Matheology: If God is infinite, how can anything else exist?

In The Mind’s Road to God, St. Bonaventure explains how the created world reflects its Creator, and these reflections help us lift our minds to Him. I’ve sometimes found certain mathematical truths to inspire similar reflection. Here’s one example.

A quarter century ago, when the internet seemed new, I hung out on some usenet lists. One day, an atheist posed, with as straight a face as one might perceive muster online,✗I write this because some other atheists called him a troll. I don’t doubt them; he gave plenty of reasons for the reader to conclude this. In any case, I’m not interested in building a strawman of atheism, but rather to illustrate St. Bonaventure’s point. the question, The answer, of course, is that God is immaterial, and an infinite immaterial thing does not take up “space”. We can illustrate this via numbers, which are themselves immaterial.

This argument does not require you to believe that numbers exist, by the way;✗ I myself don’t believe that numbers exist in and of themselves, not in a material sense, anyway. To put it as briefly as possible: nouns exist in and of themselves; adjectives do not. You will not find a “red” in the world; you find red things. In the same way, numbers are not nouns; they are adjectives: you will not find a “two” in the world; you find two things.

To elaborate on this somewhat: mathematics is a language we use to describe real things. Mathematics is internally consistent; that is, it is logically perfectly accurate when describing itself, but the moment you turn it loose on the real world, things no longer work. As an example, find me a truly perfect circle in the material world: you cannot. There will always be some sort of imperfection. The imamterial mathematical models we construct for real-world phenomena (e.g., Physics) are true in that they are logically consistent, and beautiful in how well they model the phenomena, but almost always they are ultimately wrong, if only slightly, leading to occasional revision. it merely illustrates the point that multiple infinite objects can coexist, overlap, etc, precisely because they are immaterial.

Second question

Mathematicians work with sets of numbers. The definition of a “set” is a little hard to get right (one reason that multiple set theories exist) but you can think of it as a way of identifying numbers.

We’re interested in two particular sets of numbers: rational and irrational.

Numbers I have ❤️‍🩹’d

Let’s review some familiar sets.

• The “natural“ numbers, denoted ℕ, consist of 0, 1, 2, … .
  Some people call them “counting” numbers, perhaps because they are good at modeling the ability to “count in nature”: you can count 5, 10, or more than a billion people; and there was a time where God would have counted zero of us.✗Some mathematicians exclude 0 from the natural numbers, illustrating an important point about mathematics: definitions matter. A difference in definition can lead to a difference in results.
  Sometimes, the differences in definition matter enough that different schools will converge on a consensus definition, which makes communication easier. An example would be the definition of a prime number, which is any natural number with exactly two natural divisors. Thus,
  • 2 is prime because it has exactly two natural divisors: 1 and 2;
  • 4 is not prime because it has three natural divisors: 1, 2, and 4; and
  • 1 is not prime (this is important!) because it has only one natural divisor: itself.
• A negative number is one that, when added to a particular natural number, gives you zero. For example, -2 + 2 = 0.
  Negatives are good at modeling things like debt or spending: you may earn $20, but if you already have to spend $20, your balance is effectively $0; the debt is a “negative” quantity.
• The integers, denoted ℤ, consist of 0, ±1, ±2, … . In other words, you get the integers by including negative numbers with the natural numbers. (When mathematicians write ±1 we mean 1 and −1.)
• The “real” numbers, denoted ℝ, consist of all values we can in principle measure along a line.
  In the diagram above, we can see that a is a “real” number because we can measure it along a line. Precisely which real number it is, we’ll return to in a moment.
  Real numbers need not be restricted to a quantity of distance; one can also include direction. Think of “two steps forward, one step back.”

We can now describe rational and irrational numbers.

