American Journal of Social Issues & Humanities (ISSN: 2276-6928) Vol.1(2) pp. 82-94 Nov. 2011 Available online http://www.ajsih.org 2011 American Journal of Social Issues & Humanities THE ROLE OF APRIORI, EMPIRICAL, ANALYTIC AND SYNTHETIC IN PHILOSOPHY OF MATHEMATICS. Abstract ETTA, EMMANUEL EFEM, PhD, AND KYRIAN, A. OJONG, PhD DEPARTMENT OF PHILOSOPHY, UNIVERSITY OF CALABAR Email: Emmanuel_etta@yahoo.com Philosophy is said to be a reflective discipline of the humanities or of the social sciences as the case may be, which investigates all claims to knowledge through it s peculiar sources and tools. However, in spite of the foundational relevance of philosophy in almost every academic field and the historic fact that philosophy is the mother of all disciplines, little or nothing is usually known about their functions in relation to other disciplines. Consequently, this paper is an attempt to explore the place of philosophic tools such as the apriori, empirical, analytic and synthetic in philosophy of mathematics as a latest dimension in philosophy. Key words: Empirical; Apriori; Synthetic; Epistemology; Analytic Introduction Philosophy of mathematics as a foundational study presupposed epistemology. Hence, this paper examines critically, from an epistemological perspective the place of apriori, empirical, analytic and synthetic tools of philosophy and their function in philosophy of mathematics. It is also true however, that certain philosophical problems become salient only when the appropriate area of mathematics is taken into consideration (Mancosu 2008:2).In light of above, A.N. Whitehead (1948)
while supporting the complementary disciplinary influence between philosophy and mathematics, observed that, in the age of Galileo, Descartes, Spinoza, Newton and Leibniz, mathematics was an influence of the first magnitude in the formation of philosophical ideas (35). The apriori is associated with the rationalist school of thought. It holds that reason can alone attain certainty about true knowledge without or independent of experience. According to J. R. Brown (2008), even though mathematics is a priori, it need not be certain, because the mind s eye is subject to illusions and the vicissitudes of concepts for concepts formation jus s the empirical senses are (23). However, Whitehead opines that, the certainly of mathematics depends upon its complete abstract generality. But however, contends that, we can have no apriori certainty that we are right in believing that the observed entities in the concrete universe forms a particular instance of what falls under our general reasoning (28). It is pertinent to reconcile or juxtapose this is position with his earlier argument, where the same Whitehead opined, that the originality of mathematics consists in the fact that mathematical science connections between things are exhibited which, apart from the agency of human reason are extremely unobvious (27). The empiricists are of the view that true knowledge is attained through human sense experience or observation of the physical world.here, Whitehead attempts to point out how the empirical is related to mathematics. For him the generality of mathematics is the most complete generality consistent with the community of occasion which constitutes our metaphysical situation (31).In like manner, E. Robson and J. Stedall (2008) in their work The Oxford Handbook of the History of Mathematics, strongly support the positive relevance of the empirical science to mathematics. Hence, they explain that, at presence the general opinion among historians of mathematics is that such a view of mathematical concepts as time, place, and context-independent is not very fruitful if one wants to understand the historical development of mathematics. Rather, a much more rewarding approach is to focus on concrete practices of AJSIH Vol.1 No.2. (Nov.2011) 82-94 Etta, Kyrian 83
mathematics, acknowledging that despite its universal character, mathematical knowledge is produced by mathematicians who live, interact, and communicate in concrete social settings(5). The analytic proposition means a statement which says something about a subject but what the predicate says about the subject is already contained in the subject, hence it does not say anything new. Its truth is internal and deductive. Synthetic proposition on the other hand is a statement which says something about the subject. The predicate here says something new about the subject that was not already contained in the subject. It is in other words known as external or inductive proposition. The question as to, whether the propositions mentioned above have relevance to mathematics forms the basis for this work. Here, I shall try to show that the above mentioned tools are relevance to the attainment of mathematical knowledge. This is rather historic. Hence Pythagoras is quoted by Whitehead, as having insisted on the importance of the utmost generality in reasoning. Consequently, he divined the importance of number as an aid to the construction of any representation of the conditions involved in the order of nature (32). This task shall be accomplished by first of all giving the meaning of each of the tools, acknowledging theorists and their postulation on each of the tools, and as these relate to mathematics. A brief analysis shall be given from where I shall show that the various tools are truly means to attaining mathematical knowledge. The roles of each tool in philosophy of mathematics would be articulated, and then a conclusion. Methodology Apriori Apriori knowledge is knowledge that is known independently of experience (that is it is nonempirical, or arrived at beforehand, usually by reason)(wikipedia 2010:7).The apriori is a term which is associated with the rationalists school of thought in epistemology as a branch of philosophy. The rationalists claim that the mind has the power to know with certainty various truth about the universe AJSIH Vol.1 No.2. (Nov.2011) 82-94 Etta, Kyrian 84
