The Similarity Between Philosophy and Mathematics - Essay Example

Relationship between Philosophy and Mathematics Introduction Philosophy and math are both argumentative disciplines that employ a series of stages in a bid to obtain conclusions and solutions to their arguments. The mode of presentation of the argument by the two disciplines is, however, varied despite the logicality they are widely known for. This paper examines the methodologies applied by the two disciplines and establishes the disparities and similarities between them. The two subjects bring together the most fundamental and widely acceptable intellectual skills. As mathematics provides the knowledge and ability to tackle quantifiable problems, philosophy imparts the ability to analyze the mathematical quantifiable data. The issues, data and assumptions raised by the mathematics are deeply analyzed by philosophy to create an articulate understanding. This combination of mathematics and philosophy provide a formidable foundation upon which to build in the course of career development and pursuit. Historically, the two have a strong link, as the logic is a strong branch of both the subjects. The work of logic in mathematics-symbolic logic- and the applied logic in philosophy provides a natural bridge with which the two subjects closely link. The other area of similarity is the fact that those undertake courses in either of two can pursue their career in a wide range of areas. Such fields of career pursuit include the computer science, journalism, financial and investment analysts, civil and diplomatic services among others. In addition, the similarity between mathematics and philosophy is what they present at the end of their argument, that is, conclusion. The major concerns are the being, existence and the truth of the presented solution. In order to establish the truth of the final solution and settlement of an argument, whether philosophical or mathematical, highly relies on the connectedness of the sense, reference and the ‘name’ of the phenomenon or the object. The sense and reference in the philosophy establish a relation between objects or names or names of the objects (Zalta 42). Philosophy for instance distinguishes the meaning of ‘a=b’ and ‘a=a’ in its conclusion. Philosophy establishes that ‘a=a’ is analytic in its nature while ‘a=b’ has a valuable extension, which must be explored, and the meaning established. Mathematics, on the other hand, does not go further to offer explanations, if any, about the cognitive meaning and value of the two statements. In a bid to reach a satisfactory statement  through computation, mathematics employs a number of conventionally recognized signs and symbols for generality and identity of certain mathematical parameters, dimensions of figures and at times, the final solutions to the initial problem. Such signs and symbols of identity include theta and alpha. Conversely, these conventional signs and symbols do not apply anywhere in philosophical arguments. In essence, the philosophical arguments can be effectively expounded in oral form without writing regardless of how complex the argument is. It is, however, admissible that complex mathematical problems would only be understood through explanation on a piece of paper, using the widely accepted and understood signs and symbols. As observed by Wittgenstein, mathematics differs with philosophy in the sense that mathematics is a logical method obtained through repeated applications of the operations (187). An example is a case of the number four (4), which the exponent is provided to an operation performed four times in a row. The discipline of philosophy, on the other hand, does not employ such repetitions in the argument to reach a satisfactory conclusion. Only one-time arguments within the content and context of the premises are used to generate conclusions. In addition, as much as both the disciplines use logic largely, the manner of application of the logical argument is different in the two disciplines. Philosophy is open to the application of both inductive and deductive reasoning when establishing the conclusion and so the truth. In mathematics, the case is contrary since the reasoning applied is the deductive form, through which the final answer is obtained. The use of axioms is widely accepted and applicable in the construction of a mathematical argument. This indication is not fully self-supporting in terms of how it should proceed in a logical argument. On the other hand, in philosophical arguments the axioms and the assumptions necessary for advancing the argument and proceed in logic. Such makes itself manifest in a philosophical argument, since the logic in philosophy has the capability of taking care of itself. In addition, mathematics makes use of external laws to guide its procedures of its logical argument and calculations. On the other hand, philosophy does not need the external laws and rules to inform how to proceed with its logical argument. This is because there is nothing external to the philosophical logic. Neither does philosophy require any laws of inference to dictate what follows what in an