• apokrisis
    4.5k
    Numbers do number-like things. What are the important things left unsaid?
  • fishfry
    469
    Numbers do number-like things. What are the important things left unsaid?apokrisis

    Saying that a number is anything that's number-like is a circular definition. No better than the poster above who said that a quantity is anything that's quantitative. You are defining a thing in terms of itself. It's not a definition.

    Even worse, being "number-like" is neither necessary nor sufficient for something to be considered a number! Some examples:

    * Matrices can be added, subtracted, multiplied, and sometimes divided. In fact the set of nxn matrices for fixed n forms a ring, an important algebraic structure. But matrices are not regarded as numbers.

    * The set of permutations on a given finite set forms a group. An example would be the six permutations of the set {a,b,c}. Geometrically this is the group of symetries of an equalateral triangle.

    Permutations can be combined via the operations of composition: you do one permutation followed by another. With this operation, the set of permutations is a group. You can multiply and divide. But nobody ever calls a permutation a number.

    * If we have a pair or an arbitrary collection of algebraic objects such as groups, rings, fields, vector spaces, or modules, we can form their direct sum and their direct product; and in the case of modules, their tensor product. These sums and products obey various algebraic identities and are considered part of the study of algebra.

    But nobody ever calls groups, rings, fields, etc. "numbers." They're algebraic objects but they are not numbers.

    These examples show that there are things that are "number-like" that are not numbers; and things that are not numbers that can nevertheless be added and multiplied.

    I make two assertions:

    1) In math, there is no general definition of number. Of course there are perfectly clear definitions of certain classes or types of number: integers, reals, quaternions, p-adics, transfinite ordinals, etc. But there is no general definition that says, "A number is such-and-so."

    I was surprised at the, let's say, passion of some of the responses to this tame and factual assertion. Many interesting points were raised. I'll try to respond in detail to each post in the next couple of days.

    2) It's surprisingly tricky to give a good definition of number. I hope my examples bear that out. But if they don't, I have a lot more examples!
  • apokrisis
    4.5k
    Saying that a number is anything that's number-like is a circular definition. No better than the poster above who said that a quantity is anything that's quantitative. You are defining a thing in terms of itself. It's not a definition.fishfry

    No it's not. It's defining something in terms of its relational qualities rather than in terms of its supposed essences.

    The whole point is that "number" is elusive as an "abstract object" because that is wrongly to seek some constant essential thing that is more primary than the relations that ensue. So the right way to look at it is to switch to a contextual, constraints-based, metaphysics where objects are defined in terms of structures of relations.

    But if you want to complain, write a letter to the Department of Category Theory. Let them know it is from the Department of Set Theory. That will help them make a speedy decision just where to "file" your complaint. ;)

    Matrices can be added, subtracted, multiplied, and sometimes divided. In fact the set of nxn matrices for fixed n forms a ring, an important algebraic structure. But matrices are not regarded as numbers.fishfry

    Not even complex ones?

    I was surprised at the, let's say, passion of some of the responses to this tame and factual assertion.fishfry

    Or maybe you are wrong?

    It's surprisingly tricky to give a good definition of number. I hope my examples bear that out.fishfry

    It seems curious that you can both claim numbers don't have a good definition and then so easily rule lots of things in or out as numbers.

    I wonder what criteria you use?
  • fishfry
    469
    The list of posts I need to respond to is growing faster than my available time to respond. I'm plugging along though. Like the guy writing his autobiography. He takes a day to write about each minute of his life. He gets farther and farther behind every day. Yet if he lives forever, he'll document every minute.

    Really? What is number theory about? What is Principia Mathematica about?tom

    Ok good questions. First to clarify what I'm talking about, my totally humble thesis is:

    There is no general definition of number anywhere in mathematics. Of course there are perfectly clear definitions of particular types of numbers. Integers, reals, quaternions, p-adics, transfinite ordinals, and so on. But nowhere in mathematical literature will you find anyone who ever says: "A number is defined as such and so."

