## What's in a number?

For the discussion of math. Duh.

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majikthise
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### Re: What's in a number?

You'd never leave off the second "f" when stating a theorem (if it was indeed a two way implication), so why do it for definitions?
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### Re: What's in a number?

A number is an element of a completion of the naturals.

Will that do?
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### Re: What's in a number?

majikthise wrote:You'd never leave off the second "f" when stating a theorem (if it was indeed a two way implication), so why do it for definitions?

Because of what auteur said: definitions are always if and only if, so the extra "f" isn't needed. This is not the case with theorems.

A number is an element of a completion of the naturals.

Will that do?

What do you mean by completion? Intuitively, the reals are more "complete" than the naturals, but do you want to include complex numbers? Quaternions? Do you just mean "generalization"? If you do, then we get the same problems as we saw earlier: matrices are generalizations of the natural numbers, but most people don't think of them as numbers.
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### Re: What's in a number?

Yakk wrote:Yes, you can model complex numbers using ZF(C) sets.

That doesn't tell you what is a number, and what isn't a number.

Well it does tell us what is a complex number and what is not. We also have a definition for real numbers, rational numbers and natural numbers. Unless I have missed sum that would seem to be sufficient. In fact we could define "a number" as being "a member of a field" to cover these... unless we need to include quaternions, or similar.
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NathanielJ
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### Re: What's in a number?

Frimble wrote:In fact we could define "a number" as being "a member of a field" to cover these... unless we need to include quaternions, or similar.

To include quaternions, we could just define a number to be a member of a normed division algebra. That grabs octonions too.
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auteur52
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### Re: What's in a number?

I have been thinking about this a little, and the closest meta-definition I could come up with (which is far from perfect) is that a number is a mathematical object with little structure in and of itself. I think this would rule out calling functions "numbers", as well as polynomials. However, just how much structure a real number or an integer has is very debatable, but I think what I said is close to encapsulating what makes the integers, rationals, reals, complex numbers, p-adics, quaternions, octonions, ordinals, etc. all "numbers".

Dew_N_O
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### Re: What's in a number?

If we could define numbers and quantities in a categorical, absolute sense (A is a number and B is not) without resorting to numbers and quantities, that sounds important. But for this very reason, if these definitions are tautological, I don't think we could gain anything important from them. Frankly I think this all is a bit absurd to try and define these concepts so rigorously, because all you'd be doing (at best) is defining them in terms of other principles and words, and it's hard to conceive something deeper than quantity that preserves all the information of quantity. Quantity is a robust and efficient beast and seems to defy any sort of reductions. What's more, even IF we managed to reduce it in a nontrivial way, these new principles which subsume it would just demand a new explanation.

It's more interesting to consider the quality of numbers from a more fluid standpoint, and this discussion has been doing a good job of that. What makes something pass from non-quantitative to quantitative? What makes something go from not being a number to being a number? Well...for that latter question, it seems that we like numbers to be commutative beasts. They aren't just elements of an arbitrary ring. Those that do not qualify are odd, obscure things like matrices and quaternions and transformations. They should have exactly one dimension. Multi-dimensional structures somehow seem to have more than quantity to them. They have a *direction* to them, which is somehow qualitative, an order to the numbers. You can't say you have M sheep, where M is a matrix. That isn't merely absurd, it's *unimaginable and ambiguous and removed from pure quantity*. That adjective isn't well-defined for taking sets to sets. How is the matrix acting on the sheep? It's not merely counting them, it's applying something to them and seeing the result. It's structuring the set of sheep. You won't be left with sheep, you will be left with something different from your set of sheep if you have M sheep. M sheep is not any amount of sheep. Complex numbers seem to straddle the line, because well-defined they certainly are, and they do seem to be counting *something*. I don't know. This is where our rigorous definition of numbers hits the wall of social definitions. A lot of us took calculus and pre-calc, and a whole lot more, in which the complex numbers were explained as just a bigger field than the reals, which solved polynomial equations and helped us with mathematical frameworks. If we hadn't had that, complex numbers might seem a lot more like structured objects (i.e. matrices/functions/integrals) than numbers.

