Is there any mathematical meaning to words like: Event or Cause or Effect etc If you read any text on Physics then you would reorganize after enough time that  Yes language of physics (Nature) is mathematics. I'm letting this discussion to opinion-based nor some philosophy of science.
So when I say the word energy I know How do I say the same thing mathematically or velocity or position etc. But I found some of the words that play a central role but I didn't found any mathematical words to describe them Like in special relativity.

A reference frame is a way of labeling each event with its location in space and the time at which it occurs.

I'm here concern with the word event. What do we mean by an event? The emission of the photon is an event, the transition of an electron is an event, electron kick out of the atom is an event. If there is a thing, there is an event. The example seems plausible, but How do we define an event (Mathematically if possible)?
I have a similar problem with the word cause and effect. See the following sentence.

To be able to make sense of cause and effect, only events in our past can influence us, and we can influence only events in our future.

This doesn't make sense to me what we mean by influence. I know what it meant in a daily sense, If a bullet gets through me, it's going to influence me, But really I don't get it mathematically when do I say something influencing me.
Can anyone help me through this?
 A: Physics uses mathematics as a tool, a language as you state, in order to describe nature, phenomena, and to be able to predict phenomena.
Mathematics is a discipline where  starting with a  self consistent  system  of mathematical functions and   a few statements called "axioms",  theorems can be proven, and any possible arithmetic consequence with the particular set up can be calculated.
Take Euclidean geometry, the first complete example of a mathematical theory, at the time physics and mathematics were not separated. It has its axioms, and a series of theorems can be proven , and a theorem can become an axiom turning the axiom into a theorem. It is a closed mathematical system, where one can prove or disprove a statement , true or false.
After the time of Newton physics used mathematics as a tool, and organize measurements and observations , and become predictive of new situations. To do this, new axiom-like definitions have to be used to get a theory of physics. Starting with Newton's time, these have been called "laws", "principles", "postulates" to pick up from the myriads of mathematical solutions, those solutions that were descriptive and predictive of data.
Important consequence is that one can never "prove" a physics theory, only validate it, because it is as valid as its axioms and axiomatic statements; if new data disagree , the theory  is falsified.
Also axiomatically, one has to define what "cause and effect" "event" etc mean in the theory built up using mathematics. It is futile to ask for a mathematical definition of "event", it can only be defined for the particular problem under calculation. It can be the merging of two black holes, or the scattering of a proton on a proton. It is  a metalanguage to the mathematics used.
"before" and "after" also are assumed from the everyday language axiomatically, taking time as a running coordinate. The are axiomatic concepts, and cannot be dressed in a mathematical format , except in particulular cases (t>5 or t<5).
A: Mathematics starts with undefined entities like "points", and some particular defined relationships among those entities, and looks at the implications that have to be true given those entities and the chosen axioms about them.
Physics is the same way. We can explain some of what we observe in the real world by making particular assumptions about assumed entities. But we cannot possibly define the starting assumptions.
So we have charge, mass, space, time, etc. We can assume relationships among them that imply relationships which fit our observations, but we can't say anything about the original assumed entities except the relationships, which are both predicted and observed.
I think mathematically an "event" occurs at a particular time and place. There are various ways to measure times and locations, but I expect that if one of them separates two events at different times and places, and another measurement says they are at exactly the same time and place, one of them is wrong.

The emission of the photon is an event, the transition of an electron is an event, electron kick out of the atom is an event.

This is probably a convenient approximation. Probably emission of a photon takes an interval of time, and it happens in a bigger space than a point. Maybe we want a mathematics that is less precise than using infinitesimal points.

I have a similar problem with the word cause and effect.

Physics as it's done today finds correlations between events. This could often be interpreted as acceleration of charges at a later defined time and distance, by charges at an earlier time and distance. This is a reasonable interpretation, but the math does not say that the earlier events influenced the later ones. It only reports the correlation. Cause and effect is an idea that we impose on the math. It's compatible with physics but not decided by the physics.
Consider a particular event that has a charge present. Acceleration of the charge is correlated with charges at particular other events. All observers agree about which events correlate with the acceleration at this event. But they disagree about how to label those events. Special relativity is only about the labeling.
