How do you think about time as an axis/dimension? I'm studying mathematics, but (obviously?) have an interest in physics.
I've been thinking, how do people in physics think about time as an axis or dimension (as they would be called in mathematics)?
Since to me it seems that in many (mathematical expressions of) physics concepts the time dimension is not "explicitly seen". It's stated, but not visually graphed (like in $\mathbb{R}^3$ one'd know that there's three axes and only three) and in some calculations time may be a result or time may vanish.
 A: You need to distinguish between relativistic and non-relativistic scenarios.
In relativity spacetime is a four dimensional pseudo-Riemannian manifold. We can choose any four dimensional coordinate system, but we will always find three of the coordinates are spacelike and one is timelike. We do not treat time any differently from the spacelike dimensions, though to keep equations dimensionally consistent we normally use the coordinate $ct$ rather than just $t$, where $c$ is the speed of light. Note however that the timelike dimension need not be a time in the sense that it records what is shown on a clock. For example in the Kruskal-Szekeres coordinates, commonly used to describe the Schwarzschild geometry, the $v$ coordinate is timelike but does not correspond to anything recorded on a clock.
In non-relativistic physics we commonly treat time as something that a three dimensional system evolves in, and we don't regard it as a dimension in the same way as the spatial dimensions. However you need to bear in mind that this is a low energy approximation made for convenience.
A: The first point I would like to make is that there is no known theory of time. Period. What we do have is a very good understanding of how to measure time. Our clock making skills have been improving steadily over the past centuries and clocks are, probably by far, the most precise metrology devices that have ever been produced by man. 
Lacking a good theory we have to keep close to what we know, which is this: Time is that which a suitable clock shows. What is a suitable clock? It's a clock that agrees to the highest possible degree with other clocks independently of their environment. 
So what is a clock? A clock is a special kind of system in perfectly constant disequilibrium. 
One of the easiest to construct clocks is a gravity clock with an upper and a lower reservoir and a flow regulating device between them. If the upper reservoir is being held at a constant level and the flow regulator is good, such a (e.g. water) clock or an hour glass filled with sand can be fairly precise. The next higher level of precision can be achieved by using periodic systems with a high Q (quality factor). In the past we have been using planetary motion, mechanical pendulums, piezoelectric crystals and various forms or atomic clocks. The planetary clocks rely on an enormous amount of stored kinetic energy for their operation while all lab-sized clocks, be they mechanical, electric or atomic need an energy source and a very precise mechanism that can supply exactly the same amount of energy with each period that gets lost. The latter mechanism has to operate in such a way as to not to reduce the quality factor of the periodic system too much, which requires quite a bit or artistry of design by the metrologist. 
Why is this important? It is important because it immediately identifies the crucial difference between spatial coordinates and time. Spatial coordinates can be measured with a simple, passive and, to be honest outright "dumb" rod made from baryonic matter. Time measurements, on the other hand, require a sophisticated active machine which is powered by an energy source and which therefor has two thermodynamic reservoirs at different temperatures. 
Hence time is by no means even remotely similar to space and every theory that pretends that it is, misses 100% of the difference. 
Now, the good news is that there is no such theory, even though there are quite a few theory books that pretend that there is. 
One can classify our physical theories into two types. One type assumes that there is one global time, that's classical mechanics and non-relativistic quantum mechanics. In classical mechanics and non-relativistic QM time is a very special one dimensional parameter that exists before and independently of everything else. We certainly do not pretend that it has similar properties as the spatial dimensions that are also part of these theories. 
The other type is more precise and acknowledges that moving clocks have a history dependent reading. This would be special relativity, general relativity and, so I hope, relativistic field theories, although I have not seen an easy to understand analysis of the moving observer problem in QFT. If one reads the original works by Einstein, he gave clocks a central operational role in defining time. Time is not an abstract parameter. It's a locally measured physical observation using a special device that moves with the observer. 
Where all of this went slightly wrong in the teaching of relativity was with the introduction of Minkowski diagrams and the introduction of metrics. Those are tools to describe the mathematical dependencies between space and time, but they are not space and time themselves. Unfortunately people have a bad tendency to mistake one for the other. My high school physics teacher pointed out to me that many students forget about the definition of force by Newton's second law as soon as they see a force gauge. In their minds force becomes something that is being defined by the elongation of a spring, rather than the acceleration of a mass. The same thing seems to happen when we introduce students to metrical theories where time suddenly becomes a linear object in a tangent space that is equal to space. In reality, of course, space is still defined by passive rods while time always stays defined by actively ticking clocks. 
The question that remains to be answered is how nature transports "time" from one observer system in a consistent manner into all other systems. Why can we synchronize two clocks, then move them far apart and, up to relativistic motion effects, keep them extremely well synchronized for a long (cosmological!) time? That is the real question in my opinion. How does nature do this transport? There are a few ideas out there along the lines of quantum entanglement, but I am not going to be able to relay them appropriately. I have to admit that I do not understand them conceptually well enough to say anything about that, yet.
What is important is to keep in mind that the nature of time is inherently linked to the existence of non-equilibrium, which is inevitable for any kind of physical dynamics. Once in equilibrium, dynamics stops, and with that the ability to differentiate the past from the future stops. All clocks stop or, at the very least, they become babbling physical systems that go forth and back around a thermodynamic average without any memory of their past state. If we can explain the difference between this state and dynamic states with a well defined directionality in terms of quantum mechanics, then we will have a valid theory of time. 
