Nobody really disagrees with Galileo's idea of the "great book" of the universe:
Science and mathematics have come a long way since that time, but this is still essentially true in a very deep way. On the other hand, the analytic philosophers have gone in a strange direction that is disconnected from the actual world we live in. Not to indict the whole tradition, but even where they provided needed grounding for formal systems generally. No, the problem is that some of them made a bit of a religion of their views culminating in Logical Positivism. For all its diversity, the American academy is dominated by this failed tradition in many fields. It turns out that when you push analysis to its limits, it doesn't say anything final as would be necessary to realize their program. The methods of science and the language of mathematics have clear limitations that suggest the impossibility of complete knowledge and that the whole ediface is a bit fragile. You can pretent that the foundations are any more solid that the historic religious texts. The nature of space and time itself it being viewed as derivative and dependant on something more fundamental. These foundations are so far from ordinary experience that we cannot really claim they are intuitive or grounded in the phenomenology of every day experience. We first extend out snses with instraments and our perceptive capacities with more and more advance signal processing. They paint a highly accurate picture of a world that is different only in detail from what we see around us. So even if it is unnatural, it is intuitive in some abstract mathematical sense and it allows us to predict events across the observable universe with remarkable precision.
Maybe this needs to be simplified to make the main point. Although we can't and probably never will be able to verify that we have the final mathematical description at the fundamental level and that this claimed fantastic accuracy can be attained for every context, that doesn't make it a matter of blind faith to claim that in principle it is. The real world is much to messy to fully characterize the initial conditions, and so we only have this fantastic accuracy for special contexts where we can rule out most other effects. But, because the actual work and observations are so densly intertwined and complementary that you can't really tinker with the edges. Any new theory has to match the old theory in all the ways we have measured it implicitly and explicitly over a lot of time in many fields.
I claim that the pragmatic tradition offers the way out as it seeks to ground knowledge in experience as well as math and logic. To include embodyment in our descriptions of what knowledge is and how it works. I want to start from echos of the greeks in the foundations of pragmatism is Peirce, and his conception of the categories.
Peirce realizes that Kantian and other categorizations are necessarily derivative of the human being as a particular kind of being that not only uses signs, but through the development of semiosis is able to use signs to "think about" signs. The third part of the sign, what it means, is also a sign that can be further developed in thought (semiosis). Peirce points back to some of the scholastics rather than explicitly anticipating developments to come, but you can see that his way of working is completely compatible with scientific developments of the 20th century. The relativities of physics is part of that, but even moreso the best work in cognitive science and cognitive biology. When I survey current thinking in many fields, I find that knowledge of Peirce's categories is rare and it would be useful in clarifying their thinking about foundations.
References to language and semanticists of the analytic tradition are notable for this lack when it is precisely these fields that have a responsibility to know the foundations and philosophical groundings of their field. These are the same fundamentalists who insist they have proof of God's absence, of the impossibility of spirit. Peirce comes close by insisting on the unreliability of intuition, but he is pointing to the pragmatic insistence of grounding in experience and not negating existential claims (c.f. Kierkigaard) to have experience of spirit. I see nothing wrong with postulating God as the Unity of spirit or consciousness in the world. Keeping in mind that this is similar to postulating the existence of extraterestrial intelligence. Some people think we have incontrovertable evidence of all of these things as well as the correct theory of the Kenedy assasination and 9/11. I remain skeptical as is the tradition of science and philosophy that I respect.
Let us treat the title of this chapter as a similar postulate that says that not only is the book of the universe written in the language of math, but that there is an exact mathematical description of the world. That in some sense, the ontology, the existence of the world is fundamentally mathematical. That's why a lot of physicists talk about the universe as simulation on a quantum computer. In my view this is independent of the God Postulate not because of any "god of the gaps theory", but because something like spirit or consciousness is necessary for a complete description. You can get a long way towards the goal of showing that nothing like this is needed with a more complete mathematical interpretation of information theory in physics, but I don't think it is enough. Reguardless of the evidence for and against each of these very deep postulates of metaphysics and ontology, there is a lot of evidence that a lot of modern traditions are deeply ideological on the answers. Peirce is refreshing in his relentless skepticism of ideology even as he insists that all of his work is grounded in a seemingly ideosyncratic set of categories.
The Three Categories
The natural analogy to Peirce's reverance for the number three is the Pythagorean philosophy of harmonics. In my view, this more mystical treatment of math as underlying if not totally determining the world is more natural that Platonic idealism. Combined with Democretous (atoms) you almost have the post-modern wave-particle duality. Resonance is much more important in physics than perfect circles, and the circle is sort-of a source generator for all oscilation (resonance). The complex and deep harmonies that produce the standard model of physics from some as yet to be discovered symetric master theory (math) and all of what we see in the emergent layers of life on earth.
Another analogy for me is one of my favorite design philosophers, Buckminster Fuller. Fuller is all about solid geometry since he is interested in building structures and he identifies the minimal rigid structure in the tetrahedron. It isn't mysticism, it is fundamental structural pattern. Peirce is looking at epistemology, how we know what we know and how we extend what we know, and he finds that he must first understand the sign.
Like Fuller, he is looking for the minimal structural element of his domain and he finds the three categories in the minimal structural element of the sign. There are always three things in a sign, what appears, what it refers to (Saucer stops here) and what it means. The analytic tradition treats this semantic space inconsistently so they can never really distinguish clearly between different modal status of claims because that status is in the third part, how we interpret a statement, how we enact a response. I respond "12:30" the "Do you have the time" because of previous enacted commitments that allow me to project that the correct interpretation of the query is a request for the current time if you have it. Modern linguists easily show that words aren't even phrases (parsable) without a context. Different levels of knowledge about the subject are continually brought to bear on an evolving context. New and updated signs are coming in from the outside and the interpretant is evolving internally. In the pragmatic tradition as developed by Lewis, the real, is contained in this evolving interpretant which is made collective in a language community. It becomes objective, but it is never complete because all interpretants are provisional and only tending towards a final interpretant shared by the community. Lewis' given is the object of the sign, what it points to.