The fact that it has been proven that certain results in quantum mechanics cannot be explained by the use of real numbers and one must use complex numbers to describe what Nature is doing makes complex numbers a part of reality for me.
But suppose some piece of mathematics was not relevant for physics or indeed anything physical. Perhaps a certain differential equation simply had no relevance for physics, although we could study it mathematically the same as others that were physically relevant. Would that make it unreal? Are the ordinals real, if they have no physical application? The surreal numbers? Suppose that some particular natural number is so large that it will never appear in any equation or physical analysis undertaken by a human. Should we regard it as imaginary?
I think that in a sense, for one to regard a number as imaginary, it must be unseen by the physical world - for it to be seen will require that it is in fact real, and perhaps complex.
Let R int C =/= emptyset => there exists some e and d such that e and d are in R and C, it follows that one of these must be i or -i.
I think what you say raises interesting points about the definition of what constitutes reality. The human imagination uses a “real” substrate (the matter and physics that make up the brain), to conjure up an infinity of unreal objects. The possible thinking up the impossible. Can we and should we have a useful definition of “real”. ?
Further to the above- Max Tegmark’s mathematical multiverse ideas are an attempt at answering what is real. No doubt that could be challenged too but it provides possibly the widest definition of reality I know of.
The fact that it has been proven that certain results in quantum mechanics cannot be explained by the use of real numbers and one must use complex numbers to describe what Nature is doing makes complex numbers a part of reality for me.
https://www.nature.com/articles/s41586-021-04160-4
But suppose some piece of mathematics was not relevant for physics or indeed anything physical. Perhaps a certain differential equation simply had no relevance for physics, although we could study it mathematically the same as others that were physically relevant. Would that make it unreal? Are the ordinals real, if they have no physical application? The surreal numbers? Suppose that some particular natural number is so large that it will never appear in any equation or physical analysis undertaken by a human. Should we regard it as imaginary?
I think that in a sense, for one to regard a number as imaginary, it must be unseen by the physical world - for it to be seen will require that it is in fact real, and perhaps complex.
Let R int C =/= emptyset => there exists some e and d such that e and d are in R and C, it follows that one of these must be i or -i.
0 - sqrt(1) = -1
Take sqrt( 0 - sqrt(1)) => sqrt(-1) => i => -i
For each 0,1 real.
Unfortunately, your comments seem to make little sense.
By the negation of the multiplicative identity we have the above to be true - that is in that the complex may be derived from the reals.
I think what you say raises interesting points about the definition of what constitutes reality. The human imagination uses a “real” substrate (the matter and physics that make up the brain), to conjure up an infinity of unreal objects. The possible thinking up the impossible. Can we and should we have a useful definition of “real”. ?
Further to the above- Max Tegmark’s mathematical multiverse ideas are an attempt at answering what is real. No doubt that could be challenged too but it provides possibly the widest definition of reality I know of.