# Exploring the Limits of Knowledge in Mathematics and Science

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## The Unanswerable Questions

There are certain inquiries in existence that elude definitive answers. Questions like: Is there a God? Is time merely a human construct? What lies at the boundary of the Universe, and what exists beyond? Which came first, the chicken or the egg?

(For the record, it’s the egg.)

Profound questions regarding the essence of science, reality, and the (non)existence of deities often lead to these unresolvable mysteries. In the early 20th century, European philosophy saw the rise of the **logical positivists**, who asserted that questions were devoid of meaning if their answers could never be empirically verified.

These logical positivists dismissed discussions of ethics and morality (which seek to establish normative claims about reality) because such claims could not be verified. There exists no experiment that can determine the "right" action; thus, the concept becomes meaningless. To them, metaphysics was akin to magic.

However, the decline of the logical positivists' intellectual dominance can be attributed, in part, to their overly restrictive definitions of rational discourse. Just because we cannot empirically **verify** theories about moral norms or the existence of a deity does not mean we cannot engage in insightful and enriching conversations on these topics.

Moreover, methodological concerns arose regarding the very idea of **verification** itself. Karl Popper, for instance, constructed an entire philosophical framework around the inherent **impossibility** of verifying statements like "All swans are white." To validate such a claim, one would need to observe every swan **in existence**—an evidently impossible task. Consequently, Popper advocated for a falsificationist approach: **one can never prove anything, only disprove it.**

## The Complexity of Scientific Foundations

The relevance of these discussions to the idea of axiomatically resolving our problems is significant, as the situation is far more intricate than it appears.

Scientific foundations are anything but solid. We lack clarity regarding the assumptions we can justifiably make about the world, and scientific theories are far too intricate to categorize as strictly true or false based solely on a finite set of data. The challenge lies in the fact that theories are constructed upon a hierarchy of other theories, leading to an infinite regress of ideas—“turtles all the way down.”

Consider climate models as an illustration. We are aware (or believe) that the Earth is experiencing warming, largely attributed to human activity, yet the error margins in our models are considerable. How much warming is occurring, by when, and which specific behaviors are responsible? What are the anticipated consequences concerning sea levels and weather patterns?

Many climate models use probabilistic approaches on a global scale, formed from a patchwork of local models that account for specific regions or limited variables. These are based on theories regarding temperature, fluid dynamics, and acidity, and one could trace these back to quantum mechanics (and perhaps further) if one desired to delve into the specifics of how individual chemical interactions impact global climate.

Visualize science as a skyscraper; if a single brick is misplaced at the base, the entire structure may collapse—but we won’t realize it until the construction is complete. Currently, science appears reliable (at least much of it does) and seems to lead us to genuine truths about our world. But what if it turns out that our fundamental theories of matter, heat, or acidity are slightly (or completely) erroneous?

What then?

Perhaps science resembles not so much a skyscraper with a flawed foundation, but rather an intricate castle suspended in the air, or a house built on piles in a marsh. This metaphor aligns with Popper's own perspective:

“The piles are driven down from above into the swamp, but not down to any natural or ‘given’ base; and if we stop driving the piles deeper, it is not because we have reached firm ground. We simply stop when we are satisfied that the piles are firm enough to carry the structure, at least for the time being.”

## The Limits of Mathematical Certainty

You might challenge the preceding argument, thinking, “Surely, all science can ultimately be distilled into mathematics? If climate science reduces to quantum theory, and quantum theory is grounded in mathematics, which is merely logical reasoning based on a few fundamental principles… then isn’t science more of a ‘castle on solid ground’?”

This perspective is understandable, yet it overlooks the complexities of what mathematics entails, particularly in light of **Gödel’s incompleteness theorems**.

Kurt Gödel, a mathematician, philosopher, and logician of the early 20th century, made significant advancements in logic from a remarkably young age—publishing groundbreaking results at just 25 years old.

To frame Gödel's findings, it's vital to grasp that **all mathematics is a theoretical framework founded on axioms**. These axioms are known as the **Peano axioms**:

- 0 is a natural number.
- For every natural number x, x = x. (Reflexivity of equality)
- For all natural numbers x and y, if x = y, then y = x. (Symmetry of equality)
- For all natural numbers x, y, and z, if x = y and y = z, then x = z. (Transitivity of equality)
- For all a and b, if b is a natural number and a = b, then a is also a natural number. (Closure under equality)
- For every natural number n, S(n) is a natural number. (Closure under S)
- For all natural numbers m and n, m = n if and only if S(m) = S(n). (Injection of S)
- For every natural number n, S(n) = 0 is false. (No natural number has 0 as its successor)
- If K is a set such that: (a) 0 is in K, and (b) for every natural number n, if n is in K, then S(n) is in K, then K contains all natural numbers.

(Source: Wikipedia)

If these axioms seem confusing, don't worry; what matters is that they affirm basic truths of mathematics. For instance, that 0 exists or that x = x for any x. These axioms are generally accepted in mathematics, but they are not necessarily the only possible truths; other axiom sets could be used. So far, these axioms seem logical and reflective of reality, leading to a robust mathematical system.

