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. — Metaphysician Undercover

That's silly. I can count the captains of the starship Enterprise even though they're imaginary.

https://screenrant.com/star-trek-movies-shows-enterprise-captains-kirk-picard/

I can count the harpooneers on the Pequod even though they're imaginary. What ever are you talking about?

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. — Metaphysician Undercover

This is "not even wrong." As Truman Capote once said of an inferior writer: "That's not writing, that's typing."

What is your definition of number?
— fishfry

It is a value representing a quantity. — Metaphysician Undercover

A moment ago you Meta-splained why there can be no general definition of number, and now you give one. Don't you even read your own posts?

But quantity is only one thing numbers represent. Integers (zeri, positive and negative whole numbers) represent signed or directed quantities. Rational numbers represent ratios of whole numbers. Irrationals and complex numbers, p-adics and hypperreals, each represent some other aspect of number-hood. Aristotle said that the reason bowling balls fall to earth is that earth is the "natural place" of a bowling ball. Surely you are aware that we have more modern explanations now. Why do you deny the historical development of our understanding of the concept of number?

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

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. — Metaphysician Undercover

Not following that convo, but do I take it that you deny complex numbers? Do you likewise deny negative numbers, zero, rationals and irrationals? Is your physics likewise stuck in the days of Aristotle? Why exactly do YOU think bowling balls fall down?

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. — Metaphysician Undercover

You're a trivial sophist with no insight or awareness of intellectual history.

Of course I'm wrong mathematically, I'm arguing against accepted mathematical principles. — Metaphysician Undercover

If only you were, we could have a conversation. But you have no actual principles or arguments, only nihilism and denial.

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? — Metaphysician Undercover

Guess we're done here. Again. I'd like to say something more substantive, but what can I say to someone who rejects the role of numbers as expressing order, or numbers as used in quantum physics, or even fractions for dividing up a pumpkin pie? What words besides nihilism fairly describes your mathematical perspective?

fractions for dividing up a pumpkin pie — fishfry

Not in the real world. I eat the whole pie all at once.

Math needs to correspond to reality!

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.

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?
— Metaphysician Undercover

Agree. — TonesInDeepFreeze

That was a while ago. But you're still asking!

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. — Metaphysician Undercover

There you even correctly posted yourself that you surmise that I agree that if we count 2 objects then there exist 2 objects.

The context here has been of a shelf that has books on it. I've said more than once that if you count 1 book or 2 books, then, yes, there are books on the shelf.

From this thread, this is the context in which we are talking about a shelf that has books on it:

To have a count of one, there must be an object which is counted. In order for the count to be a valid count, there must be something which is counted. — Metaphysician Undercover

Do you agree that there must be some of these things (objects) which are classed as "books", for us to have a true count.
— Metaphysician Undercover

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

Do you agree that there is no activity of counting if there is no objects counted?
— Metaphysician Undercover

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

Therefore the number 5 loses its meaning if it does not refer to five of something counted, books in this case. — Metaphysician Undercover

The context is not a shelf with no books, but a shelf with 5 books.

In everyday understanding, when we count, we associate one thing with 1, then the next thing with 2, etc. — TonesInDeepFreeze

There I keep in context of having at least one book on the shelf.

To have a count of one, there must be an object which is counted. In order for the count to be a valid count, there must be something which is counted. — Metaphysician Undercover

Again, the context is that there are books on the shelf.

The count of two is justified by the existence of two such objects — Metaphysician Undercover

Again, the context is that there are books on the shelf.

Now:

If no books are counted, do you consider this to be a count? — Metaphysician Undercover

I already addressed that. If there are no books, then it's not a non-empty count. It's not the kind of count we're talking about in your example.

do you agree that it is necessary that there is a thing counted
— Metaphysician Undercover

To have a count (in sense (1)), you need something to count. (Except in the base case, there is the empty count.) — TonesInDeepFreeze

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.

