MATHEMATICAL EXPLANATION: WHY IT MATTERS

MATHEMATICAL EXPLANATION: WHY IT MATTERS

(From The Book Philosophy of Mathematical Pratice ByPaolo Mancosu)

to complete the task of Science Philosophy Course

Lecturer : Dr. LailaFitriana, M.Pd.

Heri Satriawan

POSTGRADUATE OF MATHEMATICS EDUCATION

FACULTY OF TEACHER TRAINING AND EDUCATION

UNIVERSITAS SEBELAS MARET

SURAKARTA

2018

CHAPTER 1

INTRODUCTION

A.    Background

The last two decades have witnessed a significant increase of attention to mathematical explanation, both in science and mathematics. This has led to novel joint work between philosophers of science and philosophers of mathematics.The philosophical analysis of mathematical explanations concerns itself with two different, although connected, areas of investigation.

The first area addresses the question of whether mathematics can play an explicit role in the natural and social sciences (mathematical explanation of scientific facts). A mathematical explanation of scientific facts based on a clear phenomenon with detailed information and minimizing the significance of error by explanation. The second offering with the problem of whether mathematical explanation occurs in mathematics itself (mathematical explanation of mathematical facts). Mathematical explanation of mathematical facts based on formal evidence and informal evidence. Formal evidence is basically the logical form of the set of premises and axioms. Whereas, informal proof is a form of proof that aims to conclude a new statement based on a previously known statement. The difference in informal proofs is more likely to be in the form of paragraphs than in logical forms. Thus, these entries survey contributions to both fields, it shows their relevance to the history of philosophy, mathematics, and science, it articulates their connections, and points to the expected philosophy by deepening our understanding of the topic. Mathematical applications occur in a natural phenomenon studied science is then described using mathematics or when the problem to be solved by science is answered by using mathematical techniques.

Based on consideration, the evidence of mathematical explanation can be generalized rationally, those that are not so the 'generalization' stipulations which fail to illuminate those areas of mathematics which have already been developed are not rationally acceptable. The process of generalization rationality in mathematics, also need a mathematical explanation. Only significant generalizations can explain rational changes in mathematics. Thus, it should be clear how the need for a mathematical explanatory theory appears in generalization. The explanatory theory requires that the explanatory evidence be generalizable. To uncover these problems, then all will be explained in this paper.

B.     Problem Formulation

Based on the background issues, the problems discussed can be formulated as follows:

1.      Why is the matter of mathematical explanation of scientific facts to mathematical explanations of mathematical facts?

2.      What is the connection of explanation and generalization in mathematics?

C.    The Aim of Research

Based on the problem formulation, the aims of the research are:

1.      To know the matterof mathematical explanation of scientific facts to mathematical explanations of mathematical facts.

2.      To know the connection of explanation and generalization in mathematics.

CHAPTER 2

FINDINGS AND DISCUSSION

5.1 Mathematical Explanations of Scientific Facts

The mathematical explanations of natural phenomena are widely recognized in the literature. However, until now very little attention has been devoted to them. The mathematical explanations of natural phenomena articulate how the explanation of how mathematics relates to reality, for example the explanation of the application of mathematics to reality. Illustrative example: a bunch of sticks thrown into the air with many turns and falls. When the stick falls freely, the stick is more near the horizontal than near the vertical. Broadly speaking, the facts of geometry can explain the cause.

The mathematical explanation of empirical phenomena can not be used to infer the existence of a mathematical entity, since existence is presupposed in the factual description to be explained. In other words, the existence of a mathematical entity can not be observed by the human senses. Example: "there are two cows in the field" the existence of number two can not be explained in a physical statement, since clear reference to number two can be explained by using the standard of the numeric number. The mathematical explanations of the empirical facts have not been sufficiently studied, as much detailed case study is needed to understand the various uses of explanations that mathematics can play in an empirical context.

Philosophical advantage may come from at least three different directions. First, towards a better understanding of the application of mathematics to the world, that is how mathematics can help in scientific explanations. Second, the study of mathematical explanations of scientific facts will serve as a causal explanation (causation). Thirdly, the arguments and forcing nominalist discussions take a stand on how to explain the clarity of mathematics in empirical science

5.2  From Mathematical Explanations of Scientific Facts to Mathematical Explanations of Mathematical Facts

In an interesting note to her paper Leng says:

Given the form of Baker and Colyvan’s argument, one might wonder whyit is mathematical explanations of physical phenomena that get priority. For ifthere are, as we have suggested, some genuine mathematical explanations [ofmathematical facts] then these explanations must also have true explanans. Thereason that this argument can’t be used is that, in the context of an argument forrealism about mathematics, it is question begging. For we also assume here thatgenuine explanations must have a true explanandum, and when the explanandumis mathematical, its truth will also be in question. (2005, p. 174)

This comment reflects the general use to which indispensability arguments. just as standard indispensability arguments address those who are realists abouttheoretical entities in science, so here the intended audience for the argumentwould consist of those who are realists about a certain realm of mathematicalentities (say, the natural numbers) and in addition are not already committedto a foundational position (such as predicativism) which forbids entertainingthe entities being postulated by the explanationhave been put.

The indispensability argument is not endorsed for mathematics. The original form of the indispensability argument relied on a form of confirmational holism. This left the argument open to the objection, raised forcefully by Maddy, that scientific practice proceeds otherwise or to the objection that other accounts of confirmation block the conclusion (Sober, 1993). In response, advocates like Colyvan and Baker have argued that explanatory considerations lead to platonism even if we drop conformational holism. But, as pointed out, nobody really has an account of mathematical explanations of scientific phenomena.

