Introduction: On what is not
To write about what does not exist - and this is what we intend to do in this issue of Delta - we had to work out a clear understanding of the notion of existence. At the first attempt to define this notion we discovered that the meaning it has in each of the three areas represented in our journal is quite different.
Contrary to appearances, the easiest case turned out to be that of mathematics (although this does not necessarily imply that it is easy to verify the existence of something!) Whenever a mathematician states that an object exists, he means that no intellectual manipulation with its defining properties leads to contradiction, i.e. no two contradictory features can be deduced.
Let's explain: intellectual manipulation is simply an argument based on logic, whereas two contradictory features are features such that the definition of each of them can be obtained from the definition of the other by placing a negation sign in front of it. What is a logical argument, you may ask? This we leave without further explanation. Indeed, we do not want to fall into the trap of regressus ad infinitum - going infinitely back with non-terminating explanations.
In spite of this restriction (the question cannot be investigated at infinite depth), mathematical existence is a relatively simple notion: existence = absence of contradiction. Therefore we can state with absolute certainty that there is no odd number divisible by 4. Less obvious examples will appear in the articles that follow.
The notion of existence in physics seems to be similar, yet the meaning of contradiction is quite different. An object does not exist as long as experience does not corroborate the properties previously deduced from its mathematical model. Existence is thus much narrower a notion than in mathematics.
The construction of a mathematically consistent model for a physical regularity does not guarantee its existence. It is also required that the known facts do not contradict the properties of the model. It's almost as if a mathematical theorem were considered true as long as there is no counterexample to prove the contrary. The results, however, are quite satisfactory: due to this approach there is no absolute vacuum, no absolute zero (defined as the state of complete inertness), no perfect gas (satisfying Clapeyron's equation), even if each of these objects has a consistent mathematical model.
Astronomy is even more restrictive. To have the astronomer admit the existence of an object not only must there be a consistent mathematical model, not only must its deduced properties be not contradicted by observation, but in addition there must be no other conceivable model in sight that might explain the facts obtained by observation equally well. This is why from the astronomers' point of view there is no black hole in the center of our galaxy. It could well be there and this would explain many a fact, but whatever can be observed (and, alas, not much can) can be explained in several other ways too.
Thus we have different levels at which existence of objects, phenomena, forces or regularities can be accepted. Obviously, other disciplines of science and other areas of life provide still other meanings for existence. Nevertheless, one common feature must be stressed: whenever something is considered to exist or not within a given discipline, this is due to an arbitrary, although generally approved, convention. What is more, in each of the domains invaded by the human mind the notion of existence itself has undergone many evolutions. This, however, is quite another story.