Mathematical Truth

"The objection I see to Coherentism is that there appear to be some things that won’t change.....2 + 2 will always be 4 regardless of whether we are counting real objects or just imagining them".

This is a really interesting avenue. The "truth" of mathematics is something which I have been pondering. Is pure mathematics something that we discover or is it something that humans invent? If the former, then I can see how mathematics can be taken as a (natural) foundational type belief, if the latter then either coherentist or perhaps rational foundationalist.

If you were to ask most secondary school children who have studied mathematics at a basic level "Is it true or false that the sum of the angles in a triangle always add up to 180 degrees?" I suspect (hope?) most would say it was true. For them, this knowledge would be similar to 2+2=4 in terms of never changing, in a sense it would be a foundational belief.

However, when you draw a triangle on a curved surface, the sum of the angles do not add up to 180 degrees. Imagine drawing a triangle on the surface of a globe, with one corner at the north pole, and the other two corners on the equator. In this case the sum of the angles is not 180°. It turns out that the sum of the angles of any triangle is less than 180° if the geometry is hyperbolic, equal to 180° if the geometry is Euclidean, and greater than 180° if the geometry is elliptic.

In this sense is is it not therefore conceivable that mathematical knowledge is not always unchanging?

I'm currently reading a biography of Paul Dirac by Graham Farmelo called 'The Strangest Man'. Dirac was a British theoretical physicist who some regard as the successor to Einstein. Dirac was a genius. He was also probably autistic, according to Farmelo, and was not apt to pontificate on philosophical questions, or at least not until the end of his life. I get the impression from the book that, from a relatively young age Dirac held the view that any satisfactory explanation of the way the world is, in mathematical terms, needs to be elegant or beautiful because that is the way the world is. He would reject mathematical theories that he did not regard as beautiful largely on the grounds of their lack of aesthetic appeal.

Dirac therefore seems to have held a foundational type belief that the laws that govern the world are not random or inconsistent or inexplicable or ugly. He also believed that it was inconceivable that mankind would not continue to progress.

For all of these foundational type beliefs I remain largely skeptical because I want to ask the question "how do you know that it's true?"