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### Re: What's in a number?

auteur52 wrote:I have been thinking about this a little, and the closest meta-definition I could come up with (which is far from perfect) is that a number is a mathematical object with little structure in and of itself. I think this would rule out calling functions "numbers", as well as polynomials. However, just how much structure a real number or an integer has is very debatable, but I think what I said is close to encapsulating what makes the integers, rationals, reals, complex numbers, p-adics, quaternions, octonions, ordinals, etc. all "numbers".

Given that a real number is a map from Q+ to Q with very specific properties... (or from N to Q as a sequence, or N to 2 as a cut).

Footnote:
Q+->Q: "epsilon-delta" real.
N->Q: "cauchy sequence" real.
Q->2: "cut" real
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### Re: What's in a number?

auteur52 wrote:a number is a mathematical object with little structure in and of itself.

The integers are full of structure. The standard mathematical object claimed to have little structure is a graph, and I don't think anybody would call graphs numbers (unless they were working with homotopy cardinality or groupoid cardinality, again).

Anyway, I'm going to stand by "a number is whatever mathematicians call a number," since it covers not only the constructions we currently call numbers but anything that we might later call a number. Mathematicians in the 1700s couldn't have predicted the existence of the quaternions, and mathematicians in the (insert appropriate time frame) weren't thinking about the p-adics.

auteur52
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### Re: What's in a number?

t0rajir0u wrote:
auteur52 wrote:a number is a mathematical object with little structure in and of itself.

The integers are full of structure.

Well I meant not the structure of Z, for instance, but the structure of one of its constituents. I think in some intuitive, but very hard to pin-down, way the number "5" has less structure than a function. I definitely admit that this is all on very shaky ground, but it's hard to say much about a specific number's structure without using the group/ring/field/algebra structure. A function has a little more going on inside just itself, apart from its place inside a function space.

But I still agree we shouldn't be trying to define what a "number" is.

LLCoolDave
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### Re: What's in a number?

Note that his claim was that an Integer doesn't have much structure. He never claimed that the set of all Integers had little structure. Depending on the way you define them, that claim may not subjectively hold true. Although I don't yet agree with the statement, it certainly feels like it captures the spirit of our intuitive definition of a number best from all the suggestions so far.

edahl
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### Re: What's in a number?

How about a collection of objects satisfying a given set of operations with a given set of properties (ie. the field axioms), and perhaps some ordering axioms? When you're not counting sheep, the real interest is in my opinion what properties the "numbers" satisfy under certain operations. Some arbitrary count of objects doesn't tell you much without some relation to some greater structure or concept. A number n is just a unique identifier for the concept of "a collection of n objects." I guess you can also define it as the concept of some distance on the real line, in which case you have a set of numbers ideal for measuring.

edahl
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### Re: What's in a number?

majikthise wrote:You'd never leave off the second "f" when stating a theorem (if it was indeed a two way implication), so why do it for definitions?

It's true that the "standard" is to let if mean iff in definitions. It may be annoying at first, but when you know it it's just the way it is. It saves space. Especially if you don't like writing 'iff', and write 'if and only if' instead.

auteur52
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### Re: What's in a number?

edahl wrote:It's true that the "standard" is to let if mean iff in definitions. It may be annoying at first, but when you know it it's just the way it is. It saves space. Especially if you don't like writing 'iff', and write 'if and only if' instead.

Which is what you should do if you're writing something that's going to be published. "Iff" is not a word, it is basically just short hand to use on chalkboards.

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### Re: What's in a number?

auteur52 wrote:I have been thinking about this a little, and the closest meta-definition I could come up with (which is far from perfect) is that a number is a mathematical object with little structure in and of itself. I think this would rule out calling functions "numbers", as well as polynomials. However, just how much structure a real number or an integer has is very debatable, but I think what I said is close to encapsulating what makes the integers, rationals, reals, complex numbers, p-adics, quaternions, octonions, ordinals, etc. all "numbers".

