Logic and Empiricism

This is a topic that comes up again and again, so I thought I'd say a few things. First of all I must deal with statements "mathematical construct" and "Is maths reality?/maths is not reality?". These terms are used very vaguely, so rather than trying to break it down I'll explain how maths is used in theoretical physics, with examples and use it to discuss these statements.1. Mathematical statements about physical quantities
An example of this kind of statement is "momentum is a one form". In mathematics you can commonly switch between coordinates. An example is the 2-D plane where you can use Cartesian coordinates or polar coordinates: Now various things can live on the plane, functions and vectors for example. When I switch between different coordinates vectors change in a specific manner. One-forms are objects that change in the opposite manner (in a sense).
Physics often require transformations between coordinates. An example would be moving between the coordinates of somebody on a train and somebody on the ground. When you do these transformations momentum changes like a one-form. Hence we say momentum is a one-form. In a similar sense we'd say area is a real number. When a surveyor says a field is 100.24 m^2 nobody goes "A 100.24!, what is a 100.24?, it doesn't exist. Maths only models reality!" So I'll take it that these kind of statements aren't controversial.2. Mathematical definitions of new physical quantities
This would be related to your example of an electric field, which would be a vector valued function satisfying Maxwell's equations. Energy is simlarly defined and yet most people have no contentions about it. Quarks would be similar.These are the things that often recieve the question "Are they real?".
At the end of the day it's like asking is a cereal bowl "really" a cereal bowl or just a collection of atoms. Are the words "cereal bowl" just a useful linguistic-mental tool to describe this collection of atoms and their interactions or does it reflect some deep truth about what is actually there?
I don't think that scientists don't care about philosophical issues, it's just that the "Is it real?" one is very old and doesn't really go anywhere and is applied unevenly (e.g., nobody stops to ask is velocity just mathematical construct).
Compare it to much deeper issues such as how much of our universe's physics is forced/implied by symmetry.

Well extra-dimensions usually have a direct effect on particle interactions. I'd have to have somebody give me an example of an entity from a physical theory that has no observable consequences or presence and yet is a required as an axiom for the theories foundation. Again dark matter and dark energy have detectable effects that characterise themselves. Dark Matter itself having been detected recently in the bullet cluster.I honestly can't think of an example where there is some entity that cannot be observed and yet the theory needs it at its basis.

Ah, I see what you mean. Usually no, physical constraints work in such a way that different theories rarely give the same answers to a question.I'll answer for the extreme first. If you have two theories which give the same answers for every possible question that exists within their domain of explanation then they always turn out to be actually the same theory, simply expressed in two different mathematical languages. It is simply that the languages emphasize different things.
An example would be wave-mechanics and matrix-mechanics. Both are actually Quantum Mechanics, with different primary entities. Think of a object-oriented language and a verb-orientated language both describing a goat going up a mountain.The less extreme case, that they agree on somethings, is rarer. You'd be talking about the point in time when we have two theories where their area of agreement has been tested and verified, where as their area of disagreement remains untested. As I said this is rare and even still it is not much of concern for the following reason.
Either:
(a)One theory is always an emergent of another. Let's say I had a String Theory that agrees completely with the Standard Model of particle physics on low energy scales. Is the Standard Model right or is String right? Is it truely point Particles or is it Strings? The thing is that the Standard model naturally tells you that it gives way to some other structure at high energies. It states "Particles behave as points up to a certain energy scale", this is also the statement String makes and that is the statement you have empirical support for.(b) They talk about the same stuff, they only give it different rules. An example would be General Relativity and Brans-Dicke Theory. They don't agree, however they still talk about the same stuff.
i.e. "curved spacetime", e.t.c.
So you don't have the question of "Which theory's objects actually exist?"Edited by Son Goku, : Minor punctuationEdited by Son Goku, : Spelling