### What's in a number?

Posted:

**Mon May 25, 2009 12:28 am UTC**I've seen "what is the definition of a real number", but what about just "a number"? How does one define the concept of a number? I'll post my idea a bit later.

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Posted: **Mon May 25, 2009 12:28 am UTC**

I've seen "what is the definition of a real number", but what about just "a number"? How does one define the concept of a number? I'll post my idea a bit later.

Posted: **Mon May 25, 2009 12:40 am UTC**

A number is simply a quantity, amount, or value (excluding any units). I guess...

Posted: **Mon May 25, 2009 12:46 am UTC**

But, in that case, how is a complex number a number. it is not an amount. In that sense it is completely arbitrary, which doesn't seem right.

Posted: **Mon May 25, 2009 12:58 am UTC**

I had this worked out a half hour ago, now I can't remember what it was... I was just thinking: "an abstraction relating to real or abstract quantities." (my other one was better)

Complex numbers are quantities in a sense that they are vector sums of two real numbers: one on the Real line and another on the Imaginary line. They are not ordered, as the Real or Rationals, for example, but they are quantities.

Complex numbers are quantities in a sense that they are vector sums of two real numbers: one on the Real line and another on the Imaginary line. They are not ordered, as the Real or Rationals, for example, but they are quantities.

Posted: **Mon May 25, 2009 1:57 am UTC**

oh, i was thinking in a physical sense. You wouldn't say something like, "Mary has 2+3i apples..." it wouldn't make much sense. It would be cool if we lived in a universe in which it did, though.

Posted: **Mon May 25, 2009 1:59 am UTC**

Engineers use complex numbers to represent voltages all the time, so that's not a valid reason to discredit complex numbers. You could make the same claim about negative numbers or real numbers.

Anyway, this isn't a very useful question. The word number encompasses a lot of related constructions that are used for different purposes. The history of the expansion of our definition of "number" has been related to the need to solve certain problems, and our current need for tools to solve certain problems is now so large that there is no reason to use the word "number" to restrict them.

I'd like you to answer these questions, which will help you decide what problems you want the word "number" to solve:

- Do you consider the p-adic numbers to be numbers?

- Do you consider the octonions to be numbers?

- Do you consider polynomials to be numbers?

- Do you consider elements of a field to be numbers? A commutative ring? A ring?

Anyway, this isn't a very useful question. The word number encompasses a lot of related constructions that are used for different purposes. The history of the expansion of our definition of "number" has been related to the need to solve certain problems, and our current need for tools to solve certain problems is now so large that there is no reason to use the word "number" to restrict them.

I'd like you to answer these questions, which will help you decide what problems you want the word "number" to solve:

- Do you consider the p-adic numbers to be numbers?

- Do you consider the octonions to be numbers?

- Do you consider polynomials to be numbers?

- Do you consider elements of a field to be numbers? A commutative ring? A ring?

Posted: **Mon May 25, 2009 2:14 am UTC**

Hmm... with the concept of "number" being such an abstract term, i wonder at the likelihood that an alien race would develop the same mathematics as we have done. Is math truly universal?

Posted: **Mon May 25, 2009 2:19 am UTC**

Brwagur wrote:i wonder at the likelihood that an alien race would develop the same mathematics as we have done.

Zero. Mathematics is fundamentally a tool: we invent the mathematics that is relevant to us, even if that relevance takes decades to be fully understood. An alien race with fundamentally different priorities from us will invent fundamentally different mathematics. Perhaps it's reasonable to suspect that a sufficiently intelligent species consisting of distinct organisms will invent the natural numbers as a way to count groups of themselves, but I don't think it's reasonable to suspect that alien forms of life will consist of distinct organisms. What if an intelligent species consists of a colony of extremely complex cells in an aquatic environment? Counting would be much less important than understanding the geometry of their environment; they'd probably invent differential equations before defining integers. And an even more alien species might invent even less recognizable mathematics.

andrewxc wrote:"an abstraction relating to real or abstract quantities." (my other one was better)

That only changes the question to "what is a quantity"?

Posted: **Mon May 25, 2009 6:32 am UTC**

I feel a "number" is more of a meta-mathematical term than it is a pue mathematical concept. As such, I think we can get away with a less precise definition. I'd suggest: "A Meta-Mathematical Description for elements of commonly used Algebraic Structures (such as, for Example the Natural Numbers or the Real Numbers [Which can be defined purely on their properties, without any relation to their frequent use as "numbers]) that are, or at least were in their origin or construction, closely related to the concept of meassuring quantities in the physical world or other concepts already labeled as numbers."

