Formal Systems
Formal Systems and Axiomatic Foundations
Abstract
In a formal system, the difference between a hypothesis (like the Riemann Hypothesis, for example) and a theorem is that a hypothesis could turn out to be false in theory, but a theorem cannot. For example, until 1994, when Andrew Wiles proved it, Fermat’s Last Theorem was in fact misnamed, for prior to 1994, just like the Riemann Hypothesis, Fermat’s claim should have been called Fermat’s Last Hypothesis, not a theorem. In this paper, we explain that what makes theorems so valuable in reality is that if a hypothesis turns out to be false in reality, we do not know why it is false. It could be because the hypothesis is false in theory. However, if a theorem turns out to be false in reality, then we know with absolute certainty that because it cannot be false in theory, one of the underlying axioms is guaranteed to be false in reality, and this is the only possible reason how a theorem could turn out to be false in reality – one of the axioms is true in theory, but false in reality.
Formal Systems:
Formal systems in mathematics are frameworks consisting of a set of axioms and rules of inference. These systems are used to derive statements and prove theorems within a consistent logical structure. Examples include:
Peano Axioms for Arithmetic: These axioms define the natural numbers and their basic properties. Within this formal system, statements like "2 + 2 = 4" are derived and hold true as long as the axioms are satisfied.
Set Theory: Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) is another foundational system, used to construct much of modern mathematics.
Violations in Reality:
In practical applications, the axioms and assumptions underlying formal systems may not hold perfectly, leading to discrepancies between theoretical results and real-world observations. This is a key topic in the philosophy of mathematics and the application of mathematical models:
Induction Principle: One of the Peano axioms states that properties holding for the number 0 and preserved under succession hold for all natural numbers (mathematical induction). In real-world scenarios, the assumptions of continuity and infinite succession may not apply.
Counting and Measurement: While "2 + 2 = 4" is true in abstract arithmetic, real-world counting and measurement often involve practical limitations and contextual factors. For example, physical entities like moons or objects might not adhere strictly to mathematical abstractions due to constraints like indivisibility, spatial distribution, or observational limitations.
Example: Mars' Moons
Abstract Arithmetic: In the abstract world of arithmetic governed by Peano's axioms, "2 + 2 = 4" is an invariant truth.
Real-World Application: When counting moons, the context and constraints matter. If the moons are considered indistinguishable and we apply simple addition, we expect "2 + 2 = 4". However, practical issues like grouping, physical constraints, or observational limitations might lead to different practical interpretations.
Bell's Inequality and Quantum Mechanics
Another example of the limitations of classical mathematical systems in describing physical reality is seen in quantum mechanics:
Bell's Inequality: Bell's theorem and the associated inequalities are fundamental results in quantum mechanics, demonstrating that no local hidden variable theories can reproduce all the predictions of quantum mechanics. Experiments have shown violations of Bell's inequalities, supporting the non-classical correlations predicted by quantum mechanics.
Axiom of Separation in ZFC Set Theory: In the context of entangled photons, the axiom of separation (also known as the axiom of subsets) in ZFC set theory is violated. The axiom of separation states that for any set and any property, there is a subset containing exactly those elements of the original set that satisfy the property. However, entangled photons exhibit correlations that cannot be neatly separated into distinct subsets without violating the principles of quantum mechanics. This means that the classical notion of separability does not apply, highlighting a fundamental limitation of classical set theory in describing quantum phenomena.
Broader Implications
Model Theory: This area explores how different models of a formal system can satisfy or fail to satisfy certain statements. A model where "2 + 2 = 4" fails could be constructed by altering the axioms or the interpretation of the terms.
Empirical Mathematics: The field that addresses the application of mathematical models to real-world problems often encounters situations where theoretical results need adjustments or approximations to fit empirical data.
Non-Standard Models: In certain models of arithmetic or set theory, statements that are universally true in the standard model might not hold, providing insights into the flexibility and limitations of formal systems.
Conclusion
The interplay between formal systems, their axioms, and real-world applications is a rich and well-explored area in both mathematics and the philosophy of mathematics. The idea that a statement like "2 + 2 = 4" might not hold in a specific real-world context due to the violation of underlying axioms or practical constraints is an example of how abstract mathematical truths interact with empirical realities. Additionally, the violation of Bell's inequalities in the context of entangled photons underscores the limitations of classical set theory, such as the axiom of separation, in fully capturing the nuances of quantum mechanics. This exploration helps refine our understanding of both the power and the limitations of formal mathematical reasoning.
Theory-Induced Blindness
However, and this is the most important thing we must realize, when a logical claim – such as Bell’s Inequality – guaranteed to hold true in algebra theory, does not hold true in reality, we know with absolute certainty that one of the axioms is being violated in reality. This is the key difference between a theorem and a hypothesis in math – if the hypothesis does not hold true in reality, it could be because it is false in theory, or because one of the axioms is violated in reality – there is no way for us to know. But if we are talking about a proven theorem, it cannot turn out to be false in theory, and if a theorem is false in reality, then we know with absolute certainty that an axiom is false.
Until we identify the flawed axiom and correct the theory to accurately model reality, the known-to-be-false underlying theory must not be used. Continuing to use such flawed, false assumption-dependent axiom-based theories is what Daniel Kahneman defines as theory-induced blindness. This highlights the critical importance of continuously validating theoretical models against empirical evidence to avoid the pitfalls of adhering to outdated or incorrect theoretical frameworks.
In this sense – before we fantasize about crazy hypotheses, like dark matter, or the holographic universe – necessary to address discrepancies between theoretical predictions and real-world outcomes, we should always, first figure out which axiom is being violated, and fix it, before attempting to come up with a hypothesis (like dark matter) that is singularly unlikely to turn out to be true in reality, on account of the fact that it was made up to account for a known and false axiom in ZFC set theory, or another false axiom.
This approach ensures that our scientific and mathematical theories remain grounded in verifiable reality, reducing the risk of developing elaborate but ultimately unfounded hypotheses to explain phenomena that could instead be resolved by revisiting and correcting our foundational assumptions. By rigorously validating and refining our axiomatic systems, we can build more accurate and reliable models of the natural world, fostering true scientific progress and understanding. Just like Einstein used Riemannian geometry to model the real-world curvature of space-time – a task unsuitable for Euclidean geometry – rooted in a false assumption that the shortest distance between two points is a straight line – an assumption that does not hold true in reality where space-time is, in fact, curved, and not straight.