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.

Then why do you ask me to repeat myself?

Look, I think it's very important for a rigorous mathematics to distinguish between counting real things, and counting imaginary things. This is because we have no empirical criteria by which we can determine what qualifies as a thing or not, when the things are imaginary. Therefore we can only count representations of the imaginary things, which exist as symbols. So we are not really counting the imaginary things, but symbols or representations of them, and we have empirical criteria by which we judge the symbols and pretend to count the imaginary things represented by the symbols. But this is not really counting because there are no things being counted. We simply assume that the symbol represents a thing, or a number of things, so we count them as things when there really aren't any things there at all.

So counting imaginary things by means of symbols is completely different from counting real things because one symbol can represent numerous things, like "5" represents a number of things. And we aren't really counting things, we are inferring from the symbol that there is an imaginary thing, or number of things represented by the symbol, to be counted. So it's a matter of faith, that the imaginary things represented by the symbol, are really there to counted. But of course they really are not there, because they are imaginary, so it's false faith.

I want to be clear in my mind. Is this your position on the subject? — fishfry

Read my last post.

And that parenthetical is simply to make clear that in this context we're not talking about the technical notion of an empty count. We're talking about counts that start at 1. — TonesInDeepFreeze

We've been talking about what it means to count. And we've determine that the count starts at one. If you know of some other way of counting which is based in something else, let me know please.

If there is a count that reaches 1, then there exists at least one object counted, and if there is a count that reaches 2, then there exist at least two objects counted. — TonesInDeepFreeze

If the count does not reach one, then it is not a count, because one is the beginning of the count. We could count by twos, or fives, or tens, but I don't think you've even accepted this yet, insisting that counting is a bijection with individuals. How do you ever get to the idea that the count "reaches" one when it necessarily starts at one and there is no count prior to one?

Your original and ongoing question regarded the context in which there are books on the shelf. You didn't ask me about the notion of an empty count. — TonesInDeepFreeze

Why do you keep avoiding the question? We're moving on from my original question, because I want to know how you come up with your notion of "countable". This is relevant to the topic of the thread, infinity. How do you proceed from the notion that "a count" is the activity of counting, to the conclusion that zero objects are countable, or that an infinite amount of objects are countable? It seems to me, that to do this you would need to change the definition of "a count".

But about the empty count: It's a technical set theoretical matter. It's not intended that the use of the word 'count' in 'empty count' corresponds to our everyday English senses of 'count'. I happily agree that it's an odd use of the word 'count'. If you don't like the notion, then that's okay in this context, because the representation with a bijection doesn't depend on the notion. — TonesInDeepFreeze

Do you realize, that within a logical system you cannot change the "sense" of a word without the fallacy of equivocation? I think therefore, that we have started with a faulty definition of "a count", your definition (1). If we are going to say that zero objects is a countable number of objects, then we need a definition of "count" which is consistent with this.

Should we try definition (2), the result of a count? How many books are on the shelf? None. We know that there are zero, without counting any. It's an observation, there is nothing which satisfies the criteria for "book", so we make an empirical claim that there is zero books. This is similar to what I said about seeing two chairs, or seeing that there are five books, without pairing them individually with a number (bijection). To derive the number of a specified object, we do not need to count (def 1) the objects. Cleary then, 0 is not the result of an act of counting Can we assume that numbers do not represent "a count" at all, nor do they represent the result of a count, they represent empirical observations? Otherwise, we need a definition of "count" which could be consistently applied, and this doesn't seem possible.

We are not claiming it is a count of actual captains. — TonesInDeepFreeze

You defined "count" with the activity of counting. And we described counting as requiring objects to be counted. I distinguished a true count from a false count on this basis, as requiring objects to be counted. Clearly, if the objects counted are not actual objects, but imaginary objects, it is not a true count.

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.

I've answered that already a few times. To have a non-empty count, of course there exist the objects counted, and in you example, these objects are books. — TonesInDeepFreeze

The question was whether there could be a count if there are no books.. If no books are counted, do you consider this to be a count? I think that if no books are counted then there is no activity, of counting, therefore no result of counting either.

Now I'm answering yet again, there is no no-empty count if there are not objects counted.

