Mathematical Concept: What are their notions?

By Marsigit
Yogyakarta State University

Bold, T., 2004, notified that the essential components of mathematics covers the concepts of integer numbers, fractions, additions, divisions and equations; in which addition and division are connected with the study of mathematical propositions and the concept of integers and fractions are elements of mathematical concepts. He stated that the concept of equation can fall under mathematical propositions, but it is also related to issues concerning mathematical certainty; and integer numbers are statements about certain properties of bodies; hence, the concept of natural number is explained by study about this particular property of bodies or “notion of unity” or “quantitative property”. He clarified that there are two necessary elements involved with explanation of what a mathematical statement assures: quantifiable bodies and quantitative property of bodies. According to him, “quantifiable bodies” is mathematical statements initially involve only bodies that have capacity to be quantified by the mind; in which, once the concept of quantity is achieved, all bodies are quantifiable and consequently, it will be impossible for an empty mind that only interacts with bodies like sand and air to form the concept of number. While “quantitative property of bodies” is that once quantifiable bodies are present, mathematical statements do not affirm just any properties of those bodies, but only about that particular quantitative property; a mathematical statement is about quantitative properties of bodies, and quantitative bodies are in the bodies. That is how mathematical statements connect with the world.

Bold, T., 2004, further indicated that the second necessary element for interpretation of mathematical concepts is man’s ability of to abstract, that is the mind’s ability to abstract the quantitative property from bodies and use it without the presence of bodies. Due to the fact that all of mathematics is abstract, he believes that one of the motives of intuitionists to think mathematics is a sole product of the mind. He added that a third important element is the concept of infinity; while infinity is based on the concept of possibility. Accordingly, infinity is not a quantity, but a concept based on unrestricted possibility; it is a character of possibility. Next he claimed that the concept of fraction is just based on abstraction and possibleness. According to him, the issue involved with rational and irrational numbers is completely irrelevant for interpretation of concepts of fraction as Arend Heyting is overly concerned. As far as mathematical concepts are concerned, rational numbers as n/p and irrational numbers as q are just a matter of different ways of expression. The difference between them is issue within mathematics to be explained by mathematical terms and language.

On the other hand, Podnieks, K., 1992, claimed that the concept of natural numbers developed from human operations with collections of discrete objects; however, it is impossible to verify such an assertion empirically and the concept of natural number was already stabilized and detached from its real source viz. the quantitative relations of discrete collections in the human practice, and it began to work as a stable self-contained model. According to him, the system of natural numbers is an idealization of these quantitative relations; in which people abstracted it from their experience with small collections and extrapolated their rules onto much greater collections (millions of things) and thus idealized the real situation. He insisted that the process of idealization ended in stable, fixed, self-contained concepts of numbers, points, lines etc and ceased to change. While the stabilization of concepts is an evidence of their detachment from real objects that have led people to these concepts and that are continuing their independent life and contain an immense variety of changing details.

According to Podnieks, K., 1992, when working in geometry, a mathematician does not investigate the relations of things of the human practice directly, he investigates some stable notion of these relations viz. an idealized, fantastic "world" of points, lines etc; and during the investigation this notion is treated subjectively as the "last reality", without any "more fundamental" reality behind it. Further he claimed that if during the process of reasoning mathematicians had to remember permanently the peculiarities of real things, then instead of a science viz. efficient geometrical methods, we would have an art - simple, specific algorithms obtained by means of trial and error or on behalf of some elementary intuition. He summed that Mathematics of Ancient Orient stopped at this level and Greeks went further. According to him Plato treats the end product of the evolution of mathematical concepts that is a stable, self-contained system of idealized objects, as an independent beginning point of the evolution of the "world of things"; Plato tried to explain those aspects of the human knowledge, which remained inaccessible to other philosophers of his time.

Jones, R.B.,1997, elaborated that in the hands of the ancient Greeks mathematics becomes a systematic body of knowledge rather than a collection of practical techniques; mathematics is established as a deductive science in which the standard of rigorous demonstration is deductive proof. According to him, Aristotle provides a codification of logic which remains definitive for two thousand years; while the axiomatic method is established and is systematically applied to the mathematics of the classical period by Euclid, whose Elements becomes one of the most influential books in history. Jones insisted that the next major advances in logic after Aristotle appear in the nineteenth century, in which Boole introduces the propositional (boolean) logic and Frege devises the predicate calculus. This provides the technical basis for the logicisation of mathematics and the transition from informal to formal proof. On the other hand, Russell's paradox shakes the foundational advances of Frege, but is quickly resolved.