    I regard this as a philosophical curiosity, worthy of discussion. What surprises me is people claiming that my thesis is factually false.

    Now to falsify my thesis, all anyone needs to do is post a reference to a math textbook or published paper where someone defines what a number is, in a way that other mathematicians have adopted (ie not specialized to that one paper).

    Post that reference and my thesis stands refuted.

    All the verbiage in the world that is NOT such a specific reference does NOT falsify my thesis.

    I trust this is perfectly clear to fairminded philosophers. I am not making any metaphysical statement about numbers. I'm saying that in the mathematical literature, there is no general definition of number.

    To falsify a universal statement it is both necessary and sufficient for you to supply a single counterexample.

    And if you DON'T, my thesis stands till you do.

    * Now you did reference two interesting topics. First, number theory at the elementary level is about the study of the integers, and sometimes just the positive integers. These two classes of numbers have perfectly good definitions. At higher levels. algebraic number theory is the study of the algebraic integers. Analytic number theory is the use of the techniques of analysis: limits, calculus, etc., to study integers or algebraic integers.

    In all of number theory, there is no definition of number.

    * Now the Principia is cool. It's one of my favorite things. In the Principia, Newton describes the fundamentals of calculus; and uses his techniques to prove that the planets and the apples on the trees obey a universal law of gravitation that can be described by a simple equation.

    Now that's cool as hell. He was one smart cookie that Isaac.

    But I can tell you for a fact that in all of the Principia there is no definition of what a number is. Why would Newton concern himself with a matter as trivial as that? He was laying out how the universe works, not quibbling about definitions.

    As an aside, Newton had worked out calculus using a symbolic formulation; but he wrote the Principia using the geometry of the ancients. He knew that he was proposing radical ideas, and he wanted to use familiar mathematics. He didn't let his new notation get in the way of the acceptance of his physical ideas.

    Another reason Newton deliberately obfuscated the Principia was, in his own words, "... to avoid being baited by little Smatterers in Mathematicks."

    I can totally relate!


    Sorry but you are bullshitting to an extraordinary level.tom

    I wish you could explain this to me. I stated that in all of math there's no official definition of a number.

    Why does this push your buttons? Why would you even doubt me? I've done the research. It's not hard to verify. I think there might be a Stackexchange thread about it.

    Why do you think I'm bullshitting about this? And really, if you think of it, why WOULD I bullshit about something like this? What's in it for me?



    You think numbers are defined in terms of quantity?tom

    No, you didn't read far enough back in the thread. Someone proposed the definition, "A number represents a quantity." I proposed the counterexample of the imaginary unit i, which does not denote a quantity in any sensible interpretation of the word. And I also asked that poster to define quantity for me. So that I would know for myself whether i could be somehow interpreted as denoting a quantity.

    So I was asking the question as part of my challenge to their proposed definition of a number.

    You understood me to be claiming that a number represents a quantity, but of course that is NOT a belief I hold. Complex numbers, p-adics, transfinite ordinals. I would never say a number represents a quantity.


    i represents the square root of -1.tom

    Agreed, though it's more precise to say that i represents "a" square root of -1, because -i is another one, and i and -i can't be distinguished. There's no sense of positive or negative in the complex numbers.


    i is a number.tom

    Yes of course. I believe in the complex numbers.

    i exists.tom

    Yes it does. It not only has existence within formal mathematics; it occurs everywhere in the real world. If you want to do quantum field theory or just turn left at the corner, you are using the number i.

    Simples.tom

    Absolutely. Since we share the same mathematical ontology, I wonder what I said to upset you.
  • tom
    1.5k
    There is no general definition of number anywhere in mathematics. Of course there are perfectly clear definitions of particular types of numbers. Integers, reals, quaternions, p-adics, transfinite ordinals, and so on. But nowhere in mathematical literature will you find anyone who ever says: "A number is defined as such and so."fishfry

    At least Frege, Russell, and Whitehead defined what a number is. There are probably several others.