/

I'll say it one more time in this forrm: 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.

And that reflects the representation with a bijection. If a natural number n is in the range, then there must be at least n objects in the domain.

You don't read my posts adequately to register in your mind what I wrote, let alone understanding them.

And you're even more ridiculous, since the question of whether there are objects counted - already answered by me - is answered right in the representation with a bijection itself. You can see for yourself that the two books are right there in the domain of the bijection.

/

I'm asking you if you believe there is such a thing as an empty count — Metaphysician Undercover

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.

I mentioned the empty count only to avoid a pedantic, technical hitch. I am not talking about empty counts in the context where there are books on the shelf.

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.

/

you are counting hypothetical doors, symbolic representations of doors — Metaphysician Undercover

But it's still counting.

If you present this as a true count of actual captains of an actual starship, you'd be engaged in deception. — Metaphysician Undercover

Ah, you resort to the strawman. We are not claiming it is a count of actual captains.

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.

Why do you keep avoiding the question? — Metaphysician Undercover

Why do you keep saying I've avoided the question when I have not, when, indeed, I have answered several times and with copious explanation and detail? Possible answers: (1) You are dishonest, (2) You have cognitive problems.

within a logical system you cannot change the "sense" of a word without the fallacy of equivocation — Metaphysician Undercover

In a formal system, a terminology can be defined only once, so it is not possible to have equivocation.

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. — Metaphysician Undercover

I didn't use the phrase "countable number of objects".

None. We know that there are zero, without counting any. — Metaphysician Undercover

Correct. I told you that the pedantic technical mention I made does not pertain to the everyday English sense of 'count'.

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

That is an extraordinary statement, even for you.

One more time:

We are in a context of everyday English. Then, I have given a mathematical representation of that everyday English sense. A mathematical representation:

A count is a bijection f from a finite set onto a set of successive positive numbers that includes 1. The result of the count is the greatest number in the range of the bijection. The count induces an order on the domain by: x precedes y iff f(x) < f(y).

{<'War And Peace' 1> <'Portnoy's Complaint' 2>} is a count.

The domain is {'War And Peace' 'Portnoy's Complaint'}.

The range is {1 2}.

The result is 2.

The order induced is {<'War And Peace' 'Portnoy's Complaint'>}.

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. — Metaphysician Undercover

I'm unmoved by your argument. I can't respond at all. I don't think you've said anything meaningful here. You can't count the harpooneers on the Pequod without engaging in deception? If I'm building a house and the architect shows me the plans and I count the doors, I'm not really counting the doors? This is an intellectual point you want people to take seriously?

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. — Metaphysician Undercover

You have a better idea? You reject it wholesale? You disagree with the famous measurement of the magnetic moment of the electron, the most accurate physical experiment ever done, accurate to within 7.6 parts in $10^{13}$? When shown this result you say, "Pish tosh, those quantum mechanics don't know jack."

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

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

Read my last post.

Read my last post. — Metaphysician Undercover

I'm thinking that I've read your last post.

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.

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

Then why do you ask me to repeat myself? — Metaphysician Undercover

He didn't.

Then why do you ask me to repeat myself? — Metaphysician Undercover

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.

And where the question is coming from is that since your understanding of the concept of number is stuck back about 2500 years ago, it makes me wonder if your understanding of physics is similarly ancient.

But that's actually not what my most recent remark meant. When I wrote,

I'm thinking that I've read your last post. — fishfry

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

Look, I think it's very important for a rigorous mathematics to distinguish between counting real things, and counting imaginary things. — Metaphysician Undercover

Actually you couldn't be more wrong about that. Pure math doesn't care what you count. If you count chickens, you're a farmer. If you count harpooneers in Moby Dick, you're a professor of English literature or a high school student reading the Cliff notes. If you count molecules you're a chemist; if you count quarks your a physicist. But if you study the act of counting itself, utterly without regard to the thing being counted, then you are a mathematician.

