[terryf]

Thank you for the thorough explanation!

I see your point with abstract expressions, but the connection I'm failing to make is - if they are completely abstract, then how can the equation be related to the physical world?

In essence, the equation defines the relations between some values - but if any or some of the values can be replaced by real measurements, then surely the relations between real measurements are not interchangeable?

[jasode]

if they are completely abstract, then how can the equation be related to the physical world?

Math equations do not have to be related to anything in the physical world. It's just that humans have found that some patterns in nature seem to match some equations. Why or how math describes the real world is a deep philosophical question![1]

When G.H. Hardy[2] was exploring certain aspects of number theory and properties of prime numbers, it didn't have any relevance to the real world. 100 years later, we now use the math concepts in public key cryptography.

Does Isaac Newton's "F = ma" of "force = mass times acceleration" describe the "reality" of gravity and how planets move in the solar system? Well -- it does -- and it doesn't. It seemed to match reality well enough to find the planet Neptune -- purely by using math. But Newton's math equations didn't seem to match the "reality" of Mercury's orbit.

Then Einstein came up with another set of math equations that matched the "reality" of Mercury's shifting orbit.[3] But now today's physicists notice that Einstein's formulas don't match the "reality" of galaxies rotating faster than the his formulas predict. Today's physicists are trying to come up with another set of math equations to match that reality (or find the elusive "dark matter" so that Einstein's equations remain unchanged.)

(Einstein's equations do not make Newtons's equations obsolete. Newton's math (that doesn't perfectly match reality) is still good enough to guide Apollo rockets to the Moon and back. It's just not good enough to fully describe how Mercury wiggles around the Sun.)

Math is not reality. Math is its own topic that can stand apart from "physics" or "engineering". (Hence, math is the ultimate abstraction.) It's just sometimes convenient for humans to map some mathematics to some realities. It's often helpful for us to do that. Sometimes it backfires. Some Wall Street guys "mapped" math equations to model the financial behavior of mortgages and they turned out to be wrong.

Put another way: If math is used to model the physical world, it is only an approximation for it.

[1]https://www.google.com/search?q=the+unreasonable+effectivene...

[2]https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology#Cr...

[3]http://physics.stackexchange.com/questions/26408/what-did-ge...

[naasking]

> Math equations do not have to be related to anything in the physical world.

Not to derail this into philosophy, but that position is conjecture. There are people who adopt a position where all mathematical concepts actually derive from physical structures, ie. a kind of mathematical physicalism. These people hold that the reduction of math to the physical may be obscure, like a reduction of economics to physics, but still real.