This is an interesting approach, saying, if I may rephrase, that a number should be simple. Seeing it as an attempt to kill the monsters that my defn brought up for torajirou, it certainly does that. But it evades a fact, that numbers must have an interaction amongst themselves to have their nature.

Consider, for example, the number one. Some number ones are such when you add them to themselves, you get 0, whereas other number ones are such that, no matter how many times you add them to themselves, you never get 0. So, unless you specify where the number lives, and how it interacts, you haven't specified the object. You could certainly argue that there are ways to specify the number that describe where it lives (specifying one as a dedekind cut, for instance) but that would be necessarily ambiguous in some cases I expect ie certain field extensions, and the numbers therein, would be hard to describe unambiguously across all field extensions.
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### Re: What's in a number?

Yakk wrote:Given that a real number is a map from Q+ to Q with very specific properties... (or from N to Q as a sequence....

Well, an equivalence class of such things. But in any case, it's not really. When we talk about algebras of functions, the function-ness of the elements of the algebra tend to keep their importance, even when we are thinking only about algebraic properties (one reason for this is because thinking this way often makes it very easy to characterize subalgebras). On the other hand, with the real numbers, we almost never think of them as being cuts, or equivalence classes of cauchy sequences, because that way of thinking is not usually terribly useful. Frequently people do the construction once, to prove that there exists a complete ordered field, and then forget all about it and treat real numbers as objects without internal structure.
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### Re: What's in a number?

t0rajir0u wrote:
auteur52 wrote:a number is a mathematical object with little structure in and of itself.

The integers are full of structure. The standard mathematical object claimed to have little structure is a graph, and I don't think anybody would call graphs numbers (unless they were working with homotopy cardinality or groupoid cardinality, again).

That's not what he meant. The ring of integers is full of structure, but any particular integer is not. In fact, the only structural properties one would usually ascribe to an integer are things like its prime decomposition, which it gets from being a member of this ring, and not anything intrinsic to itself. Compare this to a member of a different ring, say the ring of entire functions, which will have a lot of structural properties unrelated to its presence in that ring. Perhaps this is the reason we call integers (and members of many other rings) numbers, but we would be unlikely to call a function a number, even when we are thinking about it as an element of a ring.
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### Re: What's in a number?

The claim that definitions should be stated as "if and only if" has always struck me as confused. A definition is not a logical argument. It's...a definition. If you're not happy with, "A set with a binary operation is a group if blah blah blah" don't make it "if and only if". Make it "By a group, we mean a set with a binary operation such that blah blah blah". That makes it clear that if someone comes up to you and says, "Well, how about this thing. It's a set with a binary operation but doesn't satisfy blah blah blah. Is it a group?", the correct response is not, "No, it's if and only if." The correct response is, "I just told you what the word 'group' means. Why are you asking me this?"

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### Re: What's in a number?

Hmm.

"You have addition and multiplication, with some reasonably nice properties (including a(b+c) = ab+ac).
You have a one. 1*b = b*1 = a.
Numbers are things that are defined in relation to some 1."

This doesn't deal with "even integers" as numbers. But, in a sense, they are defined in relation to the absent 1.

---

Grp(X) if B(X) is B(X) -> Grp(X).

theorem(X) if Grp(X) is Grp(X) -> theorem(X)

If we have C(X) -> Grp(X) (or Grp(X) if C(X)), then we are perfectly happen: theorem(X) holds for all theorems that work under groups.

This means you can have a loose definition of Grp(X), and things still work.

Of course, the problem is if instead of proving theorem(X) if Grp(X), you prove theorem(X) if B(X) (which is very natural).

This leaves open the possibility that the thing you are talking about (Grp(X)) is tighter than what you defined it to be (B(X)).

It also opens you up for multiple possible things you know to be true about things that are groups. You could have
Grp(X) if A(X).
Grp(X) if B(X).
Grp(X) if C(X).
and have three different theories of what a Grp is.

This isn't that much like how we use Grp(X) if B(X), I'll admit. But it sort of describes what the clause could be read as.
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### Re: What's in a number?

