By Elazar Meroz
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Additional info for Mathematical, philosophical, religious and spontaneous students' explanations of the paradox of Achilles and the tortoise
Stated that we could not mnœive the infinite and whenever we r e f e n d to the infinite it was nothing more than a statement that we were unable to conceive due to ouf own inability. More moderate empiria'sts, such as John Locke, held viewo that were more in line with the theories of Zeno and An'stotle- Locke's view was that we cauld not have a real idea of what the infinite was, however we were able to recognize the possibility of inaeasing perpetually and dividing space and time without limit Essentiallywhat he meant was that we c w l d not really t h M of infinity as a consummated thing in our rninds.
This could be done as follows: sinœ Al-A since &C,W A,,&+,- then angle ADIA1- O. In general, AIDI DI then the angle &CwtAn~=ADIAt which is non -zero therefore O for all n. Tharefore the sequenœ San= AAl + Al& + ApAs +.. AJL1 b inaeasing yet < A 0 for al1 n . We see that San b a convergent series whose lim%is AO. The sum of the series cannot be larger or smaller than AO. From this constructionwe can derive the generally known fomub for the sums of an infinite series S = al (lq). We can use this formulation to explain the paradox of Achilles and the Tortoise.
What constitutes knowfedge is the interest of the area of philosophy called epistemology. Epistemology a s k questions such as: 'what are the sources and aims of scientific knowledge? 'These questions can be asked in a general way or can be examined for sorne spacific domain of scientific study. There are generally two categories for the origins of knowledge, rationalism and empiricisrn. "The central point of mntroversy be-n rationalism and empiricism was the extent to which understanding of the world c w l d be amved at - by a priorimeans by the exercise of pure reason which is characteristic of mathematics (Moore, 1990.