May need some more work, but i really don't feel that a Mathematician needs to have a precise definition of a number.

May need some more work, but i really don't feel that a Mathematician needs to have a precise definition of a number.

Posted: **Mon May 25, 2009 7:09 am UTC**

LLCoolDave wrote:that are, or at least were in their origin or construction, closely related to the concept of meassuring quantities in the physical world or other concepts already labeled as numbers.

That still doesn't answer the question of what a quantity is. But to the extent it makes sense, this definition boils down to "a number is whatever mathematicians call a number," which is a perfectly fine definition. They say mathematics is what mathematicians do.

LLCoolDave wrote:i really don't feel that a Mathematician needs to have a precise definition of a number.

No disagreement there. That's why we have precise definitions of rings, fields, and algebras instead.

Posted: **Mon May 25, 2009 4:48 pm UTC**

A curmudgeon (like me) objects to anything other than the naturals being called "numbers". Whether or not zero is a number is a matter of debate.

I think most practical-minded mathematicians are happy with calling C "numbers". As soon as you move to more general constructions and throwing out axioms, they start gradually objecting to use of the term "number".

Real answer: it doesn't matter. "number" is (usually) not used as a mathematically precise term by mathematicians, so it doesn't much matter what exactly it means. There are exceptions to this of course, but when people are using it in a mathematically precise fashion, they will tell you what they are taking it to mean ahead of time.

I think most practical-minded mathematicians are happy with calling C "numbers". As soon as you move to more general constructions and throwing out axioms, they start gradually objecting to use of the term "number".

Real answer: it doesn't matter. "number" is (usually) not used as a mathematically precise term by mathematicians, so it doesn't much matter what exactly it means. There are exceptions to this of course, but when people are using it in a mathematically precise fashion, they will tell you what they are taking it to mean ahead of time.

Posted: **Mon May 25, 2009 5:24 pm UTC**

t0rajir0u wrote: "a number is whatever mathematicians call a number"

This summarizes my position. Generally, I feel debating semantics is not a useful exercise for anyone. If you're looking for a set of properties that a certain object should have if and only if it is to be called a "number", best of luck.

As long as everyone knows what everyone else it talking about, we could replace the word "number" with "pancakes" and nothing would change. Well. I might get hungrier when doing math. I do love pancakes.

Posted: **Mon May 25, 2009 5:34 pm UTC**

A number is some ring that contains the natural numbers as a sub ring?

... That doesn't work -- because the integers modulo 2 really should be numbers.

... That doesn't work -- because the integers modulo 2 really should be numbers.

Posted: **Mon May 25, 2009 5:58 pm UTC**

An ad-hoc designation given to elements of various algebraic structures?

Posted: **Mon May 25, 2009 6:05 pm UTC**

Yakk wrote:A number is some ring that contains the natural numbers as a sub ring?

... That doesn't work -- because the integers modulo 2 really should be numbers.

And the naturals aren't a ring.

Posted: **Mon May 25, 2009 6:33 pm UTC**

Frankly, if a collection of objects has an addition, that's quite nice, a multiplication, that is still niceish but perhaps not as nice as the addition, and they distribute over each other in the nice way most of the time, then I'd call them numbers. And invite them to morning tea, because they're so damn nice.

Posted: **Mon May 25, 2009 7:02 pm UTC**

auteur52 wrote:Yakk wrote:A number is some ring that contains the natural numbers as a sub ring?

... That doesn't work -- because the integers modulo 2 really should be numbers.

And the naturals aren't a ring.

Bah, sub semi ring.

Jesting, the issue is that we don't call matrices numbers. And matrices have nice addition, not quite as nice multiplication, and they distribute nicely most of the time.

(Ok, we do call some sub-sets of the matrices numbers under certain angles and views.)

Posted: **Mon May 25, 2009 7:03 pm UTC**

auteur52 wrote:And the naturals aren't a ring.

Clearly you're just defining multiplication and addition incorrectly!

Posted: **Mon May 25, 2009 7:06 pm UTC**

jestingrabbit wrote:Frankly, if a collection of objects has an addition, that's quite nice, a multiplication, that is still niceish but perhaps not as nice as the addition, and they distribute over each other in the nice way most of the time, then I'd call them numbers. And invite them to morning tea, because they're so damn nice.

