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?

.

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. — Metaphysician Undercover

You neglected adding, "And you don't care!" :scream:

If I spent my time brooding over this issue I'd not get much math done. It's good there are gurus like you who are willing to navel gaze into this profound mystery and lay the foundations while we flitter about, inconsequential moths circling your flame.

The winner takes first place and the runner up takes second place — fishfry

Or, as Jerry Seinfeld reminds us, taking Silver in the Olympics just means you're the best of all the losers.

while we flitter about, inconsequential moths circling your flame. — jgill

If a flame be a dumpster fire.

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. — Metaphysician Undercover

You're the only one who thinks it's a misnomer. Everyone else considers "counting up to ten" to be counting (you also called it "counting", by the way). 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"?

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.

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.

Yes, i call it "counting" — Metaphysician Undercover

You call it "counting" even though you consider it a misnomer to call it "counting" (since there is "nothing being counted")?

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, — Metaphysician Undercover

You introduced this distinction solely to make the point that one side of the distinction is not real counting. But sure, let's avoid ambiguity and equivocation over the two distinct types of counting (P.S. you think one of them isn't real counting).

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". — Metaphysician Undercover

You're repeating yourself. This is just another way of saying that real counting must involve "determining a quantity". 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"?

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"?

The point is to avoid equivocation which is a logical fallacy. — Metaphysician Undercover

That's not your point, though. Your point is not merely to avoid equivocation; your point in drawing the distinction between the two senses of "counting" is to discount the sense of "counting the natural numbers", "counting from one to ten", or "counting imaginary things" as not a true sort of counting. You have attempted to argue that the only true sort of counting is "determining a quantity" and/or counting "actual objects". For example:

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

Since one sense of "counting" involves counting real things, then why not call this "real counting"? — Metaphysician Undercover

You have not been using the term "real counting"; I have. My use of "real counting" does not denote "counting real things". "Real counting" denotes genuine counting, as opposed to non-genuine counting.

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. Your only "argument" has been that true counting must involve "determining a quantity" and/or counting "actual objects". However, you have still provided no justification for this so-called "argument" (i.e. stipulation).

Or, as Jerry Seinfeld reminds us, taking Silver in the Olympics just means you're the best of all the losers. — TonesInDeepFreeze

Or in a two-person race, the loser finished second, and the winner finished next to last.

And it's even classified as Vital. There must be 10K pages on math on Wiki. I wonder how many are added each day? — jgill

What's the smallest positive integer that Wiki doesn't have an article on? That would make it deserving of its own article!

The original paper is in Jean van Heijenoorts's 'From Frege To Godel'. — TonesInDeepFreeze

I see that I can buy a copy, but I didn't find a pdf. I'm wondering if you could summarize. Did von Neumann anticipate the categorical approach to set theory way back on 1925? Wouldn't be surprised, just curious to know, but not curious enough to buy the book. I have half a dozen physical books already stacked up to read. It's so much easer to buy books than to read them. Someday we'll just be able to upload via neural interface.

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.

Let's get this straight. I am not talking physical referents here. I am talking space and time, which are conceptual. — Metaphysician Undercover

Now that is very interesting. When you say space and time, I've understood you to be referring to those words as understood in physics. The space and time of the physical world. Which makes sense. You would be claiming that 4 comes before 5 in terms of physical space or time.

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. I confess you've lost me and perhaps lost your own point as well. If space and time are abstract conceptual things, then why can't numbers be also?

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. — Metaphysician Undercover

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.

Numbers no longer refer to physical objects being counted, they refer to these abstract concepts of space and time. — Metaphysician Undercover

Ok, but now I'm confused by your claim that space and time are no longer physical things, but rather conceptual things. Why can't numbers be conceptual things inspired by physical things too?

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. — Metaphysician Undercover

You've swapped out mathematicians for logicians, and I'm not sure I can accept that. I'm talking about mathematical practice, which goes far beyond logic. Logicism's dead, right?

