That doesn't make sense automatically because formalism is a program for foundations, platonism is an ontological claim. And idk what post of MU it is. — Lionino

Ontological assumptions are what foundations are made of, and Platonism provides the assumptions required for formalism, the idea of pure form.

Meaningless word games. The fingers on your hand are a physical instantiation of the number 5. Positive integers have the property that the smaller among them may be physically instantiated. 12 as in a dozen eggs, 9 as in the planets unless an astronomical bureaucracy demotes Pluto. That's one for the philosophers, don't you agree? The number of planets turns out to be a matter of politics, not math or astrophysics. — fishfry

I don't see what this all has to do with your claim that a concept like a number, 5, could have a physical instantiation . Fingers are fingers, and are therefore physical instantiations of fingers, not of numbers, not matter how many of them you have. Wittgenstein took up this issue in the Philosophical Investigations, showing why there is a lot more involved with learning a language than simple ostensive definition. Abstraction is very complex, and with complex concepts like number, an explanation of what it is about the thing which is being shown, which is being referred to with the word, is a requirement.

A person cannot simply look at the fingers on a hand and apprehend the concept 5. An explanation about quantity, or counting is required. The concept 5 is learned from the explanation, not from the ostensive hand, therefore the hand is not a physical instantiation of the number.

Judged by who? Politicians? Academic administrators? Philosophers? How about by their fellow mathematicians? That's the standard of what counts as math. — fishfry

It can be judged by anyone. The issue though, is that many, like yourself refuse to make such a judgement. You say that there is no truth or falsity to mathematical axioms, they are simply tools which cannot be judged for truth. Since mathematicians tend to think this way, they are not well suited for judging truth or falsity of their axioms. But I've shown how axioms can be judged for truth. If an axiom defines a word or symbol in a way which is inconsistent with the way that the symbol is used, then it is a false axiom.

So for example, if a mathematical axiom defines "=" as meaning "the same as", yet in applied mathematics the mathematicians use "=" to mean "has the same value as", then the axiom makes a false definition. This axiom will be misleading to any "pure mathematician" who uses it to produce a further conceptual structure with that axiom at the base, just like if anyone else working in speculative theories in other fields of science starts from a false premise. False propositions are fascinating, sometimes leading to theories which are extremely useful, because they are designed for the purpose at hand.

They're meta-false, as I understand you. They're not literally false. If the powerset axiom is false, you get set theory without powersets. You don't get some kind of philosophical contradiction. You are equivocating levels. — fishfry

Sorry, I don't understand what you mean by "meta-false". I am talking about "literally false". False to me, means not corresponding with reality. For example, if someone says that in the use of mathematics, "=" indicates "the same as", but in reality, when mathematicians use equations, "=" means "has the same value as", then the person who said that "=" indicates "the same as" has spoken a falsity. Do you agree that this would be an instance of "literally false"?

A model, not a description. Is that better? — fishfry

That doesn't help. Numbers form discrete units, and discrete units cannot model an idealized continuum. There is an inconsistency between these two, demonstrated by those philosophers who argue that no matter how many non-dimensional points you put together, you'll never get a line. The real numbers mark non-dimensional points, the continuum is a line. The two are incompatible.

So you can bet that Trump is going to use this attempt as a weapon against Biden, — Wayfarer

That's foregone, the moment the shot rang out:

Biden incited them to shoot Trump. Isn’t that how it works?

“We’re done talking about the debate, it’s time to put Trump in a bullseye."

- Joe Biden — NOS4A2

a mathematician is an explorer trying to find a path extending knowledge in a particular direction or discovering new directions. — jgill

I would say that this is a type of problem solving, wouldn't you? The problem being worked on is not necessarily a practical issue. Philosophy is like this too, as well as speculative theorizing, there is a wide range to the types of problems. Sometimes, problems are being worked on without any obvious practical implications.

I see an out. In this para you have stated your aim about the real numbers and the number 5. I don't think I have any interest in this topic. I know it's important and meaningful to you, but it isn't to me. Perhaps I'm to dim to grasp all these philosophical subtleties such as you raise. If so, so be it.

But secondly, and I'd be remiss if I didn't add, that I have formally studied the real numbers and the number 5. That doesn't make me right and you wrong, by any means. What it does mean is that I'm not likely to ever defer to your opinions about the real numbers or the number 5. — fishfry

Well, "the real numbers", and "5" being an instance of a real number, was your example. I agree that by some accepted principles of mathematics, the axioms of set theory, etc., 5 is an instance of a real number. This I believe to be the influence of Platonism which assumes that a number is an object. I disagree with this, and think that a number is a concept, and conceptions are quite different from objects. The way that one concept relates to another for example is completely different from the way that one object relates to another.

You might think that it doesn't matter whether a number is an object or not. You might think that within the confines of the logical system of "the real numbers", a number can be whatever the mathematician who states the axiom wants it to be. My argument is that numbers are used billions of times a day by human beings, and according to that usage there is some truth and falsity about what a number is. Therefore when an axiom makes a statement about what a number is, and it's not consistent with how numbers are actually used, the axiom can be judged as false.

When it suits my argument. I'm a formalist as well at times. — fishfry

Like I explained earlier, formulism is just a specific type of Platonism. It takes Platonist principles much deeper in an attempt to realize the ideal within the work of human beings, while other Platonists allow the ideal to be separate from human beings.

Mathematical philosophies are tools, nothing more. Conceptual tools, frameworks for thinking about the development and structure of math. They aren't "true" or "false," they're just models, if you will. — fishfry

Do you not look at mathematics, and mathematicians as real human beings, carrying out activities in the real world? If so, then don't you think that there is such a thing as true and false propositions about what those mathematicians are doing? If you follow, and agree so far, then why wouldn't you also agree that mathematical philosophies, as tools, or models, ought to be judged for truth and falsity? If a mathematical philosophy provides false propositions about what mathematicians are doing, offering this philosophy as a tool for understanding the structure and development of math, it is likely to mislead.

Problem solvers and theory builders. The theory builders don't solve problems at all. They create conceptual frameworks in which others can solve problems. — fishfry

As I explained to jgill above, theory building is a form of problem solving, it just involves a different type of problem. There are many different types of problems which can be categorized in different ways.

LOL. 1 + 1 and 2 are each representations of the same set in ZF, with "1" and "2" interpreted as defined symbols in the inductive set given by the axiom of infinity; and likewise "+" is formally defined. — fishfry

Yes, this is the problem, axioms of set theory are false, in the way described above.

BUT! Are you telling me that you don't believe in the physical instantiation of the natural number 5? Just look at the fingers on your hand. I rest my case. — fishfry

I said that 5 is not an instance of a real number. Also, I would say that the fingers on my hand are not an instance of the number 5, they are an instance of a quantity of five. You see, this is the problem of mixing up the ideal with the physical. "The natural number 5" is an ideal, a type of Platonic object called "a number". There is no physical instantiation of numbers, they are by definition ideal. So we need to refer to the use of "5" to see its meaning, and then we can find a physical representation for its meaning. In the context of usage of the natural numbers my understanding is that 5 represents a specific quantity, and the fingers on my hand provide an example of this specific quantity.

If we say that the numeral 5 represents a number, which goes by that name, 5, we have no meaning indicated to assist us in finding a physical example of the number five. All we have is that there is a type of thing called a number, and one of them is named 5. In order for numbers such as 5 to be used in practise, we need to provide something more, otherwise we're stuck with the interaction problem of idealism, these ideal things have no bearing on the real world. But if we give the number 5 further meaning, such as "a specific quantity", to allow it to be useful in the world, then the ideal, the number 5 becomes redundant, and completely useless. Why not just say that the numeral "5" means a specific quantity, and be done with it. Well I'll tell you why not. The numeral "5" is assumed to represent a number, 5, which is an abstract, Platonic object, for another purpose. The other purpose is mathematical philosophy, building structures and frameworks to be used as tools for understanding the development of math. However, as explained above, rather than assisting understanding, it misleads.

Why me? — fishfry

You are free to abandon me anytime you want.

If I'm understanding you, I agree. I don't think the mathematical real numbers refer to anything in the world at all. They describe the idealized continuum, something that we have no evidence can exist. — fishfry

If you truly believe this, then how would you validate your claim that the number 5 is an instance of a real number. Do you see that when you talk about "a real number", and "the real numbers", you validate the claim that "the real numbers" refers to a collection of individual objects? And that is contrary to what you say here. And do you see that in set theory, "numbers" also must refer to individual things, and this is contrary to being a description of "the idealized continuum".

Your view of intentionality strips out the essence of intention and swaps it for causality; which of no use when we analyze the intentions of someone. — Bob Ross

I strongly disagree with this. Our most reliable access to a person's intention is through observations of the actions which that person causes. This is because often if we ask a person what their goals were when they acted they do not answer honestly, they might just make something up. Furthermore, the issue I described already is that the person often does not even accurately know one's own intentions when actions are carried out. This is the case with habit. This boosts the inclination to make things up. Therefore the most accurate way to analyze the intentions of someone is through the actions which they cause.

The intention is wrapped up, inextricably, with the action; and what is caused is an effect. — Bob Ross

Exactly, what is caused is an effect, the effect of the person's intention. The effects of a person's intention are observable and analyzable. Because of this we can produce a reliable science of intention. On the other hand, if we ask a person what one's goals were, we generally do not acquire reliable information.

What is intentional is what is related to the intention; and the intention is the end which is being aimed at. — Bob Ross

That is your preferred definition of "intention" because it is most consistent with the convention which associates intention with purpose. What I am saying is that if we define "intention" as the cause of one's actions instead, this provides us with a more scientific approach toward understanding purpose, aims, and goals. This is because, as I described in the last post, a person's actions are often not consistent with the person's goals. There are many reasons for this inconsistency, the force of habit, the force of mental illness, and the common example of faulty reasoning. In many cases, the person's determination of the means to the desired end, is faulty.

Because of these factors, which produce inconsistency or incoherency between one's actions and one's goals, and the fact that for moral/legal purposes the person's acts must be considered "intentional" even when the acts are not conducive to the desired end, we need to associate "intention" with the act rather than with the end which is aimed for. This indicates that "intention" ought to be associated with the act rather than the goal.

