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Nobody needs equations

Adonai Schlup Sant'Anna

Abstract

Equations are ubiquitous in most mathematical activities. Nevertheless, in this paper it is shown how to do standard mathematics without any equation at all. More than that, it is proven there is a foundational framework for standard mathematics where equations cannot be even written, in the sense they are not formulas. The proof of those claims is very simple, almost obvious. I use this framework to suggest a way to deal with certain notions of indiscernibility between `objects', with special emphasis on some aspects of quantum mechanics. Finally I compare this approach to quasi-set theory, an unnecessarily complicated formal work designed to deal with violation of Leibniz Principle of the Identity of Indiscernibles.

Nobody needs equations

Abstract

Equations are ubiquitous in most mathematical activities. Nevertheless, in this paper it is shown how to do standard mathematics without any equation at all. More than that, it is proven there is a foundational framework for standard mathematics where equations cannot be even written, in the sense they are not formulas. The proof of those claims is very simple, almost obvious. I use this framework to suggest a way to deal with certain notions of indiscernibility between `objects', with special emphasis on some aspects of quantum mechanics. Finally I compare this approach to quasi-set theory, an unnecessarily complicated formal work designed to deal with violation of Leibniz Principle of the Identity of Indiscernibles.

Paper Structure

This paper contains 7 sections, 2 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. ZF
  3. Models of ZF
  4. A variation of ZF with no identity
  5. Some remarks
  6. Quantum non-individuality
  7. Acknowledgments

Key Result

Proposition 1

Any model of ZF is a transitive model of $Z\Phi$.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Definition 3