• fishfry
    408
    a structuralist definition, in that numbers are whatever it takes to get certain number-like operations - like those that preserve certain global symmetries, such as commutativity or associativity.apokrisis

    The quaternions are numbers whose multiplication is not commutative. The transfinite ordinals are numbers whose addition is not commutative. How weird is that, right?

    Good idea and a very natural attempt; but arithmetic properties aren't sufficient. Weirder still, there are numbers that lose associativity as well, such as the octonions. Octonions come up in physics so these are not only of abstract mathematical interest.


    I intend to go back to your first post on the subject and respond in detail to your comments on mathematical structuralism and category theory, so I hope you can be a little patient. I want to start at the chronological beginning of your posts on the subject and I can't do that tonight.

    In short though, mathematical structuralism is more subtle than just listing arithmetic properties like associativity.The kinds of properties that they use in category theory are ... well, they're kind of weird and nonintuitive when you first see them. The structural relations they have in mind are various types of universal mapping properties. It's hard to do justice to what this means in a simplified format but I might take a run at it once I get into responding in detail to your earlier post on structuralism.
  • tom
    1.3k
    There is an advantage to this approach. Mathematicians are not constrained by a definition of number, which allows them to discover new types of numbers all the time.fishfry

    If you don't like the fact that numbers are defined in terms of set theory, and further properties deduced from there, I guess you won't like the fact that numbers are also defined in terms of field axioms.

    And no, new types of number are not "discovered all the time".
  • fishfry
    408
    If you don't like the fact that numbers are defined in terms of set theorytom

    I certainly can't understand how you would have gotten that impression. Many specific types of numbers are defined within set theory. But there is no general definition of what a number is in set theory or in any other foundational approach.

    I guess you won't like the fact that numbers are also defined in terms of field axioms.tom

    Not a bad idea. But the field axioms don't say anything about numbers. It's true that many types of numbers satisfy the field axioms, such at the rationals, the reals, the complex numbers, the integers mod p, and all the finite fields of the form p^n.

    However, the set of rational functions in one variable satisfies the field axioms, but rational functions are not numbers. Rational functions are quotients of polynomials. It's not hard to show that they can be added, subtracted, and multiplied. It's a standard, somewhat nontrivial exercise to show they can be divided. So the field axioms aren't sufficient to define what we mean by a number.

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

    And no, new types of number are not "discovered all the time".tom

    The quaternions (discovered in 1843 by Hamilton), transfinite ordinals and cardinals (Cantor 1874-1890's), the p-adics (Hensel, 1897), and the hyperreals (Hewitt, 1948) are a few examples that come to mind. These are very recent developments in the history of math. People didn't used to believe in zero, negative numbers, rational numbers, real numbers, or complex numbers. Each time someone discovers a new type of number, mathematicians have to expand their own ideas about what constitutes a number.
  • apokrisis
    2.9k
    Good idea and a very natural attempt; but arithmetic properties aren't sufficient. Weirder still, there are numbers that lose associativity as well, such as the octonions.fishfry

    Wasn't that my point. You can still be a numerical object even if you don't qualify for the full checklist of arithmetic properties. So the octonions lack a particular property. But they still count as part of the algebra family which takes numbers as their relational objects.

    It probably helps that we can see why octonions lack this further constraint. We get the feeling they would express this property if only they could, as that is the general direction they are headed. But their own nature prevents fulfilling that goal. Therefore they qualify to be part of the family even if they can't tick every last box of some ideal definition.

    Do you know much about Bourbaki structuralism - the three mother structures of algebra, topology and order - or category theory structuralism?

    As I've been saying, you seem to want a definition founded on the mathematical objects rather than the mathematical relations or structures. But that just seems an antiquated notion.

    And what is true in philosophy of maths is true of metaphysics generally. Nature is becoming understood in terms of its global structure rather than its local atoms.

    That has been the recurring sticking point in any discussion we've had. You just presume the correctness of a reductionist or atomistic metaphysics. You then seem to have no understanding of the alternative view that is that of the structuralist, process philosopher or systems scientist.

    In short though, mathematical structuralism is more subtle than just listing arithmetic properties like associativity.The kinds of properties that they use in category theory are ... well, they're kind of weird and nonintuitive when you first see them. The structural relations they have in mind are various types of universal mapping properties. It's hard to do justice to what this means in a simplified format but I might take a run at it once I get into responding in detail to your earlier post on structuralism.fishfry

    Maybe just focus on that. Structures are the objects now. Then morphism is how structures have a relational structure that allows acts of mapping.

    So the essential property that founds the whole business is closure or symmetry. Hey, just like physics!

    (And then I should add that the fundamental question becomes how could closure emerge? What constrains the openness that seems the alternative?)
  • fishfry
    408
    you seem to want a definitionapokrisis

    I don't want a definition. I merely pointed out that there isn't one in math. You agree with this by now, yes?
  • apokrisis
    2.9k
    Don’t be a dick and misquote me.

