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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The mathematician only cares about 5. — fishfry

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Here's the quote from that post:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Symbols are not imaginary.

Wait, NOW you believe in ordinals? — fishfry

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The mathematician only cares about three. — fishfry

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

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

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

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

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

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

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

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

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

Then why do you ask me to repeat myself?

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

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

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

Read my last post.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Not sure how that was unclear. — Book273

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Then you said, "an object is implied".

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

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

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

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

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

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

Are you holding up? — jgill

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

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

So I don't say

'War And Peace' is implied.

But I do say

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Stipulation does not make truth.

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

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

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

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

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

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

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

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

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

What is your definition of number? — fishfry

It is a value representing a quantity.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

It requires more than innocence to be a saint.

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

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

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

* ({1,2,3,4,…},<)({1,2,3,4,…},<) and

* ({1,2,3,4,…},≺)({1,2,3,4,…},≺)

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

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

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

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

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

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

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

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

You're failing to distinguish between cardinals and ordinals.

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

The point is that we were talking about a count, which is a measure of quantity, not an order. To use numbers to indicate an order is a different matter. So to demonstrate the use of numbers in ordering now, is to equivocate, because an order does not necessarily imply a count

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

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

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

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

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

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

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

People, Trump and co. really could win.Then the fun will be over. — baker

Been there, done that.

It’s uncivilised for sentient beings to undergo involuntary pain and suffering – or any experience below hedonic zero. — David Pearce

Involuntary pain and suffering is most often derived in accidental ways, mistakes and not knowing the potential source, and how to prevent it. I do not think it is possible to eliminate the possibility of such pain and still remain living beings. Nor do I think we ought to attempt to eliminate the possibility of such involuntary pain, as this possibility is what inclines us to think, and develop new epistemological strategies for certainty.

What I think is a far more significant and important issue is the matter of voluntarily inflicting pain and suffering on others. If your goal is to manipulate the human being towards a more civilized existence, then the propensity for human beings to mistreat others is what you ought to focus on, rather than the capacity for pain. See, you appear to be focused on relieving the symptoms, rather than curing the illness itself.

In everyday understanding, when we count, we associate one thing with 1, then the next thing with 2, etc. Literally. We say the numbers, one for each object as we count the objects. Mathematically. this is expressed as a function from the set of things counted to a set of numbers: — TonesInDeepFreeze

We might say the numbers "one for each object as we count the objects", but that does not mean that we associate "2" solely with the object pointed to when "two" is said. In reality we associate "2" with having counted two objects, so the first object is also associated with the spoken "2". It is imperative to the count that the other object counted is remembered, and is an integral part of the meaning of "2'" when it is spoken. If the other object is not remembered as a part of the 2, then we could go back to the first object and say "3", but that's not a valid count.

It is the very point that you can count more than one way.

You can count 'War And Peace as the first, then 'Portnoy's Complaint' as the second. Or you can count 'Portnoy's Complaint' as the first, then 'War And Peace' as the second. In either case, both counts show that there's a first and second, thus there are two. — TonesInDeepFreeze

Right you can count the objects in any order that you want. Therefore "pairing", or bijection, which represents the count as assigning a specific order to the objects is a false representation of counting. In instances when there is a small number of objects we can look at them and see the number of objects, without giving them any order at all. So ordering them, or "pairing" them is accidental to the count, it is not an essential aspect of counting. We simply do it as an aid, to ensure that we are not making a mistake and producing a false count.

You ought to accept and understand this fact, because it is fundamental to many forms of measurement, and how we actually count something in reality. When we weigh something, we do not pair a different part of the object with each gram counted, and when we measure the electrical potential we do not pair each part of it with a volt. This is clear evidence, that in general practice, counting something is not a matter of pairing objects with numbers. In modern practice, we deal with billions, trillions, and numbers so high, that if counting something was a matter of pairing, we'd never get done counting any of these astronomically high numbers which we deal with.

The sequence a,b,c,d,e is a sequence of five letters. e is letter five. — jgill

That's an arbitrary designation, dependent on a stipulation that there is a left to right order to the sequence. "a" could just as easily be letter five, or we could assume an ordering which makes any of the letters number five. The point being that even though we order things when counting, (first, second, third, counted, etc.) because it facilitates distinguishing between what has been counted, and what has not been counted, helping to ensure certainty, ordering is not essential to counting. We can count things without ordering them.

If I'm not mistaken, in another thread, you were using the word 'refer' in the sense of 'denote'. So if not 'denote' what exactly do you mean by 'refer' in this thread? — TonesInDeepFreeze

"Refer" is more general than denote, such that to denote is a specific type of referring. So when we say that a word refers to something, whether that something is a thing, an activity, an idea, a concept, or whatever, it means that we must direct our attention toward whatever it is which is referred to, in order to understand the use of the word.