which outward observation can never give us. According to this rationalist school, absolutely certain knowledge of general principles gained independently of observations is called apriori knowledge (Randall & Buchler 1971:76). John Harman Randall and Justus Bachler(1971) posited further that by apriori, in a version interpreted by the empiricist, it meant generally statements known to be true, that is, with certainty yet without empirical testing. Joseph Omoregbe(1990) in his book knowing Philosophy, argues that apriori is a claim by the rationalists that reason alone can and does attain knowledge without any reference to experience. Such knowledge according to the rationalists, acquired by reason alone independently of experience is known as apriori knowledge (42). This apriori for J. Omoregbe is in contrast with the aposterior knowledge of the empiricist which holds that all knowledge derives from experience Empirical According to J. P. Ekarika(1986), in his book Philosophy; introduction to philosophy, preliminary notions in logic, metaphysics and theory of knowledge, an empirical knowledge is knowing something by our given senses or sense experience. It is a way of knowing the physical world and things by seeing, hearing, touching, smelling and testing (31). This position is similarly supported by Randall and Buchler when they said that an empirical knowledge is knowing something external and objective, the world of fact or nature (87). Analytic This term is used to stand for or explain statement which says something about things within the same statement. When in a proposition or statement, the predicate says something about the subject that is already contained in the notion of the subject, the proposition is said to be analytic. The predicate does not say anything new about the subject since what it says about it is already contained in the notion of the subject. Analytic statements are necessarily true, and cannot be denied AJSIH Vol.1 No.2. (Nov.2011) 82-94 Etta, Kyrian 85
without contradicting oneself (43). Alan Ryan(1970) posits that since analytic statements have its truth or falsity within itself, it is an internal statement because knowledge is derivable within or in the internal logic of the statement. Consequently, he adds that analytic knowledge is deductive (28). Synthetic According to Leila Haaparanta (2009)in one sense, propositions are said to be contents of propositional attitudes like beliefs, knowledge, will, among others, the class of these attitudes is somewhat vague(565).synthetic propositions on the other hand are statement in which what is affirmed in the predicate is not already contained in the notion of the subject. This type of proposition is called synthetic. The predicate in this case affirms something new about the subject. This proposition is not necessarily true; there can be true or false hence there is no contradiction involved in denying those (143). According to Alan Ryan(1970), synthetic statements are known to be external in the sense that the truth or falsity of such statements are or rest on the correspondence of the statement to an outside world. Consequently, he maintained that a synthetic statement of knowledge is inductive (28). In light of the above, this paper asserts that these tools as explained are used by philosophers of mathematics, to arrive at certain goals. The roles they place in philosophy of mathematics are thus foundational, epistemological, methodological, historical and justificatory as I have already articulated and or pointed out. Relationship between apriori, empirical, analytic and synthetic tools and mathematics The task of this subsection as it relates to the topic of this work is to investigate various theoretical postulation by different philosophers and or scholars on apriori, empirical analytic and synthetic propositions, to enable us ascertain their place or function in philosophy of mathematics. AJSIH Vol.1 No.2. (Nov.2011) 82-94 Etta, Kyrian 86
A priori proposition as defined in an earlier section of this work, states that reason alone is capable of attaining certainty of truth independent of experience. Our question is, can reason attain certain mathematic truth? The answer is affirmative. Hence, according to Condorcet (1988), reason was basic to human nature, and with reason it was possible to guarantee universal agreement on questions about the world as well as about matters of mathematics. He adds that reason enabled us to discover the intricate, abstract truths of mathematics, and to apply these to our understanding of the working of the universe (10). In other words, this school of thought is of the view that reason (a priori) is a sure method for acquiring mathematical knowledge. The above position is substantially supported by John Hospers (1997) when he said that mathematics is a matter of pure reason, not of experience. He illustrated by saying that when we add 2 + 2 which is equal to 4 we arrive at it not by experience, but through reason (132).To corroborate the above, J. R. Brown (2008),avers that mathematics is apriori, not empirical. This is because empirical knowledge is based on sensory experience. That is, based on input from the usual physical senses: seeing, hearing, tasting, smelling, touching. But seeing with the mind s eye is not included on this list. It is a kind of experience that is independent of physical sense and, to that extent, apriori (22). The empirical school holds that knowledge truth certainty is gotten, justified, from empirical observable evidence, through our sense organs. In relation to mathematical knowledge acquisition, J. S. Mill contends that mathematical knowledge rests on empirical evidence. Explaining further, Mill maintained that arithmetical knowledge is simply a generalization of observed instance in which sets of physical objects are counted (Lehman 1979:155). According to Putnam (1979), to give satisfactory explanations of empirical data, physicians must make mathematical assumptions. These assumptions for him are justified by the same empirical AJSIH Vol.1 No.2. (Nov.2011) 82-94 Etta, Kyrian 87
