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1. #PARADOX ANCIENT SPACE DISCOVERIES LOCATIONS PLUS#
2. #PARADOX ANCIENT SPACE DISCOVERIES LOCATIONS SERIES#

One of the many representations (and formulations) of Zeno of Elea's paradox relating to the. Under this line of thinking, it may still be impossible for Atalanta to reach her destination. If each jump took the same amount of time, for example, regardless of the distance traveled, it would take an infinite amount of time to cover whatever tiny fraction-of-the-journey remains. How could time come into play to ruin this mathematically elegant and compelling "solution" to Zeno's paradox?īecause there's no guarantee that each of the infinite number of jumps you need to take - even to cover a finite distance - occurs in a finite amount of time. It doesn't tell you anything about how long it takes you to reach your destination, and that's the tricky part of the paradox. This mathematical line of reasoning is only good enough to show that the total distance you must travel converges to a finite value.

#PARADOX ANCIENT SPACE DISCOVERIES LOCATIONS PLUS#

convergent series: one entire "thing" can be obtained by summing up one half plus one fourth plus one eighth, etc. And this method applies to both finite and infi­nite sets.By continuously halving a quantity, you can show that the sum of each successive half leads to a. If each ele­ment finds its part­ner and none remains alone (we call this a bijec­tion), we can then say that the two sets are equal. His answer lies in the way he com­pares two sets: instead of count­ing the num­ber of ele­ments of the lat­ter and com­par­ing them (which can­not be done with an infi­nite set), the method con­sists in try­ing to match an ele­ment of the first set with an ele­ment of the sec­ond. Georg Can­tor asked him­self a sim­ple ques­tion: can we com­pare two infi­nite sets? Can one be “big­ger” than the other? In the end, infin­i­ty is just a tool among many oth­ers that math­e­mat­ics puts at our dis­pos­al to mea­sure, cal­cu­late and under­stand our environment.īut a Ger­man math­e­mati­cian from the end of the 19 th Cen­tu­ry went much fur­ther than any­one else at the time to manip­u­late infin­i­ty, or more pre­cise­ly infi­nite sets. There is no need to under­stand or visu­alise infin­i­ty to use it.

#PARADOX ANCIENT SPACE DISCOVERIES LOCATIONS SERIES#

This series is called a geo­met­ric series.

How­ev­er, this para­dox was solved much lat­er by one of the branch­es of math­e­mat­ics that stud­ies infi­nite sums of num­bers: the series.Īdding 1/2+ 1/4+ 1/8+ 1/16+… is like adding 1/2+ 1/2 2+ 1/2 3+ 1/2 4+… Since it takes an infi­nite num­ber of steps to cross the dis­tance between the bow and the tar­get, Zeno con­clud­ed that it was impos­si­ble for the arrow to reach its des­ti­na­tion in a finite time. Indeed, one can always divide its remain­ing path by two and there will always remain a por­tion of path to cov­er (1/2, 1/4, 1/8, 1/16, 1/32), ad infini­tum. The result is his famous “arrow para­dox” that, accord­ing to him, should nev­er be able to hit its tar­get. Zeno of Elia (~450 BC) tried to show the “phys­i­cal” impos­si­bil­i­ty of infin­i­ty, not by mea­sur­ing it but by using it to divide things into small­er and small­er elements. And it is math­e­mati­cians who took it upon them­selves to observe it a lit­tle clos­er. Of course, infin­i­ty is first and fore­most a math­e­mat­i­cal con­cept.

It is more a philo­soph­i­cal posi­tion that dif­fer­en­ti­ates one school of thought from another. Of course, none of these state­ments are sup­port­ed by an “exper­i­ment” or a “mea­sure­ment” by cur­rent sci­en­tif­ic stan­dards. It is through infin­i­ty that we per­ceive our own existence. For Her­a­cli­tus, on the oth­er hand, it is time that is infi­nite.

From it are born an infi­nite num­ber of worlds that fill the vol­ume of the Uni­verse. were already won­der­ing about the amaz­ing prop­er­ties of infinity.īut is infin­i­ty only a con­cept? A weird idea that math­e­mati­cians play with? Or does it have a con­nec­tion with the world around us? Is any­thing real­ly infi­nite?įor Anax­i­man­der, infin­i­ty is the found­ing prin­ci­ple of real­i­ty. We know that the philoso­phers of the first mil­len­ni­um B.C. Unfor­tu­nate­ly, the writ­ings that attest to this are all the rar­er because they are old. It seems that ques­tions about infin­i­ty are almost as old as human­i­ty.