    I said you seem to think that a definition would be in terms of the mathematical objects involved, and not the structure of relations needed to produce them via a holistic system of constraints.

    A reply that makes some contact with the relevant philosophy of maths would be appreciated if of course no longer expected.
  • fishfry
    408


    I did not write the quote you attributed to me. What is your attitude problem?

    I stated originally that there is no general definition of number in math. Nobody has provided a counterexample and you now seem to agree. That's all I said. I actually can't imagine why you are going on about structuralism, which has nothing to do with what I said.

    I'm not making any point about philosophy. I'm making a statement about math. There is no general definition of number in math. This is uncontroversial and widely known. You are going off on wild tangents that don't bear on what I said and that don't falsify what I said. If you choose category theory as your foundation, there's still no general definition of number.

    Perhaps you would consider starting a thread on mathematical structuralism. It's an interesting topic. It has nothing to do with what I said, which is that there is no general definition of number in math.
  • apokrisis
    2.9k
    I did not write the quote you attributed to me. What is your attitude problem?fishfry

    I get a bad attitude pretty fast when someone like you plays cute with a quote. If you leave off the important part of what my sentence said, that is flat out misrepresentation. Expect a swift kick in the arse.

    Nobody has provided a counterexample and you now seem to agree.fishfry

    Keep trying the same trick. You are looking worse and worse.

    If you choose category theory as your foundation, there's still no general definition of number.fishfry

    Your inability to discuss the foundations of maths is noted.
  • fishfry
    408
    Your inability to discuss the foundations of maths is noted.apokrisis

    Your complete misunderstanding and lack of comprehension of category theory and mathematical structuralism was evident to several other posters the last time we discussed this. I was hoping in the limited time I have each day to post here that I would gradually work through your earlier posts in this thread and help you sort out some of your ideas. But you are simply too rude and annoying for me to bother any more.

    I'm done responding to your posts on this site. It would be for the best if you'd simply stop responding to me. Regardless I will no longer respond to you.
  • apokrisis
    2.9k
    Your complete misunderstanding and lack of comprehension of category theory and mathematical structuralism was evident to several other posters the last time we discussed this.fishfry

    Here we go. You and your circle of imaginary friends.

    I'm done responding to your posts on this site.fishfry

    Funny. I was waiting for you to start.
  • cruffyd
    3
    Glad to be on this forum.

    Forgive any apparent crude or unintelligible thoughts. As to the question of whether numbers exist:

    Who can question whether number exists in the mind? Whether they have concreteness is simply to say, 'do they exist in reality as they exist in mind?' Obviously not. Temporal construction is such that inequality defines its nature. Equality, on the other hand, can only be outside of temporality. One might say: 'equality can only exist eternally.'

    The question then arises, 'what is the nature of number?' Conjecturally, one might say, number is a series of equal values (quantity). Hence, Pythagoras' and other ancient mathematicians' inclination to render number as equal, whole values. If this is an accurate description of number, then it follows, the concept of number is tied to the idea of a 'unity' value (unit measure).

    The question can then be asked, 'where does the concept of 'unity' come from? Again, conjecturally speaking, unity may only be understood as an eternal concept. So, the question of whether number exists, is tied to the answer to the question of whether eternity exists.
  • apokrisis
    2.9k
    The question then arises, 'what is the nature of number?' Conjecturally, one might say, number is a series of equal values (quantity). Hence, Pythagoras' and other ancient mathematicians' inclination to render number as equal, whole values. If this is an accurate description of number, then it follows, the concept of number is tied to the idea of a 'unity' value (unit measure).cruffyd

    Yeah. The historical view is a good way to get at it. There was a reason why the Greeks were so horrified by the notion of an irrational number. That very reaction betrays the underlying belief about what a definition might be.

    And so we have "oneness" as the central object of arithmetic. And we have "a dimensionless point" as the central object of geometry.

    The Greeks were discovering what the maximally invariant mathematical objects looked like - the ones that had irreducible identity despite all possible operations that might attempt to change that in one of the mathematical families.

    Having established the highest symmetry identity operations, then maths could progress by identifying and relaxing the various constraints that ensured the existence of these ideal objects - the 1 and the point.

    Geometry could go non-euclidean and topological. Numbers could loosen to include negatives and irrationals.

    Bourbaki's talk about the three mother structures makes me wonder where order structure fits in to the Ancient Greek story. I guess Aristotle's work on the logic of hierarchies - the [genus [species]] relation - does describe what is the most primitive notion of set theory.

    Temporal construction is such that inequality defines its nature. Equality, on the other hand, can only be outside of temporality. One might say: 'equality can only exist eternally.'cruffyd

    I get what you mean but in that direction can lie hard Platonism. However the altermative I prefer is another long story.
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