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Krantz described a number of mathematical discoveries which characterize mathematical uniqueness from all other disciplines, whereby he indicated that mathematics discipline is distinguished from all others by its characteristic feature of proof. Krantz incorporates the idea of critical thinking in mathematics as a major factor that helps mathematicians to prove their work. He documents several works exemplifying the idea of proof and its benefits in mathematics including Euclid, Eudoxus, Pythagoras work among others. He discussed other works which were shown to be against proof in the mathematic study. Further, he endeavored to show how such proof in mathematics has been integrated and advanced in other disciplines in recent centuries. In his conclusion after the discussion of proof under different perspectives, he suggests that proof in mathematics will be perpetuated through expansion through the systems of interdisciplinary work.
Significance of the article
The content of the article predominantly indicated that its main purpose was to prove to the audience that mathematical almost all mathematical statements have proof, and this phenomenon began in ancient times before the Pythagoras theorem was invented. Krantz further wanted to portray that proof in the field of mathematics is timeless based on the fact it’s applied in numerous other disciplines such as physics, chemistry, computer science, and biological sciences just to mention a few of them. The author further offers the idea that mathematicians’ idea and attitude is not to dwell on the theoretical part, hence, any mathematical work should have a proof for it to be worthy substantiated. Generally, many of the mathematical proofs that the author describes in the article are supported by works of other experts which he acknowledges in his work. According to the author’s opinion, he points out that the Babylonians had the idea of the Pythagoras theorem before its invention by Pythagoras, as they had already developed several diagrams which epitomized such ideas (Biggs, 225).
However, he asserts that the theorem is of incredible significance in the mathematical study as it embodies the idea of mathematical proof, that is proving of geometrical statements in particular (Sierpiński, 118). While the author consents of the Pythagorean contributions in the understanding of the geometrical work, he further finds much more contribution in the advancement of mathematical understanding especially in rational work. Krantz indicated that Pythagorean triples were one of the numerous proofs which not helped mathematicians to transcend from the previous theoretical work, but the work set the foundation for numerous other works that followed such as those of Euclid which entails determination of the lengths of line segments. He posits that Pythagorean triple principles laid the foundation of the recent algebraic work, which is now widely applicable in many areas of study. However, the author did not entirely consent to the subject matter that every mathematical work has to be proven. In this connection, he explores Errett’s and Nicola’s work which is strongly against this argument.
Modern application of Pythagorean triples
The idea of the Pythagorean triple has descended from one generation to another, and today, it has found wide application in many areas in society. Ultimately, this has been used in computerized projection. This is well applied through the use of concepts of projective geometry (Biggs, 347).
The author showed in the article that experts from a number of disciplines are becoming much more inclined to use proof in their fields. He adds that they are employing it in high levels of complexities which have helped in the linking of individuals from different professions. There are many changes that have occurred in the learning institutions and the global community that may contribute to a new application of the Pythagorean triple and the general use of proof in the present and in the future time. Overtly, there has been an augmented division of subjects which is believed result to in increased integration of mathematical proof such as those of Pythagorean triple in the newly created disciplines (Sierpiński, 15).
Pythagoras’ work was invented several decades back before the publication of this particular article; however, it appears that since its invention, it has contributed a lot to the advancement of the mathematical discipline. The writer’s gist of the work was advocating the importance of proof in the discipline of mathematics which he believed to have been founded in Pythagoras work, and that will be perpetuated in the pure mathematics as well as the proliferation of other disciplines.
Biggs, Norman. Discrete Mathematics; London: Oxford University Press, 2003.
Sierpiński, Waclaw. Pythagorean Triangles. New York: Dover Publications, 2003.