The Question That Has Divided Philosophers and Mathematicians for Centuries

In the early 1800s, astronomers carefully observed the motion of Uranus. According to the laws of Issac Newton, Uranus should have followed a predictable path around the Sun. But there was a problem. Uranus was not exactly where Newton's equations said it should be. The deviations were tiny, but they were real. Astronomers wondered: Was Newton's theory wrong? Were the observations inaccurate, or was something else affecting Uranus? Two mathematicians, John Adams and Urbain Le Verrier, independently tackled the problem using mathematics. They reasoned that if an unknown planet exists beyond Uranus, its gravity could be pulling Uranus slightly off course. Working backwards from the observed deviations, they calculated the approximate mass of the unknown planet, its distance from the sky, and where it should appear. It's astonishing to see that no one had seen the planet, no telescope had detected it, and it was completely invisible to human eyes. In 1846, astronomer Johann Gottfried Galle pointed a telescope at the location predicted by Le Verrier. Neptune was found less than a degree from the predicted position.

The story reveals that the Newton equations were originally invented to describe the motion of falling apples, cannonballs, and planets. Yet those same symbols on paper revealed the existence of a world that nobody knew existed. Mathematics was not merely describing observations after the fact; it was uncovering a hidden part of reality.

It raises a philosophical question.

Did mathematicians invent these truths, or were they discovering something that already existed?

Humans created numbers and invented symbols like +,-,*, and =. Mathematics is a language we designed to describe patterns. Just as humans invented English, they invented mathematics, but mathematical truths existed before humans. For instance, even if no humans existed, 2+3 = 5 would still be true. Likewise, the value of π would still describe the circles. It suggests that mathematics is an objective feature of reality.

Nature seems to obey mathematics, even though trees, stars, and animals don't consciously know mathematics. Complex numbers, once considered purely imaginary, became essential in quantum mechanics. Non-Euclidean geometry, developed without a practical purpose, later became central to the theory of gravity.

Maxwell developed equations describing electricity and magnetism. While studying his equations, he noticed something unexpected. The equations predicted electromagnetic waves travelling at a specific speed, and that speed turns out to be the speed of light. It means light is also an electromagnetic wave. Nobody had planned this discovery. The mathematics revealed a fact about reality before experiments confirmed it. The equations knew something that the scientists didn't.

Let's revisit history to see how ancient people thought about mathematics.

Before the Greek civilisations, such as the Egyptians and Babylonians, used mathematics mainly for practical purposes like measuring land, building monuments, collecting taxes, and astronomy. They knew many mathematical techniques, but they rarely asked why mathematical rules were true. The Greeks transformed mathematics into a logical system based on proof. Instead of simply knowing that a rule worked, they wanted to demonstrate why it must always work.

Pythagoras founded a philosophical and religious community in southern Italy. His followers believed mathematics was the key to understanding the universe. They were not merely mathematicians; they formed a way of life centred on numbers. They believed everything in the universe is fundamentally based on numbers. They noticed numerical patterns everywhere, like musical harmony, geometry, astronomy, and architecture. Numbers were not just tools to them. In fact, they thought of them as an essence of reality.

One of the greatest discoveries involved musical instruments. If a vibrating string is shortened by half, two-thirds, or three-fourths, it produces an octave, a perfect fifth, and a perfect fourth, respectively. Simple numerical ratios produce harmonious sounds. It made them believe that beauty and harmony arise from mathematics. The Pythagoreans even gave symbolic meanings to numbers. For instance, 1 is unity, 2 is diversity, 3 is harmony, 4 is justice, and 10 is perfection. They regarded number 10 as sacred because 1 + 2 + 3 +4 = 10. Their fascination with numbers combined mathematics, philosophy, and mysticism.

Irrational numbers came as a surprise to Pythagoreans. Everything changed when they studied the diagonal of a square. Imagine a square with a side length of 1. Using the Pythagoras theorem, a² + b² = c². The length of a diagonal is √2. The astonishing discovery was that √2 can't be written as a ratio of two whole numbers. It means that not every quantity can be expressed by simple ratios. This was a profound crisis for all the Pythagoreans, who believed all reality was built from whole number relationships. This discovery was one of the greatest turning points in the history of mathematics because it showed that reality is richer than anyone had imagined.

