Existence of Mathematical Objects
The philosophy of mathematics is a subject that has not for ones bored philosophers even as they take on different stances concerning the various presented concepts. One of the concepts that has brought intense debate since the previous century is the issue of existence of mathematical objects. Any individual who has indulged in deep thought concerning mathematics’ nature has most likely been surprised by its objects’ status. A common question that would arise is whether the objects that are involved in mathematics including functions, sets, and numbers, among others, are discovered or created. One would wonder whether such objects should be thought in the same way as planets and stars are, such that their existence and character is completely independent of human activities and investigations. On the other hand, one would wonder whether such objects should be thought of as objects of fiction, whereby their character and existence depend on what they are made of by their authors. Plato’s position on the existence of mathematical objects has greatly influenced the philosophy of mathematics through what is currently referred to as “Platonism” and this paper tends to agree with his take on the existence of such objects as projected by the provided evidence. As such, this paper upholds that Mathematical objects exist as evidenced by the independence of numbers, abstractness, and the ontological nature of numbers.
Do Mathematical Objects Exist?
Ontological Nature of Numbers
One of the positions that has been embraced in the justification of existence of mathematical objects is the ontological nature of numbers. With reference to the Platonic view, Friend asserts that, mathematical objects exist. This is evidenced by the ontological nature of mathematical objects including sets and real numbers. According to Platonists, almost all mathematical objects are naturally part of a collection, including the cyclical group of order 20, the sets, and real numbers. As such, if model theory terminology were to be utilized, it could be said that almost all mathematical objects form part of mathematical domain elements. As such, mathematical objects that are within the different domains maintain different relationships among them and uphold certain properties. Such mathematical properties which distinguish the objects are also deemed to be mathematical ontology items. This take highlights the mere idea that numbers exist based on the categorization that such numbers are accorded, like other objects in the physical realm are. Different philosophers in agreement with the platonic approach have taken a rather different approach to the subject, suggesting that fundamental items in mathematics ontology should be referred to as structures and not objects. According to this approach, it is suggested that mathematical structures for the subject matter of mathematics. As such, the single mathematical entities such as the complex numbers form places or positions on within such structures. Proponents of this new approach suggest that a greater dependence is exhibited between such positions of structures than they are exhibited between mathematical domain objects as suggested by object Platonists. Proponents of structural Platonism, structuralisms, maintain that different places of the structures uphold various structural properties.
These properties are properties that are common among systems that are definitive of a specific structure. As such, each place’s identity is defined by its structural properties. The aspect of properties as brought forth by both structural and object Platonists reveals that for objects to be termed as existent, they must uphold certain properties, which can be used to define them. The fact that various mathematical objects have different properties which allow them to be categorized differently is a clear indication that such object are in existence. Case in point, the square number “1” (one) is “1”, and the square of number “0” (zero) is “0”. Such numbers share properties in that their squares give them the same number. Such properties a definitive of existence of number one and number zero in the first place. As such, mathematical objects exist as evidenced by the ontological nature of numbers.
Abstractness of Mathematical Objects
Another stance adopted by the platonic justification of mathematical object as existing is the view that such object are abstract. From the philosophical definitions, the term abstract refers to art, and thus, no clear way is in place for one to critically address the meaning of abstractness. In literature, different philosophers have used the term abstract differently. Plato believed that there exists a different world beyond the world that people see and know, in which things occur in their concrete status, and that the world that people see and the objects that seem to exist are only abstracts of the objects that exist in another dimension. As such, it is from this take that mathematical objects have been viewed by Platonists as existent ion their abstract nature. The term abstract as used in justification of mathematics is deemed a collective concept that comprises of other concepts.