The numeral '5' has meaning. The number 5 is not the numeral '5'. — TonesInDeepFreeze

The number 5 is a concept, therefore it has meaning, like any other concept.

The fact that 5 is a count doesn't contradict that 5 also is a number no matter what it happens to count. — TonesInDeepFreeze

My point is that 5 must count something, or else we forfeit its meaning. There is no sense to saying that there is a count of five which does not have five things.

5 is the successor of 4. 4 is the successor of 3. 3 is the successor of 2. 2 is the successor of 1. 1 is the successor of 0.

No matter what the numbers count, they exist by virtue of successorship or by being 0. — TonesInDeepFreeze

This is simply not true. Numbers are defined by quantity, not order. If you want to define numbers by order, then you assign temporality as the difference between 1 ,2,3 and 4. But this is not at all how numbers are used. We might assign numbers to units of time, like first second, third, fourth, but it's really not true to say that numbers derive their value from order or succession, rather than from quantity.

of course are different, but nothing is "invalidated". Saying the pairings are "invalidated" is not even sensical. — TonesInDeepFreeze

As I explained, what is invalidated is your representation of the count as a pairing. The pairing you described is not a valid representation of a count, for the reasons explained.

You're doing it again! We do not use '2' to name a book. '2' does not denote a book. — TonesInDeepFreeze

If your count is nothing but a pairing, then that is all you are doing, assigning a number to a book, naming a book, with a number. This is why your representation of a count, as a pairing, or bijection, is incorrect. That is not what a count is.

We can switch them so that we have:

{<'Portnoy's Complaint' 1> <'War And Peace' 2>}

But the greatest number in the range is still 2. — TonesInDeepFreeze

If you switch them, then your original pairing is invalidated. Which pair is the true representation of the count? It can't be both at the same time. But in a true count, neither book is paired with 1 or 2, because a count is not a paring. There are two books, and neither one is number 1 or number 2, they are equivalent as books.

That doesn't contradict that when we see discrete objects then we may count them. — TonesInDeepFreeze

Sure, we can count discrete objects (units), that's what I've been arguing is necessary for a count, to have discrete units which are counted. What is incorrect, for the reasons explained, is your representation of a count, as an act of pairing a discrete unit with a number. Do you understand those reasons given?

How we use the concept of counting is a matter of practical approach, such as putting the water in a beaker with lines and counting the lines in the beaker to the point the water level ends or whatever. Whatever difficulties there may be conceptually with that, they don't negate the more basic notion of counting by bijection. — TonesInDeepFreeze

Again, this is completely untrue. If we want to know what a number is, within a count, then we must produce a true representation of what a count is. To simply produce a false representation of a count, for the sake of supporting your claim of what a number is, is to just beg the question with a false premise.

You present as so confused that I wonder whether you are posting as some kind of stunt or dumb cluck character. — TonesInDeepFreeze

You present yourself as someone who has not yet learned how to count.

The way I see it is that as human beings, we are first and foremost, animals. That's what defines us, although we like to separate ourselves from the other animals, to say that we're somehow a special type of animal. Let's say that specialness as "civilised". If we're already civilised, then what could it even mean to suggest making us more civilised? If civilised is a general category, then all sorts of particular instances qualify as civilised, and what would make one "more" civilised than another? If we are not yet civilised, then what really does "civilised" mean? Distinguishing us from the other animals, is not even justified now. Unless we answer this type of questions, assuming that such and such is "more civilised", is simply an unjustified assumption.

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

A number (we're talking about natural numbers in this context) is a count in sense (2). That doesn't preclude that a number is a mathematical object. — TonesInDeepFreeze

That's right, it doesn't preclude that the number is a mathematical object. But the point is that your definition (2) stipulates "the result of counting". So correct use of "5" is dependent on the count of the books, that there are five books. Therefore the number 5 loses its meaning if it does not refer to five of something counted, books in this case. Anytime we use "5" regardless of whether you think it refers to a mathematical object or not, it necessarily refers to five distinct units, or else you are using it incorrectly.

We better dispense with that notion. It's nuts. A number is not a book. — TonesInDeepFreeze

I'm not saying a number is a book, that's nonsense. But when we use "5" it is necessary that there are five distinct units indicated in that usage or else you are using "5" in an unacceptable way. Do you agree?

