The Mathematician's Perspective on Infinities: A Closer Exploration
When we dive into the realm of infinities, particularly from the viewpoint of mathematicians, we encounter a range of interpretations and discussions that deepen our understanding of mathematical principles.
Dependence on the Mathematician's Approach
The way a mathematician perceives and works with infinities often depends on their personal philosophy and methodological approach. For instance, someone like Gottfried Wilhelm Leibniz had a very dear and fundamental standpoint on this topic. Leibniz believed that for anything to be considered as a legitimate reality, it must be grounded in provable evidence. This is a viewpoint that extends beyond just the mathematical community, but also into philosophical and theological discussions.
An Illustration from Leibniz
Leibniz's famous quote, "The best of all possible worlds", reflects his desire for things to be impeccable and absolutely provable. However, in the case of infinity, he saw a significant challenge. If infinity is to be considered a reality, it must be proven not just logically, but through methods that do not fall into the realms of philosophy or theology. This can be a daunting task, which brings us to another key point of discussion.
Challenges in Proving Infinity
Infinity, by its very nature, is inherently unprovable. If we attempt to prove something infinite, it would require either infinite time, which is practically impossible, or it would necessitate philosophical and theological leaps. This makes the idea of infinity in mathematics a complex and somewhat controversial topic.
Historical Insights
It is interesting to note that ancient civilizations such as the Mesopotamians, Egyptians, and Greeks discovered and exploited mathematical concepts primarily through practical applications rather than through the constructs of modern set theory. According to Thom Alexander and Archie Thom (1988), their mathematical understanding was often based on metrology and geometry. This is a significant departure from the first-order logic definitions we use today.
Relevance to Modern Mathematics
The contrast between ancient and modern approaches to mathematics raises questions about the validity and origin of our current mathematical constructs. The ancient methods, which were more empirical and practical, should give us pause for thought when considering the logical foundations of modern mathematics. This comparison highlights the need to carefully examine the true foundations of math and avoid placing overly romantic or irrational beliefs on simpler or more practical methods.
The Logical Validity Debate
Modern mathematics often takes a deductive approach, following what is known as the axiomatic method. However, some argue that mathematics is actually more inductive and empirical in nature, rather than deductive. This view challenges the prevalent belief that mathematics is solely the realm of pure, logical deduction. Instead, it suggests that mathematical operations may have been invented and discovered through practical necessity and empirical observation, similar to how ancient civilizations approached mathematics.
The Impact on Physics and the Academic Scene
The acceptance of unproven or vaguely philosophical concepts in mathematics has had a profound impact on the fields of physics. As Sabrina Hossenfelder points out in her book, the current state of physics is often labeled as "Lost in Math." This indicates a blind adherence to mathematical models that lacks empirical validation, leading to a type of philosophical beauty that is more akin to art than to science.
The Need for Caution
It is crucial to maintain a clear distinction between the map (the mathematical model) and the territory (the real world). When we confuse these two, it can lead to a misalignment between theory and reality, as seen in contemporary physics. This misalignment is partly due to the influence of unproven mathematical concepts that are treated as if they were concrete realities.
Conclusion
Overall, the true foundations of mathematics are more complex and empirical than often believed. Ancient civilizations discovered mathematical concepts through practical means, which adds a layer of complexity to our understanding of mathematical and scientific progress. This exploration of how insights from the past can inform our understanding of the present is crucial for the future development of both mathematics and physics.
By delving into these historical and philosophical perspectives, we can develop a more rigorous and balanced approach to the study of infinities and other theoretical concepts in mathematics and physics.