Vandana Singh’s wonderful science fiction short story Infinities introduced me to the German mathematician Georg Cantor. In her award-winning story, Singh describes a mathematical epiphany that involves a higher order landscape as an explanation for the seemingly random distribution of prime numbers. Her description of this epiphany and Cantor’s theory of transfinite numbers both seem to jibe with my Truth Function theory. This led me to dig deeper into Cantor’s work. What I found regarding how Cantor’s theories relate to my ideas compelled me to write this post. Thanks to Vandana Singh’s science fiction story – I believe I now have a deeper understanding of where my ideas might fit into the world of mathematics.
Let’s start at the beginning. The aforementioned German mathematician Georg Cantor proved that real numbers are “more numerous” than the natural numbers. This proof implies the existence of an “infinity of infinities”. In other words, some infinities are different from other infinities. According to Cantor, transfinite numbers are larger than finite numbers, yet not necessarily absolutely infinite. This ‘now’ accepted idea was originally regarded as shocking and we can see why – how can there be a number between finite and infinity? This doesn’t seem logical. Like many currently accepted scientific conclusions such as quantum mechanics, it doesn’t square with our intuition.
Infinity means without limit. Finite means limited. So according to Cantor, transfinite numbers are unlimited in quantity but somehow less than infinite numbers. How can a thing be unlimited yet less than infinite? Well it turns out that I may have come up with an intuitively logical solution to this illogical result. By the way – my idea also provides a logical solution to the illogical results yielded by another puzzling idea known as – quantum theory.
In the Truth’s Trek posts a solution to infinite infinities can be visualized using what I call, The Riddle of the Ray. The Riddle of the Ray is a thought exercise that answers the question, “how can a ray and a line both be the same length even though a ray is a half line?” My solution involves the introduction of a hidden dimension. In the Truth posts I show how this result provides an intuitively logical solution to quantum mechanics. I believe that this result may also provide an intuitively logical solution to Cantor’s transfinite numbers. Let me explain how…
The ‘Riddle of the Ray’ provides a solution to the paradox of transfinite and finite numbers because it shows how a ray (transfinite in length) can be compared to a line (infinite in length). A ray is sometimes called a half line because two collinear rays comprise one line. Therefore a ray should be half the length of a line. The paradox is that a ray is infinite in length – as is a line – which means they are both the same length. Cantor proved that these types of different size infinities do indeed exist. My solution to the Riddle of the Ray provides a possible solution to this paradox. It explains why we perceive rays or transfinite sets as less than infinite and greater than finite. The solution involves a hidden dimension. I propose in the Riddle of the Ray that a ray is actually a bent line. The trick is that the bend is not in one of the known three dimensions of space. The ray is bent in a hidden or unperceived direction.
Side Note: hidden dimensions may sound nonsensical on the face of it – however remember that string theory proposes 11 dimensions of space. Yes, hidden physical dimensions are not new to theoretical physics.
When we tilt the ray along this hidden dimension it is revealed that a ray is in fact a line. This then resolves the paradox. Rays or transfinite numbers are extra-dimensional. So when viewed from our perspective of 3 dimensional space – they appear to be both infinite and less than infinite – or transfinite. However when viewed from a different perspective – they are revealed to be infinite. And circling back to Vandana Singh – this notion coincides with the mathematical epiphany describe in her science fiction story Infinities. In this story the main character is shown a higher order landscape which reveals an explanation for the seemingly random distribution of prime numbers.
It is great when ideas with different origins fit together. My theory of the Truth Function, Cantor’s transfinite numbers and Singh’s science fiction story Infinities all seem to lead us to the same conclusion:
Physical and mathematical paradoxes can be explained by extra, hidden dimensions of space.
Is it coincidence or are we on to something? I don’t know what you think – but personally – I think we are on to something. I believe these ideas buttress my proposal that extra dimensions explain why we observe quantum behavior at Planck scales and how this leads us to the conclusion that everything we experience here in three dimensions of space and one dimension of time – actually exist in pure form as dimensionless information beyond the event horizon of our universe.