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For example if we accept that $3=5$ then we can prove true statements such as $8=8$ as well as false statements that $8=10$. One way this is phrased is "from falsehood anything follows," known as the principle of explosion according to Google.

But how would we then prove very unrelated concepts such as "the moon is made of cheese" or something similar to this? Or is "anything" really only within the context of the system we're working in?

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  • $\begingroup$ See the related post: question-about-consistency-of-axiomatic-systems. $\endgroup$ Mar 12 '18 at 8:09
  • $\begingroup$ The intuitive meaning is quite simple: from an inconsistent "system" of premises we can derive everything as conclusion, and this means that the "system" is unable to discriminate what is true from what is false. $\endgroup$ Mar 12 '18 at 10:11
  • $\begingroup$ @MauroALLEGRANZA Is it possible to derive any result from a single false premise alone? For instance if I said $3=5$ can I prove any $x=y$? $\endgroup$
    – user539262
    Mar 12 '18 at 13:42
  • $\begingroup$ NO; a contradiction must be something like: $P \land \lnot P$. If we are working in the context of arithmetic (or in a context - like everyday life ? - where arithmetic is implicitly assumed) $3=5$ is not per se contradictory: it is false, because we know that $3 \ne 5$, i.e. $\lnot (3 = 5)$. And if we know that $\lnot (3=5)$ holds, when we assume that $(3=5)$ we are implicitly asserting a contradiction. $\endgroup$ Mar 12 '18 at 13:45
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"Anything" is certainly only within the context of the system we are working in - the only propositions we can talk about, and thus the only ones we can prove or disprove, are ones that are well-formed statements in our language. But that doesn't preclude the system we are working in from having a very expressive language. While I doubt we could seriously consider the moon being made of cheese to be a mathematical statement in any context, we don't necessarily need to confine ourselves to only some narrow, well-trod mathematical territory like arithmetic. Any system that is based in classical logic (or many other types of logic, for that matter) will have ex falso / explosion.

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Anecdote (maybe true, maybe urban myth).

Someone once challenged Bertrand Russell to prove that "if $1=2$ then you [i.e. Russell] are the Pope".

Russell replied, "Either the Pope and I are one person or we are two people. If $1=2$ then in either case we are one person. Therefore, I am the Pope."

[The point of the challenge: Russell had a very low opinion of religion.]

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  • $\begingroup$ You can use this argument to prove that the moon is made of cheese : Let $M$ be a material that compose the moon. The set $\{ \text{Cheese} \} \cup \{ M\}$ has either $1$ or $2$ elements. If it has $1$ element, then $M = \text{Cheese}$. If it has $2$ elements, then it has $1$ element (because $1 = 2$), and it follow that $M = \text{Cheese}$. That prove that if $1=2$, every material that compose the moon is Cheese. $\endgroup$
    – Tryss
    Mar 12 '18 at 4:39
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"But how would we then prove very unrelated concepts such as "the moon is made of cheese" or something similar to this? Or is "anything" really only within the context of the system we're working in?"

The latter.

Or you can attempt the former if you're willing to do a lot of work, for example find the density of cheese and density of moon. Then Prove that the two numbers are equal. And then do the same with the moon's spectral analysis and that of the cheese, basically all physical characteristics. Then you can say that with overwhelming evidence, the moon is made of cheese.

Note that the proof that the moon is made of cheese (or that the moon is made of rocks for that matter) is no longer in the realm of "pure mathematics". That's physics and that requires empirical evidence... so we can only say with a high degree of certainty that the moon is made of rocks but we can never be 100% sure.

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