Taginfinity
Of Infinity and Beyond
What are the problems and solutions with infinity in mathematics?The concept of infinity has plagued a great many proofs, both formal and informal. I think that there are two foundational problems at play in most people’s thinking about infinity that causes issues. The first problem people have with infinity is that they treat it as if it were a single value. Because infinity is bigger than all possible natural numbers, people assume that it is bigger than any number, and therefore there is nothing beyond infinity. Therefore, people have the concept that if I have two infinities, then I still have the same number. They believe that 2 * infinity = infinity. However, using that logic can quickly lead to contradictions. This problem is exacerbated by much mathematical notation. People often will Read More ›
1. Why Infinity Does Not Exist in Reality
A few examples will show the absurd results that come from assuming that infinity exists in the world around us as it does in mathDoes infinity exist in reality? There are, surprisingly, scientists who think infinity is a possibility even though they are unable to point to any example of infinity in reality. The great mathematician David Hilbert claimed that “the infinite is nowhere to be found in reality.” Nevertheless, the mathematical theory of infinity developed by Georg Cantor is beautiful. Hilbert was in awe of Cantor’s beautiful theory and said “No one shall drive us from the paradise which Cantor has created for us.” An assumption of the infinite leads to weird counterintuitive results. In this and the following four articles, various ludicrous properties of the infinite are explored. We’ll see, for example, that the entire Library of Congress is encoded somewhere in almost every Read More ›
Gregory Chaitin on the Great Mathematicians, East and West
Himself a “game-changer” in mathematics, Chaitin muses on what made the great thinkers stand outIn this week’s podcast, “The Chaitin interview I: Chaitin chats with Kurt Gödel,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on the almost supernatural awareness that the great mathematicians had of the foundations of reality in the mathematics of our universe: https://episodes.castos.com/mindmatters/Mind-Matters-124-Gregory-Chaitin.mp3 This discussion begins at 8:26 min. A partial transcript, Show Notes and Additional Resources follow. Robert J. Marks: There are few people who can be credited without any controversy with the founding of a game changing field of mathematics. We are really fortunate today to talk to Gregory Chaitin (pictured) who has that distinction. Professor Chaitin is a co-founder of the Field of Algorithmic Information Theory that explores the properties of Read More ›
Can We Add New Numbers to Mathematics?
We can work with hyperreal numbers using conventional methods. It could start in high schoolSometimes mathematics is moved forward by the discovery of new formulas and solutions to problems. However, sometimes mathematics grows by adding new kinds of numbers to the number system. In the early days of mathematics, it was thought that whole numbers were the only kind that existed. Sure, there were fractions, but fractions are merely ratios of whole numbers. It was thought that every possible number could be written in terms of whole numbers. These numbers were called rational numbers because they could be written as a ratio. There is a story about a Greek philosopher, Hippasus who discovered, roughly 2500 years ago, that certain numbers (specifically the square root of two) could not be written in terms of ratios Read More ›