You can’t implicate someone as intentionally doing something they entirely did not foresee happening just because it resulted from an act of intention towards something else. — Bob Ross

Of course you can. For example, if someone thinks that burning the front lawn is a good way to cleanup the yard, and lights it on fire, then lighting the fire is intentional, regardless of whether the yard actually gets cleaned up, or if the whole neighbourhood gets burned. What is significant is that the fire started from an intentional act. What the person's actual goal was when lighting the fire is insignificant. And even if the fire is started by carelessly throwing away a cigarette, that is an intentional act, so the person is responsible for the damage caused by the fire.

I don’t understand what you mean by a “conscious act” which is not intentional (in the traditional sense of intentionality); and this seems to be the crux of your argument. If I consciously decide to do X, then I intentionally did X—even if X is the end I am trying to actualize. — Bob Ross

You don't understand because you restrict "intention" to the end on your understanding. Take my previous example of tossing a burning cigarette. Suppose the person just does it by habit, having no goal in mind when the action takes place. The person does not consciously develop the goal of tossing the butt, just does it. That is what I mean by a conscious act which is not intentional (in your sense). I believe there are many such haphazard, whimsical type acts, which the average person does every day, which cannot be said to be goal-directed. You might try to say that the act itself is the goal, but it cannot be truly expressed the way you say, "X is the goal", because there is no goal, just the urge to act in a specific way. So anytime the answer to "why did you do that", is "I felt like it", this is an example. It's very common in the way that people converse (speaking being a conscious act), many times we speak without thinking, no deliberation at all. And after speaking, in these situations, I cannot say that it was my goal to say what I said, it just came to my mind in the circumstances. Young children are also more prone toward acting this way, before they learn to control themselves.

Of course not. If I take your position seriously, then it would be; because your view attaches the intentionality of an act to all causality related effects. — Bob Ross

Yes, that is what I am arguing. We ought to associate intentionality with the act itself, which is the means, rather than with the end. Intention is a cause, and what is caused is action. Within the mind, there is a process of reason which links the end to the means, and the decision is made that a particular act is required to bring about a specific end. So the relation between the means and the end is a product of the mind, and this may be mistaken.

The convention (as derived from Aristotle) is to associate "intention" with the end. But when we analyze the nature of "an intentional act", we see that intention causes an act, which is understood to be the means to an end, and intention does not necessarily cause the end (as the case of mistakes). Therefore, we can establish a direct relation between intention, (as cause), and the means, but we cannot establish a direct relation between intention and the end. So despite the convention, which is to associate intention with the end, we'd have a more true representation if we associated intention with the means, instead.

Before we dive into this, I need you to define what you mean by “intention”; because you are using it in very unwieldy ways here. — Bob Ross

I am using "intentional" to signify something which is cause by an act of intention. "Intention" refers to that part of a being which causes activity, which is commonly represented as the free will. This is slightly different from the convention, which associates "intention" with the aim, or purpose of a freely willed act. I am using it in this way, in an attempt to demonstrate that we can produce a better representation of the nature of intention, if we associate it with activities rather than the common convention which is to associate it with a thing intended.

I referred briefly to human responsibility for one's acts and one's mistakes, because the fields which deal with these acts, morality and law, are more advanced in this subject. They recognize "intentional acts". Intentional acts are supposed to be acts which are guided by an aim, or purpose, directed toward an end, but since it's often difficult for an observer to identify the goal, we often do not require that in designating an act as "intentional" in the fields of morality and law. Plato would call the intended goal "the good" toward which the act is directed, and Aristotle termed it as "that for the sake of". This is the goal of the intentional act. Intentional acts then, are understood as directed toward those goods which appear to the mind of the being.

However, this perspective runs into a problem exposed by Plato, and later discussed more extensively by Augustine. This is the question of how a man can know what is good, yet act otherwise. Quite often, the human mind apprehends a good, but does not act accordingly. This creates the issue of what exactly does direct the conscious actions which are not consistent with the apprehended good. The common explanation is that the actions are directed toward some other good. But such an "other good" is often not identifiable, and this is very evident in the case of habitual actions. So when we associate "intention" with "the good", end, or goal, we have a whole category of actions from conscious agents which cannot be classed as "intentional". These are actions of habit, and apparently random acts, which cannot be associated with any goal or end.

That is the reason why I propose that we could obtain a better understanding of the acts of conscious agents if we associate intention directly with the act, rather than with the aim of the act.

The point is that what one knows is relevant to what one is aiming at. — Bob Ross

Yes, knowledge and the aim are closely related. Reason, of some sort, tends to determine the aim, and even the goals of confused or "irrational" people are determined through some sort of faulty knowledge. The problem though is that many acts carried out are not consistent with any reasoned goal. This was the argument Plato brought against the sophists who claim to teach virtue, insisting that virtue is a type of knowledge. There is a definite separation between virtue and knowledge because virtue requires control over those habitual acts which are carried out without guidance from a reasoned aim, knowledge.

Was is intentional is not solely about the causation that occurs from a given act: it is more fundamentally about what the person is aiming at. — Bob Ross

This is what I am disputing. You get that idea because the convention is to associate intention with the aim. But what I am saying is that this convention is based in a faulty description of intentional acts. When we stipulate, that to be intentional, it is required that the act is associated with an end, then we leave a whole lot of actions of the conscious agent which cannot be categorized. They are not caused by determinist causes, nor are they directed toward an identified goal. So, I propose that we bring these acts into the category of "intentional", and this requires that we change the meaning of "intentional" to include acts which are not directed toward a specific goal.

I just think you're working yourself up over nothing. I'm losing interest. Can you write less? This is tedious, I find nothing of interest here. — fishfry

If you desire to avoid the long posts, I think, by the end of my reply here, that I have isolated the primary point of disagreement between us. It is exposed in how you and I each relate to what is referred to by "the real numbers", and what is referred to with "5" in the context of "the real numbers". And further, how this relates to the extension/intension distinction.

Therefore, I think you might just read through my post and reply to the aspects which are related to this issue. However, the issue of what mathematics is, how you and I would each describe "what mathematicians do", might also be important and relevant.

Pure math is math done without any eye towards contemporary applications. That's a decent enough working definition. Mathematicians know the difference. — fishfry

The issue though, is that even supposedly "pure" mathematicians work toward resolving problems, and problems always have a real world source or else they are really not problems, but more like amusements. A mathematician working in pure abstractions works with abstractions already produced, and may not even know how real world problems have shaped the already exist abstract structure. Even if we attempt to step aside from existing conceptions, and 'start from scratch' as philosophers often do, we are guided by our intuitions which have been shaped and formed by life in the world. And intuition comes from the subconscious into the mind, so we cannot get our minds beneath it, to free ourselves from that real world base. And since it is from the subconscious, we have no idea of how the real world effects it.

Mathematics is whatever mathematicians do in their professional capacity. — fishfry

I agree, but the description of what mathematicians do, is very difficult to get an agreement on. It's not a circular definition, but a proposal of how to produce a definition. So to actually provide the definition of mathematics, we need that description. It will be very difficult for you and I to agree on such description. You will probably place as the primary defining feature, (the essential aspect), of what mathematicians do, as working with abstractions. I will say, that description is problematic because then we need some understanding of what an abstraction is, and what it means to "work" with this type of thing. This almost certainly will lead to Platonism because we've already assumed as a premise, the existence of things called "abstractions".

Therefore I look at what mathematicians are doing as "solving problems". That's what they do, and there is a specific type of problem which they deal with. You are most likely not going to like this proposal for a description of what mathematicians are doing, because it eliminates the distinction between "pure" mathematics and "applied" mathematics. In the way described above, there is no such thing as "pure" mathematics. However, my starting point has the advantage of applying equally to all mathematicians, by applying the initial assumption of pragmaticism. Instead of saying "mathematicians are working with abstractions", we say "mathematicians are working with symbols (language), to solve problems. This way we avoid the messy ontological problem of "abstractions" It is only when we start sorting out the different types of problems which mathematicians work on, do we get the divisions within mathematics.

This is a standard complaint. If math follows from axioms, then all the theorems are tautologies hence no new information is added once we write down the theorems. But that's like saying the sculptor should save himself the trouble and just leave the statue in the block of clay. Or that once elements exist, chemists are doing trivial work in combining them. It's a specious and disingenuous argument. — fishfry

This is not the point at all, and you are not paying respect to the difference between the two distinct fields, mathematics, and mathematical logic, so your analogy is not well formed. If the field of mathematics is represented by the sculptor, then the field of mathematical logic is represented by the critic. Whenever the critic mistakenly represents what the sculptor is doing, then the critic is wrong. When mathematical logic represents mathematicians as using = to symbolize identity, the logic is wrong.

We agreed long ago that 1 + 1 and 2 are not the same string; and many people have explained the difference between the intensional and extensional meanings of a string. Morning star and evening star and all that. — fishfry

Fishfry, wake up! Was it getting late there or something? There is no physical object involved! There is no star! I think we've been through this before. The intensional/extensional distinction is completely irrelevant in this case because everything referred to is meaning (intensional). There is nothing extensional, no objects referred to by "1+1", or "2". That is the heart of the sophistic ruse. This intensional/extensional rhetoric falsely persuades mathematicians. It wrongly misleads them due to their tendency to be Platonist, and to think of mathematical abstractions as objects. As soon as meaning is replaced by objects, then "extensional" is validated, the sophist has succeeded in misleading you, and down the misguided route you go. In reality, there is only meaning referred to by "1+1", and by "2", everything here is intensional, and there is nothing extensional.

This is why I was very steadfast on the previous issue, to explain that "5" is not "an instance of a real number". It is that type of nomenclature, that type of understanding, which leads one into allowing that there is a place for extensional definitions in mathematics. Really, "5" in that example is just a part of that conception called "the real numbers". It receives it's meaning as part of that conception. there are no extensional objects referred to by "the real numbers", and "5" is just an intensional aspect of that conception. When you apprehend "the real numbers" as referring to a collection of things, instead of as referring to a conception, then you understand "5" as referring to an instance of a real number, instead of understanding it as a specific part of that conception. Then you may be misled into the "extensionality" of real numbers, instead of understanding "the real numbers" as completely intensional.