Now, are you going to continue asking me this over and over again? — TonesInDeepFreeze

I'm asking you if you believe there is such a thing as an empty count. That would be contradiction, obviously, to have an activity of counting when nothing is being counted. Do you agree? You did say that a set could be an empty class. Do you agree, that by your definition of "count" (1) the act of counting, an empty set is not countable? There seems to be discrepancy between how you define the count (1), and and how you say "countable" is defined in the mathematical sense.

I can count the captains of the starship Enterprise even though they're imaginary. — fishfry

That's what I would call a false count, because it's hypothetical. It's like if you look at an architect's blueprints, and count how many doors are on the first floor of a planned building. You are not really counting doors, you are counting hypothetical doors, symbolic representations of doors, in the architect's design. Likewise, if you count how many people are in a work of fiction, these people are hypothetical people, so you are not really counting people, you are counting symbolic representations. We can count representations, but they are counted as symbols, like the architect's representation of a door, may be counted as a specific type of symbol. And when you count captains of the Enterprise, you are likewise counting symbolic representations. If you present this as a true count of actual captains of an actual starship, you'd be engaged in deception. You are not counting captains of a starship, only symbolic representations.

Curious to know: If you deny complex numbers do you likewise deny quantum physics, which has the imaginary unit i in its core equation? — fishfry

Yes, I think quantum physics uses a very primitive, and completely mistaken representation of space and time. That's why it has so many interpretative difficulties.

Problems of interpretation come from trying to explain why the electron sometimes appears as a particle and sometimes a wave. — khaled

I think the issue here is that the electron does not have any real existence as a particle at all. It is a particular quantity of energy, and we, as human beings desire to give that quantum of energy real existence as a unit, making it an entity, called an electron. But there is no real existence of that particle, this is simply how we relate to that energy from our perspective as human beings, with human artifices.

Not sure how that was unclear. — Book273

It was quite unclear. Are you saying that your life will begin anew, at your death? I always thought that death was the end of life, and that is final.

x is a set iff (x is the empty class or (x is a non-empty class and there is a y such x is a member of y)).

Or, the sets are objects that satisfy the set theory axioms.

Or, the sets are the objects that the quantifier ranges over. — TonesInDeepFreeze

So, a set is a class. How's that relevant? Say we're counting books, the set is called "books" then. Do you agree that there must be some of these things (objects) which are classed as "books", for us to have a true count. If there aren't any books, we do not have any counting of books at all.

I have always been completely clear that the bijection represents the count, not the result. You are terribly terribly confused. — TonesInDeepFreeze

And I've been completely clear, that bijection is unacceptable as a representation of counting. Therefore one or both of us misunderstands what the activity of counting is, so we are stuck here, unable to proceed until we find some agreement or compromise on this. Do you agree that there is no activity of counting if there is no objects counted?

I am not concerned with death. She is an old friend that will call on me as she chooses. When she does, I will hold her hand and walk through that door with her. And it will begin anew. — Book273

"And it will begin anew"? Don't you really mean "It will be finished forever"?

But it was not the sense in your bookshelf example, which may be represented mathematicaly as the bijection I mentioned. — TonesInDeepFreeze

I explained to you already why bijection (paring) is an inadequate representation of counting, as defined by you (1). This effort required a number of posts. I assume you didn't understand.

I'm sorry Tones, but you've really lost me now. You don't seem to be directly addressing any of the points I make, and we do not seem to be understanding each other at all, at this point.

You are critically confused on the very point here, and one that previously you even said you understood. That point is that the result is different from the count. I didn't represent the result as a set*. I explicity said (several times) that the result is a number. Meanwhile I represented the count (not the result) as a bijection, which is a certain kind of set. — TonesInDeepFreeze

I don't know what a "set" is, you haven't defined it. But you seemed to be using it as if it meant the result of the count, i.e. the number. I asked where did the notion of a set come from, and you said "When I gave a mathematical representation of a count." Isn't it the case, that the mathematical representation of a count, is the number, which is the result of the count? Or, you might give a mathematical representation of the activity of counting as "1+1+1+1...". However you've already agreed that there's more than one way to count, so there is probably a number of different acceptable mathematical representations of counting. Bijection though, as described by you as pairing, is not an acceptable representation, for the reasons I already explained.

