Meanwhile, I allow that anyone is welcome to stipulate their own terminology and even to propose an entirely different framework for consideration of mathematics. — TonesInDeepFreeze

When you reject such, and insist on the other, it's dogmaticism.

What dogmatism do you think you have witnessed? — TonesInDeepFreeze

Aside from whatever SophistiCat might say, it is not the case that formalism regards the incompleteness theorem in that way.

(1) Sentences are not true in a language. They are true or false in a model for a language.

(2) Sentences are not proven just in a language, but rather in a system of axioms and rules of inference.

(3) There are different versions of formalism, and it is not the case that in general formalism regards truth to be just provability.

(4) Godel's theorem in its bare form is about provability and does not need to mention the relationship between truth and proof (though a corollary does pertain to truth). The theorem is about provability, which is syntactical. Even if we had no particular notion of truth in mind, Godels' theorem goes right ahead to show that for systems of a certain kind, there are sentences in the language for the system such that neither the sentence not its negation are provable in the system. — TonesInDeepFreeze

Thanks. I have learned from this thread to avoid discussion of this topic in future. — Wayfarer

Wise decision, the dogmatic don't provide reasonable discourse.

Sextus Empiricus against the dogmatist's criteria of truth:

At the end of Sextus’ discussion in PH II, he clearly signals, as one would expect, that he suspends judgment on whether there are criteria of truth:

You must realize that it is not our intention to assert that standards of truth are unreal (that would be dogmatic); rather, since the Dogmatists seem plausibly to have established that there is a standard of truth, we have set up plausible-seeming arguments in opposition to them, affirming neither that they are true nor that they are more plausible than those on the contrary side, but concluding to suspension of judgement because of the apparently equal plausibility of these arguments and those produced by the Dogmatists. (PH II 79; cf. M VII 444) — Stanford Encyclopedia of Philosophy

According to Chisholm, there are only three responses to the Problem of the Criterion: particularism, methodism, and skepticism. The particularist assumes an answer to (1) and then uses that to answer (2), whereas the methodist assumes an answer to (2) and then uses that to answer (1). The skeptic claims that you cannot answer (1) without first having an answer to (2) and you cannot answer (2) without first having an answer to (1), and so you cannot answer either. Chisholm claims that, unfortunately, regardless of which of these responses to the Problem of the Criterion we adopt we are forced to beg the question. It will be worth examining each of the responses to the Problem of the Criterion that Chisholm considers and how each begs the question against the others. — Internet Encyclopedia of Philosophy

You claim that the elements of the diagram have a "spatial order". Effectively, this is to say that the elements "have the order they have" without specifying what that order is. — Luke

I specified the order. It is a spatial order, the one demonstrated by the diagram. Why is this difficult for you to understand? When a diagram shows us an arrangement of dots, it shows us the spatial order of those dots, where the dots must be on a spatial plane to fulfill the order being demonstrated. What is the diagram? An arrangement of dots. What does it demonstrate to us? An ordering of those dots. Someone could proceed with that diagram to lay out the same pattern with other objects, with the ground, or some other surface as the plane. Just because fishfry called it a "random" arrangement doesn't mean that it does not demonstrate an order. Fishfry used "random" deceptively, as I explained already.

You have provided no explanation as to why the diagram has the spatial order it has instead of any other possible spatial ordering of the the same elements. — Luke

Yes I did explain that. There was a process which put those dots where they are, a cause, therefore a reason for them being as they are and not in any other possible ordering. That is why it is not true to call it a random arrangement, unless you are using "random" to signify something other than no order.

This is an important constituent of the distinction between actual order, and possible order. A distinction which fishfry rejected as not with principle. But there is such a principle, which fishfry simply denied, that things must have an actual order, to have existence. I believe it's called the principle of sufficient reason. And this principle renders "the set", as being a unity composed of parts, without any inherent order, as an incoherent notion. We could say that there are many possible orders which the parts could have, but if they do not have an actual order, the supposition of "unity" and therefore "set" is unjustified.

How does this account for the order of rank of military officers, or of suits in a game of bridge? Or the order of values of playing cards in Blackjack? Or the order of the letters of the alphabet? Or monetary value? — Luke

Examples like that is how fishfry convinced me otherwise.

You claimed that the diagram has an inherent order. Specify that order. — Luke

I believe I already did. It's a spatial order, each dot has its own specific position on the plane. To change the position of one would change the order, requiring a different diagram. So that order is inherent to that diagram.

Specify that order. Which dots are the start and end points of that order? — Luke

There is no need to specify a start and end. After giving me examples of order which is not a temporal order, you cannot now turn around and insist that "order" implies a known start and end. "Order" is defined as "the condition in which every part, unit, etc., is in its right place". That's why the diagram has an inherent order. If any of the dots were in a different place it would not be the diagram which it is, because some part would be in the wrong place for it to be that particular diagram.

But the deeper point is that YOUR CONCEPT of a set has inherent order, and that's fine. But the mathematical concept of a set has no inherent order. So you have your own private math. I certainly can't argue with you about it. — fishfry

Right, and do you also see now, that the mathematical concept of a set is incoherent? I hope so, after all the time I've spent explaining that to you.

The sets {a,b,c}, {c,b,a}, and {a,c,b} are exactly the same set. Just as Sonny and Cher are the same singing group as Cher and Sonny. They're the same two people. You and I are the same two people whether we're described as Meta and fishfry or fishfry and Meta. If you can't see that, what the heck could I ever say and why would I bother? — fishfry

Now, do you see that Sonny and Cher, Meta and fishfry, as individual people, have spatial temporal positioning, therefore an inherent order? I am here, you are there, etc.. We can change the order, and switch places, or move to other places, but at no time is there not an order.

So, you propose a set [a,b.c], [c,b,a], or phrase it however you like. You have these three elements. Do you agree that the three things referred to by "a", "b", and "c", must have an order, just like three people must have an order, or else the set is really not a set of anything? There is nothing which could fulfill the condition of having no order.

I know, you'll probably say it's abstract objects, mathematical objects, referred to by the letters as members of the set, therefore there is no spatial-temporal order. But even this type of "thing" must have an order as defined by, or as being part of a logical system, or else they can't even qualify as conceptions or abstract objects. Without any order, they cannot be logical, and are simply nothings, not even abstract objects. It appears like you want to abstract the order out of the thing, but that's completely incoherent. Order is what is intelligible to us, so to remove the order is to render the concept unintelligible. What's the point to an unintelligible concept of "set"?

Whatever man. You're talking nonsense. The Sun, the Moon, and the stars are the same collection of astronomical objects as the Moon, the stars, and the Sun. — fishfry

Yes. Now do you see that these three things have order, regardless of the order in which you name them? And all things have some sort of order regardless of whether you recognize the order, or not. If there was something without any order it would not be sensible, cognizable or recognizable at all. In fact it makes no sense whatsoever to assume something without any order, or even to claim that such a thing is a real possibility. So to propose that there could be a complete lack of order, and start with this as a premise, whereby you might claim infinite possibility for order, you'd be making a false proposition. It's false because a complete lack of order would be absolute nothing, therefore nothing to be ordered, and absolutely zero possibility for order. But you want to say that there is "something" which has no order, and this something provides the possibility of order. By insisting that there is no order to this "something" you presume it to be unintelligible.

But why can't I have two conceptual, abstract spheres? — fishfry

If you think about what it means to be a conceptual abstract sphere, the answer ought to become apparent to you. What makes one sphere different from another is their physical presence. If you have two distinct concepts of a sphere, then they are only both the exact same concept of sphere through the fallacy of equivocation. If you have one concept of an abstract sphere then it is false to say that this is two concepts. It is simply impossible to have two distinct abstract concepts which are exactly the same, because you could not tell them apart. It's just one concept.

It is not I who is making the dumb propositions.

erhaps “numerical” wasn’t the right word. The context of the post and the preceding discussion indicates that a “before and after” ordinality was implied, of the sort Tones recently mentioned: — Luke

Before and after, are temporal terms. Fishfry had rejected the notion that "order" is based in spatial-temporal relations, and wanted an order based in quantity. But a quantity based order is what produces the problem I first referred to. If "2" refers to a quantity of objects in the context of "order", then it does not refer to a single object, the number 2. In any case, what distinguishes one thing from another, allowing for individuals, and quantity itself is spatial relations. So we're back to spatial relations as the bases of order. It's very clear that the subject was order of any sort, when "no inherent order" was mentioned.

It was my suggestion that "order" is fundamentally temporal, but fishfry produced examples of an order based on a judgement of better and worse, to discount that theory. So we really haven't agreed on any specific type of order yet. This is probably because we haven't agreed on what type of existence the things which are supposed to have no inherent order, but are capable of being ordered, have.

Yes, it certainly looks like that faulty brick in the foundations of math will surely cause the giant structure to collapse. I wish I had known this before becoming a mathematician. :cry: — jgill

I don't foresee any imminent collapse, but the structure ought to be dismantled because it supports the faulty worldview which is prevalent today, which is a sort of scientism. Fishfry perceives that physics has reached a sort of dead end in its endeavours, but refuses to acknowledge that the dead end is brought about by the principles employed (mathematics included) rather than the unintelligibility of the world itself.

The use of "infinity" which is the topic of this thread (believe it or not) is a very good example. I apprehend, that at the base of the idea of infinity in natural numbers, is the desire, or intention to allow that numbers can be used to count anything. There will never be something which cannot be counted because the numbering system has been designed to allow that the numbers can always go higher. This gives the appearance that everything is measurable.

The only drawback is that this renders infinity itself, (the principle which provides this capacity, that everything is measurable) as immeasurable. So in reality everything is measurable except the principles we use to measure with. What this means is that to understand the nature of measurement and infinity, we must place these into a different category from the category of things which are measurable, and understand it on those terms, as immeasurable. If we attempt to bring infinity into the category of things which are measurable, as is the trend in modern mathematics, because we want mathematics to enable science to be applicable everything, even the thing which by its own design is immeasurable, then we introduce contradiction (the immeasurable is measurable) and therefore unintelligibility into our principles. That is where we are today, we have allowed unintelligibility to inhere within our principles of measurement. As fishfry said "math has experienced a loss of certainty".

Fishfry and I really agree that the big picture is very hazy (uncertain). But when it comes down to determining the specific points where the haziness arises from, fishfry refuses to follow. It's like seeing smoke on the horizon and wondering why it's there. But when I point to the fire, fishfry refuses to acknowledge a relationship between the fire and the smoke on the horizon. Maybe if fishfry would accept the possibility of a relationship, a closer look would reveal smoke rising from the fire.

If not specified, then at least strongly implied in the same post: — Luke

I really don't see how the qualification "numerical" is relevant , or even meaningful in the context of dots on a plane. So I don't see why you think it was implied. Fishfry is not sloppy and would not have forgotten to mention a special type of order was meant when "no inherent order" was said numerous times.

You wrote 'Russel' twice. It's 'Russell'. — TonesInDeepFreeze

Sorry "Bertie", as fishfry says.

Tracking recent points Metaphysician Undercover has either evaded or failed to recognize that he was mistaken. — TonesInDeepFreeze

I believe all the relevant points were addressed. You don't seem to know how to read very well.

I wasn't born wearing a hat but I can go buy one. The idea that a thing can't gain stuff it didn't have before is false. But I've already shown you how we impose order on a set mathematically. We start with the bare (newborn if you like) set. Then we pair it with another, entirely different set that consists of all the ordered pairs of elements of the first set that define the desired order. — fishfry

The problem is that your demonstration was unacceptable because you claimed to start with a set that had no order. A newborn is not a thing without order, so the newborn analogy doesn't help you.

So we start with the unordered set {a,b,c}. — fishfry

You are showing me an order, "a" is to the right of "b" which is to the right of "c". And even if you state that there is a set which consists of these three letters without any order, that would be unacceptable because it's impossible that three letters could exist without any order.

If you insist that it's not the letters you are talking about, but what the letters stand for, or symbolize, then I ask you what kind of things do these letters stand for, which allows them to be free from any order? To me, "a", "b", and "c" signify sounds. How can you have sounds without an order? Maybe have them all at the same time like a musical chord? No, that constitutes an order. Maybe suppose they are non-existent sounds? But then they are not sounds. So the result is contradiction.

I really do not understand, and need an explanation, if you think you understand how these things in the set can exist without any order. What do "a", "b", and "c" signify, if it's something which can exist without any order? Do you know of some type of magical "element" which has the quality of existing in a multitude without any order? I don't think so. I think it's just a ploy to avoid the fundamental laws of logic, just like your supposed "two spheres" which cannot be distinguished one from the other, because they are really just one sphere.

