Oct 10, 2012

Elegi Menggapai "Mathematical Objectivity"

By Marsigit
Yogyakarta State University

Rand A. et al theorizes that objectivism recognizes a deeper connection between mathematics and philosophy than advocates of other philosophies have imagined.

The process of concept-formation 1 involves the grasp of quantitative relationships among units and the omission of their specific measurements.

They view that it thus places mathematics at the core of human knowledge as a crucial element of the process of abstraction.

While, Tait W.W (1983) indicates that the question of objectivity in mathematics concerns, not primarily the existence of objects, but the objectivity of mathematical discourse.

Objectivity in mathematics 2 is established when meaning has been specified for mathematical propositions, including existential propositions.

This obviously resonates with Frege’s so-called context principle, although Frege seems to have rejected the general view point of Cantor and, more explicitly, Hilbert towards mathematical existence.

Tait W.W points that the question of objective existence and truth concern the 3 axiomatic method as it posit for ‘concrete’ mathematics, i.e. axioms of logic and the theory of finite and transfinite numbers and the cumulative hierarchy of sets over them; in which we can reason about arbitrary groups, spaces and the like, and can construct examples of them.

The axiomatic method seems to run into difficulties. If meaning and truth are to be determined by what is deducible from the axioms, then we ought to require at least that the axioms be consistent, since otherwise the distinctions true/false and existence/non-existence collapse.

Tait W. W claims that there is an external criterion of mathematical existence and truth and that numbers, functions, sets, etc., satisfy it, is often called ‘Platonism’; but Plato deserves a better fate.

Wittgenstein, at least in analogous cases, called it ‘realism’; but he wants to save this term for the view that we can truthfully assert the existence of numbers and the like without explaining the assertion away as saying something else.

From the realist perspective 4, what is objective are the grounds for judging truth of mathematical propositions, including existential ones, namely, derivability from the axioms.

However, to hold that there is some ground beyond this, to which the axioms themselves are accountable, is to enter the domain of super-realism, where mathematics is again speculative; if the axioms are accountable, then they might be false. 5

Tait W.W (1983) emphasizes that our mathematical knowledge is objective and unchanging because it’s knowledge of objects external to us, independent of us, which are indeed changeless.

From the view of realists 6, our mathematical knowledge, in the sense of what is known, is objective, in that the criterion for truth, namely provability, is public.

He further states that the criterion depends on the fact that we agree about what counts as a correct application—what counts as a correct proof.

Nevertheless, there is such agreement and the criterion is the same for all and is in no way subjective.

As long as there remains an objective criterion for truth, namely provability from the axioms, it is inessential to this conception that there always remain some indeterminate propositions.

However, there is a further challenge to realism which seems to cut deeper, because it challenges the idea that provability from the axioms is objective. 7

1 Rand, A, Binswanger, H., Peikoff, P.,1990, “Introduction to Objectivist Epistemology”, Retrieved 2004
2Tait, W.W., 1983, “Beyond the axioms: The question of objectivity in mathematics”, Retrieved 2004
4 Ibid
5 Ibid
6 Ibid.
7 Ibid.

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