it's meaning from the axioms, then there is meaning which inheres within, according to its use in the axioms.

And now you are making the same point, as if just a few days ago you weren't strenuously disagreeing with this point of view.

But in any event, welcome to my side of the issue. Set has no definition. Its meaning comes exclusively from its behavior as specified by the axioms. — fishfry

What we do not agree on is what "inherent order" means. i really do not see how you get from the premise, that "set" is not defined, but gets its meaning from its use, to the conclusion that a set might have no inherent order. In order for the word "set" to exist, it must have been used. Therefore it is impossible for "set" not to have meaning, and we might say that there is meaning (order, if order is analogous to meaning, as you seem to think), which inheres within. Wouldn't you agree with this, concerning the use of any word? If the word has been used, there is meaning which inheres within, as given by that use. And, for a word to have any existence it must have been used.

Not at all. Bricks are the constituents of buildings, but all the different architectural styles aren't inherent in bricks. There are plenty of sets that aren't numbers. Topological spaces aren't numbers. The set of prime numbers isn't a number. Groups aren't numbers. The powerset of the reals isn't a number. Just because numbers are made of sets in the formalism doesn't mean every set is a number. — fishfry

It appears like you misunderstood. I didn't say every set is a number, to the contrary. I said that if we proceed under the precepts of set theory, every number is a set. Therefore we cannot say that "number" is undefined because "set" is now a defining feature of "number", just like when we say every human beings is an animal, "animal" becomes a defining feature of "human being".

Meta I find you agreeing with my point of view in this post. — fishfry

Didn't it strike you that I was in a very agreeable mood that day? Now I'm back to my old self, pointing out your contradiction in saying that things could be in the same set without having any meaningful relation to each other, other than being in the same set. You just do not seem to understand that things don't just magically get into the same set. There is a reason why they are in the same set.

Maybe at some point we'll discuss the supposed empty set. How do you suppose that nothing could get into a set?

So you would ban the teaching of Euclidean geometry now that the physicists have accepted general relativity? — fishfry

Actually I do not agree with general relativity, so I would ban that first.

Would you ban Euclidean geometry from the high school curriculum because it turns out not to be strictly true? — fishfry

You keep saying things like this, the Pythagorean theorem is not true, now Euclidian geometry in general is not true. I suppose pi is not true for you either? Until you provide some evidence or at least an argument, these are just baseless assertions.

There is no criterion. In fact there are provably more sets than criteria. If by "criterion" you mean a finite-length string of symbols, there are only countably many of those, and uncountably many subsets of natural numbers. So most sets of natural numbers have no unifying criterion whatsoever, They're entirely random. — fishfry

On what basis do you say they are a unity then? You have a random group of natural numbers. Saying that they are a unity does not make them a unity. So saying that they are a "set" does not make them a unity. This is where you need a definition of "set" which would make a set a unity.

I just proved that most sets of natural numbers are entirely random. There is no articulable criterion linking their members other than membership in the given set. There is no formal logical definition of the elements. There is no Turing machine or computer program that cranks out the elements. That's a fact. — fishfry

Then you have no basis to your claim that a set is a unity. And you cannot treat a set as a unified whole. If a set is supposed to be a unified whole, then you cannot claim that "set" is not defined.

According to Simmias' argument there is nothing prior to the body that directs its parts. The body is self-organizing. — Fooloso4

These two ideas, that there is such a thing as the soul, and that each part of the body is itself a "self-organizing" entity, is what Socrates demonstrates are incompatible. If there is such a thing as "the soul", it is what directs the parts, to make a unity, a whole, the body, therefore the parts are not self-organizing.

Right, and that is the problem with your argument. Not only do you assume that all the parts together must be arranged, but for the same reason each of the parts individually must be arranged. If the soul arranges all of the parts together what arranges each of the individual parts? It can't be the soul because then the soul would be the cause of the body. — Fooloso4

Huh? This makes no sense. The argument leads to the conclusion that the soul must be prior to the body, then you conclude "It can't be the soul because then the soul would be the cause of the body." When the logic tells you that the soul must be the cause of the body, what premise tells you that the soul can't be the cause of the body, so you may conclude that the logic is flawed? When the logic gives you a conclusion which you do not like, due to some prejudice, that is not reason to reject the logic, it's reason to reject your prejudice.