Also: "How is it possible that a set can be ordered in one way and in a contrary way at the same time, without contradiction?" Like ordering schoolkids by height, or alphabetically by name. Two different ways to order the same set. I can't understand why this isn't obvious to you. — fishfry

Those are different ways, but not contrary ways.

A set has no inherent order. That's the axiom of extensionality. — fishfry

Huh, all my research into the axiom of extensionality indicates that it is concerned with equality. I really don't see it mentioned anywhere that the axiom states that a set has no inherent order. Are you sure you interpret the axiom in the conventional way? I have no problem admitting that two equal things might consist of the same elements in different orders. We might say that they are equal on the basis of having the same elements, but then we cannot say that the two are the same set, because they have different orders to those elements, making them different sets, by that fact.

You can't visualize two identical spheres floating in an otherwise empty universe? How can you not be able to visualize that? I don't believe you. — fishfry

Are you serious? If I can imagine them as distinct things, I know that they cannot be identical. That's the law of identity, the uniqueness of an individual,. A fundamental law of logic which you clearly have no respect for.

I did not make this up. I'm not clever enough to have made it up. I read it somewhere. Now you've read it. — fishfry

The argument of Max Black fails because pi is irrational. There is no such thing as a perfectly symmetrical sphere. The irrationality of pi indicates that there cannot be a center point to a perfect circle. Therefore we cannot even imagine an ideal sphere, let alone two of them.

No, on the contrary. Mathematicians love festering falsities. And of course even this point of view is historically contingent and relatively recent. In the time of Newton and Kant, Euclidean geometry was the true geometry of the world, and Euclidean math was true in an absolute, physical sense. Then Riemann and others showed that non-Euclidean geometry was at least logically consistent, so that math itself could not distinguish truth from falsity. Then Einstein (or more properly Minkowski and Einstein's buddy Marcel Grossman) realized that Riemann's crazy and difficult ideas were exactly what was needed to give general relativity a beautiful mathematical formalism. The resulting "loss of certainty" was documented in Morris Kline's book, Mathematics: The Loss of Certainty. Surely the title alone must tell you that mathematicians are not unaware of the issues you raise. — fishfry

Yes, it seems like mathematics has really taken a turn for the worse. If you really believe that mathematicians are aware of this problem, why do you think they keep heading deeper and deeper in this direction of worse? I really don't think they are aware of the depth of the problem.

Hopefully others will correct me if I'm wrong but, as I understand it, the point iof the diagram of "dots" is that the elements of the set have no inherent numerical order or sequence. Otherwise, you should be able to number the elements from 1 to n and explain why that is their inherent order. — Luke

Actually "numerical order" (whatever that is supposed to mean in reference to a diagram of dots) was not specified. It was simply asserted that the elements have no inherent order.

I don't speak for fishfry, but the second 'you' above appears to include both of us. But I have never said, implied, or remotely suggested, that truth and falsity are irrelevant to pure mathematics. So you are lying to suggest that I did. — TonesInDeepFreeze

Are you taking lessons from Luke on how to make strawman interpretations? Read the quoted passage. Fishfry and Tones are in denial of the logical fallacies, and "you" (directed at fishfry only) talk about truth and falsity not being relevant to pure mathematics. How is that sentence so difficult for you to read properly?

You ignore what I said. That is your favorite argument tactic: — TonesInDeepFreeze

Yes, I am finding that to be the best tactic in dealing with the type of nonsense you throw at me.

What is "THE INHERENT" order you claim that the dots have? — TonesInDeepFreeze

The one in the diagram. Take a look at it yourself, and see it.

Start with what people say in everyday language. — TonesInDeepFreeze

I have no problem with what people say in everyday language, about the number of students in the class, the number of chairs in the room, the number of trees in the forest, etc., where I have the problem is with what mathematicians say about numbers alone, without referring to "the number of ..."

Sets of cardinality greater than 1 have more than one ordering. — TonesInDeepFreeze

Look at it this way Tones. As you describe sets, order is an attribute, or property of the set. How is it possible that a set can be ordered in one way and in a contrary way at the same time, without contradiction? Fishfry resolves this by saying that a set has no order, so order is not a property of a set. But then it appears like fishfry wants to smuggle order in, with some notion of possible orders. However the set is already defined as not having the property of order, therefore order is impossible.

Between you and fishfry, the two of you do not even seem to be in agreement as to whether a set has order or not. Fishfry says that a set has no inherent order. You say that not only does a set have order, but it has a multitude of different orders at the same time. See what happens when you employ contradictory axioms? Total confusion.

Nobody is claiming math is absolute truth but you. — fishfry

Wow, that's an even worse interpretation of what I'm saying than TIDF's terrible interpretation. I argue to demonstrate untruths in math, and you say I'm claiming math is absolute truth. This thread has gone too far. I think you're cracking up.

Don't you think he was recognizing and responding to exactly the point you are making? — fishfry

No, I have read a fair bit of Russel and he was in no way responding to the same points I'm making. More precisely he was helping to establish the situation which I am so critical of. Remarks like that only inspire mathematicians to produce more nonsense.

Try understanding the axiom of extensionality. — fishfry

We've been through the axiom of extensionality you and I, in case you've forgotten. It's where you get the faulty idea that equal to, means the same as.

One example that comes to mind is the famous counterexample to the identity of indiscernibles, in which we posit a universe consisting of two identical spheres. You can't distinguish one from the other by any property, even though the spheres are distinct. This example also serves as a pair of objects without any inherent order. — fishfry

Contradiction again. If you cannot distinguish one from the other, you cannot say that there are two. To count two, you need to apprehend two distinct things. But to say that you cannot distinguish one from the other means that you cannot apprehend two distinct things. Therefore it is false to say that there are two. So you are just proposing a contradictory scenario, that there are two distinct spheres which cannot be distinguished as two distinct spheres (therefore they are not two distinct spheres), hoping that someone will fall for your contradiction. Obviously, if one cannot be distinguished from the other, they are simply two instances of the same sphere, and you cannot say that there are two. And to count one and the same sphere as two spheres is a false count.

Reality? I make no claims of reality of math. I've said that a hundred times. YOU are the one who claims math is real then complains that this can't possibly true. — fishfry

Coming from the Platonic realist who claims the reality of "mathematical objects".

But math makes no claims as to the truth of "this." — fishfry

Right, just like the statement from Russel. That's why there is a real need for metaphysicians to rid mathematics of falsity. The mathematicians obviously do not care about festering falsities.

.

A contradiction is a statement and its negation. — TonesInDeepFreeze

Contradiction may be implied. Here's Wikipedia's opening statement:
'In traditional logic, a contradiction consists of a logical incompatibility or incongruity between two or more propositions."

The problem is that you refuse to recognize that an arrangement of points on a plane, logically implies order, therefore "an arrangement of points on a plane without order" is contradictory.

This was in reference to my question, Why don't you treat math like chess, and accept it on its own terms? — fishfry

Don't you see that I said math is not like chess. Therefore I do not treat math like chess. I answered your question.

Then your complaint is with the physicists, engineers, and others; and not the mathematicians, who frankly are harmless. — fishfry

Obviously not, as you've already noticed,

Again, your complaint is with those mis-applying math or applying math to bad ends. — fishfry

No, my complaint is with the fundamental principles of mathematicians, As explained already to you, violation of the law of identity, contradiction, and falsity. You, and Tones alike (please excuse me Tones, but I love to mention you, and see your response. Still counting?), are simply in denial of these logical fallacies existing in the fundamental principles of mathematics, and you say truth and falsity is irrelevant to the pure mathematicians.

Particles? Dots? What are those? In math, the elements of sets are other sets. There are no particles or dots. Again, you confuse math with physics. — fishfry

In case you forgot, you posted a diagram with dots, intended to represent a plane with an arrangement of points without any order. This is what I argued is contradictory, "an arrangement... without order". And this was representative of our disagreement about the ordering of sets. You insisted that it is possible to have a set in which the elements have no order. You implied that there was some special, magical act of "collection" by which the elements could be collected together, and exist without any order. What you are in denial of, is that if the elements exist, in any way, shape, or form, then they necessarily have order, because that's what existence is, to be endowed with some type of order.

You tell me, just imagine a plane, with points on the plane, without any order, and I tell you I can't imagine such a thing because it's clearly contradictory. If the points are on the plane, then they have order. And you just want to pretend that it has been imagined and proceed into your smoke and mirrors tricks of the mathemajicians. I'm sorry, but I refuse to follow such sophistry.

I can't argue with the fantasies in your head. Set theory is what it is. — fishfry

Why not give it a try? I can argue with the fantasies in your head, demonstrating that they are contradictory. So please explain to me how you think you can have a collection of elements, points, or anything, and that collection has no order. Take this fantasy out of your head and demonstrate the reality of it.

There are no dots. I don't know what dots are. I tried to give you a visual example but perhaps that was yet another rhetorical error. I should just refer you to the axiom of extensionality and be done with it, because in truth that is all there is to the matter. — fishfry

The dots. I believe, were supposed to be a representation of points on a plane. The points on a plane, I believe, were supposed to be a representation of elements in a set. And you were using these representations in an attempt to show me that there is no inherent order within a set. So, are you ready to give it another try? Demonstrate to me how there could be a set with elements, and no order to these elements.

I've explained to you the problem. You describe the set as a sort of unity. And you want to say that the parts which compose this unity have no inherent order. Do you recognize that to be a unity, the parts must be ordered? There is no unity in disordered parts. Or are you going to continue with your denial and refusal to recognize the fundamental flaws of set theory?

First thing I really want to know what are the bad things that you think mathematicians and scientists are going to cause to happen? — TonesInDeepFreeze

Come on TIDF, don't you see that as a ridiculous question? If one could predict the bad things that were going to happen, before they happened, then we could take the necessary measures to ensure that they don't happen. It's like asking me what accident are you going to have today. It's a matter of risk management. If the mathematics employed in any given situation is faulty, the risk is increased. The biggest problem, I think, is the complete denial of the faults, from people like you. This creates a false sense of certainty. That's why it's like religion, you completely submit to the power of the mathematics, with your faith, believing that your omnibenevolent "God", the mathematics would never mislead you.

Suppose the number 2 is not distinct from the numeral '2'. Suppose also that the number 2 is not distinct from the Hebrew numeral for 2. Then both the numeral '2' and the Hebrew numeral for 2 are the same. But they are not. — TonesInDeepFreeze

The symbols are not the same, nor ought they be said to be the same, or to say the same thing. They ought not be said to say the same thing, because different cultures have different ways of looking at the world. Where's the problem with that? If someone translates a passage of philosophy from ancient Greece, we ought not say that the translation says the same thing as the original. Something is always lost in translation. Likewise, we ought not say that the numeral 2 says the same thing as the Hebrew symbol. This would be very clear to you if you would consider all the different numbering systems discussed on this forum, natural, rational, real, cardinals, ordinals, etc.. The same symbol has a different meaning depending on the system. If we do not keep these distinguished, and adhere to the rules of the specific system, we have equivocation.

That's a picture of dots in a disk. It's not an ordering. — TonesInDeepFreeze

If you refuse to acknowledge that there's an order to those dots, then I don't see any point in proceeding with this discussion.

. That means for you to state which dots come before other dots, for each dot. — TonesInDeepFreeze

Order is not necessarily temporal. And, modern physics looks at time as the fourth dimension of space. So if you cannot see order in an arrangement on a two dimensional plane, I don't see any point in discussing "order" with you.

The symbols do represent how many individuals there are. What do you mean by “directly”? — Luke

If you follow what is taught in math, the symbol "2" represents a mathematical object which is called a number. The number represents how many individuals there are.

Do you recognize that the word 'tree' is not a tree? — TonesInDeepFreeze

Of course, the word "tree" might be used as a symbol, to represent a tree.

But you fail to recognize that the word 'two' or the symbol '2' are not the number 2. — TonesInDeepFreeze

You misunderstand. What I am asking is why can't the symbol "2" be used to represent a quantity of two individuals, just like the word "tree" is used to represent a tree? Why must the symbol "2" represent a mathematical object, the number two, and the number two represents a quantity of two individuals? We don't say that the word "tree" represents a conceptual object, tree, and this concept represents the individual tree.