What math teacher hurt your feelings, man? Was it Mrs. Screechy in third grade? I had Mrs. Screechy for trig, and she all but wrecked me. It's over half a century later and I can still hear her screechy voice. I hated that woman, still do. When I'm in charge, I'm sending all the math teachers to Gitmo first thing. — fishfry

Again, you are not distinguishing between "mathematics", and the "mathematical logic" which the head sophist preaches. One is the artist, the other the critic. My beef is not with mathematics (the art), it is with mathematical logic (the critic). I see mathematical logic as sophistry intended to deceive. And I will explain the reason why i say there is an intent to deceive.

Mathematics has a long history of exposing us to problems which we just cannot seem to solve. These are issues such as Zeno's paradoxes, and other apparent paradoxes discussed at TPF, which generally amount to problems with the conception of infinity, the continuity of space and time, etc.. What mathematical logic does, is create the illusion that such problems have been solved. So, the intent to deceive is inherent within the conceptual structure, which makes these problems solvable. It deceives mathematicians into thinking that they have solved various problems, by allowing them to work within a structure which makes them solvable. The problem though is that the basic axioms (extensionality for example) are blatantly wrong, and designed specifically so as to make a bunch of problems solvable, regardless of the fact that incorrect axioms are required to make the problems solvable.

Whatevs. I can't follow you. And I've already noted that the difference between pure and applied math is often a century or two, or a millennium or two. — fishfry

Future application is not the issue here. The issue is that mathematicians work toward problem solving, by the very nature of what mathematics is. The problems are preexistent. Therefore mathematics by its very nature is fundamentally "applied". If you remove problem solving from the essence of mathematics, then it would be random fictions. But mathematics is not random fictions, the mathematicians always follow at least some principles of "number", already produced.

Now what do I mean by "essentially the same?" Well now we're into structuralism and category theory. Sameness in math is a deep subject. I'll take your point on that. — fishfry

What I think, is that there is really no such things as sameness in math, and this is better described as a misleading subject. Mathematics actual deals with difference, and ways of making difference intelligible through number. Similarity is not sameness, but difference which can be quantified. To me, "essential the same" just means similar, which is different.

Even so, 5 is one of the real numbers. What do you call it if not an instance? What WOULD be an instance of a real number? — fishfry

This appears to be the substance of our difference, or disagreement. If you do not like long posts, we could just focus on this specific issue. The issue is whether "the real numbers" refers to a conceptual structure, or whether it refers to a group of things, numbers. I believe the former, and the fact that "numbers" is plural is just a relic of ancient tradition. From my perspective, "5", in the context of "a real number" is just a specific part of that conception. Then the relations are purely intensional, and there is nothing extensional here. If however, you apprehend "the real numbers" as referring to a group of things called "numbers", then "5" refers to one of those things, and there is the premises required for extensionality.

Explicity stated in any textbook in mathematical logic. — TonesInDeepFreeze

This is exactly the problem. Notice you refer to "any textbook in mathematical logic", rather than any textbook in mathematics. If you look at a textbook in mathematics, you'll find "=" defined in the way of my reference, "equals"... "indicates two values are the same". So there is a discrepancy between what "=" means in mathematics, and what "=" means in mathematical logic. And, since mathematical logic is supposed to provide a representation of the logic used in mathematics, we can conclude that mathematical logic has a false premise. The proposition of mathematical logic, which states that "=" indicates identity, or that it means "is" or "is the same as" in mathematics, is a false premise. This is a false representation of how "=" is used in mathematics. As I explained in other threads, if "=" indicated "is the same as", then an equation would be absolutely useless, because the left side would say the exact same thing as the right side. (Incidentally, this is what many philosophers have been known to say about the law of identity, it is a useless tautology). In ontology though we see the law of identity as a useful tool against sophistry like yours.

You agree with me about pure math. — fishfry

I agreed with you about "pure math", for the sake of discussion, so that we could obtain some understanding of each other. But I will tell you now, as came up one other time when we had this discussion, I do not agree that there is such a thing as "pure math" by your understanding of this term. So I agree that if there was such a thing as pure math, that's what it would be like. However, I think your idea of "pure math" is just a Platonist/formalist fantasy, which is a misrepresentation of what mathematics is. In reality, all math is corrupted by pragmatics to some degree, and none reaches the goal of "pure math". You criticize me to say, it's not a goal, it's what pure math is, but I say that's false, it is a goal, an ideal, which cannot be obtained. Therefore "pure math" as you understand it, is not real, it's an ideal.

I think the issue being exposed here is a difference of opinion as to what mathematics is. Since this is a question of "what something is", the type of existence it has, I think it is an ontological issue. Would you agree with this assessment? For example, the head sophist refers to "mathematical logic", and I find this defined in Wikipedia as the study of the formal logic within mathematics. So we have a distinction here between the use of mathematics (applied mathematics), and the study of the logic used by mathematicians (mathematical logic). "Mathematical logic" would be a sort of representation, or description, of the logic used in mathematics. What you call "pure mathematics", I believe would be something distinct from both, applied math and mathematical logic, as the creative process whereby mathematical principles are developed. But I think that this process is not really "pure", it's always tainted by pragmatics and therefore empirical principles.

The issue I have with the head sophist is with the way that mathematical logic represents the use of the = symbol as an identity symbol. In applied mathematics, it is impossible that "=" is an identity symbol because if both sides of an equation represented the exact same thing, the equation would be absolutely useless. This I've explained in a number of different threads. In reality, as any mathematics textbook will show, "=" means "has the same value as". Therefore we can conclude that any mathematical logic which represents "=" as an identity symbol is simply using a false proposition. When a "textbook in mathematical logic" states that "=" is an identity symbol, this can be taken as the false premise of mathematical logic.

You have conceded my point regarding math. I have no other point. — fishfry

I have conceded the point regarding what you call "pure math". However, I am now qualifying this concession to say that "pure math" is just an unreal ideal. There is no such thing as pure math. It's a term which people like to use in an attempt to validate their ideals. In reality though, such ideals are fiction, so all that I have really conceded, is that within the fictitious conception which you call "pure math", this is the way things are. Of course, I'm not going to argue about the way things are in your work of fiction, but I will argue about the way that your fiction bears on the real world.

Tens of thousands of professional pure mathematicians would disagree. — fishfry

Sure, there are thousands of people who might call themselves "pure mathematicians". In reality though, these people are not engaged in "pure mathematics", as I believe you understand this to mean. As I said above, all mathematics is tainted by pragmatics (applications), and there is no such thing as "pure" mathematics.

This is very evident in our discussion of the meaning of "=". In what you call "pure mathematics", we might say that "=" signifies "is the same as". This would remove the basic fact that what mathematicians work with are values. To make the mathematics "pure" we must remove this content, what the mathematicians work with, values. We remove the inherent nature of the thing represented by the symbols (i.e. that the symbols represent values) to allow simply that the symbols represent things without any inherent nature, no inherent content. Then we might claim the left side of the equation represents the exact same thing as the right. However, this type of equation would be totally useless. We could do nothing with an equation, solve no problems.

Furthermore, there would be a disconnect, an inconsistency between the mathematicians practising "applied" math, who use "=" to represent "is the same value as", and those "pure" mathematicians creating mathematical principles which were inconsistent with the applied mathematics. Since the supposed "pure mathematicians" actually produce principles which are compatible with, and are actually used in applied mathematics, we can conclude that the supposed "pure" mathematics is not really pure, and the principles they are using and developing do not really treat "=" as meaning "is the same as". That's just a misrepresentation, supported by the misrepresentation that these people are doing "pure" mathematics.

Any two set theorists will give {0, 1, 2, 3, 4} as the definition of 5. That's due to John von Neumann, who invented game theory, worked on quantum physics, worked on the theory of the hydrogen bomb, and did fundamental work in set theory. Now there was a guy who blended the applied with the pure. — fishfry

I can't say I understand everything you wrote following this, but it mostly makes sense to me. I'll have to work on these ideas of "mod 4", and "cyclical group".

Can you give an example? I might have not followed you. — fishfry

What I mean, is that if you recognize that two things are different from each other, then that difference has already made a difference to you (in the subconscious for example) by the very fact that you are recognizing them as different. So for example, if you see two chairs across the room, and they appear to be identical, yet you see them as distinct, then the difference between them must have already made a difference to you, by the fact that you see them as distinct. So to say that the difference between them is a difference which doesn't make a difference must be a falsity from the outset. We might even say that they are identical in every way except that they are in different locations, but this very difference is the difference which makes them two distinct chairs instead of one and the same chair.

A type of number. No, don't agree. Real numbers and complex numbers and quaternions are types of numbers. The real number 5 is an instance of a real number hence an instance of a number. It must be so, mustn't it? — fishfry

I knew you wouldn't agree, but i wouldn't agree that the real number 5 is an instance of a real number. The problem I think has to do with the statement "a real number". "The real numbers" is a conceptual construct in itself. This conception dictates the the meaning of "a real number". So in reality any supposed instance of "a real number" is just a logical conclusion drawn from the dictates of "the real numbers". In other words its not a distinct or individuated thing, which would be required for "an instance", it's just a specific part of "the real numbers". Can we agree that the real number 5 is a specific real number?

= Equal sign ... equals ... Indicates two values
are the same -(-5) = 5
2z2 + 4z - 6 = 0

https://www.techtarget.com/searchdatacenter/definition/Mathematical-Symbols

it is explicitly stated that '=' is interpreted as 'is' in mathematics. — TonesInDeepFreeze

Explicitly stated by you, the head sophist

Mathematics adheres to the law of identity, since in mathematics, for any x, x=x, which is to say, for any x, x is x. — TonesInDeepFreeze

Sorry Tones, but "for any x, x=x" does not say "for any x, x is x", unless "=" is defined as "is". And, in mathematics it is very clear that "=" is not defined as "is".

Not necessarily. If the side effect is not easily foreseen, then we typically don't consider it intentional; or we might say that it was intentional insofar as the person was aware that there was a chance of it happening and accepting those odds. However, in the case that it is foreseeable or was foreseen (with high probability)(all else being equal), then I completely agree it was intentional: it as indirectly intended, which entails it was not accidental.

