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Paper #: SCP-001
Abstract: This paper covers the history of the progress of scientific and mathematical thought. First, it details traditional theories about Axiomatic models of mathematics and physical sciences; then, leads into modern advances into Non-Axiomatic mathematical theory, and how that has had an effect on research topics in the regions of physics, chemistry, and biology. The philosophical repercussions of the discipline of Non-Axiomatic mathematics are then considered, especially regarding the interdependency of a biologically-dependent organism within a Non-Axiomatic physical system and its comprehension of Non-Axiomatic mathematical theory. Recent experiments into a return to and re-formalisation of Axiomatic mathematical theory are discussed. Finally, the results of these experiments are analysed, followed by discussion and justification for the need to return to Axiomatic mathematical thought.
NOTE: This paper is targeted towards those with comparatively little backing knowledge of mathematics and the sciences. It therefore builds up concepts from an entry-level understanding, and may be too simplistic an analysis for some readers. For more complex research topics and in-depth discussion, please consult subsequent scientific papers in this series.
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Table of Contents
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1. Introduction
"Look at all the order around you," he said. And from that, he deluded honest men to believe that reality was a straightjacket affair and not the happy romance as men had known it.
It is not presently understood why men were so gullible at that particular time, for absolutely no one thought to observe all the disorder around them and conclude just the opposite.
It is necessary to begin this document with some degree of levity, for the remainder of this paper will have little to entertain you. The history of the Sciences and Mathematics is a barbarous one, with academic thought being stifled by traditionalist institutions since the dawn of time. Historically, most of those with a scientific mind believed that prediction of physical phenomena was practical and desirable, and the pursuit of knowledge to be the purest and noblest objective in life. Those of a more religious bent claimed that all was in the eyes of god, predetermined and immutable, a holy and static state of being that none could interfere with. Both sides, essentially, believed that the universe should, on some underlying and fundamental level, make sense.
It does not.
After a point, both of these theories unravel into the same muddled pile of thread. Physics collapses into the vague uncertainties of quantum mechanics, while pure philosophy rarely holds water beyond the abstract. Both cling desperately to concrete notions, either of Reducibility or Truth, and so are unable to fully explain the wider gamut of reality.
2. Traditional Mathematical and Scientific Thought
2.1. Axiomatic Models of Mathematics
2.2. Axiomatic Models of Physical Sciences
3. Modern Mathematical and Scientific Thought
3.1. Modern Understanding of Reality
There are four main properties of our universe that do not abide by traditional physical thinking. These are that our universe is currently, in varying degrees:
- Non-Deterministic,
- Non-Causal,
- Non-Self-Consistent, and
- Non-Axiomatic.
While the repercussions of these properties can rarely be seen on a macroscopic scale (and when they are, are typically objects of particular research benefit), on a nanoscopic scale, they can easily be observed in all matter.
Before continuing, some definitions are required.
In a Deterministic system, all future system states are predictable from past system states. In simpler terms, in a Deterministic system XXXXXXXXXXXXXXXXXX
In a Causal system, phenomena will only transpire as a result of physical causes; that is to say, spontaneous phenomena cannot occur. This does not relate to Chronological causality, where effect must follow cause; in this context, a Causal system may have effect preceding cause. Non-Causal systems are inherently dangerous due to their unpredictable nature.
While rare, it is possible for a Non-Causal system to still be Deterministic: using Non-Axiomatic Mathematical methods, some predictions can be made about phenomena which have no external cause. For example, if a system is both Non-Causal and Self-Consistent, then someone may observe the Non-Causal events occurring, then travel back in time and pass that information to themselves. Due to the Self-Consistent nature of such an environment, these actions could not alter past events, and therefore the Non-Causal events can be known as Deterministic.
Self-Consistency can have one of two meanings. If a system is Axiomatic, then a Self-Consistent system's Axioms have both the mathematical properties of axiomatic Consistency and Completeness. This means that every statement that can be asserted for the system must be either provably true or provably false. Thus, physical contradictions and contrapositives cannot occur. In a Non-Axiomatic system, then Self-Consistency refers to the Generalised Novikov self-consistency principle. In less formal terms, this means that it is impossible for physical paradoxes to arise, both chronologically-based and otherwise. While these formal definitions are different, their consequences are identical, as shown in papers by Borwein and Tao.
While this may superficially appear to have the same consequences as a system being Causal, they do not have the same meaning. In a Self-Consistent model, paradoxes cannot arise of any kind, while in a Causal system paradoxes can still arise. In a Self-Consistent model, for example, it is impossible to change the past; any "changes" are simply self-fulfilling future causes. In a Causal system, one may alter past events through a cause which occurs in the present; thus, the computationally explosive effect of historical alteration may still occur, making any predictions about the system's eventual stability (or non-stability) far more difficult. Because of this, when chronological consistency is desired, it is more important that an environment have Self-Consistency enforced on it rather than Causality.
In an Axiomatic system, all physical phenomena are reducible to some set of laws.
3.2. Physically Empirical Non-Axiomatic Models of Mathematics
3.3. Non-Axiomatic Models of Physics
The Heisenberg uncertainty principle indicates that certain pairings of physical properties of matter cannot simultaneously be certain. For instance, as the certainty of the position of a particle becomes more clear, its momentum reduces in an inverse-proportional manner. More formally:
(1)Where $\sigma_{x}$ denotes the standard deviation of position, $\sigma_{p}$ denotes the standard deviation of momentum, and $\hbar$ denotes the reduced Planck constant, that being the quantum of angular momentum.