    Hang on, there's even a Wikipedia page:

    https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

    Post that reference and my thesis stands refuted.fishfry

    Follow the Wikipedia links, do some Googling, you are refuted.
  • fishfry
    469
    At least Frege, Russell, and Whitehead defined what a number is. There are probably several others.tom

    Yes that's an interesting point. Philosophers and logicians have struggled to define what a number is. Mathematicians don't really care that they haven't got a precise definition. Mathematicians expend zero energy going down that rabbit hole. It's not mathematically productive.

    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.



    I have repeated the same thing several times, yet you are still misunderstanding what I'm saying.

    There are perfectly clear definitions of specific types of numbers such as naturals, integers, quaternions, etc.

    And in fact each foundational approach has its own definition. In set theory the natural numbers are defined as the von Neumann ordinals. In category theory there's a natural number object, as @apokrisis mentioned earlier.

    The page you linked to is the von Neumann definition of the finite ordinals, which has the nice advantage that it can be easily extended to transfinite ordinals.

    In category theory there's a concept called a natural number object. This defines the natural numbers structurally, as @apokrisis noted earlier. The benefit of the latter approach is that it avoids the so-called "junk theorems" of the von Neumann approach. For example in standard set theory, 2 ∈ 3 is a valid theorem. No sensible person would claim it means anything. The categorical approach gets rid of that type of problem.

    My thesis is entirely agnostic of foundational approach, a point @apokrisis does not sufficiently appreciate. There is no general definition of number in set theory, category theory, homotopy type theory, Martin-Löf type theory, intuitionist type theory, any of the various constructivist ideas, or any other foundational approach. There are many foundational approaches these days. My statement applies to all of them.

    There is no general definition in math that tells us what a number is. There are plenty of definitions of specific types of numbers. There are even (distinct but closely related) definitions of specific types of numbers in different foundations. But there is no general definition of number.
  • apokrisis
    4.5k
    My remark is entirely agnostic of foundational approach, a point apokrisis does not sufficiently appreciate.

    There is no general definition in math that tells us what a number is.
    fishfry

    Or maybe I'm just pointing out that the idea of "numbers" speaks to a family resemblance. No single definition could hope to pin down "numbers" in some exact sense. But there is a constraints-based definition, or 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.

    So you wouldn't expect a hard and fast definition of something general. As a foundational fact, all we can talk about is a family resemblance which emerges as we enforce greater and greater constraint in terms of bounding symmetries. Then in the limit, we arrive at associative division algebras.

    So the octonions are not associative but they answer to the weaker constraint of being alternate. They still deal in numbers, but a slightly less constrained version.

    The general definition is thus inherently fuzzy. The most highly constrained or specified kind of number system isn't broad enough to capture the weaker kinds that are possible with some of the constraints relaxed or even absent. And yet at some point the "general definition" is so weak that it doesn't then capture the essential operations that do define being part of that general family.

    This is a familiar issue when framing scientific law. Philosophy of science says exactly the same thing.

    So you may be agnostic about foundational approaches. But philosophy of maths can't afford to be. And I think the same understanding of the trade-off between the general and the particular applies across all the rational disciplines.

    You are simply looking for the wrong kind of "general definition". Structualism is needed because that focuses on generic relations of which numbers then can be the various possible kinds of object.
  • Akanthinos
    1k
    There is no general definition in math that tells us what a number is.fishfry

    Why wouldn't ZFC count?
  • fishfry
    469
    "Is" can be a tricky word.tim wood

    Bill Clinton made that very same argument to try to wiggle out of a sex scandal. In the end he lost his license to practice law and was impeached (but not convicted).

    You ask "what is quantity?" Quantity is the general name for an idea that is always particular, and that refers to anything that can be quantified.tim wood

    You and @apokrisis seem to feel that "a number is anything that's number-like" and "quantities is whatever can be quantified" represent valid definitions. What happens if the biologists get hold of this trick? A fish is whatever is fish-like. A cat is whatever is cat-like. A virus is whatever is virus-like. And the deepest question of all: life is whatever is life-like.

    This sounds like a fast path to meaninglessness to me.