evidence which justifies the non-mathematical assumptions which enter into such explanations. For the factionalist as represented by Vaihinger and Korner (1979), it is empirical verification that justifies the use of mathematics. Put differently, this school of thought has its basic opinion that a theory including its mathematical postulates, are justified by the evidence provided that the observable consequences of the theory is verified. And also, that mathematical principles such as the axioms of real number theory are known to be true because they have been strongly confirmed empirically (156). In other words, proof is essential in knowing. Consequently, James R. Brown argues that with a proof, the result is certain; without it, belief should be suspended. He explains further that, sometimes mathematicians believe mathematical propositions even though they lack a proof. But however opines that, without a proof, a mathematical proposition is not justified and should not be used to desire other mathematical propositions (10). In a nutshell, what the empirical or empiricist school is putting across here is that empirical observation, sense experience or physical data, evidence verification are all means to attaining and justifying any knowledge that can be said to be mathematically true and or valid. The analytic school is of the opinion that the truth and falsity of a statement or knowledge claim is in the language structure, that is, internal, deductive, and that its truths is necessary. This position of the analytic school has been or is part of the views upheld by philosophers of mathematics. Hence, according to A. J. Ayer(1979), analytic proposition do not provide any information about any matter or fact. In other words, they are entirely devoid of factual content. In claiming that mathematical propositions are devoid of factual content, philosophers such as Ayer and Hempel, were making two claims. Firstly, that mathematical propositons are not confirmable or verifiable by reference to sensory observation. In other words, we do not know propositions such as 3+2=5 because of anything that we have observed. Such statements are neither confirmable nor AJSIH Vol.1 No.2. (Nov.2011) 82-94 Etta, Kyrian 88
refutable by reference to observations. There are no sensory observations which anyone needs to make in order for men to know that 3+2=5. Secondly, they were saying that mathematical propositions have no ontological import. That is, even though statements such as 3+2=5 are true, this does not mean that anything exists. Such statements would be true even if nothing existed. This means such statements do not correspond to any external reality. Nor are they true because of such correspondence (27). To further prove that analytic statements with regards to its claim of necessary truth, are relevant to the mathematician or philosophers of mathematics, it will be pertinent to recognize John Hospers (1997) in this regard. Thus, in his book introduction to philosophical analysis, he opined that mathematics like logic, consists of necessary truths by this, he means that when we say 1+4 = 5, the truth and falsity of it is internal and necessary, thus it will be a contradiction to deny its truth (133). According to this school of thought as further buttressed by Korner(1979), mathematical principles are known through an inferential process which accords with the hypothetic deductive pattern of reasoning. For him, mathematics proceeds from a set of hypothesis and assumptions about particular experimental or observation set-ups plus theoretical assumptions, and further observational consequences are deduced by valid forms of argument (149). Furthermore, John Hospers (1997) in supporting the above position, maintained that there are laboratories for discovering truths about nature, but there are no mathematical laboratories, because mathematicians don t need them, nor do they need test tubes and Bunsen burner to discover that 3 x 12 = 36; this is because the mathematician has a way of figuring it out, deductively, as it is done in logic (132). This analytic school which claim internality is otherwise deductive in my opinion was given a summary treatment by Descartes a soaring speculator as well as a mathematical philosopher of the modern period in the history of philosophy. He postulated that, it is clear that mathematics must be AJSIH Vol.1 No.2. (Nov.2011) 82-94 Etta, Kyrian 89
the sole and adequate key to unlock the truths of nature. Our next question is what is this mathematical method for Descartes? According to Descartes, his method is deduction. By deduction he means a chain of necessary inference from facts intuitively known, the certitude of this conclusion being known by the intuitions and memory of their necessary connection in thought. That is, it is rational deduction of consequences (Burtt 1980:106). The synthetic proposition, which claims to know truth externally and inductively, has also been noted, to have relevance in mathematical method of acquiring knowledge. This claim is spotted in the words of Descartes when he said, Now mathematics is just that universal science that deals with order and measurement generally. That is why arithmetic and geometry are the sciences in which sure and indubitable knowledge is possible. They deal with an object so pure and uncomplicated that they need make no assumptions at all that experience renders uncertain, but wholly consist in the rational deduction of consequences. This does not mean that the objects of mathematics are imaginary entities without existence in the physical world. For whoever demises that objects of pure mathematics exists, can hardly maintain that our geometrical ideas have been abstracted from existing things (107). In a nutshell, what Descartes meant in the above quotation is that mathematical knowledge is acquired through external and inductive methods which have to do with appeal to data in the observable physical world, otherwise synthetic. Having examined the various theoretical postulations about the apriori, empirical analytic and synthetic propositions, the other issue that occupies one s mind is, does these propositions, based on the theoretical postulations about them, in relation to mathematical method of attaining true AJSIH Vol.1 No.2. (Nov.2011) 82-94 Etta, Kyrian 90