In this case he did more than just turn it around. Simmias' argument did not include a separate soul. Socrates does not deal with Simmias' argument because the result would be that the soul does not endure. — Fooloso4

Saying that the soul is like a harmony, or attunement, is to assume that there is such a thing as "the soul" which is being talked about. .Socrates simply demonstrates that if there is such a thing, it is not like a harmony, and separate. Simmias could have insisted that there is no such thing as the soul, and it makes no sense to talk about the soul, but of course Plato, as the author of the dialogue, is dictating what the characters are saying.

Directing the parts does not mean creating the parts. The soul does not cause the body. — Fooloso4

You don't seem to be grasping the issue. The body only exists as an arrangement of parts, you said so yourself, above. Therefore the thing which directs the parts is necessarily prior to the body, as the cause of it. Not even modern physics has an understanding of fundamental particles, so we cannot say how a body comes into existence, only that the body has no existence until the parts are arranged properly. We cannot say that the fundamental parts are bodies because we do not understand what these parts are. and if we assume that they are bodies, then they would be composed of an arrangement of parts, which would also be composed of an arrangement of parts, ad infinitum.

Although, as Apollodorus pointed out to me, 'the argument from harmony' is actually dismissed in the dialogue. — Wayfarer

Socrates' argument is that the soul is not like a harmony, it is more like the cause of the harmony.

Socrates’ argument does not depend on the pre-existence of soul. Even if the soul's pre-existence is not assumed, Simmias’ analogy still fails. — Apollodorus

That's right, Socrates' argument doesn't depend on the pre-existence of the soul, but he uses the proposed harmony analogy to demonstrate that the pre-existence of the soul is a necessary conclusion. That's why he proceeds at 95 to say that proving that the soul existed before we were born does not prove that it is immortal, (because he believes to have proven the soul's pre-existence) only that it has existed for a very long time. He says that entering the human body might be the beginning of its destruction, and it might perish with the death of the human body.

That is not Simmias' argument. Note the following: — Fooloso4

That's right, it's not Simmias' argument, it's Socrates' argument I am talking about. That is Socrates' way, to take another's argument, put it in his own words, and turn it around to produce the opposite conclusion as the one produced by the person who proposes the argument. This is how he demonstrates the faults in the arguments of others, and shows what the real conclusion ought to be.

That is not what Simmias' argument says. And according to Socrates' argument, the soul does not cause the body that is strung and held together by warm and cold and dry and wet and the like — Fooloso4

Yes, Socrates does argue this. The soul directs the parts, which creates a harmony. I gave you the quotes 94 c-e.

You write very well. That must be why I like to engage with you, not that I want to troll you.

Since no two things are identical in spatiotemporal reality, do you also reject the number 2? — Luke

Yes that' the mathematical Platonism I reject. I believe we had a lengthy discussion on this in the other thread, you and I. The number 2 is an unnecessary intermediary between the symbol, and what the symbol represents, or means, in use. Of course you might use the symbol "2" to represent the number 2, but then you are writing fiction.

Sets can contain other sets. In fact a set is "something" in addition to its constituent elements. It's a "something" that allows us to treat the elements as a single whole. If I have the numbers 1, 2, and 3, that's three things. The set {1,2,3} is one thing. It's a very subtle and profound difference. A set is a thing in and of itself. — fishfry

This is what I was asking about earlier, what allows for that unity if not some judgement of criteria, making the elements similar, or the same in some respect., a definition. This is a very important ontological question because we do not even understand what produces the unity observed in an empirical object.

Suppose you arbitrarily name a number of items and designate it as a set. You have created "a thing" here, a set, which is some form of unity. But that unity is completely fictitious. You are just saying that these items compose a unity called "a set", without any justification for that supposed unity. In its simplest from, this is the issue of counting apples and oranges. We can count an apple and orange as two distinct objects, and call them 2 objects. But if we want to make them a set we assume that something unifies them. If we are allowed to arbitrarily designate unity in this way, without any criteria of similarity, then our concept of unity, which some philosophers (Neo-Platonist for example) consider as fundamental loses all its logical strength or significance.

The harmony is the tuning. — Fooloso4

A harmony is a group of notes played together, like a chord, which are judged as sounding good. This is why I do not like your interpretation of the work. The tuning is what creates or produces the harmony as cause of it. It is not the harmony.

