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The philosophy of mathematics begins with Pythagoras, who later developed Pythagoras Theorem. While believing that mathematics gave us the key to understand reality he proved that mathematics is known a priori that is, without appeal to sense experience. He started talking to a slave boy, and by a series of questions elicited from him a method of constructing a line, using a special case of Pythagoras' theorem.…

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- Subject: History
- Type: Math Problem
- Level: Masters
- Pages: 12 (3000 words)
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- Author: liamquigley

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When I talk about the diagonal of the square, or the nine-point circle, or the Euler line, I am not talking about the often rather sketchy and highly imperfect drawing on the blackboard, but about something which underlies all particular exemplifications of squares and diagonals, nine-point circles, or Euler lines, and is independent of each of them" 2. The very fact that we use the definite article, and talk of the square, the nine-point circle, etc., bears witness to this; and by the same token, it would be absurd to ask where the square was, or to ask when the nine-point center came to be on the Euler line, or to suggest that Pythagoras' theorem might hold for you but not for me. So Plato's answer to the question "What is mathematics about" is that it is about something timeless, spaceless and objective 3.

Among the five postulates which Euclid wanted us to grant the fifth one is "If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles. ...

aight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles. "These were generally taken to express self-evident truths. This is somewhat surprising, in that the first three are not really propositions at all, but instructions expressed in the infinitive, and the last too complex to be self-evident no finite man can see it to be true, because no finite man can see indefinitely far to make sure that the two lines actually do meet in every case. Many other formulations of the fifth postulate have been offered, both in the ancient and in the modern world, in the hope of their being more self-evidently true"4 . Among them the most notable was "In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides" 5.

Fig 1.1 6

The alternative formulations of the fifth postulate of the theorem are less cumbersome and may be more acceptable than Euclid's own version, but none of them are so self-evident that they cannot be questioned. The importance of Pythagoras proposed theorem can be seen from the fact that Pythagoras' theorem is far from being obviously true, something that should be granted without more ado, it does not need any further justifications. "In fact, none of the other alternative formulations was felt to be completely obvious, and they all seemed in need of some kind of further justification. The philosophers Wallis and Saccheri in search of a better justification, devoted years to trying to prove the fifth postulate by a reductio ad absurdum, assuming it to be false and trying to derive a contradiction. The attempt failed, but in the course of it he unwittingly discovered ...Download file to see next pagesRead More

Among the five postulates which Euclid wanted us to grant the fifth one is "If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles. ...

aight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles. "These were generally taken to express self-evident truths. This is somewhat surprising, in that the first three are not really propositions at all, but instructions expressed in the infinitive, and the last too complex to be self-evident no finite man can see it to be true, because no finite man can see indefinitely far to make sure that the two lines actually do meet in every case. Many other formulations of the fifth postulate have been offered, both in the ancient and in the modern world, in the hope of their being more self-evidently true"4 . Among them the most notable was "In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides" 5.

Fig 1.1 6

The alternative formulations of the fifth postulate of the theorem are less cumbersome and may be more acceptable than Euclid's own version, but none of them are so self-evident that they cannot be questioned. The importance of Pythagoras proposed theorem can be seen from the fact that Pythagoras' theorem is far from being obviously true, something that should be granted without more ado, it does not need any further justifications. "In fact, none of the other alternative formulations was felt to be completely obvious, and they all seemed in need of some kind of further justification. The philosophers Wallis and Saccheri in search of a better justification, devoted years to trying to prove the fifth postulate by a reductio ad absurdum, assuming it to be false and trying to derive a contradiction. The attempt failed, but in the course of it he unwittingly discovered ...Download file to see next pagesRead More

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