You can't say some accidents are intentional: that's like saying some orange squares are not orange. — Bob Ross

I'm talking about things totally unforeseen. I agree that we do not commonly call them "intentional", but in the sense that they are the direct effect of the intentional act, just like the desired end is the direct effect of the intentional end, there is fundamentally no essential difference between them. That is why we are just as responsible for our mistakes as we are for our correct actions. Do you agree that you are responsible for your mistakes, as they are the results of your intentional acts?

The hammer hitting your thumb was not intentional whatsoever prima facie in your example. The act of swinging the hammer, intending to bring about the end of hitting the nail into something, was intentional. — Bob Ross

I don't agree with this. The first premise is that the act of swinging the hammer was intentional. Do you agree? You claim that the effect of the act can be separated from its cause, to say that the cause was intentional but the effect was not intentional. The point being that when things are set in motion by an act of intention, and we allow that more than just the immediate act itself is intentional, that an effect is also intentional, then we need consistency, and say that all the effects are intentional.

What we are talking about is misjudgment, mistake. The fact that a person misjudges the effects of one's actions does not make the effects any less intentional. It just means that the person made a mistake in judgement. A mistake in judgement does not remove intentionality from the act, nor does it remove intentionality from the effects of the act.

Now, let's say you foresaw that the hammer might hit your thumb and new this with 20% probability and still decided to carry it out: we would say that you intentionally swung the hammer knowing it may result in an accident, but we would NOT say that you intentionally caused that accident. Now, let's say you foresaw with a 99% probability that you were going to cause the accident instead of what you really intend, then we might say you intended it because of the probabilistic certainty that you had of bringing it about. It depends though, because we might say you are just stupid and didn't realize that it doesn't make sense to carry it out with that high of a probability; or we might say you are unwise (unprudent) for doing it anyways out of (presumably) passion or desire to hit the nail. — Bob Ross

This does not make any sense to me. A judgement as to the probability of success of one's intentional acts, is not useful toward determining whether the effect of that act is intentional or not. Suppose I flip a coin, and the probability is 50/50. No matter what the outcome is, that outcome was intended, because I flipped the coin for the purpose of having an outcome, and the particular outcome which occurs is irrelevant to that intent. Likewise, when I make any intentional act, the goal is that the act will have an effect. It's true that I desire a specific outcome, like when I bet on the coin toss, but the fact that one outcome is more desirable than others, does not make that outcome more intentional than the others. Does it make any sense to say that when I bet on heads, if it lands heads, that was intended, and if it lands tails that was not intended?

My main point is just that accidents, by definition, cannot be intentional. That's categorically incoherent to posit. — Bob Ross

I do not agree with this. I think that we simply misuse "intentional" to say that the desired outcome is intentional, and the undesirable outcomes are not intentional. Each effect is essentially the same, of the same type or category, the effect of an intentional act. It is inconsistent, and therefore incoherent, to say that one effect of the intentional act is intentional, and another effect is not intentional.

Now explain this to me ONCE AND FOR ALL. Are we talking about pure math and set theory? Or are we talking about the physical world of time, space, energy, quantum fields, and bowling balls falling towards earth? — fishfry

I don't understand you. I gave you an example of how equivocation of "same" has a considerable effect. Of course it has no effect in "pure mathematics", because by definition "pure" mathematics maintains its purity, and the purity of its definitions. Pure mathematics is not applied, and therefore has no effect in relation to the physical world where "same" means something else.. We live in the physical world, our cares and concerns involve the world we live in, it is impossible that anything in the fantasy world of "pure mathematics" could actually concern us. This is known as the interaction problem of idealism. However, in reality we apply mathematics and this is where the concerns are.

You seem to misunderstand the issue completely. You appear to understand that there is a difference between the use of "same" by mathematicians (synonymous with equal), and the use of "same" in the law of identity (not synonymous with equal). You said that this difference has no bearing on anything you know or care about. The things included in the category of what you know and care about, are not limited to principles of pure mathematics, because you live and act in the physical world. The law of identity applies to things in the physical world which we live and act in.

So, to make myself clear, I do not claim that there is a problem with using "same" as synonymous with equal, within the conceptual structures of mathematics. The problem is in the application of mathematics, as inevitably it is applied, and this use of "same" is brought into the world of physical activity, and taken to be consistent with the use of "same" when referring to physical objects. That is where the problem occurs. Sophjists such as Tones enhance and deepen the problem by arguing that the use of "same" in mathematics(synonymous with equal) is consistent with the use of "same" in the law of identity (not synonymous with equal).

You can not have it both ways. — fishfry

This is exactly the issue, the reality of the situation is that we do have it both ways. There are two very distinct ways for understanding "same". You can dictate "you cannot have it both ways" all you want, but that's not consistent with reality, where we have both ways of using the term. If you think that we ought to reduce this to one, (insisting "we cannot have it both ways"), the two cannot be combined, or reduced to one, because they are fundamentally incompatible (despite what the head sophist claims). This means that we have to choose one or the other. If we choose the one from pure mathematics, then we have nothing left to understand the identity of a physical object in its temporal extension. If we choose the one from the law of identity, then we simply understand "equal" as distinct from "same", and the problem is solved. Obviously the latter makes the most sense, and doing this would support your imperative dictate: "You can not have it both ways."

No. You don't understand how math works, and you continually demostrate that. — fishfry

It is very clearly you are the one who does not understand how math "works". Math only works when it is applied. "Pure mathematics" does nothing, it does not "work", as math only works in application. You are only fooling yourself, with this idea that pure mathematics is completely removed from the physical world, the world of content, and it "works" within its own formal structures. That is the folly of formalism which I explained earlier. To avoid the interaction problem of Platonist idealism, the formalist claims that mathematics "works" in its own realm of existence. But the claim of "works" is sophistic deception, and the formalist really digs deeper into Platonism, hiding behind the smoke and mirrors of the sophistry hidden behind this word "works". That is when the term "mathemagician" is called for.

You finally said something interesting. Is the 5 in your mind the same as the 5 in my mind? I think so, but I might be hard pressed to rigorously argue the point. — fishfry

I believe, the concept of "five" in my mind is completely different from, though similar to, the concept of "five" in your mind. There is a number of ways to demonstrate the truth of this. The first is to get two different people to define the term, and see if they use the exact same expression. Another way is to look at what "five" means in different numbering systems, natural, rational, real, etc.. Another is from the discussions of mathematical principles in general. There is always difference in interpretation of such principles. You and I have significant differences, You and Tones have less significant differences.

Nevertheless, the differences exist and are very real. There is a principle which I've seen argued, and this is to say that this type of difference is a difference which does not make a difference. Aristotle called these differences accidentals, what is nonessential. The problem with that expression though, "difference which does not make a difference", is that to notice something as a difference, it is implied that it has already made a difference. So this argument is really nothing other than veiled contradiction.

Anyway, this is the issue with identity, in a nutshell. When we ignore differences which we designate as not making a difference, and say that two instances are "the same" on that basis, we really violate the meaning of "same". The meaning of "same" is violated because we know that we are noticing differences, yet dismissing them as not making a difference, in order to incorrectly say "same". Therefore we know ourselves to be dishonest with ourselves when we say that the two instances are the same, by ignoring differences which are judged as not making a difference. So when you say that you think the 5 in my mind is the same as the 5 in your mind, I think that this is an instance of dishonesty, you really know that there are differences, and if pressed to argue such a claim, you'd end up in contradiction, dismissing the obvious differences as not making a difference.

Is an apple an instance of fruit? Apples don't have a peelable yellow skin. 'Splain me this point. By this logic, nothing could ever be a specific instance of anything, since specific things always differ in some particulars from other things in the same class. — fishfry

Right, particulars are instances, specifics are not. The concept "red" is not an instance of colour, it is a specific type of colour. A particular red thing is an instance of red, and an instance of colour, exemplifying both. The concept "apple" is not an instance of fruit, it is a specific type of fruit. A particular apple is an instance of both. The concept "5" is not an instance of number, it is a specific type of number. A group of five particular things is an instance of both.

When I arrive home in the evening, it makes quite a big difference to me if I return to the same residence or just one that's "equal" to it in value. — fishfry

Hey fishfry, do you not remember what you said to me? You said " I don't make a distinction between "same as" and "is equal to." In math they're the same. If you have different meanings for them, it does not bear on anything I know or care about." Now you've totally changed your position to say "it makes quite a big difference to me", if the taxi driver took you to a house which had an equal fare as yours, but was not the same house.

I think it would be more accurate to say "The apparent unintelligibility is due to a thing's matter or potential." — Ludwig V

I don't quite get what you mean here. Let's say there's something about reality which appears to be unintelligible. If we assign a name to that aspect, aren't we saying that there is actually something there which is unintelligible, and we've named it. This is to take a step further than simply that it appears as unintelligible.

I don't think that's quite right. It is true that if the lamp is on, it has the potential to be off, and if the lamp is off, it has the potential to be on. But that's not the same as having the potential to be neither off nor on. — Ludwig V

The point is that potential defies the laws of logic. That's why modal logic gets so complex, it's an attempt to bring that which defies the laws of logic into a logical structure.

The point I made, derived from Aristotle, is that whenever the lamp switches from on to off, or vise versa, there is necessarily a period of time during which it is changing (becoming). In other words, it is impossible that the switch from one to the other is instantaneous, and this is proven logically. In this 'mean time', the lamp is neither on nor off, and this defies the law of excluded middle. Dialethists would hold that it is both on and off, defying the law of noncontradiction.

Aristotle uses the concept of "potential" to explain his choice for defying the law of excluded middle rather than defying the law of noncontradiction. For him, the concept of "potential" is required to explain how something changes form having x property (being on), to not having x property (being off). "Potential" is a requirement of such a change, the thing cannot change without having the potential for change. However, this is a temporal concept, and the conclusion is that actualization requires a duration of time. So there is always a period of time between having x property (being on), and not having x property (being off).

