Are Numbers Real?

[polymath257]

The answer to the question depends on what you mean by real.

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.

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The definition of a circle is, via Euclid, "A circle is a plane figure bounded by one line, and such that all right lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its center."

That is already the ideal, we don't draw anything else from a simple definition.

Pi is simply the ratio of the diameter to the circumference; it just happens to be a transcendental number, which means that it is a decimal fraction that never repeats. It is not an approximation; we use approximate values for it depending on the required precision for what we are doing.

That definition only works in Euclidean geometry. In other geometries, there may not be only one such ratio. Nonetheless, we can define pi in several other ways and reach the same number. But in any case, we have to assume properties of the 'real' numbers to even define what the circumference is supposed to mean, let alone what the possible ratios are.

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We create maths. We discover ways in which particular maths has applications when applied to reality.

Remember that maths need not have any bearing on reality at all. To say that you "discover" a new mathematical system would be metaphorical only; it would be like exploring the platonic plane of abstract concepts.

Maths makes statements which are either true or false within their own framework. They are true, essentially, because we say they are true. They are a logical result of applying the rules which we say are true. Verifying they are true is a matter of making sure the rules have been correctly followed.

So (concerning right triangles) it is only true that the square of the hypotenuse is equal to the square of the sum of the other two sides because we say it's true. This is not a fact that we discovered about right triangles? Is this what you're saying?

It's more that it is true because we say certain axioms are true. No, it is not a fact about all right triangles. As an example, there is a triangle on a sphere which has three right angles: pole to equator, 90 degrees along the equator, then back up to the pole. That is a perfectly legitimate triangle in spherical geometry. And the sum of its angles is 270 degrees, not 180. In such a triangle, even defining which is the hypothesis is problematic (it is a equilateral triangle after all). And the Pythagorean equality fails badly.

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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? Is the value of pi just a rough approximation based on empirical observation? Is there any possible universe in which the value of pi is different?

Depends on your definition of the number pi. Different potential definitions can give different results, depending on underlying assumptions.

Pi is a real number, which means it is an 'object' in an abstract formal system. That system is defined by its axioms and rules of inference. One 'nice' definition of pi is 4 times the integral from 0 to 1 of 1/(1+x^2). No circles need be harmed in this definition. but to define the integral requires a fair amount of work if you are starting from the axioms. But to even get the real numbers requires some work if you are starting from the axioms of set theory.

Now, the usual abstract formal system we use has shown itself useful for making models in physics. That is why that model is preferred.