Math Structures: Proving Existence w/ Accurate Solutions?

[hddd123456789]

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 :)

 

[Number Nine]

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.

 

[lucid_dream]

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.

 

[micromass]

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.