What the lamp problem does not take into account, is that period of time between being on and off, during which it is changing. Assuming that the amount of time required to change from on to off, and vise versa, remains constant, then as the amount of time that the lamp is on and off for, gets smaller and smaller, the proportion of the time which it is neither, gets larger and larger. So at the beginning, when the time on and off are relatively long periods, the time of neither seems completely insignificant. But as the off/on actualization rapidly increases, the time of being on and off soon becomes insignificant in comparison to the time of being neither. The time of neither approaches all the time

A lamp, by definition, is something that is on or off, but not neither and not both. There are things that are neither off nor on, but they are not lamps and the point about them is that "off" and "on" are not defined for them. Tables, Trees, Rainbows etc. — Ludwig V

You only say this, because the time of change in which the lamp is neither on nor off is so short and insignificant that it appears to be nil. Aristotle demonstrated logically that it cannot be nil. So when we say things like "lamps are a type of thing which must be on or off, and cannot be neither", this is a statement about how things appear to be, and this facilitates much of our talk about such things. But when we get down to the way that things actually are, the way that logic tells us they must be, we can see that this way of allowing appearances to guide our speaking is actually misleading.

I don't think that's quite right. The LEM does not apply, or cannot be applied in the same way to possibilities and probabilities. "may" does not usually exclude "may not". On the contrary, it is essential to the meaning that both are (normally) possible - but not both at the same time. — Ludwig V

I don't understand this. If a thing neither has nor has not the specified property, the excluded middle principle is violated (unless it's an inapplicable category). Potential itself neither is nor is not, and that's why we say it refers to what may or may not be. So "may or may not be" refers to the property we judge as in potential, and this says it neither is nor is not attributable to the thing.

And are there 'vast differences' between Plotinus and Plato? I readily grant at every juncture that your knowledge of the texts greatly exceeds my own, but I had thought it well-established that Plotinus saw himself as no more than a faithful exegete of Plato. — Wayfarer

Yes there are vast differences between Plotinus and Plato. Plotinus goes far beyond Plato in his theory of Forms, to propose a hierarchical order, emanation. and even assigning a position to "matter".

I think it is a matter of two different ways of using the word, "intentional". In one sense, we say that an accident is not intentional. However, in another sense, when something is the effect of intention, we say it is intentional regardless of whether the effect is accidental.

For example, I swing the hammer at a nail, and accidentally hit my thumb. The act of swinging the hammer was intentional, regardless of whether I hit the nail or my thumb. So whether the nail is hit or my thumb nail is hit, is irrelevant to the fact that the act which results in one or the other is an intentional act. So even though it is my thumb which is hit, the act which has that effect is intentional.

What we have therefore is a separation between the act and the effect of the act. This is the separation between the means and the end. Both "means" and "end" refer to what is intended, but the effect of the chosen means (the act) is not necessarily consistent with the intended end, so the effect of the action may be unintended. Since the act is intended (swinging the hammer) yet the effect (hitting my thumb) is not intended, we must assume a separation between cause and effect, a lack of necessity in that relationship, to allow that the one is intentional and the other is not.

All eels have very interesting life cycles. They begin life in the ocean, and stay there many years, often traveling thousands of kilometers. Then they travel up fresh water rivers and streams for many more kilometers. There they stay for many years, to feed and grow. Then they head back to the ocean traveling thousands of kilometers to spawn. All the while, they metamorphose through numerous stages suited to what they are doing at the time.

The substance of these questions has been before you repeatedly and you make no substantive answer. — tim wood

Please allow me to clarify, if I wasn't clear enough for you last time. You and I have a completely different understanding of the nature of "a relation". We could not even find grounds to start any agreement, to converse. Consequently you'll understand "relation" in your way, and I'll understand "relation" in my way. Since "order" is a specific type of relation, any discussion about order, between us, will be rife with misunderstanding. Furthermore, I have no inclination to stoop to your level, and utilize your meaning of "relation", so that you might actually understand me, because I see it as nothing but childish closed mindedness.

I don't make a distinction between "same as" and "is equal to." In math they're the same. If you have different meanings for them, it does not bear on anything I know or care about. — fishfry

This is exactly the problem, failing to distinguish between "same as" and "equal to". Because you do not believe that there is a distinction to be made here, you will not notice the effects of such a failure, and you will insist that it doesn't bear on anything you care about. Insisting that it doesn't bear on anything you care about will allow you to be mislead, even tricked by intentional deception (as you were by the sophist's employment of "identity of indiscernibles"), and you may never ever even notice it.

Here is a simple example of where the difference bears in a substantial way, though I am sure there are more complex examples. In quantum physics, a quantum of energy is emitted as a photon, and an equal quantum may be detected as a photon. Since these two quanta of energy are equal, they are said to be "the same" photon. That is the mathematician's use of "same", equal quanta implies one quantum, a photon. By the law of identity "same" implies temporal continuity, such that the photon exists, with that identity, for the entire period between emission and detection. Equivocation between these two senses of "same" inclines some people to believe that the photon exists, as the same "particle" for the entire period of time between emission and detection. However, the electromagnetic energy is observed to exist as waves in the meantime.

This produces significant theoretical problems. Some claim a contradictory wave/particle duality theory, in which the energy travels as both waves and as particles at the same time. Furthermore, since the photon of energy emitted is assumed to be "the same photon" as the photon detected, and it's path cannot be determined, it is claimed to have multiple paths all at the same time. All of this sort of problem is due to equivocation of "same". The mathematical "same", an equal quantum of energy is emitted and detected, is confused with "same" by the law of identity, to conclude that a distinct quantum (particle) of energy, known as the photon, has continuous existence between the time of emission and the time of detection.

You say that this issue doesn't bear on anything you know or care about, but until you recognize and understand the issue you'll never know how it bears. Furthermore, I saw how the head sophist, persuaded you to see a mathematical axiom differently, through reference to the identity of indiscernible, so I know that it really does bear on things that you care about.

No, orderings are not "contradictory properties." Technically, an order on a set is another set, namely the set of pairs (x,y) for which we mean to denote that x < y in the ordering. The ordering is distinctly and noticeably separate from the set it applies to. — fishfry

What is said about a thing is distinct from the thing itself. Contradiction is not in the thing itself, it is in what is said about the thing. To say that a thing has contradictory orderings is contradiction. The contradiction "is distinctly and noticeably separate from the [thing] it applies to".

That distinction has no meaning or relevance in my understanding of the world. "equal" and "the same as" are entirely synonymous. — fishfry

You are in denial, just like the sophist. "Equal" means to have the same value within a system of valuation, "same" means identical, not different. Notice that "equal" is a qualified sense of "same" the "same value", meaning identical value, whereas "same" refers to identity itself without such qualification. Two distinct things are said to be equal, being judged according to a specified value system. Two distinct things are not the same. Please tell me that you understand this difference.

Would you agree that "number" is a general abstraction and that 5 is a partcular instance of number? Isn't that the most commonplace observation ever? — fishfry

This is colloquial vernacular insufficient for logical rigour. The proper classification is like this. The abstraction "number" is more general, and the abstraction "5" is more specific, just like "animal" is more general, and "human being" is more specific, or "colour" is general and "red" specific. Neither is a "particular instance".

One might however say that there is a particular instance of the abstraction "5", and the abstraction "number", in your mind, and another particular instance in my mind. But that would be an ontological stance which would be denying common Platonism. Platonists would say that what I just called particular instances, are really just parts of one unified concept "5".

Red is not an instance of the concept of color? How do you figure that? — fishfry

Fishfry, do you not understand what "instance" means? Here, from OED, "an example or illustration of". How do you think that a specific colour, red, is an example or illustration of the concept of colour? Red cannot exemplify "colour", because all the other colours are absent from it. That's why we go from the more general to the more specific in the act of explaining. Referring to the more specific abstraction, "red" is an instance of "specifying", it is not an instance of "colour".

"The axiom given above assumes that equality is a primitive symbol in predicate logic." — fishfry

How do you propose that this indicates that equal implies "same as"? I cannot follow your association. What the head sophist calls "identity theory" is simply an axiom of identity which is inconsistent with the law of identity. The sophist dictates that "=" means identical to, and this is the first principle of the sophistry referred to as "identity theory".

Humans, more than most animals, are "animated by purpose". — Gnomon

This is debatable. Humans, with intellect and will, provide themselves the freedom to choose from a multitude of options in their activities. Animals seem to be driven towards very specific goals, without the capacity to choose. Each is "animated by purpose", and I do not see how you could argue one is more so than the other.

If the lamp is neither off nor on at 12:00 (and still exists) then it must be in a third state of some kind. — Ludwig V

This issue was actually resolved a long time ago by Aristotle, in his discussions on the nature of "becoming". What he demonstrated is that between two opposing states (on and off in this case), there is a process of change, known as becoming. This process is the means by which the one property is replaced by the opposing property. If we posit a third state between the two states, as the process of change, then there will now be a process of change between the first and the third, and between the third and the second. We'd now have five distinct states, and the need to posit more states in between, to account for the process of change which occurs between each of the five. This produces an infinite regress.

So what Aristotle proposed is that becoming, as the activity which results in a changed state, is categorically different from, and incompatible with states of being. Further, he posited "matter" as the potential for change. "Potential" refers to that which neither is nor is not. As what may or may not be, "potential" violates the law of excluded middle. So in the example, when the lamp is neither on nor off, rather than think that there must be a third state which violates the excluded middle law, we can say that it is neither on nor off, being understood as potential. This is the way that I understand Aristotle to have proposed that we deal with such activity, which appears to be unintelligible by violation of the law of excluded middle, neither having nor not having a specified property. The unintelligibility is due to a thing's matter or potential.

Accidents are never intentional; but evolution doesn't operate on accidents. — Bob Ross

"Accidents are never intentional" needs clarification. An accident can be the unintended result of an intentional action. As the result of an intentional act, it is cause by intention, just like the intended goal is also the result of the intended act, and caused by intention. So, from the point of view of efficient causation, there is no difference between the accident, and the successful end. However, since the accidental effect is said to be unintended, we need another way to talk about this type of effect of an intentional act, so it's called accidental. Therefore we have two types of effects of intentional acts, those intended, and those not intended.

