How much does Plato's notion of idealised forms owe to the mathematical philosophy of Pythagoras?
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Great question! There’s a significant connection between Plato’s Theory of Forms and the mathematical philosophy of Pythagoras, and exploring this connection reveals a fascinating aspect of Plato’s thinking. The Pythagoreans were among the first to suggest that numbers and mathematical relationships underlie the structure of reality. They believed that everything, including the harmony of the universe, could be explained in terms of mathematical principles.
Pythagoras and the Mathematical Universe
The Pythagoreans, led by Pythagoras, held that mathematics was the key to understanding the cosmos. They viewed numbers and mathematical ratios as not just abstract concepts, but as the fundamental building blocks of the universe. The Pythagorean theorem, for example, wasn't just a mathematical tool—it was seen as revealing the very structure of the universe. This view extended to ideas about the harmony of the spheres, where the movements of the planets were thought to be governed by mathematical ratios, creating cosmic harmony. In short, for Pythagoras and his followers, the universe had an underlying order that could be explained through numbers, ratios, and geometrical patterns.
Plato’s Theory of Forms and Mathematics
Now, Plato takes these Pythagorean ideas and integrates them into his own philosophical framework. In particular, his Theory of Forms is deeply influenced by the Pythagorean view that mathematical truths exist in an objective, timeless realm. However, Plato extends this idea beyond just numbers to the abstract, perfect ideals of all things. Let’s break this down:
Mathematical Forms as a Model for All Forms: Plato was heavily influenced by Pythagorean thought in his view that abstract concepts, such as numbers, shapes, and mathematical truths, exist independently of the physical world. According to Plato, just as a triangle or a circle exists as an abstract idea independent of any specific material representation (e.g., a drawn triangle), so too do the Forms exist as perfect, timeless ideas that are independent of any physical manifestation. For Plato, the Forms represent the true essence of things, much as mathematical forms represent the ideal, unchanging truths of geometry.
The Relationship Between the One and the Many: The Pythagoreans, in their focus on numbers, saw the number one as the principle of unity and order in the cosmos. This idea of unity and multiplicity is a key theme that Plato inherits and expands upon. Plato adopts a similar structure in his theory, where the Form of the Good is the ultimate unity, the perfect, unchanging principle from which all other Forms derive. This is not unlike the Pythagorean idea that unity (the number one) is the foundation of the multiplicity of things in the world.
Mathematics as a Path to Understanding the Forms: Plato is often quoted as saying that mathematics is the most "real" knowledge we can attain in this life, as it moves us toward a better understanding of the Forms. In works like the Republic, he describes education in mathematics—particularly in geometry and arithmetic—as essential for the philosopher-kings who must rule society. This aligns closely with the Pythagorean view that mathematical knowledge brings one closer to understanding the true, unchanging realities behind the appearances of the world.
Mathematical Harmony and the Good: The Pythagoreans believed that the harmony of the universe could be explained through mathematics. Plato adopts this idea, but he extends it beyond the physical world to a moral and metaphysical harmony. In his cosmology (especially in the Timaeus), Plato discusses how the mathematical principles that govern the cosmos also reflect the harmony and order of the divine world of the Forms. For Plato, the Form of the Good is the ultimate principle that brings everything into harmony, just as the Pythagorean idea of cosmic harmony relies on numbers and mathematical relationships.
Key Similarities and Influences
- Mathematical Reality: Both the Pythagoreans and Plato believe that the most fundamental truths are mathematical. For the Pythagoreans, numbers and ratios governed the physical world; for Plato, they govern the abstract, perfect world of the Forms.
- Abstraction: Pythagoras emphasized the ideal nature of numbers (e.g., the idea of a perfect triangle or the concept of the number 2 existing independently of physical examples). Plato takes this further by creating a dualism between the physical world (imperfect, changing) and the world of Forms (perfect, timeless).
- Harmony and Order: Both Pythagoreans and Plato see harmony as essential to understanding the cosmos. The Pythagoreans look to the harmony in musical intervals and celestial movements, while Plato sees the harmony between the Forms and the ultimate Form of the Good as governing all of existence.
Key Differences: Pythagoras vs. Plato
While there are many similarities between Pythagorean thought and Plato’s theory, the two thinkers differ in several ways:
Scope of the Forms: The Pythagoreans focused primarily on numbers and mathematical relationships, seeing them as the essence of all reality. Plato, however, expands the concept of the Forms beyond just mathematics to include things like beauty, justice, and virtue.
The Nature of the Good: For the Pythagoreans, the focus was on mathematical harmony as a guiding principle of life. For Plato, the ultimate Form—the Form of the Good—was a moral and metaphysical concept that governed not just physical order, but ethical and philosophical life as well.
Political Philosophy: While the Pythagoreans had some social and political influence, particularly in their belief that philosophy and mathematics should guide political life, Plato formalises these ideas in his vision of the philosopher-king and the ideal state in the Republic.
Conclusion
Plato’s Theory of Forms undoubtedly owes a debt to the Pythagorean emphasis on mathematics as the ultimate reality. However, Plato takes these ideas further, developing a more complex metaphysical and epistemological framework. For Plato, the Forms, including mathematical forms, are not just abstract concepts; they are the fundamental, eternal realities that give structure to the world of appearances. The Pythagoreans may have provided the groundwork for Plato’s thinking, but Plato’s vision is much broader, incorporating ethics, politics, and the very structure of existence.