Single most important object in mathematics?

[tgt]

For me it would have to be the set of complex numbers. What's yours?

 

[mathman]

Mathematics starts with the integers, so I would consider them the most important.

 

[turbo]

The concept of zero, hands-down.

 

[Dragonfall]

The von Neumann hierarchy.

 

[tgt]

Mathematics starts with the integers, so I would consider them the most important.

The concept of zero, hands-down.

Well, the complex numbers include both of them (and also the reals).

 

[Office_Shredder]

From the empty set you can construct everything.

 

[AUMathTutor]

I second the empty set.

 

[maverick_starstrider]

Mathematics starts with the integers, so I would consider them the most important.

To be accurate math starts (in the standard construction) with the natural numbers and sets. From there you derive integers, then rationals, then reals, then complex, then quaternions, then octernian, etc.

 

[pbandjay]

Oh I'm totally all about Graham's number!

 

[flatmaster]

How many dollars I have.

 

[dx]

The ideas of 'set' and 'function'.

 

[g_edgar]

the proof, Euclid I 47

 

[lormanv]

The field of algebraic numbers.

 

[Dragonfall]

So what's the LEAST important object in mathematics?

 

[maverick_starstrider]

*insert offensive generalization about hygiene, women, etc.*.. oh snap!

 

[user111_23]

1 is by far the most important.

 

[chiro]

For me it would have to be the set of complex numbers. What's yours?

To me its the idea that representation of any kind is abstracted to a mathematical quantity. This fact alone says a lot about the power of mathematics.

 

[ƒ(x)]

most important: equals sign?

 

[VeeEight]

I third the empty set.

 

[camilus]

Im going to have to read about the empty set.

The most important thing in mathematics is the concept of numbers. And from there stems everything else.

But SPECIFICALLY, my absolute favorite is Euler's formula and his Identity.

$$e^{ix}= \cos (x)+ i \sin (x)$$

$$e^{i\pi}+1 =0$$

The most beautiful relationship ever.

 

[n!kofeyn]

From the empty set you can construct everything.

I thought the empty set is just a convention, unless this was a joke, but I've seen seconds and thirds of the empty set.

I guess the most important object in terms of foundational object would be the set of axioms that all of mathematics is based upon. My personal favorite object is the integral. Although there are many forms, I really enjoy studying integration theory: Riemann integration, complex integration, Lebesgue integration, Feynman's path integral (although I haven't learned this yet), etc. One of my professors once said that "integrals always know" because of their ability to detect or give information about their integrands.

 

[Office_Shredder]

I thought the empty set is just a convention, unless this was a joke, but I've seen seconds and thirds of the empty set.

The empty set is the set containing no elements. How is that a convention? Using the standard set theory axioms, given the empty set you can 'construct' the natural numbers, and hence basically all of math

 

[n!kofeyn]

The empty set is the set containing no elements. How is that a convention? Using the standard set theory axioms, given the empty set you can 'construct' the natural numbers, and hence basically all of math

Well this is how James Munkres describes it in his Topology book. I have also seen other books take it as convention. I'm not familiar with foundational set theory, but from what I've read, the empty set is just assumed to exist, either by definition, convention, or axiom. I realize that it is defined as the set containing no elements, but how do you know such an object even exists?

 

[Office_Shredder]

Well this is how James Munkres describes it in his Topology book. I have also seen other books take it as convention. I'm not familiar with foundational set theory, but from what I've read, the empty set is just assumed to exist, either by definition, convention, or axiom. I realize that it is defined as the set containing no elements, but how do you know such an object even exists?

In foundational set theory it's assumed to exist as an axiom. How do you know any object exists? Nothing exists in math without assuming something, so I wouldn't call it a weakness

 

[n!kofeyn]

In foundational set theory it's assumed to exist as an axiom. How do you know any object exists? Nothing exists in math without assuming something, so I wouldn't call it a weakness

I didn't say it is a weakness. Many objects in math are shown to actually exist, although everything is based upon the basic assumed axioms. I think my point is that what I seem to understand is that the empty set is just assumed to exist by axiom because it makes things easier to talk about. In other words, it isn't absolutely necessary, but it is a convention that simplifies the discussion of some things.

 

[Office_Shredder]

I didn't say it is a weakness. Many objects in math are shown to actually exist, although everything is based upon the basic assumed axioms. I think my point is that what I seem to understand is that the empty set is just assumed to exist by axiom because it makes things easier to talk about. In other words, it isn't absolutely necessary, but it is a convention that simplifies the discussion of some things.

Ok, but then nothing is necessary because you can just change the axioms to make sure they don't exist. So I'm still unsure how the empty set is special in this regard.

 

[AUMathTutor]

The empty set is important because it can be used to construct everything, and is itself both something (a set) and nothing (because what is a set but its extent).

It's just a really cool idea.

 

[AmberCam]

The Cartesian Co-ordinate System

 

[Pinu7]

Logical Quantifiers.

Mathematicians are too lazy to type out "for every" and would rather go to a mathtype window to make an upside-down "A" so their theorems are unreadable to those that do not know logical quantifiers.

 

[maverick_starstrider]

Logical Quantifiers.

Mathematicians are too lazy to type out "for every" and would rather go to a mathtype window to make an upside-down "A" so their theorems are unreadable to those that do not know logical quantifiers.

Although it takes like 10 minutes to learn the symobolism.

 

[kramer733]

Notations are the most important things in mathematics. With good notation, you can clearly define objects and thus advance through math more.

 

[Char. Limit]

The concept of equality.

 

[wisvuze]

the concept of an "isomorphism"! ( there are tons, but this has got to be in the top bunch )

 

[pwsnafu]

Axioms, definitely. Mathematics is nothing more than applied logic.

Otherwise, I'll fourth (fifth?) the empty set.

*insert offensive generalization about hygiene, women, etc.*.. oh snap!

In physics, the ideal woman is a point particle. :tongue:

Ok, but then nothing is necessary because you can just change the axioms to make sure they don't exist. So I'm still unsure how the empty set is special in this regard.

So if the empty set doesn't exist, what happens when I take the set of integers and remove all of its elements? What about the singleton set {a} and want to exclude an element?

 

[disregardthat]

Mathematicians, of course.