argument. Such procedures are provided and very clear in the structure of the propositions, contrary to mathematical arguments. What are, however, required are the elementary proportions with successive use of the operations that regard all the propositions applied. Hence, there is a need to establish and differentiate the kinds of meanings for some terms and statements used in any form of argument. Reference, as a building block of the eventual truth about any argument, mathematics does not seem to pay much attention to it. In fact, mathematics does much of assumptions about the reference by simply allocating signs and symbols to the objects considered as the parameters. Reference, according to Wiggins “..is the specific object to which a word refers, for instance for the word chair, the reference is the chair” (36). The philosophical argument would first establish that each word used in a statement has a reference and analyze the statement to verify its consistency with reality. This then ignites the question of the correspondence theory of truth, which demands that, and conclusive position after an argument is that, which corresponds with the reality in the environment. The other two aspects of the statement are the sense and signs of each word used in the statement. Zalta states, “…..the word sense is the way and manner in which it refers to the object while the object sign or name is used to designate a proper name that refers to the specific member of a class or a group of objects” (491). Because there are varied ideas for sense; including the mood, thoughts and feelings, philosophy and philosophical argument requires that, the sense and the reference of the sign are distinguished. This is majorly because these ideas of sense normally go along with the memory of objects, and, as such, disparity in perception arises from qualitative differences of instant experience. It also depends on the quality of the sense organ by an individual. Such are the inner and subtle details that mathematics though logical in its approach to solution provision, does not encompass. It then follows that mathematics is, but a system of symbolic logic superficial in nature, majorly employing the deductive form of reasoning to obtain the final statement in the form of a solution. This further indicates that, without the idea of existence of such actual references, the sense of the world and objects become even more and more confusing in the eye of philosophy. Mathematics does not pay much attention to such deeper aspects of knowledge that philosophy seeks to obtain and factor in its argument and final statement proposition. As earlier mentioned, the concerns for sense, sign, and the reference of a word in a statement, especially philosophical, is to help arrive at the eventual truth. Through this, the discipline of philosophy does not merely stop at presenting a final statement in the form of a conclusion to its argument, but proceeds to seek if it is indeed the truth. The reference of a statement is only sought when we inquire after its true value by researching. Through the very act of research, we get to stumble on the reference upon learning and to establish the truth (Frege 442). The other important concern in philosophy, which fails to come out clearly in the logic of mathematics, is the relation. The equality is very challenging question to the reality, particularly between the objects, and between the signs and the objects. For instance, as explained by Wittgenstein, ‘a=a’, and ‘a=b’ have different cognitive values (62). In these statements, ‘a=a’ is analytic while ‘a=b’ has the valuable extensions of our knowledge and as such can never be established in prior (62). It, therefore, follows that if we only regard the signs or names then the argument ‘a=b’ may be true. However, in philosophically, “….an argument as a thing to itself and indeed one in which each thing stand to itself, but no other thing” (Zalta 87). In this case, it is evident that ‘a=b’ would not refer to any subject matter. “…differentiation occurs where we distinguish the sign ‘a’ from the sign ‘b’ only as an object by the means of shape, but not by the manner in which it designates something” (Wiggins 46). In such a case, the cognitive value of ‘a=a’ is essentially the same as that of ‘a=b’, provided the a-b is true. However, when the difference between these two signs is presented, and it corresponds to the difference in the mode of presentation, then a difference also arises in the cognitive value of the expressions. For instance, in the case of an equilateral triangle with the intersection points of two similar opposite sides, it would be right to conclude that the reference for the points of intersection are the same, but not the senses (Wiggins 57). It shows that philosophically, the ‘sign’ or ‘name’ has its specific and definite reference to a particular object. Philosophy also requires that designations be given proper name, since designations at times consist of many words and signs, so this would help for brevity purposes. As explained by Frege, the interesting twist is that the regular connection between a sign, its