We are at an impasse. — Fooloso4
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
How so? — fishfry
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
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
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
There is no need for outside agency. This view is much closer to our scientific understanding of physiology and homeostasis. — Fooloso4
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
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
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 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
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
. In addition to those above there is the problem of the identity of Socrates himself. — Fooloso4
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're wrong. I just demonstrated a specific example, one that is not only famous in theoretical mathematics, but that is also important in every field that depends on infinitary probability theory such as statistics, actuarial science, and data science.
I know you have an intuition. Your intuition is wrong. One of the things studying math does, is refine your intuitions. — fishfry
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
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
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
There is in this theory no outside agent or principle acting: — Fooloso4
The tuning is not the act of tuning, it is the ratio of frequencies according to which something is tuned. — Fooloso4
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
Whether the body requires something else acting on it is never discussed. — Fooloso4
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
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
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
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
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
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
The soul, according to his argument, brings life to the body. — Fooloso4
[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
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
There is no “Form of Harmony” in Plato for the simple reason that what we call “harmonious” in Modern English, is “rightly-ordered” or “just” (depending on the context) in Plato. So, the corresponding Form would be Justice, not “Harmony” which does not exist.
In Plato, the proper functioning of a whole, be it a city or a human, is not harmony but justice or righteousness (dikaiosyne). Dikaiosyne is the state of the whole in which each part fulfills its function: — Apollodorus
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
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
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
So "2" cannot refer to two distinct but same things? — Luke
You cannot have 2 apples or 2 iPhones, etc? — Luke
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
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
Second, in accord with Socrates' notion of Forms something is beautiful because of Beauty itself. Something is just because of the Just itself. Something is harmonious because of Harmony itself. Beauty itself is prior to some thing that is beautiful. The Just itself is prior to some thing being just. Harmony itself is prior to some thing being harmonious. In each case there is an arrangement of parts.
The question is, why did Socrates avoid his standard argument for Forms? It is an important question, one that we should not avoid. — Fooloso4
I'm missing your point also. What's your gripe about the innocuous Riemann sphere? :chin: — jgill
"2" can also refer to two distinct but same things, such as "things" of the same type or category. — Luke
But all categories/classifications are equally as fictitious and man-made as the sets and orders you reject. — Luke
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
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
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
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
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
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
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
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
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
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
And now you are making the same point, as if just a few days ago you weren't strenuously disagreeing with this point of view.
But in any event, welcome to my side of the issue. Set has no definition. Its meaning comes exclusively from its behavior as specified by the axioms. — fishfry
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
Meta I find you agreeing with my point of view in this post. — fishfry
So you would ban the teaching of Euclidean geometry now that the physicists have accepted general relativity? — fishfry
Would you ban Euclidean geometry from the high school curriculum because it turns out not to be strictly true? — fishfry
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
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
According to Simmias' argument there is nothing prior to the body that directs its parts. The body is self-organizing. — Fooloso4
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
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
Directing the parts does not mean creating the parts. The soul does not cause the body. — Fooloso4
Although, as Apollodorus pointed out to me, 'the argument from harmony' is actually dismissed in the dialogue. — Wayfarer
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 is not Simmias' argument. Note the following: — Fooloso4
That is not what Simmias' argument says. And according to Socrates' argument, the soul does not cause the body that is strung and held together by warm and cold and dry and wet and the like — Fooloso4
Since no two things are identical in spatiotemporal reality, do you also reject the number 2? — Luke
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
The harmony is the tuning. — Fooloso4
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
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
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
In set theory everything is a set. — fishfry
Sets whose elements are sets whose elements are sets, drilling all the way down to the empty set. — fishfry
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
The concept of "set" itself has no definition, as I've pointed out to you in the past. — fishfry
There is no set of ordinals, this is the famous Burali-Forti paradox. — fishfry
There is no general definition of number. — fishfry
You see you're at best a part-time Platonist yourself. — fishfry
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
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
But Meta, really, you are a mathematical Platonist? I had no idea. — fishfry
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
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
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
That incorrectly makes it appear that I said, "Incorrect: We should not use 'least' if we don't mean quantity." — TonesInDeepFreeze
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
This is NOT true of sets in general, but it IS true for ordinals, and that's what makes the construction work. — fishfry
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
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
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
Cardinal equivalence is a relation between two sets. It's not something a set can have by itself. — fishfry
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
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
Incorrect: We should not use 'least' if we don't mean quantity.
It is typical of cranks unfamiliar with mathematical practice to think that the special mathematical senses of words most conform to their own sense of the words or even to everyday non-mathematical senses. The formal theories don't even have natural language words in them. Rather, they are purely symbolic. Natural language words are used conversationally and in writing so that we can more easily communicate and see concepts in our mind's eye. The words themselves are often suggestive of our intuitions and our conceptual motivations, but proofs in the formal theory cannot appeal to what the words suggest or connote. And for any word such as 'least' if a crank simply could not stomach using that word in the mathematical sense, then, if we were fabulously indulgent of the crank, we could say, "Fine, we'll say 'schmleast' instead. 'schmardinality' instead'. 'ploompty ket' instead of 'empty set' ... It would not affect the mathematics, as the structural relations among the words would remain, and the formal symbolism too. — TonesInDeepFreeze
df: K is a cardinal iff K is an ordinal and there is no ordinal j less than K such that there is a bijection between K and j.
There is no mention of 'cardinal' or 'cardinality' in the definiens. — TonesInDeepFreeze
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
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
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
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
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
No. Cardinal equivalence is logically prior to ordinals in the sense that every ordinal is cardinally equivalent to some other sets. At the very least, every ordinal is cardinally equivalent to itself.
When you use the word "cardinality" you are halfway between cardinal numbers and cardinal equivalence, so you confuse the issue. Better to say that cardinal equivalence is logically prior to ordinals; and that (in the modern formulation) ordinals are logically prior to cardinals. — fishfry
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
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 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
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
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
Cardinality is inherent. — fishfry
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
So yes, cardinality is already inherent within the ordinals. Each ordinal has a cardinality. I — fishfry
Not at all. Not "more or less," but "prior in the order," if you prefer more accurate verbiage.
You insist on conflating order with quantity, and that's an elementary conceptual error. In an order relation x < y, it means that x precedes y in the order. x is not "smaller than" y in a quantitative sense. I can't do anything about your refusal to recognize the distinction between quantity and order. — fishfry
The modern definition is the von Neumann cardinal assignment. Von Neumann defined a cardinal as the least ordinal having that cardinality. — fishfry
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
Right, but a lyre is not a living thing. It is not capable of self-movement or self-attunement.
Wayfarer makes an important point: — Fooloso4
With all his talk of opposite forms Socrates neglects to consider Harmonious /Unharmonious or — Fooloso4
The question is why Socrates neglected this argument? — Fooloso4
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
The analogy with the lyre is not with a lyre that needs to be tuned but that is tuned, that is, in harmony. — Fooloso4
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
God is supposed to be a necessary being. Something is necessary if it is true in every possible world. — Banno
Logic is needed in order to have the discussion, not as a consequence of the discussion. — Banno
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
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
The tuning of a lyre exists apart from any particular lyre. — Fooloso4
It is this relationship of frequencies that is used to tune a particular lyre. — Fooloso4
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
But pi is not a particular real number? How can I have a conversation with you? — fishfry