Introduction

Mathematics and science are deeply interconnected, with math providing the founfation for scientific theories. However, there exists an huge gap within mathematics—a gap that ensures we will never touch complete certainty. Certain true statements remain unprovable, potentially challenging our belief in scientific discoveries.

Historical context

Russell’s paradox and formalism

In 1874, Georg Cantor introduced the concept of sets, and a branch of mathematics named set theory, which studies well-defined collections of things. Despite his attempts to establish a rigorous mathematical framework based on set theory, Cantor acknowledged the inherent difficulties in defining limit precisely. By the end of the 19-th century, a huge debate among mathematicians centered around the foundations of mathematics. On the one hand were the intuitionists led by Luitzen Egbertus Jan Brouwer, who emphasized the constructive aspects of mathematics, arguing that mathematical truths are not discovered but created. On the other hand were the formalist championed by David Hilbert, they aimed to prove the consistency of mathematical systems through formal methods. But Bertrand Russell discovered a paradox in the late spring of 1901 while working within the framework of naive set theory. Russell’s paradox arises when considering the set $R$ of all sets that do not contain themselves. Is $R$ a member of itself?

  • If $R \in R$, by definition of $R, R \notin R$.
  • If $R \notin R$, by definition of $R, R \in R$.

Russell’s paradox exposed a fundamental contradiction in naive set theory, a logical contradiction comes from self-reference. One common approach to addressing this paradox involves refining how sets are defined. Please consider the formula $\phi(x)$ as $x \not\in x$. The set $\lbrace x\in S: x \not\in x \rbrace $ is not contradictory because it only includes elements in $ S $ that are not members of themselves 1, it avoids including itself within the set. However, this story continues.

Hilbert’s program

Hilbert’s program, formulated by the German mathematician David Hilbert in the early 1920s, aimed to address the foundational crisis in mathematics by establishing a new system for mathematical proofs 2. The construction of a system of proof, relies on the assumption that certain axioms are true. For example, a straight line segment could be drawn joining any two points in Euclidean geometry. And two sets are equal if and only if they contain the same elements in set theory. These axioms serve as basic building blocks for constructing mathematical systems. Theorems can be derived from axioms using a set of inference rules. It is a structured framework used to determine the validity of propositions or statements. At the International Congress of Mathematicians held in Bologna, Italy, in 1928 David Hilbert returned to the second one of the 23 problems, asking 3

  • Is mathematics complete?
  • Is mathematics consistent?
  • Is mathematics decidable?

Completeness means that every true mathematical statement could be proven by using an assumed formal system. And consistency means that the system does not lead to any contradictions; in other words, you can’t prove both a statement and its opposite. If the final one is true, it suggests that whether any given mathematical statement is true or false could be determinied. Hilbert was convinced that all the answers of those questions were yes, as he saying before, Wir müssen wissen, wir werden wissen (We must know, we will know).

Gödel’s incompleteness theorems

Gödel’s incompleteness proof

In 1931, Gödel addressed Hilbert’s first question with his incompleteness theorem 4. This theorem established that it is impossible to construct a formal system, using the axiomatic method, for any branch of mathematics containing arithmetic that encompasses all its truths. This meant that no matter how carefully mathematicians chose their axioms, there would always be some truths that escaped formal proof 5. Gödel’s incompleteness proof is achieved by assigning unique Gödel numbers to symbols and statements, constructing a self-referential statement that asserts its own unprovability and using logical contradictions to demonstrate that it must be true but unprovable. The key point is still the thing called self-reference. In 6, the author uses an analogy to illustrate this similar thought, if editor claims that every concept is defined in that dictionary, then it must be logically circular. This is like defining the first word using a second word, and then explaining the second word with a third, continuing indefinitely. And then Gödel’s Second Incompleteness Theorem shows that no consistent formal system includes basic arithmetic can prove its own consistency. These groundbreaking results revolutionized the fields of mathematics and logic, profoundly impacting the philosophy of mathematics.

Undecidability in mathematics

How about addressing the last question of decidability in mathematics? Interestingly, this question is actually related to Turing machine. Turing imagined an entirely mechanical computer, it consists of an infinite tape divided into cells, a head that can read and write symbols on the tape, and a set of rules that dictate the machine’s actions based on the current state and the symbol it reads. The decidability of a proposition may be considered equivalent to the famous Turing halting problem in computational theory 7. The most famous example of an undecidable problem is Halting Problem, which asks whether a given Turing machine will halt (stop running) or continue to run forever based on a given input. Alan Turing proved that there is no algorithm can solve the Halting Problem for all possible Turing machine and input pairs. Furthermore, the problem of undecidability even appears in complex physical systems. Cubitt et al. shows that determining whether a quantum system has a spectral gap (a gap in energy levels) is undecidable. This means that there is no algorithm could determine for all possible Hamiltonians whether they are gapped or gapless 8. Now, we could answer those questions from Hilbert, the truth is we could not know, we will not know.