This sounds nice, but I'm still not really comfortable with calling continuous functions, endomorphisms, polynomials, and matrices "numbers". However, I'm also in the camp that what we call them doesn't matter at all.

Posted: **Mon May 25, 2009 7:24 pm UTC**

auteur52 wrote:jestingrabbit wrote:Frankly, if a collection of objects has an addition, that's quite nice, a multiplication, that is still niceish but perhaps not as nice as the addition, and they distribute over each other in the nice way most of the time, then I'd call them numbers. And invite them to morning tea, because they're so damn nice.

This sounds nice, but I'm still not really comfortable with calling continuous functions, endomorphisms, polynomials, and matrices "numbers". However, I'm also in the camp that what we call them doesn't matter at all.

Okay, but we end up using them very much like numbers. Take matrices. The complex numbers can be realised as a sub algebra of the two by two real matrices, as can the dual numbers, split complex numbers and probably a couple that I'm not thinking of at present. So... we have so many matrices fitting into number sets, we solve matrix equations, we evaluate the exponential of matrices etc etc. We might not call them numbers, but they have a lot a lot of number-like properties if you ask me. If you were to ask me to explain matrices really vaguely, I'd probably come up with a statement like "matrices are numbers made up of other numbers" or something silly like that. Now sure, its not like this is a great description, but is the line really "nice arithmetic properties and simple enough to be written down with at most two real constants, unless we're talking about quaternions or octonions", or are we just drawing a very arbitrary line in some very arbitrary sand?

I am also of the "who cares?" persuasion, but wanted to have my say regardless.

Posted: **Mon May 25, 2009 7:50 pm UTC**

t0rajir0u wrote:Brwagur wrote:i wonder at the likelihood that an alien race would develop the same mathematics as we have done.

Zero.

Perhaps, but I suspect there would be significant overlap. Certainly we wouldn't expect them to come up with the same definitions and theorems in the same order that we have, but it seems reasonable that natural definitions would come up again. For example, I wouldn't be at all surprised if we encountered an alien race and discovered that they had classified the finite simple groups, and knew what the monster group was. (I would be very surprised if their name for it translated as "monster group".)

Posted: **Mon May 25, 2009 8:30 pm UTC**

jestingrabbit wrote:Okay, but we end up using them very much like numbers. Take matrices. The complex numbers can be realised as a sub algebra of the two by two real matrices, as can the dual numbers, split complex numbers and probably a couple that I'm not thinking of at present.

Yes, but the mathematical analogy is just as strong in the other direction: you wouldn't call the ring of functions on an algebraic variety "numbers," but if you take the analogy between that ring and the integers seriously enough you get Spec.

skeptical scientist wrote:For example, I wouldn't be at all surprised if we encountered an alien race and discovered that they had classified the finite simple groups, and knew what the monster group was.

Really? I think this is a very interesting question. Our approach to mathematics is based on reductionism: break things up into their constituent parts and study each part separately. That's why the representation theory of finite groups is so appealing: representations can be completely understood in terms of irreducible representations. But this is manifestly not true of the relationship between groups and simple groups. I could see an alien race that approached mathematics from a non-reductionist perspective: rather than studying finite simple groups, perhaps they would prove classification results about group actions instead.

I also think it's possible that, from the perspective of the classification of finite simple groups, the definition of a group is slightly off: perhaps a different definition would reveal the sporadic groups to be part of another infinite family. There's already some work in this direction (placing the sporadic groups into families that also involve the classical groups), but it doesn't seem to involve changing definition. Does anyone know anything about the classification of "finite simple Hopf algebras," for example? (I do agree that the definition of a group is natural enough: any alien species that talks about automorphisms in any category would be aware of groups sooner or later.)

Posted: **Mon May 25, 2009 9:04 pm UTC**

any alien race that starts counting things will probably discover groups. the question is, is mathematics the way it is as an artifact of the way our minds work, or is there something fundamental about mathematical operations. it's an interesting consideration.

Posted: **Tue May 26, 2009 12:49 am UTC**

Well, it seems likely that we won't know for a very long time. Here's hoping, though.

P.S. I'd be more than slightly amused if they showed up and demanded to know the value of R(5,5).

P.S. I'd be more than slightly amused if they showed up and demanded to know the value of R(5,5).

Posted: **Tue May 26, 2009 1:27 am UTC**

It seems that the mind is something one must be non-sober to think about... thinking about math being an artificial construct is making my teeth hurt.

