Circular Reasoning and Impossible Equation

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Discussion Overview

The discussion revolves around the implications of circular dependencies between two variables, x and y, in the context of formulating equations. Participants explore whether it is possible to write meaningful equations when each variable depends on the other, examining both theoretical and practical aspects of such relationships.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that circular dependencies make it impossible to derive finite values for x and y, suggesting that any equation formed would only yield identities rather than unique solutions.
  • Others argue that it is still possible to write equations, such as x = y, which can convey useful information despite not allowing for the determination of specific values for the variables.
  • A participant introduces specific equations, x = 2y and y = x - 1, asserting that these are independent and can yield a unique solution for the ordered pair (x, y).
  • Another participant reiterates the idea that while equations can be written, they may not provide finite solutions, emphasizing the distinction between unique solutions and infinitely large solution sets.

Areas of Agreement / Disagreement

Participants express differing views on the nature of equations formed under circular dependencies. Some maintain that such equations lack finite solutions, while others contend that meaningful equations can still exist, leading to an unresolved discussion.

Contextual Notes

The discussion highlights the complexity of defining relationships between variables in the presence of circular dependencies, with varying interpretations of what constitutes a sensible equation.

e2m2a
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Suppose there is a problem such that in order to know a variable x, you have to know a variable y. But in order to know variable y, you have to know the variable x. Because of this circular dependency, wouldn't it be impossible to write any sensible equation containing x and y?
 
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You can definitely write an equation but you cannot get some finite set of values for x and y only for which that equation will be true it will be true for a large number of ordered pair (x,y). It will be some thing like identity and not an equation!
 
Let'sthink said:
You can definitely write an equation but you cannot get some finite set of values for x and y only for which that equation will be true it will be true for a large number of ordered pair (x,y). It will be some thing like identity and not an equation!
That makes sense. Thanks.
 
What if ##x = 2y## and ##y = x - 1##?
 
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They are two independent equations and will give you the solution for a unique ordered pair (x, y).
 
e2m2a said:
Suppose there is a problem such that in order to know a variable x, you have to know a variable y. But in order to know variable y, you have to know the variable x. Because of this circular dependency, wouldn't it be impossible to write any sensible equation containing x and y?
Take, for example, x=y. From this you can determine neither x nor y. Nevertheless, the equation is very sensible, i.e. contains a lot of useful information. For instance, if x is your position and y is the position of your wallet, and you are a tourist lost in Rio De Janeiro, you will be very happy to know that x=y. :smile:
 
Let'sthink said:
You can definitely write an equation but you cannot get some finite set of values for x and y only for which that equation will be true it will be true for a large number of ordered pair (x,y). It will be some thing like identity and not an equation!
fresh_42 said:
What if ##x = 2y## and ##y = x - 1##?
Let'sthink said:
They are two independent equations and will give you the solution for a unique ordered pair (x, y).
A unique solution is quite different from a solution set that is infinitely large.
 

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