In reality we simply use the word "tree" to represent a tree, and we use the symbol "2" to represent a quantity of two individuals. There is no conceptual object, or mathematical object in between. So if someone states as a premise, that "2" represents a mathematical object, the number two, this would be a false premise.

The number does not represent how many individuals there are.

The number is how many individuals there are. — Luke

Well no, this is not true. The number is how many individuals it is said that there are. The number is supposed to be what the numeral stands for. It is conceptual, and a representation of a particular quantity of individuals. Being universal, we cannot say that it is actually a feature of the individuals involved, but a feature of our description, therefore a representation. That's why the OED defines "number" as "an arithmetical value representing a particular quantity and used in counting and making calculations." If the number is not a representation of how many individuals there are, but actually "how many individuals there are", there would be no possibility of error, or falsity. If I said "there are 2 chairs", and the supposed mathematical object, the number 2 which is said to be signified by the numeral "2" was "how many individuals there are", rather than how many there are said to be, how could I possibly lie?

You said your teacher insisted that "the numeral is not the number" and that you couldn't understand it. But you also said that you had no problem with basic arithmetic. My point was that you must have understood that "the numeral is not the number" in order to do basic arithmetic. — Luke

No, as I explained. The numeral 2 represents how many objects there are. We could also call that symbol the number 2, which represents how many objects there are. There is no need to assume that the number 2 is distinct from the symbol, to do basic arithmetic..

But "1" or "2" are the number of individuals, not the individuals. — Luke

If that were the case, I'd be fine with it, but it's not what I was told. I was told that "1" and "2" are numerals, symbols, and there is also something else, called the numbers 1 and 2. The numbers are distinct from the numerals, as what is represented by the numerals. So, I was told that "1" and "2" are symbols, which represent the numbers 1 and 2, and the number represent how many individuals there are. Why not just say that the symbols "'1" and "2" represent how many individuals there are, directly?

That is one of the best, most risible, evasions of a challenge I've ever read. What is "the order they actually have" as opposed to all the others? Saying that they have the order they "actually" have is not telling us what you contend to be the order nor how other orderings are not the "actual" ordering. You are so transparently evading and obfuscating here. — TonesInDeepFreeze

Fishfry posted the order, it's right here:

What more do you want?

When confronted with the challenge of points in a plane, a reasonable response by you would be "Let me think about that." But instead you reflexively resort to the first specious and evasive reply that comes to you and post it twice with supposed serious intent. That indicates once again your lack of intellectual curiosity, honesty or credibility. — TonesInDeepFreeze

Look, if the dots exist on a plane, they have positions on that plane, and therefore an exact order which is specific to that particular positioning. They do not have any other order, or else they would not be those same dots on that plane. Take a look at that posting of fishfry's and see the order which the dots have, on that plane, and tell me how they could have a different order, or no order at all, and still be those same dots on that same plane.

If you cannot apprehend this simple fact, then tell me what is so difficult for you.

You were presented with points in a plane, without being given a stated particular ordering. — TonesInDeepFreeze

How can you not see that 'points in a plane without a particular ordering' is a blatant contradiction? If the points exist on a plane, then they each have a particular position on that plane, as demonstrated in fishfry's post, and it is impossible that they have no particular order, because the particular order has been posted. Can you grasp this fact?

One could just as well say 'unstated'. — TonesInDeepFreeze

The problem is, that it is stated. It is stated that they exist on a plane. Therefore each point has a position on that plane unique to itself. Not one of these points makes a line, nor occupies a section of the plane, they each have a specific position. Therefore there is necessarily an order to these points, their positions on that plane, according to what is stated. To give them no order you'd have to remove them from their positions on the plane.

Suppose we just assume a multitude of points, without any spatial reference, no dimensions or anything, just points. Then we have the question of what distinguishes one point from another. It is stipulated that there is a multitude of points. If there is no spatial reference, therefore no space separating one point from another, what makes them distinct from one another? How can we assume a multitude of points when we posit no principle whereby one point is distinguished from another point? And if we posit a principle of separation other than space, (suppose one is later in time than another, or something like that), then isn't this a principle of order. it is impossible to posit a multitude of points without implying order.

It isn't I who is evading the issue. All those people who simply assume that it is possible for a multitude of points to exist without any order, are the one's evading the issue, because such a scenario is logically impossible.

It is not the case that there are not orderings. The point though is that there is not a single ordering that is "THE actual ordering". There are many orderings and they are actual even though 'actual' is gratuitious. — TonesInDeepFreeze

Again, look at fishfry's post:
Do you not see that there is an actual order to those dots on the plane? How could there be "many orderings" if to give them a different order would be to change their positions? Then it would no longer be those dots on that plane. And if your intent is to abstract them, remove them from that plane, then they are no longer those dots on that plane. Why is something so simple so difficult for you to understand?

First, of course, is that we may take a collection of dots as given, without stipulating that a particular person placed the dots herself. — TonesInDeepFreeze

OK, but do you agree that something must have caused those dots to be where they are, i.e. given them that order?

Second, let's even suppose that "actual order" is a function of a person placing the dots. Say that Joe places the dots in temporal succession and Val places the dots in a different temporal succession. But that both collection of dots look exactly the same to us. So there's "Joes actual (temporal) order" and "Val's actual (temporal) order", but no one can say which is THE actual order of the collection of dots we are looking at without Joe and Val there to tell us (if they even remember) the different order of placement they used. — TonesInDeepFreeze

I am talking about their spatial ordering, their positioning on the plane, like what is described by a Cartesian system. Do you not apprehend spatial arrangements as order?

Meanwhile, you're not even familiar with the distinction between semantics and syntax and the notion of model theoretic truth. — TonesInDeepFreeze

That's not true. I simply don't accept it as a realistic notion of "truth", and don't want to waste my time discussing it.

In any event, can you please respond to my point about chess? Surely if you learned to play chess, or any other artificial game -- monopoly, bridge, checkers, baseball -- you were willing to simply accept the rules as given, without objecting that they don't have proper referents in the real world or that they make unwarranted philosophical assumptions. If you could see math that way, even temporarily, for sake of discussion, you might learn a little about it. And then your criticisms would have more punch, because they'd be based on knowledge. I wonder if you can respond to this point. Why can't you just treat math like chess? Take it on its own terms and shelve your philosophical objections in favor of the pleasure of the game. — fishfry

I didn't answer, because it's not relevant. Philosophy is not a game in which you either accept the rules of play or you don't,, neither is theoretical physics such a game, nor is what you call "pure mathematics" (or as close to "pure" as is possible). In these fields we determine, and create rules which are deemed applicable. So your analogy is not relevant, because the issue here is not a matter of "will you follow the rules or not", it's a matter of making up the rules. And there's no point to arguing that people must follow rules in the act of making up rules because this is circular, and does not account for how rules come into existence in the first place.

It makes no sense to anyone else either. This is well known. Especially in terms of quantum fields being "probability waves." That makes no sense to me. Physics has perhaps lost its way. Many argue so. You and I might well be in agreement on this. — fishfry

Ok, we've found a point of agreement, physics has lost it's way. Do you ever think that there must be a reason for this? And, since physics is firmly based in mathematics, don't you see the implication, that perhaps the root of the problem is actually that mathematics has lost its way.

Ok. I get that. And I've asked you this many times. You don't want to play the game of math. So then why the energetic objection to it? After all if someone invites me to play Parcheesi and I prefer not to, I don't then go on an anti-Parcheesi crusade to convince the enthusiasts of the game that they are mis-allocating their time on a philosophically wrong pursuit. So there must be more to it than that. With respect to a perfectly harmless pastime like Parcheesi or modern math, one can be for, against, or indifferent. You have explained why you are indifferent; but NOT why you are so vehemently against. — fishfry

Physicists, engineers, and others, applying mathematics in the world have a huge impact on the world in which I live, unlike Parcheesi players. Despite arguments that mathematical objects exist in some realm of eternal truth where they are ineffectual, non-causal, I think it is undeniable, that the mathematical principles which are applied, have an impact on our world. I believe it is inevitable that bad mathematics will have a bad effect.

That people vehemently support and defend fundamental axioms which may or may not be true, refusing to analyze and understand the meaning of these axioms, simply accepting them on faith, and applying them in the conventional way, in new situations, with little or no understanding of the situation, or the axioms, to me is a clear indication that bad results are inevitable.

Makes no sense. It's perfectly clear that you can order a random assemblage of disordered points any way you like, and that no one order is to be preferred over any other. — fishfry

You do not seem to be making any effort to understand this fundamental principle, which is the key to understanding what I am arguing. A group of particles, or dots (we cannot really use "points" here because they are imaginary) existing in a spatial layout, have an order by that very fact that they are existing in a spatial arrangement. Yes, they can be "ordered any way you like", but not without changing the order that they already have. The order which they have is their actual order, whereas all those others are possible orders.

Do you understand and accept this? Or do you dispute it, and know some way to demonstrate how a spatial arrangement of dots or particles could exist without any order? It's one thing to move to imaginary points, and claim to have a specific number of imaginary points, in your mind, which have no spatial arrangement, but once you give them a spatial arrangement you give them order. Even if we just claim "a specific number of points", we need to validate that imaginary number of points without ordering them. This is what Tones and I discussed earlier. How can we count a specific number of points without assigning some sort of order to them? To count them we need to distinguish one from the other by some means or else we do not know which ones have been counted and which have not been counted. So even to have "a specific number of points", imaginary, in your mind, requires that they have an order, or else that specific number cannot be validated.

Well yes, the random number generator I used was actually determined at the moment of the big bang, if one believes in determinism. But you're making a point about randomness, not about the order of the points. You are not persuading me with your claim that a completely random collection of points has an inherent order. — fishfry

Yes, I'm making a point about "randomness" because you are using the term "random" to justify your claim that a bunch of dots in a spatial arrangement could have no order. You simply say, the points are "randomly distributed" and you think that just because you say "randomly", this means that there actually could be existing dots in a spatial assemblage, without any order. But your use of the term does not support your claim. There was a process which placed the dots where they are, therefore they were ordered by that process, regardless of whether you call that process "random" or not.

You don't want to read the Wiki piece on order theory. — fishfry

I looked at the Wikipedia entry, and it does not appear to cover the issue of whether existing things necessarily have an order or not. So it seems to provide nothing which bears on the point which I am trying to get you to understand.

Actually it doesn't make initial sense. Moving from one letter to the next is always a whole step, except from B to C and from E to F. And then double flats move you down a letter except from C to Cbb and from F to Fbb, and double sharps move you up a step except from B to B## and from E to E##. — TonesInDeepFreeze

All this, what you say, comes later, it's not "initial". What is "initial" is that you learn a specific fingering, and it sounds good, therefore it makes sense. The part that doesn't make sense is when you move deeper into the theory. This is just like mathematics. The initial practice makes sense, learning addition, multiplication, pi, Pythagorean theorem, etc.. All these simple procedures make perfect sense, you learn a procedure, apply it, and it works. However, then there is layers of theory piled on after the fact, and this is where the sense gets lost, because the theory doesn't necessarily follow what is actually the case.

You could see the quantity of objects but not the number of objects? — Luke

Right, I don't look at two chairs and see the number 2 there.

You must have already understood that the number is not the numeral in order to do simple arithmetic. Otherwise, the addition of any two numbers (i.e. numerals) would always equal 2 (numerals). — Luke

No, the numeral represents a quantity, and a quantity must consist of particulars, or individual things. So "2"" represents a quantity, or number of individuals, two, and "1" represents a quantity of one individual. What is added or multiplied is the quantity or number of individuals. The number is of the individual, a predication, and what is added or subtracted is the individuals, not the number. Have two individuals, add two more individuals, and you have four individuals. See, the operation is a manipulation of individuals, not a manipulation of some imaginary "numbers". And, the fact that we can make the individuals imaginary, such that the manipulation of individuals involves imaginary individuals, does not change the reality that the individuals are what is manipulated, not the numbers.

I didn't need a teacher to make me aware that numerals are not numbers. '2' and 'two' refer to the same thing. But '2' is not 'two'. So whatever they refer to is something else, which is a number, which is an abstraction. Rather than be a benighted bloviating ignoramus (such as you), I could see that thought uses concepts and abstraction and our explanations, reasoning and knowledge are not limited to always merely pointing at physical objects. — TonesInDeepFreeze

When I was seven years old I had no idea what an abstraction is, or what a concept is. I didn't understand this until much later when I studied philosophy. This is why mathematics really is like religion. We are required just to accept the rules, on faith, follow and obey, without any real understanding. 'There really is a number two there, accept and obey'. 'There really is a God there, accept and obey'.

