I think most here would know that math is not complete, consistent or decidable. (https://www.youtube.com/watch?v=HeQX2HjkcNo) But I'm going to leave that aside as it's pretty high level math for me and I never run into those problems in my life.
My personal problem with math that prevents me from seeing it as "discovery of fundamental principles claimed to exist in an objective reality" is natural numbers. It's impossible for me to clearly find the number 1 (for example) anywhere. The boundaries between one unit and 0.X or 1.Y seem arbitrary and chosen to help us create models. Religion and philosophy deal with this, but it's also apparent in digital signal processing. Or medicine: are the numerous bacteria in our bodies us, or not? Is the heat radiation from my body me? What about the fact that 1 + 1 rarely if ever equals 2? Meaning that two things together physically interact to create more than the sum of both parts. From celestial bodies to a replicated database the complexity goes through the roof when you have more than one thing.
My personal problem with math that prevents me from seeing it as "discovery of fundamental principles claimed to exist in an objective reality" is natural numbers. It's impossible for me to clearly find the number 1 (for example) anywhere. The boundaries between one unit and 0.X or 1.Y seem arbitrary and chosen to help us create models. Religion and philosophy deal with this, but it's also apparent in digital signal processing. Or medicine: are the numerous bacteria in our bodies us, or not? Is the heat radiation from my body me? What about the fact that 1 + 1 rarely if ever equals 2? Meaning that two things together physically interact to create more than the sum of both parts. From celestial bodies to a replicated database the complexity goes through the roof when you have more than one thing.