Are mathematical models sufficient to create general artificial intelligence?
I am not sure if it is possible to represent e.g. emotions or intuition using mathematical model. Do we need a new approach in order to solve this problem?
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Mathematical models are essentially highly formalised knowledge. When it comes to computer engineering, there is literally no other choice - anything you can write code for, or design a machine for, will have an associated mathematical model. That model may not be fully explored or comprehended analytically by theorists, it may be just too complex (and driven by calculations) or even mathematically intractable. However, that doesn't make it non-mathematical, just more driven by empirical results than theory.
We don't have complete models for how AGI definitely work, nor tight enough definitions of general intelligence to base maths from where we can say "if we implemented a framework based on this maths, we could build an AGI". Right now, exploration and experiments based on intuition of what might work are far ahead of such theory.
The theoretical work behind e.g. neural networks is chipping away at the problem, and there are more general over-arching theories about intelligent rational behaviour available e.g. the equations of AIXI. AIXI doesn't cover emotions and intuition directly, but does attempt to cover knowledge and how a rational agent would approach understanding the world in general from scratch. It is possible that an embodied system driven by a software implementing something like AIXI could exhibit intuition and emotions in an emergent fashion, but whether or not that would happen in practice is not at all clear from the theory. AIXI is just one theory/model out of many, and I am not qualified to analyse it in depth, but its creators have strong pedigree in AI research, so it is as good as any IMO if you are interested in starting to research AI from a theoretical perspective.
Despite all the unknowns, the success of deep learning and the loose analogies between artificial neural networks and biological ones, makes it look likely that neural networks or something like them will form a component of an AGI. The current state of the art for more narrow problem solving, learning from examples or experience through backpropagation of error gradients, might not be the most important or key component. However, whatever the structure, whether it is a small extension of existing systems, or involves some new science, it will be describable mathematically.
The doubts in the OP are eligible, mathematical formulas are not the strongest language mankind has invented. They can only prove other formulas created before, but not bring wisdom to the world. Apart from the AIXI formula, the self-referential axiom of Kurt Gödel is an example, what mathematics can't do. Even if the formula of Gödel is right, is is not the pathway to Artificial General Intelligence. And yes, we need a new approach to solve the problem. The best language for expressing complex issues is normal English and additionally screenshots of working programs.
It remains to answer the question what exactly AGI is. One answer is, that it is case-based reasoning system with the feature to reason about his own sourcecode. Jürgen Schmidhuber calls it a Gödel Machine.