Posted: **Tue May 26, 2009 1:46 am UTC**

Less so if they want R(6,6).skeptical scientist wrote:P.S. I'd be more than slightly amused if they showed up and demanded to know the value of R(5,5).

I think this discussion is fascinating. I'd think that the math any aliens would find important is something we'd at least recognize as being similar to a field we already know about. But maybe t0r disagrees with this. Would an independent mathematical community have devised topics that don't fit into our hierarchy of algebra, analysis, combinatorics, topology, logic, etc?

Posted: **Tue May 26, 2009 2:38 am UTC**

Number is a radial category. The natural numbers are the central elements of the category (esp. so for the small natural numbers) and everything that we refer to as numbers are connected to these because they share some properties of the naturals, with the centrality of each type of number indicated by the order in which we learn them in school. Unlike most radial categories, the connections between different types of number are less metaphorical but perhaps more abstract.

Posted: **Tue May 26, 2009 3:49 am UTC**

Buttons wrote:Would an independent mathematical community have devised topics that don't fit into our hierarchy of algebra, analysis, combinatorics, topology, logic, etc?

Maybe it's just that I think the translation process would be less trivial than this discussion is treating it as. Suppose that alien mathematicians devise a notion of field of sets that is not quite the definition of a topology, nor the definition of a measurable space, nor the definition of a Boolean algebra, but that is used for a subtly different purpose. Suppose further that these aliens don't state things in terms of axioms; perhaps they do mathematics by metaphor instead. How easy would it be to recognize that what they're talking about is a kind of field of sets?

(One of the bigger points I am trying to make is that even if mathematical truths themselves are "universal" (whatever that means), mathematical method may not be.)

Edit: I just realized I didn't actually answer your question. I think it's perfectly reasonable to hypothesize that alien mathematicians would come up with interesting combinatorial and algebraic structures for certain purposes that we wouldn't think of but that would relate to things that we might recognize; I'm thinking of less mainstream things like block designs, which include Steiner systems, which include finite projective planes, which include projective planes over finite fields. So it may be that they never decide that finite fields are interesting but they study block designs or something like them intensively, so in some way their theorems (if they even think in terms of theorems!) say things about finite fields. Again, I don't think the translation process would be trivial.

Posted: **Tue May 26, 2009 8:49 am UTC**

a number is an axiomatic definition that we find useful. You can't really break it down any further than that. If you try to define it in terms of quantity or amount you'll be playing a self referencing game that can't be won. What's a number? An amount. What's an amount? a number. I'm rather reminded of the time a rebellious youth smoked a joint and said wow man, what is math? there are structural properties about the universe that seem to be simply givens. Remarkable how it tends to work out though eh?

Posted: **Tue May 26, 2009 7:24 pm UTC**

Haeche wrote:a number is an axiomatic definition that we find useful.

But this definition is too vague to be correct. We make plenty of axiomatic definitions that nobody would call numbers.

Posted: **Tue May 26, 2009 7:47 pm UTC**

This was one of the first questions I was asked upon starting my degree. The answer I gave was that a number was the cardinality of a set, to which my tutor responded "Very good, but what about pi". He then went on to tell me that I could finish the degree and without ever needing to know the answer.

However if you know how to define any real number using sets, then it is fairly simple to define a complex number in terms of sets, just by defining it in terms of real numbers.

ie. A complex number is a member of the Feild of ordered pairs of real numbers upon which the following binary operations are defined:

(a,b)+(c,d)=(a+b,c+d)

(a,b)(c,d)=(a.b-c.d,a.c+b.d)

I believe this definition is sufficient as all other properties of complex numbers can be derived from these.

Basically numbers, real or otherwise are considered to be sets (assuming you are using ZF or similar).

Ed. Replaced "Groups" with "Feild" throughout.

However if you know how to define any real number using sets, then it is fairly simple to define a complex number in terms of sets, just by defining it in terms of real numbers.

ie. A complex number is a member of the Feild of ordered pairs of real numbers upon which the following binary operations are defined:

(a,b)+(c,d)=(a+b,c+d)

(a,b)(c,d)=(a.b-c.d,a.c+b.d)

I believe this definition is sufficient as all other properties of complex numbers can be derived from these.

Basically numbers, real or otherwise are considered to be sets (assuming you are using ZF or similar).

Ed. Replaced "Groups" with "Feild" throughout.