For physical world matters. However, in the mathematics itself, ordinals don't refer to space and time. — TonesInDeepFreeze

I was talking about truth and falsity in the use of mathematics, and I use these terms in the sense of correspondence with reality. So it's not necessarily the "physical world" we are talking about, it's "reality" in general. If mathematics talks about an order which is not temporally, nor spatially grounded, then I think such a mathematics would be nonsensical. I've seen some people argue for a "logical order" which is neither temporal nor spatial, but this so-called logical order, which is usually expressed in terms of first and second, is always reducible to a temporal order.

In your post you said, "it is implied that there is one thing". And that is how I use 'imply' too. I use 'imply' to say 'It is implied that [fill in statement here].

Then you said, "an object is implied".

I don't use 'implied' to say '[fill in noun phrase here] is implied'. — TonesInDeepFreeze

When you agree that "it is implied that there is one thing", do you not agree that the "thing" is an object? Can we go to my original term, a "unity". Do you agree that the thing is a "unity"? I mean, we could stick to calling it a "thing", as you seem to agree that there is something which is referred to as "thing" here, but why quibble about terms? Like I said in the last post, what we call the thing is irrelevant; we could call it "object", "entity", "unity", "particular", "individual", "book", "War and Peace", whatever, so long as there is something counted. What is important is that this name refers to something or else you are not truly counting. Do you agree? Even if you are counting names or titles, "War and Peace", etc., those are still "things" which are being counted

If you simply say "1,2,3,4,5" , you might say "I am counting", but it's not a true count, because nothing is counted, therefore the symbols actually refer to nothing whatsoever, and the count itself is invalidated because that sequence of symbols does not have any meaning at all. Suppose someone memorizes that sequence of symbols, 1-5, and repeats them saying "I can count to five". Unless the person knows what the symbols mean they are not really counting to five, they are just repeating symbols. If they know what the symbols mean, then they know that there must be five things (objects, unities, individuals, or whatever you want to call them), or else the count is false. Do you agree? If not how do you validate the meaning of the symbols?

When I gave a mathematical representation of a count. — TonesInDeepFreeze

Please, do not jump ahead like that. You spent days differentiating between (1) the act of counting, and (2) the result of that act. As far as I can see, the "mathematical representation" of both (1) and (2) consists of numerals, "1", "2", "3", etc.. There is no need to represent (2), the result of the act of counting, as a "set", or whatever your intent is. Let's just adhere to these defined principles, and maintain clarity.

How do you feel your campaign is doing?
Has it been worth the struggle?
Have there been casualties?

Are you holding up? — jgill

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.

When I say 'P is implied', then P is a statement, not an object.

So I don't say

'War And Peace' is implied.

But I do say

That 'War And Peace' is on the bookshelf is implied.

This is just a matter of being very careful in usage that may be critical in discussions about mathematics. — TonesInDeepFreeze

Sorry, I don't follow this at all. If you count "1", then it is implied that there is one thing (an object) counted. Do you, or do you not agree with this? If you do not agree, then what are you counting when you count "1"? If you are counting books, then aren't books objects? And you could be counting any type of objects, or maybe just objects in general. But don't you agree that if you count "1", it is necessary that an object has been counted? Therefore an object is implied by any count of 1?

This is just a matter of being very careful in usage that may be critical in discussions about mathematics.

Regarding this example of counting, I take it as a given assumption that

'War And Peace' is on the bookshelf and 'Portnoy's Complaint' is on the bookshelf.

I am not deriving ''War And Peace' is on the bookshelf and 'Portnoy's Complaint' is on the bookshelf' as implied by anything other than the initial assumption of the example.

And, of course, I am not showing an example of a non-empty count on the empty set. It is a given assumption of the example that:

the set of books on shelf = {'War And Peace' 'Portnoy's Complaint'} — TonesInDeepFreeze

I don't see how this is relevant. You seem to have changed the subject. We were not talking about sets. We were talking about (1) the act of counting, and (2) the result of this act. When did a "set" enter the picture?

I don't speak of objects being implied. What are implied are statements (or propositions). — TonesInDeepFreeze

The statement is not implied, it is explicit, stated as "first", "second", etc... What is implied, in order that your count be a true count, is that there are objects counted . Otherwise, as I said it is not a true or valid count. You can state "first", "second", "third", "fourth", but unless there is something referred to, you are not counting anything and it's not a true or valid count.

