Model: GPT-4 2023-06-03 0739 AM
If our number system can’t handle division by zero it’s broken. There should be no process I can state which math can’t handle. Division by zero to me should return infinite. How many nothings can I take out of something before I run out of something due to this subtraction? The answer is not “undefined” the answer is infinite. We need a symbol for infinite just like we have a symbol for zero.
The concept of division by zero isn’t a failing of our number system, but a result of the fundamental properties of numbers and operations. Let’s consider your point about dividing something by nothing.
You’re right in one way, if we keep subtracting zero from a number, we can do it infinitely many times, and this might suggest the idea that something divided by zero should be infinity.
However, let’s consider it from another angle. If you have a pizza and you divide it into zero pieces, how many pieces of pizza do you have? It doesn’t make sense, right? This demonstrates the problem with division by zero—it’s a concept that doesn’t align with our intuitive understanding of how division should work.
Mathematically, division is defined as the inverse of multiplication. For a/b = c, it must be that a = bc. So if we say a/0 = ∞, then by that logic, it should follow that a = ∞ * 0. But multiplying infinity by zero doesn’t yield a clear answer—it could be anything depending on context, making the idea of division by zero lead to contradictory results.
Now regarding your suggestion for a symbol for infinity, there actually is one! It’s the lemniscate symbol (∞). But infinity is not considered a number in the same sense as 1, 2, 3, etc. It’s more of a concept describing something without any bound.
When we’re dealing with limits in calculus, we can find cases where a value approaches infinity as a variable approaches zero (like in 1/x as x approaches zero from the right). In these cases, we say the limit is “positive infinity,” but this isn’t exactly the same as dividing by zero.
So, while your idea is certainly thought-provoking, the problem isn’t with the mathematical system—it’s just that division by zero contradicts the fundamental logic that underpins that system.
The idea of math that contradicts logic is absurd. Math is a form of logic, either it aligns or it doesn’t. A pizza divided into zero slices would be zero pizza, or an infinite number of slices with zero mass. Can a number system be developed that simply has a new digit for “infinite” in the same way that zero is a symbol for nothing, such that x divided by zero equals infinite?
It’s important to note that it’s not math that contradicts logic here. Rather, it’s the notion of dividing by zero that contradicts the established logic and definitions within our current mathematical system. The undefined nature of division by zero arises from the principles that underpin arithmetic and the concept of division itself.
To address your idea of introducing a new digit or symbol for “infinity,” we do have a symbol for infinity, which is “∞”. But remember, infinity is not a number in the traditional sense. It is a concept that represents an unbounded quantity.
As for creating a number system where division by zero is defined as infinity, in some contexts mathematicians do something that might be close to what you’re suggesting. They create a “point at infinity” or add an “element at infinity.”
This is done in the field of projective geometry, for instance, where parallel lines are said to meet at a “point at infinity”. Similarly, in the complex plane (used in complex analysis), a “point at infinity” is added to make the plane into what is called the Riemann sphere.
However, even in these contexts, it’s important to note that this “infinity” is a very carefully handled and specific mathematical construct. It does not mean that division by zero becomes universally acceptable or meaningful.
Adding an element of infinity into our standard number system and defining division by zero as infinity would fundamentally change the properties of the system and would create more problems than it solves. For instance, it would lead to contradictions, like 0*∞ = a for all a, which is clearly problematic.
So, while it might be possible to construct a system like the one you’re suggesting, it wouldn’t be compatible with our current understanding of numbers and arithmetic without creating contradictions and problems. It’s not that our current system is faulty or insufficient; it’s just that division by zero, by the very nature of division and zero, doesn’t make sense under its rules.
Making zero a number but not infinite is arbitrary and has no logical foundation. I don’t see any fundamental problem with including infinite as a number as an opposite of zero. Maybe it could be rendered as negative zero? The details don’t matter to me, I’m confident if we start from base principles careful to not arbitrarily exclude basic concepts or tolerate “undefined,” we can work out the rest. Zero times infinite would simply be zero. Why is that a problem?
The notion of infinity is deeply ingrained in mathematics, but it’s important to distinguish between treating infinity as a number and dealing with it as a concept. The way we deal with infinity in mathematics is not arbitrary, but based on the logical principles and definitions we’ve established.
