The treatment of real numbers as 'objects'

Question

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.

Answer 1

You might be interested in reading Russell's "Introduction to Mathematical Philosophy" which asserts: "The number 3 is something which all trios have in common, and which distinguishes them from other collections." That is, (natural) numbers are attributes that apply to collections. (Although one must be careful to avoid talking about the set of all sets of size 3)

This perspective has its source in Frege's "The Concept of Number" which I found a bit trickier to read, but has this excellent quote (from the translation by Michael S. Mahoney):

"The unimaginability of the content of a word is no reason, then, to deny it any meaning or to exclude it from usage. That we are nevertheless inclined to do so is probably owing to the fact that we consider words individually and ask about their meaning [in isolation], for which we then adopt a mental picture. Thus a word for which we are lacking a corresponding inner picture will seem to have no content. However, we must always consider a complete sentence. Only in [the context of] the latter do the words really have a meaning. The inner pictures which somehow sway before us (in reading the sentence) need not correspond to the logical components of the judgment. It is enough if the sentence as a whole has a sense; by means of this its parts also receive their content."

Frege says this to shed light on concepts like infinitesimals, but I personally see it as a good perspective for not concerning ourselves with the precise meaning of numbers alone, but just with the meaning of sentences with numbers in them.

Answer 2

I picture numberlines as an abstraction of relationships across space. Discussed here: The Unreasonable Ineffectiveness of Mathematics in most sciences

We begin with simple abstractions, and create more complex structures of abstraction based on analogy and metaphor: Relationship between real quantities and numbers For instance imaginary numbers arise from considering geometrical relationships across rotations, then turn out to be very helpful for keeping track of phases of waves. It must be noted many people objected to and resisted imaginary numbers. Just as people object to whether infinities are meaningfully real. I would argue it is about salience landscapes for organising experiences, and the coherence of a structure relating ideas, which makes the issue of whether mathematical ideas are real a category error.

I would describe mathematics as a Language Game, or rather a type of set of those. That's a term from Wittgenstein, where he described how difficult it is to concisely and generally define the word 'game' and yet we have a strong feeling we will know one when we see it. Our use of abstractions is like this, we make useful groupings, that also have fuzzy edges to them, which allows for innovation in usage.

Things like the meaning of zero, how we treat 1/0, how we use indices, and the square root of minus one, were not simply discovered, they were defined and the implications of definitions explored, then the wider community agreed to definitions in the basis of which are powerful. Have a look at the Wikipedia page on 'undefined' in mathematics; very often useful inferences for real-life things being modelled can be drawn from behaviour close to an undefined point, even though the mathematics fails in specific places. Very often those points tell us something in our model is wrong, like the appearance of Singularities in blackhole maths tells us Relativity is not a complete enough description for inside the Event Horizon. In different circumstances depending on what is modelled, we treat 'undefined' in different ways, eg a tachyonic field tells us spontenous symmetry breaking has happened.

The area of thinking about higher dimensions, explicitly pushes is to contemplate things beyond what we can experience directly. I argue here mathematics is exactly a tool for visualising them, usually by finding tricks or tactics to simplify things, but visualise or get an intuitive grasp of none-the-less: Is it possible to visualize higher dimensional space? It's fairly likely there are higher dimensions, with the holographic principle widely considered the way to grapple with the Black Hole Information Paradox, AdS-CFT Correspondence, & M-Theory (String Theory) sidelined not because it doesn't work but because it is too flexible to make specific predictions.

We begin with simple experiences and ideas, and through mimicry and teaching, we generate structures of abstractions, which relate to our Modes of Life - our shared purposes, which allow meanings to be passed from mind to mind: According to the major theories of concepts, where do meanings come from?