The title of this blog entry: in as much as it assumes there is such a “thing” as a beginning it assumes a certain relationship between the discrete and the continuous.
Which comes “first”, discrete or continuous, how do they relate? Does the discrete “generate” the continuous as in parent/child?…or is there a “smooth” transition, a continuous path, a circle, from continuous to discrete and back?
Which comes FIRST? The question begs the question since the concept of “first” already presumes the discrete…so what can we do?
Given this situation the best answer must be, “what does Nature tell us.” What do we see (hear, feel, smell, taste) around us? Where is the perfectly, absolutely impermeable boundary in nature? Where is the “continuous”? Likewise, where is the boundary that cuts the space in two…where is the “discrete”?
Nowhere.
Asking that question amounts to asking, “where is time?” It’s a nonsense question.
There is no such thing as a perfectly impermeable boundary. So there is nothing that cuts our world in two either. The closest thing to it is a “horizon”. It retreats as we try to attain it, to approach it….never to be crossed. It is a perfect model of infinity….the only perfectly, absolutely impermeable “boundary”. But the “other” side is inaccessible…there, but not “there” as in “here it is”. We can’t get there to say “here it is”. Even using the term “boundary” in reference to infinity is a contradiction…to infinity there is no boundary, absolutely none.
Infinity is what is beyond contradiction. It cannot be established by “contradiction”. We cannot use ideas like “fixed point”, “premise”, “define”, etc., since for “infinity” they all presume what it is we seek to “define”. Where is the end of the circle? In Nature.
So again, there is no such thing as a perfectly impermeable boundary in Nature…everything is penetrated by something. There is the relationship between “for all x” and “there exists x”, between the continuous and the discrete. “everything is penetrated by something” cannot be “broken down” into “components”…it defies “categorization”…it is the fundamental structure of Nature…it defies the reductionist approach to understanding. Take it apart and it is no more. It is the horizon.
To build a structure of ideas on the idea of a perfectly impermeable boundary is therefore likewise nonsense. Any structure built by “axiom” without reference to Nature is hollow…it fails before it starts since it starts with a falsehood: that we can just “assume” a beginning…a beginning by “wish”…a convenient fiction. “Platonic Universes” are nonsense. There is no such “perfect” world….everything in Nature has “holes”.
So let’s start there…everything has holes. Not holes like ϕ, the empty “set”, since “set” by construction already implies a boundary….∈, the membership relation. Let’s start by recognizing that the relationship between T and F in Aristotelian logic is not symmetrical….that F can only throw out a limited number of cases while T is unbounded. Remember the black swan. Falsifiability is itself a foundation of sand.
Can we break the relationship between T and F “down” any further? No, together they form a system of “medium vs. boundary with holes”.
Some might say we should characterize F as fixed points…points of reference. That is a bad model. It reverses the roles of T and F. It implies a reversal of “power”. There is always a way around a “fixed” point, so how “fixed” can it be?…only conditionally “fixed”. A much better model for F is a boundary with holes. T flows through and hence “around” no matter what. Proof by contradiction is the wrong approach to a true understanding of our condition as humans. We live in a world characterized by conditional boundaries, “membranes with holes”. So let’s start with that model.
The only reasonable “beginning” is to explore the relationship between “continuous” and “discrete”…between the “aether” and “the boundary with holes”….between the “real world” and our understanding of it.
References:
“Clearing Up Mysteries – the Original Goal”, by E. T. Jaynes https://bayes.wustl.edu/etj/articles/cmystery.pdf
21st Century Paradox 