    Now I think you are confused in that you think a definition somehow "is" what something is.tim wood

    No I'm not confused on that at all. There were cats long before a biologist said that a cat is "a small domesticated carnivorous mammal with soft fur, a short snout, and retractile claws." The thing clearly precedes its definition.

    A definition is more like a classifier. I'm working in a factory and my job is to stand at a conveyor belt and throw things into one bin or the other: cat and not-cat. A definition is a set of criteria that let me unambiguously do that. The definition lets me recognize things that are cats; and things that are not-cats. In other words I'll throw the cats into the cat bin and the not-cats into the not-cat bin with as close to 100% accuracy as possible. That's what a definition is.

    Now today my job is to identify quantities versus non-quantities. So my definition is, a quantity is anything that can be quantified. But that's no help! You've just given me a different syntactic form of the same word. You have NOT provided me with classification criteria. So "quantities are things that can be classified" is not a definition, nor is "a number is something that's number-like." You haven't told me how to sort the objects into the bins.

    Or you apparently think that the definition of number, or quantity, will tell you what these things are.tim wood

    A good definition does let me determine whether a given object is or isn't the thing in question. If an object comes down the conveyor belt and it's a small furry domestic animal etc., I know it's a cat.

    This approach or understanding - actually utilization - gets a lot of the world's work done, but it isn't remotely true. A definition is simple an agreed description, for some purpose.tim wood

    It's a strong description. It's a description that fits 100% of the things we wish to include, and none of the things we wish to exclude. If a description satisfies that criterion, it's a definition. "Quantities are things that can be quantified" is no help at all. It's a description but not a definition.

    As to i, it's the square root of -1, it's a number, and it exists (keeping in mind you probably have at best a partial idea of what "existence" means, and of what I mean by it).tim wood

    To be sure, I have no idea what you mean by existence. I would say that i exists because it exists in math according to the formal rules; and also because we see many instantiations of the i in the physical world. That latter point isn't obvious to everyone but for example anytime you make a 90 degree counterclockwise turn, you are instantiating the number i in the world. And of course i comes up in physics and engineering all the time.

    Definitions, then, are functional.tim wood

    Yes I agree with that. A definition is whatever you can write down on an index card for me that will allow me to recognize cats and numbers and quantities as they come down the conveyor belt. Functional. Good word for it.

    And if any thing is going to be discussed in terms of its definition, or any understanding of what that something is, then it's best to start with some explicit expression of that definition or understanding. That's just good navigation. And of course it's negotiable, if that's appropriate.tim wood

    To sum up, or rather to get back to basics, you claimed that numbers represent quantities. The number i represents a phase angle in electromagnetism or a quarter turn if you're in the plane. But I don't see those as quantities. So I have to ask again, what is a quantity? Are you claiming that the number i represents a quantity? That I do not agree with. I don't see it.
  • fishfry
    469
    Why wouldn't ZFC count?Akanthinos

    There's no definition of number in ZFC. In ZFC we have a definition of natural numbers, and we can make definitions of the integers, rationals, reals, complex numbers quaternions, transfinite ordinals and cardinals, hyperreals, and many other types of number.

    But there is no general definition of number. If a thing comes down the conveyor belt and I have to say if it's a number or not, of course 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. ZFC offers no help in this regard.
  • fishfry
    469
    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
    1.5k
    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
    469
    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
    4.5k
    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
    469
    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
    4.5k
    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
    469


    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
    4.5k
    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
    469
    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
    4.5k
    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
    7
    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
    4.5k
    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.
  • cruffyd
    7
    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.apokrisis

    Thanks for the mention.

    My knowledge of mathematics and its history is only rudimentary. Perhaps I slightly misunderstand the role of the Pythagoreans in relation to irrational number. It seems, by most accounts, that the Pythagoreans were not 'horrified' by the idea of irrational number, but rather of this idea becoming generally known by the 'uninitiated.' By Hipparsus' own account, Pythagoreans may have been well aware of irrational numbers and accepted their existence.