So the numeral does not denote a book, but rather it denotes the number that is paired to the book in the bijection (or, in everyday terms, in the pairing off procedure we call 'counting'). — TonesInDeepFreeze

Strictly speaking this (bijection in the way you describe it) is not a valid count. Suppose we say that there are two books. "War and Peace" is numbered as 1, and "Portnoy's Complaint" is numbered as 2. The relation between "Portnoy's Complaint" and the number 2 is not a simple pairing. This is evident from the fact that if we remove "War and Peace", there is no longer two books, and the pairing is invalidated. You might still use "2" to name the book, but it is not a valid count of two, because there is only one book.

So we cannot say that "Portnoy's Complaint" is paired with 2. That is a false representation because it does not include the necessary requirement of another book. "Portnoy's Complaint" can only be paired with 2 in a valid count, if there is another book paired with one. Furthermore, neither Portnoy's complaint nor "War and Peace" need to be paired with either 1 or 2, for there to be a valid count of 2. Do you recognize this point? There is no need for a pairing to have a valid count. We can have two objects, and say that there are two, without naming either as one or two, they are simply two.

This latter point is something which is very important to understand, especially when we count things like electrons which are difficult to distinguish from one another. We can have a count of 2 without establishing the principles required to distinguish one from the other. We can say that there are two electrons in the same orbit, without the need of distinguishing one from the other. We have principles which say they are distinguishable, but we need not distinguish them. Likewise, we can talk about 12 volts, without the need to distinguish and label each unit of electrical potential, as 1,2,3, etc..

So it is very clear that your method of representing "a count", as pairing a number with a unit (bijection) is a totally inadequate representation of what a count really is.

We don't say "''1' denotes 'War And Peace' and '2' denotes 'War And Peace' together with 'Portnoy's Complaint'". That's crazy. — TonesInDeepFreeze

You think it's crazy, but it's what's required to have a valid count. If "2" denotes "Portnoy's Complaint", unconditionally, and you have no other books, then obviously your count of 2 books is invalid. If you deny this requirement them you allow for invalid counts. You look at your bookshelf, number "Portnoy's Complaint" as 2, and bring it in to me, telling me you have two books in your hand, because "Portnoy's Complaint" is identified as two books. That's what's really crazy.

Right, but do you agree that it is necessary that there is a thing counted, a book in this case? So as much as the numeral "1" "denotes" what you call the number 1, which is a property of "the count", it must also refer to the one book, or else the count is not a true count. For "the count" to qualify as a true count, there must be something which is counted. If "1" does not refer to the book, as well as what you call the number, then there is nothing being counted, and therefore no count, because if there is nothing being counted, this does not qualify as a count.

Therefore, we cannot dispense with the fact that "1" must refer to the object being counted, a book, as well as what you call the number 1, or else we have annihilated "the count" as false because we cannot have a count with nothing being counted. But we cannot annihilate the count, because that is what gives logical coherency to the numbers.

If this is not clear to you, imagine that you go to count the books, and you count the same book over and over again, 1, 2, 3, 4, etc., such that you could have an infinite number of books, by counting the same book over and over again. That is not a valid or true count. To have a true count, "1" must refer to the first book, "2" refers to the first and second together, "3" refers to those two with a third, etc.. If we remove the need to have distinct objects being counted, then the count is not a valid or true count.

But to return to the earlier example of playing chess, one can fanatically aspire to improve one's game and play to win even though one will invariably lose. — David Pearce

I don't agree. One cannot play to win if the person knows that winning is impossible. In a similar way one cannot get the same enjoyment from winning if the person knows that losing is impossible. So there must be the real possibility of losing (suffering) if winning is to be enjoyable.

Two invincibly happy (trans)humans can play competitive chess against each other and both improve their game. Honestly, I don't see the problem! — David Pearce

The point is that one must lose, and suffer from the loss, if the other is to win and obtain the enjoyment of winning. If we remove the winning and losing from the game, we can't call it a competitive game.

Rather, what needs questioning is the widespread assumption that the "raw feels" of suffering are computationally indispensable. If the indispensability hypothesis were ever demonstrated, then this result would be a revolutionary discovery in computer science: — David Pearce

The issue, in my mind, is not whether suffering is indispensable, but the question of whether we can have gain without the possibility of suffering. If it is the case, as I believe it is, that all actions which could result in a gain, also run some risk of loss, and loss implies suffering, then to avoid suffering requires that we avoid taking any actions which might produce a gain. But if gain is necessary for happiness, and this is inevitable due to biological needs, then the goal of happiness cannot include the elimination of suffering. Therefore the goal of eliminating suffering must have something other than happiness as its final end. What could that final end be? If eliminating suffering is itself the final end, but it can only be brought about at the cost of eliminating happiness, then it's not such a noble goal.