I am making the point that in order theory, one order is as good as another. An order is ANY relation that's reflexive, antisymmetric, and transitive. That's what an order is in order theory. I didn't make this up, it's on Wikipedia and as someone with a (little) bit of mathematical training, I can confirm that Wiki got this one right. https://en.wikipedia.org/wiki/Order_theory

Of course you are correct that the natural order of the positive integers is 1, 2, 3, ... but that is not the ONLY possible order relation on them, there are many others.

For example, a self-contradicting premise is not allowed. — Metaphysician Undercover

As long as my order is reflexive, antisymmetric, and transitive, it's allowed. And for the umpteenth time (umpteenth is an ordinal!!), a contraction is a statement P such that both P and not-P can be proved from a given set of axioms. A contradiction is NOT merely something that offends your intuition. In math we get quite accustomed to having our intuitions challenged and corrected.

So there are fundamental rules as to the criteria for "order" which cannot be broken. — Metaphysician Undercover

Yes there are. Reflexive, anti-symmetric, and transitive. Those are the rules. Or sometimes we required a strict order for convenience, and deny reflexivity (ie < rather than <=) but that's a small point.

But there ARE rules, and the rules are documented on Wikipedia, and I've pointed them out to you.

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. — Metaphysician Undercover

Even that's not true. I can put an order on red, green, blue. Say lex order: blue, green, red. Or length order: red, blue, green. In each case the order is reflexive, anti-symmetric, and transitive.

I'm sorry your intuition is challenged on this point, but a big part of learning mathematics is having our intuitions challenged, so that we come out the other end with better intuitions.

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. — Metaphysician Undercover

I have no idea why you've swapped in logicians. I'm talking about mathematicians. I'm explaining to you how mathematicians define order. I can't help your naive intuitions, I'm trying to dispel them in favor of more clarifying concepts.

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. — Metaphysician Undercover

The only thing faulty is your intuition about what an order relation is.

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

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.

Yes that is my point as to how counting order is different from counting a quantity. — Metaphysician Undercover

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.

To count a quantity requires particular things, but to count an order requires only time. — Metaphysician Undercover

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.

However, time is something in the world, and that's why I don't believe in what you call "the pure concept of order". — Metaphysician Undercover

Your belief in mathematics is not required by mathematics. Mathematics can exist in the world side-by-side with your willful ignorance and obfuscation.

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? — Metaphysician Undercover

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 $3 + 5 i$ 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.

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.[/quote}

What makes 6 an even number is that it's divisible by two; or equivalently, that it's residue class mod 2 is zero. That's how we recognize 6 as an even number.

Here's a more striking example that even you will have to concede. I can recognize 45385793759385938534 as an even number without knowing ANYTHING about its quantity or order. I merely have to note that the low-order digit is even, and appeal to the theorem that a number is even if and only if its low-order digit is.

— Metaphysician Undercover

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. — Metaphysician Undercover

There is no particular attributed that's a defining feature of number. The concept of number is a historically contingent opinion of mathematicians. Zero didn't used to be a number, neither did $5 + i$, and neither did $\aleph_{47}$. Today they're all regarded as numbers.

There is no general definition of number; nor is there any particular defining property by which we can say, "This thing is a number," and "That thing isn't." The concept of number is whatever the mathematicians of a given era agree is a number.

That's how it is.

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

There's no general attribute that uniquely characterizes a thing as a number. What is or is not a number is a matter of historically contingent opinion of mathematicians.

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. — Metaphysician Undercover

I think we're at a point of diminishing returns in this convo. You're flailing and not saying anything I find interesting enough to even argue with.

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. — Metaphysician Undercover

Why don't we table this till next time. I've made my point and all you have is mathematical ignorance. That's all you ever have. You've even fatally undermined your own thesis by agreeing that space and time aren't even the space and time of physics, but are rather "conceptual," while denying the same status to numbers.

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. — Metaphysician Undercover

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.

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". — Metaphysician Undercover

You're flailing and no longer even trying to make a coherent point.

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. — Metaphysician Undercover

That's manifestly false.

So, if "a set" is like the kids in the classroom, then it must have an order to exist as a set. — Metaphysician Undercover

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

https://en.wikipedia.org/wiki/Axiom_of_extensionality

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. — Metaphysician Undercover

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.