How do we see science?

The correctness of mathematics is equivalent to the correctness of its axioms. It is assumed in mathematics that axioms are self-evident truths from which all mathematical theories are derived. However, the incompleteness, inconsistency, and undecidability inherent in mathematics remind us that even the most fundamental axioms may have their limitations. These limitations could unexpectedly influence various fields of natural science in ways we may not anticipate. Regarding the question “How do we see science?” I have some further thinkings as follows.

The problem of induction

Inductive reasoning, by its nature, lacks strict logical rigor; it serves as a summary of scientific empirical facts to make broader generalizations or predictions which could rise numerous issues. For instance, observing that the sun has risen every day in the past does not logically ensure that it will rise tomorrow. This is because such an inference assumes that the future will resemble the past, which is itself an unproven assumption. It leads to Hume problem articulated by the Scottish philosopher David Hume. Although we rely on inductive reasoning to navigate the world, there is no rational basis for the belief that the future will be consistent with the past. This presents a paradox, despite the lack of rational justification, inductive reasoning seems both indispensable and reliable in practice.

Bayesian approach to belief and evidence

Employing the views of reinforcement learning, we could envision world as an enigmatic black box—a realm where our initial understanding is starkly devoid of knowledge. Through persistent trial and error, we receive feedback from environment, gradually deepening our comprehension of reality. For instance, in ancient times, as people pushed stones, they observed that exerting greater force resulted in faster motion. Through relentless experimentation, this insight was reinforced. In the virtual realm of computing, such logic manifests as foundational code, while in the tangible world, it is encapsulated by mathematics, physics, and the broader domain of science.

These perspectives resonate closely with Bayesian epistemology, prior probability represents the degree of belief in a hypothesis before new evidence is taken into account. As evidence is gathered, the prior is updated to the posterior probability, which reflects the revised belief in the hypothesis after considering the new evidence. We strive to approximate the natural laws, perpetually reaching for the profound mysteries of the universe. Nevertheless, the quintessential nature of these truths eludes us, shrouded in obscurity much like the form casts the shadow itself.

Challenges and criticisms

Even within the formalistic framework of logic, there persist certain doubts that challenge the very foundations upon which it stands. Someone has raised questions about law of excluded middle. The principle of excluded middle typically states that in any statement, it is either true or false, with no other possibilities. Is it possible for there to exist a proposition where we cannot determine its correctness or incorrectness, meaning that the law of excluded middle does not apply? Another one is about topos theory, it introduces a novel perspective on truth and logic. Unlike classical logic, where statements are simply true or false, topos theory accommodates a more nuanced view where the truth of a statement can vary depending on the context 9. This approach has implications for the foundations of scientific cognition, offering an alternative prespective to view science.

Discussion

Science is not just a collection of facts, it is a dynamic path to understanding. Moreover, science is also about curiosity and the willingness to explore and ask questions about the world around us. It’s about being open to new ideas and evidence, and being ready to revise our understanding as new information becomes available. It is conceivable that we may never reach the ultimate shores of truth, our journey along the path towards enlightenment will persist unceasingly.

References

  1. Irvine, A. D. & Deutsch, H. in The Stanford Encyclopedia of Philosophy (ed Zalta, E. N.) Spring 2021 (Metaphysics Research Lab, Stanford University, 2021). ↩︎

  2. Zach, R. in The Stanford Encyclopedia of Philosophy (eds Zalta, E. N. & Nodelman, U.) Winter 2023 (Metaphysics Research Lab, Stanford University, 2023). ↩︎

  3. Hilbert, D. Probleme der Grundlegung der Mathematik. I, 135–141 (1929). ↩︎

  4. Gdel, K. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für mathematik und physik 38, 173–198 (1931). ↩︎

  5. Gödel, K. On formally undecidable propositions of Principia Mathematica and related systems (Courier Corporation, 1992). ↩︎

  6. Cui, W. Can Science Reveal the Origin of the Universe? European Journal of Applied Sciences–Vol 12 (2024). ↩︎

  7. Turing, A. M. et al. On computable numbers, with an application to the Entscheidungsproblem. J. of Math 58, 5 (1936). ↩︎

  8. Cubitt, T. S., Perez-Garcia, D. & Wolf, M. M. Undecidability of the spectral gap. Nature 528, 207–211 (2015). ↩︎

  9. Johnstone, P. T. Topos theory (Courier Corporation, 2014). ↩︎