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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The harmony is the tuning. — Fooloso4

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

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

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

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

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

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

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

In set theory everything is a set. — fishfry

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

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

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

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

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

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

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

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

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

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

There is no general definition of number. — fishfry

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cardinality is inherent. — fishfry

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

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

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

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

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

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

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

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

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

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

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

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

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

Wayfarer makes an important point: — Fooloso4

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

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

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

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

The question is why Socrates neglected this argument? — Fooloso4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Can we please stop now? — fishfry

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

How do I find out where my likes are?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

But it's still a formal game. — fishfry

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

News to me. — fishfry

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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