The world of maths can be seen as a language. Like any language it can be used to say things that are true, and things which are false. However there is also a view that goes back to people such as Plato, where there is some pre-existing reality that numbers come from.
In this view, its not just that maths can be used – as a tool, invented by humans – to describe certain aspects of the universe, its more properly described as a fundamental part of the universe itself. Looking at the likes of equations in physics, I find it easy to sympathise with this view. There is a symmetry around the equals sign in the middle of the equation – move things around it and you can predict how things will be billions of miles away, or right in front of us at scales so small that all our normal logic and reason no longer work. The maths carries on working just fine in places we can’t even dream of visiting, which can sometimes astound those using it, almost as if its a kind of magic. Stephen Hawkins asked what it was that “breathes the fire into the equations”.
It is also clear that some types of maths can be as far from truth as you can get, something we learn early on in maths classes as a cross next to our answer. Famously statistics is an area where maths can even be used to deliberately deceive, although as often they are simply meaningless because all factors and their effects are not known or understood. This ‘maths as a language’ view can easily hide from us the other more fundamental type of maths, and we loose something if we cannot tell them apart.
One example is “imaginary numbers”. Here you have the normal number line with zero in the middle, minus numbers to the left, plus numbers to the right. You then have a question, “what is the square root of -1 ?”. No normal numbers can be multiplied by itself to get a minus number, and so early mathemstitians invented a new type of numbers, which they called “imaginary” (or “i”). Gauss showed that this was in fact just a new number line, running up and down rather than left and right, but otherwise pluses one way, minuses the other. When you use both numberlines together as a two dimensional plane, you get “complex numbers”, and just like an equations on the normal number line, there are simple rules for moving around this plane.
I suspect that most early mathematicians who used complex numbers considered them a purely intellectual excercise, a natural consequence of extending the numbers we use for the real world beyond the limits they were designed for. However as time went on, it became more and nore apparent that these imaginary numbers were necessary in describing all kinds of real phenomena mathematically. We would not have technologies such as mobile phones or GPS without them.
From what I can tell, most in physics (and even maths) would see complex numbers as a useful mathematical trick, which happens to produce valid results. However I can’t help wondering whether they represent something more fundamental, an aspect of the universe that somehow doesn’t fit with our default epistemology.
Another area is Cantors infinities. Cantor argued that not all infinities are equal, that if you compared a set of all ‘whole’ numbers, with one of all ‘real’ numbers (i.e. including all possible decimals), then the second set had an infinite number of members for each member in the first set. In his view this made them different – although he said they were of a different order, not size. I find it difficult to accept Cantor’s arguments, but the maths that comes from it is used regularly by physicists in the real world. So does nature really have different types of infinity, and does this actually mean anything ? For my money this is veering close to statistics, but I think its a useful example.
Dimensions are an area where maths can describe completely different realities that seem very alien. We’re used to three spatial dimensions, Einstein added time as a fourth dimension. Since then, physicists and mathematicians have put all our maths together and come up with string theory, which has more dimensions. At one stage there were a few of these theories which seemed incomplete, but when reformulated as an 11 dimensional model, there were some compelling hints that this represented something real about how the universe really is (M theory). However most of string theories biggest problems stem from how far away it is from anything we can measure. You can tweak it so much and get so many different results that it becomes meaningless. Nonetheless, it seems likely that there is something here that the ‘basic’ maths (in reality very complicated maths) is pointing to, with extra dimensions of some form being a real possibility, whether spatial, temporal, or something else we can’t even imagine.
Strangely, the sheer number of possible ways that string theory can be tweaked seems to have resulted in physicists being drawn towards the idea of a multiverse, where every possible configuration that can happen has happened. This despite the fact that we have no evidence for any universe other than our own.
I may be stretching things here, but it seems to me that we have some strong clues in maths that the central areas of our default modern epistemology are actualy just a small corner of the big picture of reality. I also think we benefit more from stepping back and imagining what the big picture may look like, than constantly narrowing areas of study into smaller areas of specialisation.