There is a classical problem in Antiquity in geometry, the problem of the squaring of the circle. The basic problem is whether a square can be constructed that has the same (not approximate) area of a circle.
In 1882, the Leidermann-Weierstrass Theorem established that Pi is a transcendental irrational number--not only is Pi irrational, e.g. inexpressible as a series of repeating rational integers, but that the square root of Pi cannot be a solution to a quadratic equation with rational coefficients. [For example, the square root of 2 is the solution to x^2 = 2, so the square root of two is not transcendental.] Note that the square of an irrational number can give rise to either an irrational number or a rational number, for example, the square root of two, while irrational, when squared equals two, whereas the square root of pi gives rise to an irrational number. [In 1761, Johann Lambert Heinrich proved that pi was irrational, and speculated that pi was transcendental, but this was not proved until over a century afterwards.] [Note: to be accurate, if Pi is transcendental, then no root of Pi (square, cubed, etc.), to whatever integer power, will yield a rational number.]
The geometric point in all of this is that one can geometrically construct a square with an area of 2. One takes a line segment two unit long, and makes a right triangle with its apex in the center of the line segment. Using the Pythagorean (a^2 + b^2 = c^2), where the two sides are the same length (x^2 + x ^2 = 4 transforms to 2 * x^2 = 4 transforms to x^2 = 2, and x = sqrt (2).) As above, one constructs a similar triangle below, resulting in a square with an area of two. Interestingly, the points on the square, although they can be represented, cannot be assigned any exact Cartesian coordinates as the length of the line segment is irrational. The approximate value of the square root of two is 1.41, but you can see that the next digit (4) lies between 1.41 and 1.42. Because the series is irrational, and can never be expressed as a repeating sequence of rational numbers, there is always another digit, and the position of a line of length sqrt 2 can never described precisely, even though we can draw it. However, in the case of a square with the area of Pi, no such square can be described, because Pi is irrational, and so the figure cannot be constructed.
The consequence of this result is that the area of a circle cannot be expressed as either a square or even a sum of squares (although we could approximate it through a fractal structure of recurring squares decreasing in size, which would have to "proceed to infinity" to capture the area of a circle). Likewise, a curved line can never be represented by a series of straight lines (although it can be approximated by a series of ever diminishing line segments).
Now, there are at least two forms of symmetry in nature, curvilinear and linear symmetry. We can represent a line in space with Cartesian coordinates (x,y), and construct linear equations for such structures, which represent such structures (ax + b = 0). Likewise, we can use polar coordinates (an angle and a length of radius) to describe curved structures in space (r*(phi) = a). Because the circle cannot be squared, a curvilinear representation can never be analytically translated into a Cartesian representation (draw a circle on your etch-a-sketch). While a point in space can be described interchangeably between Cartesian and Polar coordinates, an equation in Cartesian coordinates cannot be translated analytically into an equation in polar coordinates, or vice versa [again, recall that approximation is possible]. It is the property of the whole, its symmetry (straight or curved), not the property of a part, that makes the whole capable of representation. The symmetry of the whole (line segment or curvilinear segment) can only be captured in one of the two coordinate systems.
We tend to think of "space" as being independent of a system of representation (such as Cartesian or polar coordinate systems) as well as "things" in "space" being independent of a system of representation. Yet a circle can only be analytically represented in polar coordinates, and a line segment can only be represented in linear coordinates. It might be pointed out that there is no such thing as a perfect circle in nature, so if we use linear coordinates, there is not a problem. But it can also be pointed out that there is no such thing as a perfectly straight line segment in nature either. It is not a case of one system of representation being "natural" while another adapted for a special case. The choice of the system of representation depends upon utility: some problems may only be solved in a particular system of representation due to the unique symmetry of the object of study. Our systems of representation are fundamentally incommensurable with each other, and sensible "reality" incommensurable with either system. The point is that when we think about mathematical representations of things in space, we are thinking about our system of representation, not something outside our system of representation. When we talk about representations of things in space, we are talking about the grammar of our system of representation, not the "thing-in-itself". Our notion of "space" cannot be divorced from the system of representation used to describe "things" in "space." Further, our system of representation can always and only be an approximation of the so-called "real thing".