Once again you utterly fail to understand the nature of mathematics yet wield your ignorance like a cudgel.

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. — Metaphysician Undercover

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. And to the extent that it ever is, the things that mathematicians count are ALWAYS imaginary. We count the rational numbers (same quantity as the naturals) or the reals (a higher cardinality). There's no claim that these things "exist" like rocks or planets or even quarks. How you fail to understand this yet regard yourself as having insight into the philosophy of mathematics, I can't figure out.

Therefore we can only count representations of the imaginary things, which exist as symbols. — Metaphysician Undercover

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.

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. — Metaphysician Undercover

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.

How do you not get this?

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. — Metaphysician Undercover

It's hard to take this line of thought seriously since mathematical practice so obviously falsifies your claim.

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. — Metaphysician Undercover

Well that's my first point above. To a pure mathematician there is no difference between counting 120 rocks and counting the 120 distinct permutations of five objects. Why don't you understand that?

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. — Metaphysician Undercover

Nonsense. Abject bullpucky.

So it's a matter of faith, that the imaginary things represented by the symbol, are really there to counted. — Metaphysician Undercover

No it's not. One need not reify abstract things in order to talk about them. YOU continually try to reify things that need not and should not be reified. I'm coming to see that this is your core error.

But of course they really are not there, because they are imaginary, so it's false faith. — Metaphysician Undercover

Yeah right.

So we are not really counting the imaginary things, but symbols or representations of them — Metaphysician Undercover

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

But this is not really counting because there are no things being counted. — Metaphysician Undercover

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"?

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.

Given that intuitionists who reject the existence of actual infinity also reject the law of excluded middle, they are likely to disagree with the assumption that the universe must either be finite or infinite.

For the intuitionist, truth is synonymous with verification of some sort, meaning that according to this stance there are no unknowable true propositions. This implies that if it is unknowable in principle as to whether the universe is finite or infinite, then there is no transcendental matter-of-fact as to which is the case.

Also, it should be mentioned that the commonest use-case of potential infinity involves an agent querying nature for the value of an unbounded variable and accepting the received response (if any). Therefore it is false to claim that denial of actual infinity entails denial of external reality.

I'd say it's ordering, not counting. — Metaphysician Undercover

Wait, NOW you believe in ordinals?

I'm going to skip responding to your points. Earlier you said numbers were for quantity and I pointed out that there are other kinds of numbers (in the finite case, the exact same numbers viewed differently) for order. I made my point. Then we got off onto other things. You say you don't think there are pure mathematicians, that you can't count abstract things, that you don't believe in experimental science (since you apparently reject the example I gave) and so forth. I'm out of enthusiasm to continue. Till next time.

:up:

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.

So we are not really counting the imaginary things, but symbols or representations of them
— Metaphysician Undercover

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 — Metaphysician Undercover

I didn't call it counting; you did. You said: "So we are not really counting the imaginary things, but symbols or representations of them."

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.

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. — Metaphysician Undercover

Why?

It's called "counting sheep" not "ordering sheep". Are you saying we cannot count imaginary things but we can order imaginary things?

How do you account for The Magnificent Seven, The Famous Five, 12 Angry Men, etc.?

They're not called The Magnificent Seventh, The Famous Fifth and 12th Angry Men.

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.

Symbols are not imaginary. — Metaphysician Undercover

That's right. I never said they were.

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

Oh dear. Did you not read that section of the thread, where I described the difference between quantity and order? — Metaphysician Undercover

Quite possibly I didn't. I only read the posts that generated my mentions. I did not read anything else in the thread. And as I believe I admitted, my cardinal/ordinal remark was only in reference to a single sentence you wrote, and not at all in reference to the larger context of the discussion, of which I was and still remain ignorant.

That also explains why I ran out of steam for engaging in further convo. I made a small point, that there are ordinals as well as cardinals. I was not intending to engage at any deeper level.