One of the most important concepts related to abstractness is the concept of “non-spatio-temporality”. When referring to this concept, items are not required to stand entirely beyond the scope of spatio-temporal relations. As such, being non-spatio-temporal entities, mathematical objects can stand in relations, which are solely, non-formally temporal, with spatio-temporal entities. Another concept involves “acausality”, which asserts that an item cannot exert a strict influence that will cause a certain effect on any other item, and neither can it be directly influenced through a strict causal influence from another item. As such, any relation that is deemed strict causal is one that can only be established between spatio-temporal entities. Case in point, a child as a spatio-temporal entity can through a stone, which is also a spatio-temporal entity, and cause it to move from the child’s position to a new position. Nevertheless, when it comes to mathematical objects and structures, such as numbers, one cannot engage in a physical activity such as throwing them.
The concept of “eternality” has also been used to justify the abstract existence of the mathematical objects and structures. In this case, such objects and structures are either defined as omnitemporal, which means that they exist at all times, or atempporal, which means that such objects exist outside any temporal relation networks. The concept of “changelessness” under the abstractness of mathematical objects asserts that the intrinsic properties of the objects do not change. In this case, the intrinsic properties of the objects refer to the properties that are held independent of the objects’ relations to other items. The last concept in this case is “necessary existence”, which asserts that the item had to exist. Given that the mathematical objects have most if not all of these features, which are central to abstractness, then it could be termed that such objects are abstract and therefore in existence.
Independence of Mathematical Objects
According to Gold and Simons, all mathematical objects are independent in nature, an aspect that makes them existent. As such, most philosophers agree that certain objects exist independent of human constructions, and mathematical objects including numbers, are independent of human constructions and that one would not be mistaken to presume that they exist. The common account in establishing independence is that if an object M is independent of an object N, then even in circumstance where N did not exist, M would still exist. As such, this notion of independence aims at maintaining that mathematical objects would exist even in cases where physical or mental rational activity were absent, an aspect that makes it independent of such activities, yet existent. As such, proponents of this idea maintain that mathematical objects would still have the features that they currently have even if the rational activities that currently exist were not present, or if other entirely different rational activities were in existence. As such from this perspective, it is held that the mathematical activities are the determinants of how the mathematical real is established into properties, objects, and relations.
Anti-realists have come out strongly in their opposition of the existence of mathematical objects. Their claims revolve around the notion that mathematics is a social construct and thus such does not exist. To address this issue, a metaphysical account is established to define the relationship between mathematical and spatio-temporal realms. According to the established account, the metaphysical gap that exists between mathematical and spatio-temporal realms in impenetrable. Thus, these realms cannot have causal interaction between them. It is important to observe that this is the adopted notion in suggesting that mathematical objects are abstract. As such, anti-realists suggest that existence of such an impenetrable gap makes any ability of human beings to obtain mathematical knowledge and justify formed mathematical beliefs completely mysterious and out of order. The gap separates any form of interaction between the two realms, such that no understanding can be established of one realm from the other. Proponents of this challenge against Platonism believe that human beings do not hold any clear understanding of the mathematical realm and thus any established knowledge or understanding of the realm is unjustifiable and unrealistic. Nevertheless, Platonists believe that such a challenge does not hold any water as supposing that the mathematical believes and knowledge of human beings are justified is natural. For instance, adding one apple to a basket with another apple would make them two apples. These apples are mathematical entities and adding them together, which is a mathematical function, gives a realistic number of two apples.
It is clear that the debate around the existence of mathematical objects is unique, given that concepts of mathematics do not physically exist and that justification does not involve any physical experimentation. Platonists’ approach to the issue concerning existence of mathematical objects provides a clear way of justifying such existence. According to Platonists, mathematical objects exist as evidenced by the independence of numbers, abstractness, and the ontological nature of numbers. The ontological nature of mathematical objects refers to fact that almost all these objects belong to a certain collection, which is defined by certain properties. This makes such objects existent since their properties define their existence. On the other hand, Platonists take on the idea of abstractness to define existence of mathematical objects. Abstractness involves various concepts, forming one cluster, which involve various features that define their relations with spatio-temporal entities, which make them existent regardless of the fact that such objects are non-spatial-temporal. Lastly, the independence of mathematical objects is brought out in the existence of each of the objects such that they can still exist in the absence of the mental and physical activities that currently exist.