How do we know particles exist? Aren't they just waves? In the book "A Briefer History of Time" Stephen Hawking wrote:

The unpredictable, random element comes in only when we try to
  interpret the wave in terms of the positions and velocities of
  particles. But maybe that is our mistake: maybe there are no particle
  positions and velocities, but only waves. It is just that we try to
  fit the waves to our preconceived ideas of positions and velocities.
  The resulting mismatch is the cause of the apparent unpredictability.

Are there evidences that disprove this hypothesis? 
If true, would it eliminate most of the apparent quantum paradoxes, and necessity to "Shut up and calculate!" for those who attempt to interpret quantum physics with common sense?
Edit: I assume that S. Hawking is aware of Standard Model, and he considers this statement as a legitimate hypothesis. Are there evidences that prove that it's not? In other words, is it a philosophical or scientific question?
 A: It's not clear what sort of evidence could prove or disprove this idea. And that makes it philosophical, not scientific.
If someone were to develop an experiment by which we could distinguish between the two ideas, then the situation would change.
A: Hawking or not, the displayed statement is wrong. Wave theory was known two centuries ago and is very rigorous and predictive. It cannot explain the data for microscopic elementary particles.
What we call "elementary particles" at the microscopic level display properties of macroscopic type particles, as in this image of proton antiproton annihilation into pions: 

and properties of wave type interference, of the kind we see with macroscopic waves,  in innumerable different experiments, as this electron build up experiment, single  electron at a time. The build up shows an interference pattern.

The only consistent mathematical theory that encompasses both observations is that the quantum mechanical nature of "particles" is a probability function which displays spatial and timelike wave properties, mathematically as classical waves do, and particle properties under  different experimental conditions.
Wave mechanics cannot do it, i.e. fit all the experimental data, otherwise it  would have been done.
A: read "What is a particle" by Rovelli and Colosi and also just any book about quantum fields on curved space. The truth is that our particle interpretation rests on an artifact of being in a flatish spacetime. 
Sean Carroll's book on general relativity has a section at the end that is pretty accessible (albeit follows the book "Quantum Fields in Curved Space by Birrell and Davies rather closely") and outlines the notion that particle number is not conserved in a general spacetime. Thus it makes sense that Hawking would make a statement like this since his research prompted many of these ideas.
A: Contrary to what the cosmologist says, there are not waves [1], but just particles [2]. Everything around us has been found to be made of particles. The corresponding branch of physics is particle physics.
For a basic introduction to our current understanding of the structure of matter and the known particles
http://public.web.cern.ch/public/en/science/standardmodel-en.html
For a discussion of the wave-particle duality misconception and why there is no any real wave but particles
http://statintquant.net/siq/siqse3.html#x42-60003
[1] The wavefunction used in some formulations of quantum mechanics is not a physical wave, but an unobservable mathematical function.
[2] Quantum particles are not Newtonian particles. Quantum particles are not tiny balls and its motion is not described by classical mechanics.
