Mathematical Platonism is the concept that describes the existence, abstractness, and independence as adjectives for mathematics routine practices. The philosophy insinuates that some objects exist independently of external forces, such as language and thoughts (Park, 2018). However, the truth is not asserted regarding the formal existence and the role of Platonism in pure mathematics. Besides, the concept of independence is shallow and does not make sense to true realists because of its idea that objects occur regardless of the existence of other practices. Studying Platonism provides alternative thinking that reality extends beyond the physical world and includes objects not part of the casual and spatiotemporal order studied in biological sciences (Ruloff, 2020). The study will help understand how mathematical Platonism influences the naturalist’s theories school of thought because there is little or no doubt on mathematical philosophies. Frege’s argument that mathematical language is quantifiable gives Platonism a lifeline and true meaning (Paul, 2020). However, little is known about the objectiveness of the mathematical concepts, the level of accessibility, the metaphysical problems it seeks to address. This paper will discuss realism and true realism as concepts that help understand mathematical Platonism and answer the question of existence, abstractedness, and independence.
Keywords:Existence, abstract, independence, Platonism
Mathematical Platonism
Platonism is a mathematics philosophy insinuating that some objects exist independent of external variables, such as people, thoughts, and routine practices. Mathematical Platonism is described in three conjunctions, including existence, abstractness, and independence, which are the traditional variables of the topic. In 1953, Gottlob Frege posted a strong argument that the language of mathematics tends to refer to and quantify the mathematical objects and the corresponding theories are true (Park, 2018). However, the approach cannot be valid unless the first expression comes after and gives meaning to the statement. Consequently, the mathematical theory of Platonism has developed objections based on Frege’s argument (Ruloff, 2020). Classical semantics states that the singular language of mathematics refers to the first order of quantifiers and objects that range over such entities. Similarly, the truth indicates that most of the sentences accepted in mathematics are true. Thus, the mathematical concepts are epistemologically inaccessible and metaphysically problematic that attract long debate on Platonism.
Definition of Platonism
Mathematical Platonism is that mathematical adjectives replace the idea that arises from existence, abstractness, and independence. For instance, the formal structure of reality can be ‘∃xMx,’ where ‘Mx‘ represents the predicate ‘x is a mathematical object’ and is acceptable in all objects studied in pure mathematics. Second, the mathematical abstract of abstractedness occurs when an object is deemed abstract in the case of spatiotemporal and casual inefficacious. Lastly, claims of independence are the least clear compared to the first two claims because they ascribe to the autonomy of an object where regardless of the occurrence of other practices. However, the claim of sovereignty is still schematic and not well structured. Thus, mathematical Platonism indicates that objects exist abstractly, independent of intelligent agents such as language and believes.
Significance of Mathematical Platonism
Mathematical Platonism puts pressure on the physicality idea that reality is exhausted. Exhaustion is due to the physical objects that provide alternative thinking that reality extends beyond the physical world and includes objects that are not part of the casual and spatiotemporal order studied in biological sciences (Plebani, 2017). In addition, if mathematical Platonism is true, it puts pressure on the naturalist’s theories school of thought because there is little or no doubt on mathematical philosophies. Further, an in-depth understanding of the mathematics of Platonism will establish the knowledge of abstract objects leading to rediscovery, which many naturalist theories struggle to accommodate. The philosophical consequences are similar to the mathematical Platonism, but the forms are suited to accommodate such effects (Park, 2018). The resemblance is due to the mathematical concept of owning the rights as a scientific tool. Few philosophers are against the Platonism ideas with strong core claims whose scientific credentials are as strong as mathematics.
Mathematical Significance of Platonism
Platonism defends specific mathematical ideas such as the classical first-order language, impredicative definitions, and Hilbertian optimism, which are the core values in mathematical philosophy. However, according to working realism, more classical methods support mathematical reasoning but require philosophical defense based on Platonism. Therefore, while Platonism is a philosophical view, realism is a view within mathematics about the methodology of disciplines, making the two concepts distinct (Park, 2018). Furthermore, there is a positive correlation between the two views because realism receives strong acceptance from mathematical Platonism. In addition, Platonism ensures that mathematical concepts are discovered rather than invented because there is no need to restrict the idea to construction methods that establish a powerful argument (Park, 2021). Therefore, Platonism has mathematical significance because it indicates that mathematics is about independently occurring objects with unique answers, which motivates the Hilbertian optimism.
