Yogyakarta State University
Posy C. sets forth that according to structuralist the basic element of mathematics shouldn't be arbitrarily picked, yet nothing dictates its choice and the basic units are structures, not actually objects.
This 1 leads to structuralism to perceive that to be a natural number is to be a place in the sequence; accordingly, if mathematics is totally abstract, why should it have any applicability?
Structuralists 2 argues that mathematics is not about some particular collection of abstract objects but rather mathematics is the science of patterns of structures, and particular objects are relevant to mathematics only in so far as they instantiate some pattern or structure.
Benacerraf 3, as a structuralist says:
When it comes to learning about numbers, they merely learn new names for familiar sets. They count members of a set by determining the cardinality of the set, and they establish this by demonstrating that a specific relation holds between the set and one of the numbers. To count the elements of some k-membered set b is to establish a one-to-one correspondence between the elements of b and the elements of A less than or equal to k. The relation "pointing-to-each-member-of-b-in-turn-while-saying-the-numbers-up-to-and-including-" esta blishes such a correspondence.
Benacerraf 4 concludes that there is no one account which conclusively establishes which sets are the "real" numbers, and he doesn't believe that there could be such an argument.
According to him, any "object" or referent will do as long as the structural relations are maintained.
Benacerraf 5 argues that Frege's belief of some "objects" for number words to name and with which numbers could be identical, stemmed from his inconsistent logic.
Since all objects of the universe were on par, the question whether two names had the same referent always had a truth value; however, identity conditions make sense only in contexts where there exist individuating conditions.
Benacerraf 6 claims that if an expression of the form "x=y" is to have sense, it can be only in contexts where it is clear that both x and y are of some kind or category C, and that it is the conditions which individuate things as the same C which are operative and determine its truth value.
According to Benacerraf 7, Frege fails to realize this fact. It is a thesis that is supported by the activity of mathematicians, and is essential to the philosophical perspective underlying category theory.
Benacerraf 8 concludes that numbers could not be sets at all on the grounds that there are no good reasons to say that any particular number is some particular set, for any system of objects that forms a recursive progression would be adequate.
Benacerraf 9 concludes that a system of objects exhibits the structure of the integers implies that the elements of that system have some properties which are not dependent on structure.
Accordingly, it must be possible to individuate those objects independently of the role they play in the structure; however, this is precisely what cannot be done with numbers.
Benacerraf 10 argues that numbers possess outside of the properties of the structure are of no consequence to the mathematician, nor should they therefore be of concern for the philosopher of mathematics.
Accordingly, there is an activity which theorizes about the unique properties of individual numbers separated from the progressive structure.
According to Structuralist 11, arithmetic is the science exploring the abstract structure that all progressions have in common merely in virtue of being progressions.
Arithmetic is not concerned with particular numbers, and there are no unique set of objects which are the numbers.
Number does not have a singular reference, because the theory is elaborating an abstract structure and not the properties of individual objects.
In counting, we do not correlate sets with initial segments of the sequence of numbers as extra-linguistic entities, but correlate sets with initial segments of the sequence of number words.
The recursive sequence 12 is a sort of yardstick which we use to measure sets; questions of the identification of the referents of number words should be dismissed as misguided in just the way that a question about the referents of the parts of the ruler would be seen as misguided.
According to Structuralist 13, the mathematical description, model, structure, theory, or whatever, cannot serve as an explanation of a non-mathematical event without an account of the relationship between mathematics per se and scientific reality per se.
A mathematical structure can, perhaps, be similarly construed as the form of a possible system of related objects, ignoring the features of the objects that are not relevant to the interrelations.
A mathematical structure is completely described in terms of the interrelations; a typical beginning of a mathematical text consists of the announcement that certain mathematical objects such real numbers are to be studied.
In some cases, at least, the only thing about these objects is that there are certain relations among them and/or operations on them; and one easily gets the impression that the objects themselves are not the problems.
The relations and operations are what we study.
1 Benacerraf in Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics” Retrieved 2004
2Posy, C. ,1992, “Philosophy of Mathematics”. Retreived 2004
3Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics”, Retrieved 2004
6 Benacerraf in Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics” Retrieved 2004
8 Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics”, Retrieved 2004
11In Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics”, Retrieved 2004
12Benacerraf in Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics” Retrieved 2004
13Shapiro in Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics” Retrieved 2004