Is it possible to build math backwards?

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Discussion Overview

The discussion explores the concept of performing mathematical operations in reverse, questioning whether it is possible to construct a mathematical framework where traditionally difficult operations become easier. The scope includes theoretical considerations and conceptual clarifications regarding inverse problems and the foundational axioms of arithmetic.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that performing operations "backwards" is often more challenging than doing them "forwards," citing examples like matrix inversion and differentiation versus integration.
  • One participant introduces the idea of "inverse problems," explaining that these problems can be more complex due to multiple potential solutions or the absence of real solutions.
  • Another participant suggests that the difficulty of inverse operations may not be fundamentally necessary and speculates on a hypothetical scenario where integration is easier than differentiation, which could lead to a different educational approach.
  • There is a correction regarding the foundational axioms of arithmetic, with participants discussing the reliance on addition and multiplication rather than subtraction and division.

Areas of Agreement / Disagreement

Participants express differing views on the nature of mathematical operations and their inherent difficulties. While there is some agreement on the challenges of inverse operations, the discussion remains unresolved regarding the possibility of constructing a mathematical framework that reverses these difficulties.

Contextual Notes

Participants highlight the dependence on definitions and the context of mathematical operations, particularly in relation to the axioms of arithmetic. The discussion does not resolve the complexities associated with inverse problems or the foundational aspects of mathematics.

daniel_i_l
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In many areas of math doing something "backwards" is much harder that doing it "forwards", for example - inverting a matrix is harder than multiplying it, taking a derivative is easier that an integral...
But why should this be, I mean, there's nothing fundamental in the way that we happened to set things up. Is it possible to "build math backwards" so that all the things that are hard fo us now are easy and vice versa? If not then why not?
Thanks.
 
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I think you are talking about the "inverse" problem. A "direct" problem, means that we are given a specific formula, such as "f(x)= x6- 2x5+ 3x+ 4" and asked to evaluate f(2). An "inverse" problem is the other way: "If f(x)= 10, what is x" does not give us a formula. It is more difficult for several reasons: there may be more than one solution (if you did the original problem one of them is easy!), for some polynomials there may be no (real) solution, and, in fact, it may not be possible to write a solution in terms of roots or other standard ways of writing numbers.
 
daniel_i_l said:
In many areas of math doing something "backwards" is much harder that doing it "forwards", for example - inverting a matrix is harder than multiplying it, taking a derivative is easier that an integral...
But why should this be, I mean, there's nothing fundamental in the way that we happened to set things up. Is it possible to "build math backwards" so that all the things that are hard fo us now are easy and vice versa? If not then why not?
Thanks.
That question reminds one of the arithmetic axioms, which rely ONLY on addition and subtraction. They DO NOT specifically include subtraction or division.
 
I think you meant "rely ONLY on addition and multiplication".
 
But why should this be, I mean, there's nothing fundamental in the way that we happened to set things up

Yes quite possibly so. Imagine for example that (in some perpendicular universe) things were reversed and integration just happened to be substantially easier than differentiation. Then it's highly likely that integration would then be taught prior to differentiation and differentiation would commonly be thought of as "anti-integration" rather than the other way around. See what's happened, even when we've reversed the relative difficulties your "inverse processes seem harder" observation still holds.
 
HallsofIvy said:
I think you meant "rely ONLY on addition and multiplication".
You are exactly correct; I did mean "Addition and Multiplication". The laws of arithmetic which we learn formally during the first two years of "high school algebra" rely only on addition and multiplication.
 

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