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. — Metaphysician Undercover

It can have any order as long as the order satisfies the properties of a partial or total order.

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. — Metaphysician Undercover

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?

You've never heard "the object of the game"? — Metaphysician Undercover

That's a different meaning. You're being silly now, unserious.

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". — Metaphysician Undercover

You're wrong. If you have a set, and you impose an order on the set, and in that order there's an element that's less than every other element, that order may be called the first element. That's the definition.

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. — Metaphysician Undercover

Ad hominems is all you've got left, I see. I hope you will not mind if I'm done here, nothing new can be said at this point.

.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. — Metaphysician Undercover

Project much?

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. — Metaphysician Undercover

I distinguish them just fine.

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?
. — Metaphysician Undercover

It's been fun chatting. I'm done with this topic. Till next time.

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. — Metaphysician Undercover

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".

I trust you will have no problem with this as it avoids any equivocation.

The original paper is in Jean van Heijenoorts's 'From Frege To Godel'.
— TonesInDeepFreeze

[,,,] I'm wondering if you could summarize. — fishfry

It's too many technicalities to easily summarize. Roughly speaking, primitives:

2-place operation
(x y)
"pairing"

2-place operation
[f x]
"value of the function f at argument x"

constant
A

constant
B

predicate
I-object
"is a function"

predicate
II-object
"is an argument"

predicate
I-II-object
"is a function that is itself also an argument"

Then a lot of axioms with those.

Did von Neumann anticipate the categorical approach — fishfry

It doesn't seem to me to be a precursor to category theory, but I don't opine.

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?

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".

I surely have not denied that "5" has conceptual meaning. — Metaphysician Undercover

Hi, I didn't read the rest of your post yet but I realized I needed to clarify what I wrote last night.

4 is indeed the cardinal number 4 or the ordinal 4, and 5 is the cardinal or ordinal 5. So 4 and 5 do indeed represent quantities, or orders. I misspoke myself, or rather I failed to adequately address your point.

What I was saying about order is that the usual or standard order on the entire set of natural numbers is not the only possible order. So if I decide to reorder the naturals as 1, 2, 3, 5, 4, 6, ..., where 5 comes before 4, it is still the case that 5 represents a set of five elements. So each individual natural number can be seen as representing a quantity, or an order. For example in the Peano construction, 5 is the successor of 4, which is the successor of 3, etc.

So it's the SET of natural numbers that have no inherent order. But an individual natural number does represent a quantity or order.

On the other hand, rational, real, and complex numbers can't be seen as representing quantity in the same way that natural numbers do. Quaternions are little known, but it turns out that game developers use quaternions because they're the most natural tool for representing 3-D rotations. So you can add "rotation" to quantity and order. Every number represents some quality of interest, but there are more of those than just quantity and order.

Hope this clarifies a point of confusion that I didn't adequately address last night. I'll get to the rest of your post later.

It doesn't seem to me to be a precursor to category theory, but I don't opine. — TonesInDeepFreeze

Thanks much.

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, — Metaphysician Undercover

If the number 2 means "a quantity of two", then how could counting the natural numbers be "expressing an order", as you claim?

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. — Metaphysician Undercover

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 measuring (determining a quantity), rather than counting.

That is, 2 comes after 1 in either case.

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. — Metaphysician Undercover

Yes ok. I misspoke myself when I was unclear that a SET can be given an arbitrary order, and that no one order is to be preferred above any other; but that nevertheless you are correct that the natural numbers individually are either cardinals or ordinals, referring to quantity or order. You're right about that I should have been more clear.

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. — Metaphysician Undercover

You're right, 5 refers to the class of sets having 5 elements, or it refers to a canonical representation in set theory of the number 5, or in Peano arithmetic it refers to the successor of 4. Quantity and order are essential aspects of natural numbers. I was wrong not to realize earlier that I should have noted that.

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. — Metaphysician Undercover

Have I clarified my earlier inaccuracy enough yet? 5 refers to fiveness, quantity or order. I agree.

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. — Metaphysician Undercover

Does our disagreement on this point go away now that I've clarified my inaccuracy?