sense, and its inference is of a kind such that to the sign, there corresponds to a definite sense and to this in turn, there is a definite reference (394). However, the complication arises in this relationship since to the given reference or object; not a single sign belongs (Wiggins 62). Further, the same sign has a number of varied expressions in different languages, or worse still in the same language. Contrary to the philosophy, mathematics’ lack of the consideration for language inputs into the ‘sign’ does not complicate matters this far as it does in the philosophy. In the need to establish the truth, it may be granted that any expression so far has a sense. This, however, may not establish that every sense has a reference. “….such statements as ‘celestial body most distant from the Earth’ have a sense but it is highly doubtful if it has a reference’ (Wittgenstein 221). Also concerning our everyday life speeches and talks, one may be heard remarking that the other person’s speech had a lot of senses. The question of reference is normally not talked about concerning one’s speech. These are, but pieces of evidence that the world has no customary reference so far. This search for the seemingly elusive truth in another way differentiates philosophy and mathematics. Philosophy is much inclined on the search for reality; a phenomenon that even the discipline itself contends is difficult if not impossible to obtain. For this reason, philosophy has always seemed as the field of abstraction. In essence, by trying to get realistic and practical, philosophy has ended up achieving the opposite, while mathematics, which has remained as, simply concerned with the figures and their manipulation has been understood as practical and more realistic as compared to philosophy. This is because the truth is “….the points in which we acquire the truth value of a statement and where the reference for its components is involved” (Kim &Sossa 413). As further elaborated by Kim & Sossa, “….the truth is, therefore, an arbitrary fancy or perhaps, a mere play upon an object which can be more exactly discussed only in the concept of components and relation” (346). To explain further, mathematics may put it as simple as “5 is a prime number” and then proceeds to work with it in a logical manner to reach a satisfactory statement of conclusion, in the form of an answer. One might be tempted to think that this subject to the predicate proposition is the truth, like the sense to reference. Closer examination and study of this proposition, however, indicates that nothing has been added to or said about the simpler statement apart from the basic form in which it is presented. The closer examination would reveal that this statement contains only a thought and the relation of the thought to the truth does not in any way match the sense and reference truth. By reading this statement, one never advances from one level of knowledge and truth to another, but simply moves about the same position. The proposition can, however, be used in its form in mathematics, and this is indicative of the low level of growth in the mathematics occasioned majorly by the confinement of the discipline within some external laws and rules. In this case, the discipline of philosophy remains progressive. In the same understanding, it is worth noting that the solutions provided through mathematical arguments are conclusive and acceptable as the only ‘right’ the premises used to reach it. This also has a lot of the connection with the deductive argument, which is regimented within certain predetermined lines of thinking. Mathematics, compared to philosophy is less liberal to the human mind. The final statement provided by philosophical argument is not normally conclusive, and other conclusions can as well be generated, following the structure of an argument and with the premises established. Conclusion So far, one glaring similarity between philosophy and mathematics is that they both employ structural, logical arguments to reach conclusions. Evidently, given the above discussion, there are indeed many disparities in the methodology of approach by the two disciplines. In the approach to evaluation, philosophy is more self-independent hence does not apply external laws and the convention. Mathematics, apart from applying these external laws, is more precise and sounds factual in its symbolic logic. Philosophy, on the other hand, has been understood to involve a lot of ‘unnecessary’ detail and complexities, which renders it difficult to understand. This is what some authorities have referred to as the ‘philosophical hair splitting’ Works Cited Frege, G. On the Sense and Reference. Ed. Geach, P and Max, B. Oxford: Basil Blackwell, 1990. Print. Kim, J and Sosa, E Fictional Entities: A Companion to Metaphysics, Oxford: Blackwell, 2009. Print Wiggins, D. Necessary Existents, Cambridge: Cambridge University Press, 2002. Print Wittgenstein, L Tractatus Logicophilosophicus. Ed. Kenny A. Williston VT: Willey, Blackwell, 2002. Print Zalta, E., 1983, Abstract Objects: An Introduction to Mathematics and Axiomatic Metaphysics, Dordrecht: 2006. Print Read More