That about sums it up. Math is like religion, a whole bunch of bullshit which we are told to accept on faith.

Sorry Tones, but we're so far apart on these principles of truth and falsity, that I see no place to start, or any point to it. I look at truth as corresponding with reality.

It's like saying that learning to play a musical instrument is tremendously difficult at first so people should just give up. — fishfry

I don't think the analogy is good. I learned to play a musical instrument, and it always made sense to me, right from the start. I learned philosophy and it always made sense to me. The point is that I proceeded because it made sense to. If mathematics requires self-deception, then this does not make sense to me, and so I will not proceed. Many people would not become athletes because there is much physical pain involved in the practice. This might be similar to my refusal to learn math because physical pain, and self-deception may both be viewed as harmful. Some people though will put up with the physical pain because they see a greater good in being an athlete. Maybe you put up with the self-deception of mathematics because you apprehend a greater good.

It's true of virtually EVERYTHING that at first, the subject makes no sense. You just do as you're told, do the exercises, do the homework, do the problem sets without comprehension, till one day you wake up and realize you've learned something. It must be that you've learned nothing at all in your life, having given up the moment something doesn't make immediate sense to you. — fishfry

Again I don't agree with this. Many things I've learned made sense to me right from the start. Even learning the numerals, how to count, and simple arithmetic, addition, subtraction multiplication, division made sense to me right from the start. It was only later, when they started insisting that there existed a number, distinct from the numeral, that things started not making sense.

I had a similar experience later with physics. We learned basic physics, then we learned about waves, and got to experiment in wave tanks. We learned that waves were an activity within a medium and we were shown through diagrams how the particles of the medium moved to formulate such an activity. All of this made very much sense to me. Then we were shown empirical proof that light existed as waves, and we were told that light waves had no medium. Of course this made no sense to me.

When you learned to play chess, or any game -- bridge, poker, whist -- do you say, "Oh this is nonsense, no knight REALLY moves this way," and quit? Why can't you learn a formal game on its own terms? If for no other reason than to be able to criticize it from a base of knowledge rather than ignorance? If you've never seen a baseball game, it makes no sense. As you watch, especially if you are lucky enough to have a companion who is willing to teach you the fine points of the game, you develop appreciation. Is that not the human activity called LEARNING? Why are you morally opposed to it? — fishfry

When I learn a game, I must learn the rules before I play. If the rules are such that I have no desire to play the game, I do not play. It's not a question of whether the game makes sense or not, so the analogy is not a good one.

Finally, even your basic objection to unordered sets is wrong. Imagine a bunch (infinitely many, even) of points randomly distributed on the plane or in 3-space. Can't you see that there is no inherent order? Then you come by and say, "Order them left to right, top to bottom." Or, "Order them by distance from the origin, and break ties by flipping a coin." Or, "Call this one 1, call this one 2, etc." — fishfry

Finally, you decided to address the issue. If there are points distributed on a plane, or 3d space, the positioning of those points relative to each other is describable, therefore there is an inherent order to them. If there was no order their positioning relative to each other could not be described..

You say that they are "randomly distributed", to create the illusion that there is no order. But the fact is that they must have been distributed by some activity, and their positioning posterior to that activity is a reflection of that activity, therefore their positioning is necessarily ordered, by that activity.

So just adopt the formalist perspective. There are only numerals and the rules for manipulating them. It's a game. What on earth is your objection? Were you like this when you learned to play chess? "There is no knight!" "The Queen has her hands full with Harry and that witch Meghan!" etc. Surely you're not like this all the time, are you? — fishfry

If you think you can interpret the rules as we go, then I'd advise you not to play any games with me.

What is the inherent order of the points in this set? Can you see that the points are inherently disordered or unordered, and that we may impose order on them arbitrarily in many different ways? Pick one and call it the first. Pick another and call it the second. Etc. What's wrong with that? — fishfry

The inherent order is the exact spatial positioning shown in the diagram. If any point changes location, then the order is broken. Is that so difficult to understand? A spatial ordering is not a matter of first and second, that is a temporal ordering.

Is it possible for an axiom to be false? Please explain. Don't refer to inconsistency. :roll: — jgill

For sure it's possible, the difficulty would be to demonstrate falsity, and this would require reference to some sort of inconsistency. What else could demonstrate falsity other than a reference to some form of inconsistency?.

An axiom is expressed as a bunch of symbols, so it must be interpreted. Interpretation requires that it be related to something else, and here we can have inconsistency and contradiction. So the author of an axiom will intentionally avoid internal inconsistency, or contradiction, but to understand, or employ the axiom it must be related to something external to it. If in interpretation, there is a contradiction with another principle then one or both must be false. If the other is a principle one holds to be true, then the axiom must be viewed as false.

Take the axiom of extensionality for example. Here's how Wikipedia states it:
".Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B.
(It is not really essential that X here be a set — but in ZF, everything is. See Ur-elements below for when this is violated.)"

Further, Wikipedia says it is interpreted like this:
" To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that A and B have precisely the same members. Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members."

Notice there is an exchange of "equal" and "same". As I've argued in other threads, if we adhere to the law of identity, this is a false use of "same". To resolve this issue, one might deny the law of identity, or insist on a faulty interpretation of that law. I think that approach is futile, so we must look directly at the axiom of extensionality and see what "equal" means in that context. If we can interpret in a way which does not employ "same" we might avoid the falsity.

Which axioms of finite set theory do you think are false? — TonesInDeepFreeze

Read above.

I think you are just not cut out for mathematical abstraction and should pick another major. — fishfry

I found that out at about tenth grade, despite living in an extremely mathematically inclined family. Prior to that though, I had difficulty even in grade school, when the teachers insisted on distinguishing numbers from numerals. Where are these "numbers" that the teacher kept trying to tell us about, I thought. All I could see is the numerals, and the quantity of objects referred to by the numeral. But the teacher insisted no, the numeral is not the number. So it took me very long to figure out that the numeral was not the "number" which the teacher was talking about, and that the number was just some fictitious thing existing in the teacher's mind, so I shouldn't even bother looking for it because I have to make up that fiction in my own mind, for there to be a number for me to "see". Of course, I chose philosophy as a major instead, because philosophy provides a solid, grounded understanding of abstraction, rather than simply insisting on the existence of fictitious "numbers" existing in people's minds, and trying to convince people to create those fictitious things in their minds.

Enough. You win. You wore me out. — fishfry

Simply put, I'm right and you're wrong. Nah, nah, let's go back to grade school. You should have chosen philosophy instead of math, if you wanted to learn the truth about abstraction.

Pick another major. — fishfry

I have picked another major, philosophy. That's why I'm discussing this in a philosophy forum. Care to join me? Or will you simply assert that mathematics is far superior to philosophy, then run off and hide under some numbers somewhere when the unintelligibility of your principles is demonstrated to you?

You may well have a philosophical point to make, but you are preventing yourself from learning the subject. And it's learning the subject that would allow you to make more substantive rather than naive and obfuscatory objections. — fishfry

You seem to have a very naive outlook. How do you propose that one proceed toward "learning the subject", when the most basic principles in that subject do not make any sense to the person? To me, the distinction between a numeral and a number is fundamentally unintelligible, as a falsity, because it requires producing a fictitious thing in my mind, and then talking about that fictitious thing as if it is a truth. Therefore proceeding into "learning the subject" requires an initial step of dishonesty, self-deception, then deceiving others in talking about this issue I have deceived myself about. I am not prepared to make that step of dishonesty. Making that initial step of self-deception is the first step toward misunderstanding, not toward understanding.

We prove from axioms. — TonesInDeepFreeze

If an axiom is false then the proof is unsound.

You also told us that you assume numbers are objects. — Luke

Notice, the quoted passage says numbers are assumed when "you" count. And, it's your count that I argue is false. .

You are back to your pathetic strawman misinterpretation, for the sake of ridicule.

It's not that sets don't have orderings. It's that sets have many orderings (though in some cases we need a choice axiom or an axiom weaker than choice but still implying linear ordering). So the point is that there is no single ordering that is "the ordering". — TonesInDeepFreeze

If we assume that a set necessarily has an ordering, but it could be one of many possible orderings, by what principle can we say that each of these many possible orderings constitutes the same set? What type of entity is an "element", such that the identity of a unity of numerous elements is based solely in the identity of its parts with complete disregard for the relations between those parts? Isn't this a sort of fallacy of composition?

You assume that numbers are objects but argue that numbers are not objects? Sounds about right given your confusion. — Luke

Do you understand the meaning of the word "if"? I don't think it's me who's the confused one.

By the axiom of extensionality, a set is entirely characterized by its elements, without regard to order. So the set {a,b,c} is the exact same set as {b,c,a} or {c,b,a}. — fishfry

That "elements" may exist without an order is the falsity I've explained to you already. And if we say that "element" indicates an abstraction, then it is a universal, not a particular, and to assume that an abstraction is a particular is a category mistake.

Let's assume a special type of "element", created, or imagined specifically for set theory. This type of element can exist in a multitude without that multitude having any order. Each of these elements would have no spatial or temporal relation to any other element, or else there would be an order, according to that relation. We could say that they are like points, but without a spatial reference, so that we cannot draw lines between them etc., because there is no order to them. But if they were like points, without spatial relations constituting order, there would be no way to distinguish one from another

Unlike points though, there is something which distinguishes one element from another, so that in the set of (a,b,c,), "a" does not represent the same thing as "b" does. Can I conclude, that the distinct elements are separated from one another, and distinguished one from another, by something other than space? To make them distinct and individual, they must have separation, but the separation cannot be spatial or else they would have an order, by that spatial relation.

Do you see, that from this premise alone, we cannot give any order to any set? To give a set an order would be a violation of the fundamental meaning of "element" which allows that elements can exist as particulars without any spatial temporal; relations. To be able to talk about an order within a set, would require that we transform the elements into something other than "elements", something which could have spatial or temporal relations and therefore an order. Remember, even quantity requires spatial-temporal separation between one and the other, to distinguish separate individuals.

Now we want to layer on the concept of order. To do that, we define a binary relation, which I'll call <, and we list or designate all the true pairs x < y in our set. So for example to designate the order relation a,b,c, we would take the base set {a,b,c}, and pair it with the set of ordered pairs {a < b, a < c, b < c}. Then the ordered set is designated as the PAIR ({a,b,c,}, {a < b, a < c, b < c}). I hope this is clear. — fishfry

No, sorry, it's not clear at all. You have imagined distinct "elements" which exist without any spatial or temporal relations, thereby having no order, though they are somehow distinct individuals. Now you want to add order. You have already defined order out of the set, to add it in, is blatant contradiction.

The basic takeaway is that a set has no inherent order. We impose an order on a set by PAIRING the set with an order relation. — fishfry

What I need, is a clear explanation of what an "order relation" is. What type of relation are you attempting to give to these elements, which gives them an order, when you've already stipulated the premise that they have no order?

The point is, that to give them existence without order requires a special conceptualization which I described above. Now if we want to proceed with that conceptualization, and now bring in principles of order, we must do so in a consistent way. So, we need to describe what separates one element from another, since it's clearly not space, and what makes it distinct as an element, in terms which do not give it a relationship to the others, to allow that the multitude of them do not already have any order, Then we need a principle by which order can be initiated within this non-ordered type of separation.

A set is a collection of elements, regarded as an individual thing, a set. — fishfry

Clearly, for a group of things to be regarded as an individual thing, "unity" is implied. And, it is quite clear that for a group of parts to form a unity, it is necessary that the parts exist with some sort of order. So this statement directly contradicts you assertion that a set has no inherent order.

Perhaps it's the distinction between a bunch of athletes and a team, or a collection of birds and a flock. I'm sure some philosophers have found ways to describe this. A set is a collection of elements, along with the concept of their set-hood. That's the best I can do! — fishfry

Do you see the contradiction? You describe a "set" as a thing, a unity, like a team, but then you say that there is no inherent order to this unified thing. Do you see how ridiculous this is, to say that there exists a unified thing, composed of parts, but there is no order to the parts? How in the world are we supposed to conceive of a unity of parts which have no order? To say that they are a unity is to say that they have order.

A set is inherently without order and without any kind of structure. — fishfry

Take a look again. You are proposing a type of unity, a "set", without any structure. By what principle do you say that it is a collection?