And then again, we have no evidence that the mathematical real numbers are even a decent model for time. The real numbers are continuous, but nobody knows if time is. — fishfry

I think that this is what the so-called "paradox" of supertasks is all about. What is revealed is that at least one or the other, space or time, or both, must not be continuous. I think that's what @Michael has been arguing since the beginning. Tones attempted to hide this behind sophistry by replacing the continuity of the real numbers with the density of the rational numbers.

The real issue is that if one of these, space or time, is not continuous, then it cannot be modeled as one thing. There must be something else, a duality, which provides for the separations, or boundaries. But I don't think anyone has shown evidence of such a duality, so we have no real principles to base a non-continuous ordering system on.

It's a shame we use the word "approach," because many are confused by that. — fishfry

I'd say this is similar to Tones' use of "identity" in set theory. We take a word, such as "approach", which clearly does not mean achieving the stated goal, and through practise we allow vagueness (to use Peirce's word), then the meaning becomes twisted, and the use of the word in practise gets reflected back onto the theory. So we have the theory stating one thing, and practise stating something different, then the meaning of the words in the theory get twisted to match the practise. Practise says .999... is equal to 1, so "approach" in the theory then takes on the meaning of "equal". Practise says that two equal sets are identical, so "equal" in the theory takes on the meaning of "identical". These are examples of how theory gets corrupted through practise when the words are not well defined.

I believe that's what The Lounge is for. The deep stuff gets booted off the main page, being for most, undistinguishable from shit.

dissing me with passive aggressive faint praise as a way to diss the other poster — TonesInDeepFreeze

I agree, it's no compliment to say that you're higher in the order of virtue than tim is. I should have just spoke the truth, tim is even lower down than you are. And both of you make a snake appear like an angel.

Sorry tim, I have no interest in engaging with you here in the Lounge. You have demonstrated that you are very steadfast with an extremely closed mind. Banter with TIDF is at least somewhat amusing. That sophist actually has a sense of humour and some degree of conscience, which you seem to be fully and completely lacking in. Furthermore, Tones actually listens to what I say, and sometimes makes an attempt at understanding, whereas you simply dismiss it as "toxic", "dishonest", and "a waste of time". No point in wasting time if it doesn't come with entertainment.

As I said, the context dictates their order, and context is singular. An object does not exist in a multitude of distinct contexts at the same time, despite the fact that the context may change over time. I covered change and temporality in my reply to fishfry. You might go back and read that.

Yikes. — TonesInDeepFreeze

The sense of humour leaves the head sophist exposed, revealing no control over the inclination to equivocate.

As they can relate in multiple ways, it would seem, according to you, they can have more than one order. Thus you say they have one order and no other, and yet many. — tim wood

This is not more than one order, it is just different aspects of one order, like one object has numerous properties, but the properties are all just different aspects of the one object. That we say a thing has volume, weight, colour, etc., is just a feature of how we describe the thing. For example, saying B is prior to A in the smallness scale, and A is prior to B in the largeness scale, does not mean that the two objects have more than one order, it's just a feature of the way we describe things, i.e. our way of imposing a conceptual order.

You appear to be mixing up the natural order which things have by being the things which they are, in the circumstances which they are in, and the conceptual order which we artificially impose on the thing in abstract understanding. That is the same mistake the head sophist makes, failing to distinguish the concrete thing itself, from the conception of it.

Judging by what you argued in the other thread on "purpose", you believe that the only kind of order is conceptual order. Until you realize that this idea is faulty, you will never understand the natural order that things have simply by being the things that they are, in the circumstances that they are in. Discussion seems pointless right now.

Very nice. How toxic of you, MU. But note that what I "put down" is just what you put down, I merely asking you to make sense of it. — tim wood

I never spoke about "one and one", nor about "many". I have no idea what you are making reference to, or how you draw the conclusion that "one and one" are a requirement for "many".

I'm not trying to be toxic, only I have no idea of what you are trying to express. What I said was an expression meaning that I am not understanding what you are saying.

@fishfry
I think you and I agree substantially on the difference between abstractions and physical objects, and that the elements of a set are always abstractions and never physical objects. So you might avoid a long reply on that subject. It's only the head sophist who disagrees with us on this, claiming that the elements of a set may be concrete objects.

We do have significant disagreement concerning your claim to a proof that "X=X", when X signifies a set, means that X is the same as itself by virtue of the law of identity. You have not provided that proof in any form which I could understand.

It appears, then, that one and one and no other is actually a many. — tim wood

Sorry tim, I'm not picking up what you're putting down.

It is not any more a contradiction for a set to have more than one ordering than it is a contradiction for a person to own more than one hat. — TonesInDeepFreeze

Having X hat does not exclude having Y hat, that's obvious. The two do not contradict. But if X order contradicts Y order (e.g. John is closer to the front of the stage than Paul, contradicts Paul is closer to the front of the stage than John), then X order excludes Y order.

Your analogy is not relevant because having one property clearly does not exclude the possibility of having another property, but having the property of one order clearly contradicts having the property of the contradictory order.

I think it's time for you to go back to grade school and learn some fundamentals of logical thinking.

Set consisting of three balls colored red, white and blue. They also have differing weights. What is THE order? Just curious. — jgill

Show me your balls and I will tell you their order.

And exactly what order is that? — tim wood

However those objects relate to other objects, the context, or environment they are in, dictates their order.

Ok this is interesting. My quote was, "Sets are subject to the law of identity." So that if X is a set, I can say X = X without appealing to any principle of set theory.

Tones convinced me of that. Now you say that he only sophisted me. How so please? If X is a set, how is X = X not given by the law of identity? You have me curious.

You think I'm a victim of Tones's sophistry. That is an interesting remark. — fishfry

What is the case, is that "X=X" is an ambiguous and misleading representation of the law of identity. This is because "=" must mean "is the same as", to represent that law, but it could be taken as "is equal to". Notice that in the axiom of extensionality it is taken to mean "is equal to". Therefore when Tones takes "X=X" to be an indication of the law of identity there is most likely equivocation involved.

Set theory is a mathematical structure. The analogy is:

Set theory is to group theory as a particular set is to a particular group.

But a set is a mathematical structure too, since the elements of sets are other sets. — fishfry

So, do you recognize, and respect the fact that group theory is separate from, as a theoretical representation of, the objects which are said to be members of a specified "group"? And, I'm sure you understand that just like there is a theoretical representation of the group, there is also a theoretical representation of each member of the group. In set theory therefore, there is a theoretical "set", and also theoretical "elements".

So when Tones says that a set may consist of concrete objects, this is explicitly false, because the set is the theoretical representation, and the elements of the set are theoretical representations as well. Through such false assertions, Tones misleads people and earns the title of sophist.

When Tones speaks about the set "George, Ringo, John, Paul", these names signify an abstract representation of those people, as the members of that set, the names do not signify the concrete individuals. You, Fishfry, have shown me very clearly that you know this. So there is an imaginary "George", "Ringo" etc., which are referred to as members of the set. The imaginary representation is known in classical logic as "the subject". We make predications of the subject, and the subject may or may not be assumed to represent a physical object. Comparison between what is predicated of the subject, and how the object supposedly represented by the subject appears, is how we judge truth, as correspondence.

What is important to understand in mathematics, is that the subject need not represent an object at all. It may be purely imaginary, like your example Cinderella. This allows mathematicians to manipulate subjects freely, without concern for any "correspondence" with objects. Beware the sophist though. I believe that when the sophist says that the members of a set may be abstractions, or they may be concrete objects, what is really meant if we get behind the sophistry, is that in some cases the imaginary, abstract "element", may be assumed to have a corresponding concrete object, and sometimes it may not. Notice though, that in all cases, as you've been insisting in discussions with me, the elements of the sets are abstractions, as part of the theory, and never are they the actual physical objects. Failure to uphold this distinction results in an inability to determine truth as correspondence. And that is the effect of Tones' sophistry

This is true about kids in playgrounds, NOT mathematical sets. You have informed me that you don't like real-world analogies so I no longer use them. Mathematical sets have no inherent order. — fishfry

I'll return to the schoolkids example briefly to tell you why I didn't like it. Using that example made it unclear whether "schoolkids" referred to assumed actual physical objects, or imaginary representations. That's why "real-world analogies" are difficult and misleading. The names, "George", "Paul", etc., appear to refer to real-world physical objects, and Tones even claims that they do, but within the theory, they do not, they are simply theoretical objects. If we maintain the principle that the supposed "schoolkids" are simply imaginary, then they have no inherent order unless one is stipulated as part of the rules for creating the imaginary scenario. Set theory ensures that the elements have no inherent order, but this also ensures that the elements are imaginary.

A temporal extension. You are saying it only applies to things that exist in time? Meaning not sets? I don't think that's right. Any set is identical to itself and also equal to itself by virtue of the law of identity. — fishfry

This is wrong, and where Tones mislead you in sophistry. A set is not identical to itself by the law of identity. The set has multiple contradictory orderings, and this implies violation of the law of identity. We allow that "a thing", a physical object has contradictory properties with the principle of temporal extension. At one time the thing has a property contradictory to what it has at another time, by virtue of what is known as "change", and this requires time. But set theory has no such principle of temporality, and the set simply has multiple (contradictory) orderings.

Tones did explain that to me, but not via sophistry. He asked me to prove the transitivity of set equality. Once I attempted to do that, I realized that I needed not the axiom of extensionality, but its converse. And that converse is true by way of the law of identity from the underlying predicate logic. This I discovered for myself when Tones pointed me to it. — fishfry

As I said, the reference was to the identity of indiscernibles, not the law of identity. You recognize that these two are different. The proof was not by way of the law of identity. If you still believe it was, show me the proof, and I will point out where it is inconsistent with the law of identity.

I tell you that a set has no inherent order; and that the set of natural numbers in its usual order; and the set of natural numbers in the even-odd order say -- 0, 2, 4, 6, ...; 1, 3, 5, 7, ... is exactly the same set. It is a different ordered set, because in an ordered set, the order is part of the identity of the set. In a plain set, it's not. This is how mathematicians play their abstraction game. — fishfry

We agree on this very well. The principle we need to adhere to, is that this is always an "abstraction game". If we start using names like "Ringo" etc., where it appears like the named elements of the set are concrete objects, then we invite ambiguity and equivocation. And if we assert that the elements are concrete objects, like Tones did, this is blatantly incorrect.