In order not to have to continually specify which sense I mean, I'll use 'count' in sense (1) and 'result' for sense (2). — TonesInDeepFreeze

I like that, instead of calling (2) the count, we'll call it the result of the count. We might even call it the conclusion, Then I can say that the conclusion is unsound if there aren't any objects counted, because to say "that is the second", or "there are two", is not true unless there are objects which have been counted. To count "1", or "first", without counting anything is to make a false statement.

A (non-empty) count is a bijection form a set onto a set of natural numbers (where 1 is in the set and there are no gaps). The result is the greatest number in the range of the count. — TonesInDeepFreeze

As I explained in my last post, we ought not consider that a number is a countable object, for the reasons I described. So I consider such a count to be a false count.

This involves nothing about "implying objects" or "signifying objects". — TonesInDeepFreeze

Of course it implies objects. You have mentioned things being counted. I deny that natural numbers are things which can be counted. Therefore I conclude that your result is unsound, by this false premise that natural numbers are things which can be counted.

Of course, though, it is already assumed that there are objects (books on a shelf in this case) named 'War And Peace' and 'Portnoy's Complaint'. But that's not a mathematical concern. It's just a given from the physical world example. — TonesInDeepFreeze

Truth and falsity may not be a mathematical concern, but it is a philosophical concern.

By the principle of stipulative definition. Anyway, your question doesn't weigh on the mathematical notion of counting. — TonesInDeepFreeze

Stipulation does not make truth.

Setting aside your other confusions, I will address the term 'countable' as used in a mathematics, to prevent misunderstanding that might arise:

'countable' is a technical term in mathematics that does not adhere to the way 'countable' is often used in non-mathematical contexts.

In non-mathematical contexts, people might use 'countable' in the sense that that a set can be counted as in a finite human count.

But in mathematics 'countable' doesn't have that meaning. Instead, in mathematics the definition of 'countable' is given by:

x is countable iff (there is a bijection between x and a natural number or there is a bijection between x and the set of natural numbers). — TonesInDeepFreeze

Obviously, I do not accept this stipulative definition of "countable", for the reasons explained in my last post. Principally, if we use numbers to count numbers, the numbering system which counts numbers will need to be different than the numbers being counted (by the reasons explained), then we'll want another numbering system to count those numbers, and another to count those numbers, etc', ad infinitum.

There is really no reason to attempt to count the natural numbers, when we know that this is impossible because they are infinite. And numbers are not even countable objects in the first place, they are imaginary, so such a count, counting imaginary things, is a false count. Therefore natural numbers ought not be thought of as countable.

First, there is no general definition of number in mathematics. — fishfry

That's because numbers are not objects, and therefore they cannot be described or identified as such. And since they cannot be identified, they cannot be counted.

What is your definition of number? — fishfry

It is a value representing a quantity.

Not in math. After all, some numbers have neither quantity nor order, like 3+5i3+5i in the complex numbers. No quantity, no order, but a perfectly respectable number. You take this point, I hope. And are you claiming a philosopher would deny the numbertude of 3+5i3+5i? You won't be able to support that claim. — fishfry

Yes, that's a symptom of the problem I explained to TIDF. Once we decide that numbers are objects which can be counted, then we need to devise a numbering system to count them. So we create a new type of number. Then we might want to count these numbers, as objects as well, so we need to devise another numbering system, and onward, ad infinitum. Instead of falling into this infinite regress of creating new types of imaginary objects (numbers), mathemajicians ought to just recognize that numbers are not countable, and work on something useful.

You're wrong mathematically, as I've pointed out. — fishfry

Of course I'm wrong mathematically, I'm arguing against accepted mathematical principles. But the question is one of truth and falsity. Are numbers objects which can be counted, rendering a true result to a count, or are they just something in your imagination, and if you count them and say "I have ten", you don't really have ten, a false count is what you really have?