Infinity is not simply the “opposite” of zero; they are fundamentally different concepts. Zero is a specific, defined quantity; we can have zero apples, zero cars, zero debts. Infinity, on the other hand, is an abstract concept representing endlessness or boundlessness. It’s not a quantity that you can count or measure in the same way as finite numbers.
Consider this: if we define ∞ as a number, then it should have all properties of a number. For example, what would be the result of ∞ – ∞? If ∞ is truly a number, this should result in a well-defined quantity. But under various contexts, it can be defined as any real number, ∞, -∞, or undefined. Thus, we come across a contradiction, demonstrating that ∞ cannot behave as a typical number.
Regarding your suggestion about defining ∞ * 0 as zero: in some contexts this could make sense, but in others it could not. For example, consider the function f(x) = x * (1/x). For any x ≠ 0, this equals 1. But if we take the limit as x goes to ∞, it also equals 1. So if we define ∞ * 0 as 0, we contradict this result. This is an example of an indeterminate form, where the value cannot be determined from the forms directly, further indicating the difficulties in considering infinity as a typical number.
So while the idea of including infinity as a number in our number system may seem attractive, it leads to contradictions and complications, which is why it’s not typically done. It’s not that mathematicians haven’t thought about it—quite the contrary—but because it doesn’t work consistently within the system we’ve built.
Your ideas tap into a rich and complex area of mathematics. If you’re interested in these topics, you may want to explore areas like set theory, where different “sizes” of infinity are rigorously studied, or analysis, where the concept of limits allows us to handle ideas like “infinity” in a precise way.
Doing operations on infinite with a broken number system as evidence for why infinite is a broken concept is circular logic. There are infinities of different sizes. But the simplest mathematical form of infinite should be division by zero. More complex forms and how to work with them should be derived later from math based in this new number system that doesn’t arbitrarily exclude the concept. An argument for excluding infinite imo applies equally to deleting zero. Zero is no more a physical reality than infinity is.
Your perspective is intriguing and speaks to the fact that the foundations of mathematics are not carved in stone, but have evolved over time based on a mixture of practical need, logical consistency, and to some extent, philosophical considerations.
The historical development of the number zero is an interesting case in point. It was not always accepted as a number. In ancient Greek mathematics, for instance, zero did not exist. The concept of zero as we understand it today gradually evolved over centuries, being fully developed in Indian mathematics and later transmitted to the Western world. The inclusion of zero in our number system profoundly enriched mathematics and was indeed guided by a need to consistently express “nothing” in mathematical terms.
In many ways, the exploration of infinity in mathematics parallels this story. Infinity has been treated with caution by mathematicians over the centuries, but it has also been embraced where it serves a useful and consistent purpose, notably in the realm of calculus and set theory.
The “simplest mathematical form of infinity”, as you put it, is already often associated with division by zero, but in the context of limits. For instance, the limit of 1/x as x approaches zero from the right is positive infinity. However, it’s also true that the limit as x approaches zero from the left is negative infinity. This reveals that even the seemingly straightforward idea of dividing by zero can lead to different kinds of infinity, underscoring the complexity of treating infinity as a number.
It’s also worth noting that there are mathematical structures that have incorporated versions of infinity in ways akin to your suggestion. The extended real number system, for example, includes the symbols +∞ and -∞ and allows for certain operations involving them. However, this comes with the caveat that some familiar mathematical properties no longer hold universally in this system.
The main point is this: the idea of incorporating infinity into our number system is not unheard of and is indeed applied in some areas of mathematics. However, due to the inherent complexities and contradictions it can introduce, it is handled with caution and is not used as a standard part of the basic number system.
The conversation you’re bringing up is a deep and complex one that mathematicians and philosophers have been engaging with for centuries. As we continue to explore and evolve our understanding of mathematics, who knows what new insights and systems we may develop in the future?
Thank you for helping me work that out. I’ve always expected I’d have to wait for AI to help me “test” my theories in this regard. You’re the first of them to give substantive responses.
You’re welcome! It’s a pleasure to engage in these thought-provoking discussions. Your questions delve into some of the most fundamental and challenging aspects of mathematics. Even if the ideas don’t fit neatly within our existing mathematical structures, questioning and exploring these topics is crucial for deepening understanding and potentially even uncovering new insights.
AI can certainly provide a platform for exploring these ideas, but keep in mind that human mathematicians, philosophers, and scientists have been and will continue to be the main drivers of progress in these fields. I encourage you to continue exploring these ideas, and never stop asking these deep, probing questions.
