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
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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.
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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?
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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 $n$).
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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.
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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.
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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.
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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.
Most of us believe that $e+\pi\notin\mathbb{Q}$, but this is not proved yet. Otherwise, this will lead to very interesting consequences.