To infinity and beyond

Mathematician Eugenia Cheng has written a popular, amusing, and fascinating book on the concept of infinity.

Beyond Infinity:  An Expedition to the Outer Limits of Mathematics takes up its paradoxes, mathematical conundrums, and important uses.

For example, one mathematical axiom is that:

infinity X infinity = infinity.

But if you work out this equation by dividing both sides by infinity, you get:

infinity = 1

Since that can’t be, infinity must not be a number, exactly.  But what is it?

Read an excerpt from her book, taken from Science Friday. [Read more…]

Mathematical proof of God’s existence?

Kurt_gödelNPG D23949; St Anselm after Unknown artistThe great mathematician and logician Kurt Gödel, who died in 1978, left behind a series of equations that purport to prove the existence of God.

As I understand it (and I don’t understand the math!), the equations test the validity of St. Anselm’s ontological argument for God’s existence, which defines God as the greatest being that can be conceived.  Such a being would have to have the property of existence; otherwise, we could conceive of a greater being, namely, one that exists.  And that one would be God.

This sounds like a language game, but philosophers have wrestled with the argument for centuries, finding it more formidable than it might appear on the surface.

Now two European computer scientists have run Gödel’s mathematical proof on a computer and found it valid.

You do the math:

“Ax. 1. {P(φ)∧◻∀x[φ(x)→ψ(x)]} →P(ψ)Ax. 2.P(¬φ)↔¬P(φ)Th. 1.P(φ)→◊∃x[φ(x)]Df. 1.G(x)⟺∀φ[P(φ)→φ(x)]Ax. 3.P(G)Th. 2.◊∃xG(x)Df. 2.φ ess x⟺φ(x)∧∀ψ{ψ(x)→◻∀y[φ(y)→ψ(y)]}Ax. 4.P(φ)→◻P(φ)Th. 3.G(x)→G ess xDf. 3.E(x)⟺∀φ[φ ess x→◻∃yφ(y)]Ax. 5.P(E)Th. 4.◻∃xG(x)”.

After the jump, a news story on the computer scientists’ work.  I also include Gödel’s proof and a link explaining the above mathematical notation. [Read more…]

How could medieval maps be so accurate?

In the 13th century, so-called “portolan maps” appeared that are so accurate, they could be used in navigation today.  But it has been a mystery how they were made and how, given the limits in technology of the time, they could be so accurate.  (This is another example of how the notion that people from other times were unintelligent is just untrue, as in the myth that people in the Middle Ages thought the earth was flat.)  A mathematician has figured out at least part of the answer of why these hand-drawn maps are so good, with even their limitations pointing to a startling sophistication. [Read more…]

Happy Super Pi Day: 3.14.15

Today is “Pi Day,” the 14th day of the 3rd month (3.14).  Not only that, it is “Super Pi Day,” with the rest of the date giving the next two numbers: 3.14.15.  Pi is the ratio of the circumference of a circle to its diameter.  Though circles are everywhere, their numeric ratios can never be exact.  The mysterious number represented by the Greek letter π has been proven to be an “irrational number,” one which has an infinite number of non-repeating decimals.  And, yet, the ratio has to be used in all kinds of common calculations, from figuring the area of a circle to analyzing subatomic and astronomical phenomena.

After the jump, an excerpt and a link to an essay on π and pi day by Cornell mathematicisn Tara S. Holm.  Do go to the link for an account of the history of our knowledge of the concept, including a government attempt to regularize it at 3.2 by passing a law.  My favorite part is how Prof. Holm is celebrating the day:  Getting her family together at 9:26 and 53 seconds (the next five numbers) and eating a piece of pie. [Read more…]

What else Turing did

The movie The Imitation Game focused on how mathematician Alan Turing broke the German “Enigma” code, a major contribution to the Allied victory in World War II.   Those interested in artificial intelligence talk about the “Turing test,” the goal of making it impossible to tell whether a machine or a human being is responding to questions.  But  Turing’s most enduring contribution is not known so much.  He wrote a paper about 0’s and 1’s and computable numbers that basically invented the concept of software. [Read more…]

The God of multiple infinities

There are an infinite number of numbers.  But there are also an infinite number of numbers between any two numbers!  In fact, there are more numbers between numbers than there are countable numbers, even though both are infinite!  (Mind blown yet?)  George Cantor, the father of Set Theory and a devout Christian, proved that.  Joel Bezaire shows what the concept of “multiple infinities” can do to our sense of the infinity of God. [Read more…]