Teaching Irrational Numbers Through Trigonometry

Philosophy and all the academic disciplines are sensitive to the aim of sound reasoning – except for the dialectical tradition which sanctions contradictions and antinomies (Heraclitus, Nicolas of Cusa, Hegel, Marx, Vaihinger, Simmel, Rex, and Dahrendorf). A brief overview is presented of conflicting theoretical stances within the various academic disciplines before an assessment is given of the positive and negative meaning of ‘reductionism.’ Against the background of historical lines of development the multiple terms employed in this context are mentioned and eventually positioned within the context of the normativity holding for logical thinking. It is argued that the logical contrary between logical and illogical serves as the foundation of other normative contraries, such as legal and illegal and moral and immoral. Through the discovery of irrational numbers the initial Pythagorean conviction that everything is number reverted to a geometrical perspective that generated a static metaphysics of being which challenged the ideas of plurality and motion. This development uncovered the problem of primitive terms in scientific discourse as an alternative for those theoretical attempts aimed at reducing whatever there is to one single mode of explanation. Zeno’s paradoxes are used to demonstrate an alternative understanding of the difference between the potential and the actual infinite as well as the nature of (theoretical) antinomies. It is argued that genuine antinomies are inter-modal in nature (such as is found in the attempt to reduce movement to static positions in space) and therefore differ from logical contradictions (such as a ‘square circle’ which merely confuses two figures within one modal aspect). Although every antinomy does entail logical contradictions, the latter do not necessarily presuppose an antinomy. The implication is that logic itself has an ontic foundation – as is seen from the nature of the principle of sufficient reason (ground) and the principle of the excluded antinomy – and therefore only acquires meaning on the basis of a non-reductionist ontology. When the method of immanent critique unveils genuine antinomies, the way is opened for meaningful intellectual interaction between different philosophical stances. In distinguishing between contradiction and antinomy philosophers are actually challenged to contemplate the implications of a non-reductionist ontology, such as avoiding the stance of monistic isms.

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Rational and Irrational Numbers

Arithmetical Wonderland ◽

10.5948/9781614441199.009 ◽

2015 ◽

pp. 133-154

Keyword(s):

Irrational Numbers

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Irrational Numbers arising from Certain Differential Equations

Canadian Mathematical Bulletin ◽

10.4153/cmb-1977-021-x ◽

1977 ◽

Vol 20 (1) ◽

pp. 117-120

Author(s):

M. Ram Murty ◽

V. Kumar Murty

Keyword(s):

Differential Equations ◽

Simple Proof ◽

General Theorem ◽

Irrational Numbers ◽

Suitable Modification

Niven [3] gave a simple proof that π is irrational. Koksma [2] modified Niven's proof to show that er is irrational for every non-zero rational r. Dixon [1] made a similar modification to show that π is not algebraic of degree 2. In this note, we prove a general theorem which gives Niven's and Koksma's results as easy corollaries. A suitable modification in our proof also gives Dixon's result.

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