## Large numbers and abstraction in mathematics – a musing

It’s been some time since I’ve written on this blog.  So it’s appropriate I talk about large numbers.  I currently teach sections of a mathematics class for liberal arts students.  This means that they are not typically Science, Technology, Education or Mathematics (the so called STEM majors).  It has its ups and downs, one of the ups being a flexible curriculum.  This semester I wanted to work on two themes, one was geometrical.  Starting with the geometry of experience 2d/3d I moved onto the geometry of 4d (in a very  limited way) by analogy.  There’s a great deal you can do if you work on thinking like a Flatlander, though it’s not easy to do so since we interpret visual data spatially.

The second theme of the semester and the one that is relevant to this post is finite to infinite.  We have experience with finite numbers, they describe quantities, lengths, percents etc.  Our knowledge of the finite equips us to generalize or abstract to the infinite.  Large numbers appear in our daily lives, often from the news.  Millions, billions and trillions are common place.  If we’re interested in science we often hear of yet larger numbers, numbers that describe facts or processes about our solar system, galaxy or the entire universe.  At some point we reach numbers which seem to contain, by their immensity, the entire universe.  It’s estimated that there are around $10^{80}$ protons in our universe and upwards of $10^{125}$ bits of information.  Are these numbers the largest that have any physical relevance?  We can certainly continue to add zeroes as long as we’d like. We can create notations that allow us to write down incredible numbers but  what do they represent?

After my students spent some time playing with numbers that described the astronomical I spent a week teaching them the basics of RSA encryption (blog post forthcoming).  This relies on the difficulty in factoring very large numbers (RSA modulus) into their prime factors.  It’s relatively easy for computers to multiply a couple of large primes and obtain a large number however.  Some of the more secure RSA encryption uses primes that are over 300 digits in length.  Division isn’t as hard as factoring and so a naive concern might be that some folks with ill intent would simply keep a list of large primes and then just check them against the RSA modulus.  No worries here, the Prime Number Theorem allows us to place an estimate on the number of 300 digit primes, which is around $10^{297}$.  There are too many of these for this universe to contain!  Using millions of these primes every second wouldn’t even dent this list, this enormous list of enormous numbers.

I was a bit surprised by this number. Two avenues of thought resulted.  The first concerned what we mean when we do arithmetic on numbers of this size. The second concerns the existence of numbers.  Dealing with the first, when we learn to do arithmetic we do so in an experimental setting.  We have piles of counters to check that the addition or subtraction of numbers is consistent with the manipulation of quantities.  We can be confident that 2+5=7 because we can check it.  This suggests some shortcuts for things like 124+453 because we can manipulate the digits using these facts: 4+3=7, 20+50=70 and 100 +400=500 so we have 577. This sum could still be checked in principle, we don’t.  We’re well convinced by this time and carry on.

What does it mean to perform the following sum?

$3x10^{184} + 5x10^{184}$

This can  certainly still be thought of as a quantity of something, some number of permutations or positions in some game etc.  It no longer refers to a physical quantity however.  Not a number of atoms, not a number of stars or organisms etc.  Let me say that I don’t doubt that the operations used here are consistent with the operations used on integers that can represent heaps of things.  Rather I’m reminded how students are often taken back when they learn about $i=sqrt{-1}$ which bears the unfortunate name of imaginary. In my teaching experience this seems to be too much for most and they become rather suspicious of what we mathematicians are doing.  This might be the first time they’ve come up against the concept of a number that’s not a quantity.  Of course there have been others, but this is the first they’ve noticed.  I make a point of pointing out that the irrational numbers are just as imaginary given the limits of measurement.  Irrational numbers are, in some sense, conveniences to smooth out the dense but pock marked number line of rational values.  Calculus would be something awful without the continuum.  Much in the same way that plenty of branches of mathematics would bemoan the loss of the complex plane.

Integers of any size seem benign.  The integers above are very large and it’s easier to come up with larger ones (the number of books in Jorge Borge’s The Library of Babel) that dwarf them.  I wonder though whether this familiarity with the digits of these large numbers make us slow to see their necessary abstraction.  We can talk about, prove results about, access and find numbers of 300 digits that are prime.  But we can never have them all.  There are too many, it would take too long and there’s too much information for the universe we live in.  So what’s a number?  I think I’ll find myself heading back for another attempt at Frege’s text.

I’m also reminded, though I can’t find the quote at the moment, of something Brouwer is supposed to have remarked.  I don’t understand a great deal about intuitionist mathematics but I remember reading something to the effect that he wrote about the application of mathematics to science.  The mathematics he was doing, which had slightly different foundations, would result in consequences incompatible with mainstream mathematics and science.  Brouwer said something along the lines of “it doesn’t matter if mathematics can or can’t be used in science, it’s not the job of mathematics to support science” or something to this effect.  I’ll be editing this paragraph for certain.

The second avenue of thought is about the existence of these numbers and I’ve already said a little.  In what sense do these large numbers exist?  I must caution the reader that I’m no philosopher and can only blunder about with these ideas and questions.  I repeat, in what sense do they exist.  How do you introduce them?  What are the examples of them?  I can show you examples of 7 and presumably, after a while, you see that the common thing among the piles or heaps I show you is their quantity which has the cardinality of 7.  How do I show you

44422180169906282030865231707744178205474059724209874047561419422925718096057797267971937899588020604289210726207367610233576023602937600936234

Where do I point?  What about numbers such as $10^{10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}$ which cannot be written out in full in this universe.  What about Graham’s number?  We can work with these numbers, use them, prove things to be true and false about them but they cannot reside in our universe, they don’t describe physical things or they are just too big to fit in our tiny universe.

No conclusions, only thoughts.  Yours are appreciated.