Now, returning to Gödel.

Gödel’s first significant contribution was the **first incompleteness theorem**, which states that regardless of the axioms chosen as the foundation of mathematics, there will always be mathematical truths that remain unprovable. This implies that mathematics can never be "complete." The **second incompleteness theorem** asserts that one cannot determine if a given system—such as mathematics based on the Peano axioms—is entirely self-consistent using that system alone. Both of these theorems stem from pure mathematics and logic, indicating that mathematics itself can never be complete.

One cannot select axioms that ensure mathematics is definitively self-consistent, nor can one prove every desired proposition within mathematics. Certain truths (like the Peano axioms) simply **are**, irrespective of proof. They are **axiomatic**.

## Lakatos: Merging Popper with Gödel?

Imre Lakatos, a Hungarian philosopher of science who conducted much of his work in London, was initially a student of Popper and considered himself a strict adherent of Popperian falsification. However, like all good things, this phase passed, leading Lakatos to develop his philosophy of science.

For Popper, science was a continuous endeavor of deducing which hypotheses were incorrect based on evidence. He argued that **we have no reason to believe any of our theories are truly correct**, but we can adopt them and proceed as if they are, until proven otherwise.

The challenge arises in discerning how one **knows** a theory is false; it's not always clear.

Take, for example, the Gran Sasso experiment, which measured the speed of neutrinos and suggested they were traveling **faster** than light. This seemed to contradict the theory of relativity, which posits a universal speed limit. Surprisingly, few scientists accepted the findings despite the apparently reliable data. Eventually, the scientific community's skepticism was justified when a faulty clock was discovered, which had inaccurately timed the neutrinos' journey.

What a relief!

The Gran Sasso incident teaches us that nothing can be definitively **falsified**, as we cannot be sure our theories regarding the instruments we employ are correct. How can we be certain that the various models forming global climate predictions are accurate? How can we verify the foundational axioms of mathematics, particularly when Gödel reminds us that we cannot ascertain self-consistency from mathematics alone?

Enter Lakatos.

For Lakatos, a scientific theory (or “research program”) consists of a sphere with two layers. It contains **a “hard core” of statements that are treated as axiomatic** and remain unquestioned unless absolutely necessary, alongside **a “protective belt” of hypotheses that can be flexible and adjusted to align with the data**.

Take Newtonian physics as an example: the hard core includes Newton’s three laws of motion and the law of universal gravitation, while the protective belt encompasses all other factors (such as the masses, positions, and velocities of objects) that enable the Newtonian framework to function.

A historical application of Lakatosian thinking can be seen in the discovery of Neptune. Newtonian physicists noted that Uranus's orbit deviated from the expected elliptical path, prompting questions about Newton's theories. Should they discard **Principia** and start anew?

Instead, physicists chose to protect the hard core—Newton's laws and gravitational theory—believing that an external factor must be influencing Uranus's orbit. By analyzing the observed perturbations, they hypothesized the existence of an undiscovered planet.

Subsequently, two astronomers, Urbain Le Verrier of France and John Couch Adams of England, identified the new planet that accounted for Uranus's wobble—this planet was named **Neptune**.

Hooray! Newtonian mechanics were vindicated, and Newton's research program continued.

Until, centuries later, quantum mechanics fundamentally challenged the basic axioms of Newton's hard core.

## Axiomatically Addressing Problems

Science is a complex endeavor, as are philosophy and mathematics. None of these disciplines are as neatly defined as we might believe; they all rest upon certain assumptions that are often difficult to prove and sometimes even harder to disprove.

Among philosophers of science, I find Lakatos particularly compelling. His notion of axiomatically addressing issues that could otherwise be problematic, while employing more adaptable and lower-stakes hypotheses to safeguard these core facts, seems to accurately represent the nature of scientific inquiry. However, Lakatos's true strength lies not only in his description of science but also in his insights into human intellectual behavior and cognition.

Consider religion, a frequent subject of my reflections. Isn't it true that some individuals regard the existence of a deity as axiomatic, forming part of their “hard core,” and support this belief with a flexible set of **ad hoc** hypotheses, such as “God is mysterious,” “God is love,” or “God is just”?

Or consider economics. The foundational tenets of Marxism may constitute the hard core for some revolutionaries, who might defend these principles by constructing a protective belt to address any critiques. A common protective hypothesis might be, “They were not **real** Marxists!” to justify the core tenets of communism.

I do not intend to criticize the religious or Marxist perspectives; instead, I highlight that this pattern is a universal aspect of how we all engage in science, philosophy, and mathematics. We articulate the Peano axioms in mathematics and proceed from that foundation, despite Gödel's assertion that they will always remain insufficient and incomplete (as will any alternative axioms we might select). We presume that induction will hold true in science because we axiomatize the uniformity of nature. We cherish our favorite philosophies and philosophers—our hard cores—and shield them from criticism.

When we axiomatically address problematic aspects, construct our intricate castles in the sky, or cease driving piles into the swamp indefinitely, we are not committing a philosophical transgression.

We are simply embodying the human experience in an inherently perplexing world.