Just my opinion but I think you are wrong; they are two of the same.

My (layperson's) understanding is that science does not claim to find The Truth, but rather that while in search of Truth (big T) it is finding that which is 'true enough' (small t), meaning true enough to produce reproducible results. From those 'truths' we get technologies which work in the physical world, but those truths are always open to reevaluation and refutation, providing the refutation is able to adequately explain all of the current and previously observed phenomena better than the previous 'truth' did.

Isn't that basically how we moved from a purely Newtonian physics to the world of Quantum physics? Things like black body radiation were just not adequately explained by Newtonian physics. Though it had for centuries been true enough for observed phenomenon it proved not to be so for the realm of the very small.

..without diving into philosophy, Mathematics 'exists' it could be said in a kind of mental or Platonic space..external to ordinary space (the mathematical language), whereas Science studies physical/material objects (which we agree by general consensus are 'real' with a different certainty to numbers)

like..show me 'where' a number is..
.
.

By "hypothetical", I didn't mean unreal. I just meant to say that any possible reality can be fully modeled and understood mathematically, but determining which model most closely resembles our reality requires experimentation. For example, using mathematics, I can give you a formula for the kinetic energy of a particle moving in n-dimensional space, but a physics experiment may be required in order to determine what n actually is (though even physicists don't always agree on it).


I agree with that.

I also agree that they are one in the same in that they both require creativity and intellect. In that sense, things such as music composition are also the same. Anyways, the same brain produces math, science, and music. Isn't that neat?

Formally speaking, one dictionary defines them as follows:
Mathematics - the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically.
Science - systematic knowledge of the physical or material world gained through observation and experimentation.

We could also look at the way these words are used.
Example sentences:
"X requires mathematics."
"X is a science."

Based on these two sentences, it seems to me that science and mathematics are not even the same type of thing.

^..manipulation of symbols..
(good post btw)

Great thread!

My thinking on it is this, and I agree with a lot of what has been said.

Science definitely deals with probabilities. A good scientist rarely, if ever, would say "x happens when y." Or, "Z happens because of q." The proper terminology is "smiles appear to happen because of cake, or are likely to happen when sex."

So, as has been said, science deals with what it likely to be true, but never deals with truth. One would need to actually see the future to be able to tell that the Earth will keep orbiting the Sun, or that the laws of thermodynamics will continue as they have (which, no mathematical proofs exist to explain the laws of thermodynamics, they just seem to work).

Math deals in more absolutes, but is seldom concretely true. Take, for example the value of pi. To derive the value of pie, one needs the diameter or radius of a circle to compare to the circumference. But where do the values for the ratio come from? Pi. So, the incompleteness theorem applies here, and to almost every proof in math. A mathematical proof is still dependent on outside information which is not provable.

Finally, math and science mingle, but never connect. Math is used as a language to describe physical phenomena. The math is not the physical phenomena. Many plants branch at the angle of Phi (about 130-ish degrees, Phi being an irrational number), but Phi is not a part of the plants.

I hope I've contributed something! That's how I understand things, but I need to chew over some more of these posts, for sure.

The connection to probability was good. The professor that I work under recently gave an interesting interpretation of the difference between theorists and experimentalists. He said something along the lines of: "Theorists use computer to compute the equations underlying physical interactions, but the physical system studied by an experimentalist is like the original computer of these equations." He didn't elaborate very much, but he laid out an interesting intepretation in which the equations are the reality, and computers and physical systems are two different ways of studying these equations. To connect to what you said, the equations that theorists attempt to deduce mathematically are automatically "calculated" by the physical systems that experimentalists study, though the computation generated by the physical system always comes with noise and probability when measured.