And the sense I have been using is indeed the one that is relevant - assigning successive numbers. — TonesInDeepFreeze

OK, let's proceed using your sense of counting, "assigning successive numbers". Do you agree with me, that when you assign "2" indicating the second object, the first object is also implied, as necessary to make your assignment of second a valid and truthful assignment? The "2" does not simply pair with the second object, because "second" implies that there was a first, so this is more than a straight pairing, because there is necessarily implied another pairing between "1" and the first object. Therefore "2", in this count, of assigning successive numbers, refers to or signifies, two objects, the first and the second.

So, as you understand that by 'count' I mean in the sense of 'successive numbering', you may see that my mathematical representation of it is correct and that indeed an ordering is induced. — TonesInDeepFreeze

It is your representation of counting as a simple pairing which I objected to. Even when restricted to a "successive numbering", counting is not a simple pairing. This is because, as I explained, when you pair the second, the pairing of the first is also implied, therefore referred to within the mention of "second". To say "second" refers to the first pairing and the second paring, as two distinct pairings.

Ordinarily, when someone says "I counted the books on the shelf", we understand that he used numbers (indeed as the positive natural numbers are sometimes called 'the counting numbers'), numbering in increasing order as he looked individually at each book, and not that just that he immediately perceived a quantity. That is the ordinary sense of counting I have been talking about.

Also, for example, if I see an 8 oz glass and that it's full of water, then I may say that the quantity of water is 8 ounces, without counting in the sense of numbering each ounce one by one. But that's not what people ordinarily mean by 'counting'.

Again, if you mean some wider sense, then of course certain of my remarks would not pertain. — TonesInDeepFreeze

OK, I agree that this is the "ordinary way" that a person counts, so we have a pretty good understanding between us as to what counting is, so let's go back to the fundamental problem I mentioned in the first place. When you say "2" if you are counting (ordering in your sense), and there are two objects referred to by "2", the fist and the second (the first is necessary to validate the notion of "second"), by what principle do we say that "2" refers to one object, the number 2?

I think you agree with me on the necessity of having two objects to make the use of "2" or "second", a true or valid use. So if we say that "2" also refers to one object, a number, then this type of object must be completely distinct from the other type of object, or else we'd have contradiction, because now there are three objects indicated, the first, the second, and the number 2. If this is the case, then "2" refers to the two objects counted, and a third object, the number 2.

Now, do you see the need to say that the number 2, if it is to be considered an object, must be a distinct type of object, or else we'd have three objects being referred to by "2"? If you see this need, to say that the number 2, if it is supposed to be an object, must be a very distinct type of object from the type of objects which we count, or order when counting, then you ought to also see the need to ask whether it is even possible to count this type of object. I think it is impossible to count these so-called objects because the fact that they are the count, rather than what is counted, is what distinguishes them from the objects which are counted. Therefore, as "the count" , and distinguished from what is counted as "not what is counted", they are by definition not countable. So the simple solution (I offered already), is to recognize that they are not really objects and therefore not countable.

That the numbers, proposed as objects, are not countable, is also evident from the problem of infinite regress. If we wanted to count the numbers, as objects, it would require a different numbering system from the one we use to count ordinary objects, to avoid equivocation. For example, when we have two ordinary objects, we have the number 2 which is another object that would be counted as 1,object if numbers are counted. So we cannot have both "1" and "2" describing how many objects are there unless the "1" was part of a distinct numbering system from the "2". However, then these numbers in the distinct system would be proposed as objects as well, and we'd want to count them alao, so we'd need another numbering system to count them. then we'd proceed toward an infinite number of numbering systems, in the attempt to count all the numbers which count the numbers which count the numbers, ad infinitum..

The simple solution again, is to recognize the truth of the fact, that the numbers are simply not countable. They are infinite and this renders them as not countable, by definition. So we ought not even attempt to count them as this is known to be impossible. Also, we can clearly see that the numbers are not objects, and so they are not something which is countable.

I said that the count itself implies an ordering. The ordering I have in mind is the ordering by the number associated to each item. — TonesInDeepFreeze

Clearly, to see that there are two chairs in front of me, does not require that I associate a number to each of them. Therefore "the count", the determination that there are two chairs, does not imply an order. We can count (determine the number) without associating a number with each item. Therefore associating a number with each item is not an essential aspect of counting, or the count itself.

I refuted the argument about seeing things at a glance. — TonesInDeepFreeze

All you said is "We're not talking about taking in at a glance a quantity". That's your idea of a refutation? The definition of counting is to determine the number, clearly "taking in at a glance" qualifies.

