Introduction
Everyone who has taken math in school is familiar with the real numbers -- for example, \(-3\), \(1/2\), \(-\sqrt{2}\), \(\pi\), and so on; but not many people know where these numbers come from. Similarly, whether or not the numbers in question literally exist is a topic of debate.
In this page it's argued that real numbers literally exist. The standard construction of the real numbers is given as well. Numbers literally exist, in the mind of God. Stop reading here if this is too difficult to believe. This page is the most difficult in the whole series of pages where we prove the MVT. It is important to prove that numbers (and other mathematical objects) literally exist, for if they are merely useful fictions, then the mean value theorem would not be literally true :)
Nominalism
Mathematical nominalism (the belief that mathematical objects do not exist) is believed implicitly by most people. If you walk up to a random person on the street and ask them whether or not they think numbers exist, its likely that, if they give a response, it will be along the lines of "we invented numbers to describe reality". This would mean numbers only exist in the mind, i.e. they do not literally exist (and hence statements about mathematics are false, even though they might be true within the story of mathematics)
One large problem with nominalism is that it does not explain the applicability of math for real-world application. We use mathematical sentences to develop scientific theories (\(F=ma\) for a simple example) and we have reason to believe that these scientific theories give an accurate picture of our world -- that is, the mathematical sentences themselves are true. But if the mathematical sentences (which quantify over mathematical objects) are literally true, then the objects in question must exist. (This is a form of the indispensability argument from Quine and Putnam.)
One response to this argument is to claim that mathematics is genuinely dispensable for modern science, meaning that mathematical objects can be eliminated without affecting the success, accuracy or power of a given theory. The claim is that empirical theories can be nominalized -- that the same theory can be re-stated in a way which doesn't make rely on existential quantification of mathematical objects (i.e. a theory without implicitly or explicitly stating that a mathematical object exists.) This presents a problem, because it's hard to see how this could be true unless the nominalization is actually performed for each theory. Similarly, if we can show that there's even a single theory that can't be nominalized, this whole response fails.
The creator of this response, Hartry Field, performed the nominalization of Newton's law of gravity, presenting it without numbers (the book: Science Without Numbers, I'm not a nominalist but it's interesting to see how he did it), but notably it has been argued (by Malament in 1982) that quantum mechanics cannot be nominalized -- if this is true it'd knock this response down. Malament's response has been counter-signaled by Mark Balageur in 1996, where he argues that QM can be nominalized; Balageur's response has received another response by Otávio Bueno in 2003 to claim that Balageur's nominalization strategy doesn't succeed.
Platonism
Nominalism notwithstanding, mathematical platonism holds to the literal existence of mathematical objects, which are abstract & independent of intelligent agents. They do not exist in the mind, nor do they exist physically, but exist in another reality separate from these two, which Frege called the third realm but Christians know them to exist in the mind of Christ the Λóγος.
Nominalists might respond to the existence of abstract objects by questioning our ability to have knowledge about them. If mathematical objects are abstract (so they exist outside of space-time) then how would human beings, which exist within space-time, be able to tap in & gain knowledge about them? This is the epistemological argument against mathematical platonism. The response for Christians is very simple. By being made in God's image, we