Hear me out. I know this is going to be tad bit long but I assure you this is thought provoking. ( I mean I still have a hard time convincing myself if this doubt is even well founded)
- Most fundamental shape in mathematics :
Firstly we don't know the most fundamental shape of all and consequently the dimension for existence. Now I say this because firstly a circle although a shape, is mathematically defined as a polygon of infinite sides, right?
- Cyclical nature of construction of a line:
Secondly, now when you want to construct a line in a single dimension, what you do is that you take a point in 2 dimensions of infinitesimal size and glue a bunch of points together, in order to amount to a given length( note that here when the size although infinitesimal is fixed one tends to replicate those sizes and then actually make it amount to that length), but when we see that the solid circle that we are calling our point here can't really be though of as the most fundamental shape, then we can actually see that this infinitesimal point of the smallest length comprehensible still can be called as an infinite sided polygon and therefore we can never really define the point of any dimension rather we would be stuck in cycles, and we can never really define a point because for that we would need some line segment and basically we can't define even an infinitesimal point, right?
PS: I am still very much confused if at all this thought is well founded, and if it is, what might this imply.
Your exploration touches upon interesting concepts in mathematics and philosophy. The idea that fundamental shapes might be elusive due to the infinite divisibility of points is reminiscent of Zeno's paradoxes, where motion and change are questioned through the division of space and time into infinitely small increments.
In mathematics, points are often considered as primitives, without further decomposition. However, your perspective brings attention to the challenge of defining a point without invoking other geometric concepts.
This philosophical questioning has led to various mathematical theories and discussions about the foundations of geometry, such as non-Euclidean geometries and set theories. It's a deep and ongoing exploration in the philosophy of mathematics.
While your thoughts might not provide a definitive answer, they reflect a curiosity and awareness of foundational questions in mathematics. Exploring these ideas further could lead to a richer understanding of the philosophical underpinnings of mathematical concepts.
Defining a point without relying on other geometric concepts poses a challenge because a point is inherently relational in geometry, and its understanding often involves reference to other elements. Analogizing, or drawing parallels between concepts, becomes a fundamental tool in grappling with this challenge. Analogies, akin to equations, allow us to explore the nuanced outcomes of interactions by substituting different elements into the relational structure.
In this context, analogies serve as a bridge, connecting the abstract notion of a point to more tangible geometric ideas. The diversity of outcomes, stemming from varied initial conditions, mirrors the versatility inherent in analogical reasoning. Much like equations with different inputs, analogies enable us to navigate through the manifold possibilities arising from the relationships between geometric elements.
Considering analogies as equations, the varied forms they can take highlight the adaptability of geometric concepts. Whether it's diverging into distinct shapes or converging to a common foundation, these outcomes contribute to a richer understanding of geometry. The interplay between common roots and diverse manifestations underscores the interconnected nature of geometric principles.
Therefore, the challenge lies not only in defining a point independently but also in appreciating the spectrum of relationships it can forge with other geometric entities. Analogies become a linguistic and conceptual tool that allows us to express and comprehend the intricate dance of geometry, showcasing both the shared rules governing interactions and the plethora of outcomes that arise from nuanced initial conditions.
Aryan Verma Syed Fahad Ahmed Yazen Alawaideh Robert A. Phillips
Mathematically speaking point is a geometric object with no dimensions.
Philosophically speaking we encounter the notion of "cyclical" or "recursive" definitions. We may define a point as having no dimensions, but then we may ask, "What is 'no dimensions'?" and find ourselves in a circular explanation.
We have to settle on maths because without points circle does not exist but without a circle visible point exists.
En mathématiques, la forme la plus fondamentale est souvent considérée comme l'axiomatique. Les axiomes fournissent la base logique à partir de laquelle d'autres propositions et théorèmes sont déduits. Quant à la définition d'un point, elle peut être vue comme un concept fondamental, bien que certains philosophes des mathématiques aient exploré des questions liées à la nature des points et à la possibilité de cycles infinis dans la définition. Cependant, dans le cadre mathématique classique, les axiomes jouent un rôle central pour établir une base solide.
Certainly! When considering analogies as equations, it's helpful to understand that equations represent mathematical relationships between variables. Similarly, analogies express relationships and proportions between different elements or concepts. In equations, you have symbols representing quantities and mathematical operations specifying their relationships. Analogies, on the other hand, use comparisons to highlight similarities or relationships between different ideas or objects. Both equations and analogies serve to convey a connection or proportionality, but they do so in different contexts – mathematical versus linguistic or conceptual.
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