Plato was one of history's greatest philosophers and a student of Socrates. He deeply admired mathematics and believed it revealed eternal truths. Plato proposed that there are two worlds. One is the physical world. The world we see around us includes trees, mountains, animals, and people. Everything here changes and eventually perishes. The other is the world of forms. This is an invisible, perfect realm containing an ideal version of things. For instance, every circle drawn on paper has imperfections, but the perfect circle exists as an ideal form.

Mathematics was not invented, according to Plato. It already exists in the perfect realm. When any mathematician proves a theorem, he discovers a truth that has already existed. This is known as mathematical platonism. Plato argued that our senses deceive us, but mathematics is known through reason. A stick may look bent in water, but mathematical truths never change. For example, the angles of a Euclidean triangle sum to 180°. These truths are independent of opinion or observation.

Plato's philosophy greatly influenced later scientists. If nature follows mathematical laws, then by studying mathematics, we can understand nature. This idea inspired generations of mathematicians and physicists.

Archimedes and Galileo Galilei transformed mathematics from a purely intellectual pursuit into the foundation of modern science. They made mathematics seem magical because they could predict nature with astonishing precision.

Archimedes is presented as the greatest mathematician of the ancient world. He once helped a king determine whether his crown had been adulterated with a cheaper metal. The king suspected a goldsmith had mixed silver into the crown. Archimedes discovered that when an object is immersed in water, it displaces a volume equal to its own volume. By measuring displacement and comparing densities, he could determine whether the crown was pure gold.

His famous statement, "Give me a place to stand, and I will move the Earth," refers to the mathematics of levers. He showed that mathematics could explain why things happen. For instance, a lever balances because of a precise mathematical relationship between force and distance.

Galileo created modern science by challenging both ancient and religious authority. For nearly 2000 years, people believed Aristotle's claim that heavier objects fall faster than lighter objects. People accepted it because Aristotle was regarded as the greatest philosopher. Galileo noticed a fallacy in Aristotle's claim through an ingenious thought experiment.

Following Aristotle's claim, a 10kg stone falls faster than a 1kg stone. The heavier the object, the faster it falls. Galileo asked, what happens if we tie the two stones together. Suppose a heavy stone falls fast and a light stone falls slowly. Now tie them together with a string, which creates a new object. Looking at it from the perspective of light stone. Since the light stone falls more slowly, it should hold back the heavy stone. So, the combined object should fall more slowly than the heavy stone alone. According to Aristotle's idea, a heavy stone plus a light stone is slower than the heavy stone. Now look at it from the combined object perspective. The two stones together make an even heavier object. If heavier objects fall faster than the light stone. The heavy stone plus the light stone should fall faster than the heavy stone alone.

The new object(heavy stone + light stone) can't be slower or faster than the heavy stone at the same time. It means the original assumption that heavier objects fall faster is inconsistent. Ignoring air resistance, all objects fall with the same acceleration, regardless of their mass.

Galileo's most famous idea is that the universe is written in the language of mathematics. He argued that without mathematics, we can't truly understand nature. Observations alone are not enough. We must express natural laws mathematically. He shaped the structure of Science through insistence on experiments, observations, mathematical equations, and evidence.

Why do equations written on paper match the behaviour of the Universe?

Archimedes and Galileo didn't answer this question. They demonstrated that it is true.

René Descartes also followed the footsteps of Galileo. He doubted everything, and he realised one thing that can't be doubted. Even if he was being deceived, he must exist to be deceived. Thus, he concluded, I think, therefore I am. He unified two completely different branches of mathematics, geometry and algebra. Algebra dealt with equations, and geometry dealt with shapes. People considered them separate subjects. Descartes showed that they are actually the same.

Imagine a point on a sheet of paper. Instead of describing it verbally, assign it coordinates: (x,y). Every point can now be represented by numbers. Likewise, every equation corresponds to a geometrical curve. For example, x =  y² is not merely an equation. It is the parabola it defines. Geometry becomes algebra and algebra becomes geometry. This is one of the greatest conceptual breakthroughs in mathematics.

Using analytic geometry, curves could be derived with equations. It implies that the physical motion could be written mathematically. Without Descartes, Newton's achievement would have been far more difficult. It demonstrates that scientific revolutions often build on earlier conceptual revolutions.

