• apokrisis
    4.2k
    No. It's not. That's the point. i is a number but it's not a quantity.fishfry

    Perhaps this is pedantic, but even in terms of rotations in the complex plane i does have a couple of associated quantities with its notion of multiplication. It represents an anti-clockwise rotation of 90 degrees and a magnitude of 1 in terms of the size of complex numbersfdrake

    Fdrake is right. If we want to ask what i quantifies, it quantifies the number of dimensions that a number is constrained by. So i is a widget to rotate a real number into an orthogonal direction that turns the number line into a number plane.

    The number line stands for the most constrained notion of continuity. Complex numbers relaxes that strong constraint and allow numbers to wander in two dimensions. And the numbers still behave like numbers - objects that meet the functional criteria of associative division algebras.

    We can continue to relax the number of dimensions in play. We could consider a three dimensional number. But now it doesn’t behave arithmetically. It is not a suitable object of algebraic structure - a fact of undoubted physical significance when it comes to why space winds up being three dimensional.

    Then with quarternions, we have four dimensions and a bounce back to a large amount of algebraic structure. Five, six and seven dimension again see that structure disappear. Then the octonions provide a last echo.

    So i is a good example of structualism at work. We can define some basic relational properties that numbers are meant to have. The associative division algebras do that. And then we can see how the “hard structure” emerges as constraints are added.

    As we constrain the dimensionality that defines the continuous space in which discrete mathematical objects are meant to move, we can see the role those constraints play in actually defining the mathematical properties those objects are understood to have.

    The limits maketh the objects and not the other way round.
  • tim wood
    1.1k
    I pointed out that it's very difficult to define in general what a number is. You suggested that a number is something that can be quantified or that represents or is a quantity. I gave as a counterexample the number i, which is a number but is not and does not represent a quantity.

    You said a quantity is something that can be quantified. I don't find that helpful because it doesn't tell me what a quantity is. If you tell me a cat is a furry domesticated mammal with retractile claws, that's a lot more helpful than saying that a cat is anything that's cat-like.
    fishfry

    Read the English, fishfry!! What is your native language, please? If you cannot understand what you read, how are the rest of us to credit anything you write?!

    like 2 apples is a quantity.fishfry
    Two apples is not a quantity! Try to wrap your mind around that. As to i being a quantity, you say it isn't. I suspect you're mistaken, but I'll raise the question elsewhere. As to a definition of number, there's no shortage of them. The problem you're having is that you're confused about what definitions are, what they're used for, and how they work. Research it! I will offer this: I think you're making a category error in being confused about what, exactly, is being defined. If your reading truly reflects your thinking, then you have your work cut out for on this. Good luck! .I'd go for it, but you have so clearly exhibited an odd (and annoying) problem with reading that I am stopped from writing further.
  • fdrake
    1.3k


    I do of course agree with you point that 2i is a quantity of two i's, like 2 apples is a quantity. So the question reduces to asking exactly what is a quantity. @tim wood brought up the idea of quantity a while back so I asked him what is a quantity, and so far I have not gotten an answer.

    By saying that the magnitude of i is 1, what I meant wasn't that there was a single i, an answer to how many 'i's are there - but that a vector that starts at the origin in the complex plane and points to i has length 1. This allows for there to be real numbers of 'i' as the 'number of i's in a complex number, so to speak.

    More generally, speaking about complex numbers like z=4+pi*i. Pi isn't exactly an answer to 'how many' i's are there in z since its interpretation is severed from counting numbers in a few ways. The first way it's severed is that z is not a multiple of i in anything like the counting number sense (there are pi-4i i's in z), so we cannot chunk z into i sized bits through division. an 'i sized bit' doesn't even make sense as imaginary numbers don't enter into the notion of size for complex or imaginary numbers.**

    The second way the interpretation of the magnitude of z is severed from the interpretation of a real number or fraction is that z has two senses of magnitude inherent in it. There's the real part and the imaginary part (which individually work exactly the same as real numbers and usual counting in terms of 'how many' questions, to the extent that irrational numbers can be said to be answers to 'how many' questions) or there's the polar form of the radius and angle - requiring two descriptors of magnitude to specify the quantity ('number') rather than the single one for scalars. Polar form and Cartesian form for complex numbers also have differences in interpretation since the polar form contains an unbounded quantity (radius) and a bounded one (the angle), and Cartesian form is done in terms of two unbounded quantities (the magnitudes of real and imaginary parts). They also mean different things (polar form and Cartesian form) even though they are just different ways of talking about the same thing (naming complex numbers).