I think it's rather pointless because you do not seem at all inclined to make any effort toward understanding. — Metaphysician Undercover

I would say that I've made a considerable effort the past several years to understand your point of view. But I agree that I prefer to make the effort in small doses; and on this thread, I reached my local limit. I'm sure we'll do this again in some other thread. But in truth you made so many strange statements here that I saw no basis to continue. 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. This is perfectly consistent with your 2200 year old view of math. You have a 2200 year old view of physics as well. I don't wish to argue that point with you.

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. — Metaphysician Undercover

Thank you for the kind words. I did not see this, as it did not contain an @ before my handle. As I said, I've only looked at posts that contain a mention of my handle.

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. — Metaphysician Undercover

But since I'm here, let me note that this could not be more false. I already gave the counterexample One can order the natural numbers $\mathbb N$ with the linear order $\lt$, the usual order; or $\prec$, the "funny order" in which everything is the same as the standard order except that $n \prec 3$ for all $n \neq 3$. This is a purely abstract order relation on the natural numbers. There is no spacial or temporal referent involved. One abstract set, two distinct abstract orders. Absolutely no referents in the physical world (that we yet know of) but of critical importance in mathematical logic, proof theory, and various other abstract branches of math.

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?

So the truth of a determined quantity depends on the criteria for what qualifies as an object to be counted, — Metaphysician Undercover

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.

while the truth of a determined order is dependent only on our concepts of space and time. — Metaphysician Undercover

Please show me space or time in the $\prec$ order on the natural numbers.

So, in the case of quantity, truth or falsity is dependent on the truth of our concept of distinct, individual objects, — Metaphysician Undercover

Those harpooneers don't exist, yet there are three of them; four if you count Ahab's personal harpooneer Fedallah. They can be counted sure as the planets. Even more surely, since there's no committee removing harpooneer-hood from Queequeg, as there is for removing planet-hood from poor Pluto. I remember being on vacation, sitting in the Portland airport reading the New Yorker, and discovering that Pluto was no longer a planet. Counting is not as sure a thing as you'd think. The "truth of our concept of distinct, individual objects," as you put it, turns out to be subject to the vote of a committee. Such is reality these days. A lot more tenuous than you'd think.

but in the case of ordering, truth or falsity is dependent on the truth of our concepts of space and time. — Metaphysician Undercover

I ask again for you to please show me space and time in $\prec$. It's an alternate ordering of the natural numbers, an alternate order type in fact, but there is no space or time involved. It's purely abstract.

Since we think of space and time as continuous, non-discrete, — Metaphysician Undercover

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.

we have two very different, and incompatible uses of the same numerals. — Metaphysician Undercover

Of course quantity and order are two distinct aspects of the "same numerals" in the finite case. In the transfinite case we use different numerals; $\aleph_0$ for the cardinal representing the natural numbers; and $\omega$ to represent the exact same set with its usual order $\lt$.

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.

I really do not think there is any type of order which is not based in a spatial or temporal relation. — Metaphysician Undercover

You have just been shown one, namely $\prec$. But you haven't actually "just" been shown one. I showed you this example several days ago.

You know, to sum this all up, I understand that it's difficult to give a coherent account of abstract objects. But that doesn't mean they're not important. You use the former to utterly reject the latter, and that forces you into positions that are impossible to defend.

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

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

TonesInDeepFreeze was equivocating, or at best, creating ambiguity between quantity and order, using "2" to mean "second", when counting a quantity of two. — Metaphysician Undercover

There is no equivocation or ambiguity in what I said, Your confusions and implacable dedication to remaining ignorant of mathematics and dreadfully misunderstanding it are your own.

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.

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. — Metaphysician Undercover

How is an imaginary thing a self-contradicting set of ideas?

if you go back and reread TIDF's discussion of counting a quantity, you'll see the equivocation with order — Metaphysician Undercover

That is the second time you've made that false claim. Moreover, I did not couch anything as "counting a quantity".