I’ve heard it said that many biologists are frustrated physicists who couldn’t handle the math. I’ve never been very good at math – I reached my limit as an undergrad when I took (and passed!) a differential equations course, experiencing considerable pain and suffering in the process. I’ve had considerable success in my career as a biologist, and derived much pleasure from the things that I’ve worked on, but I’ve always been aware that there is a barrier beyond which I cannot go. That barrier is defined by mathematics.

For some reason, I’ve been fascinated with numbers for most of my life. In their most basic sense, numbers were invented by humans to keep track of their stuff. Humans can perceive directly only small numbers of objects – you can immediately see if a group contains one, two or three things, and possibly four or five, but if the group size gets much larger than that, you have to resort to counting, unless you have some other way to improve your perception. That’s why pips on playing cards are arranged in patterns, for example. So if you’re a shepherd, and you want to be sure that the same number of sheep you let out to pasture in the morning returned to the pen in the evening, you need a way to assess the herd size at those two times. You also needed a unique symbol that represents that herd size, that you can scratch on a piece of bark so you can compare the morning’s count with the evening’s. Those symbols are numbers.

Like with many things invented by humans, people began to fiddle with numbers in their free time, and not always in a pragmatic way. They discovered certain numbers had properties that allowed them to be classified – even numbers, odd numbers, prime numbers, perfect numbers, etc. Even more basically, some began to wonder just how many numbers there were. As many as the stars in the sky, or the grains of sand on the beach? How about as many as both together? These thoughts open the door to the world of very large numbers.

These days, increasingly large numbers are being tossed around to keep track of our stuff. At this writing, the U.S. national debt is around 60 trillion (6.0 x 10^{13}) dollars, and growing exponentially. As of last year, it was estimated that 2.5 quintillion (2.5 x 10^{18}) bytes (exabytes) of data were being created each day, worldwide (http://en.wikipedia.org/wiki/Big_data). The radius of the observable universe has been estimated to be about 3 x 10^{23 }miles (http://www.pbs.org/wgbh/nova/physics/blog/2012/10/how-large-is-the-observable-universe/). Contained in the volume defined by this radius are 1 x 10^{97} fundamental particles (http://www.mrob.com/pub/math/numbers-19.html#le078_521) . Numbers any bigger than that one certainly are not useful for keeping track of our stuff, because there is no more stuff than that to keep track of.

But numbers can get bigger still. How big? So big that we have to invent new numeration systems even to be able to write them down (http://www-users.cs.york.ac.uk/~susan/cyc/b/big.htm) .

Consideration of systems like these leads to the conclusion that, theoretically, numbers go on forever. This leads to the concept that we have named infinity. But does infinity really exist? Or put another way, is there any physical entity, be it stuff, space or time that exists without limit? Over 2,000 years ago, one of our greatest thinkers, Aristotle, considered this question and decided that, while the potential for infinity (or infinity in the mathematical sense) did exist, actual infinity did not (http://people.ucsc.edu/~jbowin/BOWA-2.1.pdf) . Modern science seems to validate his conclusion, at least for now, by defining finite limits (the Planck scale, the speed of light) for physical observation (http://www.phys.unsw.edu.au/einsteinlight/jw/module6_Planck.htm) . So it seems that physicists have limits just as biologists do!

Much more than I can cover here has been written on these topics. Two of the best sources for further exploration are Robert Munafo’s large number pages (http://www.mrob.com/pub/math/largenum.html#intro) and Jonathan Bowers’ discussion of various systems for representing very large numbers (http://www.polytope.net/hedrondude/bnc.htm). Bowers, in particular, has defined the limits of numbers that humans can perceive, even mathematically.