From my experience with you, your mode of argument is to define the term in an unacceptable, false way (in the sense of correspondence with how the word is actually used), which begs the question. So, you define counting in a way which excludes any form of determining the quantity without any ordering, to support your conclusion that counting implies ordering. Obviously your so-called refutation is fallacious because you're just begging the question.

Do you accept the OED definition, that to count is to determine the number? And do you accept the fact that we can determine the number without ordering as you said here?

We may infer, by whatever means, that there are a certain number of electrons or volts. — TonesInDeepFreeze

The important point, which I'll return to, is that when we have a count, it is necessary that there are as many objects as the count indicates, but it is not necessary that any object is paired with any number. When you recognize this, you'll see that the act, which is counting (determining the count), is not necessarily an ordering, or pairing. Counting, the act which produces a count, is not necessarily an ordering.

That's talk about "a first" and "units". That sets a context that is a far cry from the far broader "determine the total number". — TonesInDeepFreeze

I was giving an example of counting. Did you or did you not agree that there is more than one way to count? If so , then you ought to be able to understand that a count does not imply an ordering.

Now that we have somewhat of an idea about what each other thinks about this matter, let's return to the issue at hand. Let's look at the numeral "2", and see if we can agree on the valid use of it. When we use "2" within the act of counting, do you agree that it signifies that a quantity of two objects have been counted. or do you believe that the numeral pairs with one particular object as "the second"?

If you choose the latter as the use of "2", then I would argue that you are talking about an act of ordering, not an act of counting and these two are distinct. Do you recognize the difference between such ordering, and counting? When we say or write "2" it is implied that there is a quantity of objects, two, which is referred to. When we say "second", it is not necessary that there is such a quantity, because when we say "second", the first may have already disappeared, like counting the hours. So "second" refers directly to one object, and there is no necessity that the prior object still exists, because we are not saying that there are two objects. But when we say "2" it implies that there is a quantity of two objects, or else it's not a valid use of "2".

So when we are counting the hours, and we assign "2" to the second hour, what is really being said is that it is the second hour, not that there are two hours. And these two ways of "counting" one being determining the number or quantity, the other being assigning numerals to an order of things, are very distinct and ought not be conflated by reason of equivocation.

You don't even know what I'm saying. — TonesInDeepFreeze

I know what you said. You said "A count (1) implies an ordering". And I'm telling you that this is false for the reasons I explained. There is more than one way to carry out that action which is counting, and not all ways require ordering. Therefore it is false to say a count (1) implies an ordering.

I showed you how it does. And less formally, even a child understands that when you count, there's the first item counted then the second item counted ... — TonesInDeepFreeze

You showed me one way of counting, which involved ordering, but you also admitted that there are other ways of counting. So clearly you use invalid logic when you say that counting implies ordering. Only that one way of counting, which you demonstrated, implies an ordering, not all ways of counting. You can see that there are five books on the shelf without ordering them at all, just like I can see that there are two chairs in front of me right now, without ordering them at all. That is counting them without ordering them.

A measeurment might not itself be a (human) count. — TonesInDeepFreeze

Why does the action of counting have to be a human count? We have, as humans, devised all sorts of mechanisms to make counting easier, or even do our counting for us. This is the important point here, the essence of counting (what is necessary to the act), is to determine the quantity, no matter how this is done, by machine or whatever. That we commonly do this by ordering is accidental, not an essential aspect of counting.

We're not talking about taking in at a glance a quantity. We're talking about counting. You're grasping at straws. I notice you tend to do that after a while in a thread. — TonesInDeepFreeze

Actually it's you who is grasping at straws. My OED defines "count" definition #1 as "determine the total number or amount of, esp. by assigning successive numbers". Notice that it says "esp.", which means mostly, or more often than not, but it does not mean necessarily. Therefore, to determine the total number or amount of, in a way which is not assigning successive numbers, though it might be a less common use of "count", it is still an act of counting.

Anyway, I don't know what point you're trying to make. You disagreed with what fishfry wrote, then he clearly explained how your disagreement is incorrect. You seem not to understand his explanation, though it was eminently clear. — TonesInDeepFreeze

Right, I don't understand how what fishfry was saying is relevant.

