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On an elementary school in the US, there was a teacher who used to give the pupils math exercises he invented himself. One day, they calculated the cathetus of a right triangle and magically, they got the result to be $\sqrt{-4}$ . The news quickly spread through the media and, eventually, the mathematicians and philosophers had to deal with it.

What are the possible standpoints they could have taken?

*Sure, it doesn't make any sense but that's kind of the point. Even the notion that things behave in a logical way in our world has been kind of empirically observed.

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closed as primarily opinion-based by Alexander von Wernherr, Mołot, JDługosz, Hohmannfan, Frostfyre Feb 15 '17 at 12:56

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ I imagine the response from the STEM community being "Cool. How can we use this?" $\endgroup$ – frodoskywalker Feb 15 '17 at 9:56
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    $\begingroup$ And how do you plan to build a world with this question? As the question currently is phrased, I would argue that it is off topic for worldbuilding and, even if it weren't, it would be too broad to yield any good answer. $\endgroup$ – Mrkvička Feb 15 '17 at 10:01
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    $\begingroup$ @Mrkvička A question about "how can logic and math influence the actual world" is an interesting one, on-topic with that. Also, there is only one answer: not much. $\endgroup$ – PatJ Feb 15 '17 at 10:19
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    $\begingroup$ @PatJ I agree that "how can logic and math influence the actual world" is an interesting, on topic, question - but I disagree with that it would be what OP is asking. Sure, I may have misunderstood the question, but I read it as "How would mathematicians and philosophers react if they realized they were wrong in one particular case" $\endgroup$ – Mrkvička Feb 15 '17 at 10:48
  • $\begingroup$ I really don't get why this is marked pob. All three current answers basically say the same thing. I provided references to previous occurrences of that happening. $\endgroup$ – PatJ Feb 15 '17 at 15:18
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Assuming that there are no mistakes or errors (like the division by 0 in the proof that 1=2)

A whole slew of mathematicians would examine his working, work out exactly which axioms he was using and then describe in excruciating detail why his initial assumptions are wrong in the real world.

A good example of a mathematical paradox arises in set mathematics, where it's possible to mathematically prove that you can disassemble a pea, rotate the pieces in real space, and then reassemble them into two peas. It blatantly doesn't work, yet is mathematically sound.

In the real world it doesn't work because peas are actually made of discrete units, not a continuous euclidian solid. If your teacher's proof is purely mathematical then something similar will apply.

If, however, your teacher has found a way to use mathematics to magically warp the nature of reality then all bets are off. Expect Mathemagicians to rise to be the dominant world power as they prove that arrows cannot possibly reach them and they can make millet from thin air.

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  • $\begingroup$ But that's different kind of paradox, which actually makes sense. What I'd introduced was meant to be something that truly doesn't make any sense. But you're right: contradiction implies everything so the situation isn't real. Though, as I'm saying, the point of the question actually is to discover how could a non-logical word look like. $\endgroup$ – Probably Feb 15 '17 at 21:12
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It has happened before

Well, the $\sqrt{-4}$ length hasn't happened before. But "things that were deemed impossible mathematically" has.

The first example of this that comes to mind is Russell's paradox (you may have heard about its derivation: the Barber paradox). It showed the base axioms of Cantor's naive set theory where unsound. The only thing to do was to change the axioms.

An other example (in a sense closer to yours) was when the Pythagoreans discovered $\sqrt{2}$ to be irrational. They did not believe irrational numbers could be constructible.

From Wikipedia (emphasis mine):

Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.” Another legend states that Hippasus was merely exiled for this revelation. Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable–a foundation of their theory.

It won't matter much to non-mathematicians

Well, it will probably matter to string theorist too, those guys do weird stuff.

A mathematical demonstration is not going to change the world. The beauty of mathematics and logic is that they're abstract. If you find a contradiction in it, you can change your abstraction.

I hope your math teacher is not on a boat when he makes his discovery.

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We believed our planet was the center of the Universe, until somebody proved the contrary.

We believed vacuum was impossible, until somebody proved it wasn't.

Paradox comes from παρά (against) and δόξα (opinion), meaning something which is against the current opinion.

Easily said, if facts prove an opinion is wrong, the opinion changes (or you can try to alter the facts to suit your opinion...)

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