A set is entirely characterized by its elements; but a set is more than just its elements. It's the elements along with the collecting of the objects into a set. — fishfry

What is this act which you call "the collecting of the objects into a set"? Wouldn't such an act necessarily create an order, if only just a temporal order according to which ones are collected first? I'm trying to figure out how you get around the need for the elements to have an order. I mean, it's one thing to assert, "I've got this collection of elements and they have no inherent order" (to which I'd say you're lying), and another thing to demonstrate how you've collected a group of elements into a unified whole, without them having any order.

I welcome you to provide a non-circular reason for why "determining a quantity" is (true) counting and why "reciting the natural numbers in ascending order" is not (true) counting. — Luke

I explained this already. Your "ascending order" is based on quantity, therefore your supposed "count" of ascending order means nothing unless it is determining a quantity. This is why "numbers" as objects are assumed, so that when you count up to ten you have counted ten objects, (numbers).

Without this assumption that the symbol represents a quantity, you have a meaningless order of symbols which cannot be said to be "ascending". But if you allow that the symbol represents a quantity, then you have an ascending order. However, if the symbol does represent a quantity, then there must be objects which are counted, to validate the use of the symbols. Therefore, it is proposed that numbers are the objects which are counted, to validate the fact that the symbol must represent a quantity.

Therefore your "reciting the natural numbers in ascending order" is nothing but an act of determining a quantity of numbers. And, if numbers are not true objects, as I argue is the case, then this is not a true act of counting at all.

To determine a quantity is equally to make reference to an ascending order. — Luke

This is not true, as I argued with TIDF earlier in the thread. There are many ways to determine a quantity without referencing an ascending order.

In a logical proceeding, it is imperative that the symbol employed maintains the same meaning, to avoid the fallacy of equivocation. If "beating" means something different when used to describe beating eggs, from what it means when used to describe beating drums, and we proceed with a logic process, there could be a fallacious conclusion. For example, after the eggs are beaten, the internal parts are all mixed up into a new order, therefore if I beat the drums the internal parts will become all mixed up into a new order.

You might argue that “counting” in the sense of reciting the natural numbers in ascending order is not the proper meaning of the word, but why is it not? Why is “counting” in the sense of determining a quantity the only proper meaning of the word? These are both counting. — Luke

I explained this already, I think more than once. It is my opinion that there is no such thing as numbers which serve as a medium between the numeral (symbol) and its meaning, or what it represents. So this sense of "counting", which you describe, or define, as "reciting the natural numbers in ascending order", has a false description, or definition. This false definition would act as an unsound premise.

When we "count" in that sense, we are making an expression of symbols. As fishfry has explained, there is no inherent order to those symbols. To say that a particular order is "ascending order", is simply to make a reference to quantity. Therefore the meaning of that sense off "counting" is derived from, or based in that other sense, which is determining a quantity. So determining a quantity is the primary, and proper sense. If we remove that reference to quantity then there is no basis for any specific order, and you cannot say that "counting" involves an "ascending order", because "ascending" is not justified.

What I've been trying to tell fishfry, is that there is better sense of "order" which is not based in quantity, but it is temporal. If we refer to a temporal order, then we need some reason other than quantity to support any proposed order, showing why one symbol ought to be prior in time to another, when we "count" in the sense of expressing an order. This reason for ordering in this manner would provided the alternative name.

I think you need to reread my post. I have no desire to respond to your misinterpretation.


Thanks for the clarification fishfry, but here's a couple more things still to clear up.

To me, the following statements contradict each other.

But the set of natural number may nonetheless be ordered in many alternative ways. — fishfry

But no set has order. That's the axiom of extensionality. Will you kindly engage with this point? — fishfry

Which is the case, no set has order, or a set may be ordered in many different ways. Do you apprehend the contradiction? Which is it, ordered in different ways, or not ordered?

Let me go back to my question from the last post. What exactly constitutes "the set"? Is it the description, or is it the elements which are the members of the set. If it is the description, or definition, then order is excluded by the definition. But if the set is the actual participants, then as I explained already they cannot exist without having an order. If the supposed participants have no existence then they cannot constitute the set.

That's why I ask, which is it? Can a set be ordered, or is it inherently without order? Surely it cannot be both.

So I can use the phrase mathematical, but not mathematical objects? But mathematical is an adjective and mathematical object is a noun. You've still not answered the question.

But are you saying that if I call 5 a "mathematical concept" you're ok with that, but NOT with my calling it a mathematical object? Ok, I can almost live with that. Although to me, it's a mathematical object. — fishfry

Let's look at "concept" as a noun, as if a concept is a thing. Do you agree that a concept is the product, or result of conception, which is a mental activity? There's different mental activity involved, understanding, judgement, conclusion, and effort to remember. Would you agree that the effort to remember is what maintains the concept as a static thing, So if a "concept" is used as a noun, and is said to be a thing, it is in the same sense that a memory is said to be a thing. Would you agree that if a mathematical concept is "a thing", it is a thing in the same sense that a memory is a thing?

I propose instead that we reserve the term "counting" for counting the natural numbers and counting imaginary things, and that we should use the term "measuring" (instead of "counting") for "determining a quantity". — Luke

Now you just have a vicious circle. What does the numeral "2" refer to? The imaginary object which is the number 2. What is the number 2? The imaginary object which the numeral "2" refers to. See, vicious circle.

If you give the number 2 meaning, a definition, to validate its existence as a conceptual object, you might say that it means a quantity of two, but then you justify my argument, that counting is counting a quantity of objects, and "2" refers to two objects, not one object, the number 2. If you give the number 2 meaning by saying that it is the number which comes after 1, then you justify my argument that what you are doing is expressing an order, rather than counting. Either way, you'd be validating what I called justification, and you refused to acknowledge as justification. Or, would you like to give the number 2 some other type of definition, to validate its existence as a conceptual object? Prefer just remain within your vicious circle?

Furthermore, you ought to see that there is no need to assume "an object", or "number", as the intermediary between the sign "2", and its definition. When we say "square" there is no need to assume a conceptual object which is a square, as an intermediary between the word "square", and its definition, "equilateral rectangle".

But now you are saying that space and time have "conceptual" meaning; at the same time you deny that 5 or other numbers can have conceptual meaning. — fishfry

I surely have not denied that "5" has conceptual meaning. To say that the numeral "5", when it is properly used, must refer to five distinct particular things, is to give it conceptual meaning. It is a universal statement, therefore conceptual. I am not saying that it must refer to one specific group of five, as a name of that group, I am saying that it could refer to any group of five, therefore it is a universal, and this indicates that the "5" in my usage refers to a concept, what you've called an abstraction, rather than any particular group of five.

For example, if I said that to properly use "square", it must refer to an equilateral rectangle, or "circle" must refer to a plane round figure with a circumference which has each point equidistant from its center point, I give these terms conceptual meaning, because I do not say that the words must refer to a particular figure, I allow them to refer to a class or category of figures.

Even if I said that "5" must refer only to one particular group of five, or that "square" must refer only to one particular figure, it could still be argued that this is "conceptual meaning", because to understand this phrase "must refer only to one particular", is to understand something conceptual. In reality any meaning assigned to word usage is conceptual, so this position you've thrust at me, that I deny the conceptual meaning of 5, is nonsense. What I say is that the conceptual meaning given to "5", in some situations, namely that it refers to a type of object called a number (as described by platonic realism), ought to be considered as wrong. Do you accept the fact that concepts can be wrong? For instance, your example of "justice". A group of people could have a wrong idea about what "justice" means. Likewise, a group of people could have a wrong idea about what "5" means.

How about "inspired by" rather than grounded? As in Moby Dick being a work of fiction nevertheless inspired by a real historical event. Of course we get our concept of number from real, physical things. Nobody's denying that. — fishfry

Why would you want to make this change to "inspired" rather than "grounded"? Logic is grounded in true premises, and this is an important aspect of soundness. If your desire is to remove that requirement, and insist that the axioms of mathematics need not be true, they need only to be "inspired", like a work of fiction, the result would be unsound mathematics. Sure this unsound mathematics might be fun to play with for these people whom you call "pure mathematicians", and I call "mathemagicians", but unsound mathematics can't be said to provide acceptable principles for a discipline like physics.

Well the "first" element of a total order is an element that is less than any other element. Some orders have a first element, such as 1 in the positive integers. Some orders don't. There's no first positive rational number.

That's what first means. — fishfry

OK, I assume that "less than" refers to quantity. So we're right back to my original argument then. Numerals like "1", "2", "3", "4", refer to a quantity of objects, "3' indicating a quantity which is less than that indicated by "4", and "first" indicates a lower quantity. How do you propose to remove the quantitative reference to produce a pure order, not grounded in a physical quantity?

Now that's funny, as we got off onto this conversation by pointing out to you that numbers can indicate order as well as quantity. But of course ordinals are different than cardinals. Two distinct ordinals can have the same cardinal. — fishfry

If you are grounding your definition of "order" in "less than", as you have, then numbers simply indicate quantity, and your "order" is just implied. It is not the case that "2" indicates "first" in relation to "3" and "4", it is the case that "2" indicates a quantity which is less than the quantity indicated by "3" and by "4". And by your premise, that the "first "is the one which is less than the others, you conclude that "2" is first.

Therefore "order" as you have presented it is not indicated by the numbers, only quantity is indicated by the numbers. Order is indicated by something other than the numbers, it's indicated by your premise that the numeral signifying a quantity less than the others, is first.

red, blue, green. Three words ordered by length. There is no time involved. You are stuck on this point through stubborness, not rational discourse. The player who finishes first in a golf tournament is the one with the lowest score, NOT the one who races around the course first. — fishfry

You clearly haven't followed what I've been saying, and I realize that I did not make myself clear at all. The point is that if we remove the reference to a quantity of individual objects, from numerals, then the ordering of numbers requires a spatial or temporal reference. You seemed to believe that we could remove the quantitative reference, and have numbers with their meanings understood in reference to order only. Clearly, "less than" does not provide this for us. And your example of the length of the word here, is a spatial reference.

Your other example, of the best score being first is only made relevant through a quantitative interpretation. How is 3 better than 4? Because it's less than. So you have not removed the reference to quantity as the necessary aspect of numerals, to provide a purely ordinal definition. Therefore I am still waiting for you to prove your claims.

I have already given many counterexamples such as rationals, reals, complex numbers, p-adics, hyperreals, and various other exotic classes of numbers studied by mathematicians. What quantity or order does 3+5i3+5i represent?

There is no general definition of number in math. That's kind of a curiosity, and it's kind of an interesting philosophical point, and it's also factually true. — fishfry

You have shown me absolutely nothing in the sense of a number not dependent on quantity for its meaning.

I've made my point and all you have is mathematical ignorance. — fishfry

If your point is that "order" is defined by " less than", and this is supposed to be an order which is independent from quantity, then you've failed miserably at making your point.

You haven't seen them in the playground at recess. Of course that's only when I was a kid. These days I gather they don't let the kids run around randomly at recess. — fishfry

Obviously, "in the playground" is not "in the classroom", and you're clutching at straws in defense of a lost cause.

If you don't know that sets have no inherent order, there is no point in my arguing with your willful mathematical ignorance. — fishfry

Instead of addressing my argument you portray me as mathematically ignorant. It's not a matter of ignorance on my part, it's a refusal to accept a mathematical axiom which is clearly false. So I'd correct this to say that this is an instance of your denial, and willful ignorance of the truth, for the sake of supporting a false mathematical axiom.

No that is not true. It's entirely contrary to the concept of set. A set has no inherent order. An order is a binary relation that's imposed on a given set. If I have a set and don't bother to supply an order relation, then the set has no order. Sets inherently have no order. That's what a set is. You can sit here all day long and make up your own definitions, but that's of no use or interest to anyone. — fishfry

Show me that set which has no order then. And remember, there is a difference between a thing itself, and the description of a thing. Therefore to describe a set which has no order is not to show me a set which has no order.

I think you need to make clear what "set" means. Does it refer to a group of things, or does it refer to the category which those things are classed into? The two are completely different. Take your example of "schoolkids" for instance. Does "set" refer to the actual kids, in which case there is necessarily an order which they are in, even if they are running around and changing their order? Or, does "set" refer to the concept, the category "schoolkids", in which case there are no particular individuals being referred to, and no necessary order? Which is it that "set" refers to, the particulars or the universal? Or is "set" just a clusterfuck, a massive category mistake?