• The “rational” numbers, denoted ℚ, consist of
  • 0,
  • ±1/1, ±1/2, ±1/3, …
  • ±2/1, ±2/2, ±2/3, …
  • ±3/1, ±3/2, ±3/3, …
  • …
  In other words, you get the rational numbers by taking ratios of integers, with the exception that 0 cannot appear in the denominator. Some rational numbers are equivalent; for example, 1/1 = 2/2 = 3/3 = …
• The “irrational” numbers are precisely those real numbers that are not “rational”.✗You may be wondering why these numbers have the name “irrational”, which means "not reasonable". We can blame the Pythagoreans. They first thought rational numbers sufficient to describe reality, until one of their school discovered that the real number a apparent in the triangle above, also known as √2, is not a ratio of two integers. Thus, if real numbers exist, or if the phenomena they model exist, then irrational numbers must also exist, or rather the phenomena they model. But in a truly bizarre intellectual twist, the Pythagoreans didn’t like irrational numbers, which is why they have the name “unreasonable”.

How the major sets of numbers relate to each other

Notice that

• Every natural number is also an integer, by definition of an integer.
• However, not every integer is also a natural number: -1 is not natural.
• Every natural number is real, because it measures a distance: 1 foot, for example.
• Every integer can be viewed as real when we consider direction in addition to distance: walking five feet forward is 5, and walking five steps backward is -5.
• Every integer is equivalentI’m using “equivalent” instead of “equal” for an important reason; they don’t actually mean the same thing. However, for all intents and purposes, the lay reader can consider “equivalent” to be interchangeable with “equal”. to a rational number:
  1 = 1/1 .
  In fact, every integer is equivalent to multiple rational numbers:
  1 = 1/1 = 2/2 = 3/3 = …
• However, “most” rational numbers are not equivalent to integers. Rational numbers like 1/2 lie “between” two integers.
• Every rational number a/b is real, inasmuch as you can obtain it by measuring out a segment, calling that length 1/b, and placing a copies of that segment next to each other.
• However, “most“ real numbers are not rational. The diagram above shows that √2 is real, but a slightly technical proof shows that it is not rational.✝In brief,
  1. Suppose that √2 = a/b, where a and b have no common factor.
  2. Rewrite the equation as b√2 = a. (Clear denominators.)
  3. Rewrite again as b²×2 = a². (Square both sides.)
  4. Notice that Equation 3 tells us that a² is, by definition, even. Since odd numbers don’t square to give us even numbers — it is easy to convince yourself of this, and not hard to prove✞
     1. Suppose that a is odd. By definition, a = 2c + 1 for some integer c.
     2. By substitution and a little algebra, we rewrite a² as (2c + 1)² = 4c² + 4c + 1.
     3. Rewrite again as 2(2c² + 2c) + 1.
     4. Let d = 2c² + 2c.
     5. By substitution, a² = 2d + 1.
     By definition, a² is odd. — a itself must be even.
  5. Rewrite a = 2c.
  6. By substitution, equation 3 becomes b²×2 = (2c)².
  7. Rewrite the equation as b²×2 = 4c². (Simplify the square.)
  8. Rewrite again as b² = 2c².
  9. Notice that the equation 8 tells us that b² is, by definition, even. Just as a² being even forced a to be even, b² being even forces b to be even.
  10. We have found that a and b are both even. Pretty cool, right? In fact, no: Statement 1 assumed that a and b have no common factor! We have a contradiction!
  An important principle of logic is that if you make an assumption, construct a valid argument from it, and arrive at a contradiction, the assumption must be false. Hence, Statement 1 is false. In other words, √2 ≠ a/b, where a and b have no common factor. Any fraction can be reduced to a ratio of integers with no common factor, so √2 is irrational.

A “technical” way of writing this is That is shorthand for saying, “ℕ is ‘contained’ in ℤ, but not equal to it; ℤ is ‘contained’ in ℚ, but not equal to it; and ℚ is ‘contained’ in ℝ, but not equal to it.”

Back to the question

Our question was, We can answer in the following way: