A long time ago I read an introduction to a book in which the author suggested (with some seriousness) that reality was made of numbers, or at least math. He then offered an example to show that people considered math more fundamental than anything. The following is a paraphrase of that example:
Imagine you have one apple on a table. You then take an apple out of a bag and put it on the table next to the first apple. How many apples are now on the table? There should be two. But imagine there are now three. What do you do?
Most people would check for trickery or assume they miscounted the number of apples originally on the table. If this keeps happening, they may assume some weirdness like teleporting apples. They may even start to doubt their eyes or sanity. What they would not do is doubt that 1+1=2. It would not even occur to them to doubt that.
Math is so fundamental that it trumps physics and even our very perceptions. Let’s look at another example of how fundamental this attitude towards the primacy of math is.
The Engineer and the Machine
Let’s say a factory engineer observes the reliability of a machine after so many hours of operation. She starts by collecting data on the defect rate at different points in time. For example, she may find the following:
Armed with this data, and assuming Time determines Defects, she now wonders if there is a rule she can apply to predict unobserved defect rates. For example, she never measured the defect rate at 15,000 hours, or even at 1.7 hours and would like to predict those.
In assuming an answer to this question, she is assuming there is a rule that relates Time to Defects. Naturally, this rule is assumed to be mathematical. Not only is the machine assumed to follow a mathematical rule, but the defect prediction is assumed to be a mathematical problem that is independent of the machine. In short, the problem becomes to find the mathematical structure of the data.
To find the defect rate, she simply finds the relation that governs the data. There are tons of mathematical techniques that do this, but this example was meant to be simple so the pattern may be obvious. In this case, she finds Defect = 0.002 x Time
The Subject is Irrelevant. The Math is Not
How was this found? It had nothing to do with the machine. It was purely the relations obtaining among the data. We could have been talking about cola consumption and violent crime, and the process and formula would have been the same. The engineer could have delivered the data to a mathematician, removed all explanations, and simply asked for a formula. Heck, she could have plugged those six numbers into any curve fit program, and gotten that formula.
What’s more, anyone armed with the formula could predict the defect rate, and would need to know nothing about machines to do it. In fact, if we called the formula f, and simply asked for the value of f at 15, this person could provide an answer to the machine defect rate and at no point would mention of a machine have entered into it. The person would have provided an answer to the machine’s defect rate without knowing it. This kind of knowledge does not sit well with some people. For an anlgous situation, check out Searle’s Chinese Room thought experiment.
In short, once the data has been gathered, this turned into a purely mathematical problem.
So when we seek certain types of knowledge — possibly most types of predictive knowledge — we are seeking nothing more than the mathematical structure of data, with no reference to the subject or source of this data. Put another way, most non-trivial knowledge is mathematical knowledge, and not knowledge of a subject per se.
Is reality math?
Bye Bye Data
It gets better or worse, depending in your perspective, for the mathematical structure not only trumps the subject matter, but it may even trump the data itself! Let’s start by visualizing a formula by plotting a point for each data pair. Once the points are plotted, we can try to fit a shape to all the points and this shape then becomes a picture of our function. Here is a picture of a different process:
Courtesy of Wikipedia
The curve passes through all the points and we are good.
But what if the data were different? For example, take this plot:
Courtesy of Wikipedia
Notice how the line does not connect with the dots. Connecting the dots would produce a crooked, disjointed line, and a more complex formula. So what do we do? Well standard practice is to go with the straight line above, and the resulting function, even though it does NOT fit all the data. We assume the data is “noisy”. That is, we assume the observations do correspond to the line, but due to imprecise measurements, observation error, maybe even the operation of a hidden variable, it wasn’t measured as such. In some extreme cases, we may even discard some points (or re-assign them) as outliers.
Let that sink in. When the data does not match the a priori mathematical assumption, we basically reject the data. Again, we doubt our perceptions before we doubt the math. The assumption is that reality behaves according to smooth and relatively simple mathematical laws, and we refuse to let reality get in the way of this assumption.
Now the situation is not as cut and dried. Sometimes with enough anomalous data, we have to take notice. Interestingly enough, when we do, its usually to find another smooth, simple mathematical function that fits this data. Also in some cases, knowledge of the subject helps, as certain classes of processes are known to have specific classes of formulas, and this can guide us in fitting a formula and even in deciding if any data is to be accepted or rejected. In some cases, we may even grit our teeth and accept disjointed functions (splines). But the evidence has to be compelling. That is, we assume smooth, simple mathematical reality unless there’s compelling evidence otherwise. And if we grudgingly accept a more complex model, we assume there is a simpler one we haven’t discovered yet — that basically this model isn’t “really” what is happening.
But is there a chicken and egg problem here? Is it that math is supreme, or is it that math is merely an abstraction of the parts of reality that matter to us? If it is the latter, then arguing reality is mathematical is tantamount to arguing that reality behaves in ways previously observed. Even in cases where abstract math has shown surprising real life applications, we can argue that even the most abstract math is ultimately derived from a study of the concrete, being as it is a study of relations or patterns. Since reality is all about relations or patterns, it would stand to reason they should obey these laws.
Is there anything special about math?
How do you explain the effectiveness of math in describing reality?
Can you point to a priori math concepts you encountered?
Can you think of areas where math failed?