So lately I've been digging into geometry and I always seem to come back to the same question, how can we really test to make sure our math values can correctly apply to world that we live in? First, lets assume that we live way back in time and our scale of measurements where limited (for example, we couldn't go past 1 decimal point) For example, if we construct a simple wheel (lets say with a measurement of "1") and take a piece of string, wrap it around the wheel and make a cut where the string just starts to overlap. We can measure that ~3.1 units long. Now, if we apply the same concept in math we end up with an irrational number 3.1415.... So, as our scale of measurements got better and allowed us to measure things better, we can measure out to more decimal points. Now, say we can measure out to 2 or 3 decimal points. Where is the point that someone said "Alright, the numbers are close enough. I can for sure say that in the world we live in, the ratio of the circumference of a circle to it's diameter is equal to pi." It just seems like it would be a huge assumption to make back in the day. Again, we take it for granted now, yes we can take really really really small measurements, but we can't be 100% sure. Always bothered me a bit. Kind of a side note,is it possible that our number system won't hold up in the future? For example, when I learned a bit about quantum physics you can have a particle be super positioned and we found out that observing particles will change their behavior. Is it possible that our whole math system is becoming "out of date"? Or does the language need to be expanded? Oh and another thing, it seems like in order for us to stumble upon math and understand it, we first needed to apply experiments in the world we lived in first. After we got a good grasp of things, we then didn't need physical models to test things out, math was self supportive, or so I believe. I'm just curious if recently (within the past 50 years or so) if we had to do physical experiments before a new branch of mathematics where created. It seems that at first applied math was a must, but then we where able to discover and practice theoretical math more. Thanks.