If Descartes gave mathematics a new language, Newton gave it an extraordinary power. Seventeenth-century scientists understood many individual facts, like objects fall, planets orbit around the sun, and the moon circles Earth. Nobody possessed a single mathematical framework explaining all these motions. Newton wanted one set of laws governing the heavens and the Earth. He realised ordinary arithmetic and geometry were inadequate for describing continuously changing quantities. He developed calculus to answer questions like:

How fast is an object moving at any instant?

How does velocity change?

How can we calculate the path of a planet?

Calculus became the mathematics of change. This is one of the greatest intellectual achievements in history.

Newton then discovered that the same force causing an apple to fall also keeps the motions in orbit. He expressed it through the laws of universal gravitation. The same rule applies to heaven and earth, and the mathematics governing both of them would be the same. He unified nature in an unprecedented way. He didn't just describe observations. His mathematics predicted phenomena. Scientists could calculate planetary positions, comet orbits, tides, and eclipses before observing them. The predictive power strengthened the idea that mathematics reveals genuine features of reality.

Descartes and Newton invented new mathematical methods to uncover patterns in nature that seemed to have been there all along. These methods discovered the objective truths about the universe with astonishing accuracy.

For centuries, scientists believed that if enough information were available, every event could be predicted exactly. This view reached its peak with Issac Newton. According to Newtonian physics, every object follows exact laws, and every event has a definite cause. In principle, if you knew all positions and velocities, you could calculate the future. The universe was often compared with a perfectly running clock, but everyday life suggested something different.

Why is the weather difficult to predict?

Why can't we predict exactly when someone will get sick?

Why do insurance companies rely on averages rather than certainty?

These questions led to probability. It is the mathematics of uncertainty. It doesn't predict the outcome of a single event with certainty. However, it tells us how likely different outcomes are. It provides certainty about the distribution of possibilities even when individual events remain uncertain.

Modern probability began in 1600 with gamblers asking practical questions. Blaise Pascal and Pierre De Fermat played a key role. They analysed the game of chance and showed that seemingly random events could be understood mathematically. This is also a great intellectual achievement because randomness had often been viewed as beyond mathematical analysis.

Statistics differ slightly from probability. Probability starts with a mathematical model and predicts likely outcomes, whereas statistics starts with data and tries to infer the underlying pattern. Imagine measuring the heights of 10,000 people. No two people are the same, yet when all measurements are plotted, they form a smooth bell-shaped curve. It is also known as the normal distribution. Many characteristics naturally cluster around an average, like height, blood pressure, measurement errors, etc. Most observations lie near the centre, while very high and very low values are rare. The bell curve demonstrates that random individuals can collectively display remarkable mathematical order. For example, gas molecules move randomly, yet gases obey precise physical laws.

Probability became indispensable in many fields. In classical physics, particles were thought to have exact positions and motion. With the development of quantum mechanics, scientists such as Werner Heisenberg and Niels Bohr showed that at the quantum level, only probability can be predicted. For instance, quantum theory predicts the probability of where an electron may be detected rather than an exact path. This challenged the earlier belief that complete certainty was always possible.

In biology, probability played a key role in explaining inheritance. Gregor Mendel chose pea plants to formulate the laws of inheritance.

How did he formulate these laws?

Mendel first grew plants that always produced the same trait generation after generation. For example, tall × tall will always be tall and dwarf × dwarf will always be dwarf. These are called true-breeding pure lines. Their genetic make-up is Tall = TT and dwarf = tt.

Normally, pea plants self-pollinate. Mendel prevented this by removing the male parts from one flower. He collected pollens from another plant and placed them onto the female part(stigma). This ensured controlled crosses like TT × tt.

He observed the first generation. All of them were Tt. They were all tall, and no dwarf plants appeared. This showed that tall is dominant and dwarf is recessive. These are known as the first-generation plants represented by F1.

