The Existence of the Pythagorean Theorem: A Philosophical Dilemma

The relationship among the sides of a right triangle has always existed, independent of human minds. However, it was the advent of mathematical proofs that gave it a concrete form and understanding. This raises a philosophical question: did the Pythagorean theorem exist before an intelligent mind formally proved it, or did it only come into existence with such a proof?

The Nature of Mathematical Truth

The nature of mathematical truths, especially theorems like the Pythagorean theorem, is a topic of much debate. From an idealist perspective, as advocated by philosophers like Plato, mathematical truths exist independently of human perception. For a Platonic idealist, the properties of a right triangle, including the theorem stating that the square of the hypotenuse is equal to the sum of the squares of the other two sides, are eternal and more real than the ephemeral physical world. However, this viewpoint is not what I support. I argue that all mathematical concepts and their properties are inventions of the human mind. Some theories may be quickly forgotten, while others like the Pythagorean theorem have endured across millennia, highlighting their dependence on the minds that understand and use them.

Existence of theorems and Proofs

A theorem's existence is tied to its proof. From my perspective, a theorem truly exists when a proof is provided by an intelligent mind. Before this, it can be considered a conjecture if someone questions its validity. Consider the question, 'Does there exist a function that assigns provability to every possible statement within a mathematical system?' This problem is known as the Entscheidungsproblem, and it was ultimately proven that such a function does not exist, thanks to the works of mathematicians like Alan Turing.

The Independence of Mathematical Truth from Consciousness

One might argue that the proof, not the theorem itself, is the core of its existence. Indeed, the theorem's truth or falsehood is independent of any conscious mind. The relationship between the sides of a right triangle is an eternal truth of the physical world, even if it is not immediately apparent. However, the theorem only becomes formalized and recognizable when stated by an intelligent mind. This was the case with the Pythagorean theorem, which was stated and proved by the ancient Greek mathematician Pythagoras, and its eventual dissemination across the world.

It is also important to consider the nature of physical space. Space is only flat when it is empty, but the physical world is far from empty. Therefore, in the physical world, the theorem is more of a guideline for our expectations rather than a reflection of reality. Our expectations and understanding of the theorem only became apparent with its formalization and proof.

A Case for Mathematical Existence independent of Mind

While I acknowledge the human-centric nature of mathematical proofs, I also believe there is an underlying truth waiting to be discovered. The Pythagorean theorem is just one of countless mathematical truths that exist beyond human discovery. Although we may never discover all of them, the idea of mathematical truths existing independently of human minds is both appealing and profound. The theorem may not be a true reflection of the physical world, but it is a testament to the power of human intellect and its ability to uncover eternal truths.

Conclusion

While the question of whether the Pythagorean theorem existed before an intelligent mind proved it may be a philosophical cold comfort, it highlights the fascinating interplay between human intelligence and mathematical truths. The theorem itself is a testament to the enduring power of human thought, and its formalization and proof allow us to better understand the world around us.