OK, I assume that "less than" refers to quantity. — Metaphysician Undercover

Sigh. Less than refers to whatever is on the left side of x < y if '<' denotes a strict order relation.

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? — Metaphysician Undercover

Ok. You're right, up to a point. Natural numbers refer to quantity or order. But the set of natural number may nonetheless be ordered in many alternative ways.

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. — Metaphysician Undercover

I hope I've clarified my exposition here. I see that I caused myself trouble by not being more clear earlier.

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. — Metaphysician Undercover

Sorry that one lost me.

You clearly haven't followed what I've been saying, — Metaphysician Undercover

Not for lack of trying.

and I realize that I did not make myself clear at all. — Metaphysician Undercover

Ok let's skip all this and hopefully go forward with my admission that natural numbers individually do refer to quantity or order; but (imperative you get this) the SET of natural numbers may be reordered at will.

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. — Metaphysician Undercover

I agree that natural numbers have a quantitative reference. I don't know why I obfuscated that earlier. My bad.

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. — Metaphysician Undercover

You swapped out temporal for quantity. Sneaky sneaky.

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

Quaternions? Transcendentals? p-adics?

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. — Metaphysician Undercover

Or you've failed miserably to understand my point.

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

You're hanging on to the other end of the straw.

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. — Metaphysician Undercover

I based my statement on your general mathematical ignorance, and the way you use it as a weapon in debate.

Show me that set which has no order then. — Metaphysician Undercover

{a, b, c}.

Or $\mathbb N]/math] without any particular order being implied.<br />$

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. — Metaphysician Undercover

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

I think you need to make clear what "set" means. — Metaphysician Undercover

LOL. For purposes of our discussion, anything that satisfies the ZF or ZFC axioms. The very first of which is the axiom of extensionality which says that sets have no inherent order, being completely characterized by their elements.

https://en.wikipedia.org/wiki/Axiom_of_extensionality

Does it refer to a group of things, or does it refer to the category which those things are classed into? — Metaphysician Undercover

A set is a mathematical set.

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? — Metaphysician Undercover

Schoolkids are not mathematical sets. I'm using them purely as an analogy.

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? — Metaphysician Undercover

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.

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

I see. Allow me to try again.

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. — Metaphysician Undercover

If you give the number 2 meaning, a definition, to validate its existence as a conceptual object, you might say that it means the number which comes after 1, but then you justify my argument, that counting (e.g counting the natural numbers) is expressing an order, and “2” refers to one of the numbers in that order, 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. — Metaphysician Undercover

If you give the number 2 meaning by saying that it means a quantity of two, then you justify my argument that what you are doing is measuring, rather than counting.

In plain terms, your argument is like saying that there is a difference in meaning between beating a drum and beating eggs, therefore we shouldn’t use the same word “beating” for both of these, and eggs are not genuinely beatable. But of course you can “beat” both eggs and a drum despite the difference in meaning, and despite your protestations, and attempts to restrict language, to the contrary. We don’t “beat” eggs the same way that we “beat” a drum, but neither do we “beat” a drum the same way that we “beat” eggs. One meaning is not superior to the other.

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

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

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

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

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

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

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

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

It is my opinion that there is no such thing as numbers which serve as a medium between the numeral (symbol) and its meaning, or what it represents. — Metaphysician Undercover

I have no idea what this is supposed to mean or how you think it relates to anything I've said.

To say that a particular order is "ascending order", is simply to make a reference to quantity. — Metaphysician Undercover

To determine a quantity is equally to make reference to an ascending order. Counting in one sense is no different to counting in the other sense. They are both the same sense of "counting".

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

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

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

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

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

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

I explained this already. Your "ascending order" is based on quantity, therefore your supposed "count" of ascending order means nothing unless it is determining a quantity. — Metaphysician Undercover

I explained this already. Your “quantity” is based on ascending order, therefore your supposed “count” of quantity means nothing unless it is reciting an ascending order.

This is why "numbers" as objects are assumed, so that when you count up to ten you have counted ten objects, (numbers). — Metaphysician Undercover

And, if numbers are not true objects, as I argue is the case, then this is not a true act of counting at all. — Metaphysician Undercover

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