In everyday life we have a clear notion of what it means for something to be larger than something else. Usually we would evaluate something's size based on it's volume. However, is there a formal method for extending this everyday notion of size to numbers and the realm of mathematics in general?
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3The short answer is that what it means for a number to be "larger" than another is unclear, and so if you use such a term, you should specify. In the case of real numbers, we usually treat "larger" as meaning greater, with respect to the standard ordering. In other contexts, though, you should specify. – qaphla Jun 25 '15 at 20:02
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2A metric doesn't automatically induce the notion of what is bigger or smaller; that is why complex numbers are not ordered, for example :-) – Ant Jun 25 '15 at 20:04
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Generally we mean the latter, that $3 > -10$ because $3$ is numerically greater than $-10$. I don't have to get away from the Euclidean metric for this to start to complicate. Consider the points $(2,3)$ and $(3,2)$ in $\mathbb{R}^2$. Which one is larger? Or, should I say they are of the same "size" because they have the same distance from the origin?
I believe you're confusing the notions of ordering (given two elements, I can tell you which is greater) and size. Certainly in a metric space (Euclidean or not) I will have a concept of distance, but I'm not sure how you're using the word "size" to relate this to your question.
Sloan
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You're right that my use of the word size is ambiguous. But this is the root of the question. Is there any formal method for extending the everyday use of the word "size" into the realm of mathematics? – Zach L Jun 25 '15 at 20:21
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I promise I'm not trying to be pedantic, I want to get at exactly what your question is so I can do my best to answer it. In the everyday use of the word size, is $3$ bigger than $-10$ or the other way around? – Sloan Jun 25 '15 at 20:24
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On a personal level, I would consider -10 to have a larger size because it has more of an impact when used. What I mean is that if we add or subtract -10, it produces a larger change than adding or subtracting 3.
But with multiplication 0 produces much more of a change than 1, but I would still consider 1 to be "bigger". So I am struggling to define what it means for numbers to have size.
However, in everyday life, I would confidently say that an elephant is bigger than a mouse, not vice versa, and this is what I mean when I reference the everyday use of the word size.
– Zach L Jun 25 '15 at 20:37 -
Your first usage (where $-10$ is larger than $3$) relates to distance from the origin. In any metric space you have a distance and can compute distance from the origin. You have to accept though that two different items could have the same distance (for insance, $10$ and $-10$). This is very common in mathematics. The second usage (where $-10$ is smaller) relates to ordering. Some sets are totally ordered (you give me two items, I will tell you which is greater). Some sets are only partially ordered (called posets). There is also much math to read about here. I hope this helps! – Sloan Jun 25 '15 at 20:40
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On some sets there exists an ordering $<$ which satisfies
(1) $a \not < a$
(2) $a<b$ and $b<c \implies a<c$
When we say $a$ is larger $b$ than we typically mean $a>b$. It would be unusual to say larger when dealing with metric spaces, we would probably say further and as in your example say $-10$ is further from $0$ than $3$ if $m(-10,0) >m(3,0)$ where $m$ is a metric.
RowanS
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So here you have introduced a point of reference--0--with respect to which "farther" is measured. It might be argued that the issue of "size" is different in that no such point of reference is required. – ben Jun 25 '15 at 20:34
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Taking your question to the absolute level of generality, you can define any reordering of the integers that you want and then just use the usual metric on that reordered set of integers, and you will obtain a unique ordering defined by that metric (up to the direction of the ordering).
user2566092
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