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When we learn mathematics we are given these visual representations of numbers, as things become more advanced we learn the idea of a 'number line', my question is how valid is the idea of a number line in the definition of numbers? For example if we think of negative numbers as negative units and positive numbers as positive units (like in school) how valid is this as formal view of numbers?

To do so seems trival, as it seems to contradict our number line view, so what is the absolute definition of these objects, does the idea of a number line 'define' the number or is simply a representation that lets us make sense of operations such as addition and multiplication? Could we learn the idea of mathematics without any such number-line?

Confused
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    See e.g. Marcus Giaquinto, Visual Thinking in Mathematics (Oxford UP, 2007) – Mauro ALLEGRANZA Nov 16 '22 at 12:45
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    There is no single definition of what a number is other than the theory itself. A number line is one application of the theory or a model of the theory. One of many. – David Gudeman Nov 16 '22 at 15:10
  • As @DavidGudeman notes, a number line is a model of the numbers, in particular and usually with an order structure. there is no need to define the numbers through the number line. – emesupap Nov 16 '22 at 15:17
  • @DavidGudeman it's hard to say what a number really is, except it exists, and it relates with other numbers in certain ways? – Confused Nov 16 '22 at 18:39
  • @user1007028, I don't even know what you mean by "what a number really is". It's a number. That's what it really is. Perhaps you are making some sort of generic reductionist assumption--that there must be a way to reduce numbers to something else that is not a number, like reducing the physical world to perceptions or reducing intelligence to behavior. If that's your meaning, then I'm afraid I can't be sympathetic. In my view practically all philosophical reductions are wrong, and most are pretty obviously wrong. – David Gudeman Nov 16 '22 at 18:59
  • @DavidGudeman I guess 'is' is quite a bad reasoning, I guess I mean what defines it, but that is something that is hard to define. – Confused Nov 16 '22 at 20:08
  • "what is the absolute definition of these objects" What is the absolute definition of the word 'game'? We don't have one & we don't need one, to recognise one when we see it (Private Language Argument). I would describe mathematics as a structure based on increasingly abstract metaphors: https://philosophy.stackexchange.com/questions/94460/relationship-between-real-quantities-and-numbers/94462#94462 Group Theory or set theory or abstract algebra can be called more fundamental, but they are better understood as interfacing between domains, like the idea of energy does in physics – CriglCragl Nov 16 '22 at 23:05

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The most up-to-date account of the number line that I'm aware of is the theory of proper classes and the lemniscate, modulo the surreal numbers. For example, +∞ can be "constructed as" a specific surreal left/right set-gap, but is not taken strictly as a number.

All numbers "proper" do have a "normal form" modulo the surreal class, and the mechanics of identifying the class are equisemiotic with an infinite line under the intended interpretation. Whether this linearity is a minimal-but-necessary semiosis I'm not sure; there are nonlinear logical semiotics (e.g. in Frege and C. S. Pierce) but those aren't quite antilinear, so to say. Kant did think of logical functions as enumerable (he has 12 categories determinately, for example) but held our human mathematics to involve significant and fundamental geometric translation options.

Kristian Berry
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There are all sorts of numbers but we can focus on natural numbers and real numbers. Natural numbers are intuitive concepts emerging from our experience of considering, exchanging, trading discreet quantities of things, such as for example, apples. The formal, mathematical version is essentially an elaboration on our intuitive notions.

Real numbers?

Fractions and therefore the idea of rational numbers also came through the bare necessities of exchanging goods, and sometimes, half or a quarter etc. of the usual quantity. The idea of negative numbers came intuitively out of the need to represent debt. So not only natural numbers, but also integers and rational numbers are intuitive conceptual constructs that emerge directly out of our everyday experience of the real world.

Real numbers? It is usually said that the first idea of real number was the idea of irrational number that mathematicians discovered when trying to find numerical solution to quadratic equations. However, the idea that the set of abscissa of points of an infinite straight line is the same as the set of real numbers demonstrates that we also have an intuitive notion of real numbers, essentially as distances in space and intervals in time.

The particularly interesting twist, though, is that we don't know how to construct all real numbers from rational numbers, even though we understand, again intuitively, that we could in principle find rational numbers infinitely close to any real number.

So the visual representation of numbers is perfectly valid as far as anyone can tell. It is valid probably because our idea of numbers come in large part from our experience of our visual perception of the world, being apples or distances.

Arithmetic is just the systematisation of our intuitive notions about numbers. As such, arithmetic essentially came an effort to represent formally our intuitive concepts. No intuition, no arithmetic.

Speakpigeon
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