For this reason, we cannot actually talk about the "real thing". It is itself an idealization, not something we can observe, describe or represent. It is an abstraction, the "real length" of the board less the imprecision of our system of measurement, the "real shape" of the board, absent the limits of our representational system. It is for this reason that Plato indicated that the knowledge of the sensible world constitutes, doxa, opinion, and not true knowledge, and it was for this reason that he held that the sensible world was an intermediate world between being and non-being. The sensible world is only knowable through a system of representation (e.g. a system of meaning) while it can only be described in approximation. Because there can never be an "ultimate" description of the sensible world, only an incessant series of better approximations, there can be no ultimate knowledge of the sensible universe. Further, whatever approximate knowledge that could be had could only be found through ideas, which correspond to something like our system of representation. These ideas are not logically derived from the sensible world (they may be causally derived from the sensible world), they are the product of the human mind and spirit. In other words, mathematics is not empirically falsifiable, even though the nature of the empirical world only conditions the utility of a particular mathematical approach (e.g. the use of polar coordinate or Cartesian coordinates based on natural symmetry.)
The bottom line for naturalism is that the idea (or "apparent idea") of a universe of inert things in space obeying mechanical transformation rules/laws cannot possibly be the case. The thing can never be identified with the representation of the thing. The system of representation is not given by things-in-themselves, but in the forms of human life. For example, Godel's Incompleteness theorem established that human arithmetic for natural numbers cannot be reduced to a set of axioms plus associated transformation rules. In other words, arithmetic does not and cannot be reduced to a set of axioms and a set of algorithms, whereby a computer could be programmed to derive all the truths of arithmetic. In arthimetic, there are some theorems which cannot be established as true (e.g. the system is incomplete), and therefore, it cannot be demonstrated that mathematics is logically consistent with itself (as the truth of all theorems cannot be proved or disproved). [If a formal system contains a logical contradiction, anything can be proved from it.] Notwithstanding, we assume that it is the case that arithmetic is consistent. As a consequence, machines using recursive algorithmic processes cannot "do" arithmetic, they can only serve as instrumental tools in the activity of arithmetic. [A computer may "prove" something in mathematics, but only mathematicians can deem the solution "valid".]
For example, we can consider the following article from Der Spiegel:
Clearly, Godel's ontological proof for the existence of God can be formalized, and a computer algorithm can establish its logical validity [to the satisfaction of logicians], but this does not establish the existence of God. It merely establishes that if one accepts the premises of the Godel's proof, then the consequence, God's existence, logically follows. It is the initial leap of faith, such as faith in the consistency of mathematics, that makes mathematics a viable human activity, and it is the consequences of that faith, not merely in terms of utility, but also beauty and elegance, that confirm the soundness of the initial faith commitment (without, of course, justifying it). Likewise, it is the faith in the intelligibility of the world that makes faith in social institutions such as organized scientific inquiry possible, while it is the consequences of that initial faith which have brought about subsequent research funding.
Chronologically, the sensible world precedes the human construction of systems of representation. We have not been here from the beginning, no matter what your cosmology. Moreover, a system of meaning, like mathematics, cannot be reduced to a set of inputs and mechanical transformation rules, so it is impossible for there to be a de-personalized and de-socialized activity of mathematics. It would be one thing if human beings emerged and then created a virtual reality with computers and scientific equations. This is not how it went down. Instead, there was a world, and human beings emerged (co-emerging with language, which is trans-personal and the basis of human ratio) subsequently. There is an interesting isomorphism between human systems of representation and the world of experience, e.g. the world of experience can be represented in approximation by mathematics. But we cannot ultimately "explain" why our mathematical tools "work". We have faith that they work, and there are useful results that stem from the fact that they work. Presumably, we have these tools and not other possible tools because other possible tools would not be useful, but this is an a posteriori explanation. It does not reach the question of why any abstract tools were possible in the first instance. We can't explain how our means of description describes, anymore than we can prove the completeness of arithmetic. The means of description is simply the given, just as the universe itself is only a given.
One can only approach this task in terms of a fractal analogy. Just as we have a community of human agents who use language and mathematics to describe a virtual world, we can imagine a community of persons who use language to create a real world. Because we are created, as fractal fragments in a sense, from these persons, we can approximate this process. This can only be understood as an analogy, because we lie embedded within the universe, while these persons lie outside the universe, in a different order of being. Our words cannot describe them, but rather reflect them, reflect their word. For this reason, the universe is intelligible, as there is an analogy between our systems of meaning and the system of meaning that gave rise to the universe.
Otherwise, we are trapped in our anthropocentric circle of representations which can only function within our anthropocentric activities. We cannot talk about a sensible world because it lies precisely, and unreachably, outside of our system of representations limited by its underlying assumptions and approximations. We cannot talk about an intelligible world because such a world can only exist in contrast to some other world (e.g. sensible). Moreover, it is not even clear that we can talk about a world in this case, we can only speak of a narrow human community composed of certain conventions of language and activity, which comes into periodic conflict with other human communities. In this case, the only thing that remains to hold the community together is authority, the authority of language and reason. But, within this community, it is this authority that lacks any legitimacy, which by nature can only come by virtue of a relationship with something outside itself.