NB: This began as a reply to The difference between mathematics and physics but sort of outgrew its parent.

To consider mathematics capable of providing truth is to become somewhat over-enamoured with its far-reaching and sometimes bewildering power to do stuff. From a certain perspective, mathematics in its lower forms (basic arithmetic, simple differentials) is what enables everyday life to occur, from speedometers on a car to a grocer adding up apples in a brown bag. When contrasted with the fantastical possibilities of upper math, dealing with everything from how the fundamental laws of another universe might operate to the very nature of numbers themselves, it is easy to see math as the purveyor of universal truth, the ultimate fair hand by which discovery can be earned if we are just smart enough to figure out the next theory, the next case, the next generality. Math can never be 'unfair' to anyone, as all it demands of its user is that they comprehend it well enough to properly wield its true power, which it rewards richly by facilitating the construction of simpler mathematical tools to further the understanding of the entire human race. The intricacies of calculus are unknown to most, but its applicability and significance in the development of modern society is undeniable.

With all these wonderful qualities it is easy to be romanced by the sheer scope and strength of mathematics, which as with all human infatuation leads inevitably to amorous (albeit good-intentioned) inflation of the object's virtues. Like heartstruck poets we do elevate our love, our admiration and desire, out of the earthly realm of mortal matter and hoist it upon the pedestal of 'objectivity'; the ultimate divinity of information, whereupon it may survive unscathed should aliens annihilate our universe and replace it with a slightly faster one. Maths, we say, must exist everywhere, for it is fundamental to reality itself. However, no matter the beauty of mathematics, it's nature is best described by returning to the grocer analogy. Let us return to our infant education, and imagine that the grocer has three apples, all of which are ripe and juicy fresh. The grocer knows he has three apples because he understands mathematics, that numbers stack to become larger numbers. From this perspective it could be argued that without numbers nothing could exist, because you simply would not be able to express (in terms of information theory) the extant nature of anything. There would be no apples, no bag, no grocer, no universe. However, intrinsic to the properties of numbers is that they are quantities - they are (fundamentally) indications of amount of stuff. Without a 'thing' to count (be it apples, atoms or abstract 'entities') numbers cant exist either. Maths does not exist in potentia throughout the multiverse, because where nothing exists, math cannot exist either as you have nothing to so much as write a number down on. Thus to label physical reality subordinate in import to a method of expressing it's behaviour is folly, in the same way that King's English does not define Shakespeare. They are equal dance partners in the same metamathematical ballet in the sky.

Returning inexorably to our grocer analogy, our grocer's apples have become spoiled during our brief diatribe. The grocer is still aware he has three apples, but is now less happy about it as all his customers have long gone, and he is cold and lonely and has forgotten to lock up. In order both to pass the time and to properly honour his fallen fruity comrades he resolves to write a book, nay a saga, to be the definitive work on his apples that could ever exist. None who read this book could fail to understand an apple, even had they never before laid eye nor tooth upon one. As his quill hovers above the parchment, his mind struggling fully to express their taste, their suppleness, their delicate crunchiness twinned with unfathomably tender flesh, he suddenly realises that he is completely and utterly illiterate, and thus unable to exposit his epitaph as he should like. Woe! Our grocer decides to simply invent some method of telling people about apples, some....ooohh I dunno some symbology to represent their various properties, a symbology (like math, which he understands perfectly) will be self-supporting enough to be evident even to an apple layperson. Enervated and renewed he labours long into the night, until finally his new language is complete and his Book of the Apple is truly perfect. All his customers agree that he really is spot on, and a lot of their friends come to try them and yet are not surprised when they taste them for they have also read the Book, and the Book (for the grocer is truly a master of his craft) is the perfect description of his apples.

The point of this lengthly missive is that the grocer has invented a language to describe his apples, and for the purposes of me not having to fill this writeup with caveats (...dammit!) the language is perfect and not subject to misunderstanding. Later research by the grocer's favourite son, Emilio, reveals that simple manipulation of the language can express almost any fruit in existence, except the ones which have infinities, which just puzzle the hell out of Emilio, who cant see a way to fit them into his store room. But apples the Book - the language - describe perfectly. The point of all this enthusiastic narrative is that the language perfectly describes the apple, but it is not itself an apple.

Put more bluntly, you cannot eat a book.

Even if you were a strange creature that could, you cannot digest pure information, merely the matter or particle used to convey or store that information in a coherent pattern. However amazing and beautiful the language, it is simple a means to allow real things to happen, such as Emilio forming a global fruit exporting empire to last a thousand years. The language, while extensive and beautiful, is not 'real'. It merely describes a real thing, and without apples to talk about it could never have come into being. Should it fall through a wormhole (...geddit?) into another universe without fruit, the language would be without purpose and utterly meaningless, even if its own existence continued.

In tardy conclusion, mathematics is language, a tool, nothing more. It is the formalisation and rationalisation of the coincidences that formed the laws of our universe, but it is not the laws themselves. The universe does not run on maths, it runs on physics. Mathematics is simply the tool which physics requires to do its work, as a gardener uses a shovel to create Emilio's white marble driveway. However perfect and necessary and endlessly adaptable, the shovel remains bound to the thing it was built to create, without which is it without purpose. Similarly, without the shovel the gardener just has a dirty great pile of posh gravel, and time is money on this job. We can therefore rename our dancers Physics and Maths, for without the other each would fall to ruin, yet together they can create beauty as yet undreamt by man.