Is Mathematics Invented or Discovered?

Is Mathematics Invented or Discovered?
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Is Mathematics Invented or Discovered?
Introduction
The debate on whether mathematics is invented or discovered has gone on for a long time. It is fueled by the ability of mathematician to develop a concept with no application in mind. After many years, the idea continues to be proven in various fields of science. The models or formulas that are developed display a connection with reality. There are some people who have argued that the mathematical concepts are invented by human beings, and then they are applied in the description of the real world phenomena. Other people have also argued that the exactness of the mathematical formulas in describing the occurrences in the world implies that mathematical concepts exist hence human beings are just discovering them. The debate on the philosophy of mathematics continues to rage on today, and the right answer continues to elude people. There exists no clear dichotomy between invention and discovery when discussing mathematical concepts. Their existence can be attributed to both invention and discovery.
Nature of Mathematical Objects
Mathematical Objects Exist
There are various philosophers that have advanced theories that support the existence of mathematical objects. Illustrious thinkers that include Roger Penrose, Emmanuel Kant, Plato, and Mill have all supported the existence of mathematical objects. The thinkers believe that the concepts exist in an abstract realm, and the role of human beings is to discover the concepts.

The existence of mathematical objects is discussed under rationalism and empiricism as well as logicism, intuitionism, and naturalism.
Rationalism and Empiricism
Rationalism provides that there are two types of knowledge; empirical knowledge that exists as a result of experience and priori knowledge that refers to knowledge that arises as a result of reason and not previous experience. Therefore, empiricism is much more concerned with the material universe while priori knowledge is logical (Baker, 2017). The priori knowledge is used to describe the existence of a phenomenon in the universe. Mathematical concepts form part of the knowledge because they refer to things that do not claim the universe. For example, when a person claims that “triangles have three sides,” they referring to something that does not exist in the universe hence there is no claim of experience. However, the world contains physical entities that capture the shape of the triangles.
Mathematics is used to capture the natural world. People derive formulas out of the need to describe the phenomenon that exists in the world. The formulas are then modified to capture the reality of the world upon making further discoveries (Baker, 2017). Therefore, mathematical concepts are closely linked to the phenomenon that already exists in the world. For example, Isaac Newton developed the mathematical concepts that are used to calculate the force of gravity out of natural world experience. Archimedes, a Greek mathematician also came up with the Archimedes principle out of the need to explain displacement. Empirical knowledge and logical knowledge have to interact with a person to discover a mathematical concept. It, therefore, confirms that mathematical concepts exist in abstract forms and are subsequently discovered and refined by a human being through a process of logical reasoning. The mathematical concepts are then used to describe natural world phenomena that already exist.
Empiricism denies that mathematical concepts can be known priori. Mathematical concepts are products of empirical research. According to John Stuart Mill, the process of arriving at a mathematical concept is similar to the process of establishing facts in sciences (Paseau, 2017). Mathematical concepts embody empirical research, and this makes them indispensable when in the field of empirical sciences. One is not able to come up with a mathematical formula out of empty reasoning, they have to interact with different phenomena in the world, conduct research, and then develop a universal formula that will apply to a specific phenomenon. Research contributes to the stupendous accuracy of the mathematical formulas that are formulated by the mathematicians. Science utilizes numbers to make any of the explanation of the fact hence the necessity of the existence of numbers. Therefore, mathematics is used to give the best explanations of the human experiences in the world, and this attribute makes them be similar to any other science that utilizes empirical research and knowledge.
Logicism, Intuitionism, and Naturalism
The argument that mathematical objects exist and are not invented by humans is further strengthened by the arguments logicism, intuitionism, and naturalism. Logicism supports that mathematics can be known as part of knowledge priori but goes further to argue that the knowledge only constitutes a small part of the general knowledge in logic (Paseau, 2017). . The theory further observes that the knowledge in mathematics is analytic. The argument thus provides that mathematical objects are established through the same method that is used to establish facts in science. Mathematicians have to go through the experience, conduct analytical research, and then develop the mathematical objects. The mathematical concepts are thus derived from logical concepts that already exist. The derivation is a process of making the abstract concepts to have a clear meaning.
Intuitionism and naturalism further provide that there cannot be non-experience mathematical facts. The construction of mathematical concepts heavily relies on objects that already exist thus what mathematicians do is the provision of a logical construction of a formula that is used to confirm their existence. It is difficult to construct a mathematical concept from a non-existing phenomenon. The mathematical objects are thus to inform human beings on the empirical objects that they perceive in the world.
Mathematical Objects do not Exist
There are various arguments that have been advanced that provide explanations that mathematical objects do not exist and are thus invented by mathematicians. Arguments put forward by fictionalism, nominalism, and ontological realism help in providing arguments that mathematical objects are invented by the mathematicians.
Fictionalism, Nominalism, Ontological Anti Realism
The schools of thought that argue for the non-existence of mathematical objects provide that the mathematical formulas that are developed by the mathematicians are untruths that cannot be proven but are used for the representation of reality. According to Field, numbers are not a compulsory part of science hence can be ignored when describing experiences or constructing scientific theories (Boccuni & Sereni, 2016). Mathematics is thus a body of truths that have an independent entity. Its existence does not depend on the real world experiences. Through the argument, it can be deduced that mathematical concepts are entities that are invented. In fact, Fields argues that mathematical concepts are a body of falsehoods that do not represent anything in reality.
