As such, you can construct infinitely many "worlds" and theories using it.
It is not that mathematics maps to some real world observable phenomena; it is that some gifted observers of the real world phenomena have successfully formalized the observation into a formal mathematical expression. If you want your math concoction to have any bearing to the real world you start with the real world.
Now, as I said, in math you can construct infinitely many "worlds", it sometimes happens that SOME of the constructed worlds do correspond to yet not-formalized or not-discovered phenomena.
Case in point: non-Euclidean geometry (Lobachevsky) turned out to be a fitting model for general relativity.
The real challenge is in formalizing the real world observable phenomena .. sometimes there are simply too many parameters we can not properly model and we are forced to leave them out, resulting in an approximative model. How good is approximative model is open to interpretation .. while "better than nothing" can also result in misplaced adherence and belief in a flawed model.
"While certain mathematical models—like Ohm’s Law—map well to observable phenomena..."
Sounds like there was a mathematical model before the phenomena.
The "model" is just an equation containing multiplication, it can be model of anything (like F = m*a, D = v*t, etc.) it is through context (observation of phenomena) where variable/letters get assigned meaning and interpretation in the real world.
If you have not read: Eugen Wigner - THE UNREASONABLE EFFECTIVENSS OF MATHEMATICS IN THE NATURAL SCIENCES
You certainly express it differently, but I think what I was trying to get at is that the mathematical model is a formalism, and an approximation that works in the real world, is found to be applicable, and does not exist as some Platonic form waiting to be discovered.
If we take such discussions beyond pragmatics we get into what I call the metaphysical mire, with ill-posed ideas. Similar I think to what Wittgenstein called language games - maybe the only clear thing he ever said, but I could be wrong about that.
In my take on things, mathematics is an invented tool.
In a past life, I could wring the I out of a circuit having R, just by applying E. ;-)
I have encountered a few brief references to Wigner regarding this topic, but have not read him. I am an old guy now, and probably will not get around to it. Certainly he sounds very pertinent to this discussion based on short summaries of his work that I have come across. I have read a fair bit of philosophy in general, for a layperson with only a little formal training in the discipline, and only some 101 level physics.
You have it backward.
Mathematics is a formal (symbolic) system.
As such, you can construct infinitely many "worlds" and theories using it.
It is not that mathematics maps to some real world observable phenomena; it is that some gifted observers of the real world phenomena have successfully formalized the observation into a formal mathematical expression. If you want your math concoction to have any bearing to the real world you start with the real world.
Now, as I said, in math you can construct infinitely many "worlds", it sometimes happens that SOME of the constructed worlds do correspond to yet not-formalized or not-discovered phenomena.
Case in point: non-Euclidean geometry (Lobachevsky) turned out to be a fitting model for general relativity.
The real challenge is in formalizing the real world observable phenomena .. sometimes there are simply too many parameters we can not properly model and we are forced to leave them out, resulting in an approximative model. How good is approximative model is open to interpretation .. while "better than nothing" can also result in misplaced adherence and belief in a flawed model.
Not sure I have it backwards, since that was pretty much what I was trying to say, although you have said it well, and perhaps more precisely,
The sentence:
"While certain mathematical models—like Ohm’s Law—map well to observable phenomena..."
Sounds like there was a mathematical model before the phenomena.
The "model" is just an equation containing multiplication, it can be model of anything (like F = m*a, D = v*t, etc.) it is through context (observation of phenomena) where variable/letters get assigned meaning and interpretation in the real world.
If you have not read: Eugen Wigner - THE UNREASONABLE EFFECTIVENSS OF MATHEMATICS IN THE NATURAL SCIENCES
You certainly express it differently, but I think what I was trying to get at is that the mathematical model is a formalism, and an approximation that works in the real world, is found to be applicable, and does not exist as some Platonic form waiting to be discovered.
If we take such discussions beyond pragmatics we get into what I call the metaphysical mire, with ill-posed ideas. Similar I think to what Wittgenstein called language games - maybe the only clear thing he ever said, but I could be wrong about that.
In my take on things, mathematics is an invented tool.
In a past life, I could wring the I out of a circuit having R, just by applying E. ;-)
I have encountered a few brief references to Wigner regarding this topic, but have not read him. I am an old guy now, and probably will not get around to it. Certainly he sounds very pertinent to this discussion based on short summaries of his work that I have come across. I have read a fair bit of philosophy in general, for a layperson with only a little formal training in the discipline, and only some 101 level physics.