This is an old idea (e.g. Euclid). The "modern" part of it is that the definitions are not assumed to be true outside the neighborhood. This is simple and powerful because the results are larger worlds that can be compared to others and to phenomena and experiment (and without setting up dogmas and religions).
Because of the way human minds work, there will be tendencies to think the definitions are "actually true" (and so the logic inside the boundaries) if they and the conclusions are appealing. But the form of this knowledge helps keep us saner if we are diligent about drawing the maps and boundaries correctly.
Much of science has this character, and the model helps to understand what it means to "know" something scientifically. Science is a negotiation between "what's out there?" and what we can represent inside our heads via phenomena on the one hand and the "boundaries and neighborhoods" on the other. Einstein's nice line about "math vs. reality" hits it perfectly ("As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.").
Newton's Principia was a huge step along these (and many other lines). He completely separates out the math part in the first large parts of the work. And only then does he start looking to see how the math models map onto observed phenomena.
To say it again, Science is partly about being very careful about how the definitions map to "out there".
In terms of context, if you are aware that you are in contexts -- the first step! -- and aware and careful about the ones you are using -- the next steps! -- there is a chance that "reasonable thinking" might happen.