Yakk wrote:"You have addition and multiplication, with some reasonably nice properties (including a(b+c) = ab+ac).
You have a one. 1*b = b*1 = a.
Numbers are things that are defined in relation to some 1."

What do you mean "in relation to"? Because it sounds like you are just talking about being an element of a ring (or perhaps a semiring, or some other ring generalization), and I don't think every such object qualifies as a number, because members of rings of functions are generally thought of as functions but not numbers. I can't see anyone calling the square root function a "number".
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### Re: What's in a number?

Define the function sqrt by it's relationship to 1? (uniquely)

Numbers from C are hard to describe in relationship to 1, I'll admit.

(And while you can define the squareroot function pretty easily in the ring where (f*g)(x) = f(x)*g(x) and (f+g)(x) = f(x)+g(x), then identifying a sub-ring of these functions as "being the real numbers" isn't that alien.)
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t0rajir0u
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### Re: What's in a number?

I'm going to stand by my claim that the distinction between function and number isn't important: it's actually important to think of functions as numbers (as far as proving theorems about a function space as if you were in a "smaller" metric space) and to think about numbers as functions (as far as developing modern algebraic geometry). For those who aren't "in the know," the philosophy is that a number is the same thing as a function on the primes that sends [imath]n[/imath] to the value of [imath]n \bmod p[/imath]. The values of this "function" uniquely determine [imath]n[/imath] by the Chinese Remainder Theorem.

Again, mathematics is a tool. You work with the perspective most suited to your particular problem, and depending on the problem you may want to think of a lot of functions as numbers or a lot of numbers as functions.

Actually, I'm going to go ahead and use that as a tentative definition: elements of some set [imath]S[/imath] are being viewed as numbers (and note that this is no longer an intrinsic property of any particular set) if the emphasis is on binary operations [imath]S \times S \to S[/imath] instead of on evaluation maps [imath]S \times A \to B[/imath]. How's that sound?

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### Re: What's in a number?

Sure -- there is a correspondence between the function "_1 mod p" and "p". Saying that "_1 mod p" "is" the prime p isn't that radical.

But what about functions that don't naturally or neatly map to 'traditional numbers'? Calling them numbers seems less natural (as it is harder to find a correspondence to traditional numbers).
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### Re: What's in a number?

Yakk wrote:Saying that "_1 mod p" "is" the prime p isn't that radical.

That's not quite what the analogy is saying. There is a correspondence in algebraic geometry between ideals in polynomial rings and varieties, and the point of the analogy is that the standard construction only allows you to deal with certain types of commutative rings and certain ideals, which is unnatural, so you "complete" the correspondence by thinking of ideals in arbitrary commutative rings as geometric objects. This is radical: the set of "points" of a ring become its set of prime ideals, and its "subvarieties" are its maximal ideals, but there's also a lot of topological data added in that I'm not qualified to talk about.

A toy example: the prime ideals in [imath]\mathbb{C}[x][/imath] are the linear functions [imath]x - a, a \in \mathbb{C}[/imath], which are in one-to-one correspondence with the points of [imath]\mathbb{C}[/imath]. One thinks of [imath]\mathbb{C}[x][/imath] as the ring of functions on [imath]\mathbb{C}[/imath] in the following way: a polynomial [imath]p(x)[/imath] is evaluated at a point [imath]a[/imath] and returns a value [imath]p(a)[/imath]. But this is equal to the value of [imath]p(x) \bmod (x - a)[/imath]. (Conclusion: Lagrange interpolation is a special case of the Chinese Remainder Theorem.) In other words, finding an equivalence class modulo a prime ideal is the same thing as "evaluation at a point." So one thinks of [imath]n \bmod p[/imath] as the function [imath]n[/imath] evaluated at the point [imath]p[/imath], which I think is a great change in perspective; it's a geometric perspective on a number-theoretic object, and such changes in perspective are invaluable.

(There's an even stranger change in perspective making the rounds these days; apparently the integers behave like a 3-manifold and primes behave like knots.)