    To the extent that the idea behind whole numbers may be tied to an eternal, or extra-temporal idea, they fall into the same category as what we are calling 'irrational,' hence the question at the outset of this discussion. To the extent that a number such as pi can be concretely understood as to its function, it could be called 'rational', loosely speaking, of course.

    Again, in returning to the question of 'existence', numbers have more existence as concept than as concrete. Defining existence itself can become problematic. One school of thought mentions as many as three other types of existence apart from the concrete, or physical, one of which is conceptual.
  • tim wood
    1.4k
    Bill Clinton made that very same argument to try to wiggle out of a sex scandal. In the end he lost his license to practice law and was impeached (but not convicted).fishfry
    He made a grammatical point, and in this he was correct.

    You and apokrisis seem to feel that "a number is anything that's number-like" and "quantities is whatever can be quantified" represent valid definitions. What happens if the biologists get hold of this trick? A fish is whatever is fish-like. A cat is whatever is cat-like. A virus is whatever is virus-like. And the deepest question of all: life is whatever is life-like.fishfry
    This criticism might have some merit if that were what we were doing. But we weren't, so it doesn't.

    So my definition is, a quantity is anything that can be quantified. But that's no help!fishfry
    I should think not; keep in mind I did not offer a definition of "quantity." You asked what quantity is, and I answered. I thought it was a pretty good answer - to the question asked!

    To be sure, I have no idea what you mean by existence. I would say that i exists because it exists in math according to the formal rules; and also because we see many instantiations of the i in the physical world.fishfry
    Question: does i exist in some, or any, sense or way that is different, in any way, from the way that other numbers exist? Question: Where did you see an i?

    To sum up, or rather to get back to basics, you claimed that numbers represent quantities. The number i represents a phase angle in electromagnetism or a quarter turn if you're in the plane. But I don't see those as quantities. So I have to ask again, what is a quantity? Are you claiming that the number i represents a quantity? That I do not agree with. I don't see it.fishfry

    If i is not a number, then what is it? If numbers do not represent quantities, then what do they represent? You're free to agree or disagree with whatever you like; in this case, you might have done some research. I did. Mathematicians appear to classify i as a number.
  • fishfry
    469
    He made a grammatical point, and in this he was correct.tim wood

    My remark was intended as lighthearted. What the meaning of "is" is was very big in American popular culture during that particular scandal. This is the only time I've heard that question raised since the Bubba and Monica affair. Cigars, stains on dresses.

    If Americans want to know how we ended up with a monstrously crude man as President like Trump, I'd say the bar was set low when the American people made Bill Clinton a two term president and never held him accountable. Trump and Bubba used to be golfing buddies. You think they talked about women's rights?

    I think my interest in American politics is off topic here so I'll let it go. But when you say it depends on what the meaning of "is" is, you can hardly be aurprised that the first thing anyone would think of is Bill Clinton and the intern. It's the 20 year anniversary of that scandal right now. So it's in the air.

    This criticism might have some merit if that were what we were doing. But we weren't, so it doesn't.tim wood

    Ok. So when you say that a quantity is that which can be quantified, you are NOT saying that a cat is what which can be cat-like.

    I confess to not understanding why anyone would regard this as a sensible response. But I'm sure that's more due to my philosophical ignorance. But if I'm ignorant, this would be a point where you could educate me. When you say, "A quantity is that which can be quantified," what actual information are you imparting? To me it just likes you changed the form of a word without adding meaning.

    I should think not; keep in mind I did not offer a definition of "quantity." You asked what quantity is, and I answered. I thought it was a pretty good answer - to the question asked!tim wood

    We definitely disagree and I am curious to understand your reasoning. You said that a quantity is that which can be quantified. I don't recognize that as the answer to any question I asked. I'm sure the communication problems are all on my side, but I'd like to bridge them if that's possible.

    Question: does i exist in some, or any, sense or way that is different, in any way, from the way that other numbers exist? Question: Where did you see an i?tim wood

    My university training is in mathematics, although my post-university career involved following math only at an amateur level online. But I absolutely regard i as a number. To me the number i is as concrete as the number 6. It just refers to something different than what 6 does.