Object Realism
Object realism assumes that a mathematical abstract exists and acts as a conjunction of the existence and abstractness. Unlike nominalism, which defines the view that there are no abstract objects, object realism is abstract, and a universe exists (Plebani, 2017). However, object realism leaves out independence because it is logically weak in the concept of mathematical Platonism. The argument of object realism is stronger than mathematical Platonism because many scientists believe in non-physical objects as long as they are dependent on material things (Paul, 2020). For instance, physicalisms accept the law, poems, and corporations as a mystery of epistemic access to non-physical items and the processes constituting them. Other scientists believe that the philosophy of mathematics revolves around the objects outside the Platonism theory, such as traditional intimism. These ideas support the existence of mathematical objects that are dependent on mathematicians and their activities.
Truth Value Realism
True value realism statement involves unique objects with actual value independent of its logical order in the current mathematical theories. The idea also notes that mathematical objects are deemed accurate and vice versa, making the concept a metaphysical view that is non-ontological (Park, 2021). However, despite indicating that every mathematical statement has an actual value, it does not conform with the Platonist ideas that the actual values occur in the ontology of known mathematical objects. Mathematical Platonism promotes the concept of natural value realism by enhancing the platform where the mathematical objects acquire their truth value (Paul, 2020). However, mathematical Platonism does not include the true value realism unless further clauses are provided.
Mathematical Platonism motivates truth realism because it gets the actual value of objects and endorses at least one branch of mathematics, such as arithmetic. Nominal believers commit to the odd-sounding idea that the ordinary math statements that; there are multiples of itself between 10 and 20. In this case, it is essential to differentiate the mathematical language Lm and the nominalists and other philosophers who believe in the Lp format. Nominalist believes prime numbers exist, but no abstract objects are made in the Lp format (Park, 2021). Thus, they indicate that the actual value of the Lm values is in a fixed manner that does not make mathematical sense. Furthermore, for the existence to have the desired effects, it must occur in the Lp language used by the mathematicians and accepted by the nominalists contrary to the purpose of the claim (Paul, 2020). Philosophers argue that true realism ideologies are ideal for Platonism debate because it gives clear and tractable evidence.
Mathematical Platonism involves the ideas of existence, abstractness, and independence as adjectives for mathematics routine practices. The philosophy insinuates that some objects exist independently of external forces, such as language and thoughts. Similarly, the truth is not clear about the existence and the role of Platonism in pure mathematics. This paper agrees with the naturalist’s concept that independence is shallow because of its idea that objects occur regardless of the existence of other practices. Studying Platonism helps to understand its role in explaining how mathematical objects extend beyond the physical world. In addition, the study discusses how mathematical Platonism influences the naturalist’s theories school of thought because there is little or no doubt on mathematical philosophies. Frege argues that mathematical language is quantifiable, and the objects exist in true realism regardless of language and thoughts. Mathematics of Platonism implies that mathematical entities are valid and live and are abstract, independent of rational human activities.
References
Park, W. (2018). Philosophy’s loss of logic to mathematics. Studies in Applied Philosophy, Epistemology and Rational Ethics, 43. Web.
Park, W. (2021). On abducing the axioms of mathematics. Studies in Applied Philosophy, Epistemology and Rational Ethics, 161-175. Web.
Paul, T. (2020). Mathematical entities without objects. On realism in mathematics and a possible Mathematization of (Non) Platonism: Does Platonism dissolve in mathematics? European Review, 29(2), 253-273. Web.
Plebani, M. (2017). Does mathematical Platonism meet ontological pluralism? Inquiry, 63(6), 655-673. Web.
Ruloff, C. (2020). Theism, explanation, and mathematical Platonism. Philosophia Christi, 22(2), 325-334. Web.