It was later "purified" to reveal semantic difficulties with language itself. On the other hand, one of the most complex and mystifying paradoxes is the Banach-Tarski Theorem, an actual theorem, which allows the decomposition of a bowling ball and reassembly into a giant planet earth-sized bowling ball - and with just a finite number of pieces!
I nfinity is fraught with issues that require the most careful study. This chapter is about them; it is about primitive concepts and the evolution to modern ideas. There are many excellent references, some of which are at the end of the first reading. I n most cases the readings will be presented in the form of Acrobat PDF documents. To read and print them you will need the Adobe Acrobat Reader.
Nicholas of Cusa in the middle of the 15 th Century was a brilliant scientist who argued that the universe was infinite and that the stars were distant suns. By the 16 th Century, the Catholic Church in Europe began to try to stamp out such heresies. Giordano Bruno was not a mathematician or scientist, but he argued vigorously the case for an infinite universe in On the infinite universe and worlds Brought before the Inquisition, he was tortured for nine years in an attempt to make him agree that the universe was finite.
He refused to change his views and he was burned at the stake in Galileo was acutely aware of Bruno 's fate at the hands of the Inquisition and he became very cautious in putting forward his views. Although the circumference of A A A is twice the length of the circumference of B B B they have the same number of points.
Galileo proposed adding an infinite number of infinitely small gaps to the smaller length to make it equal to the larger yet allow them to have the same number of points. He wrote:- These difficulties are real; and they are not the only ones. But let us remember that we are dealing with infinities and indivisibles, both of which transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness. In spite of this, men cannot refrain from discussing them, even though it must be done in a roundabout way.
However, Galileo argued that the difficulties came about because He then gave another paradox similar to the circle paradox yet this time with numbers so no infinite indivisibles could be inserted to correct the situation.
He produced the standard one-to-one correspondence between the positive integers and their squares. On the one hand this showed that there were the same number of squares as there were whole numbers. However most numbers were not perfect squares. Galileo says this means only that In [ 25 ] Knobloch takes a new look at this work by Galileo.
In the same paper Leibniz 's careful definitions of the infinitesimal and the infinite in terms of limit procedures are examined. Leibniz 's development of the calculus was built on ideas of the infinitely small which has been studied for a long time. He gave quite rigorous methods of comparing areas, known as the "Principle of Cavalieri ".
If a line is moved parallel to itself across two areas and if the ratio of the lengths of the line within each area is always a : b a : b a : b then the ratio of the areas is a : b a : b a : b. Roberval went further down the road of thinking of lines as being the sum of an infinite number of small indivisible parts. He introduced methods to compare the sizes of the indivisibles so even if they did not have a magnitude themselves one could define ratios of their magnitudes.
It was a real step forward in dealing with infinite processes since for the first time he was able to ignore magnitudes which were small compared to others. However there was a difference between being able to use the method correctly and writing down rigorously precise conditions when it would work.
Consequently paradoxes arose which led some to want the method of indivisibles to be rejected. The Roman College rejected indivisibles and banned their teaching in Jesuit Colleges in The Church had failed to silence Bruno despite putting him to death, it had failed to silence Galileo despite putting him under house arrest and it would not stop progress towards the differential and integral calculus by banning the teaching of indivisibles.
Rather the Church would only force mathematicians to strive for greater rigour in the face of criticism. He chose it to represent the fact that one could traverse the curve infinitely often.
Three years later Fermat identified an important property of the positive integers, namely that it did not contain an infinite descending sequence.
He did this in introducing the method of infinite descent I call this method of proving infinite descent The method was based on showing that if a proposition was true for some positive integer value n n n , then it was also true for some positive integer value less than n n n.
Since no infinite descending chain existed in the positive integers such a proof would yield a contradiction. References show. J Benardete, Infinity Oxford, A Moretto, Hegel e la "matematica dell'infinito" Trento, R Rucker, Infinity and the mind Prinveton, N. A A Davenport, The Catholics, the Cathars, and the concept of infinity in the thirteenth century, Isis 88 2 , - J E Fenstad, Infinities in mathematics and the natural sciences, in Methods and applications of mathematical logic, Campinas, , Contemp.
J Holtsmark, The concept of infinity from Aristotle to the scholastics, Phys. J Hrubes, The genesis of the philosophical-mathematical concept of infinity Czech , Sb. Siwan 15 1 , B 13 -B E Knobloch, Galileo and Leibniz : different approaches to infinity, Arch.
Exact Sci. Werk und Wirkung 1 Berlin, , - Magdeburg 33 2 , 61 - Remarkably, Hardy and Ramanujan managed to do exactly that, during their brief but intense collaboration.
Against all odds they found a very precise formula for P n , which is too complicated to give here in full. We state only their asymptotic result, namely that for large integers n , the number P n is well approximated by. To prove the statement. We can generalise this for any number m :. The proof of Hardy and Ramanujan of their formula for P n is complicated, and few professional mathematicians have examined and appreciated all its intricacies.
Nevertheless, due to their work and that of others to follow we now have very explicit information about the value of P n for any n.
And because they gave us a proof, we will never have to doubt this result. Pursuit home All sections. Share selection to:.
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