    6, you see, does generally represent a quantity. Six ducks in a row, six eggs in half a dozen, six bullet items in your PowerPoint slide. The number 6 is instantiated in everyone's every day experience all their life.

    Now the number i, as it turn out, is every bit as pervasive and a normal part of our daily lives. However people don't recognize this, because the number i is taught very poorly in high schools around the world.

    Forget that crap about "the square root of -1," which always sounds like bullshit because they just got through telling you that there is no square root of -1.

    Think instead of i being a gadget that keeps track of how many counterclockwise turns of 90 degrees you make. Say you start facing east. You then turn north. Call that i. Then you turn west. You are now facing directly opposite the way you started. In other words ... i^2 = -1, and this notation is simply an expression of something very simple. If we make two quarter turns to the left, we are now facing in the exact opposite direction of where we started.

    Now one more turn is -i, and one more turn aft that is ... 1.We're facing east and we just discuvered that i^4 = i^0 = 1.

    So i is a number. but it is not a quantity. What it is, is an instruction to make a quarter turn left. That's what numbers can sometimes be. Representations of geometric transformations.

    A general complex number is z = a + bi where a and b are real. An alternate and more insightful notation is polar representation. If z is a complex number then we can write z = re^(it) in complex exponential form, there t is the angle made by the line segement between the origin and z, and the positive x-axis.

    In trig form this is the same as saying z = r(cos it, sin it). This rotates the oringinal vecor through and angle of t, and it scales it by a factor of r.

    If you plug in t = pi/2 and r = 1you get the special case of z = i. In fact the case r = 1 is very important because as t goes from 0 to 2pi you get all the points on the unit circle.

    So every time you turn left -- at a traffic intersection, on a street corner if you're walking. or if you're just standing in your living room spinning around conterclockswise: You are instantiating the complex number i. Every time you turn through an angle of t, you end up at a particular point on the unit circle.

    That's not all. The number i is an essential part of modern physics and engineering. Having a symbolism for something being 90 degrees out of phase is very handy. So i can be defined in formal math, and it comes up in physics. It's a number, and it is instantiated in the world.

    So YES, i is a number. But NO, i is not a quantity. The number 6 is a quantity. It's 6 of something. But i represents no quantity. I represents a quarter turn in the plane. And geometric rotations and scalings of the plane happen to have very algebraic properties.

    You don't even need a magic "square root of -1" to do this. There's a particular subset of 2x2 matrices whose entries are real numbers. They are an isomorphic copy of the complex numbers. So nobody has to believe in anything "imaginary." If you believe in the real numbers, then you'll agree to believe in 4-tuples of numbers arranged in a 2x2 array, along with the usual array operations of matrix addition and multiplication.

    One more example. The area of a circle with radius 1 is pi, right? Now is that a quantity? A quantity of what?

    We determine the area in multivariable calculus by defining the two-dimensional Riemann sum. We fill up the circle with little squares and count the squares. Then we fill it in with smaller squares. At the end of that limiting process is the area of the circle, which comes out to pi.

    But there's no quantity anymore. At each step there was a finite quantity of little squares. But in the limit, there are NOT infinitely many infinitesimal squares. Calculus abandoned that approach. Instead we just work with the limits. So at the end of this process, pi is a number but it's not a quantity of anything.


    If i is not a number, then what is it?tim wood

    I'm a math guy. Of course i is a number. I mentioned this earlier in a reply to @Tom, if you read back a few posts you might find it. I believe in the mathematical reality of all mathematical structures. [Note that this is not to say I believe in their physical reality. Only that if i can construct something in math, the it's a mathematical object and has mathematical existence. I make no general claims about the world].

    If numbers do not represent quantities, then what do they represent?tim wood

    Well now THAT is the good question!! In math, nobody bothers to ask the question because it's a question of philosophy and not math.

    In philosophy, we're seeing that it's damned hard to pin down what a number is. And it's fun to try. Or at least it SHOULD be fun to try. When it becomes less than fun I become less inclined to play.

