Single most important object in mathematics?

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Discussion Overview

The discussion revolves around identifying the single most important object in mathematics, with participants sharing various perspectives on foundational concepts, numbers, sets, and mathematical constructs. The scope includes theoretical and conceptual considerations.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants propose the set of complex numbers as the most important object.
  • Others argue that mathematics starts with the integers, suggesting they hold primary importance.
  • The concept of zero is highlighted by multiple participants as a fundamental element.
  • The empty set is mentioned as a foundational object from which everything can be constructed, with several participants expressing agreement.
  • Some participants emphasize the significance of axioms and the foundational nature of mathematical logic.
  • Euler's formula and identity are noted as particularly beautiful relationships in mathematics.
  • Logical quantifiers and notations are discussed as critical components for clarity in mathematical discourse.
  • The concept of equality and isomorphisms are also raised as important mathematical ideas.

Areas of Agreement / Disagreement

Participants express a variety of views on what constitutes the most important object in mathematics, with no consensus reached. Multiple competing perspectives remain, particularly regarding the significance of the empty set, integers, and complex numbers.

Contextual Notes

Some discussions touch on the assumptions underlying the existence of mathematical objects, particularly the empty set, and the role of axioms in mathematics. These points highlight the complexity and foundational nature of the concepts being discussed.

  • #31
Notations are the most important things in mathematics. With good notation, you can clearly define objects and thus advance through math more.
 
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  • #32
The concept of equality.
 
  • #33
the concept of an "isomorphism"! ( there are tons, but this has got to be in the top bunch )
 
  • #34
Axioms, definitely. Mathematics is nothing more than applied logic.

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

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

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

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?
 
  • #35
Mathematicians, of course.
 

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