    There's also the wrinkle which you already mentioned about the tension between irrational numbers (which are implicated in the magnitude of complex numbers in both directional and radial senses) as magnitudes and fractions as answers to 'how many x go into y' questions. Even the Gaussian integers have this problem (such as 1+i having magnitude 2^(1/2)).

    Actually looking at 'numbers', even in relatively simple cases like these, shows that there's no single sense of magnitude or quantity implicated within them - even if there are formally equivalent representations.

    **the closest approximation to this in the complex plane being dividing a complex number z=x+iy by r=sqrt(x^2+y^2) yielding u=z/r, u has magnitude 1, which is the same magnitude as i - all this says is that z lies on the unit circle and u can be obtained again by scaling by r.
  • MindForged
    433
    Kind of a horribly vague question. For one thing, "number" is going to be quite different depending on A) What kind of "number" you're referring to B) What sort of mathematics you're working in (numbers in ZFC + classical logic look quite different than numbers in Paraconsistent Mathematics), etc.

    This topic is simply too vast for me and I personally try not to think about it too much, lol.
  • Dzung
    42
    And now you don't need some purposeful and transcendent creator.apokrisis
    There are too many things going on here but I would like to start with an inquiry on the above statement: specifically is it Peircean or not? I haven't found anywhere Peirce expressed atheism or the like. Or maybe I didn't follow you correctly.
    The next maybe another inquiry about your support of "symmetry" concept, but let's set it aside for a moment.
  • Harry Hindu
    1.3k
    Numbers exist because they can establish causal relationships. Numbers can cause us to do different things.
  • apokrisis
    4.2k
    I would like to start with an inquiry on the above statement: specifically is it Peircean or not? I haven't found anywhere Peirce expressed atheism or the like. Or maybe I didn't follow you correctly.Dzung

    It is notoriously difficult to agree what Peirce actually believed about god or divinity. But he himself stressed he certainly did not follow any kind of orthodox view.

    And my point there was that he definitely did not argue for an external creator with some mission in mind for mankind. Instead, he identified the divine with the vague ground of being - the Firstness of pure unformed potential. And so the Comos is a state of logical regularity that evolved into being in a purely self-creating fashion with no purpose in mind except to be "increasingly reasonable" in its lawfulness and organisation.

    He did say he was more Buddhist on this score. :)
  • Dzung
    42
    he was more Buddhist on this scoreapokrisis
    wow, if you happen to have a public link, kindly share. I found only this content http://www.gnusystems.ca/CSPgod.htm#aq1 ("I think we must regard Creative Activity as an inseparable attribute of God." C.S Peirce.)... there maybe more pieces but let another time to connect them together.

    Just to play fair with the thread, numbers are in mathematics which is in turn sub-semiotics. I am not sure the last has been maturely explored but math is thought by Platonists as another world. Even the "semeiotic" sounds much to do back with Plato's Ideas, now with a better weapon of Synechism.
    If the multiverse-like metaphysics is accepted then we can perceive such a Cosmos that circumscribes it in. Old story while I find your posts more interesting and will switch to enquire about Peircean buddism or non-Peircean symmetry, where possible.

    For symmetry, I am not sure you have come across talks similar to these https://www.closertotruth.com/series/why-do-we-search-symmetry
    I drew a note that fundamentally deep down, symmetry is quite empirical and approximate. It's useful in many talks but we have to recognize its limitations and avoid it at extremes.
    Being aware of no symmetry from Peirce, I think if we still need to linger on it, we may want to analyze Synechism, not only Tychism.
  • apokrisis
    4.2k
    if you happen to have a public link, kindly share.Dzung

    Actually the quote I was thinking of was misleading as it wasn't connected to his evolutionary cosmology but to the more mundane thing of how his Christian contemporaries view his "scandalous affair". Buddhism wouldn't be so judgemental.