Further, Jones, R.B.,1997, noted that the Pythagoreans, were first inclined to regard number theory as more basic than geometry; the discovery of in-commensurable ratios presented them with a foundational crisis not fully resolved until the 19th century. Since Greek number theory, which concerns only whole numbers, cannot adequately deal with the magnitudes found in geometry, geometry comes to be considered more fundamental than arithmetic. Therefore, despite the inadequacies of the available number systems the desire to treat geometry numerically remains. Descartes1 , by inventing co-ordinate geometry advances an understanding of how geometry can be reduced to number. Meanwhile Mathematics continues to develop as Newton and Leibniz invent the calculus despite weakness in the underlying number system and Berkeley is one of the vocal critics of the soundness of the methods used. Jones2 claimed that not until the 19th Century do we see the foundational problems resolved by precise definition of the real number system and elimination of the use of infinitesimals from mathematical proofs. On the other side, Cantor's development of set theory together with Frege's advances in logic pave the way for Zermelo's first order axiomatisation of set theory, which provides the foundations for mathematics in the twentieth century.3

Landry, E., 2004, quoted Bolzano's that mathematical truths can and must be proven from the mere [the analysis of] concepts. Bolzano4 did this by demonstrating how mathematical rigor could be both an epistemological as well as a semantic notion; his demonstration of the dual character of mathematical rigor was the distinction between what he termed subjective and objective representations. According to Bolzano5 , meaning relates not to the subjective representation but rather relates to the inter-subjective content and as such is in no need of assistance from intuitions, either empirical or pure. Meanwhile, Hempel, C.G., 2001, argued that the validity of mathematics rests neither on its alleged self-evidential character nor on any empirical basis, but derives from the stipulations which determine the meaning of the mathematical concepts, and that the propositions of mathematics are therefore essentially "true by definition." He insisted that for the rigorous development of a mathematical theory proceeds not simply from a set of definitions but rather from a set of non-definitional propositions which are not proved within the theory; these are the postulates or axioms of the theory.

Hempel, C.G., 2001, exposed the example that the multiplication of natural numbers may be defined by definition which expresses in a rigorous form the idea that a product nk of two integers may be considered as the sum of k terms each of which equals n, that is (a) n.0 = 0; (b) n.k' = n.k + n. We may prove the laws governing addition and multiplication, such as the commutative, associative, and distributive laws (n + k = k + n; n.k = k.n; n + (k + I) = (n + k) + I; n.(k.l) = (n.k).l;n.(k + l) = (n.k) + (n.l)), as

commutative associative distributive
n + k = k + n
n.k = k.n n + (k + l) = (n + k) + l n.(k.l) = (n.k).l
n.(k + l) = (n.k) + (n.l)

Hempel6 concluded that in terms of addition and multiplication, the inverse operations of subtraction and division can then be defined; but it turns out that these "cannot always be performed"; i.e., in contradistinction to the sum and the product, the difference and the quotient are not defined for every couple of numbers; for example, 7-10 and 7/10 are undefined; and this situation suggests an enlargement of the number system by the introduction of negative and of rational numbers.

Ford & Peat, 1988, insisted that mathematical notation has assimilated symbols from many different alphabets and fonts includes symbols that are specific to mathematics; in mathematics a word has a different and specific meaning such as group, ring, field, category, etc; mathematical statements have their own moderately complex taxonomy, being divided into axioms, conjectures, theorems, lemmas and corollaries; and there are stock phrases in mathematics, used with specific meanings, such as "if and only if", "necessary and sufficient" and "without loss of generality". 7Any series of mathematical statements can be written in a formal language, and a finite state automaton can apply the rules of logic to check that each statement follows from the previous ones. According to them, various mathematicians attempted to achieve this in practice, in order to place the whole of mathematics on a axiomatic basis; while Gödel's incompleteness theorem shows that this ultimate goal is unreachable in which any formal language is powerful enough to capture mathematics will contain un-decidable statements. 8

Ford & Peat, 1988, claimed that the vast majority of statements in mathematics are decidable, and the existence of un-decidable statements is not a serious obstacle to practical mathematics. 9 According to them mathematics is used to communicate information about a wide range of different subjects covering to describe the real world viz. many areas of mathematics originated with attempts to describe and solve real world phenomena that is from measuring farms (geometry) to falling apples (calculus) to gambling (probability); to understand more about the universe around us from its largest scales (cosmology) to its smallest (quantum mechanics); to describe abstract structures which have no known physical counterparts at all; to describe mathematics itself such as category theory in which deals with the structures of mathematics and the relationships between them.

References:

1 In Jones, R.B.,1997, A Short History of Rigour in Mathematics,
2Jones, R.B.,1997, A Short History of Rigour in Mathematics,
3 Jones, R.B.,1997, A Short History of Rigour in Mathematics,
4 In Landry, E., 2004, Semantic Realism: Why Mathematicians Mean What They Say,
5 Ibid.
6 Hempel, C.G., 2001, On the Nature of Mathematical Truth, http://www.ltn.lv/ ~podniek/gt.htm
7 Ford & Peat, 1988, Mathematics as a language, Wikipedia, the free encyclopedia,
8 Ibid.
9 Ibid.