Infinity - Math bibliographies - in Harvard style

In-text: (Hart, 2014)

Your Bibliography: Hart, V., 2014. How many kinds of infinity are there?. [video] Available at: <https://www.youtube.com/watch?v=23I5GS4JiDg> [Accessed 14 February 2015].

In-text: (Korner, 2015)

Your Bibliography: Korner, K., 2015. All about Infinity : nrich.maths.org. [online] Nrich.maths.org. Available at: <http://nrich.maths.org/2756> [Accessed 14 February 2015].

As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others.

In-text: (Matson, 2015)

Your Bibliography: Matson, J., 2015. Strange but True: Infinity Comes in Different Sizes. [online] Scientificamerican.com. Available at: <http://www.scientificamerican.com/article/strange-but-true-infinity-comes-in-different-sizes/> [Accessed 15 February 2015].

In-text: (Nykamp, 2015)

Your Bibliography: Nykamp, D., 2015. Uncountable definition - Math Insight. [online] Mathinsight.org. Available at: <http://mathinsight.org/uncountable_definition> [Accessed 15 February 2015].

-In their study of matter they realised the fundamental question: can one continue to divide matter into smaller and smaller pieces or will one reach a tiny piece which cannot be divided further. Pythagoras had argued that "all is number" and his universe was made up of finite natural numbers. -He introduced an idea which would dominate thinking for two thousand years and is still a persuasive argument to some people today. Aristotle argued against the actual infinite and, in its place, he considered the potential infinite. His idea was that we can never conceive of the natural numbers as a whole. However they are potentially infinite in the sense that given any finite collection we can always find a larger finite collection. -Indian mathematicians worked on introducing zero into their number system over a period of 500 years beginning with Brahmagupta in the 7th Century. The problem they struggled with was how to make zero respect the usual operations of arithmetic. Bhaskara II wrote in Bijaganita:- -A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth. -It was an attempt to bring infinity, as well as zero, into the number system. Of course it does not work since if it were introduced as Bhaskara II suggests then 0 times infinity must be equal to every number n, so all numbers are equal. -Thomas Aquinas, the Christian theologian and philosopher, used the fact that there was not a number to represent infinity as an argument against the existence of the actual infinite. In Summa theologia, written in the 13th Century, Thomas Aquinas wrote:- The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited. This objection is indeed a reasonable one and in the time of Aquinas had no satisfactory reply. An actual infinite set requires a measure, and no such measure seemed possible to Aquinas. We have to move forward to Cantor near the end of the 19th Century before a satisfactory measure for infinite sets was found. The article [15] examines:- -Having moved forward in time following the progress of induction, let us go back a little to see arguments which were being made about an infinite universe. Aristotle's finite universe model with nine celestial spheres centred on the Earth had been the accepted view over a long period. It was not unopposed, however, and we have already seen Lucretius's argument in favour of an infinite universe. Nicholas of Cusa in the middle of the 15th Century was a brilliant scientist who argued that the universe was infinite and that the stars were distant suns. By the 16th 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 (1584). 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 1600.

In-text: (O'connor and Robertson, 2015)

Your Bibliography: O'connor, J. and Robertson, E., 2015. Infinity. [online] Www-history.mcs.st-and.ac.uk. Available at: <http://www-history.mcs.st-and.ac.uk/HistTopics/Infinity.html> [Accessed 15 February 2015].

In-text: (Pfander and Wunsche, 2015)

Your Bibliography: Pfander, G. and Wunsche, I., 2015. exploring infinity : number sequences in modern art. [online] math.jacobs-university. Available at: <http://math.jacobs-university.de/pfander/pubs/numbersinarts.pdf> [Accessed 19 February 2015].

The smallest infinity is the one you'd get to if you could count forever. The numbers 1, 2, 3, 4… are called the natural numbers, and they are the most obvious example of this size of infinity. In honor of them, anything that has this size of infinity is called "countable." -Cantor's discovery raised a question that hasn't been fully answered even today: Is there a "medium" size of infinity—bigger than the natural numbers but smaller than the real numbers? The supposition that nothing is in between the two in size is called the "continuum hypothesis," after the continuum of numbers. The question is so puzzling that it led to a genuine crisis in mathematics, and mathematicians still aren't sure of the answer.

In-text: (Rehmeyer, 2008)

Your Bibliography: Rehmeyer, J., 2008. Small Infinity, Big Infinity. [online] Science News. Available at: <https://www.sciencenews.org/article/small-infinity-big-infinity> [Accessed 15 February 2015].

In-text: (Rucker, 2013)

Your Bibliography: Rucker, R., 2013. infinity | mathematics. [online] Encyclopedia Britannica. Available at: <http://www.britannica.com/EBchecked/topic/287662/infinity> [Accessed 14 February 2015].

In-text: (Rucker, 2015)

Your Bibliography: Rucker, R., 2015. Infinity. [online] Math.dartmouth.edu. Available at: <https://math.dartmouth.edu/~matc/Readers/HowManyAngels/InfinityMind/IM.html> [Accessed 15 February 2015].

Cantor provided a stunning and instantly controversial proof that not only defined the nature of infinity, but it also revealed that multiple infinities existed, and some were larger than others. What made his achievement all the more remarkable was that he had built the entire thing out of an ancient and seemingly useless branch of mathematics known as set theory.

In-text: (Wilkins, 2015)

Your Bibliography: Wilkins, A., 2015. A brief introduction to infinity. [online] Gizmodo. Available at: <http://gizmodo.com/5809689/a-brief-introduction-to-infinity> [Accessed 15 February 2015].