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### Re: What's in a number?

Not being a mathematician, I think that a number is a concept that historically was invented/discovered to be able to discern between "one apple" and "two apples". As time has passed the concept has evolved with the theories that use it, and now we have complex numbers and even stranger things, but they still refer in one way or another to amounts, although not always amounts that have meaningful physical representations.

Also, I am not sure if the English language discerns between number and value, eg. $10_3 = 11_2 = 3_{10}$ denote the same values, but are they the same numbers? If they are not, I would say that a number is the written or spoken representation (using some set of rules) of a value, and then we could start arguing what a value is.

If I make an analogy into programming, I believe that one can classify two concepts, operations and data, where operations take operations and|or data as input and generate operations and|or data as output, whereas data is. When moving from programming to math operations->functions data->numbers.
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### Re: What's in a number?

insom wrote:Not being a mathematician, I think that a number is a concept that historically was invented/discovered to be able to discern between "one apple" and "two apples". As time has passed the concept has evolved with the theories that use it, and now we have complex numbers and even stranger things, but they still refer in one way or another to amounts, although not always amounts that have meaningful physical representations.

Also, I am not sure if the English language discerns between number and value, eg. $10_3 = 11_2 = 3_{10}$ denote the same values, but are they the same numbers? If they are not, I would say that a number is the written or spoken representation (using some set of rules) of a value, and then we could start arguing what a value is.

If I make an analogy into programming, I believe that one can classify two concepts, operations and data, where operations take operations and|or data as input and generate operations and|or data as output, whereas data is. When moving from programming to math operations->functions data->numbers.

I'm not sure the comparison to programming is a good one. Programming takes place in the real world, with programs being executed on real, physical computers of some sort. The distinction between operation and data makes sense because data is what exists at a certain time and operations are things that happen as time elapses. This matches out perception of the physical world surrounding us. However, no such distinction exists in the world of mathematics. Although we speak of "Applying a function/operator/whatever to a number/function/whatever", there's no actual process occuring. There is no number that exists in some sense or another which is changed to something else by a process of being passed through a function. This is one of the major misconceptions a lot of people seem to have when dealing with infinity and limits, and a lot of it stems from intuition about processes occuring and the vocaublary we devised for it. "As x goes to 0" or other similar expressions we heard and or used in school (and certainly practicing more formal mathematics as well!) certainly sound like a process, but there is no actual number that gets closer to 0 in some wicked pattern. Similarly, infinite sums can get quite confusing when first encountered. How can a summation that goes on and on and on have a precise value? Certainly, all instances of summation we encounter in the physical world take time and are a process, but the mathematical world has no time-dimension! Essentially, everything in Math-World is just data by your analogy, because everything just IS in Mathematics.

Of course, take this as a grain of opinion, I am well aware that there are different schools of though on the topic of philosophie of mathematics.

Also, I think most everyone would consider $10_3, 11_2, 3_{10}$ to be the same number. The concept of a number (although still not being clearly defined!) is independant of its represantation. See the classical "issue" of 0.999... = 1
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### Re: What's in a number?

insom wrote:If I make an analogy into programming, I believe that one can classify two concepts, operations and data, where operations take operations and|or data as input and generate operations and|or data as output, whereas data is. When moving from programming to math operations->functions data->numbers.

If you're going to make an analogy to programming then mathematics behaves more like Lisp or Haskell, where (as I understand it) operations are also data. Again, mathematicians frequently change perspective to make their lives easier, and a standard change in perspective is to think of operations as data (that is, to think of processes as objects).

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### Re: What's in a number?

I did not mean the analogy to include what exactly is 'done' when executing an operation, rather something more general than that, but I get your point. On the topic of operations being data, yes functions can indeed be data, especially in more functional programming languages, but the statement "data are functions" I think is less plausible. At least in any other sense than that it upon evaluation returns itself. Maybe one could say that a function/operation is a concept where the result of an evaluation depends on a number of factors, but a number always evaluates to itself, a constant.
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### Re: What's in a number?

t0rajir0u wrote:If you're going to make an analogy to programming then mathematics behaves more like Lisp or Haskell, where (as I understand it) operations are also data. Again, mathematicians frequently change perspective to make their lives easier, and a standard change in perspective is to think of operations as data (that is, to think of processes as objects).

