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I was reading linear vector spaces. When doing some exercise to prove some statements based on the properties defined for linear vector spaces, i suddenly noticed, outside the things defined, i'm using a common notion without proof. This notion also surfaced when i was studying Group theory. After giving a thought i come to the conclusion that i've used it in all systems which are modeled after euclid. I mean first i give some postulates. Then i derive statements from those postulates. Euclid also has it as "common notion" :

If equals are added to equals, the wholes are equal.

So i started wondering that if anybody challenged this or attempted to build a system without this.

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Doesn't this just say that "addition is a function"? –  Carl Mummert Jul 24 '12 at 1:14
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It is probably not exactly Euclid's axiom you're using. Euclid doesn't define "equal", but it looks as if he's using "equal" to mean "has the same measure as", which is different from what the modern "=" sign signifies, namely "is the same mathematical object as". If Euclid says that triangle ABC equals triangle ABQ, what he means is that the two triangls have the same area, not that they are the same triangle!

Viewed this way, Euclid's second common notion is part of a set of axioms for defining the properties of measure -- in 20th century terms we might instead have said that length, area, angle measure and so forth are finitely additive and (because the whole is greater than its part) always positive. That gives us nontrivial information, despite the fact that it sounds fairly empty.

On the other hand, what you're using in linear algebra is probably just the rule that for arbitrary expressions $a$, $b$, $c$, and $d$, if $a=b$ and $c=d$ then $a+c=b+d$. That hasn't necessarily to do with any concept of measure -- what it says that if we have already established that $a$ and $b$ are different ways to describe the same thing, then $a+c$ and $b+c$ are also different ways to describe the same thing. And then vice versa for $c$ and $d$. This is not a particular property of addition; indeed the whole point is that the addition operation does not know whether we write $a+c$ or $b+c$; it sees only the mathematical object that the expressions $a$ and $b$ both denote.

The same thing holds for every other operation when we're talking about equality as the "=" concept. If $a=b$ we can also conclude that $ac=bc$, or $\sqrt a=\sqrt b$ (if that's defined at all), or $f(a)=f(b)$ for every function $f$. Again, this sounds extremely banal, but notice that the same thing does not hold for Euclid's "equals". For example, it is not true in Euclid's language that if equal shapes are intersected with equal shapes, the intersections are equal (never mind here that "intersection" is not a word Euclid would have used) -- because Euclid's "equals" doesn't tell us anything about where in the places the shapes are located..

From a formal logic point of view, the rule that $a=b$ implies $f(a)=f(b)$ for all possible $f$ is just part of a common convention about which relations we consider worthy of writing with a "=" sign in the first place. We can and do reason about any number of properties that don't respect this rule; we just don't write them with a "=". (In logic, that is. Sometimes other disciplines make exceptions, which are then regularly decried as "abuses of notation").

The concept of an equivalence relation abstracts some but not all properties of "=" -- in particular just because $\sim$ is an equivalence relation doesn't mean that we can reason from $a\sim b$ to $f(a)\sim f(b)$. For example, $a\sim b$ might mean that the vectors $a$ and $b$ have the same length, which is a perfectly fine equivalence relation, but that doesn't mean that $a+c$ and $b+c$ will also have the same length.

Once we decide to consider a particular $\sim$ relation, it of course becomes very interesting to see whether a particular $f$ happens to make this rule true, but there's no universal terminology for this relation between $\sim$ and $f$. Often one will speak of $f$ as an operation on equivalence classes rather than on individual elements, and the question then becomes whether $f$ is "well-defined" at all.

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If this were not the case, how could we say that the things were equal in a sense meaningful for our purposes?

In different areas of mathematics, we have different notions of equality that generally mean 'the same as far as everything we're currently concerned with'. For example if we have two Riemann surfaces, if we were studying them from the point of view of topology we might call them equal if they were homeomorphic (or maybe homotopy equivalent), from the point of view of Differential Geometry we would consider them equal if they were diffeomorphic to each other, whereas from the point of view of complex geometry they would need to be biholomorphic. We may not be interested in the structure at all, and consider them equal if their underlying sets had the same cardinality.

If we had a notion of equality that did not guarantee equivalence for all things we were interested in, it would not be a good notion of equality to study those phenomenon.

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So its comes from the need of having a notion of "equality" which makes sense? –  Aftnix Jul 23 '12 at 21:03
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