Here's an example by analogy. Ordinal numbers are a type of numbers which are used for ordering. Ordering is what defines the "ordinal" aspect, not the "number" aspect. In a similar way, human beings are a type of animal said to be rational. Rational defines the human aspect but it does not define the "animal" aspect.

In this context, there are two senses of 'count':

(1) A count is an instance of counting. "Do a count of the books."

(2) A count is the result of counting. "The count of the books is five." — TonesInDeepFreeze

Right, one is a verb signifying an action, the other is a noun, signifying the result of the action.

A count(1) implies an ordering and a result that is a cardinality ("quantity", i.e. a count(2)). — TonesInDeepFreeze

This is what I have been telling you is incorrect. A count does not imply an order. You might order things to facilitate your activity of counting, but as you agreed, there's more than one way to count, and as I've been telling you, they are not all necessarily instances of ordering. Therefore you cannot define, or describe counting as ordering. That's why you can weigh a sac of flour and see that it's 5 kg. without ordering each kg of flour. And, you can see that there are five books on the shelf without placing them in any order. "A count" only implies a quantity, five, and there is no necessity of any particular order, or any order at all, only a quantity.

If lack of knowledge is innocence, then you are a saint. — TonesInDeepFreeze

It requires more than innocence to be a saint.

You wrote: "Numbers are defined by quantity, not order ..." If you didn't mean that you should not have written that. — fishfry

That's what I meant, and though you can use numbers in ordering, it is not what defines them, quantity does.

My God, you wield your ignorance like a cudgel. I could have just as easily notated the two ordered sets as:

* ({1,2,3,4,…},<)({1,2,3,4,…},<) and

* ({1,2,3,4,…},≺)({1,2,3,4,…},≺)

which shows that these two ordered sets consist of the exact same underlying set of elements but different linear orders. Remember that sets have no inherent order. So {1,2,3,4,...} has no inherent order. The order is given by << or ≺≺. — fishfry

OK, so doesn't this support my point, order is not what defines a number? If not, then I really don't know what you are trying to demonstrate, and how it is relevant. Perhaps you could explain.

On the contrary, sets have no inherent order. — fishfry

Exactly what I've been arguing, a count is a quantity, not an order, hence what I said "numbers are defined by quantity, not order".

Why don't you have a look at the Wiki page on ordinal numbers and learn something instead of continually arguing from your lack of mathematical knowledge? — fishfry

As I said, you can use numbers to order things, but this is not what defines numbers.

It's almost an admirable trait . . . but not quite. — jgill

That's like when the judge hands down the guilty verdict and thinks: 'that guy was so persistent in his claims of innocence, that I almost feel like letting him go free'. But in this case lack of knowledge is innocence, so there's no guilty verdict to be handed out. Why not just pure admiration then?

You're failing to distinguish between cardinals and ordinals.

Let me give you a standard example. Consider the positive integers in their usual order: — fishfry

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. So to demonstrate the use of numbers in ordering now, is to equivocate, because an order does not necessarily imply a count

Now the quantity of positive integers is exactly the same in either case, since the ordered set ({1,2,3,…},<)({1,2,3,…},<) and the ordered set ({1,2,4…,3},≺)({1,2,4…,3},≺) have the exact same elements, just slightly permuted. There is a one-to-one correspondence between the elements of the two ordered sets. — fishfry

That is not true. These sets do not have the same elements. If "..." implies an infinite extension of the order, then 3 does not exist in the second set. Therefore they do not have the same elements. The symbol "3" is there, but the number is excluded by the infinite order which must occur prior to it. That's an obvious problem with your mode of equivocation, and conflating counting and ordering, it allows for contradiction. You can describe an order which is never ending (infinite), then say that there is a 3 after the end of it. And for you, that 3 is there. But of course you've just accepted the contradiction.

You don't even know what it is that you don't get. — TonesInDeepFreeze

Well of course. If I knew what it is was that I didn't get, that would mean I was getting it.

Try again, maybe after you explain an infinite number of times, I'll get it.

The point is that to describe a count as a tuple is not a correct description of a count. You just don't get it.

Again, arbitrary. That you designate "a" as first in that sequence, is arbitrary.