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 by that item's existence. For example, a particle which had mass and traveled at the speed of light would be in violation of our current knowledge of relativity. However, there is nothing inherently and ultimately logical about relativity. It is easy to imagine an internally consistent system with different parameters than ours has. Such a system is thus not mathematically impossible to create. It is merely physically impossible to create. However, all things that physically exist must also mathematically exist.
This existence difference is reflected in the way one resolves the existence of something apparently impossible. If one is looking at something which seems logically or mathematically impossible, the proper course of action is to find where the definitions had gone wrong, since usually the problem is that two different logical descriptions or definitions have been used, or the situation has been misunderstood. If a physically impossible event is occurring, and it does not look to be a logical contradiction, the proper course of action at that point is to look for where human understanding of the laws of nature has gone wrong. Much scientific progress occurs due to events that are not supposed to have physical existence at all. Seemingly mathematically impossible events, on the other hand, are simply human mistakes requiring clarification.
Unfortunately, this formalist definition of existence, while generally functionally acceptable, runs into contradictions in generalized situations. Specifically, Gödel's incompleteness theorem shows that it is impossible to construct an entirely complete and self-consistent set of postulates from which the rest of mathematics may be derived. So while the freedom from contradictions seems to be necessary for something to mathematically exist, it is not sufficient, and a system of mathematics cannot be wholly derived from noncontradiction. Some mathematicians, called intuitionists, insist that any proof must demonstrate the existence of something by constructing an example of it, rather than by showing that it contains no internal inconsistencies. Of course, this is somewhat more difficult.
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