The Metaphysics of Mathematics

Mathematics is probably considered the science least associated with or affected by faith or religion.  It is all neutral axioms and firm logic, theorems and proofs, so how could beliefs come into it?  Nevertheless, there are many metaphysical connections and underlying belief aspects in the field of mathematics.  I will provide an example, some further observations, a few questions to ponder, and some references to delve deeper into the subject.

Example Exercise:  (using only high-school math)
Let f(x) = 1 for x = rational, and f(x) = 0 for x = irrational
Question: what is the integral between 0 and 1 of f(x)?

If you don't know, an integral is simply the area under the plot of f(x) on an x-y graph between the two limits of x, so there is no need for actual calculus (integration) here.

Depending on your metaphysical assumptions, any of the following answers may be correct:

a) If you were a Pythagorean back in the day, you would not believe that irrational numbers actually exist, so f(x) = 1 for all real numbers and the integral (area) between 0 and 1 would simply be 1.
Even today, there are people who question how real most irrational numbers are since the vast majority of them can never be described exactly, because they are an infinite series of digits in any numerical expansion; e.g. x = 0.876513234923746023700147829432764591034 is not irrational. It can be expressed as a ratio of two integers, and so is rational. Only by continuing such a string of pseudo-random digits to infinity can it be said to be irrational. Thus, a vanishingly small percentage of irrational numbers can be described in any finite form: numbers such as pi, e (the base of natural logarithms), square-root of two, 0.909009000900009000009..., and so on. So if most irrational numbers do not truly exist in any meaningful, presentable sense, one could perhaps say that the rational numbers between 0 and 1 far out weigh the irrational ones, and the integral approaches unity in the limit.

b) However, assuming that all irrational numbers do truly exist (in some sense), it is clear that there are any number of both irrational and rational numbers between 0 and 1. Indeed, we can say that there are an infinity of both type between 0 and 1. We cannot count how many, so we might assume (for now) that for every rational number there is an irrational one. Then half the numbers in that range are rational and half irrational, so that the area is one half (0.5). This might be thought of as a pseudo-average of f(x) over the 0 to 1 range.

c) Not so fast! That is not the full story. According to Georg Cantor (19th century) and most mathematicians since then, the rational numbers are "countable", which means they can be paired one-to-one with the infinite set of integers. He provided a simple demonstration of that. On the other hand, the irrationals cannot be counted in the same way, or at least no one has come up with a scheme for doing so. Thus, officially, the rational numbers are a different degree of infinity than the irrationals.

All infinite "countable" sets of things (numbers or otherwise) are considered to be at the "aleph-zero" level of infinity. The real numbers (rationals plus irrationals) cannot be counted, and are therefore considered as "aleph-one" level of infinity. In an aleph-one set, there are infinitely more items than in an aleph-zero set; not just double infinity, but "infinity squared" so to speak (although infinity is not a number so cannot be squared). If the real numbers are aleph-one, while the rational numbers are aleph-zero, then the irrationals must be aleph-one.

Therefore, rather than being the same number of rationals and irrationals between zero and one, there are infinitely more irrationals. That is, for every rational number between 0 and 1, there are an infinite number of irrationals. In this approach, the integral of f(x) is arbitrarily close to zero; i.e. the vast majority of the point in x have f(x) = 0 and the "average" over x = 0 to 1 is zero. I expect this would be the "official" answer to the question, referred to as the "non-constructivist" view, but it too is not without metaphysical issues.

d) Finally (I hope), the integral operation is often explained as summing up a huge number of narrow dx slices under a plot of the function, between the two limits of x. Thus, for example, when integrating the function g(x) = x, one would take narrow dx slices between x=0 and x=1, and add up their areas, as dx was made smaller and smaller, the integral would converge on a value of one half (0.5) for g(x). The integral calculus allows one to find the limit, i.e. the true area, without adding up the tiny slices. Thus, dx is referred to as an infinitesimal and, in the limit, there are an infinite number of dx slices between zero and one.

This works well for any continuous function like g(x), but f(x) is an "everywhere discontinuous" function. No matter how small you make dx, there will be an infinite number of rational numbers in the slice and infinitely more irrational ones. Thus, there is no way to look at a finite dx, much less an infinitesimal one to determine the "true value" of f(x) in that slice.

The concept of "real infinities" and infinitesimals caused mathematical philosophers a lot of trouble when calculus was being developed back in the 17th century. Still today, infinitesimals are suspect to many philosophers, who basically say we can use calculus because it "works" for functions like g(x), but not because it represents reality or truth. Given such metaphysical doubt, some would conclude that for the function f(x), dx is meaningless, therefore the integral is undefined; i.e. it cannot be computed.

Some Other Observations:
a) Zeno's paradox updated: movement is impossible as it requires something to be in an infinite number of locations in a finite time, and "real infinities" are impossible. Despite seeming to be silly, this paradox still causes metaphysical concerns for some philosophers.

b) Infinity is not a number, but Georg Cantor's transfinite numbers (aleph 0, 1, 2...) tries to make sense of infinite sets of different "sizes", as noted above. Cantor said there is an infinite series of ever increasing infinities (aleph-n), but I have not seen any descriptions above aleph-2, and some mathematicians question how real the higher alephs are. Hence answer (b) above.

c) Kurt Gödel's incompleteness theorems (in the 1930's?): in any mathematical system there are true statements that cannot be proven within that system; i.e. no mathematical system is "closed". Some people (e.g. John Lucas and Roger Penrose) claim this means that humans are not computers.

d) Euler's identity: 1+ e^i*pi = 0  
This combines five, seemingly unrelated fundamental mathematical objects in a relation: zero, unity, two transcendental numbers, even an "imaginary number". How weirdly beautiful is that?

e) Fractals are "fractional dimension" mathematical objects, infinite complexity arising from simplicity, deep beauty from simple equations. They have real-life applications: e.g. the shape of fern fronds, and how long is the coastline of England? (depends on what scale you use).

f) Humorous theorem: there is no such thing as a boring number!
Proof: if there were boring numbers, then the set of boring numbers would have a first member, "the first boring number", and that would be very interesting indeed, so it could not be boring.

Questions to Ponder (based on your own metaphysics):
1. Are mathematical objects (e.g. polygons, relations, theorems) invented or discovered?
2. Do mathematical objects (e.g. numbers, geometric results) really exist?
   They are not matter, energy, or information!  (this evokes Plato's metaphysical “forms”).
3. If mathematical objects exist, are they part of the universe?  If so, could math be different in
    another universe or creation?  i.e. could God have created mathematics differently?
4. Why is mathematics so effective for science? Do math-based physical “laws” actually operate
    the Universe or are they just human models?

References for Further Study:
How religion and faith (metaphysics) underpin mathematics:
 https://frame-poythress.org/a-biblical-view-of-mathematics/
A brief Reformed Christian view of mathematics:
 https://www.redeemer.ca/resound/stewarding-talent-in-mathematics/
A quite different (contrary) view:
 http://steve-patterson.com/the-metaphysics-of-mathematics-against-platonism/
A brief but thoughtful book review on the subject:
 https://reviewcanada.ca/magazine/2014/09/the-metaphysics-of-math/
If you want to get lost in the subject! (quickly gets deep):
 https://plato.stanford.edu/entries/philosophy-mathematics/

This last reference shows that the subject remains an active and serious area of professional inquiry and debate.