He self-pollinated the F1 plants. The possible combinations are Tt, tT, TT, tt. The genotype ratio is 1:2:1, and the phenotype ratio is 3 tall and 1 dwarf plant. He observed the 3 : 1 ratio in thousands of pea plants. He gave two laws:

1. Law of Segregation

2. Law of independent assortment

Law of Segregation: Each organism has two alleles for a trait. During the formation of reproductive cells(gametes), the two alleles separate, so each gamete receives only one allele. At fertilisation, the offspring receives one allele from each parent. This explains why recessive traits can disappear in one generation and reappear in the next.

Law of independent assortment: Mendel also studied two traits at the same time. For example, seed shape(Round / Wrinkled) and Seed color(Yellow/Green). A parent with genotype RrYy can produce four types of gametes, RY, Ry, rY, ry, with a probability of 25%. When two such plants are crossed, the offspring show the famous 9:3:3:1 phenotype ratio. 9 Round Yellow, 3 Round Green, 3 Wrinkled Yellow, and 1 Wrinkled Green. This demonstrated that inheritance of one trait is genetically independent of another, provided that they are not linked.

Before Mendel, many scientists believed inheritance worked by blending, where offspring were simply an average of their parents. Mendel corrected them by explaining that hereditary factors(now called genes) remain distinct and are passed on in unpredictable ways.

Probability is not a sign of mathematical weakness. It's an extension of mathematics that allows us to understand systems where exact prediction is impossible or impractical.

Is geometry fixed or dependent on the nature of space?

Around 300 BC, Euclid wrote the Elements, one of the most influential books ever written. Euclid built geometry from definitions, common axioms, and five postulates. Using them, he proved hundreds of theorems. For centuries, people believed Euclidean geometry described not only mathematics but also the structure of physical space itself.

The four postulates are simple. For example, a straight line can be drawn between two points, and a circle can be drawn with any centre and radius. The fifth postulate, however, was much more complicated. It says, given a line and a point not on that line, there is exactly one line through the point that never intersects the original line. Although it seems reasonable, it is much less obvious than other postulates.

Why was the fifth postulate a problem?

For nearly two thousand years, mathematicians suspected that the fifth postulate might actually be derivable from the first four. If it could be proved, Euclid's system would become even simpler. Many brilliant mathematicians tried and failed. The repeated failure gradually suggested something astonishing. Perhaps the fifth postulate is independent of the others.

Some mathematicians eventually asked: What if the parallel postulate is changed?

At first, it seems absurd. People believed geometry described reality itself. Changing a postulate appeared to mean denying reality, but mathematics cares about logical consistency. If a different set of axioms leads to a contradiction-free system, then that system is also a valid geometry.

Carl Friedrich Gauss, James Bolyai, and Nikoki Lobachevsky independently developed new geometries. Instead of saying only one parallel line, they assumed infinitely many parallel lines can pass through the point. This produced hyperbolic geometry, and everything remained logically consistent.

Later, Bernhard Riemann proposed another alternative. Suppose no parallel lines exist, which eventually gives elliptical geometry. Imagine walking on the surface of a sphere. The straight lines are great circles. For example, Earth's equator and Meridian. Two meridians start parallel near the Equator but eventually meet at the poles, yet the geometry is perfectly consistent.

In Euclidean geometry, the angles of every triangle add up to 180°. In hyperbolic geometry, the sum is less than 180°, and in elliptical geometry, it should be greater than 180°. Imagine a triangle on Earth, starting at the north pole, walking to the equator, and turning 90°. Walk along the equator and return to the north pole. Total sum of the angles becomes 90°+90°+90°=270°. It's a perfectly valid triangle, and it shows that the geometry depends on the nature of space.

This discovery transformed mathematics. Before this, people believed there was only one geometry. After the discovery, they are convinced that there can be equally valid geometries.

Which geometry describes our universe?

Riemman took the revolution even further. He suggested that space need not have constant curvature. Instead, different regions could have different curvatures. This idea eventually became essential for modern physics.

Many people initially viewed Non-Euclidean geometry as an abstract curiosity. Albert Einstein changed everything. In general relativity, gravity is not a force pulling objects through flat space. Matter bends space, and objects move along the curved geometry of space-time. The geometry developed by Riemann becomes the mathematical language of gravity. An idea once considered theoretical turned out to describe the real universe.

This discovery has corrected many long-held beliefs. The updated ideas are that mathematics is not limited to one geometry; axioms are choices, not necessarily self-evident truths; different mathematical systems can all be internally consistent; and observations determine which mathematical model best fits reality. It changed how mathematicians think about truth itself.

