'I foresee two possibilities. One: coming face to face with herself thirty years older would put her into shock and she'd simply pass out. Or two, the encounter could create a time paradox, the result of which could cause a chain reaction that would unravel the very fabric of the space-time continuum and destroy the entire universe! Granted, that's worst-case scenario. The destruction might in fact be very localized, limited to merely our own galaxy.'
— Doc Brown, Back to the Future 2
Everyone has heard the term paradox in one context or another: movies, books, television series and so on. However, it is often used trivially, without paying too much attention to its significance. On the surface, the idea of a paradox is, indeed, easy to grasp. However, interest in paradoxes lies in their implications rather than in the concept itself. In this post I will try to give a brief analysis on paradoxes: their definition, their importance and implications, as well as mentioning some of the most renowned examples of paradoxes in logic, mathematics, physics and (interestingly enough) philosophy.
Let us begin by proving that I’m Dracula. Here is my reasoning:
(2) Dracula is afraid of only me.
Therefore, I am Dracula
Clearly, this means that I spend my nights sucking blood and not writing my thesis or posts for this blog… Bad jokes aside, this silly jest, taken from (1), illustrates the idea of a paradox: a valid impossibility. Now, one may wonder: how can the previous reasoning be valid? There is no possible way that the writer is Dracula, right? Well, from a purely (formal) logic standpoint the statement is true. Indeed, by looking purely at the form of the statement, the reasoning seems to be valid: if everyone is afraid of Dracula, then Dracula is afraid of himself and, since Dracula is only afraid of me, then I must be Dracula.
Certainly, when considering the meaning behind each one of the premises in my reasoning, it is possible to realize that the statement is false. Dracula’s existence is not a fact, and even if it was, it is likely that some people are not afraid of him (think of Jonathan Harker or Abraham Van Helsing). All in all, the idea is that a paradox occurs when a seemingly valid reasoning leads to a contradiction or impossible occurrence.
The question that arises then is about the importance of paradoxes, which, as usual with different mathematical concepts, does not have an straightforward answer. A one liner could be that paradoxes help us to check theories and develop critical thinking. Certainly, several paradoxes have been used in mathematics and logic to prove that some of the axioms and fundamental lemmas allowed unsound statements to be proved (See Russel’s paradox); others have been used to generate discussions round different topics and situations.
One of the most common topics, and the one that inspired this post, concerns time travel and temporal paradoxes. Imagine that you travel back in time and, by accident, your grandfather ends up dying. This might be a bit morbid, but for the sake of the example, consider it. Once you travel forward to the starting point of your trip after the accident, you should not exist! Specifically, if your grandfather dies before he can conceive your mother/father, then it would not possible for you to have been born; this contradiction is commonly called the grandfather paradox and is used to explain which kind of inconsistencies could arise from changing the past, should time travel be possible. Curiously, these kind of thought experiments and hypothetical dissertations have given birth to well-thought solutions for physics problems that arise in certain contexts related to general relativity and quantum physics (see Novikov self-consistency principle, many worlds interpretation.
Perhaps not surprisingly, most uses of paradoxes seem to be separated from reality, even for scientists. However, from my personal viewpoint, thinking about these plausible contradictions have helped me to not assume every published theorem, lemma and interpretation as truth. To me, science is born from inquiries and questions. Hence, allowing contradictions to sprout from your own ideas and allowing others to break your own convictions and beliefs, helps in generating real and durable knowledge, able to withstand the whims and caprices of paradoxes.
- Raymond Smullyan, What is the Name of This Book? 1978.
- Dracula, Bram Stoker, 1897.
- Russel’s paradox, link.
- Back to the Future 2 (Recommended movie pick!).