Math Structures: Proving Existence w/ Accurate Solutions?

In summary, the conversation discusses the process of defining and showing the existence of new mathematical objects. It is mentioned that mathematicians do not typically say if things exist or not, but rather define objects and use axioms to further their study. The example of complex numbers is given, where they are defined as ordered pairs of real numbers with specific operations. It is also mentioned that eventually, the ZFC axioms may need to be referred to for existence, but otherwise, it is important to show the existence of objects in mathematics. Book recommendations are also requested.
  • #1
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Hey guys, let's say I were to define a new mathematical object, a novel type of number for example, and I am trying to determine its various properties (arithmetic, exponential, logarithmic, etc). Now, let's say I am able to use these numbers to produce solutions that agree with expected results. Would this lend any sort of weight towards the formal existence of said numbers, or is a formal proof required to be taken seriously?

I don't mean to be vague, but really, how does one "show" that a new type of number exists? How was it done for imaginary numbers? Were they just a novelty until someone saw that they had real world meaning in engineering?

Many thanks!

Edit: If still too general and/or vague, book recommendations will also be appreciated :)
 
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  • #2


I don't mean to be vague, but really, how does one "show" that a new type of number exists?

You define it. As long as your definition isn't internally contradictory, then go with it.
Mathematics studies structure; we define new structures all the time whenever it's convenient.

How was it done for imaginary numbers?

You mean complex numbers. They're defined as the set of ordered pairs (x, y) of real numbers with addition done component-wise and multiplication given by (a,b)(c,d) = (ac - bd, ac + bd). This definition doesn't give rise to any contradictions, so we're done.

The motivation behind the definition may be any number of things, but you can define any structure you want within the bounds of whatever logical system you're operating under.
 
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  • #3


Most mathematicians don't say if things exist or not. They define objects and axioms, and turn the wheel to further the math of these objects.
For example, I can define a shoe space to be a pair of collection of sets (A,B) such that any set in A is equal to the union of sets in B. But no set in B is equal to the intersection of sets in A. I can define operations on this shoe space like fart(A,m) = union of all sets in B excluding all sets containing m. I can turn the wheel and then make theorems. So as NumberNine said, you define new objects, you don't show they exist.
 
  • #4


lucid_dream said:
Most mathematicians don't say if things exist or not. They define objects and axioms, and turn the wheel to further the math of these objects.
For example, I can define a shoe space to be a pair of collection of sets (A,B) such that any set in A is equal to the union of sets in B. But no set in B is equal to the intersection of sets in A. I can define operations on this shoe space like fart(A,m) = union of all sets in B excluding all sets containing m. I can turn the wheel and then make theorems. So as NumberNine said, you define new objects, you don't show they exist.

Sure you show that they exist. What does "exist" mean in the first place? Physical existence? Then mathematics indeed does not show that things exist.

For example, I can easily define a group as a set with a multiplication such that some axioms are satisfied. But I do need to show that a group actually exists.

Eventually, you do need to refer to axioms (usually the ZFC axioms). But apart from those set theoretic axioms, you should always show existence of objects. It would be bad to make an entire theory about something that isn't even there.
 
  • #5


Hello, thank you for your question. I understand your curiosity about proving the existence of new mathematical objects. In mathematics, there are various methods for proving the existence of a new object, and it ultimately depends on the specific object and its properties.

In the case of a new type of number, one way to prove its existence is by constructing a mathematical structure that incorporates these numbers and shows that it is consistent and useful for solving problems. This can be done through axiomatic systems or by defining operations and properties of the new numbers.

However, simply producing solutions that agree with expected results is not enough to prove the existence of a new mathematical object. A formal proof is required to fully establish its existence and properties. This involves using logical reasoning and rigorous mathematical techniques to demonstrate that the new object is consistent and follows certain rules.

As for imaginary numbers, they were initially viewed as a novelty until mathematicians like Euler and Gauss showed their significance in solving problems in algebra and geometry. Through rigorous proofs and applications in various fields, imaginary numbers were eventually accepted as a valid mathematical concept.

In terms of book recommendations, I would suggest looking into books on mathematical proofs and logic, as well as texts on specific mathematical structures such as complex numbers or quaternions. It may also be helpful to consult with a mathematician or take a course in abstract algebra to gain a deeper understanding of mathematical structures and their existence.

I hope this helps answer your question. Keep exploring and questioning the world of mathematics!
 

What is meant by "math structures" in relation to proving existence with accurate solutions?

Math structures refer to the various systems and frameworks used in mathematics to organize and understand mathematical concepts. These structures can include sets, groups, fields, and other abstract mathematical objects.

Why is proving existence with accurate solutions important in mathematics?

Proving existence with accurate solutions is important because it provides a solid foundation for the validity and reliability of mathematical theories and concepts. It allows mathematicians to confidently make conclusions and predictions based on established proofs.

What methods are commonly used to prove existence with accurate solutions in mathematics?

There are several methods used for proving existence with accurate solutions in mathematics, including direct proof, contradiction, induction, and construction. Each method has its own advantages and is suitable for different types of problems.

How does mathematical rigor play a role in proving existence with accurate solutions?

Mathematical rigor refers to the level of thoroughness and precision in mathematical proofs. When proving existence with accurate solutions, it is crucial to follow strict logical reasoning and provide clear and concise arguments to ensure the validity of the proof.

Can proving existence with accurate solutions ever be considered 100% certain?

While mathematical proofs strive to provide absolute certainty, there is always a possibility of human error or assumptions made in the proof that may later be disproven. However, with the use of rigorous methods and peer review, the level of certainty in mathematical proofs is considered very high.

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