MATHS BITE: Infinity

Infinity is a difficult concept to grasp for more practical subjects such as physics, however it has become vital in mathematics where it is treated as if it were a number (but it is not the same as a real or natural number).

History

The earliest recorded idea of a mathematical infinity comes from the Greek philosopher Zeno of Elea, who came before Socrates. The Ancient Greeks distinguished the ‘potential infinity’ from the ‘actual infinity’, meaning that instead of saying that there is an infinite number of primes, they preferred saying that “there are more prime numbers than contained in any given collection of prime numbers”.

Indian mathematicians also touched on the concept of infinity; Surya Prajnapti classified all numbers into three sets and then subdivided these groups into three further orders:

  • Enumerable: lowest, intermediate and highest;
  • Innumerable: nearly innumerable, truly innumerable, and innumerable innumerable;
  • Infinity: nearly infinite, truly infinite, infinitely infinite.

The well known symbol of infinity: , was coined by John Wallis when he used it for area calculations by splitting a region into an infinite number of strips with a width equal to {\frac {1}{\infty }}.

Although the concept of infinity was known to the Ancient Greeks, defining infinity mathematically was not an easy task. In fact, it was not done until the 1800s. As a result, many 19th century mathematicians spoke violently against it; “infinity was something for philosophers to discuss”.  This skepticism was called ‘finitism’, a movement heralded by mathematicians such as Leopold Kronecker. When Georg Cantor published the first formal proof of the existence of infinity in 1874, Kronecker spoke out against it:

“I don’t know what predominates in Cantor’s theory – philosophy or theology, but I am sure that there is no mathematics there.”

It is a tricky concept to explain, or even believe in its existence. Despite these obstacles, infinity has made its way into mathematics and is used in a large variety of fields, including in the calculus developed by Leibniz and Newton (both real and complex analysis), set theory,  geometry and fractals. Without it, there would have been a halt in the development of certain areas in maths. What is the limit of 1/x as x approaches 0? Mathematicians answer this simply using infinity.

06.gifIt helps mathematicians answer the otherwise unexplainable – if indeed infinity is an adequate explanation.

What are your thoughts? M x

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3 comments

  1. I was working on the relationship between Boltzmann, Cantor and Shannon and noticed that the entropy equation resolves to Cantor’s continuum equation if you let the possibilities increase to the real number set. Have you seen any work on this?

    Take the Shannon measure of information and let the number of possible character values increase to the value of the continuum of real numbers.  The summation becomes trivial and S is equal to the log base two of the continuum, so that 2 to the power of S equals R.  This means that S, the measure of information in the continuum, equals the infinite but countable set of natural numbers, Aleph null, and more generally that the information that can be derived from the continuum is equal to Aleph null.

    Like

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