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From my limited mathematical experience, it seems as though lots more tough-to-prove conjectures end up being true than being false, e.g. Fermat's last theorem, Poincaré conjecture, etc. (for the full list, see wikipedia). Anyone have any thoughts on this?

For me, this raises interesting questions about how we think of mathematics. Mathematicians are able to come up with amazing conjectures that end up being true, though at the time of inception there was no way to prove them. It was more about intuition than any empirical method of proof.

Sorry if this is really confusing; l'm happy to clarify anything.

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    $\begingroup$ In some cases a counterexample requires less thinking to obtain than a proof. But I have no statistics for how often that happens. $\endgroup$ – Brian Tung Nov 14 '16 at 18:53
  • $\begingroup$ What's your question? $\endgroup$ – Pedro Tamaroff Nov 14 '16 at 18:58
  • $\begingroup$ @PedroTamaroff: "Anyone have any thoughts on this?" $\endgroup$ – Tdonut Nov 14 '16 at 20:43
  • $\begingroup$ @Tdonut The answer to that question is "Yes." $\endgroup$ – Pedro Tamaroff Nov 14 '16 at 21:57
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    $\begingroup$ The initial guesses that are easily falsifiable, or those falsifiable with a bit more extensive search, never grow up to become conjectures. This distorts the statistics that we see. This coming from someone who has formulated a few guesses about equations over finite fields that were disproven, when the search was extended to fields of size $2^{25}$ and above :-) IOW, +1 to Christian's answer. $\endgroup$ – Jyrki Lahtonen Nov 20 '16 at 7:06
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This is rather a philosophical question, and merits an answer of a more or less feuilletonistic nature.

Of course I could program my computer to formulate 1000 conjectures per day, which in due course would all be falsified.

Therefore let's talk about serious conjectures formulated by serious mathematicians.

Some conjectures (Fermat's conjecture, the four color conjecture, conjectures about the nonexistence of certain finite projective planes, etc.) are derived from the available data, and proofs of special cases. If such a conjecture (tentatively and secretly formulated by a mathematician) is wrong it will be less likely that it will see the light of day nowadays than a hundred years ago, since the available computational powers for producing a counterexample (if there is one) have greatly increased.

If, however, a conjecture is the result of deep insight into, and long contemplation of, a larger theory, then it is lying on the boundary of the established universe of truth, and, as a surface in ${\mathbb R}^3$ splits the neighborhoods of any of its points into two about equal parts, should turn out to be true in $50\%$ of the cases. Only conjectures fulfilling this criterion are worth a full bit $\ldots$

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