Yes, well, discussions of denying LEM don't interest me much. I'll agree with that. But I've come by it honestly. I've made a run at constructivism and intuitionism more than once. I've read Andrej Brauer's "Five Stages of Accepting Constructive Mathematics." It doesn't speak to me. The paragraph you quoted is a little above my philosophical pay grade. Perhaps you can explain its relevance to the topic at hand. — fishfry

The three fundamental laws of logic, identity, noncontradiction, and excluded middle, are inextricably tied together. Therefore one cannot discuss identity without expecting some reference to the other two. There has been some philosophical discussion as to which comes first, or is most basic. Aristotle seemed to believe that noncontradiction is the most basic, and identity was developed to support noncontradiction.

What C.S. Peirce noticed, is that if we allow abstract objects to have "identity" like physical objects do, as Tones seems to be insisting on, then necessarily the validity of the other two laws is compromised. Instead of denying identity to abstract objects, as I do in the Aristotelian tradition of a crusade against sophistry, Peirce sets up a structure outlining the conditions under which noncontradiction, and excluded middle ought to be violated.

I don't see why. If X is a set, then X = X by identity. — fishfry

You are missing the point. The law of identity refers explicitly to things, "a thing is the same as itself". A "set" is explicitly a group of things. Therefore when you say X = X, and X is a set, rather than a thing, then "=" does not signify identity by the law of identity.

There is no time in set theory. Mathematics is outside of time, or talks about things that are outside of time. — fishfry

Right, this is the point. "Time", or temporal extension allows that a thing may have contradictory properties, at a different time, yet maintain its identity as the same thing, all the while. This is fundamental to the law of identity. Without time (as in mathematics), the multiple orderings of a set, which Tones referred to, are simply contradictory properties. That is a good example of the issue Peirce was looking at.

But given particular instances of set theory; that is, sets; we can ask if they are equal to each other or not.

So I promise not to say that the universe of sets is equal to the universe of sets. Though the category theorists will probably disagree with you. — fishfry

Fine, but can you respect the fact that "equal" does not imply "identical", despite the sophistical tricks that Tones is so adept at.

You are distorting what I said. ANY particular set is a particular instance of the concept of set, as any particular apple is an instance of the concept (or category) of apple. That causes no problem. — fishfry

No, that's simply wrong. A particular apple is a physical object. A set is an abstraction. An instance of an apple is a physical object. Your supposed "instance" of a set is an abstraction, a concept. The two are not analogous, and I argue that this is a faulty, deceptive use of "instance".

An instance is an example, and understanding of concepts or abstractions by example does not work that way. Assume the concept "colour" for example. If I present you with the concept "red", this does not provide you with an instance of the concept "colour". An instance of the concept "colour" would be the idea of colour which you have in your mind, or the idea of colour which I have in my mind, expressed through the means of definition. Each of those would provide you with an example of the concept of "colour", an instance of that concept. The concept "red" does not provide you with an example of the concept of "colour". Nor does a specific "set" provide you with an example or instance of the concept "set".

What you are saying in this case is completely mixed up and confused.

This is where I think it makes sense to look for the original sense of Plato's eidos, the forms - not in some fanciful ethereal 'Platonic heaven' but in the underlying patterns of causal constraint which imposes order on possibility. — Wayfarer

It's still very important to understand the difference between formal cause (as the existing conditions of constraint), and the final cause, that for the sake of which, the good or intent of the agent who acts, under those conditions. The purpose of the act is directly related to the final cause. And in Juarrero's distinction between causes and constraints, final cause must be a proper cause, while formal cause is the conditions of constraint. A scientific understanding of the formal cause, constraints, will never reveal the final cause, therefore not the purpose either.

The crank is so mentally deficient that he can't see that it's not a contradiction that "there are 24 orderings of a set" does not imply "all those orderings are the same". — TonesInDeepFreeze

The orderings are different, and contradictory properties of the set. And, it is a violation of the law of noncontradiction for that set to have those contradictory orderings.

Now like a child with an attention disorder, the crank asks me whether the members of a set are abstractions or concretes, after I explicitly said that they can be either, and I gave explicit examples. Is the crank not able to read? — TonesInDeepFreeze

Exactly as I said, you fail to provide a differentiation between concrete objects and abstractions. Why did you say I lied about this?

If a set consists of concrete objects, then it has the order that those concrete objects have, and no other order. To say that the set has other orderings is to mix up concrete objects with abstract objects in the way of sophistry.

So which is it then? Does a set consist of concrete things, or does it consist of abstractions?

Moreover, I did not say that an element of a set cannot be a concrete thing. The set of pencils on my desk has only concrete things as members. — TonesInDeepFreeze

I know, that's the problem. For you, a set may consist of concrete things, or it may consist of abstractions, because in your sophistry you do not differentiate between the two. Then you claim that there is no order to the concrete things which compose a set, when in reality there is.

I never said that 24 orderings are the same or that they are equal. That would be a ridiculous thing to say. Indeed it is my point that they are not the same. There are 24 different orderings. Of course they are not all the same orderings. The crank is so mentally inept that he can't distinguish between (1) there are 24 different orderings that each have the property of being an ordering of a certain set and (2) all those orderings are the same. — TonesInDeepFreeze

Right, continue in your violation of the law of noncontradiction. The same set has contradicting properties, i.e. different orderings. Good one bro, I hope that's just your sense of humour again.

Design and purpose are inextricably linked, and can be used to two ways: the intentionality of an agent and the expression thereof in something, or the function something. I mean it in the latter sense when it comes to humans.

That my eye was not designed by an agent, does not entail it does not have the function, developed through evolution, of seeing. In that sense, it is designed for seeing. If you wish to use "design" in the former sense strictly, then I would just say that one should size up to their nature, and their nature dictates their functions. — Bob Ross

I think it would be better to say that your eyes have purpose, the purpose of seeing, but they were not designed by an agent. This demonstrates something very interesting about intention. Intention creates things, but not necessarily by design. So for example, abstract art is created intentionally, but the artist doesn't necessarily follow a design, and does not know what the outcome will be prior to the act of creation.

This sheds light on the nature of accidents. Accidents are created by intentional acts, but they are outside any design, and are not actually intended. Further, many intentional acts have no real end in mind, as when you kind of "go with the flow". In a party, you follow the party, and this may become what is known as "herd mentality". It is very clear that the ideas of intention and purpose cover a lot more area than simply design.

I never said any such thing. I've said the opposite — TonesInDeepFreeze

I know you never said such a thing. You mix up physical objects and mathematical objects as if there is no difference between them, and as if the law of identity would apply to both equally. That's why I call you a sophist. It was fishfry's principle, that elements of a set are not physical objects.

Actually I am wrong about that TonesInDeepFreeze showed me the error of my ways. All sets satisfy the law of identity. If I have a set X, I may write X = X by way of the law of identity. I do not need the axiom of extensionality for that. Perfectly clear to me now. — fishfry

No, Tones was referring to the principle called "the identity of indiscernibles", which is completely different from the law of identity. The law of identity makes a thing's identity the thing itself, the identity of indiscernibles associates a thing's identity with the thing's properties. These are fundamentally different principles.

The law of identity applies to sets. So this line of argument is null and void. — fishfry

No, you simply fell for the sophistry. Tones is very good at it, and apt to convince others, earning the title "head sophist".

Yes, very good. A group is any mathematical structure that obeys the axioms for groups. A set is any mathematical object that obeys the axioms for sets. — fishfry

So, a set is a mathematical structure. How do you make this consistent with the head sophist's claim that the members of The Beatles is a set, and the particles which make up a rock is a set? The sophist says "the set whose members are all and only the bandmates in the Beatles has 24 orderings". Notice that this is not stated as possible orderings, it is stated as the "orderings"

Remember your schoolkid example? You recognized that the objects which bear that name have what you called SOME order, and this is an expression of the condition which they are actually in, at any point in time. I would call this their "actual order". Can you see what the head sophist has done? The sophist has removed any distinction of an actual order, to say that the group, or set, has 24 orderings, and all these orderings are equal, or the same, being in each case a different presentation of the same set. But you and I recognize, that in reality there is "SOME order", an actual order, which is the order that the objects are actually in, at any given point in time. The sophist might talk about 24 orderings, but you and I recognize that if these 24 account for all the possibilities, only one of those possibilities represents the very special "actual order", and, that since these elements are physical objects, there must be an actual order which they are in, at any given time.

The law of identity is very important to recognize the actual existence of a thing, and its temporal extension. Through time a thing changes, and the law of noncontradiction stipulates that contradicting properties cannot be attributed to the same thing at the same time. So if a specific group has ordering A at a specified time, that is a property of that group, and it surely cannot have ordering B at the same time. The head sophist claims that the specified group has 24 orderings, all the time (as time is irrelevant in that fantasy land of sophistry). Obviously the head sophist has no respect for the law of noncontradiction, and is just making contradictory statements, in that sophistic fantasy.

That is what happens when we allow that abstractions such as mathematical structures have an identity. Inevitably the law of noncontradiction and/or the law of excluded middle will be violated. Charles Peirce did some excellent work on this subject. It's a difficult read, and you've already expressed a lack of interest in this subject/object distinction, so you probably don't really care. Anyway, here's a passage which begins to state what Peirce was up to.

The relevance of all this to the principles of excluded middle and contradiction is as follows. Peirce wrote that “anything is general in so far as the principle of excluded middle does not apply to it,” e.g., the proposition “Man is mortal,” and that “anything” is indefinite “in so far as the principle of contradiction does not apply to it,” e.g., the proposition “A man whom I could mention seems to be a little conceited” (5.447-8, 1905). If we take Peirce to have meant LEM and LNC, then it appears that he wanted to deny the principle of bivalence (according to which all propositions are true or else false) with regard to universally quantified propositions, and that he meant to claim that existentially quantified propositions are both true and false. But why think that “Man is mortal,” which seems to be straightforwardly true, is neither true nor false? And why think that one and the same proposition, “A man whom I could mention seems to be a little conceited,” is both true and false? Once we see what Peirce meant by “principles of excluded middle and contradiction,” we see that this is not what he was claiming. — Digital companion to C. S. Peirce

http://www.commens.org/encyclopedia/article/lane-robert-principles-excluded-middle-and-contradiction

Sets are subject to the law of identity. — fishfry

This is blatantly untrue, and as demonstrated above, if we assign "identity" to a set, the law of non-contradiction will be violated. The law of identity enables us to understand an object as changing with the passing of time, while still maintaining its identity as the thing which it is. Sets have distinct formulations existing all the time, which would cause a violation of the law of noncontradiction if we allow that a set is subject to the law of identity. Therefore we must conclude that sets are not subject to the law of identity. The type of thing which the law of identity applies to is physical objects. And there is obviously a big difference between physical objects and sets, despite what head sophist claims.

