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What is the simplest mathematical object? I am talking about mathematics in the most abstract way possible, and not as some concrete axiomatic theory (e.g. foundational ones, like ZFC).

After a lot of time pondering this question I came to the opinion that the answer is the equality relation.

P.S. This question is a tiny bit opinion-based, so I decided to post it under the philosophy and soft-question tags.

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closed as off-topic by John B, Dietrich Burde, Alex M., Andrés E. Caicedo, J.-E. Pin Mar 16 '16 at 14:09

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    $\begingroup$ What exactly is a mathematical object ? $\endgroup$ – Dietrich Burde Mar 16 '16 at 12:38
  • $\begingroup$ Perhaps not a "mathematical object," but I'd think the $=$ sign is fairly simple. (Not that people don't misuse it, sometimes incorrectly writing things like $5\times2=10+3=13$.) $\endgroup$ – Akiva Weinberger Mar 16 '16 at 12:40
  • $\begingroup$ What could be simpler than the empty set? $\endgroup$ – Gerry Myerson Mar 16 '16 at 12:43
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    $\begingroup$ @GerryMyerson $0$ is "simpler" than $\emptyset$ (for writing). $\endgroup$ – Dietrich Burde Mar 16 '16 at 12:47
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    $\begingroup$ A psychologically simple object might not be a logically simple one. $0$ is logically very simple, yet the Romans did not have it in their number system; the Arabs had to import it from the Indians, who were well versed into thinking about "nothingness". On the contrary, the natural numbers (starting with $1$) are psychologically very simple (even kids can understand them), but constructing them rigorously is not trivial (think of Peano). As you see, simplicity is not that simple. $\endgroup$ – Alex M. Mar 16 '16 at 12:47
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The point.

(If by "simplest" you instead mean "most fundamental", you should edit the question to reflect this distinction.)

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The simples mathematical object is your index finger. It is a physical representation of the number one. From this much mathematics follows.

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    $\begingroup$ I find the idea of the number $1$ simpler than my index finger $\endgroup$ – vrugtehagel Mar 16 '16 at 12:48
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    $\begingroup$ If I start pointing to my cow, abstraction (getting rid of unnecessary details) happens. $\endgroup$ – mvw Mar 16 '16 at 12:50
  • $\begingroup$ @vrugtehagel: That's because you live in 2016 A.D., having acquired a mathematical education and a view on the development of mathematics. Had you lived in 2016 B.C., you wouldn't have even understood what the words "the idea of the number $1$" mean. $\endgroup$ – Alex M. Mar 16 '16 at 12:51

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