Mathematics

Math­e­mat­ics is the study of crisply speci­fied for­mal ob­jects — for ex­am­ple, num­bers — and the ways of know­ing their prop­er­ties — such as proofs. We can see “logic” as the study of “which con­clu­sions fol­low with cer­tainty from which premises”. Us­ing this defi­ni­tion of logic, we can also see math­e­mat­ics as the study of log­i­cal ob­jects in log­i­cal uni­verses — en­tities whose prop­er­ties fol­low from speci­fi­ca­tions about them, rather than from ob­ser­va­tion of the real world. The num­ber 3 is a log­i­cal ob­ject be­cause its be­hav­ior fol­lows from ax­ioms about ad­di­tion and mul­ti­pli­ca­tion; Mount Ever­est is a phys­i­cal ob­ject be­cause we learn about it by phys­i­cally mea­sur­ing Mount Ever­est.

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