What is a wave? What is a particle? I am reading a David Bohm book on quantum theory. He says the idea that light is both a particle and a wave is incompatible: 
(1) we know light has particle-like properties through the photoelectric effect 
(2) we know light also has wave like properties because of slit experiments. 
He then explains why they are incompatible. 
But what is a "particle"? What is a "wave"? What do these terms mean precisely?  I know what they mean loosely. A particle means something occupying a spatial position. A wave is like a density or something defined with peaks, troughs, and nodes over a spatial region. But I want something more rigorous and more accurate than these loose definitions so that I know what I mean when I use the term. 
I'd like the definitions stated like we state axioms in math, clearly and specifically. In math, I say a vector space is closed under addition if $x,y\in V$ implies $x+y \in V$. Those are very specific claims. 
Can someone do something similarly clear and specific with wave and particle?  
 A: The concepts "particle" and "wave" started from classical physics and from the everyday use of the terms, to begin with. A particle of dust got into one's eye, and the sea had huge waves.
Physics came into its reign when mathematics was seriously used to model observations. 
For classical physics "particle" means an entity with small mass and a center of mass tracked at coordinates (x,y,z) at time t. Solutions of kinematic differential equations described the trajectory with accuracy determined by experimental errors.
For classical physics, waves are modeled by sinusoidal functions, i.e. functions that were the solution of "wave equations", could describe the behavior of sea waves, sound waves, and finally electromagnetic waves. Classically a wave is a variation of a measurable quantity like energy, or electric field, in space at a given time t, and the theoretical models were very successful  in describing the observations of periodic energy distributions in bulk matter, and even in empty space ( electromagnetic waves).
Then quantum mechanics became necessary, from the discreteness of atoms, the black body radiation spectrum, the photoelectric effect it was finally understood that there were regions in the variables measured that displayed a quantization of energy.
It so happens that the equations that successfully describe the quantum mechanical state of matter were diferential equations with sinusoidal solutions, i.e. wave equations, like the Schrodinger equation. The solutions for the hydrogen atom were able to explain the spectral series adhoc assigned by the Bohr model, IF the postulate was assumed that the wavefunction squared did not  represent the energy of the electron at (x,y,z) at time t,  but a probability distribution. i.e. if one accumulated with the same boundary conditions a large number of measurements and plotted the (x,y,z) at time t distributions one would know how probable it would be to find the electron at that location. 
As an example, this is similar to taking a census of the population of a city by age, and gauging how probable it would be that the first person you meet will be 8 years old. The wave function's function is just that, to give probabilities mathematically which are checked experimentally, and have been very accurate. 
The "wave" part  confused and continues to confuse people, because they think that the quantum mechanical entity, the electron for example, is spread out according to the solution of the Schrodinger equation. This is a misunderstanding, as the double slit interference experiments show with incoming single electrons:

Note the top photo, where the electron impinges on the screen, it is one whole electron . The probability pattern accumulated though shows clearly the interference effect that is expected by the sinusoidal form of the wave functions describing the electron when it hits the slits and goes through one or the other.
The "particle" facet of the electron is that it appears as a point at ( x,y,z_0) of the screen, and the "wave" facet is the probability distribution displayed in its trajectories.
If one is becoming a physicist it is simple to accept this fact, that the microcosm behaves differently than the macroscopic world we are used to. Bohm was stuck on classical frameworks and tried to derive the quantum mechanical probabilities from an underlying classical description. He succeeded in reproducing the same results as the usual quantum mechanical solutions, but afaik his model is complicated  and limited, and cannot be extended into second quantization where the ball game has gone  now.
A: A particle is simply a piece of matter.
A wave is a type of oscillatory motion (vibrating up and down in crests and troughs).
Not too long ago in the history of science, scientists used to think that a moving particle moves in strictly straight line (provided that no other force is acting on it). They also used to think that electromagnetic radiation do not contain any matter and are just an expression of energy. Planck then gave his famous law that every moving object in fact moves as a wave (with oscillations) and not exactly a straight line. The heavier the object, the lesser the oscillation (and greater the wavelength). Electromagnetic radiation are in fact extremely light particles (called photons) which travel through space in oscillatory motion.
A: A particle is the smallest vibration of a quantum field.A wave is like a density or something defined with peaks, troughs, and nodes over a spatial region.
