Oct 13, 2012
The Un-stability Foundation of Mathematics: Litlangs’,Bold’s, Thomson’s and Posy’s Arguments
Yogyakarta State University
Litlangs, 2004, confronted that Aristotle disagreed with Plato; according to Aristotle, forms were not entities remote from appearance but something which entered into objects of the world. Aristotle claimed that when we can abstract oneness or circularity, it does not mean that these abstractions represent something remote and eternal.
For Aristotle1 , mathematics was simply reasoning about idealizations; and he looked closely at the structure of mathematics, distinguishing logic, principles used to demonstrate theorems, definitions and hypotheses. Plato2 also reflected on infinity, perceiving the difference between a potential infinity e.g. adding one to a number ad infinitum and a complete infinity e.g. number of points into which a line is divisible. Bold, T., 2004, claimed that both the intuitionist and the formalist assured that mathematics are just inventions and do not inform us with anything about the world; both take this approach to explain the absolute certainty of mathematics and reject the use of infinity. Bold3 noted that intuitionists admit this major similarity the formalist and note the difference as a disagreement on where mathematical exactness exist; the intuitionist says in the human intellect and the formalist says on paper. According to Arend Heyting 4, mathematics is a production of the human mind; he claimed that intuitionism claims mathematical propositions inherit their certainty from human knowledge that is based on empirical experience. Bold 5 maintained that since, infinity can not be experienced, the intuitionist refuses to push application of mathematics beyond finite; Heyting declared that faith in transcendental existence, unsupported by concepts, must be rejected as a means of mathematical proof. Similarly, Bold 6 found that Hilbert wrote that for logical inferences to be reliable it must be possible to survey these objects completely in their parts; since there is no such survey for infinity a reliable inference can only be based on a finite system. According to the formalists, the whole of mathematics consists of only arbitrary rules like those of chess.
Further, Bold, T., 2004, indicated that, on the other hand, the logicists came close to proving that mathematics was a branch of logic. According to Bold 7, the logisticians want to define mathematical concepts in terms of logical concepts and deduct mathematical propositions from logical axioms; as the basic elements of logic are sets and their properties, the logicists use sets to define mathematical concepts. Bold elaborated the following:
For example, the meaning of 0 is the class of all null sets, 1 is the class of all sets with 1 members, 2 is the class of all sets with a pair of members and etc. The problem of this definition is that it does not explain the concept of number 2. We need not a definition, but an explanation. “Number 2 is the class of all pairs” presupposes the concept of 2. Such definition is a mere construction of a formal system that is consistent with mathematics. It might be necessary or sufficient for the work on mathematical propositions, but it does not give us any insight into mathematical concepts. The most exciting parts of the logicist account hides in what Russell and Whitehead’s Principia Mathematica has to say about mathematical propositions and what Ramsey and Wittgenstein said about its flaws. However, the present paper is primarily on mathematical concepts only. “What is number?”, “What is infinity?” and “Why is the absolute certainty?” are the questions that are in need of philosophical interpretation. I am not holding a directly opposing position to these theories; at some points I agree with them. However, I will simply attempt to pick up where the above schools are unsatisfactory and offer a better account on the ensuing issues. 8
On the other hand, Posy, C., 1992, found that Hilbert actually put a structure on the intuitive part of mathematics, essentially that of finitary thought and formal systems; with Gödel's work. Thompson, P.,1993, argued that the Gödelian brand of Platonism, in particular, takes its lead from the actual experience of doing mathematics, and Gödel accounts for the obviousness of the elementary set-theoretical axioms by positing a faculty of mathematical intuition, analogous to sense-perception in physics, so that, presumably, the axioms 'force themselves upon us' much as the assumption of 'medium-sized physical objects' forces itself upon us as an explanation of our physical experiences. However, Thompson 9 stated that counterintuitive has acquired an ambiguous role in our language use that is when applied to a strange but true principle; counterintuitive can now mean anything on a continuum from intuitively false to not intuitively true, depending on the strength of the conjecture we would have been predisposed to make against it, had we not seen, and been won over by, the proof; and indeed, to our surprise, we often find out, in times of paradox, how weak and defeatable our ordinary intuitions are.
