Yesterday's shouts gave us (I'm paraphrasing):

(1) "The number 4 is the same everywhere in the universe. Four supernovae are always four supernovae and they don't care whether we count them or not. This shows that math exists independently of us. Math is universal."

(2) "Mathematics seems unreasonably effective. It fits the laws of the universe like a glove. This cannot be a coincidence. Math is the language of nature."

Despite the fact that the first claim cites supernovae, there is quite a bit of darkness there. "They don't care ..." is unintelligible and simply a rhetorical filler. And what can it mean that "4 is the same everywhere"? It certainly is the same everywhere anyone goes who has mastered the use of the numeral "4". But that is clearly not what (1) tries to say. Rather, it proclaims, the "4" is with the celestial objects in question, a property of them, independently of what any user of language may or may not say about them. The concept of "4" predates human existence and is built into the universe, only later to be discovered and made accessible through language by sentient beings like us.

The second claim draws a similar picture. Mathematics and logic act as a kind of "superphysics". Mathematical rules govern the behaviour of the physical world and these must have been present in the cosmos from its inception. For if this were not true, how could we explain that what mathematical models predict is so often correct, especially if the predicted phenomena have never been observed before? And why is there no other way of doing this than by using mathematics, a language which all human beings agree on? The harmony between formalism and the world seems too perfect to be a coincidence.

It is clear by now that the two thoughts are related in the sense that they both need metaphysical assumptions to be intelligible.

The idea of an abstract realm where numbers and other intangible entities we handle mentally reside is quite old. It has been argued for and against many times. What is this realm? It is invisible to us and not explorable by experiment, i.e. scientific means. It contains mathematical entities, rules, and their logical relations. Our only access to it seems to be through our minds. But how it can be that we are capable of grasping the objects of that domain is as mysterious as the domain itself. The only merit the idea seems to have is that it provides us with a pattern of thinking about mathematical objects as just that - objects.

Do we need the metaphysical picture to use mathematics? Certainly not in simple acts of counting or calculation. To successfully count 5 stones on the beach, it is not necessary to invoke any kind of mental representation of the stones. They may, but don't have to appear before my "inner eye". The same is true of arithmetical operations. I can simply reply "7" when I am asked for the sum of 3 and 4. When asked for the result of 24*55 I may not be able to give the answer immediately (or not at all), but any mental activity which takes place before I can give the result needs not to have pictures of "24" or "55" in it. The assumption of a detached metaphysical domain of numbers which exists apart from our practices involving arithmetic is unnecessary to explain what we do when we exercise our powers of counting or calculation.

A similar criticism applies to (2). How do the laws of nature, if they exist in some metaphysical realm M, inform what goes on in the physical world? Do lumps of matter "check back" with rules (residing in M) that govern their behaviour before they change their physical status, such as when a metal ball rolls down a pinball table? Again, the picture seems to create more problems than it solves.

If, then, mathematics is not built into the world in a metaphysical sense, it can be concluded that if it can be called "universal" at all, it is at best universal in human beings, since we seem to arrive at identical conclusions - and even formalisms - if presented with identical (mathematical) problems. Why is that so? Human beings naturally have similar responses to similar situations. We share biological, social and historical traits and properties. So it is not too surprising that we should at least arrive at similar results once we are capable of reasoning using formal systems. Also, the formalism of mathematics is not uniquely determined by its application. There are many examples of equivalent ways to express mathematical relationships pertaining to a given problem. Not even the most basic terminology is uniquely determined by the phenomena or human abilities we model it on. For example in propositional logic, not ( A and B ) = ( not A ) or ( not B ). Hence we can define "and" using "or" and "not", or we can define "or" using "and" and "not". It follows that neither "and" nor "or" are in any way "fundamental" to the business of logic.

What about the "unreasonable effectiveness"? We do not arrive at mathematics independently from our experiences in the world. We tailor the glove according to what we find. Mathematical pictures of the world are designed to fit it and they are intelligible to us at the same time because we only use (formalistic) tools we understand in the first place. Good predictions for hitherto unknown situations can actually be expected. If the predictions fail, we refit the models. We do not say "that should not have happened" and continue to use the old theory. Thus the effectiveness of math only appears as too big a coincidence if we have a metaphysical picture of it to hold up against our theorizing: We imagine the metaphysical domain as having been fixed at the beginning of the world and our own practices as a discovery of that domain. Once we jettison the metaphysical picture of mathematical objects, the mystery of their "unreasonable effectiveness" disappears. Mathematics can then be seen not as a collection of objects and rules in some intangible realm, but as part of a set of human practices informed at the same time by what the world is like and what we are like.

Wigner, in his article, which can be read here: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html -- and I only now notice to my considerable amusement that the url contains the term "MathDrama" -- writes:

The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.

In the view I tried to express, this contains some confusion and some truth. I argued against the notion that we have no reason to expect accurate description and prediction of phenomena. I agreed with Wigner that the appropriateness of a theory does not entail a unique way of formalization, but not for the reason he gives. A puzzlement over the discovery that a unique coordination between keys and doors does not exist, arises, as far as I can see, only under the assumption of a pre-established metaphysical realm of mathematics.

From the same article, some more of what I read as a commitment of Wigner's to a metaphysical picture of math:

Furthermore, whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics.

The "actual world" (What is that?) "suggests" (How does it do that? How can it do that?) that we "describe entities" (Of which kind?). I think that all of the quoted terms require more reflection and examination than they are usually given, since without such reflection they inadvertently set us off onto a course towards a metaphysical concept of mathematics. (Wittgenstein remarked in a different, but in some ways comparable context: "The first step of the conjuring trick has been taken, and it was one that seemed to us quite innocent".) And as I tried to argue, that would only have merit if there was no alternative to the metaphysical picture.

"God made integers; all else is the work of man" -Leopold Kronecker

Mathematician don't discover mathematics so much as they invent it. In the course of trying to describe physical reality we've invented some interesting and useful stuff.  Pretending like it existed before we got here is an illusion.