I have looked at 3 books and it is clear that "internal" and "external" are two styles of defining something, I would like to know what they mean "generally" – that is very soft but it is clear to me that "internally" doing something is different from doing it "externally" even if the result is similar.
I've read ahead and motivating examples or even uses of these definitions are not given, so if you have any examples that come to mind please do share.
First it defines External direct sum using a $+$ inside a $\square$ of a finite number of vector spaces over the same field (I understand "external" might mean "not considering them as subspaces of something")
$V=V_1+…+V_n$ is defined component wise (as vectors themselves)
This is fine, I am happy with this.
Direct product next and we have a family $\funky{F}=\{v_i|i\in K\}$ the direct product is:
$\Pi_{i\in K}V_i=\{f:k\rightarrow\cup_{i\in K}V_i|f(i)\in V_i\}$
At first this seems horrible but really it's not too bad, $f$ is acting as a projection to coordinates really. This is a vector space itself and thinking about it I ask myself "How is that not the external direct sum (provided the family is finite)"
This is not what I was taught for sum take the plane $x=0$ and the plane $y=0$ in $\mathbb{R}^3$, their union is just an extruded $+$ shape but what I was taught was the sum becomes all of $\mathbb{R}^3$ because the sum was defined as $\{u+v|u\in V_1 v \in V_2\}$
But anyway, that's direct product.
External direct sum
This time it is denoted not by a $+$ in a square but by a + in a circle with "ext" in superscript, due to lack of knowledge of the name of this symbol I must describe it to you.
Anyway this sum is (on a family as above) $\{f:K\rightarrow\cup_{i\in K}V_i|f(i)\in V_i, f\text{ has finite support}\}$
finite support means $f(i)=0$ for all but a finite number of $i$
I dare not come up with examples because it's not finite.
Lastly Internal direct sums
This is written as a + with a circle around it, and it requires the following hold:
The join of the family is V, that is: $V=\sum_{i\in K}S_i$ which is the more conventional sum I assume, and independence of the family, that is:
$S_i\cap(\sum_{j\ne i}S_j)=\{0\}$
It notes that this second condition is stronger than pairwise disjoint, I cannot think of an example which shows this distinction and I'd like one.
That's all the book does on these, unless they are used far later. What is going on here? Also what do "internal" and "external" mean, I can't think of an example but I think it refers to a style of definition, I've come across and gotten used to similar things before.
The book is: Advanced Linear Algebra, Steven Roman, 3rd Edition, GTM # 135
Best Answer
The question is quite interesting, at least from an epistemological point of view. It is not easy to work out an all-comprehensive answer which is in the meantime meaningful enough in a broad context, but I will try to say something that hopefully can be understood, at least on a metamathematical level, whatever this may mean. Also, since you have tagged the question under "Category Theory", I will feel free to use that language, which, unsurprisingly, provides quite an enlightening way of interpreting the problem.
So, the main difference between an "internal" and an "external" approach may be summarised as follows. Suppose you have fixed a mathematical entity (or structure) $\mathcal{M}$ which you are interested in studying, for example a vector space $V$. For our purposes, I guess one should think to such a mathematical entity as a mathematically definable object, characterised by some defining properties (of course, this "definition" is deliberately vague).
Investigating the nature of $\mathcal{M}$ internally means to study $\mathcal{M}$ using solely the structure that $\mathcal{M}$ has in its own and the properties that the axioms defining $\mathcal{M}$ imply, as if $\mathcal{M}$ had suddenly become the only available working ground, a comprehensive universe into which doing your math. In particular, one is not allowed to refer to any other entity besides $\mathcal{M}$ or besides those that can be defined or built up in $\mathcal{M}$. For example, if $\mathcal{M}$ is your favourite vector space $V$, looking at its linear subspaces is an internal way of proceeding, because subspaces are substructures of your given entity. You can then do some operations on subspaces, such as taking intersections or internal direct sums, which are internal as long as they can be performed inside $V$ and their result still lies inside $V$.
On the other hand, an external study of $\mathcal{M}$ would try to grasp information about $\mathcal{M}$ basing on the connections that $\mathcal{M}$ may have with other entities, or, better, seeing $\mathcal{M}$ as within (or as related to) a wider landscape, a working universe to which one is allowed to link $\mathcal{M}$. For instance, once you have taken your favourite vector space $V$ and a bunch of subspaces $V_{i}$ of $V$, you may realise that such subspaces are themselves vector spaces and constitute mathematical entities on their own right, even if of a kind similar to the one of your $V$. In other words, both $V$ and the $V_{i}$ are objects of the category of vector spaces (over some field). Then, you may perform some operations on these $V_{i}$ seeing them as structures on their own, and not as substructures or $V$ (i.e. considering them as objects in the category of vector spaces and not as subobjects of $V$). For example, you may take the external direct sum, or the direct product of the $V_{i}$ and then, you may wonder if the results of such operations can be related to your $V$, giving some informations on it ($V$ may be isomorphic to the external direct sum of the $V_{i}$ or a quotient of them).
The best way to explain the two approaches comes thinking at Category Theory. Take a (locally small) category $\mathcal{C}$ (even better, an elementary topos). Then such a category is naturally related to the category $\mathbf{Set}$ of sets through the Hom functors. Now, one can study properties of $\mathcal{C}$ or even define some notions internally to $\mathcal{C}$, using only the given structure of $\mathcal{C}$, i.e. its objects, its arrows and its composition law, or externally transferring the properties and the definitions to the category of sets.
For instance, given objects $X,Y$ of $\mathcal{C}$, one can define their product $X\times Y$ either internally, requiring that $X\times Y$ is an object of $\mathcal{C}$ equipped with two morphisms in $\mathcal{C}$ satisfying the well-known universal property, or externally, requiring that the functor $$ \mathcal{C}(-, X)\times \mathcal{C}(-,Y) $$ is representable, i.e. requiring that, for all $Z\in\mathcal{C}$, $\mathcal{C}(Z,X\times Y)$ is a product in the category of sets. The same kind of double approach can be carried over to loads of notions commonly used in Category Theory, such as limits, colimits, subobject classifiers, exponentials etc.
It turns out that (at least in the abovementioned examples) the two approaches are equivalent, in the sense that an object (together with a family of arrows) in $\mathcal{C}$ satisfy a property internally if and only if it satisfies it externally.
However, the two ways of investigating the category $\mathcal{C}$ are sensibly different: the internal one makes assertions just about the structure of $\mathcal{C}$, without any further (set-theoretic) assumptions and can therefore be used to give a self-referring description of $\mathcal{C}$, as if it was the universe into which we set our play. On the other hand, the external point of view studies $\mathcal{C}$ as a structure constructed within a working, fundational environment (some kind of set-theory), into which (and dipendently to which) one develops the whole theory on $\mathcal{C}$.
The internal and the external point of view are a common feature in Category Theory (hence, in the whole Mathematics) and their interplay and connections are not only what makes the subject delightful but provide deep results (just to mention an example that springs to my mind, think about Giraud's Theorem characterizing axiomately a Grothendieck topos).