Summary : There's a biological reason why intelligent species are likely to live in universes in which mathematics is effective.
Eugene Wigner asked "Why is mathematics so unreasonably effective?"
Einstein was also puzzled. He wrote "What is inconceivable about the universe is that it is at all conceivable."
The problem is easily stated:
When we look at the world about us, such as a thrown spear or a running deer, we can describe some aspects of those things using numbers: size, mass, position, velocity, etc.
And, it turns out that in our universe, there are often relationships between those numbers that the human mind is capable of grasping and using to make workable predictions that, in practice, allow us to throw a spear towards where we think the deer will be in the future, and have the spear and deer both arrive at the same location at the same time. So not only is there a pattern, but that pattern tends to stay consistent, rather than (for example) changing over to an unpredictably different pattern every hour.
Why are there patterns, why do they stay consistent and why are they simple enough that our brains are capable of grasping a useful number of them? Why, of all the possible ways a universe might be set up, has our particular universe been set up in this way?
We can speculate that there is an underlying reason, to do with the process by which the way a universe works is determined. Perhaps a universe is set up by specifying some universal patterns, and therefore to talk about a patternless universe would be a contradiction in terms. But we don't know, and maybe even if we did all that would do is take the problem one step meta, and turn it into asking why the process is such that patterns are required.
But there's another approach, based upon the weak anthropic principle, "conditions that are observed in the universe must allow the observer to exist".
If, in order for a self-aware species capable of asking questions as complex as that of Wigner to evolve in a universe, it were a requirement (or a strong contributory factor) that parts of that universe be sufficiently patterned, consistent and amenable to logical analysis, then it would follow that the universes that get observed would tend to also be universes in which mathematics is effective.
But are such things really a requirement of evolution?
After all, if you ask most adults to use numbers to describe the movements of spears and deer, and to then plug those numbers into formal mathematical equations, not only would they be sitting staring at a page full of numbers long after the deer had run out of range, but many of them would end up with the wrong answer even if given plenty of time, and the deer could be persuaded to duplicate its previous run for them. Explicitly worked formal mathematics is not how human brains tackle such problems in practice, let alone how flies or single celled predators do it.
But I want to argue here that, none the less, these three criteria (patterns, consistency and comprehensibility) are vital for evolution to take place.
If there were no pattern to how bits of the universe relate to each other, information could not be stored in a way that it could be retrieved. What would life even mean, under such circumstances?
For complex life to evolve, it must be able to adapt and pass on those adaptions to a new generation faster than the environment itself alters which variations are beneficial. So, while evolution doesn't require that none of the fundamental laws (patterns) of a universe ever change, carbon-based life forms do place a limit on how fast you can change the pieces affecting the structure and interactions of chemical compounds. In other words, during the lifespan of an individual (or even an individual species) you'd expect to see insignificant variation in most bits of the universe's patterns.
Sea anemones have a simple network of nerves in their epidermis, that receive signals about where the anemone has been touched, and send out signals that trigger sheets of muscle to bend and twist their tentacles in particular ways. But, although some of them have evolved the ability to trap and eat small fish and other creatures, they don't really need to comprehend what they are doing. Their neurons don't have to model and predict the fish's reactions.
By the time you get to apes, though, they are capable of tactical deception that requires not only comprehending their environment, but also having a theory of how the minds of others vary from individual to individual:
"...food was hidden and only one individual, named Belle, in a group of chimpanzees was informed of the location. Belle was eager to lead the group to the food but when one chimpanzee, named Rock, began to refuse to share the food, Belle changed her behaviour. She began to sit on the food until Rock was far away, then she would uncover it quickly and eat it. Rock figured this out though and began to push her out of the way and take the food from under her. Belle then sat farther and farther away waiting for Rock to look away before she moved towards the food. In an attempt to speed the process up, Rock looked away until Belle began to run for the food. On several occasions he would even walk away, acting disinterested, and then suddenly spin around and run towards Belle just as she uncovered the food." (source)
Large brains take energy to run. Variations in a species which increase intelligence are negative and get selected against, unless that intelligence provides sufficient advantage to outweigh its cost. In an environment so incomprehensible that intelligence provided no improved ability to predict and interact with it, what would its advantage be?
Species wouldn't get much past filter feeding, and the capacity to ask "Why is mathematics so unreasonably effective?" would not arise in the first place.
But what if comprehending our environment well enough to hold in our minds a mental model of it that allows us to make successful predictions is not the same thing as the numeric relationship between the quantified patterns in that environment being amenable to simple mathematical operations such as multiplication and addition that most humans are capable of grasping?
Can we show that any environment that can be productively modeled by a limited intelligence must also be possible to productively model using simple mathematics applied to numerically quantified properties of that environment?
Showing that the inputs to the limited intelligence can always adequately replaced by a digitised version of that information (such as replacing analog direct visual input with the view from a high resolution computer monitor) would not be sufficient. What if the limited intelligence were making use of some internal physical process when analysing the inputs that can't be adequately replicated by a simple level of mathematics?
We need to show that the type of intelligence that evolves from simple physical processes (rather than one that's designed) will always (or, at least, is more likely to) use forms of signal processing and computation that work best on inputs from environments whose underlying patterns are sufficiently describable by a few numbers for useful advantage to be gained by modeling it in terms of those.
In other words, if a fish brushes against one of a sea anemone's tentacles, is there an evolutionary advantage to reducing the input down from "These 50 sensors detected touch" to "The area connected to tentacle #3 was touched", before making the decision to curl up #3 in order to catch the fish?
I think there is, and the reason harks back to the way that eyes often evolve. It doesn't start with a complex eye with lenses. It starts with a patch of skin that's sensitive to light. A simple "0" or "1" input. If that input turns out to be useful, future generations might propagate that mutation and a more complex light detecting system with nerves going to different patches in a concave well that acts like a pinhole camera:
In other words, intelligences produced by evolution don't start off with complex inputs (like high definition JPEG images) that take lots of numbers to summarise. They start off with simple easy to mathematically model inputs from primitive sense organs, and only if they can make something useful out of them, using a limited number of neurons, is it then likely that more energy will end up being invested in a more complex system.
Or, to put it another way, we comprehend only those things that our brains evolved the capability to comprehend, and the path of that evolution is more likely than not to have progressed along mathematically simplifyable lines, because that's the sort of inputs and processing power they started out with, that would have made efficient use of resources in primitive organisms.
If the signal processing required to map from touch inputs to tentacle movements couldn't have been simplified to logical operations performed upon numbers, could a processor capable of managing it have been arrived at in a step-wise fashion, starting from something simple, and each step being an advantage over the one before?