A Characterization of 1-, 2-, 3-, 4-Homomorphisms of Ordered by Halas L., Hort D.

By Halas L., Hort D.

We represent completely ordered units in the category of all ordered units containing at the very least four-element chains. We use an easy courting among their isotone changes and the so referred to as 1-endomorphism that's brought within the paper. Later we describe 1-, 2-, 3-, 4-homomorphisms of ordered units within the language of great powerful mappings.

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Again, if by proof of a mathematical proposition we mean establishing its logical validity on the basis of a set of assumptions accepted as self-evident, then Pythagoras did not possess a proof of the theorem which bears his name; not because such a proof was beyond the ken of his period, but because he was temperamentally uninterested in proofs of this nature, as may readily be gleaned from the methods which he used in his numerological deductions. 10 am about to venture a conjecture, and I want to take the opportunity to emphasize that by advancing this and sundry other opinions which will appear in this work / neither invoke I 29 BEQUEST Otf THE GREEKS do not invoke authority, because documents opinion rests on imagination, and one imagination is as good or as bad as another.

Try as he might, he could not free himself of the tendency to attribute most achievements in philosophy and in mathematics to either Pythagoras or his adherents. What is more, there is little evidence that he tried to free himself of that bias. Thus, we find Thales barely mentioned in the vitiated Dialogues; as to Hippias, Hippocrates and Democritus, who had either kept aloof from the School or had openly opposed it, they were treated with contemptuous silence. In the second place, despite the vociferous claims of the ^latonists and Neoplatonists, Plato was not a mathematician, ^o Plato and his followers mathematics was largely a means to end being philosophy; they viewed the technical mathematics as a mere device for sharpening one's most as a course of training preparatory to handling the larger issues of philosophy.

How much did 28 THE FOUNDERS Pythagoras contribute to the proposition which bears his name? Was he the discoverer of this property of right triangles? Was he the first to point out its far-reaching implications? Was he the first to demonstrate the theorem by logical arguments applied to the basic axioms of geometry? Well, such historical evidence as is available to us today suggests that all these questions be answered in the negative. Pythagoras could not have been the discoverer of the relation, because, in one guise or another, this property of right triangles was known and used by scholars and artisans of Oriental lands thousands of years before Pythagoras was born.

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