• fishfry
    2.2k
    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?
  • TonesInDeepFreeze
    265
    fractions for dividing up a pumpkin piefishfry

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

    Math needs to correspond to reality!
  • Metaphysician Undercover
    8.6k
    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.
  • TonesInDeepFreeze
    265
    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 objectsMetaphysician 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 countMetaphysician 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 doorsMetaphysician 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.
  • Metaphysician Undercover
    8.6k
    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.
  • TonesInDeepFreeze
    265
    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 equivocationMetaphysician 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.
  • TonesInDeepFreeze
    265
    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'>}.
  • fishfry
    2.2k
    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 ? 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?
  • Metaphysician Undercover
    8.6k
    I want to be clear in my mind. Is this your position on the subject?fishfry

    Read my last post.
  • fishfry
    2.2k
    Read my last post.Metaphysician Undercover

    I'm thinking that I've read your last post.
  • Metaphysician Undercover
    8.6k

    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.
  • TonesInDeepFreeze
    265


    To begin with in all that, what's your definition of "real thing"?
  • TonesInDeepFreeze
    265
    Then why do you ask me to repeat myself?Metaphysician Undercover

    He didn't.
  • fishfry
    2.2k
    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.
  • Luke
    1.4k
    So we are not really counting the imaginary things, but symbols or representations of themMetaphysician 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"?
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