The organic body is an arrangement of parts. They do not first exist in an untuned condition and subsequently become tuned. A living thing exists as an arrangement of parts. An organism is organized. — Fooloso4

Right. Now do you see that this "arrangement of parts", which constitutes "the organic body", is analogous to a harmony. The organic body is an harmonic arrangement of parts. Now, Socrates' argument is that the soul is what directs the parts in such a way as to be an harmonic arrangement of parts. This thing "the soul", which directs the arrangement of parts, is temporally prior to the arrangement of parts, as the cause of it. Since the arrangement of parts is the organic body, then the soul is prior to the body.

The assumption is that the mind or soul exists independently of the body. That is what is in question. All of the arguments for that have failed. — Fooloso4

Since the body only exists as an organized arrangement of parts, and the soul is the cause of that organized arrangement, then it is necessarily prior in time to it, therefore independent of the body, at that prior time.

Yes, that is the argument, but it assumes the very thing in question, the existence of the soul independent of the body, that they are two separate things. (86c) The attunement argument is that they are not. But Simmias had already agreed that the soul existed before the body. It is on that basis that Socrates attacks that argument. In evaluating the argument we do not have to assume the pre-existence of the soul. — Fooloso4

The argument is that a harmony, or "attunement", whatever you want to call it, requires a cause. The cause is prior to the harmony, in time, and therefore existed independently of it at that prior time. The body is analogous to the harmony, as an organized arrangement of parts. The soul is the cause of that organized arrangement of parts. Therefore the soul was independent of the body at that prior time.

In set theory everything is a set. — fishfry

I didn't know that, but it makes the problems which I've apprehended much more understandable. If everything is a set, in set theory, then infinite regress is unavoidable. A logical circle is sometimes employed, like the one mentioned here to disguise the infinite regress, but such a circle is really a vicious circle.

Sets whose elements are sets whose elements are sets, drilling all the way down to the empty set. — fishfry

I reject "the empty set" for a reason similar to the reason why I rejected a set with no inherent order. it's a fiction which has no purpose other than to hide the shortcomings of the theory. There are very good reasons why "0" ought to represent something in a class distinct from numbers. There are even reasons why "1" ought to be in a distinct class.

No, not at all. First, the elements of a set need not be "the same" in any meaningful way. The only thing they have in common is that they're elements of a given set. — fishfry

This may be the case, but you ought to recognize that being elements of the same set makes them "the same" in a meaningful way. Otherwise, a set would be a meaningless thing. So when you said for instance, that {0,1,2,3,} is a set, there must be a reason why you composed your set of those four elements. That reason constitutes some criteria or criterion which is fulfilled by each member constituting a similarity.

The concept of "set" itself has no definition, as I've pointed out to you in the past. — fishfry

This is a simple feature of common language use. A word may receive its meaning through usage rather than through an explicit definition. That the word has no definition does not mean that it has no meaning, its meaning is demonstrated by its use, as is the case with an ostensive definition. Allowing that a word, within a logical system, has no explicit definition, allows the users of the system an unbounded freedom to manipulate that symbol, (exemplified by TonesInDeepFreeze's claim with "least"), but the downfall is that ambiguity is inevitable. This is an example of the uncertainty which content brings into the formal system, that I mentioned in the other thread.

There is no set of ordinals, this is the famous Burali-Forti paradox. — fishfry

This I would say is a good representation of the philosophical concept of "infinite". Note that the philosophical conception is quite different from the mathematical conception. If every ordinal is a set composed of other ordinals, and there is no limit to the "amount" of ordinals which one may construct, then it ought to be very obvious that we cannot have an ordinal which contains all the ordinals, because we are always allowed to construct a greater ordinal which would contain that one as lesser. So we might just keep getting a greater and greater ordinal, infinitely, and it's impossible to have a greatest ordinal.

I think there is a way around this though, similar to the way that set theory allows for the set of all natural numbers, which is infinite. As you say, "set" has no official definition. And, you might notice that "set" is logically prior to "cardinal number". So all that is required is a different type of set, one which is other than an ordinal number, which could contain all the ordinals. It would require different axioms.

There is no general definition of number. — fishfry

This is not really true now, if we accept set theory. If "set" is logically prior to "number", then "set"
is a defining principle of "number". That is why you and I agreed that each ordinal is itself a set. We have a defining principle, an ordinal is a type of set, and a cardinal is a type of ordinal.

You see you're at best a part-time Platonist yourself. — fishfry

Correction, at my worst I am a part-time Platonist. At my best I am a fulltime Neo-Platonist.

If I put on my Platonist hat, I'll admit that the number 5 existed even before there were humans, before the first fish crawled onto land, before the earth formed, before the universe exploded into existence, if in fact it ever did any such thing. — fishfry

We do not have to go the full fledged Platonic realism route here, to maintain a realism. This is what I tried to explain at one point in another thread. We only need to assume the symbol "5", and what the symbol represents, or means. There is no need to assume that the symbol represents "the number 5", as some type of medium between the symbol, and what the symbol means in each particular instance of use. So when I say that a thing exists, and has a measurement, regardless of whether it has been measured, what I mean is that it has the capacity to be measured, and there is also the possibility that the measurement might be true.

I must say, though, that I am surprised to find you suddenly advocating for mathematical Platonism, after so many posts in which you have denied the existence of mathematical objects. Have you changed your mind without realizing it? — fishfry

If you think that I was advocating for mathematical Platonism, then you misunderstood. I was advocating for realism.

But Meta, really, you are a mathematical Platonist? I had no idea. — fishfry

A mathematical Platonist thinks of ideas as objects. I recognize the reality of ideas, and furthermore I accept the priority of ideas, so I am idealist. But I do not think of ideas as objects, as mathematical Platonists do, I think of them as forms, so I'm more appropriately called Neo-Platonist.

I agree with the points you're raising. I don't know if 5 existed before there were humans to invent math. I truly don't know if the transfinite cardinals were out there waiting to be discovered by Cantor, and formalized by von Neumann. After all, set theory is an exercise in formal logic. We write down axioms and prove things, but the axioms are not "true" in any meaningful sense. Perhaps we're back to the Frege-Hilbert controversy again. — fishfry

This is that vague boundary, the grey area between fact and fiction which we might call "logical possibility". If we adhere to empirical principles, we see that there are individual objects in the world, with spatial separation between them. If we are realist, we say that these objects which are observed as distinct, really are distinct objects, and therefore can be counted as distinct objects. We might see three objects, and name that "3", but "3" is simply what we call that quantity. Being realist we think that there is the same quantity of objects regardless of whether they've been counted and called "3" or not.

But if we give up on the realism, and the empirical principles, there is no need to conclude that what is being seen is actually a quantity of 3. There might be no real boundaries between things, and anything observed might be divisible an infinite number of times. Therefore whatever is observed could be any number of things. This is the world of fiction, which some might call "logical possibility", and you call pure mathematics. Empirical truths, like the fact that distinct objects can be counted as distinct objects, pi as the ratio between circumference and diameter of a circle, and the Pythagorean theorem, we say are discovered. Logical possibilities are dreamt up by the mind, and are in that sense fictions.

I do not mean to argue that dreaming up logical possibilities is a worthless activity. What I think is that this is a primary stage in producing knowledge. We look at the empirical world for example and create a list of possibilities concerning the reality of it. The secondary stage is to eliminate those logical possibilities which are determined to be physically impossible through experimentation and empirical observation. So we proceed by subjecting logical possibilities, and axioms of pure mathematics, to a process of elimination.

The first is true independent of any instrument. The second is true of a particular instrument. The first is about the ratio of frequencies. The second about whether those relations are achieved on a particular instrument. — Fooloso4

The second is always true regardless of the instrument. That's what I've been explaining to you, the temporal aspect of Socrates' argument. The harmony is the effect of, therefore caused by, the appropriate tuning. It does not direct the tuning. That's what Socrates is saying, a harmony does not direct the parts which it is composed of, to create itself. This is the key point, what directs the tuning is the mind with some mathematical principles, and harmony is the result, or effect of that direction. The soul is more like the thing which does the directing, therefore the cause of the tuning, rather than the result of the tuning, the result being the harmony itself, which is produced.

In the Republic the problem is not between the parts of the body and the soul but which part of the soul. The answer is reason. In addition, appetites are treated as a part of the soul and not the body. The conflict is within the soul, not between soul and body. Also the soul in the Republic has parts but in the Phaedo it is denied that it has parts. — Fooloso4

We are discussing the Phaedo here. Do you agree that Socrates' argument is that the soul is more like the thing which directs the parts, as the cause of harmony, rather than like the harmony which is the result, or effect of being so directed. If you agree that this is Socrates' argument, do you also agree with this principle in general?

That incorrectly makes it appear that I said, "Incorrect: We should not use 'least' if we don't mean quantity." — TonesInDeepFreeze

That is what you said. You said the phrase, "We should not use 'least' if we don't mean quantity" is incorrect. I asked, if you don't mean some sort of quantity then what do you mean by "least". And fishfry gave me an answer to that.