Objective reality

I am certainly no expert either, but this is the way I look at it:The roots of mathematics lie in what I would equate to almost logical rules born of observation; 1 + 1 = 2 would not only be mathematically valid, but also a tautology. More advanced mathematics are in essence based on remaining internally consistent with those underpinnings, so that while entire sections may have no manifestation (irrational numbers for instance) the field itself is firmly rooted in reality.Now what happens when we are wrong with a theory, where we have an incorrect formula? Usually we require weird constants and gyrations in order to fit our curve or answer to observations. These formula are accurate only within certain ranges, outside of which they are unreliable. A formula that is correct is correct in all cases, and usually fits nicely with other known formula.What I am getting at is that the concept of "elegance" is talking about internal consistency. Mathematics is based on being internally consistent with itself, and being based on reality the assumption becomes that reality is internally consistent as well. This appears to have born out to this point in time, and the confidence in this trend continuing is the expectation of "elegance" in formula for natural laws.

This concept goes back to the extreme basics of philosophy and the way in which we think. For example, put aside everything that you know and start from scratch:The famous line is "Cogito ergo sum" or "I think, therefore I am." Since you are thinking you can reasonably conclude that you exist in any sense that will be meaningful to yourself. The next step is to consider what other things exist, and this leads us to our senses.Starting from within our mind, we have no way of determining if our senses are reliable. Most people will readily admit that they can make mistakes, but the more subtle point is that *everything* could be a perfect illusion to our senses ("The Matrix" is often referenced here). There is a certain leap of faith required to conclude that anything we observe can be trusted to be real.Once we trust our senses to a certain extent, we can determine that there are other thinking beings in the world. This lets us distinguish between consistent and inconsistent observations; for instance if you saw a fairy and 9 other people did not, you could conclude that your senses were simply being unreliable (as they commonly are). Note that this does not strictly prove that these consistent observations are objectively real, only that if they are illusions that they are shared illusions as opposed to solitary ones.Now for the concept of objective reality; this is again something of a leap of faith. Objective reality is something that exists independent of the mind, and if you will recall up to this point your own mind is the one thing you can most reliably conclude exists. Everything beyond that is to some extent going out on a limb. The concept becomes defined in an awkward fashion; our observations indicate that things happen and exist regardless of them being observed. This again is summed up by a famous saying, "If a tree falls in a forest and no one is around to hear it, does it make a sound?" Our observations indicate that it does, but that sort of negates the whole "lack of observation" concept in the first place.So to wrap things up we can be reasonably sure that an objective reality exists in the same sense that we can be sure anything exists, including other minds. This is to say we are not completely certain, but sure in every reasonable and useful sense. Determining this objective reality is done through weeding out individual experiences which are obviously not objective, and whatever we are left with is concluded to be objective (although it could technically be a shared illusion).

Let me take another stab at explaining what I was getting at.Some parts of mathematics are based on observation of reality. We would tend to only consider the factual parts (1 + 1 = 2), but there is also the concept of internal consistency that is an assumption of both mathematics and reality. We apply it to the field of mathematics with the expectation that reality is also internally consistent and will share those results.Consider very simple logical rules from which new, complex logical statements can be derived. This derivation is based on the concept that logic is internally consistent, so that these new statements need not be based on observation at all. There is still the expectation that they will reflect reality however: For instance, if we had derived that since we have thing "A" in this room, some thing in another room we cannot observe must have the quality "not A".That would be "new information" which we arrived at purely through logic. We might have observation to back up "A=A", but the expansion of logic based upon itself providing accurate results assumes that reality is also based on itself.Invented maths are not necessarily invented to reflect reality (the observation to back it up may not exist), but they are done so to be consistent with the rest of mathematics. The fact that they do reliably reflect reality is a strong indication that reality is consistent with with the rest of reality. The expectation that undiscovered portions of reality will be consistent with the rest of reality is what makes mathematicians expanding the field confident that those expansions will have any practical application whatsoever.So if I have managed to make any sense yet, I view mathematics as starting from observations of reality but almost immediately developing independently. What makes it useful is that mathematics develops in a way that inevitably reflects reality because they are both internally consistent; they are two puzzles with pieces that only go together one way, the puzzle being completed cannot help but reflect the master even without reference.

Yes, that would be correct. Conversely some aspects of empirical reality could be discovered through extrapolation of mathematics, under the assumption that both are internally consistent with themselves. A mathematical system which is not based on reality could be constructed and which would yield results without a link to physical reality. However, maths which are consistent with other maths that are based on physical reality are, by that virtue, linked to physical reality and can be expected to yield results consistent with reality.This is after all why those axioms are not completely arbitrary.

Seems that way don't it? Right, but aren't those also consistent with other branches of mathematics with more concrete links to reality?