Imagine you are solving a geometry problem. You begin with axioms(basic assumptions), then use logic to derive conclusions. This process feels reliable because every step follows a rule. The logician asked,

What exactly is a proof?

What makes a proof valid?

Can every mathematical truth be proved?

Could mathematics ever contain contradictions?

These questions gave birth to mathematical logic. By the late 1800s, mathematics had become incredibly powerful, but discoveries exposed unexpected problems. For instance, infinite sets behave strangely, different branches of mathematics use different foundations, and some arguments seem to prove contradictions. Mathematicians worried: What if the entire subject rests on shaky foundations?

Logicians were trying to build a mathematical structure from a small set of unquestionable principles. The idea was to start with a few basic axioms, apply rules of logic, and derive every mathematical theorem. If they become successful, every theorem would be true, no contradictions would exist, and every mathematical question would have an answer.

David Hilbert believed mathematics should become perfectly rigorous. His vision had three goals:

1. Consistency: Mathematics should never prove both a statement and its opposite. For example, it should never prove 2 +2 = 4 and simultaneously prove 2 + 2 is not equal to 4. A contradiction would make the entire system unreliable.

2. Completeness: Every true mathematical statement should be provable. Nothing true should remain forever beyond proof.

3. Decidability: There should exist a procedure that determines whether any mathematical statement is true or false.

Hilbert believed mathematics could become a perfectly organised machine.

A formal system works like a game. For example, in chess, there are pieces, rules, and legal moves. Once the rules are fixed, every game follows them. Similarly, mathematics has symbols, axioms, and logical rules. Every proof is simply a legal consequence of logical steps. This idea suggested that mathematics could be treated almost mechanically.

As confidence was growing, Bertrand Russel uncovered a devastating contradiction. Imagine a barber with this rule: The barber shaves everyone who doesn't shave himself.

Does the barber shave himself?

If he does, he shouldn't.

If he doesn't, he should.

Both answers create contradictions. Russel found a similar paradox in set theory. This showed that naive assumptions about sets could lead to impossible conclusions. Mathematicians realised their foundations were not as secure as they had believed.

To avoid paradoxes, mathematicians developed more careful systems. They restricted how sets could be formed and rewrote the entire foundations of mathematics. The hope was that every contradiction could eventually be eliminated. For a time, it seemed this goal might succeed.

In 1931, Kurt Gödel published one of the most important results in the history of mathematics, Gödel's Incompleteness Theorem.

First Incompleteness Theorem: Gödel proved that in any formal system powerful enough to include arithmetic, there are true mathematical statements, but these statements can't be proved within that system. In other words, truth is larger than proof. No matter how powerful your logical system is, there will always be truths it can't establish.

Gödel's breakthrough relied on self-reference. He devised a way for mathematical statements to refer to themselves by assigning numbers to symbols and proofs. A simplified version of the key idea is: This statement can't be proved within this system.

If the system could prove the statement, it would be false. If the system couldn't prove it, then the statement would actually be true. This reveals a statement that lies beyond the system's proving power.

Second Incompleteness Theorem: Gödel went even further. He proved that a sufficiently powerful, consistent mathematical system can't prove its own consistency using its own methods. It means mathematics can't provide an ultimate internal guarantee that it is free from contradictions. To establish consistency, one must appeal to stronger assumptions outside the system.

Hilbert hoped mathematics would become a complete and self-contained logical structure. Gödel showed that this dream can't be fully realised. There will always be new questions, undecidable statements, and truths that require richer systems to understand. Mathematics is therefore open-ended, not a finished collection of facts.

If mathematical truths exist that can't be generated by formal rules, where do these truths come from?

Conclusion: Platonists argue that mathematical truths exist independently and are discovered. Formalists see mathematics as the manipulation of symbols according to rules, though Gödel showed that no single formal system can capture everything. Other philosophers view mathematics as a creation of the human mind while acknowledging that the surprising richness emerges from its structure. Each time mathematicians expand their foundation to solve previously unprovable problems, new questions appear beyond the horizon. The search for mathematical truth is therefore not a project with a final endpoint but an ongoing exploration.

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