    Clearly SOME numbers represent quantities. Other numbers represent scaling and rotations in the plane. Ordinal numbers represent order types. Cardinals DO represent quantity!.See we even have two different notions of transfinite numbers, one that represents quantity (cardinals) and one that represents order (ordinals).

    We have familiar numbers like pi where we're hard pressed to say what quantity of anything that represents. Pi is defined as a ratio, it's defined as an infinite series, it's defined as the smallest positive zero of the sine function, which we can define via an infinite series so that there's no geometry involved.

    Some numbers represent quantities and others don't. So it's a tricky thing to accurately express what a number must be in general to be considered a number. Every rule anyone thinks of has lots of exceptions.

    If i is not a number, then what is it?
    You're free to agree or disagree with whatever you like; in this case, you might have done some research. I did. Mathematicians appear to classify i as a number.
    tim wood

    I absolutely agree that i is a number. But it is not a quantity. You said it is a quantity. I want to know by what criteria to you call i a quantity. And it's wholly inadequate to say that i is a quantity because it can be quantified. Any fairminded philosopher must see this.
  • fdrake
    1.5k


    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 numbers. Even though it doesn't represent the same kind of quantity as scalars, there there are two associated magnitudes when thought of in terms of rotation and scaling of a point's position in the complex plane. You probably already know this.

    Another good example, which I believe you brought up, is that the integers mod p where p is prime form a field under multiplication mod p and addition mod p- the algebraic structure (with 0 removed wherever it's mathematically appropriate) which is most similar to folk intuitions of how numbers should work. But the idea of division as multiplication by multiplicative inverse mod p is nothing like the idea of division present in intuitions for fractions (rationals) and reals (rationals + irrationals).

    The reals (excluding weird stuff about 0) under multiplication and addition in the usual sense satisfy modern intuitions about what it means to be a number. When those intuitions are formalised, in turns out that there are other structures which aren't commensurate with folk intuitions that nevertheless satisfy the axiomatisation of a field inspired by those folk intuitions.

    Another wrinkle is introduced by the idea of an isomorphism. Say we have the set of numbers {0,1,2,3,4,5,6}, and addition and multiplication are equal to their remainders upon division by 7 (this is the integers mod p). We have seven elements, and since they satisfy the rules for addition and multiplication and consist of numeric symbols, it would be a stretch not to call the elements of this structure numbers.

    However, relabelling 0=A,1=B,2=C,3=D,4=E,5=F,6=G, the laws of arithmetic and the sense of equality being equality in remainder when divided by 7 produce curious statements like:

    A*x = A where x is in {A,...,G}
    A+x=x where x is in {A,...,G}
    B*F=C*G, which is equivalent to 1*5=2*6=12=5
    C^-1 = E (which states that 2*4=4*2=1)

    If presented with the {A,...,G} representation of these numbers, someone who hasn't been trained in mathematics probably will not recognise the manipulations as equivalent to manipulations of integers modulo p. Are they still numbers? Mathematically they're equivalent to numbers.

    You could do the opposite trick by labelling the symmetries of a square as numbers. Four reflection symmetries, three rotational symmetries... Further tricks by identifying usual group structures (like the integers under addition modulo 7 while ignoring multiplication) as their corresponding symmetric group representations...

    Really all this says is that 'what is a number' and 'do numbers exist' are to some extent independent from the concerns of doing mathematics. What matters is that the math functions as it's set up to.

    Another way of putting it: would a change in the ontological status of numbers change either the truth or the sense of 1+1=2? What about the ontological status of rotation groups: would what you believe about the ontological status of rotations change anything about the idea that if you rotate an object 90 degrees 4 times, you may as well have not rotated it at all? I don't think so, in either case.
  • tim wood
    1.4k
    But when you say it depends on what the meaning of "is" is,fishfry
    This is a perfect example of where we're bumping into each other. What I said was that Clinton made a grammatical point, about which point he was correct. (Whether he was correct to make the point, or any other "whether correct" you care to adduce, is a different question). I made no mention of dependence or of anything that "depended" on it.

