Where does math come from?

2024 Mar 9

Where do abstract things such as math come from? Since they aren’t material things, you can’t go to the store to buy them. You can’t get math by growing it or by digging it out of the earth. And are they real and/or true?

Although the ideas for abstractions might start from things in the material world, abstractions are mental products and exist only in the mind. We see the similar color of trees and emeralds and think of greenness. We observe similar shapes in the outline of an egg and the ripples from something dropped in water and think of roundness. We hear a gentle speaking voice or feel plush fur and we could think of softness.

We have names for quantities of objects and call them numbers. We notice the patterns of how numbers work together, develop these rules into a consistent system and call it math.

We can make observations about the law-like workings of material things. (This human effort of discovering such relationships in material things we call science.) Often these discoveries can best be represented in the notations and application rules of math. We call these scientific models.


Where do abstract things exist? This is the Problem of universals and is an ancient philosophical challenge. This challenge matters because it relates to truth and basic understanding. People have differing philosophical opinions about it, and the two biggest approaches are realism and nominalism:

Plato’s idea (realism) was that abstract objects are real, but they exist in a realm separate from both the material realm and the realm of consciousness. They exist independently of our physical world. However, they can be understood by those who are conscious and can be instantiated in material things.

A contrasting view is that abstractions are only the names (“nomen”) that we give to ideas we have created (nominalism). In this view, abstract objects don’t actually exist. note

Anti-realism is the negative form of this idea, and nominalism is a positive form of it.

There are two opposing world views into which these ideas could fit. note

These listed world views are specific:

  • Materialism is special in that it is quite atheistic, and therefore it is exclusive (material only, zero gods).
  • The theism is special because it stipulates one God (monotheism). This God is the ground of all being and is the ultimate cause of all things, so again the view is exclusive.

There are world views other than materialism and this specific theism.

  • They usually have multiple quantities or types of gods (non-exclusive).
  • However, these others don’t have much evidential support.

Both realism and nominalism could compatibly exist with most of the other (non-listed) worldviews - because with their generous options something compatible can be found. However, so many options means the options cannot all be true. Also, if there is truth, it becomes indistinct.

In my opinion, these others are generally ineffectual to explain the existence of all things. For instance in a theism with created gods like Thor, the gods specifically could not have created all existence. In a polytheism (like Hinduism), since many gods exist at once, no one is all powerful, and again it is the problem of an ultimate first cause. In a religion like Buddhism where spiritual beings aren’t really gods and there are supposedly many combinations of realms, it is unclear how anything caused our existence.

It seems to me that a comprehensive explanation for our existence comes only from monotheism. This leads me to reject these other worldviews.

  • In materialism - where only the material world exists and nothing else can exist.
  • In theism, specifically a theism with the view that God is the greatest possible being.

This sets up a grid with four quadrants at the extremes. Unfortunately all these combinations have fundamental problems:

  • For the materialist, since only a material world exists, then platonic realism with its genuine realm of abstract objects is impossible. There could be no such realm outside the material realm. (And besides, it would be impossible to test or verify such a realm.)
  • For a theist who has a concept of God as the ultimate cause, abstract ideas also were caused by God. Thus theism and realism are incompatible because abstractions could not have existence apart from God. note
  • The materialist view with nominalism is problematic because it cannot explain the persistent reoccurrence of abstractions such as truth or numbers. If the abstractions only exist when we have “named” them, then why do the same ideas keep returning (even if named differently)? They must be more permanent than materialism would allow. The consistent reoccurrences show these ideas are incompatible.
  • A nominalist view and theism are incompatible: In nominalism, fundamental abstract ideas would literally be the original products of humans, however, in theism all things derive from God.

A further problem is that a platonic form of what it means to be a god would therefore claim to define deity; however, this God could not be defined by anything external. If God is the greatest possible being and was the ultimate cause of all, then it follows that God also caused our abstractions.

However, if abstractions were to exist independently of God, they would define God and God could not be the greatest possible being. That is, a god would be a god only because the god conformed to the platonic form of what a god is.

God would therefore be dependent on the platonic form, would not be entirely self-existent, would not be the greatest possible being, and would not therefore even be God.

God's mind

There is another idea about universals that works on theism: Abstractions are things that are conceived in the mind of God. Because of this, they have persistent existence independent of any human mind. These abstractions have correspondence to the material world because the God that conceived them also used them when instantiating the material world.

We have internal personal evidence that as humans we can think about and work with abstractions. They persist in our minds. We use abstractions to make material things that have the characteristics we envision. (As humans created by God, we are able to do these things because we have a likeness to God - though God certainly is greater.) On this view by extension from our own experience we have credible evidence for the source of universals. We have a plausible explanation for the persistent existence of abstractions and a reason for their applicability to the physical world.

So, universals first come from the mind of God, and God made abstractions (like color, shape, heat and gravity) become real in the material world. Our minds can recognize patterns in what we observe, abstract these patterns, and develop mathematical relationships about them. And after we validate our ideas by experiments, we can have confidence in the truth of our abstractions.