• Plato's Phaedo
    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.
  • Taking from the infinite.
    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?
  • Plato's Phaedo
    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.
  • Taking from the infinite.
    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.

    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.
  • Plato's Phaedo
    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".
    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.
  • Plato's Phaedo
    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".
  • Taking from the infinite.
    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"?
  • Plato's Phaedo
    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

  • Inconsistent Mathematics
    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?
  • Taking from the infinite.

    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.
  • Plato's Phaedo
    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:

    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.
  • Taking from the infinite.
    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.
  • Plato's Phaedo
    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.

    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. .
  • Taking from the infinite.
    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.
  • Plato's Phaedo
    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.
  • Taking from the infinite.
    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.

    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.
  • Plato's Phaedo
    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.
  • Plato's Phaedo
    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.
  • Plato's Phaedo
    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 likeFooloso4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    If you go the other way, as you are doing, then the position is determined by your subjective judgement alone, not by the true relation between the system of principles and the object to be judged. So if you make a mistake, and put me in the wrong room, because your measurement was wrong, I have no means to argue against you, because it is your judgement which puts me in group 3, not the relation between your system and me.
  • Taking from the infinite.
    Incorrect: We should not use 'least' if we don't mean quantity.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Yes, this demonstrates very well the problem I described above. Because the set has a "cardinal equivalence, it also necessarily has a cardinality, and a corresponding mathematical object which you call a cardinal number. Why do you think that you need to determine that object, the cardinal number, before that object exists as the object which it is assumed to be, the cardinal number?
  • Plato's Phaedo
    Socrates does not make the proper distinction between a tuning and what is tuned. It is not more or less a tuning, it is more or less in tune.Fooloso4

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

    The main point though, is made at 93a, "One must therefore suppose that a harmony does not direct its components, but is directed by them". This point is built upon at 94b: "Further, of all the parts of a man, can you mention any other part that rules him than his soul, especially if it is a wise soul?" He then explains how the soul rules by opposing what the body wants, and if the soul were a harmony of parts such an opposition would not be possible.
    Well, does it now appear to do quite the opposite, ruling over all the elements of which one says it is composed, opposing nearly all of them throughout life, directing all their ways, inflicting harsh and painful punishment on them, at times in physical culture and medicine, at other times more gently by threats and exhortations, holding converse with desires and passion and fears as if it were one thing talking to a different one... — 94c-d

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

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

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

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

    I believe Aristotle's cosmological argument actually demonstrates that the idea of infinite time is what is incomprehensible, and this forces the need for something outside of time ("eternal" in the theological sense). So it's a matter of how one apprehends the boundaries. Is all of reality bounded by time (physicalism), or is time itself bounded?
  • Taking from the infinite.
    But you say it's "my" bijective equivalence as if this is some personal theory I'm promoting on this site. On the contrary, it's established math. You reject it. I can't talk you out of that.fishfry

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

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

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

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

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

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

    In other words, you agree that it's incorrect to say that ordinals are logically prior to cardinals. That is, unless you are just trying to hide a vicious circle by saying that a cardinal number is defined by its ordinality, and ordinality is defined by cardinality. But in the case of a vicious circle of two logically codependent things, it is still incorrect to say that one is logically prior to the other. So despite my lack of understanding of your "bijective equivalence", it is still you who is mistaken.
  • Plato's Phaedo
    His argument is that Harmony is a universal. What is at issue is the difference between the universal and particular. Harmony itself is prior to any particular thing that is in harmony.Fooloso4

    The argument is not about universals. It is a question of whether the activity required to produce, or create, an organized system of parts (the harmony), is necessarily prior to that organized system of parts. Read 93-95.
  • Taking from the infinite.
    So yes, cardinality is already inherent within the ordinals. Each ordinal has a cardinality. Ifishfry

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

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

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

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

    Isn't this circular? Doesn't "least" already imply cardinality, such that cardinality is already inherent within the ordinals, to allow the designation of a least ordinal? Then the claim that ordinals are logically prior to cardinals would actually be false, because more and less is already assumed within "ordinal".
  • Plato's Phaedo
    The soul is that which imparts life to the body in the first place (105c - d). Without the soul there would be no body.Apollodorus

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

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

    Wayfarer makes an important point:

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

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

    With all his talk of opposite forms Socrates neglects to consider Harmonious /Unharmonious orFooloso4

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

    The question is why Socrates neglected this argument?Fooloso4

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

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

    But Socrates demonstrates, by the argument we've been discussing, that this idea, "that the soul is a harmony" is false.
  • Plato's Phaedo
    The analogy with the lyre is not with a lyre that needs to be tuned but that is tuned, that is, in harmony.Fooloso4

    But a lyre does need to be tuned. It doesn't magically tune itself, and if used, it rapidly goes out of tune. So there is a very clear need to assume that there is something which tunes it. Likewise, there is a very clear need to assume that there is something which causes an organism to be organized. That's the soul.
  • Plato's Phaedo
    I do not know the tuning of the lyre, but let's say the strings are tuned in 4ths or 5ths. The standard is independent of any particular lyre, but whether this particular lyre is in tune cannot be independent of the tension of the strings of this lyre, and that tension cannot be achieved when this lyre is destroyed.Fooloso4

    Yes, and the belief that the soul is like a particular lyre being in tune (a harmony), is the belief which Socrates dismisses as faulty. So the fact that this particular instance of being in tune (a harmony) is destroyed when the lyre is destroyed, is irrelevant to what Socrates is arguing, because he argues that the soul is not like a particular instance of being in tune (a harmony).
  • Necessity and god
    God is supposed to be a necessary being. Something is necessary if it is true in every possible world.Banno

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

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

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

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

    See, it's a different sense of "necessary". It is the sense which describes how a contingent thing actually exists. A contingent thing requires the appropriate efficient cause to bring its actual existence from a mere possibility. The efficient cause is said to be necessary for the thing's existence.
  • Plato's Phaedo
    The tuning does not tune the lyre or body, the lyre or body is tuned according to the tuning. It must exist in order to be tuned.Fooloso4

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

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

    Socrates argues against the position that the soul is like being tuned, ( a harmony in my translation) for the reason I described, the soul is more like the cause of being tuned, which is the act of tuning. When a particular lyre is no longer tuned, the cause of it being tuned, the act of tuning, is no longer tuning that particular lyre, but it is still tuning other instruments..
  • Plato's Phaedo
    The tuning of a lyre exists apart from any particular lyre.Fooloso4

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

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

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

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

    The argument against the soul as a harmony, is not intended to say anything about the existence of the soul after death. That's why Socrates goes to the other dialectical argument (argument from the meaning of words) afterwards. The harmony argument shows that 'how to tune a lyre', the principle concerning the relationship between tones, is prior to 'the tuning of a lyre'. So the soul is prior to the body, by having that principle of how to create harmony within the parts of the body. So at 95 c-d he explains how the proof that the soul is prior to the man, does not prove that the soul is immortal. It may be the case that entering the body of a man is the decline of the soul, that this is the beginning of the end.
  • Plato's Phaedo
    I find the most compelling and important argument in The Phaedo is the argument against the soul as a harmony. As a harmony, continues to be the populist view today as emergence; life is something which emerges from properly aligned material parts. But Socrates' argument actually demonstrates that the soul must be prior to the body, being the cause of alignment of the parts, rather than the harmony which is the result of such alignment. This is important because it provides us with the basis for understanding the nature of free will, and other fundamental ontological principles.
  • Can it be that some physicists believe in the actual infinite?
    But pi is not a particular real number? How can I have a conversation with you?fishfry

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