Quine originally used the indispensability argument to argue that we should believe in sets because they do the best job in tracking all our commitments to abstract objects. For Quine the appeal to empirical science was essential. Maddy’s realism drops the connection to empirical science and tries to obtain the same conclusion just by focusing on pure mathematics. In Chapter 4 of Maddy (1990) we find a lengthy discussion of theoretical virtues, including explanatory ones, that play a role in ‘extrinsic’ justifications for axiom choice in set theory. And althoughMaddy herself gave up the attempt in favor of ‘naturalism’ (see Maddy (1997)), mathematical explanation can still play an important role in this debate.

For those who believe that her realism can be revived perhaps the detour through indispensability arguments that appeal to mathematical explanations might provide a more persuasive type of argument than the other varieties of ‘extrinsic’ justifications mentioned in 1990. Moreover, those who are persuaded by the ‘naturalist’ approach of her latest book will as a matter of fact have to welcome investigations into mathematical explanation as they are part and parcel of the kind of work the methodologist in this area ought to carry out. So both these options call for an account of mathematical explanations of mathematical facts.

5.3  Mathematical explanations of mathematical facts

The history of the philosophy of mathematics shows that a major conceptual role has been played by the opposition between proofs that convince but do not explain and proofs that in addition to providing the required conviction that the result is true also show why it is true. This philosophical opposition between types of proof also influenced mathematical practice and led many of its supporters often to criticize existing mathematical practice for its epistemological inadequacy. Steiner’s model of explanation, to be discussed below, although not relying on the Aristotelian opposition, aims at characterizing the distinction between explanatory and non-explanatory proofs.

Hence what distinguishes an explanatory proof from a non-explanatory one is that only the former involves such a characterizing property. In Steiner’s words: ‘an explanatory proof makes reference to a characterizing property of an entity or structure mentioned in the theorem, such that from the proof it is evident that the result depends on the property’. Furthermore, an explanatory proof is generalizable in the following sense. Varying the relevant feature (and hence a certain characterizing property) in such a proof gives rise to an array of corresponding theorems, which are proved—and explained—by an array of ‘deformations’ of the original proof. Thus Steiner arrives at two criteria for explanatory proofs, i.e. dependence on a characterizing property and generalizability through varying of that property (Steiner,1978a, pp. 144, 147).

According to Steiner, an explanatory proof always makes reference to a characterizing property of an entity or structure mentioned in the theorem. Furthermore, it must be evident that the result depends on the property (if we substitute the entity for another entity in the family which does not have the property, the proof fails to go through) and that by suitably `deforming' the proof while holding the `proof-idea' constant, we can get a proof of a related theorem. Though many of Steiner's concepts (family, deformation, proof-idea) are vague, we can construct examples which beyond any doubt would classify as explanatory proofs by his criterion.Steiner’s model was criticized by Resnik and Kushner (1987) who questioned the absolute distinction between explanatory and non-explanatory proofs and argued that such a distinction can only be context-dependent.

5.4  Kitcher about explanations and generalizations

According to Curnot generalizations are generated because they reveal a common basic principle in thinking derived from certain truths, connections and common origins that had not previously been seen, found in all sciences, and especially in mathematics. Such generalization is the most important generalization, and their invention is a work of genius. Sterile generalization consists of the expansion of nonessential cases populated by inventive people for important cases, leaving the rest on a visible analogy. In such cases, a further step towards abstraction and generalization does not mean improvement in the order of mathematical explanations. What distinguishes both is that the first is the second temporary explanation not. According to Cournot generalizations were able to express the sequence of explanations in accordance with the mathematical truths that are structured.

Mandelbrojt says that generalizations are informative and clear. Such generalizations can be obtained with the proper level of abstraction and should show the learned object in the "natural setting". Eventually both Cournot and Mandelbrojt regarded it as having a mathematical explanation.

Kitcher discusses the problem of generalization in his book The Nature of Mathematical Knowledge from 1984. In his book according to Kitcher One of the most easily visible patterns of mathematical change, so far not explicitly discussed is the extension of the Mathematical Language by generalizations. For example Kitcher mentions Reimann's redefinition of definite integrals, Hamilton's search for hypercomplex numbers, and generalization of citing of limited arithmetic. Kitcher's goal is to try to understand the generalization process that takes place and to see how generalization's search becomes rational. However, not all generalizations are significant. Significant generalization is explanation. They explain to us exactly how, by modifying certain rules which are constitutive of the use of some Language expressions.

Kitcher tries to distinguish between rational and non-acceptable generalizations. The generalization provisions that fail to explain the areas of mathematics that have been developed can not be accepted rationally. In other words, to explain the rationality of the generalization process in mathematics, we need mathematical explanations. Thus it must be clear how the theoretical need for mathematical explanations arises on generalizations. Only significant generalizations can explain rational change in mathematics and that is explanatory. From the above description then the mathematical explanation should be generalizable, with evidence generally applicable.

CHAPTER III

CONCLUSION

The conclution of the explanation about mathematical explanation: why it matters are

1.         The mathematical explanation of scientific facts is an explanation of how mathematics relates to reality and is informed in detail of its physical form, whether in the natural sciences or in the social sciences. Meanwhile, a mathematical explanation of mathematical facts based on explanatory proof and non-explanatory proof. What distinguishes an explanatory proof from a non-explanatory one is that only the former involves such a characterizing property. An explanatory proof always makes reference to a characterizing property of an entity or structure mentioned in the theorem.

2.         Generalization is informative and clear. Such generalizations can be obtained with the proper level of abstraction and should show the learned object in the "natural setting", which in general there is a mathematical explanation with general evidences.