In school we learn about numbers through physical amounts and we take two things and put them with two other things and call it four things in total.
Is this view of numbers as amounts slightly 'old fashioned'? Before we would say 'that's two' or something like 'put two and two together', maybe 'take 2, add 2 to it', but this seems to not be correct in the way to discuss real numbers in the 'object' view.3
We seem to take an object based approach and our symbols and names like 'two' or 'x' or '2' are like proper nouns. And a mathematical statement is more like talking about an object, so in 2+2=4 we take as a statement about objects and functions taking those objects as inputs, In this way in Mathematical logic we can actually extend this language to real objects that we use proper nouns for.
We can try to define the idea of 'quantity' through number, but when we study mechanics we find that '2' can have different meanings (2kg as a mass, +2m vector position) (such as an inherrent direction or an increase) depending on the context of what is measured, so we cannot call any number any one quantity.
However for much of school I would have taken that as Take 2(of anything) and add 2 (of anything) to it and you get 4 (in total). In this manner, thinking mentally about quantities of individualn things 'two' becomes almost a 'description' and every two things can be described as 'two' (as children and many people do in infomral situations).
Personally for me, as I learnt more algebra, logic and proofs I had to overcome the idea of viewing a number like '5' as not only describing 'quantity', but more as an object that exists in our mind and we discuss in a similar vain to a particular person or 'Finland'.
This issue I think comes from language we are taught for using numbers in a practical sense and the true abstract nature of objects we use.
Is there any meaningful difference here, with the 'amount' view of number as given at school/non-scientific backgrounds, or am I simply being mis-led by informal language and a lack of care to differentiate between 'concrete' and 'abstract' objects, and the use of 'numbers' in an adjective sense? Can a number act as a label of quantity and a singular object itself?
I would really appreciate, as this question is not an easy answer, if you could suggest some further reading which links the concept of number/quantity and abstraction around the issues outlined here.