The truths that are proven through the use of mathematical physics can still be proven through the use of non-mathematical physics. Mathematical concepts are therefore falsehoods that do not exist in reality but are invented by mathematicians to help in the understanding of aspects of reality (Burgin, 2017). The reality that they explain already exists naturally. There are some mathematical concepts that are reformulated hence exist for a long time while there are others that fall along the way if they cannot sufficiently be used to explain reality. It is further argued in fictionalism that there is no causal relationship between abstract objects and mathematical reasoning.
It Depends. Mathematical Objects Exist If…
The establishment of whether mathematical object exists or they do not exist is dependent on the several arguments that are posited by the different schools of thought.
Logicism, Formalism, Intuitionism
The argument of whether mathematical objects exist or do not exist majorly depends on the formulation of proofs by different schools of thought. Under the schools of thought of logicism and formalism, it can be interpreted that they proponents of the theories accept the partial existence and non-existence of mathematical objects. In logicism, it is argued that mathematical concepts can exist as part of the priori knowledge. However, logicism argues that the mathematical knowledge is a small part of the existing general knowledge of logic (Sereni, 2016). The argument, therefore, provides that there are mathematical objects that are in existence while there are others that are invented. The mathematical objects that are in existence do so in the form of abstract forms. Human beings, therefore, have to experience them and conduct empirical research that will yield results that can be used in the analysis. The analysis then establishes the facts that are used to formulate the mathematical concepts. Mathematical objects that are invented are in some instances replaced by concepts that exist. The concepts are discovered through empirical research. The mathematical concepts are then formulated to express the reality of the phenomenon that they describe.
There are mathematical concepts that do not previously exist but are invented by human beings and the used to refer to physical entities. For example, the number, 1, 2, 3… do not make sense on their own. They have to be given meaning by being related to the physical entities that are already in existence. The numbers are thus not discovered but invented by human beings. After their invention, they are given meanings through their attachment to physical entities. The mathematical concepts can thus be used to express reality. There are other instances when the mathematical concepts are derived as a result of experiences. After observing a phenomenon, a person will be engaged in the process of logical reasoning that will help them to formulate a mathematical concept that can be used to explain such phenomena. The importance of mathematical formulas in expressing realities has led to the suggestions that it is an indispensable part of science.
Empiricism, Logical Positivism
When argued from the point of empiricism, it is not possible for mathematical objects to exist before experience. Mathematical concepts are a product of a careful analysis of knowledge and experience. Therefore, the concepts are not invented but discovered from existing realities. The physical realities help in the formulation of mathematical concepts that can explain the phenomenon. The mathematical objects exist when analyzed from the point of empiricism. There has to be an interaction of empirical knowledge and logical reasoning for a person to formulate a mathematical concept.
Logical positivists provide that mathematical concepts need to have a truth value for them to be cognitively meaningful (Paseau, 2017). Their truth value is arrived at through a definite procedure. In the case of mathematical objects, they exist only if they can be determined to have a truth value. It implies that their application to the real world phenomena remains consistent at all times. The mathematical concepts, therefore, exist because they can be verified analytically or empirically through corresponding them with the existing state of affairs.
My Position
There exists no clear dichotomy between invention and discovery when discussing mathematical concepts. Their existence can be attributed to both invention and discovery. Discovery and invention play a critical role in mathematics. Mathematics works perfectly well because of the combination of invention and discovery. Mathematics is effective in the world because it is both active and passive. There are cases where scholars invent formulas that are used to describe real-world phenomena. For example, Isaac Newton came up with the calculus for the purpose of calculating both motion and change. While the formula can be attributed to invention, its effectiveness can be attributed to empirical research, and the research gives it the ability to make things to order. The empirical research contributes in making the theoretical values to agree with the real world phenomena. This attribute falls under the logicism argument that states that mathematical concepts can exist as part of the priori knowledge. However, logicism argues that the mathematical knowledge is a small part of the existing general knowledge of logic.
Human beings are responsible for inventing mathematical concepts. The concepts are arrived at through the abstraction of the elements of the world that are around them that include shapes, lines, and groups among other things. The mathematical concepts are developed for a specific purpose or for fun. After inventing the concepts, they go on to discover their connections with the real-world events. Their connections to real-world phenomena give them the truth value. Therefore, mathematical concepts can be attributed to both priori knowledge and going through the experience of realities that are situated in the world.
Conclusion
There is no clear verdict on whether mathematical objects exist or do not exist. There are different schools of thought that argue either for the existence of mathematical objects while other argues that they do not exist. The existence of mathematical objects is explained through rationalism and empiricism that provides that the mathematical concepts exist in abstract forms. Mathematicians conduct empirical research before they formulate the concepts. The concepts correspond to the physical realities of the world. The existence of mathematical objects has also been explained by logicism, intuitionism, and naturalism. Arguments through logicism provide that mathematical methods are established through the same process that is used to establish the facts in science. Therefore, the concepts are discovered as part of the things that are in existence. The non-existence of mathematical objects is explained through fictionalism, nominalism, and ontological realism. There are also arguments that support both the existence and non-existence of mathematical concepts. Therefore, there is no compelling school of thought that explains the existence or non-existence of mathematical objects.

References
Baker, A. (2017). The Philosophies of Mathematics. American Mathematical Monthly, 124(2), 188-192.
Boccuni, F., & Sereni, A. (2016). Objectivity, Realism, and Proof. Springer.Burgin, M. (2017). Mathematical Knowledge and the Role of an Observer: Ontological and epistemological aspects. arXiv preprint arXiv:1709.06884.
Paseau, A. (Ed.). (2017). Philosophy of Mathematics: Early 20th century philosophies: logicism, logical empiricism, intuitionism, and formalism. Routledge.Sereni, A. (2016). Equivalent explanations and mathematical realism. Reply to “Evidence, Explanation, and Enhanced Indispensability”. Synthese, 193(2), 423-434.