This is actually one of the major themes in the video lectures for The Structure and Interpretation of Computer Programs (SICP). You can switch between considering everything data, or everything process at a whim or mix and match as you choose depending on the application. Programming languages are merely a metaphor wrapped in syntax. They can go to both extremes where like Java everything is an object, to Lisp where everything is a process and even despite the attempt to keep them tidy and distinct the lines blur.

t0rajir0u's view is one I hold myself, that mathematical objects are nothing more then a collection of properties that can be considered in a variety of contexts. It's artificial to hold steadfast to one or the other as it would be to insist that the Rubin Vase has only faces, or only vase. Only when you've seen both can you fully understand the figure in it's entirety, and I think the same is true for math. From a personal perspective this wasn't something I really saw until I was exposed to abstract algebra and the concept of isomorphism and group actions. Is a vector anything more than an action given a name and treated as an object? When I add two numbers am I churning them through a mill to create a third or describing a motion across a space? Is Euclidean geometry the study of lines, points and circles or elements of a number field?

As for the original question posed in this thread, I'd have to argue that mathematically there is no "number", only algebraic entities with well defined properties. That being said, I think historically and colloquially the only numbers are the Real numbers. I asked around my office to get a feel for popular perception with questions like "Is Pi a number?", "If I give you two numbers can you put them in order?", "Can you add/subtract/multiply/divide numbers?" and the results were pretty clear cut. "numbers" as popularly used are elements of an ordered field containing irrationals. There's also the fact that claiming the only "real" numbers are the Real numbers just seems to justify the nomenclature and just makes good sense.
"Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration." - Abraham Lincoln

t0rajir0u
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### Re: What's in a number?

insom wrote:but the statement "data are functions" I think is less plausible.

In computer science, maybe, but not in algebraic geometry! Which is perhaps where the analogy to computer science starts to break down, although apparently Haskell is a great language for talking about category theory (or perhaps it's the other way around).

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### Re: What's in a number?

An object a of type A is a function that takes (A->B)->B by evaluation using a as an argument.

Ie, if f:A->B (f is a function that takes A to B), then an element a of A can be viewed as a function Fa, which takes objects of type A->B, defined as follows:

Fa(f) = f(a) for all objects of type (A->B).

I can express this trick in most computer languages that don't utterly suck, not just Haskell.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

MHD
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### Re: What's in a number?

A number... It depends on what reaml you use it in.

If you count stones on the beach the natural integers [imath]\mathbb{N}[/imath] is enough.

If you have a money system based on shiny stones you can have a debt but the unit is still indivisible, hence you need the integers [imath]\mathbb{Z}[/imath].

If you are cutting a cake fractions are incredibly usefull, thus you need the rationals [imath]\mathbb{Q}[/imath].

If you need to double the area of a square, or double the volume of a cube you need the real numbers [imath]\mathbb{R}[/imath].

If you are calculating electrical oscilliation it is a lot easier to use complex numbers [imath]\mathbb{C}[/imath].

A number, I would say, "Is an abstrac representation of amount/quantity".
EvanED wrote:be aware that when most people say "regular expression" they really mean "something that is almost, but not quite, entirely unlike a regular expression"

t0rajir0u
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### Re: What's in a number?

MHD wrote:A number, I would say, "Is an abstrac representation of amount/quantity".

Someone already gave this definition, and I already pointed out the problem with it: it only defers the question to what a "quantity" is.

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### Re: What's in a number?

t0rajir0u wrote:it only defers the question to what a "quantity" is.

It's magic.
EvanED wrote:be aware that when most people say "regular expression" they really mean "something that is almost, but not quite, entirely unlike a regular expression"

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### Re: What's in a number?

Number is countable, an amount, quantity and with value. I think, that's how number is defined.