knowledge hold water? The answer to the above question can be partly negative and partly positive. This is in view of the limit of each of the proposition, if taken individually as method. The empirical school holds that apriori (reason) is inadequate while the rationalists school (apriori)is of the view that the empirical school (sense experience) is limited also. The inadequacy of the analytic proposition, as not providing new knowledge has been noted in recent times, likewise the shortcomings of the synthetic propositional method. The controversy between the empiricists and the rationalists schools led to the synthetic apriori proposition of Kant, while the quarrel between the analytic and synthetic proposition has led to the emergence of synthetic analytic respectively. From the above analysis, one would see that each of the schools can only achieve success in attaining truth if it proceeds in cooperation with other methods or propositions. To state that the above propositions; apriori, empirical, analytic and synthetic, cannot and does not lead to attainment of mathematical knowledge would be an understatement. This is because, I agree with the various theorists as examined above that, if one adds 1+1 to arrive at 2, then 1 is placed physically side by side, and eventually added, it appeals to sense experience. But at other times, when one does not see or count before adding to get 2, and quickly says 1+1 = 2, it is reason working at such times. In light of above, A.N. Whitehead (1948) argues that mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about (27). This also shows that, if object of arithmetic are external and internal, inductive and deductive as the analytic and synthetic schools have claimed respectively, it follows that mathematical knowledge are arrived at through reason from that which is given in the physical observable data, visaviz. The Role of APriori, Empirical, Analytic and Synthetic Tools in Philosophy of Mathematics AJSIH Vol.1 No.2. (Nov.2011) 82-94 Etta, Kyrian 91
1. According to Descartes, his mathematical method of deduction which is otherwise analytic has the role of shifting the focus of attention onto knowledge of relation. 2. Secondly, Descartes maintained that it is a consequence of adopting his model that knowledge has to have a foundation. That is, how we acquire knowledge of the first principles, the foundation. 3. Thirdly, Descartes is of the view that his strategy was to show that his method is selfvalidating; that is, it can itself be deployed to reveal and secure the foundation of mathematical and scientific knowledge. 4. Fourthly, Descartes believed that his method was the sure way for clear and distinct knowledge (Tiles 1993:106). From my exposition of the theorists and their theoretical postulation, about how mathematical knowledge are arrived at through apriori, empirical, analytic and synthetic processes, analysis, and subsequent presentation of the roles played by the above tools, as articulated by Descartes, it would be pertinent to deduce that the roles played by the apriori, empirical, analytic and synthetic tools of philosophy in the philosophy of mathematics are as follows; (i) They play a foundational role. This is because they try to locate the first principles upon which mathematical method is built upon. (ii) They also play methodological role because they try to pin point the logic behind the attainment of mathematical knowledge. (iii) They play epistemological role. This is because through the propositional tools they try to locate and investigate the sources and certainty of mathematical knowledge. (iv) These tools also play historical role. This is due to its interest in knowing the source of the method of mathematical knowledge. AJSIH Vol.1 No.2. (Nov.2011) 82-94 Etta, Kyrian 92
(v) They play justificatory role also. This is because as each of the tools try to claim originality of its method, it is in other words justifying its claim and position so long as mathematical knowledge acquisition is concerned. Conclusion We have examined the apriori empirical, analytic and synthetic through clarification of terms, theoretical postulations by various theorists on each of the tools and how these relate to attainment of knowledge in mathematics. From the postulates presented by different theorists on these tools, it does show that apriori, empirical, analytic and synthetic propositions in philosophy have some foundational relevance in philosophy of mathematics. This is because when one relates one thing to another in order to get two without appeal to physical object before arriving at two,it is arrived at through reason. But if the two is arrived at after seeing or counting objects, this is gotten through sense experience. When the truth about mathematical knowledge is arrived at through logic in the internal or deductive arrangement, it is analytic, and when the truth about mathematical knowledge is arrived at through observation or addition of particular instances, while it is external or inductive, it is synthetic. AJSIH Vol.1 No.2. (Nov.2011) 82-94 Etta, Kyrian 93
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