I'm asking you, if you don't accept the phrase mathematical object, what phrase do you use to name or label conceptual entities that are mathematical, as opposed to conceptual entities like justice that are not mathematical? — fishfry

You just named it for me. "Mathematical" is the word I use to refer to mathematical concepts. In ethics there are ethical concepts like justice, in biology there are biological concepts like evolution, and in physics there are physical concepts like mass. Why do you think mathematics ought to be afforded the luxury of treating their concepts as if they are objects?

You have attempted to argue that counting natural numbers, or counting imaginary things, is not true counting, and that to call this "counting" is a misnomer. — Luke

Right, and the reason why I argued this is that we ought not have two distinct activities going by the same name in a rigorous logical system, because equivocation is inevitable. So, one ought to be called "counting" and the other something else. I propose the obvious, for the other, expressing an order.

And, I provided all the required justification. You just do not accept it. So in your mind it has not been justified. That's the nature of justification, regardless of how sound the argument is, if it is not accepted the proposal does not qualify as "justified".

I however, expect nothing less from you. This is consistent with your previous behaviour. No matter what explanation I provide, as to why specific words ought to be restricted in certain ways, to enhance the epistemic capacity of a logical system, you'll reject it. It's quite clear to me that you reject these proposals because they would incapacitate your principal means of argumentation, which is equivocation.

But what is the justification for your stipulation that counting natural numbers is not real counting or that real counting must involve "determining a quantity"? — Luke

The point is to avoid equivocation which is a logical fallacy. Since one sense of "counting" involves counting real things, then why not call this "real counting"?

If a flame be a dumpster fire. — TonesInDeepFreeze

That's me, the dumpster arsonist. Easiest way to dispose of garbage is to burn it. Not so good for the environment though. But neither is garbage.

Everyone else considers "counting up to ten" to be counting (you also called it "counting", by the way). — Luke

Yes, i call it "counting", but the point is that there's two very distinct senses of "counting" and to avoid ambiguity and equivocation we ought to have two distinct names for the activity, like fishfry explained with the distinct names for the numerals used, cardinals and ordinals.

Why should we care about your unjustified stipulation that counting the natural numbers is not real counting or that real counting must involve "determining a quantity"? — Luke

Don't mathematicians and other logicians harbour a goal of of maintaining validity, and avoiding fallacies such as equivocation? If it is the case, that when a person expresses the order of numerals, one to ten, and the person calls this "counting", it is interpreted that the person has counted a quantity of objects, a bunch of numbers, rather than having expressed an ordering of numerals, then the interpretation is fallacious due to equivocation between the distinct meanings of "counting".

The issue which fishfry and I have now approached is the idea of a set without any order. I have argued that this is a contradictory idea because if the set exists as a set, its members must have existence in the order which they have in the existing set. It is only by removing existence from the set that we can say the members of the set have any possible order. But then the set itself is not an actual set, it's just the possibility of a set. This would be like a definition without the necessity of anything fulfilling that definition. We could say it's an imaginary set, whereas a real set has real existing members and therefore a real existing order.

In the case of mathematics the question becomes what is supposed to be in the set, the symbols (numerals) or what the symbols represent (numbers). If it is the latter, then the set can be defined with the symbols, and the members within the set, being imaginary, have no existence, and therefore can be said to have no order, or any possible order. But such a set is necessarily non-existent and imaginary, and it cannot be used to represent any real things in the physical world, because real things have an order.

So we have a distinction to be made between two different uses of "set". We can refer to a group of existing objects which necessarily have an order, as a "set". And this type of set is "countable" in the sense that we can determine the quantity of objects within the set. And we can also can use "set" to refer to an imaginary group of objects, having no order because they have no existence. But this type of "set" is not "countable" in the sense that we cannot determine the quantity of objects within such a set. In other words, any set which is stated as having no order, but only possible orders, ought to be considered as imaginary and therefore of indeterminate quantity.

What unit of measurement is required for counting the natural numbers? Metres? Litres? Hours? Bananas? Obviously, no unit of measurement is required. You can count to ten without having to determine any unit of measurement. Therefore, counting is independent of measuring. Counting is not a "form of" measuring. — Luke

"Counting the natural numbers", as described here, is a matter of established an order. It is not an instance of counting in the sense of determining a quantity. There are no objects (numbers) being counted.

You are equivocating between these two senses of "counting". To count, in the sense of determining a quantity, is an act of measuring. To "count" in the sense of counting up to ten, is a case of expressing an order, two comes after one, three comes after two, etc.. To call this "counting the natural numbers" is a misnomer because this is nothing being counted, no quantity being determined. That is why we can theoretically "count the natural numbers" infinitely, without end, because we are just stating an order, not determining a quantity.

Perhaps you're right that meaning isn't the correct word. If I said we remove a concept from its worldly or physical referent, would that be better? We care about first, second, third, and not first base, second base, third base. So how would you describe that? I'm focusing on ordinality itself and not the things ordered. So you're right, meaning was an imprecise word. — fishfry

Let's get this straight. I am not talking physical referents here. I am talking space and time, which are conceptual. The issue is that when we remove the physical referents (required for "counting" in the sense of determining a quantity, as the things counted), for the sake of what you might call purely abstract numbers, the meaning of the numbers is grounded in the abstract concepts of space and time. Numbers no longer refer to physical objects being counted, they refer to these abstract concepts of space and time.

Now, we have only deferred the need to refer to physical existence, because if our conceptions of space and time are inaccurate, and the ordering of our numbers is based in these conception of space and time, then our ordering of the numbers will be faulty as well. You seem to think that in pure mathematics, a logician is free to establish whatever one wants as "an order", but this is not true, because the logician is bound by the precepts of "logic" in order that the order be logical. For example, a self-contradicting premise is not allowed. So there are fundamental rules as to the criteria for "order" which cannot be broken. And even if you argue that the order could be a completely random ordering of numbers, the rule here is that each thing in the order must be a number. And every time a logician tries to escape the rule, by establishing a principle allowing oneself to go outside that rule, there must be a new rule created, or else the logician goes outside the field of logic. And the point, is that if the rule is not grounded in empirical fact (physical existence) the logic produced is faulty, and the proposed rule ought to be rejected as a false premise.

There is no temporal reference. — fishfry

Surely, "first" does not mean "highest quality", or "best", in mathematics, so if it's not a temporal reference, what is it?

Ok. I agree that I'm having trouble precisely defining abstraction and I sort of see your point. But ordinal numbers are purely about order, but they're not about any particular things being ordered. How would you describe that? It's not meaningless, yet it refers to nothing in the world at all other than the pure concept of order. Which you don't seem to believe in. — fishfry

Yes that is my point as to how counting order is different from counting a quantity. To count a quantity requires particular things, but to count an order requires only time. However, time is something in the world, and that's why I don't believe in what you call "the pure concept of order".

But order is not essential to numbers, it's imposed afterward. — fishfry

If order is not essential to numbers, then something else must be, because to be a concept is to be definable according to essential properties. I propose, then that quantity is essential to numbers. Do you agree? If for example you make an order, or a category, of odd numbers, or even numbers, or prime numbers, it is something about the quantity represented by the number which makes it belong in one or more of these categories. If it's not quantity which is essential to numbers, as the defining feature of "number" then what do you think is? You've already ruled out order.

. I get that you are drawing a distinction between the mathematical formalism, in which order is secondary to the existence of numbers; and philosophy, in which order is an essential aspect of numbers. — fishfry

No, I am saying that if order is secondary to the existence of numbers, then quantity must be primary.

A schoolkid must have a height, but it could be any height. — fishfry

That's not true at all, it's the fallacy I referred to. The schoolkid must have height, and that height must be the height that the schoolkid has. Therefore it is impossible that the schoolkid has a height other than the height that the schoolkid has, and very obviously impossible that "it could be any height". To make such a claim is clearly fallacious, in violation of the law of identity, because you are implying that a thing could have properties other than those that it has, saying it could have any property. Obviously this is not true because a thing can only have the properties that it has, otherwise it is not the thing that it is.

You see it that way. I see it as providing beautifully logical clarity. We have the set of natural numbers, and we have the standard order and we have a lot of other orders, and we can even consider the entire collection of all possible orders, which itself turns out to be a very interesting mathematical object. It's quite a lovely intellectual structure. I'm sorry it gives you such distress. — fishfry

It gives me distress to see you describe something so obviously fallacious as "providing beautiful logical clarity". If you consider circumventing the law of identity as beautiful logical clarity, I have pity.

But I have not asserted that a set must have any order at all. The set NN has no inherent order at all. Just like a classroom full of kids has no inherent order till the teacher tells them to line up by height or by alpha firstname or reverse alpha lastname or age or test score or age. Why can't you see that? — fishfry

Again, you're continuing with your fallacy. A classroom full of kids must have an order, or else the kids have no spatial positions in the classroom. Clearly though, they are within the classroom, and whatever position they are in is the order which they have. To deny that they have an order is to deny that they have spatial existence within the room, but that contradicts your premise "a classroom full of kids".

A contradiction is a proposition P such that both P and not-P may be proven from the axioms. Perhaps you would CLEALY state some proposition whose assertion and negation are provable from the concept of order as I've presented it. I don't think you can. — fishfry

Above, is your CLEAR example of contradiction "a classroom full of kids has no inherent order". By saying "there is a classroom full of kids", you are saying that there is an order to these kids, they exist with determinate positions, in a defined space. You contradict this by saying they have no inherent order.

Absolutely agreed. Yes. The essence of a set of numbers is NOT in their order, since we can easily impose many different orders on the same underlying set. Just as the ordering by height is not essential to the classroom of kids, since we can impose a different order; or by letting them loose in the playground at recess, we can remove all semblance of order! Surely you must take this point. — fishfry

So, if "a set" is like the kids in the classroom, then it must have an order to exist as a set. We can say that the order is accidental, it is not an essential feature, so that the same set could change from one order to another, just like the kids in the class, and still maintain its status as the same set. However, we cannot say that a set could have any order by reason of the fallacy described above, because this is to say that it has no actual order which implies that it does not exist.

Ok. But that's not good enough. I asked how do you call mathematical objects like topological spaces. But justice and property are concepts and abstractions, yet they are not mathematical objects.

If you don't like the phrase, "mathematical object," what do you call them? Sure they're an abstraction, but that's way too general. You see that I'm sure. — fishfry

No, I don't see that at all. They are all concepts, ideas. By what principle do you say that mathematical concepts are "objects", but concepts like "justice" are not objects. I mean where is your criteria as to what constitutes a conceptual "object". I know it's not the law of identity.

An object is not a goal. An (American) football is an object, and the goal is to get it across the goal line. You would not say the football is a goal. I think you're way off the mark with your claim that an object is a goal or objective. 5 has no object or purpose. It's just the number 5. A mathematical object. An abstract object, as all mathematical objects are. — fishfry

You've never heard "the object of the game"?

No, not in the least. How can you say that? That's not even the meaning of the words in everyday speech in the real world. The winner takes first place and the runner up takes second place sometimes (as in a foot race) but not always (as in a weight lifting contest) by being temporally first. You must know this, why are you using such a weak argument? First place in golf goes to the player with the lowest score, not to the player who finishes the course first. This is a TERRIBLE argument you're making here. — fishfry

So in this context, "first" means best. Clearly this is not how "first" is used in mathematics. In mathematics, "first" has a temporal reference of prior to, as I said, not a qualitative reference as "best". Your attempt at equivocation is not very good, I'm happy to say, for your sake. Ask Luke who is the master of equivocation for guidance, if you want to learn. I think you ought to stay away from that though.

Math just has the number 5. — fishfry

.The problem obviously, is that you, and mathematicians in general, according to what you said above, haven't got a clue as to what a number is. It's just an imaginary thing which you claim is an object. It appears like you can't even tell me how to distinguish the number 4 from the number 5, because you refuse to recognize the importance of quantity. And if you would recognize that it is by means of quantity that we distinguish 4 from 5, then you would see that "4", and "5" cannot each represent an object, because one represents four objects, and the other five objects. Why do you take numbers for granted?

.

You are saying that counting is the same as measuring, but that can’t be right. Otherwise, what unit of measurement do we use to count? — Luke

Counting is not "the same as measuring", it's a form of measuring. What is required for measuring is a standard, The standard for counting is "the unit", which is defined as an individual, a single, a particular. So in measuring a quantity (counting) we must make a judgement as to what qualifies as a unit to be counted.

The point is that by abstracting the concept of order from any particular meaning, we can better study order. — fishfry

OK, so you define "order" as "having no meaning". That is your starting premise? What's the point? Any meaning you give to it will be logically invalid, as contradictory to that definition. There is nothing to study in a concept which has no meaning.