There are multiple interpretations of Quantum Mechanics. The most common interpretation is that the position and momentum of any given particle are represented as statistical clouds of potential, which collapse to a specific position (or momentum) when observed. However, to reconcile QM theory with General Relativity, it is necessary to assume an underlying specific position and momentum of particles, which are then (according to this model) simply unknown rather than indeterminate.
In this unified theory, the esoteric properties of quantum systems can be disregarded, and very specific and exact predictions can be made about systems. However, it allows for closed timelike curves, which allow for particles to interfere with themselves in a Non-Causal fashion. Thus, by modelling physical space as Axiomatic, we must allow it to be Non-Causal. This gives rise to paradoxical situations, which eliminate the Self-Consistency of the system also. As such, while this model is Axiomatic and Deterministic, it is Deterministic in a self-contradictory fashion. While it is useful for making predictions in some cases, it does not accurately describe the underlying mechanics in effect. This unified theory is known as the Luciano Unification model.
The alternative approach is to let QM theory dictate how General Relativity should be adjusted. This model involves considering all objects as statistical clouds, even on a macroscopic scale; as such, it is not a Deterministic model of reality. Furthermore, it is built primarily on physical observations which do not cleanly reduce to a set of Axioms; as such, this model is not Axiomatic. However, it does allow for an entirely Causal reality, as all future states of statistical probability clouds are only affected by previous states. Similarly, it satisfies the criteria for Novikov Self-Consistency. This unified theory is known as the Nishimura Unification model.
Therefore, there are two main models of the universe in modern physics: one which is Axiomatic and Deterministic, and one which is Causal and Self-Consistent. Where traditionally physicists would choose between the Quantum Mechanical or General Relativistic models of the universe depending on the problem, the modern physicist must alternate between the Luciano and Nishimura Unification models in order to solve any given problem.
Perhaps most importantly, there are existence proofs that the Luciano and Nishimura models are reconcilable, as shown in papers by Mutter et al. However, they can be unified in two ways. One allows for a Deterministic, Causal, Self-Consistent, and Axiomatic reality (henceforth referred to as the DCSA model); the other, as a corollary system, is none of these (the Non-DCSA model). This would mean that the disciplines of Anomalous and Non-Anomalous Physics could be formally separated, with the DCSA model being used to explain all traditional, common physical phenomena, and a unified Non-DCSA model being used for anomalous phenomena. The development of these unified DCSA and Non-DCSA models is therefore perhaps the most pressing research priority in modern physics.
4. The Interdependencies of Systems
4.1. Mathematical and Physical Interdependencies
4.2. Physical and Chemical Interdependencies
4.3. Chemical and Biological Interdependencies
4.4. Biological and Mathematical Interdependencies
Certain experiments can easily show the effect that existing within a Non-Axiomatic
5. Philosophical Discussion
5.1. Bias Within Non-Axiomatic Biological Minds
5.2. Repercussions on Epistemology
5.3. Analysis of the Modern Non-Axiomatic "Naïve Science"
5.4. Accounting For And Removing Bias
5.5. Effects on Engineering Disciplines
6. Recent Experiments
6.1. The Missing Number
6.2. Conceptual Deterministic Universes
6.3. The Library
6.4. Quantification of Anomalous Phenomena
6.4.1. Hume Theory of Proximity to Deterministic Reality
6.4.2. Xyank Theory of Proximity to Causal Reality
6.4.3. Descartes Theory of Proximity to Self-Consistent Reality
6.4.4. Gretchen Theory of Proximity to Axiomatic Reality
The now-defunct Heraclitean-Confucian Principle tells us that Axiomatisation and Scientific Knowledge are equivalent. Thus, a Non-Axiomatic may seem to be one of which no scientific knowledge can be extracted. While for a long time it was believed to be impossible to formalise Non-Axiomatic systems, novel techniques have been developed within the last century that allow for some level of tentative formality within them. To researchers already familiar with macroscopic Non-Axiomatic systems, this may seem obvious, as it provides the cornerstone of much modern research; however, a brief summary of these techniques shall be given here.
Anchoring to a certain level of Axiomaticism is hypothetically possible according to Hume theory. There are indications that development of such devices is already underway, as seen in the series of papers by Scranton, Caldmann, Rzewski et al. However, the effort to maintain the given level of Axiomaticism of such must, by necessity, decrease local Axiomaticism either by net increase in Humes or by Hilbert spacetime distortions (Hilbert et al. as mentioned in Paper #SCP-1848). Therefore, the use of constructs such as "Scranton Anchors" would likely reduce our reality's net proximity to a hypothetical baseline Axiomatic Reality.
6.5. Theories on Memetics
6.6. Generalised Theory Of Non-Axiomatic Physics
6.7. The Retrocausality Torus
For further details and experimental results, see Paper #SCP-1968.
6.8. The Physical Law Testing Chamber
For further details and experimental results, see Paper #SCP-536.
6.9. The Creation of a Non-Anomalous Particle
The Non-Axiomatic natures of each particles negated one another, producing a purely Axiomatic particle.
7. Discussion
Therefore, a return to an Axiomatic baseline reality should be possible through one of two plans:
- Expulsion of all anomalous phenomena from the observable universe, a plan that is still only hypothetically possible, and which has not reached scientific consensus of its validity, or;
- Consolidation of all anomalous phenomena into a singularity, which is statistically likely to result in net cancellation of all Non-Deterministic, Non-Causal, Non-Self-Consistent and Non-Axiomatic effects.
8. Conclusion
9. Appendices
9.1. Appendix 1: Classing of Macroscopic Anomalies by Type
Macroscopic anomalies can typically be classed as some combination of its violations of Determinism, Causality, Self-Consistency and Axiomaticism
9.2. Appendix 2: Protocols for Universal Neutralisation
9.3. Appendix 3: Fnord
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