YOU have problems with the empty set. I have no such problems. — fishfry

You also have no problem with contradiction, it seems.

An element is a set in a set theory without urelements. We say x is an element of y if we can legally write x∈y

∈

. Nothing could be simpler. — fishfry

This tells me nothing until you explain precisely what ∈ means. To me, you are simply saying that x is an element of y if x is an element of y. What I am asking is what does it mean "to be an element".

Cinderella is a particular fairy tale character. She doesn't exist either, but she is an INSTANCE of the category of fairy tale characters.

Fairy tale characters are abstract universals, and Cinderella is an abstract particular. — fishfry

If we go with this definition, you ought to se very clearly that sets, as categories, abstract universals, do not have an identity according to the law of identity. A category is not a thing with an identity.

Obviously this does not work. As you said already, elements are often sets. Therefore you cannot characterize the set as an abstract universal, and the element as an abstract particular, because they're both both, and you have no real distinction between universal and particular. There's no point in trying to justify the head sophist's denial of reality. If "Cinderella" refers to a particular, an instance of the category "fairy take characters", then that is a physical object. If "Cinderella" refers to a further abstract category, like in the case of "red is an instance of colour", then it does not refer to a particular. The head sophist seems to have convinced you that you can ignore the difference between a physical object and an abstraction, but you and I both know that would be a mistake.

That's a very good op Bob. If I remember correctly, "good" for Aristotle is a principle of utility, what is often translated as "that for the sake of which". From this perspective, an action is designated as needed, for the sake of an end, "that for the sake of which", and this is the good. So the action itself is "good" because it is the means to a further good, which is the end. However, an end turns out to be the means to a further end, and we must designate an ultimate end to avoid infinite regress, something desired for itself alone, and that is happiness.

I was going to reply that slowing down isn't stopping. I didn't realize that the slowing down was a convergent series. Perhaps slowing down can be stopping. — Ludwig V

In many cases of common language usage, "slowing down" is stopping, but that implies the end, not yet achieved. The point is that "stopping" is distinct from "stopped'. And if the slowing down never reaches the point of being stopped, then the term "stopping" is not justified. The convergent series is misrepresented as "stopping", because the end of "stopped is never achieved.

In the modern physical world of relativity, "stopped" is arbitrarily assigned according to an inertial reference frame. This implies a sort of equilibrium, or stability within that specific reference frame, but it's highly unlikely that it is a true case of "stopped", more likely very slow movement, misrepresented as "stopped". We like to round things off.

Well, we could if we wanted to do. But why would we want to? Apart from the fun of the paradox. Mind you, I have a peculiar view of paradoxes. I think of them as quirks in the system, which are perfectly real and which we have to navigate round, rather than resolve. Think of the paradoxes of self-reference. Never permanently settled. New variants cropping up. — Ludwig V

Why would we want to? Because we are philosophers seeking knowledge. Understanding is the primary objective. I look at such paradoxes as indications of a lack of understanding. The principles applied do not adequately map to the reality which they are being applied, this is a failure of our knowledge. Then we need to subject all the principles to skeptical doubt, to determine the various problems. We could just live with quirks in the system, but that's unphilosophical. Knowledge evolves, and that evolution is caused by people attempting to work out the quirks in the system.

I think it must be pretty clear that any expression of the relation is just the expression of an idea. — tim wood

Any expression whatsoever is an expression of an idea. "Moon", "earth", are expressions of ideas. Even if we use a name to refer 'directly' to an object, we are expressing an idea such as 'I am talking about that thing'.

With our E(arth) and M(oon) in mind, let's imagine a snapshot of our local space - no time passing. — tim wood

The earth and moon with no time passing is completely fictional, therefore nothing other than an idea. How is imagining such a scenario going to be helpful in distinguishing between what is real and what is ideal? You've just given us a purely ideal scenario.

Your claim, then, seems to be nothing more than a claim - a belief on your part. And I give beliefs as beliefs a pass. If you want more, you shall have to make clear how it can be more. — tim wood

You haven't addressed my latest proposition, "neither objects nor relations are real, they are all ideas". Can we agree on this? As indicated above, everything in your post supports this, the proposition that earth and moon might be independent of passing time, etc..

But what about that rock? If it's the one that is the crank's head, then it is indeed empty and there is only one ordering of the set of its particles, which is the empty ordering. — TonesInDeepFreeze

Excellent, I love it. TPF's head sophist has a sense of humour.

The elements of sets have no inherent order. — fishfry

OK, so here we have the issue. Remove the examples of real world objects (schoolkids etc.) as "the elements", and what exactly is an element? It cannot be a particular thing, because it does not obey the law of identity, so it is some sort of universal, an abstraction. But what type of abstraction is it, one which we pretend is a particular? Why is it pretended that these are particulars? Maybe so that the set can be subjected to bijection, and have cardinality. The question then is whether the elements are truly individuals, or just pretend individuals.

Sets have no meaning whatsoever, other than that they obey the axioms of set theory. — fishfry

Isn't that exactly what meaning is, obeyance of some rules? Now, we know what a set is, something which obeys the rules of set theory, the real issue though is what is an element of a set.

This was in response to your denial of the empty set. Tell me exactly -- and be extremely clear and specific, please -- tell me what other rule of set theory is contradicted by the empty set. — fishfry

It seems you are having problems understanding the inherent difficulty of the empty set. I think we'd better have clear agreement on what an element is before we approach that more difficult problem of the empty set.

I have explained to you the ontology of sets many times. They are mathematical abstractions. — fishfry

Yes, but you also claim that sets have no meaning. An abstraction with no meaning is contradictory. That's why I can't understand your teachings about set theory.

You know, I am not sure I agree that sets are universals. My understanding is that "fish" is a universal, and the particular tuna that ended up in this particular can of tuna I bought at the store today is a particular instance of the category or class of fish.

Sets are not like that at all.

I did ask you a long time ago to explain what you meant by universals, and you snarked off at me. And now you come back at me claiming that sets are universals. Explain to me what you mean by that.

The concept of a set is a universal. The set of rational numbers is a particular set, of which there is exactly one instance. — fishfry

Any abstraction is a universal because its applicable to more than one particular set of circumstances. Whatever it is that any multitude of particulars has in common, is a universal.

You appear to be suggesting a third category other than particular and universal, an abstraction which is not a universal. Care to explain?

LOL. Oh man you're crackin' me up. The set of rational numbers most definitely has a cardinality of ℵ0
ℵ
0
, because of Cantor's discovery of a bijection between the rational numbers and the natural numbers. — fishfry

Bijection is a problem, because it requires that the elements are individuals, particulars, which I argue they are not. This is why we need to clear up, and agree upon the ontological status of an "element" before we proceed.

Exactly and well put. I've given the crank that same explanation. He will never understand it, because he wants to not understand it. If he found himself understanding it one day, then he would face the crisis of seeing that he's been confused and in the dark for years and years (decades?). — TonesInDeepFreeze

I suggest we adhere to the principle you stated, the elements of a set are not things, like schoolkids, rocks or anything else. TPF's head sophist doesn't respect this principle.

...the set whose members are all and only the bandmates in the Beatles... — TonesInDeepFreeze

However, we may speak of the set of particles of the rock... — TonesInDeepFreeze

Etc..

That's where the thought experiment isn't a piece of fiction like a fantasy. Aesop's Fables are also not just a piece of fiction; we are meant to draw conclusions about how to live our lives from them. So "It's just a silly story" is not playing the game. This story wants us to draw a conclusion about how reality is. — Ludwig V

I agree, and the conclusion which needs to be drawn is that there is inconsistency between what we observe as the reality of space and time, and the way that we portray space and time through the application of mathematics.

So, instead of rejecting the idea that time is infinitely divisible, you are turning to Hume and arguing that anything can happen. Maybe you are on stronger ground here. I think some people would feel that you are importing more reality than the rules allow. But I can't be dogmatic about that because I don't really know what the rules are - and I'm certainly not going to argue with Hume - perhaps I'm just shirking a long complicated argument, because I don't think he's right, even though he has a point. — Ludwig V

I brought all that in, to say, as I've been saying since the beginning of the thread, that the rejection of the fiction, because it is inconsistent with "reality", isn't that simple. What we know as "reality" has the problem of lacking in certainty. This is what Plato demonstrates in how to deal with Zeno's problems. We cannot simply refer to what we know as "reality", because this is based on sense perception, and the bodily senses are proven to mislead the mind. This is why that sort of "paradox" persists, we dismiss them with reference to "reality" and they go away, that is until skepticism about "reality" reappears.

The issue here is that we really know very little about the nature of the passing of time.

Yes. I don't know how this would play with actual Relativity Theory. But in any case, I don't think that resolves the problem. Why? Because it doesn't actually get Achilles to the finishing line. In the case of Thomson's lamp, it doesn't get to the crunch point when the time runs out. In other words, it postpones, but doesn't resolve, the issue. — Ludwig V

The point, was not to resolve the problem, but to demonstrate how the "unreal" situation described could actually be real. So I was showing how the lamp could actually switch on and off infinitely, in the described manner, such that the allotted amount of time would not ever pass, and how it is possible that Achilles might never pass the tortoise. It is an example of time slowing down, and approaching a complete stop. Instead of the action of the lamp switching speeding up, think of the passage of time as slowing down, so it appears like the action is speeding up. Then the point which marks the limit, midnight or whatever never comes

That's why I insist that the convergent sequence is not about space or time, but about the analysis of space and time. — Ludwig V

I agree with this, but I'd describe it as how we apply mathematics to space and time.