Althoughin mathematics numbers, sentences, and phrases have commonly beenused to mean the same thing their meaning may be understood andinterpreted differently depending on the school of mathematicphilosophy employed. is one of the branchesof philosophy that deals with the study of assumptions, foundations,and inferences of mathematics (Beth, 76). The primary objective is toexplain the methodologies and the nature of mathematics. The argumentis that mathematics is significant in daily activities and thus it isimportant to understand the nature of some of its concepts andphrases. Mathematical knowledge is so unique that is acquired notthrough general knowledge but reasoning from elementary but logicalprinciples (Lakatos, 181). As such, philosophers offer exceptionalresponsiveness to both epistemological and ontological questionsregarding mathematics. Various philosophical schools presentdifferent understanding of same concepts in mathematics a phenomenonthat further makes mathematics interesting and beautiful (Rosellá,89). This study reviews different solutions pertaining interpretationof issues in mathematics and arguments that are supportive to thesethe solutions.
Varioussolutions can be raised and discussed in mathematics regardingconventional mathematical sentences, figures, numbers and conceptsrelating to the existence of abstract objects (Lakatos, 171). Forinstance, the meaning of the phrase “2 is an even number,” or thenumber “4” could be disputed by philosophers depending on theirinterpretation of the nature of mathematical phrases and numbers. Thedispute is that, is there something that can be referred to as anumber? According to antirealist philosophers, numbers do not exist(Beth, 61). Realists agree that there are numbers, but some of themargue that numbers are supposed to be mental objects. Other realistssuch as Platonism argue that numbers are intangible/abstract objectsthat are not only non-mental but non-physical (Rosellá, 83).Accordingly, the school of Platonists claims that abstract conceptsand objects are real but cannot be found anywhere in the minds ofpeople or the physical world.
Oneof the groups of realists referred to as nominalism argues thatnumbers do not exist. According to nominalists ideas about numbersare non-existence and phrases such as “2 is an even number” orthe number “4” cannot be interpreted or understood in their facevalue (Beth, 80). The argument is that meaning of the sentence isdifferent from what is supposed to mean on its face value.Consequently, the phrase “2 is an even number” does not make adirect claim about number 2. The phrase is fictional because it mightbe right about a particular story but not accurately correctexplanations of the real world (Lakatos, 147). According toantirealist, mathematics is thus fabrication, pretense or mythicaland is based on stories that cannot reflect the true nature ofsomething in the world or have any accurate interpretation.
Accordingto realist, represented by Platonist, abstract objects exist and areentirely non-mental and non-physical. They also argue that truemathematical phrases exist and are capable of providing real, factualand accurate explanations of the objects that they refer or describe(Rosellá, 67). Although number and phrases do not exist anywhere inthe known world (non-physical) and entirely non-mental, they existindependently but are true in that they can be relied on, trusted andused to describe objects. It can be concluded that numbers are actualdescriptions of the objects they represent. Platonist, therefore,maintains that mathematics entails the discourse into the nature ofdifferent mathematical concepts which are entirely abstract (Beth,68). Accordingly, other mathematical descriptions such as sets,functions, vectors, circles among others are true and accuraterepresentative of the objects they are meant to represent, and theyare acceptable although they are abstract. Accordingly, mathematicsentails the study of various abstract objects, structures, anddescriptions.
Consequently,the most progressive and compelling argument is the realistic view ofmathematics concepts (Beth, 92). The reason is that, althoughantirealist philosophers cannot accept or believe that abstractobjects exist, they admit that things like numbers, sets, circles andfunctions exist but deny the fact that such things constituteabstract objects. Correspondingly, individuals may have ideas aboutreal objects like the sun, but this idea cannot be taken to mean thatthe sun represents the idea. The individuals’ ideas about the sunand the sun itself are different things altogether (Lakatos, 147).Consequently, number 2 is an abstract object that is real but onlyexists independently from people and their thoughts. Number 2 isnon-physical and non-metal, but it is an objectively real thingexisting as an abstract object. Essentially, it is unchanging andentirely invariable. Abstract objects are not constituted by physicaland material things and thus cannot have a real association withother objects (Rosellá, 115). Mathematical statements, formulas,and phrases also constitute accurate descriptions of the objects theydescribe or represent.
Mathematicsis one of the most important disciplines that apply to numerousactivities in day to day life, and thus introspection into the natureand logic of its concepts is very significant. Philosophy ofmathematics seeks to explain the methodologies and the nature ofmathematics. Philosophers offer exceptional responsiveness to bothepistemological and ontological questions regarding mathematics.Moreover, various phrases and numbers lead to theoretical disputes byphilosophers depending on their interpretation of the nature ofmathematics concepts. The two leading contenders are realistsrepresented by Platonism and antirealists represented by nominalism.According to Platonist, abstract objects exist and are entirelynon-mental and non-physical. They also argue that true mathematicalphrases exist and are capable of providing true, factual and accurateexplanations of the objects that they refer or describe. On the otherhand, antirealists argue that numbers are non-existence. According tonominalists ideas about numbers are non-existence and phrases such as“2 is an even number” cannot be interpreted or understood intheir face value. Realists are more logical because antirealistphilosophers believe that abstract objects exist. Besides, people canhave abstract ideas of real objects just like individuals haveabstract ideas of the existing objects like the sun yet the sun isnot an idea but an existing object.
Beth,Evert W. MathematicalThought: An Introduction to the Philosophy of Mathematics.Dordrecht, Holland: D. Reidel Pub. Co, 2013. Print.
Lakatos,Imre. Problemsin the Philosophy of Mathematics.Amsterdam: North-Holland Pub. Co, 2014.Print.
Rosellá,Joan. FromFoundations to Philosophy of Mathematics: An Historical Account ofTheir Development in the Xx Century and Beyond.Newcastle: Cambridge Scholars, 2012. Print.