Reductio ad absurdum!

What does philosophy have to say of the Reductionist program?

Indeed what are the guiding principles of any Reductionist program applied to basic physics?

En route to revealing "Theories of Everything" there seems to be a tension between the search for a minimal set of objects that explain our reality and the deployment of a minimal set of guiding principles. Reductio ad absurdum is "a form of argument which attempts either to disprove a statement by showing it inevitably leads to a ridiculous, absurd, or impractical conclusion". The program applied to fundamental physics is by this dint self explanatory? A discussion of the Naturalness program and its partial success as a guiding principle can be found here:

https://arxiv.org/pdf/1710.07663.pdf

Our aims are more modest but probably no less coherent here. Quite generally I am just trying to keep a foothold where I can on the theoretical physics front which may just be just as penetrable to those not knee deep in their own cognitive dissonance in hoc with the Research Grant providers.

For the most part the maths has gotten away from me so if you are similarly jaded by the arcane mathematics that abounds lets try keep an eye on the bigger picture, that is the metaphysical questions that might still be in play. Lets try to to take this one easy.

You can do things with numbers (abstracted from collections of objects) : add, subtract, etc. that is perform operation on them with operators. By obeying the rules of the operating you can move up and down number line. You can create short-hand for multiple additions of those number (call that the x rule). You can create a shorthand of multiple multiplication - call that the exponentiation operation, ^. Mixing up these operations allows you to play with these objects efficiently. That is the law of distribution you ignore at high school taking them as obvious. 

The Algebra of Geometry

We can start to drop these assumptions and look at the implication that this freedom affords us. Beside is illustrated the non-commutative transformation operations of reflection through an axis of symmetry followed by a clockwise rotation. The order of such distinct operations (transformations) matters.


http://www.math.brown.edu/~banchoff/Beyond3d/chapter9/section02.html

Now if you extend the field of number (objects) to a plane (rather than line) of numbers (complex ones) some rules of algebra (the Fundamental rule of Algebra), become simpler in this broader space of numbers-objects and operations. So by giving greater freedom (increased number of allowed operations) to a greater number of objects we end up with greater simplicity. Below we see how to factorise a cubic polynomial using a high school long-division technique. The divisor of our cubic is a quadratic and gives rise to a linear "quotient" function with zero remainder .

http://mcuer.blogspot.com.es/2007/10/precalculus-25-fundamental-theorem-of.html


We can further factorise the quadratic even though it does not cut the Real axis. It possess rather Complex roots. We see then that any polynomial has the same number of roots as its degree. By widening the field of numbers over which we do our analysis more insight and greater simplicity is achieved.


To summarise, we say that the Algebra of Real numbers (object-solutions) that can be acted on by x,/,^ (that latter exponentiation) operators and are thus deemed to closed under these operations obeying laws of distributivity, commutativity and associativity. The object set can be extended to complex numbers (the operations generalised to complex exponents) and we note that in this larger formalism complex-valued polynomials have much simpler root rule forms than real valued ones.

Guiding Principles

Back then to the guiding principles. In mathematics the greater the number of axioms that one is required to adhere to, the more restricted the set of solutions that the resulting self-consistent theory delivers. At some point indeed there is a unique solution as only one "reality" satisfies all the axioms. Conversely as you peel away the axioms, slackening the freedom in the system, more possibles plausibly satisfy the loosened laws of your world.

A search for the most restricted set of (guiding principles) axioms allowing for multiple realities or the search for a unique reality built from a tightly bound lattice of constricting premises? More objects, more allowed operations, a proliferation of laws? I guess I am merely asserting that even high end physics has an axiomatic front to it even beyond hard core axiomitisers: solve a path integral based on an Action (principle) constructed with holomorphic complex functions do something with or without boundary conditions.

lessWrong

Perhaps the stationary action principle glorious as it is, is too restrictive? In it we assume sets of underlying fields and restrict yes to second order functional at most so that causality is respected. Not much else but for nature leaning heavily towards the path that minimises the difference between the Kinetic, T and potential energy, V of the system. A very narrow set of guiding principles that delivers "on-shell" equations with plenty of freedom to find them experimentally.

In science, the reductionist program seeks to reduce the world to a set of indiscernables that may or may not obey lots of laws or be founded on a limited set of guiding principles (axioms). The question is by which rule of simplicity are we guided? By the number of principles, number of resulting laws or numbers of free parameters linking a limited set of objects?

To the side we delineate the stuff of the universe into its Bosonic and Fermionic fundamental constituents. As collectives they each have distinct characteristic probability distribution profiles.


http://eenadupratibha.net/Pratibha/Engineering-Colleges/Engineering-Jobs/engg.phys_content4.html

By no lesser a mind than John Wheeler lets call out this indistinguishability between fundamental particles for what it is: they are one and the same particle! https://thecosmogasmicperson.wordpress.com/2017/08/20/its-all-the-same-electron/ Perhaps not.

Let's now play with a bit of logic try to pretend to axiomatise our thought processes to see if this gives us any insight the fundamentals of indivisibles.

Logical Reduction

Consider now the syllogism:

1. The (matter) Clumping of elementary indivisibles results in the loss of ("binding") mass-energy. 
2. Matter (stuff) distinguishes itself by the clumping of its constituent particles. 
3. Distinguishability (of stuff) results from the emission of mass-energy (in terms of radiation). 

The logic of the deduction seems sound enough, but a top-down argument to deduce the implications of a top-down program is perhaps tautological? Further, the interpolating is premised on a bottom-up induction: a posit that all elementary particles are indistinguishable (by definition?) and thus tied (being left invariant by some spatial or inner spatial group transformation) to each other by symmetry principles. The inductive insight (of the scientist) is the generalising symmetry principle of the indivisibles that enables us to extrapolate to all indivisibles. So was it inevitable that the inductive argument lead us to a process of reductive deduction? Or the other way around?

https://www.quora.com/

From this we can reflect on the following. That decay (from random emission) creates difference seems natural. That accretion through random emission creates difference follows from logical deduction seems a little less natural. Perhaps the answer lies in the predicate indivisibles? That fermions cannot reside in the same state renders them less indivisible than a boson that happily forms (Bose-Einstein) condensates. That we distinguish fermions from Bosons on the basis of their mixing statistics suggests the act of condensing (accreting,clumping) through boson exchange renders the primary predicate vulnerable to argumentation.