Thompson 10 claimed that the very idea that our intuitions should be both decisive and failsafe, derives historically from the maelstrom of senses which the term 'intuition' has acquired in a series of primitive epistemic theories in which some of these senses have been inherited from the large role introspection played in the indubitable bedrock of Cartesian-style philosophy, and some simply from the pervasiveness of out-moded theological convictions which seek to make certain modes of justification unassailable. On the other hand, Posy, C., 1992, insisted that Hilbert's formal system fits the theory of recursive functions. Posy 11 insisted that Brouwer was very much opposed to these ideas, especially that of formalizing systems; he even opposed the formalization of logic; Brouwer had a very radical view of mathematics and language's relationship. According to Brouwer 12, in language, we can communicate the output of mathematical construction, thus helping others recreate the mathematical experience; however, the proof itself is a pre-linguistic, purely conscious activity which is much more flexible than language. Brouwer thought formal systems could never be adequate to cover all the flexible options available to the creative mathematician; and thought that formalism was absurd. Posy 14 noted that, in particular, Brouwer 13 thought that it was crazy to think that codified logic could capture the rules for correct mathematical thought. Brouwer 15 showed particular rules of logic are inadequate with the most famous of the law of the excluded middle.
Thompson, P.,1993, noted that Brouwer that the common un-circumspect belief in the applicability of traditional logic to mathematics was caused historically; he next stated that by the fact that, firstly, classical logic was abstracted from the mathematics of subsets of a definite finite set, that, secondly, an a priori existence independent of mathematics was ascribed to this logic, and finally, on the basis of this suppositious apriority, it was unjustifiably applied to the mathematics of infinite sets. Furthermore, Posy, C., 1992, insisted that Brouwer hypothesized about the reason why philosophers and mathematicians included the law of the excluded middle; according to Brouwer, logic was codified when the scientific community was concerned only with finite objects. Brouwer 16 said that, considering only finite objects, the law of the excluded middle holds; however, a mistake was made when mathematics moved into the infinitary in which the rigid rules of logic were maintained without question. Brouwer 17 suggested that no rigid codification should come before the development of mathematics. Posy found that a second major distinction between Brouwer and Hilbert was that they disagreed on the position of logic in which Hilbert thought logic was an autonomous, finished science that could be freely applied to other mathematics, Brouwer argued that logic should only come after the mathematics is developed.
Litlangs, 2004, in his overview, insisted that profound questions of how varied of intellect faces difficulties in explaining mathematics internally i.e. their gaps, contradictions and ambiguities that lie beneath the most certain of procedures, leads to rough conclusion that mathematics may be no more logical than poetry; it is just free creations of the human mind that unaccountably give order to ourselves and the natural world. Litlangs 18 perceived that though mathematics might seem the clearest and most certain kind of knowledge we possess, there are problems just as serious as those in any other branch of philosophy about the nature of mathematics and the meaning of its propositions. Litlangs 19 found that Plato believed in forms or ideas that were eternal, capable of precise definition and independent of perception; among such entities Plato included numbers and the objects of geometry such as lines, points, circles, which were therefore apprehended not with the senses but with reason; he deals with the objects of mathematics with specific instances of ideal forms. According to Plato, as it was noted by Litlangs, since the true propositions of mathematics were true of the unchangeable relations between unchangeable objects, they were inevitably true that is mathematics discovered pre-existing truths "out there" rather than created something from our mental predispositions; and as for the objects perceived by our senses, they are only poor and evanescent copies of the forms.
Meanwhile, Litlangs, 2004, insisted that Leibniz brought together logic and mathematics; however, whereas Aristotle used propositions of the subject i.e. predicate form, Leibniz argued that the subject contains the predicate that is a view that brought in infinity and God. According to Leibniz 20, mathematical propositions are not true because they deal in eternal or idealized entities, but because their denial is logically impossible; they are true not only of this world, or the world of eternal forms, but of all possible worlds. Litlangs 21 insisted that unlike Plato, for whom constructions were adventitious aids, Leibniz saw the importance of notation, a symbolism of calculation, and so began what became very important in the twentieth century that is a method of forming and arranging characters and signs to represent the relationships between mathematical thoughts.
