This is a response to a question arising from my recent video on the error of the argument for god from mathematics, which Eckels raises the question of messiness. The video can be seen here:
This becomes somewhat demystified if we take Kant’s view of mathematics as our starting point and focus on Poincare’s demonstration that rational constructs can be contradictory to each other yet internally consistent and Wigner’s demonstration of limits of invariability principles and approximation.
Kant showed pure reason as the top tier organizing principle of our objective thinking. This was an innate subjective principle that, in light of modern biology, would best be seen as an evolutionary adaptation. Of course, Kant would not be aware of the evolutionary aspect. Our senses of time and space would also be purely subjective sensibilities that, along with reason, form our conditions for ordering sense data, i.e. making sense of the world. All of our representations of the world necessarily follow these conditions, although the external world is in itself quite different. On the other hand, our representations aren’t purely arbitrary either, because they are interpretations of unique sense data conditioned by the external object. This gets to what Wigner called the first tier of practical mathematics, where the world “suggests” events to our understanding. This happens with number, for example, where what we intuit from the external world “suggests” multiplicity of objects like enough to be abstracted and counted. This is the most primitive and surest of objective representation, and most likely the evolutionary adaptation early in our development that enabled us to manipulate our environment and calculate future probability. This is the ontological connection between our rational thinking and the world. It is conditioned by our forms of understanding by which we organize sense data, and stimulated (suggested) by sense data uniquely conditioned by each intuition.
In Wigner’s second tier, we are able to further abstract and manipulate these concepts according to the laws of Reason and we do so mathematically when the abstractions are numbers. At this point the apparition of miracle appears, but only if we think metaphysically. In reality, there was an approximation of the world through our understanding at the point where events suggested multiplicity of objects, the sense of which was conditioned by certain unique properties of what is intuited. It is only an approximation, though, because we impose the form on what the world provides (suggests) as content. Number is, from the beginning, an abstraction that omits much information that would favor uniqueness over likeness of these objects.
Wigner’s third tier is the invariability principles, which allow the abstractions in the second tier to be further generalized. This allows for universal laws, but with a crucial limiting factor. These abstractions can only be stretched so far until the world conflicts with the premises. In that case another limited space of invariability needs to be set off, which will likely conflict with the premises of the first, as do Newtonian physics and relativity. At these points new mathematical systems built on new premises must be invented. This builds from Poincare’s demonstration of our ability to construct inherently coherent rational systems in essential conflict with each other, which highlights the provisional nature of our universal laws. Wigner goes on to show that it is possible to construct such conflicting systems through our choice of suggested events to include in the model, and how we delimit space and time.
The point is we are stretching our faculties of reason, space and time which we originally adapted for survival on the savannahs to understand the universe. Because of the primordial joining of our subjective faculties to the events suggested by the world, there is an approximate correlation between our thoughts and reality. If there weren’t, we would never have survived the perils of the savannahs. But there is a point beyond which our stretching snaps the bands of the principles that held the understanding together.
That is the simple explanation. If we were to take a more Heideggerian approach we would necessarily reintroduce mystery in another form, but that’s a story for another day.