The point of abstraction is to take away meaning such as first base, second base, so that we can study first and second abstracted from meaning. That doesn't make abstraction meaningless, it just means that we use abstraction to study concrete things by abstracting away the concreteness. — fishfry

Of course it makes it meaningless, you just said you take away meaning from it. If you take away all the meaning from "first" and "second", you just have symbols without meaning. If you leave some sort of meaning as a ground, a base, you have a temporal reference, first is before, (prior to) second.

You are using "abstract" in a way opposite to convention. We do not "take away meaning" through abstraction, abstraction is how we construct meaning. There is a process called "abstraction", by which we remove accidental properties to give us essentials, what is necessary to the concept. We do not abstract away the meaning, we abstract what is judged as "necessary" from the concreteness, leaving behind what is unnecessary, "accidental".

Well, yes and no. Von Neumann's coding of the natural numbers has the feature that the cardinality of the number n is n. But there are other codings in which this isn't true, for example 0 = {}, 1 = {{}}, etc. So we can abstract away quantity too if we like. But that wasn't the point, Even if I grant you that cardinality provides a natural way of ordering the natural numbers, it's still not the only way. — fishfry

Sure, cardinality is not the only possible way of ordering numbers, but if the point is, as you described, to allow for any possible order, then we have to deny the necessity of all possible orders. That is to say that there is no specific order which is necessary. This removes "order" as a defining feature of numbers, because no order is necessary, so numbers do not inherently have order. Therefore order is not essential to the concept of numbers Then, we need something else to say what makes a number a number, or else we just have symbols without meaning.

We could try saying that it is necessary that numbers have an order, but the specific order which they have is not necessary, like we might say a certain type of thing must have a colour, but it could be any colour. But this will prove to be a logical quagmire because it's really just a way of smuggling in a contradiction. It is impossible, by way of contradiction, that something must be a specific colour, and at the same time is possibly any colour. It is only possible that it is the colour that it is. Likewise, it is impossible that numbers must have a specific order, but could possibly be any order, because the order that they currently have, would restrict the possibility of another order.

The point was, that if remove all order, to say that numbers are not necessarily in any order, then we must define the essence of numbers in something other than order. If this is cardinality, then cardinality is not an order.

What do you call numbers, sets, topological spaces, and the like? — fishfry

They are concepts, abstractions. I apprehend a difference between concepts and objects, because concepts are universals and objects are particulars. There is an incompatibility between the two, and to confuse them, or conflate them is known as a category mistake.

But the 5 that mathematicians study is indeed an abstract object. It's not 5 oranges or 5 planets or 5 anything. It's just 5. That's mathematical abstraction. I guess I'm all out of explanations. — fishfry

It's an idea, and ideas are not objects. I have an idea to post this comment, and this idea exists as a goal. Goals are "objects", or objectives, in a completely different sense of the word. So if you want to say that numbers, as ideas are "objects", we'd have to look at this sense of the word, goals. But it doesn't make too much sense to say that they are objects in this sense, nor does it make any sense at all, to say that numbers, as ideas, are objects in the sense of particulars, because they are universals.

There is no space or time in math. Why can't you accept abstraction? There's space and time in physics, an application of math. There's no space or time in math itself. Is this really a point I need to explain? — fishfry

Space and time are themselves abstractions, and these concepts very clearly enter into, and are fundamental to mathematics. Are a circle and a square not a spatial concept, which are mathematical? Is the order of first, second, third, fourth, not a temporal order whish is mathematical? If you seriously think that you can separate mathematical concepts from spatial and temporal concepts, then yes, this is something you really need to explain, because I've been trying to do it for many years and cannot figure out how it's possible. So please oblige me, and explain.

The mathematician only cares about 5. — fishfry

The problem is that "5" means nothing without a spatial or temporal reference. If you think that the mathematician believes that "5" refers simply to the number 5, without any further reference to give the concept which you call the number 5 meaning, then you must believe that mathematicians think that the number 5 is a concept of nothing.

How is it that we can (really) order imaginary things, but we cannot (really) count imaginary things? — Luke

If "count" is defined as determining the quantity of, then it is an act of measuring. We can't measure imaginary things. But we can describe an order without requiring that measurable things exist in that order, the order itself is imaginary.

There don't need to be any real sheep in order to make the count. One could as easily count unicorns instead of sheep. Or Enterprise captains. Or any other fictional entities. — Luke

As I said, that's an order, one imagined thing after the other, it's not a quantity.

Luke, learn how to read! The representations, (which is what we count), exist as symbols. I did not say that the imaginary things exist as symbols. You've taken the sentence out of its context so that it appears possible that I might be saying what you claim to interpret. Though context clearly shows otherwise. This is exactly what I mean, you interpret, and represent what I say, in a totally incorrect (not what I intended), strawman way, solely for the purpose of knocking it down. Your MO, to ridicule, is itself ridiculous.

You then stated that "we can only count representations of the imaginary things, which exist as symbols." — Luke

That's a false quote. I said "we are not really counting the imaginary things, but symbols or representations of them". You said they only exist as symbols, not I.

But I can't agree with your apparent extrapolation from that to an apparent rejection of all abstract math. — fishfry

What I look for, is points within abstract math where improvement is warranted.

I'm not enough of a physicist to comment. My point was only that you seemed to reject QM for some reason. I noted that you can't dismiss it so trivially, since QM has a theory -- admittedly fictional in some sense -- but that nevertheless corresponds with actual physical experiment to 13 decimal places. That's impressive, and one has to account for the way in which a fictional story about electrons can so accurately correspond to reality. Of course all science consists of historically contingent approximations. But lately some of the approximations are getting really good. Your dismissal seems excessive. — fishfry

The capacity to use mathematics to make very precise predictions does not necessarily indicate an understanding of the activity which is predictable. I often use as an example the capacity of ancient people to predict the position of the sun, moon, and planets, without understanding the orbits of the solar system. Thales apparently predicted a solar eclipse. So an ancient 'scientist' could predict the exact location the sun would rise on the horizon, and one could insist that this justifies a model which assumes that a dragon carries the sun in it's mouth around the earth from sun set to sunrise. Predicting the appearance of objects is completely different from understanding the activity involved.

FWIW I don't think anyone thinks the orbits are circular anymore. — fishfry

I know, that's the point. The concept of the magnetic moment of an electron is based in the assumption of a circular orbit, which is an idea known to be faulty. And the whole idea of "spin" in fundamental particles is not any sort of spin at all, because the particles cannot be shown to have any proper spatial area, within which to be spinning. The physicists simply apply the appropriate mathematics which produces the desired predictions, but the models which explain what the mathematics is doing are completely unacceptable, indicating that the physicists are capable of making predictions without knowing what is going on.

But you still have to account for the amazing agreement of theory with experiment. We might almost talk about the unreasonable effectiveness of physics in the physical sciences! — fishfry

That's the power of mathematics. But the experiments and the mathematics are designed for one another, so that the experiments show how good the mathematics is, and the mathematics shows how good the experimenters are. But they are only working with a very small portion of the microscale world, because of limited capacity for experimentation, and attempts to extrapolate show just how inadequate the mathematics, experiments, or both, are for producing a wider understanding.

I'm taking this from the end of your post and addressing it first to get it out of the way. As I mentioned, I didn't read any posts in this thread that didn't mention my handle. I only responded to one single sentence of yours to the effect that numbers are about quantity. I simply pointed out that there is another completely distinct use of numbers, namely order. Anything else going on in this thread I have no comment on. — fishfry

That is where I started in this thread, with the assertion that numbers are about quantity, but I've changed my mind twice since then. You got me to see the difference between quantity and order, and this difference is why I could not understand Tones' representation of counting as bijection. So, the act of "counting" may be an act of determining a quantity or it may be an act expressing an order. The two are very distinct as you say, but both are commonly referred to as "counting".

But if there are two distinct but related ways of using numerals, and each relates to the same concept "number", then we must proceed toward something further, some other idea which synthesizes the two, into one concept, "number". So I changed my mind again, in that post I asked you to reread I believe. What I said is that I think a number is "a value". This allows that the same value, expressed as "2" for example, can be assigned to a quantity of two, and also to the second in order. Remember what I said about a value. When we say "the same value", there is an equality between two distinct things without saying that the two things are the same thing (as implied by the law of identity).

Here's the quote from that post:

I think this helps to demonstrate that we cannot define numbers with counting. So, my original assumption that "2" implies a specified quantity of objects, must be false. But now we have the question of what does "2" mean? I think it is a sort of value, and by my statement above, a value we assign to empirical observations. However, if we can assign such a value to imaginary things in a similar way, we need a principle to establish equality, or compatibility, between observed things and imaginary things. This is required to use negative numbers. — Metaphysician Undercover

I may not be fully aware of the philosophical context of your use of "a priori." Do you mean mathematical abstraction? Because I am talking about, and you seem to be objecting to, the essentially abstract nature of math. The farmer has five cows but the mathematician only cares about the five. The referent of the quantity or order is unimportant. If you don't believe in abstraction at all (a theme of yours) then there's no hope. In elementary physics problems a vector has a length of 3 meters; but the exact same problem in calculus class presents the length as 3. There are no units in math other than with reference to the arbitrarily stipulated unit of 1. There aren't grams and meters and seconds. — fishfry

Surely I believe in abstraction, but all abstractions are derived (abstracted) from somewhere, unless they are completely innate. So the abstractions "quantity", and "order" must have some sort of referent themselves which give meaning to the concept. It does not make any sense, even in the context of pure mathematics, to say that there is a quantity of 5 which does not consist of five units. That's the meaning of "a quantity of 5". Even in abstraction there are necessary aspects of the concept which must be fulfilled to account for the meaning of the abstraction. If you are simply talking about "the number 5", and not a quantity of five, then we must look to see what gives "the number 5" its meaning., when it is supposed that it does not signify 5 discrete units. We might suppose that the meaning of "the number 5" is found in an order, it's the fifth. But the number 5 is not necessarily the fifth, and that's why I turned at first, to quantity to see how "the number 5" gets its meaning.

There's no time or space, just abstract numbers. I don't know how to say it better than that, and it's frustrating to me that you either pretend to not believe in mathematical abstraction, or really don't. — fishfry

This is not true, because the numbers have meaning, that's the point. You cannot use the number 5 however you please, and say "5+5=8". That is restricted by the meaning. So it's not simply "abstract numbers", it's specific numbers. Each number has its specific meaning or else all numbers would be the same. And when I look at the meaning of any specific number I find that the number either refers to spatially distinct units, 5 of them, or it refers to a temporal order, the fifth. Clearly there is time and space implied with abstract numbers, or else each number would lose its meaning which is specific to it.

You seem to want to deny the ideas themselves simply because they're abstract. That's the part of your viewpoint I don't understand. — fishfry

I do not want to deny the ideas, I want to understand them. And understanding them is what requires spatial and temporal reference. The number 5 has no meaning, and cannot be understood without such reference.

There is no need for time or space in math. I can't talk or argue or logic you out of your disbelief in human abstraction. — fishfry

An abstraction must be intelligible or else it is meaningless, useless. If it can't be understood without spatial or temporal reference, then there clearly is a need for space and time in math, or else all mathematics would be simply unintelligible.

You just phrase things like that to annoy me. How can you utterly deny human abstractions? Language is an abstraction. Law, property, traffic lights are abstractions. So is math. — fishfry

I do not deny human abstractions, I just insist that they are fundamentally distinct, different from objects. An object is a unique particular. An abstraction is a generalization. The two are very different from each other, and ought not be both classed together in the same category as "objects".

The notation is only suggestive of a deeper abstract truth, that of the idea of an endless progression of things, one after the next, with no end, such that each thing has an immediate successor. — fishfry

Do you agree that this order, "an endless progression of things, one after the next" is a temporal order?

Now the set of natural numbers N={0,1,2,3,4,…}N={0,1,2,3,4,…} has no inherent order. — fishfry

You'll have to do better than a simple assertion here. To say that the natural numbers have no inherent order, is to remove "order" as a defining feature of the natural numbers. Now we are left with quantity as the defining feature. Do you agree? There must be something which gives 5 and 6 meaning, if it's not a specific order, it must be a quantity. So we are not talking about an endless progression of things when we talk about the natural numbers in this way, we are talking about specific symbols, "1", "2", "3", etc., which represent specific quantities. Now we don't have a set of natural numbers, because we have no things, only symbols representing quantities. So the rest of your discussion of order is irrelevant. You have nothing to order, and no order to offer.