    I can't help thinking that the mother of Christianity, Buddhism, is superior to our own religion. (NEM III/2 p. 872)

    So it was more that Eastern metaphysics was in the air in his time as something exotic, but not really studied.

    Here is a more direct reference in terms of his evolutionary cosmology where he talks about its roots...

    ... tychism must give birth to an evolutionary cosmology in which all the regularities of nature and of mind are regarded as products of growth, and to a Schelling-fashioned idealism which holds matter to be mere specialized and partially deadened mind. I may mention, for the benefit of those who are curious in studying mental biographies, that I was born and reared in the neighborhood of Concord - I mean in Cambridge - at the time when Emerson, Hedge, and their friends were disseminating the ideas that they had caught from Schelling, and Schelling from Plotinus, from Boehm, or from God knows what minds stricken with the monstrous mysticism of the East. [6.102]

    Being aware of no symmetry from Peirce, I think if we still need to linger on it, we may want to analyze Synechism, not only Tychism.Dzung

    Yeah, I don't think Peirce said much about symmetry and symmetry-breaking principles. It was implicit rather than explicit at best.

    Peirce had a Victorian level understanding of phase transitions and other physical manifestations of symmetry breaking. Group theory and its fundamentality in physics was a 20th century thing, after all.
  • Janus
    5.7k
    I can identify the types of numbers I already know about: integers, reals, etc. But I can't determine in general what is a number.fishfry

    How can you identify types of numbers if you don't know what a number is?
  • apokrisis
    4.2k
    Checking further, there is this attempt at a pantheistic reading of Peirce....

    ARTHUR W. BURKS - PEIRCE'S EVOLUTIONARY PRAGMATIC IDEALISM
    https://deepblue.lib.umich.edu/bitstream/handle/2027.42/43816/11229_2004_Article_BF00413590.pdf?sequence=1

    Peirce, as a pantheist, thought God and the cosmos constituted one substance. To introduce his views we will trace the philosophic theme that runs through all four stages of his thought: the cosmos is an infinite semiotic goal-directed evolutionary process that converges on the good and the real....

    ...Peirce's evolutionary pragmatic idealism was a radically new form of pantheism. He replaced the theist's idea of a "one-shot" creation of the world by the gradual creation of the world through the evolutionary process of Tychism-Synechism-Agapism. He thought of cosmic evolution as a divine learning process. Chance, continuity, and cosmic purposes are all aspects of God, and we humans are parts of this infinite evolutionary divine system. ...

    ...When asked "Do you believe this Supreme Being to have been the creator of the universe?" he answered "Not so much to have been as to be now creating the universe",...

    ...Peirce's evolutionary pragmatic idealism is an evolutionary form of pantheism that operates in the opposite direction from emanationism and Spinozism. Whereas the latter theologies proceed from the highest level (God) on down through successively lower levels, Peirce's cosmic evolutionism begins at the simplest level of a random chaos of feelings and gradually improves under the guidance of final causality toward an infinite limit of perfection. Thus Peirce's pantheism is emanationism "turned upside down"...
  • Dzung
    42
    it wasn't connected to his evolutionary cosmologyapokrisis
    in deed your above quote is from a letter of him to Williams James. The majority of his quotes are scattered over different kinds of media but I do think all are connected, much like his philosophy about the continuum - Synechism. He also said "I do not agree with you that my papers about the evolution of the Laws of Nature are the best things I have done."[/i] (C.S.Peirce) and "I think unquestionably my best work has been my Logic.". This really helped me to grab a knot from his web.
    The information conveyed in any of his works is massive and cannot be plainly elaborated in a small article or even book. It seems his doctrines such as triadic reduction, synechism, infinitestimal and even his flatly established religion (in the same letter he also mentioned people had scoffed at his religion so he would refrain from expressing it)...have yet to be duly understood.
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