Problems that are largely believed to be true, but are unresolved Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification?
It seems that Goldbach should be true, but this is based on heuristic justification.
I am looking for conjectures that seem to be true, but where the 'why' is something other then a statistical justification, and I want know what exactly that 'why' is.
Edit: Can you please include the reason it is widely believed to be true in your posting? That is the interesting part
 A: The rank conjecture for rational elliptic curves (for every $n\gt 0$ there is an elliptic cuve over $\mathbb{Q}$ whose group of rational points has rank at least $n$).
A: The Kakeya conjecture.  It states that a set in $\mathbb{R}^d$ containing a line in every direction has Hausdorff dimension $d$.  It is solved for $d = 2$ and open for $d \geq 3$.  
The finite field analogue was recently solved by Dvir (Tao made a very nice post about it, here).  Dvir's contributions have also led to a deeper understanding and an ultimate resolution by Guth and Katz of the Erdos distance problem in $\mathbb{R}^2$ (of which Tao also has a nice post, here).
When one constructs a Kakeya set of measure zero, it is typically visualized by Bourgain's "Bush" construction or Wolff's "Hairbrush" argument.  The iterations of these constructions has full Hausdorff dimension.
I believe a great number of people would be surprised if it were false.
A: The inverse Galois problem is to determine whether every finite group is the Galois group of some Galois extension of $\mathbb{Q}$. It has been proved that all finite soluble groups and all the sporadic groups except for $M_{23}$ (whose status is unknown) appear as the Galois group of some Galois extension of $\mathbb{Q}$; likewise it has been proved that all finite groups are the Galois group of some Galois extension of other fields, $\mathbb{C}(t)$, for example.
I'd be interested to know if there is a consensus about the truth or falsity of this problem, and if so what the consensus is.
A: If for some real x > 0 both $2^x$ and $3^x$ are rational integers then so is $x$.
It is obviously true (or seems so to me at any rate); but I think it's fair to say that a proof is nowhere in sight, unless there has been some recent progress.
I seem to recall that Ramanujan proved the similar but weaker result with $2^x$, $3^x$, and $5^x$ all rational integers.
A: It is believed, that $\pi$, $e$ and every irrational algebraic number are normal to every base. 
The truth is, that none of those numbers have been proven to be normal to a single base. 
A: What about $P \neq NP$? Scott Aaronson has made some excellent points at here
A: The Riemann hypothesis is largely believed to be true, and further conjectures have been made based on its truth (e.g. statements about the distribution of prime numbers) but no one has ever proved it.
A: It is widely believed that the fundamental axioms of set theory ZFC are consistent, but this has not been proved in ZFC and in fact provably cannot be proved in ZFC itself unless ZFC is inconsistent, by the second Gödel Incompleteness theorem. 
Indeed, whatever fundamental axioms you favor, whether PA or KP or Z or ZF or ZFC or ZFC+large cardinals, then it is natural to suppose also that since you believe that those axioms are true that you also believe that those axioms are consistent, but this is provably not provable from your axioms, unless they are inconsistent. 
A: Most of us believe that $e+\pi\notin\mathbb{Q}$, but this is not proved yet. Otherwise, this will lead to very interesting consequences.
A: I believe that the Jacobian Conjecture is widely considered true (is even given as a double-starred problem in Hartshorne's Algebraic Geometry book) but it has resisted attempts.
By the way, I haven't checked in a while, so what is the current status of the Jacobian Conjecture?
A: A conjecture believed to be true is the Iwaniec conjecture. I have been studying it recently, and it states that the $p-$norm of the Beurling-Ahlfors transform is $p^*-1$, where $p^*=\max(p,p')$ and $p,p'$ are Hölder conjugate pairs. This can be written as the inequality
\begin{equation*}
\int_{\mathbb{C}}|\overline{\partial}f|^p\leq (p^*-1)^p\int_{\mathbb{C}}|\partial f|^p.
\end{equation*}
Note that the case $p=2$ is true (we have a satisfied Hölder conjugate pair). The problem is still open, but is believed to be true for various reasons. The analysis of quasiconformal mappings (via Astala's distortion theorem) implies that the $p-$norm of the Beurling-Ahlfors transform is $(p^*-1)$. The conjecture has been completely settled for so-called "stretch" functions, ie, functions $f:\mathbb{C}\to\mathbb{C}$ of the form $f(z)=g(r)e^{i\theta}$ where $z=re^{i\theta},$ and $g$ is a nonnegative locally Lipschitz function on $(0,\infty)$ such that $g(0)\equiv g(0+)=0$ and $\lim_{r\to\infty}g(r)=0$. There are various estimates of the norm that are approaching the desired bound, and include topics such as Littlewood-Paley inequalities, martingales, and Riesz-Thorin interpolation. All these factors lead people to believe the conjecture to be true since we are approaching $p^*-1$, but it is still open. 
The full Fourier restriction problem is believed to be true. This problem seeks inequalities of the form
\begin{equation*}
||\widehat{f}||_{L^q(\mathbb{S}^{n-1})}\leq C_{p,q,n}||f||_{L^p(\mathbb{R}^n)}
\end{equation*} 
for a constant $C$ depending on $p,q$ and $n$, and $\mathbb{S}^{n-1}$ is the unit sphere in $\mathbb{R}^n$. There has been progress towards this result suggesting it is true for any $p,q,n$ with famous progress made for the case $q=2$ known as the Tomas-Stein restriction theorem. 
