Reality is what we perceive it to be through our sensory perceptions. Although reality could be wrong for one person (or a few people) such as a schizophrenic. Like in the Matrix we could all be hooked up to machines that are simply feeding us sensory sensations. So reality must be an assumption. Reality is the collective perceptions of humanity. In other words, reality is purely physical. According to this definition does math exist in reality?
No, if reality is a type of collective perception then math does not exist in reality. This is because math is not perceived but conceived. A person cannot view a point of zero dimension, or a line of infinite length, or a curve of infinite smoothness. Even a number line is based on an assumption of a line which has no discontinuities. This is why no mathematical systems can exist without some form of assumptions. We have to assume something to be true which is not perceivable in reality. I term this type of pseudo-reality a "conceived-reality." Conceived-reality is the reality conceived by humanity as it relates to mathematics.
Does this mean that points and lines could not exist? No, there could be a reality in which these theorized "objects" - such as mentioned in Plato's theory of Forms - do exist and we have yet to perceive them. However, because reality is defined as the collective perceptions of humanity then mathematics, and the idealized world it embodies, does not exist in reality but instead conceived-reality.
According to this definition of reality then if everyone looked away from the moon and no measurement of its position was taken, then for that period of time the existence of the moon would be questionable. Although this seems absurd it does conform to reality. There is a chance that during this "non-measurable moon state" that some astrological event could occur which would alter the moons location in space and time. Even after looking back at the moon, and finding it where it should be, does not mean it existed the way we think it did for that elapsed time. According to probability, any number of things have the possibility of occurring so we can only say there is a high probability that nothing unexpected happened during the time that the moon was not measured.
If math is not a part of the real world then why does it define so many things in reality? This is a good question still asked to this day. I believe the answer lies in the very definition of mathematics. Mathematics is a set of self consistent truths. Because the universe is self consistent (for the most part) it only makes sense that a set of self consistent truths can be used to describe it.
So how do we use mathematics and its conceived-reality to solve problems in the real world? The answer is mapping. Essentially, every time we solve a math problem we are actually mapping properties of reality into a conceived-reality, solving the problem in this conceived-reality, and then mapping the result back into reality.