When we learn that, say, 2+2=4, is that something really true about the universe, or is it something some caveman made up? Does a cube exist as something more than just a figment of our imagination?
Most professional mathematicians have a sense that mathematics really exists. For example, in 2005 a colleague of mine and I discovered a new shape. You get this shape by gluing together 5,003,460 icosahedra in a special way. And although this shape was not known to anybody before 2005, there’s a sense in which it existed before that, implicit in the laws of logic that guide reality. In the same way, when people find some new trick for, say, multiplying numbers together, we always find that either,
- the new trick gives the same answers for the same multiplication sums as all the old tricks,
- dig deep enough and there’ll be some logical reason why the new trick is wrong.
The idea that math is discovered, not invented, is borne out by the fact that mathematics is full of strange coincidences. Patterns that appear in one area of mathematics sometimes also appear in completely unrelated areas of mathematics – and only years later is does some genius make the logical connection between the two areas. If math were merely invented, why would people working in completely different fields invent the same pattern? And if they did, why should we expect anyone to be able to make a logical connection?
It’s all very convincing to most mathematicians.
However, a counterargument dawned on me the other day. It starts like this :
Do you believe that unicorns exist?
Most people would say no. The rest of the argument gets a bit technical, but it starts like this : do you believe the function cos(x) exists? This function, besides being useful in trigonometry, is also a solution to a differential equation – an equation linking a function with its derivatives. In this case, it is a solution to
y” = -y
If that’s over your head, don’t worry too much about the technical details. The point is, y”=-y has other solutions too, for example, sin(x), or sin(x)-2cos(x), or 17.3cos(x) + 9801.252 sin(x). Do these all exist too? It would be hard to find a mathematician who would deny that the function 17.3 cos(x) + 9801.252 sin(x) – or any other solution of this differential equation – exists.
So, what about a more complicated differential equation? Say, y”=-sin(y*x+y’)/sqrt(exp(x+y’)), for example. Do all its solutions really exist as well?
If so, what if we write down a really, really, really complicated differential equation. Let’s say we write down the differential equation describing the movement of every bit of matter and energy within a cubic parsec of space. Physicists seem to believe this is possible, even if they don’t yet know how to do it.
Do all its solutions exist?
The problem with saying “yes” is that some of those solutions (surely) involve pink and purple polka-dotted unicorns frolicking in meadows on planets called “earth”. The problem with saying “no” is that you then have to decide where to draw the line – why say that solutions to some differential equations “really exist”, but not the solutions to others, and how do you decide which differential equations are which?
Probably the best answer is that there are different ways something can “exist”. The cube, the cosine function and the unicorn exist, implicit in the laws of logic that drive the universe. However, they do not exist in the physical manifestation of those laws that we are in the habit of calling “the real world.”
It also implies that a survey of all known mathematics would tell you not much at all about what kinds of things exist in mathematical reality. It would say a lot more about what kinds of logical reasoning Homo Sapiens finds useful or pretty.
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