Mathematics And Metaphysics

Mathematical Objects Are Not Real

People use math all the time in one way or another. The word mathematics comes from the Greek μάθητα, which means learning. Math provides people with a well-ordered method of thought as well as with elements that are of common use. Numbers, for instance, are mathematical objects of wide use in timing, counting and selecting things and/or people. The nature ofsuch objects, however, is not an evident one. Are those objects such as number, set and circle real? Are they merely abstracts? Were they invented, creations of the mind, or were they discovered as if they existed somewhere? It is the purpose of this paper, then, to explore these questions from a metaphysical point of view, as metaphysics is the science that deals with being as being. Themathematical knowledge necessary to follow the arguments here discussed is not vast. The logic of Aristotle will guide us through the task of exploring mathematical objects that, though simple, it is not uncomplicated.
Let us consider a set to begin with. A set could be defined as a collection of things. Collection, however, is not a precise definition. Collection is a synonym of set but it does nottruly define it. In fact, set is a mathematical undefined object. The difficulty of defining a set, however, does not reduce nor diminish in anyway its meaning and relevance in math. Set is simply a concept that seems to not to be able to stand in itself. In other words, for a set to be, it must always be accompanied by the preposition ‘of’. Set is a group or collection of things, but set in itselfis nothing either real or cognitional. Set is a universal concept, but it is not a substance. A set does not subsist either, as it is not a real being. A set can be composed by real beings but it does not make the set as such a real being. It is a holistic mathematical object, metaphysically speaking.
A set holds things (either abstract or real or both) together. It does not give any particularcharacteristic to its elements. For instance, a set of green things is a set because green things are collected under the statement ‘green things’. It does not mean that things are green because they belong to such a set. Moreover, an element that belongs to this exemplary set of green things receives its membership and keeps it as an accident. Membership of a set, therefore, can be conceived asan accident of a thing, but it only characterizes a thing (or some things) but it does not affect its (their) essence or its (their) existence.
The description of a set is not only determined by the characteristic of its elements, but also by the person who forms the concept of set. A set of green things can be made up of green big things. It is still a set of green things, but another personcould say that it is a set of big things. Though both are true, the initial intention of forming the set was to gather green things together and not big. It just happens that the green things in that set are also big.
A set of abstract things such as infinite numbers, elements in spaces of four or more dimensions and matrixes of infinite order, requires a proof of existence. The naming of the setdoes not make it existent, and one must be careful. For instance, one could say: “Consider the set of all odd numbers that can be divided by two” A careless person might start thinking about such set, and he or she could even label it without noticing that there is not such a set of odd numbers that can be divided by two. Therefore, sets’ existence must be proved mathematically first in order forthem to actually be.
Set does not subsist as a single concept. Every time a mathematician deals with sets, he or she is actually dealing with the elements that are in the set, but not with the set itself. There are many other topics referred to sets that are very intriguing, though out of the scope of this paper, such as the concept of empty set and the set of all sets.
Let us consider now...