    As to American politics, we have been in the thrall of perverted fundamentalism, well, for a long time. It is with Reagan that it takes on an overt national presence, powered by gun interests, abortion politics, rabid (ignorant, vicious) patriotism, racism, greed, self-interest, and characterized by a pathological absence of empathy and basic humanity and a lack of understanding of what America is. It is in substance mental illness, and dangerous.

    Ok. So when you say that a quantity is that which can be quantifiedfishfry
    The difficulty here is that you're thinking (I think) I defined the term. I didn't.
    You ask "what is quantity?" Quantity is the general name for an idea that is always particular, and that refers to anything that can be quantified.tim wood
    This is a proposition. Among you're options are to agree with it or disagree. Which, If you care to say? By the way, to suppose that my proposition says,
    ...that a quantity is that which can be quantified.fishfry
    is to misread the plain English of it to the extent that I must suppose that perhaps you're writing in a second or third language. Credit to you if you are! But you're misreading/misunderstanding the proposition.

    Reading the rest of your post I see that you know more math than I do. I take your point that naive/underinformed definitions can be problematic. To the point, remember this, from above?
    Fair enough, but this only means that, as you say, there is no definition. That neither stops us - anyone - from offering one, or relieves us from the obligation to try, if we want to have any sort of reasonable discussion. It might start out, "For the purposes of this discussion, in which I intend to make points a, b, and c, I provisionally define number as.... And I'll try one: number is that which has neither extension, substance, nor quality, but that expresses/represents quantity.tim wood
    Now you have me saying that i is a number (yes), that number is quantity (no), and therefore i is a quantity. What you're stumbling over is the distinction between representing something and being something. Which means our real discussion/problem is understanding each other.

    As to the quantity i: Question: is i ever an answer, in any form, to any question of how many?
  • fishfry
    469
    As to the quantity i: Question: is i ever an answer, in any form, to any question of how many?tim wood

    No. It's not. That's the point. i is a number but it's not a quantity. That's a counterexample to your idea that a number is something that is a quantity or that can be quantified. Simple as that.

    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
    469
    Really all this says is that 'what is a number' and 'do numbers exist' are to some extent independent from the concerns of doing mathematics.fdrake

    That's right. I noted that there is no general definition of number in mathematics. A well-known and true observation. For whatever reason, this simple and harmless statement triggered several people. I still don't understand why.

    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.

    But ordinals I think are the best example of numbers that absolutely can not ever be interpreted as quantities, since the same cardinal can be rearranged to represent many different ordinals.

    Number is not the same as quantity. I think that's clear.

    The reals (excluding weird stuff about 0) under multiplication and addition in the usual sense satisfy modern intuitions about what it means to be a number. When those intuitions are formalised, in turns out that there are other structures which aren't commensurate with folk intuitions that nevertheless satisfy the axiomatisation of a field inspired by those folk intuitions.fdrake

    Right. And some structures that satisfy the field axioms are most definitely NOT numbers, such as the rational functions with coefficients in a field.

    I'm not entirely sure I understood the theme or message of your post. All I'm saying is that there's no general definition of number in math; and even for logicians and philosophers, it's very difficult to pin down what a number is. I've never seen a successful definition.
  • apokrisis
    4.5k
    I noted that there is no general definition of number in mathematics. A well-known and true observation. For whatever reason, this simple and harmless statement triggered several people. I still don't understand why.fishfry

    Either people were triggered or they thought there are some good approaches worth discussing in philosophy of maths.

    Shapiro and Resnik hold that all mathematical theories, even non-algebraic ones, describe structures. This position is known as structuralism (Shapiro 1997; Resnik 1997). Structures consists of places that stand in structural relations to each other. Thus, derivatively, mathematical theories describe places or positions in structures. But they do not describe objects. The number three, for instance, will on this view not be an object but a place in the structure of the natural numbers.

    https://plato.stanford.edu/entries/philosophy-mathematics/#WhaNumCouNot
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