Introduction

The world of paradoxes

Hi, welcome to our new blog about paradoxes and their surprising prevalence. We are Henry and Nic, two mathematics PhD students who naively think they have enough time to write a blog. We hope to illuminate complex parts of the world and describe them simply - and also to have fun. We hope to write one post every week, but past performance is no guarantee of future results (and we have no track record).

Even when not directly applicable to the real world, paradoxes help us train our reasoning muscles, which we think is reason enough to think about them. We will try to find the most ridiculous cases for maximal entertainment.

To kick things off we will answer the question: what is a paradox? We will not be quoting the dictionary definition because that would just be lazy writing. Most simply a paradox is just a concept, usually an abstract or concrete statement, that is or seems self-contradictory. Paradoxes have amazed philosophers and mortals alike for millennia. They even categorized them into three main groups.

Falsidical

The recognisable characteristic of a falsidical paradox is its potential for ridiculous and comical conclusions that are false. Thankfully, the arguments are usually founded on an incorrect statement and so, by examining the reasoning in the paradox, you can usually spot the fallacy. It is important to note that you cannot refer to a falsidical paradox as a fallacy, as fallacies can lead to correct conclusions as well as false ones. One well known family of falsidical paradoxes is the statement that 2 = 1. Here is one proof by Augustus De Morgan:

Let x = 1 and therefore x^2 = x.

This leads to
x^2 - 1 = x - 1,           (subtract both sides by 1)
(x^2 - 1) = (x - 1),
(x - 1)(x + 1) = (x - 1)   (expand the bracket on the left side)

Dividing both sides by (x - 1), we conclude that 
x + 1 = 1; that is, since x = 1, 2 = 1.

The problem arises from the division by x - 1, which is zero. In this example, the dividing by zero is the error as it goes against assumptions of standard arithmetic. This mistake has caused a few issues in the past. In September 21, 1997, a crew member accidently entered a single zero into the database of the US Navy cruiser Yorktown. This was then used in a division which caused the entire system to crash including the propulsion system.

Veridical

A veridical paradox is actually true! It just seems like it should not be so. Our brain is simply not equipped to intuitively handle these types of statements. Some might discard this category as a whole and just categorize these as true statements, but we believe this to be reductive. It can be fruitful to investigate why these paradoxes trick us. Examples usually emerge from complex systems like economics or systems that are secretly high dimensional like probability/statistics. This might be able to explain why even smart / well educated people often struggle with statistics (see: all of twitter). While it is easy to lie with statistics, it is even easier to lie without them.

Antinomy

A antinomy is a paradox which is neither falsidical nor veridical. We will borrow the oxymoron from the world of literature as an example. The oxymoron juxtaposes two words of opposite meaning. See this line from Shakespeare to witness four of them in a row!

Feather of lead, bright smoke, cold fire, sick health!

The above sentence is neither false nor true - in this case the contradiction is more emotional than logical. Antinomies may also arise in logical scenarios, mostly by contradictory assumptions. These can be very subtle; it took mathematicians around two thousand years to create a set of axioms for general mathematics (see ZFC), and really we just ended up proving that you fundamentally can not have a (useful) set of axioms where you can be sure there are no contradictions.

Closing Thoughts

It is also worth noting that the boundaries on these categories are not perfectly clear, which is related to the fact that what truth even is can be unclear. Remember: the map is not the territory and we should only use these distinctions as long as they help us. Hope to see you soon for more elusive examples.

Notes

You can follow Nic on twitter @nic_kup.

Nic will write American English, Henry will write British. Sorry.