Are Numbers Real?

[Angrboda]

Without opening a whole new can of worms, let's just put the question like this: Is math something we create or something we discover? If it is something that we discover, then that implies it has "an existence" of some sort or another, independent of our perceptions.

I suppose another way of framing the question could be: Does math make truth statements? That is, does math make objectively verifiable claims? It could be argued that it does

This is a topic I have thought about A LOT. I think we both invent and discover math. Let me see if I can explain.
Suppose I ask if the game of chess was invented or discovered. I think we all can agree it was invented. But, suppose I ask if, from a particular position in chess, there is mate in 4 moves. That is a question about some truth concerning those invented rules. And we can discover such truths even though the game of chess was itself invented.

Math is primarily an investigation of abstract formal systems. In such systems, we have axioms (basic assumptions). We choose those axioms, thereby inventing a topic in mathematics. Once those axioms have been chosen, however, we discover the consequences of those axioms.

So, that right triangles obey the Pythagorean equality is a discovery from the invented system of Euclidean geometry. If you choose other axioms, say those of non-Euclidean geometry, the Pythagorean equality would fail. There is then no 3-4-5 triangle.
The number pi can be defined in several very different ways, depending on the assumptions being made. In non-Euclidean geometry, though, it is no longer the ratio between the circumference and the diameter of a circle because there is no one such ratio, but many, depending on the size of the circle. The axiom system makes a difference in the truths. More clearly, truth depends on the assumptions made.

The same goes for numbers. We have some basic intuitions concerning numbers and such things as addition and multiplication. Those basic intuitions help us choose our axiom systems, thereby inventing a subject in mathematics. We can then discover whether certain results follow from those axioms. So, for example, in ordinary arithmetic, 13 is a prime number. But, if you use Gaussian integers, it is no longer prime. There is unique factorization into primes for ordinary arithmetic, but not if you look at certain algebraic number fields.

Each system of axioms has statements that it can prove, statements it can falsify (hopefully no overlap as that gives an inconsistency), and statements that it cannot decide. The latter class of statements can either be asserted or denied in conjunction with the other axioms and still have a system that is just as consistent. We get to choose in this case, based on our intuitions and our sense of aesthetics.

So, the answer to your question is that math is invented in that we choose our axioms. But after we do so, the consequences are discovered. Different systems will give different 'truths'.

Which system is best for describing the 'real world' is yet to be determined. That is a matter of experimentation and observation.

It seems that from your examples you are suggesting that mathematics is simply the analytical content of the axioms that we choose. That seems fine as far as it goes, but in cases such as that of the Dirac quote below, we seem to somehow pack an extraordinary amount of analytical truth into a small number of axioms, from which such inferences as those that are made seems to correlate well with the real world almost in anticipation of real world truths. There is nothing necessarily contradicting the possibility that we could choose such bountiful axioms so simply and easily, but it seems to bugger the imagination that we have done so purely by chance. I know there is a good deal of fitting the axioms to the macroscopic reality of the world in mathematics today, but if you go back to, say, Euclid, and his basic postulates of geometry, so much that we can correlate at least to a reasonable approximation flows from that small set of axioms. True, when you take things like the curvature of space-time into account, or possible non-Euclidean spaces, then derivations based upon those axioms will diverge somewhat, yet for our everyday macroscopic world, they seem more than coincidentally descriptive, and our ability to pack so much into such a small number of axioms seems almost magical.

It was not until some weeks later that I realized there is no need to restrict oneself to 2 by 2 matrices. One could go on to 4 by 4 matrices, and the problem is then easily soluable. In retrospect, it seems strange that one can be so much held up over such an elementary point. The resulting wave equation for the electron turned out to be very successful. It led to correct values for the spin and the magnetic moment. This was quite unexpected. The work all followed from a study of pretty mathematics, without any thought being given to these physical properties of the electron.

~ P.A.M. Dirac

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The example of a cricle was given earlier, and no such thing as a circle exists in the so-called real world.  Our idea of a circle is an idealization that is largely a byproduct of the way our senses work, specifically with regard to granularity and sub-feature processing in the brain and eye

If our concept of circularity is based on an idealization what is it an idealization of?

By idealization I mean that it is the maximization of a specific idea. It is not the maximization of things existing in the world but the maximization of other purely constructed mental things like lines, points, curvature, and so forth. In the real world, these ideas are extracted out from data that doesn't actually contain such things, so they aren't an idealization "of" anything, they are just pure mental constructs.

Is the value of pi just a rough approximation based on empirical observation?

No, the value of Pi characterizes a relationship which can be conceived in the mind. We might derive an approximation by drawing an analogy between things which have a similarity to the mental construct, but the value of Pi as it is currently understood is purely a function of that abstraction. You could connect the dots lying along a circle as making up a fractal pattern which, like a coastline, is far longer than the metric of measuring the distance between any two points lying on the circle and the value of the ratio between the radius and the circumference of that figure would be many times greater than Pi. We 'construct' the circle as a circle in our minds by constructing a figure that consists of the separate points lying along the circle as connected by straight lines as a byproduct of our perceptual processes. We "fill in" the missing information to make a correspondence between a set of points and the idea in our mind. However, this operation is highly dependent upon the stimulus and how our perceptual systems process it. If there are 5,000 points along a circle of 6" diameter, our perception has little difficulty in reconstructing those points as belonging to a circle. If you reduce the number of points lying along a circle to five, then our perception doesn't know whether to construct our idea of the object as a circle or as a pentagon. Between these two extremes there will be a crossover region where the perception is more likely to result in the idea of a circle than a non-circle polygon, and where that region occurs depends upon the granularity of our perceptual systems. If the 5,000 points were spread around the periphery of a figure that was as far across as the physical universe is, we would not be as ready to conclude that they necessarily were a part of a circle rather than some other regular or irregular shape.

Is there any possible universe in which the value of pi is different?

Given that Pi is a relationship of a purely mental construct, it doesn't make sense to ask if there is a possible universe in which the value of Pi is different because the value of Pi doesn't exist in this universe. It exists as an analytical truth that follows from the concept of a circle which our mind constructs. If you're asking if we could have a different psychology, I'm sure we could have, but it's not a very meaningful question. I think what you are likely trying to ask is, if we assume that planar geometry accurately describes our universe, are there possible universes in which analogs to the idea in our mind, such as the drawing of a circle on a piece of paper, have different relationships, and the answer to that is yes along several possible dimensions. As pointed out already, we could have a different psychology such that the idea of a circle never resulted from an experience of seemingly contiguous points. I suspect there are evolutionary reasons why we have the psychology we do, but we certainly aren't required to have it. Second, there are geometries in which the ratio of an idealized circle's circumference to its radius is different than Pi. There is nothing that says that a possible universe cannot have a space-time which corresponds to one of these geometries, so the answer along that dimension is yes as well. Then, as noted above, our perceiving a set of points in the real world as lying along a circle depends upon our constructing a figure by drawing lines between the adjacent points along the figure's circumference. That is a simplifying assumption which is hardwired into our perception. It is not a necessary assumption, so as in the case of the pentagon and the conception of the points being perceived as lying along a fractal coastline, neither case would correspond to what we ordinarily give the value of Pi to be, so not only is it possible for there to be other values of Pi in other possible universes, it's possible to have different values of Pi within this universe itself, as our value of Pi depends upon a constructive assumption, and not upon some objective feature of the world.