I would say that I've made a considerable effort the past several years to understand your point of view. — fishfry

Yes, I've apprehend this, and I respect it. I know that's why you keep on engaging me. it's not easy to understand unorthodox and unconventional ways of thinking like mine though, so I've seen your frustration. But I do appreciate the effort. I've see the same effort to understand from jgill. I don't think TonesInDeepFreeze quite has that attitude though, and Luke just seems to be always looking for the easiest ways (mostly fallacious) of making me appear to be wrong, no matter what I say.

When you pooh-poohed the 13-digit accuracy of the measurement of the magnetic moment of the electron, you indicated a dismissal of all experimental science. — fishfry

Let me tell you something. The magnetic moment of an electron is a defining feature of how magnetism effects a massive object. Therefore it is not measured it is a stipulation based in specific assumptions such as a circular orbit. But if the electron's orbit is really not circular, then the stipulated number is incorrect.

This is a purely abstract order relation on the natural numbers. — fishfry

All I saw in you demonstration was a spatial ordering of symbols. I really do not see how to derive a purely abstract order from this. If you truly think that there is some type of order which is intelligible without any spatial or temporal reference, you need to do a better job demonstrating and explaining it.

I assure you, I am very interested to see this demonstration, because I've been looking for such a thing for a long time, because it would justify a pure form of "a priori". Of course, I'll be very harsh in my criticism because I used to believe in the pure a priori years ago, but when such a believe could not ever be justified I've since changed my mind. To persuade me back, would require what I would apprehend your demonstration as a faultless proof.

There is an issue though, that I'll warn you of. Any such demonstration which you can make, will be an empirical demonstration, using symbols to represent the abstract. So the onus will be on you, to demonstrate how the proposed "purely abstract order" could exist without the use of the empirical symbols, or else to show that the empirical symbols could exist in some sort of order which is grounded or understood neither through temporal nor spatial ideas.

I'll tell you something else though, I have opted for a sort of compromise to this problem of justifying the pure a priori, by concluding that time itself is non-empirical, thus justifying the temporal order of first, second, third, etc., as purely a priori. However, this requires that I divorce myself from the conventional idea of time which sees time as derived from spatial change. Instead, we need to see time as required, necessary for spatial change, and this places the passing of time as prior to all spatial existence. This is why I said what I did about modern physics, this position is completely incompatible with the representation of time employed in physics. In conceiving of time in this way we have the means for a sort of compromised pure a priori order. It is compromised because it divides "experience" into two parts, associated with the internal and external intuitions. The internal being the intuition of time, must be separated from "experience" to maintain the status of "a priori", free from experience, for the temporal order. So it's a compromised pure a priori.

You can't claim ignorance of this illustration of the distinction between quantity and order, since I already showed it to you in this thread. So whence comes your claim, which is false on its face, and falls on its face as well? — fishfry

I didn't deny the distinction between quantity and order, I emphasized it to accuse Tones of equivocation between the two in his representation of a count as bijection.

This also is wrong, since there is no mathematical difference between counting abstract or imaginary objects (sheep, for example, as someone noted) and counting rocks. — fishfry

That is exactly why I attack the principles of mathematics as faulty. There are empirical principles based in the law of identity, by which a physical, and sensible object is designated as an individual unit, a distinct particular, which can be counted as one discrete entity. There are no such principles for imaginary things. Imaginary things have vague and fuzzy boundaries as evidenced from the sorites paradox. so the fact that "there is no mathematical difference between counting abstract or imaginary objects...and counting rocks", is evidence of faulty mathematics.

Please show me space or time in the ≺≺ order on the natural numbers. — fishfry

As I said, all you've given me is a representation of a spatial ordering of symbols. If you are presenting me with something more than this you'll have to provide me with a better demonstration.

Who is this "we?" Surely there are many who can argue the opposite. Planck scale and all that. Simulation theory and all that. Of course we "think" of space and time as continuous if we are Newtonians, but that worldview's been paradigm-shifted as you know. — fishfry

I go both ways on this. Of space and time, one is continuous, the other discrete. But this is another reason why I think physics has a faulty representation of space and time, they tend to class the two together, as both either one or the other.

But I don't see your point. Cardinals refer to quantity and ordinals to order. The number 5 may be the cardinal 5 or the ordinal 5. The symbology is overloaded but the meaning is always clear from context; and in any event, the order type of a finite set never changes even if its order does. The distinction between cardinals and ordinals only gets interesting in the transfinite case. — fishfry

You might think, that "the meaning is always clear from context", but if you go back and reread TIDF's discussion of counting a quantity, you'll see the equivocation with order.

Then what is (represented by) an "imaginary thing"? — Luke

A faulty, self-contradicting set of ideas, which has found a place of acceptance in common parlance. Unfortunately, our language is full of these.

Nothing exists as it's representation, or else we would not call it a representation, it would be the thing itself..

If imaginary things only exist as their symbols or representations, and if we are really counting those symbols or representations, then we are really counting the imaginary things. — Luke

Symbols are not imaginary.

Wait, NOW you believe in ordinals? — fishfry

Oh dear. Did you not read that section of the thread, where I described the difference between quantity and order? It's odd that you wouldn't read those posts, because they were mostly in reply to you. Here's what I said:

The point is that we were talking about a count, which is a measure of quantity, not an order. To use numbers to indicate an order is a different matter. — Metaphysician Undercover

To say that something is a "different matter", from what we were discussing, is not to say that I do not believe in it. I'd ask you to go back and read that section again, but I think it's rather pointless because you do not seem at all inclined to make any effort toward understanding. TonesInDeepFreeze was equivocating, or at best, creating ambiguity between quantity and order, using "2" to mean "second", when counting a quantity of two.

Anyway, here is a further post I made, a few days ago:

Actually, I'm starting to get a real feel for the problem now, and I sincerely want to thank TIDF and fishfry for helping me come to this realization. I now see that there is a fundamental difference between using numerals to signify quantities, and using them to signify orders. The former requires distinct entities, objects counted, for truth in the usage, while the truth or falsity of the latter is dependent on spatial-temporal relations. So the truth of a determined quantity depends on the criteria for what qualifies as an object to be counted, while the truth of a determined order is dependent only on our concepts of space and time. So, in the case of quantity, truth or falsity is dependent on the truth of our concept of distinct, individual objects, but in the case of ordering, truth or falsity is dependent on the truth of our concepts of space and time. Since we think of space and time as continuous, non-discrete, we have two very different, and incompatible uses of the same numerals. — Metaphysician Undercover

TonesInDeepFreeze objected saying that ordering in mathematics requires no spatial or temporal relations, but I disagree with that as I think it can be demonstrated that each and every order imaginable is dependent on a spatial or temporal relation. To the right, left, or any such pattern, is spatial, and any intelligible sense of "prior" is reducible to a temporal relation. I really do not think there is any type of order which is not based in a spatial or temporal relation.

To begin with in all that, what's your definition of "real thing"? — TonesInDeepFreeze

Let' just say, it's existence is supported by empirical evidence. But we could go to the law of identity for our definition if you want.

LOL. First of all, I did actually scroll back to read your last post, and it totally failed to address the question I asked you, which was whether your claimed disbelief in quantum physics causes you to reject the most accurate physical experiment ever done, namely the calculation and experimental verification, good to 13 decimal places, of the magnetic moment of the electron. You simply ignored the question. — fishfry

Sorry, your question wasn't clear. I'll answer, though it is already answered in the other post. Physicists work with an inadequate representation of space and time. They can't even figure out whether an electron exists as a wave or a particle. When I look up the magnetic moment of an electron on a google search, I get an approximation. So much for your "most accurate" experiment.

I meant it sarcastically. As, "I have read your posts for the last time." Funny that you entirely missed that. — fishfry

See why I didn't answer your question? You don't make yourself clear.

It's perfectly true (or at least I'm willing to stipulate for sake of conversation) that the things mathematicians count are imaginary. Though I could easily make the opposite argument. The number of ways I can arrange 5 objects is 5! = 120. This is a true fact about the world, even though it's an abstract mathematical fact. If you're not sure about this you can count by hand the number of distinct ways to arrange 3 items, and you'll find that there are exactly 3! = 6. This is a truth about the world, as concrete as kicking a rock. Yet it involves counting abstractions, namely permutations on a set.

But when you say that imaginary things "exist as" symbols, you conflate abstract objects with their symbolic representations. A rookie mistake for the philosopher of math, I'd have thought you'd have figured this out by now. — fishfry

Thanks for providing support to what I am arguing. Counting possibilities, or "possible ways", is completely different from counting things, and therefore ought not be represented by the same word in a rigorous system of logic, to avoid equivocation. Furthermore, since it is a distinct activity, giving the symbols used, (numerals), a distinct meaning, we ought not even use those same symbols. If both, counting real things, and counting imaginary things (possible ways), are understood as the same way of using "counting" then the logical fallacy of equivocation will result. Since the very same numerals are used for both of these very distinct activities, such fallacy is inevitable.

To the chemist, physicists, or professor of English literature, this may well be true. But to the mathematician, it's utterly irrelevant. Mathematicians study the natural numbers; in particular their properties of quantity (cardinals) or order (ordinals). What they are counting or ordering is not important. — fishfry

This is false. What the mathematicians are counting with their use of symbols, numerals, is important, because it determines the validity of the logical system they are structuring. Mathematical systems are structured on the principles of the meaning of the symbols, which is derived from how they are, or may be used. So, the study of quantity and order, is restricted by those possibilities.

In the case of quantity for instance the mathematician is restricted by the assumption of discrete units necessary for the count of a quantity. In the case of ordering there is a more complex problem because we need to distinguish which is prior, order or unity. If we can work with all possible orders, with complete disregard for the need of discrete units to be ordered, placing order as prior to unity, then order appears to be unrestricted. But if it is necessary that there must be something which is ordered, for an order to be valid, then we have a set of restrictions which may be applied to order, derived from the principles of quantity.

Therefore, what they are counting or ordering is very important to mathematicians, because order is always dependent on a judgement of logical priority, and this judgement will be reflected in the logical structure produced. The mathematician cannot proceed without any such judgements, and pretending that no such judgements are involved turns the mathematician into a mathemajician.

Really? You don't think that counting the 120 distinct permutations of five objects is counting imaginary things? I don't believe you actually think that. Rather, I believe that if you gave the matter some actual thought, you'd realize that many of the things mathematicians count are very real, even though abstract. Others aren't. But it doesn't matter, math is in the business of dealing with conceptual abstractions. Math is about the counting, not the things. Farming or chemistry or literature are about the things. The farmer cares about three chickens. The mathematician only cares about three. — fishfry

Possible things are not real things, and this makes a big difference in how numerals are used. In the one case, we can start with the assumption of infinite possibilities, and restrict the infinite through the use of numerals. In the other case we start with what is real, actual, based on empirical observation, and the principles derived from these observations, to provide the necessary restrictions. Notice the difference. In the former case the restrictions on the possibilities for the use of numerals have no necessity, being completely arbitrary. In the latter case, we have restrictions based in real empirical evidence, and inductive reasoning.

The mathematician only cares about three. — fishfry

Sure, but how "3" is used is a judgement which the mathematician must make. We can say that it refers to the result of a count, a group of three units, or we can say that it refers to the third in an order. These two uses of "3" are fundamentally different and equivocation produces logical fallacy. If the two are conflated in equivocation the mathematician is a mathemajician. Therefore the honest mathematician must make a judgement of priority in defining what "3" means. Is it referring to a quantity or to an order?

To a pure mathematician there is no difference between counting 120 rocks and counting the 120 distinct permutations of five objects. — fishfry

That's exactly why I've argued that there is no such thing as the "pure mathematician". If there is such a thing in the world, we ought to call that person by a better name, the mathemajician, to reveal that this person actually operates with smoke and mirror illusions.

One need not reify abstract things in order to talk about them. — fishfry

Talking about things is completely different from counting things. When we count things it is implied that rigorous principle of logic are being followed. There is no such implication in talking about things.

Imaginary things only exist as symbols or representations; that's what makes them imaginary. You therefore acknowledge that we can count imaginary things. — Luke

Call it counting then if you want, but we just spent pages discussing the criteria for "counting",

Counting symbols or representations is really counting. If you're not counting imaginary sheep to help you sleep, then what would you call it instead of "counting"? — Luke

I'd say it's ordering, not counting.