well, use are defined instead

what I sort of doubt is that a list of properties shoud suffice to uniquely pin

down what the object really is. Some other  rather strange constuction could satisfy the list of properties.

with socrates there would be discussion, and I do not catch your last

sentence at all.

1. The mathematical notions ought to be determined by their properties, expressed in already known notions, which ought to be determined by previously determined notion ,etc. until we come to notions called primitive notions,cf.

https://en.wikipedia.org/wiki/Primitive_notion

The obvious impression appers, that this process is endless. Therefore this structure can be seen as inconsistent. And in fact, it is due to the final assumption, that all people understand the meta language used for the construction in the same way.  Whatever doubts appear, this axiomatic way of construction of notions works (at least, up to know) pretty well! Moreover this way helps tremandously in expressing very complex situations without sever missing the content of claims.

2. Let me start with a rethoric question: What is the true vector?  Since there is no satisfactory answer,  I am prefering vectors from the real world, instead. The notion of truth has also double meaning - the one from logic, again introduced by axioms,  and the other one treated as a value of sentences about the real world. And then we should come to the question: what is the true truth? Therefore my preference is to use the notion of true/false sentences about the real world, without any sofisticated expressions like: the value of the sentence "it will rain today" is false.

>For example we may describe vectors by stating they are as forces

This maybe where `vectors' originally came from, and it maybe a useful way to think of vectors, but it does not capture the basic structure. That is, it is useful and necessary to think more abstractly in terms of vector spaces.

>However in the second case matrices would be included, whereas in the first

Yes,  matrices of some given size can be added and multiplied by scalars, and these operations satisfy all the natural conditions you expect. One has a vector space.

There are many other examples that are not so easy to think about as 'little arrows'.

>case only true vectors.

Well, any finite dimensional  real vector space of dimension n, is non-canonically isomorphic to R^n.  You can see this by picking a basis and then constructing the isomorphism as simply giving the coordinates of a vector in the given basis.

So,  there is no clear distinction here between `vectors' and `true vectors' (in the finite dim case anyway).

>pure mathemaatics is a little too tempted in the second direction?

The amazing thing is that often a more abstract way of thinking can lead to clear results of interest in practical applications. 

My tendency is to invert the whole thing. First you have the concept of

what the object is. Then the properties it should have fall out automatically.

But maybe it works for me and not others.

> First you have the concept of

>what the object is. Then the properties it should have fall out automatically.

It can be like that in mathematics research. You have some concept that comes from examples, analogies and `gut feelings'. You then see if this makes sense and in part that involves deriving the properties - some of which you might have expected from the start.

If things don't quite work out, then you need to amend your concept. Then iterate...

Thanks all for answers

There is still another view of vectors, but involves just a property

the transformational point of view. If the vector remains the same in absolute space,

its coordinates must transform according to the transformation of the coordinate system maintaining

its magnitude..an orthogonal transformation for rotations.

Sometimes I wonder if in a given list of properties there simply does not exist

a real mathematic object with the said assumptions. In mathematics we know that this can happen.

@ Juan

The another meaning described by you as follows:

"If the vector remains the same in absolute space, its coordinates must transform according to the transformation of the coordinate system"

is developed within theory of vector fields on manifolds. The basics can be found in many books on differentiable manifolds. One of them is given at the following address

https://aetemad.iut.ac.ir/sites/aetemad.iut.ac.ir/files//files_course/william_m._boothby_an_introduction_to_differentibookfi.org_.pdf

Regards, Joachim

>its coordinates must transform according to the transformation of the coordinate...

This is one not so nice but clear way to define tensors and tensor-like objects on smooth manifolds. They all transform under some representation of the groupoid of coordinate transformations.

>Sometimes I wonder if in a given list of properties there simply does not exist

Yes, this this why in general existence theorems are needed, However, they don't always tell you how to find the objects you are looking for.