• tim wood
    678
    I think you're confusing numerals with numbers.
  • Rich
    3.2k
    As I said, numbers can take many forms. It is all how we envision them in memory.
  • tim wood
    678
    Is number a one or a many? Numerals can be whatever you want them to be. Number - I don't think so. That's why the distinction between numbers and numerals. For example, if you look at a page from a telephone book you'll see no end of numerals, but not a single number. That's one reason a definition for number is good to have before talking about them. You're talking about how you think about them; I'm asking you to say what you think they are, in such term as might facilitate discussion, or at least guide it.
  • fishfry
    469
    That's one reason a definition for number is good to have before talking about them.tim wood

    It's a philosophical curiosity that there is no definition of number in mathematics. In other words if you major in math, get a Ph.D. spend a career as a professional mathematician, you will never encounter a book or a paper that says, "A number is such and so."

    A number is pretty much anything that's number-like, and the concept is historically contingent. First numbers were only the whole numbers and the ratios. Pythagoras discovered irrational numbers. Not till the middle ages did we start regarding zero and negative numbers as numbers. Not till Cantor did we regard transfinite quantities as numbers. There are all kinds of other mathematical objects we call numbers such as the p-adics, the dual and perplex numbers, nonstandard integers and reals, and so forth.

    Yet there is not one single definition of number. It's an amorphous concept. Mathematicians "know one when they see one." I don't know if this has caught the attention of philosophers. But there is no definition of number.
  • tim wood
    678
    Yet there is not one single definition of number. It's an amorphous concept. Mathematicians "know one when they see one." I don't know if this has caught the attention of philosophers. But there is no definition of number.fishfry

    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.
  • fishfry
    469
    And I'll try one: number is that which has neither extension, substance, nor quality, but that expresses/represents quantity.tim wood

    Ok I'll play. Four questions.

    * What is quantity?

    * The imaginary unit i with i^2 = -1 ... what quantity does it represent?

    * Do you regard i as a number?

    * Does i exist?
  • apokrisis
    3.8k
    Yet there is not one single definition of number. It's an amorphous concept. Mathematicians "know one when they see one." I don't know if this has caught the attention of philosophers. But there is no definition of number.fishfry

    Probably worth mentioning that category theory and structuralism have moved past this good old set theoretic view....

    The theme of mathematical structuralism is that what matters to a mathematical theory is not the internal nature of its objects, such as its numbers, functions, sets, or points, but how those objects relate to each other. In a sense, the thesis is that mathematical objects (if there are such objects) simply have no intrinsic nature. The structuralist theme grew most notably from developments within mathematics toward the end of the nineteenth century and on through to the present, particularly, but not exclusively, in the program of providing a categorical foundation to mathematics.

    Mathematical structuralism is similar, in some ways, to functionalist views in, for example, philosophy of mind. A functional definition is, in effect, a structural one, since it, too, focuses on relations that the defined items have to each other.

    A structure is the abstract form of a system, which ignores or abstracts away from any features of the objects that do not bear on the relations. So, the natural number structure is the form common to all of the natural number systems. And this structure is the subject matter of arithmetic.

    http://www.iep.utm.edu/m-struct/

    So if it looks like algebra - it can be added, subtracted, multiplied and (perhaps) divided - then its all numbers.
  • tim wood
    678
    And I'll try one: number is that which has neither extension, substance, nor quality, but that expresses/represents quantity.
    — tim wood

    Ok I'll play. Four questions.
    * What is quantity?
    * The imaginary unit i with i^2 = -1 ... what quantity does it represent?
    * Do you regard i as a number?
    * Does i exist?
    fishfry

    "Is" can be a tricky word. 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. Now I think you are confused in that you think a definition somehow "is" what something is. Or you apparently think that the definition of number, or quantity, will tell you what these things are. 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.

    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). Definitions, then, are functional. 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.
  • tom
    1.5k
    It's a philosophical curiosity that there is no definition of number in mathematics. In other words if you major in math, get a Ph.D. spend a career as a professional mathematician, you will never encounter a book or a paper that says, "A number is such and so."fishfry

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

    Sorry but you are bullshitting to an extraordinary level.


    * What is quantity?

    * The imaginary unit i with i^2 = -1 ... what quantity does it represent?

    * Do you regard i as a number?

    * Does i exist?
    fishfry

    You think numbers are defined in terms of quantity?

    i represents the square root of -1.

    i is a number.

    i exists.

    Simples.
  • tim wood
    678
    In a sense, the thesis is that mathematical objects (if there are such objects) simply have no intrinsic nature.

    A structure is the abstract form of a system, which ignores or abstracts away from any features of the objects that do not bear on the relations.

    Sure, these work, for the people who use them - nice and functional. And if enough people agree, then whatever the problem is evaporates in a cloud of a kind of agreed ignorance. But we can still construct the question, "What is number?" It may not be a useful question that bears on the use of and utility of numbers - and in that sense it's useless, but are we all satisfied that the question is meaningless?

    Given what definitions are, I argue that for so long as alternative answers can be generated, then the question remains meaningful. Whether the answers are worth the candle is a whole other question.
  • apokrisis
    3.8k
    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
    3.8k
    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
    3.8k
    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
    758
    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.
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