Posted: **Tue May 26, 2009 8:19 pm UTC**

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.

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

Posted: **Tue May 26, 2009 11:47 pm UTC**

I think the answer to the question whether an alien race would devise mathematics in a human-comprehensible way relies much on the question whether they have to tell us about it!

I think that it is more likely that an (in some sense) intelligent race of aliens would have co-developed with artificial selection than with our mechanics-based technology (which is probably needed to get across galaxies), so they will have less of an interest in physics and thus less of a similar approach to math than one might think. Especially I think it is highly unlikely that they would have our super-formal approach - which I think you need to have if you want to be precise enough to build sufficient spaceships.

I think that it is more likely that an (in some sense) intelligent race of aliens would have co-developed with artificial selection than with our mechanics-based technology (which is probably needed to get across galaxies), so they will have less of an interest in physics and thus less of a similar approach to math than one might think. Especially I think it is highly unlikely that they would have our super-formal approach - which I think you need to have if you want to be precise enough to build sufficient spaceships.

Posted: **Wed May 27, 2009 12:47 am UTC**

Frimble wrote:The answer I gave was that a number was the cardinality of a set, to which my tutor responded "Very good, but what about pi". He then went on to tell me that I could finish the degree and without ever needing to know the answer.

There is actually some fascinating research in this area dealing with two generalizations of cardinality, one of which applies to groupoids, and the other of which applies to certain topological spaces. For example, the groupoid cardinality of the category of finite sets and bijections is [imath]e[/imath]. It is, I believe, still an open problem to find a "natural" category whose groupoid cardinality is [imath]\pi[/imath].

Frimble wrote:Basically numbers, real or otherwise are considered to be sets (assuming you are using ZF or similar).

This isn't a definition, either. You aren't making a distinction between sets we consider to be numbers and sets we don't consider to be numbers.

Posted: **Wed May 27, 2009 1:06 am UTC**

Brwagur wrote:oh, i was thinking in a physical sense. You wouldn't say something like, "Mary has 2+3i apples..." it wouldn't make much sense. It would be cool if we lived in a universe in which it did, though.

It does make sense in the physical sense, you need to take more courses in E&M .

Posted: **Wed May 27, 2009 1:26 am UTC**

t0rajir0u wrote:Haeche wrote:a number is an axiomatic definition that we find useful.

But this definition is too vague to be correct. We make plenty of axiomatic definitions that nobody would call numbers.

it's a specific axiomatic definition that we find useful.

Posted: **Wed May 27, 2009 3:30 am UTC**

LLCoolDave wrote:I feel a "number" is more of a meta-mathematical term than it is a pue mathematical concept. As such, I think we can get away with a less precise definition. I'd suggest: "A Meta-Mathematical Description for elements of commonly used Algebraic Structures (such as, for Example the Natural Numbers or the Real Numbers [Which can be defined purely on their properties, without any relation to their frequent use as "numbers]) that are, or at least were in their origin or construction, closely related to the concept of meassuring quantities in the physical world or other concepts already labeled as numbers."

May need some more work, but i really don't feel that a Mathematician needs to have a precise definition of a number.

That's more or less how I feel. I've always thought that numbers were objects derived from an axiomatic system in such a way that they allow us to express a shared abstraction of cognitive processes regarding perceived quantitative data.

Posted: **Wed May 27, 2009 9:51 am UTC**

t0rajir0u wrote:Frimble wrote:Basically numbers, real or otherwise are considered to be sets (assuming you are using ZF or similar).

This isn't a definition, either. You aren't making a distinction between sets we consider to be numbers and sets we don't consider to be numbers.

That was not supposed to be a definition. I was just saying that any number can be defined in terms of sets.

Posted: **Wed May 27, 2009 10:24 am UTC**

Off-topic but on the subject of definitions: is it just me that gets (very slightly) annoyed whenever a professor uses "if" instead of "iff" in a definition? e.g. "H<=G is a p-sylow group if blah blah blah..."

...or am I just too pedantic for my own good?

...or am I just too pedantic for my own good?

Posted: **Wed May 27, 2009 2:18 pm UTC**

majikthise wrote:Off-topic but on the subject of definitions: is it just me that gets (very slightly) annoyed whenever a professor uses "if" instead of "iff" in a definition? e.g. "H<=G is a p-sylow group if blah blah blah..."

...or am I just too pedantic for my own good?

Definitions are always "if and only if". Mathematicians don't like unnecessary repetition of words.