Litlangs, 2004, further stipulated that Immanuel Kant perceived mathematical entities as a-priori synthetic propositions, which of course provide the necessary conditions for objective experience; time and space were matrices, the containers holding the changing material of perception. According to Kant 22, mathematics was the description of space and time; if restricted to thought, mathematical concepts required only self-consistency, but the construction of such concepts involves space having a certain structure, which in Kant's day was described by Euclidean geometry. Litlangs 23 noted that for Kant, the distinction between the abstract "two" and "two pears" is about construction plus empirical matter; in his analysis of infinity, Kant accepted Aristotle's distinction between potential and complete infinity, but did not think the latter was logically impossible. Kant 24 perceived that complete infinity was an idea of reason, internally consistent, though of course never encountered in our world of sense perceptions. Litlangs further insisted that 25 Frege and Russell and their followers developed Leibniz's idea that mathematics was something logically undeniable; Frege used general laws of logic plus definitions, formulating a symbolic notation for the reasoning required. However, through the long chains of reasoning, these symbols became less intuitively obvious, the transition being mediated by definitions. Litlangs 26 noted that Russell saw them as notational conveniences, mere steps in the argument; while Frege saw them as implying something worthy of careful thought, often presenting key mathematical concepts from new angles. Litlangs 27 found that while in Russell's case the definitions had no objective existence, in Frege's case the matter was not so clear that is the definitions were logical objects which claim an existence equal to other mathematical entities. Litlangs 28 concluded that, nonetheless, Russell carried on, resolving and side-stepping many logical paradoxes, to create with Whitehead the monumental system of description and notation of the Principia Mathematica.
Meanwhile, Thompson, P.,1993, exposed the critical movement of Cauchy and Weierstrass to have been a caution or reserve over the mathematical use of the infinite, except as a façon de parler in summing series or taking limits, where it really behaved as a convenient metaphor, or mode of abbreviation, for clumsier expressions only involving finite numbers. Thompson 29 claimed that when Cantor came on the scene, the German mathematician Leopold Kronecker, who had already 'constructively' re-written the theory of algebraic number fields, objected violently to Cantor's belief that, so long as logic was respected, statements about the completed infinite were perfectly significant. According to Thompson 30, Cantor had further urged that we should be fully prepared to use familiar words in altogether new contexts, or with reference to situations not previously envisaged; Kronecker, however, felt that Cantor was blindly cashing finite schemas in infinite domains, both by attributing a cardinal to any aggregate whatsoever, finite or infinite, and worse still, in his subsequent elaboration of transfinite arithmetic. Thompson 31 insisted that although the interim strain on the intuition, at the time, was crucial to Euler's heuristic approach, this particular infinite detour had been analyzed out of his subsequent proofs of the result, which appeared almost 10 years after its discovery.
Thompson, P.,1993, clarified that Gödel's feeling is that our intuition can be suitably extended to a familiarity with very strongly axiomatic domains, such as extensions of ZF, or calculus on smooth space-time manifolds, thereby providing us with backgrounds for either accepting or rejecting hypotheses independently of our pre-theoretic prejudices or preconceptions about them. Thompson 32 indicated that the general reccursiveness, as in the Gödel and Herbrand sense, with regard to their claims to be collectively demarcating the limits of intuitive computability, is a feature of this particular problem that it is susceptible to a diversity of equally restrictive intuitive re-characterizations, whose unexpected confluence gives each of them a strong intuitive recommendation and this confluence turns out to be a surprisingly valuable asset in appraising our rather more recondite extensions of our intuitive concepts. Thompson 33 concluded that Gödel,34 with his basic trust in transcendental logic, likes to think that our logical optics is only slightly out of focus, and hopes that after some minor correction of it, we shall see sharp, and then everyone will agree that we are right; however, he who does not share such a trust will be disturbed by the high degree of arbitrariness in a system like Zermelo's, or even in Hilbert's system. Thompson 35 suggested that Hilbert will not be able to assure us of consistency forever; therefore we must be content if a simple axiomatic system of mathematics has met the test of our mathematical experiences so far.
1Litlangs, 2004, Math Theory, Poetry Magic: firstname.lastname@example.org
3Bold, T., 2004, Concepts on Mathematical Concepts, http://www.usfca.edu/philosophy/ discourse/8/bold.doc
11Posy, C., 1992, Philosophy of Mathematics, http://www.cs.washington.edu/ homes/ gjb.doc/philmath.htm
18Litlangs, 2004, Math Theory, Poetry Magic: email@example.com
29Thompson, P.,1993, The Nature And Role Of Intuition In Mathematical Epistemology, University College, Oxford University, U.K