The term "existence" is a good example of how something familiar can take on a slightly different technical meaning in a mathematical context. If a book exists, for example, it has been written, and usually published. There is a physical manifestation of that book somewhere. But if a mathematical solution to a system exists, it is not automatically implied that there is any example of that solution which is physically observable. The existence of that solution simply means that it is a possibility, logically and mathematically, for that solution to happen. According to the formalist definition, something which exists is free from internal contradictions.

If it is mathematically impossible for something to exist, a logical contradiction must result from that item's existence: true = false, say, or 0 = 4. For it to be physically impossible for something to exist, on the other hand, some law of the universe must be broken...