The Mathematics of Aliens

Do aliens exist? The evidence is varied across time and across the globe. But mathematicians who study aliens always impress me. These are individuals who firmly stand on the pillar of logical reasoning- mathematics, that is. And yet, they are considering one of the most contested ideas of humankind… Are we alone?

Mathematicians (along with a lot of other really smart people) sometimes appear crazy. Their ideas are too advanced and so, as Arthur C. Clark once coined, the ideas indistinguishable from magic. And everyone knows that magic doesn’t exist.  You can’t make something move without touching it. Except… my garage door opens every day and I’ve never touched that thing! Honestly, I enjoy viewing the world as though every scientific thing is actually magical. A new line of code that makes my life simpler? Magic! Internal combustion engines? Magic! Hot Water?!  You get the idea.

Alien_1But while I love attributing scientific advances to magic, I don’t enjoy attributing non-scientifically proven theories (or magic) as science. Unlike the scientists who founded the Jet Propulsion laboratory (as told by this Cracked Podcast), who believed in and cast spells on a regular basis, I don’t actually believe in magic. And similarly, I don’t actually believe in aliens. Not seriously anyways. Not until it’s proven.

And while all the green alien paraphernalia in Area 51 cannot convince me, mathematics might… Woodruff Sullivan and Adam Frank recently published a paper summarized in the NY Times article, “Yes, There Have Been Aliens,” which described how mathematics show that intelligent life probably existed in our universe at some point. It employs the Drake Equation which is basically the multiplication of a bunch of different probabilities. Let’s take a quick look at the details of that equation:

N=R_* \cdot f_p \cdot n_e \cdot f_l \cdot f_c \cdot L


  • R_* \; = rate of star formation. (known)
  • f_p \; = fraction of stars that have habitable planets (current research)
  • n_e \; = number of planets/star that has habitable planets (current research)
  • f_l \; = fraction that develop life (unknown)
  • f_c = fraction of life that is intelligent (unknown)
  • L = length of time to release communications (we could make a guess)

For a while f_p and n_e were expected to be the limiting factor(s) in this equation. However, as scientists discover more and more stars and planets, it seems this f_p is very close to 1 and n_e at least 1. At this point, the biggest unknown is the probability that life is formed and the probability that this life is intelligent.


The authors plug in some values into the equation to get a sense of what the values of f_l and f_c would need to be to make intelligent life unlikely.  This is kind of like saying: “If I know how many lottery tickets are winners, how many are printed, how many people play and how many tickets each player buys, then I can tell you the likelihood that you’ll win the lottery.”  In this case, the authors are say, “I can tell you the likelihood that someone won the lottery at some point in history.”  And, certainly the chances of someone winning the lottery over the entire history of the lottery’s existence are higher than the chances of me winning the lottery. (and if you want to know more about winning the lottery and how to game this system, check out Planet Money‘s episode on the subject.)

So, Sullivan and Frank are trying to show the limiting values of f_l \cdot f_c. Historically, the probability of getting a civilization on a habitable planet was pessimistically considered at one in 10 billion per planet. Sullivan and Frank show that unless f_l \cdot f_c is less than one in 10 billion trillion, life is likely to have existed. This means that even if you take the pessimistic number of 1/10 billion, then 1 trillion civilizations existed across the universe. Magic!

Before I close this inquiry into alien life, I have to point out the counter argument. The Atlantic recently posted a rebuttal article. The core of the rebuttal is that while 1/10 billion was seen as pessimistic, we don’t have any idea what the real number is.  Like, 1/10 billion trillion is really small, but it’s possible that we are the only planet with humans on it. This argument is factual, we don’t actually know the value of f_l \cdot f_c

But, wouldn’t it be cool if we did? Or maybe we should be willing to believe (in the face of scientific argument) that f_l \cdot f_c could be large enough to make intelligent life probable. Or maybe not?  I guess it depends what kind of mathematician you are.  Are you willing to believe that magic is just science we haven’t solidified yet? Or are you a mathematician who believes is magic worthless until the moment when it is definitively proved and can be reproduced ad nauseam?

About Samantha from SocialMath

Applied Mathematician and writer of
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15 Responses to The Mathematics of Aliens

  1. This article was really interesting. I, too, pretend that most science is magical. I’ve envisioned the day that my kids come to me asking how electricity works, when I’ll tell a long tale about the tiny people living inside of our chords.

    To me, however, it’s harder to believe that there isn’t life on other planets than that there is. It seems improbable to assume that ours is the only planet that has developed life. Is Earth really that unique? There are tons of planets just in our solar system, and tons more on the universe. I’m not saying that life would be something we recognize–I don’t imagine ET coming here to use telepathy on us–but I would be befuddled if we traveled the entire Universe and found that this planet made thousands of species and no other planet anywhere could produce even one.

    Of course, the biggest debate is what we consider “life.” If we find bacteria on another planet, does that count — or does it only count if it’s as intelligent as we are? That’s a bit harder to factor in mathematically 🙂

    Keep up the good work!

    • sixfootjew says:

      Cell phones are definitely magic!

    • Samantha says:

      Science is totally magical! I get in trouble with some of my more literal-minded colleagues for this belief but I love the idea of modern day magic. In so far as aliens are concerned, I agree. It’s hard to imagine that there is some so unique to our planet that we are the only one with intelligent life- which I think is kind of the goal of Drake’s Equation. It’s helping us translate the hard to understand into something more tangible (well, if you think math counts as tangible!). It’s difficult to comprehend exactly how BIG our universe is… which also feels pretty magical!

      • Haha! I think it’s so funny how people’s minds work — for me, things like Drake’s Equation make the world feel less steady and comprehensible rather than more… But then, that’s why I’m a communication major rather than a math major, right?

  2. Well seems like you have provoked my interest in maths .the article is really interesting!

  3. katiemdean says:

    This is one of the reasons I love math! It makes up everything, and can prove anything. And I also love space, so this post was really enjoyable to read. I did a calculus project once that resulted in me finding the velocity of the planets circling the sun. Math is just everywhere, and I think that’s brilliant!

    • Samantha says:

      Yeah, math is all over the place! Celestial Mechanics is a great place to find mathematics. I’m glad you enjoyed the post!

  4. Zaira Zafar says:

    Ah math! A means of proving everything, that exists and can exist, on basis of few rules. Mathematics is indeed magic, always proving everything scientifically and logically.
    Such a great article! Informative and interesting!

  5. Ms. Chin says:

    Wow, this article really made me think! I guess my problem is with how the word “magic” is being used in this context. For me, the definition of magic is related to fiction, “spells” and “wizards” and things that break the laws of physics which are immutable. I don’t think we should be applying the word “magic” to scientific phenomena we can’t understand or explain in the moment, because it is only creating a mental barrier for us to be able to explain what is happening, and to later believe the scientific explanation. That is just my opinion 🙂

    • Samantha says:

      Yeah, I think this is a very valid opinion! I guess it depends a lot on the mindset (does calling something magic mean it’s impossible?). I tend to think that scientists are wizards. Things which would be considered magic 400 years ago, are the science of today. I can literally make things move without touching them (opening my garage door, turning on a ceiling fan with a wall switch, or opening an automatic door at the grocery store) which seems pretty magical to me. But perhaps it’s just really cool science? 🙂

  6. Jay Campbell says:

    Very good article it’s shows the beauty of math and the wonders of whats out there.

  7. Maths is teacher of wonders…really love it

  8. Great article and good example that math can be creative and fun.

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