Jarle said:
How do you decide whether a number really exists, rather than just existing formally?
By prediction and measurement.
The best modern epistemologist IMHO is Robert Rosen who was a mathematical biologist and wrote great stuff on the modelling relation. His last book, Essays on Life Itself, is all about precisely this issue.
So say we are talking about the number 1. We have defined it formally as a model of "one-ness". It exists formally. But how do we find exactly 1 of anything in the real world? We take the model and use it to justify a process of prediction and test.
Is that one apple I see on the table? Well, I might have to walk all round it to be sure a second, or an infinity of apples, is not hidden just behind it, blocked from my line of sight. I might have to reach out and touch it, to be sure it is not a collection of carefully painted beetles holding some apple like formation. Or if we zoom in for a microscopic view, at some point everything dissolves into atoms (whatever they look like) and we can wonder where this exact apple starts and ends. Etc, etc. If you are talking about absolute certainty, then there are in reality an unlimited number of doubts that must be ruled out.
So can we ever really know that some physical thing is an exact example of one-ness, as formally defined? In pragmatic fashion, we can pretty quickly agree that we only have a single apple on the table. All doubt seems trivial. But doubt must always remain because reality does not seem completely measurable.
One, of course, seems the simplest number to relate to reality. Others like pi and infinity remain more troublesome.
And I think there is then another whole level to this particular debate to do with "doubt" within formal models themselves.
Quickly, axiomatic truths are in fact inevitably dichotomistic. You cannot confidently assert one thing without also creating the equally definite possibility of its opposite (thesis and antithesis as Hegel said). So you say discrete, I say continuous. You say local, I say global. You say event, I say context. You say stasis, I say flux. You say determined, I say random.
All basic concepts come in complementary pairs as all crisply definite assertions are symmetry breakings (the breaking of the symmetry of vagueness or ignorance into asymmetric polar opposites).
And so, secretly - it is rarely acknowledged, except by category theory! - that all mathematical systems have a fundamentally mixed nature. They must employ both ends of a dichotomy, even if they prefer to suppress awareness of one of the ends.
So with number theory. We have discrete numbers existing on a continuous line. Both aspects are essential to the formal model, but one aspect is suppressed.
For example, the number 1 is just taken as a discrete point (on a continuous line). This is a formal statement that seems to need no further "measuring". It just is.
But say we wanted to check? Well what we are really saying is that 1 is 1.000... Check its location on the line to as many decimal places as we like, and it will be exactly there. But of course, we also know there is a practical issue when it comes to establishing infinite facts. In practice, we can never arrive at a final count. The continuity that we have tried so hard to push out of view is here reasserting itself.
Again, the 1-ness of the number 1 is about the least troubling either in the real world, or within its own world, the realm of axiomatic formal modelling. But even for 1, there is a hidden duality behind the presumed monadic description. Counting appears based on the notion of fundamental discreteness, but for exactly this reason, it is just as much based (formally, axiomatically) on the assumed absence of fundamental continuity (and hence, in practice, by the suppression of what must also exist as part of initial state of possibility, back when axioms were being formed and reality was still intruding).
It is a case of A and not-A. You make a division and you must create two things. Both are equally real. But in your model, you just want to keep things simple and use the A. And suppress any non-A-ness. So 1 is discretely just 1, and you don't have to run round constantly stamping out threatened confusion from 1.00000... sometimes being actually 1.000001, or some other infinitesimal fluctuation.
However, when it comes to irrationals and infinities, people are more aware that the counter-balancing option of continuity is being actively suppressed (suppressed axiomatically). So they will protest and try to re-open the door to fundamental doubt. And if they go all the way back to axiom-formation, they can see doubt is justified - numbers are not real - but also that there was a reason why the formalism went with option A rather than option not-A.
Imaginary numbers are the same. To me, the notion of 2-dimensional numbers, or n-dimensional numbers, seems quite natural. A point is a constraint on a line, but a line is a constraint on a plane, which is a contraint on a volume, etc. So you can play about on the spectrum between the absolutely discrete (a zero-D point) and the absolutely continuous (an infinite, unconstrained, dimensionality).
Again, the fundamental continuity (or its suppression) becomes more obvious and so more troubling with imaginary numbers, but it is there for all numbers in a necessary fashion. To have A, you have to make not-A. To have a figure, you must have ground. To have an event, you must have context.
Which hopefully loops round to my initial points about vagueness, zero and the null set. For zero to be a local absence, it must exist in the context of a global presence. The formalism of set theory wants to throw away the {} along with its contents - treat them as a nothing as well. But it would be less confusing to accept them for what they are, a necessary part of breaking the symmetry of pure possibility. To have thesis, you must also have anti-thesis.
And vagueness then is this realm of the pure unbroken possibility, a state of infinite symmetry. Imagine a place which is neither discrete nor continuous, neither random nor determined, etc, etc. Yet can be divided into these crisp, mutually-defining, polarities.
