From: antispam@fricas.org   
      
   Janis Papanagnou  wrote:   
   > On 2026-01-27 17:31, Waldek Hebisch wrote:   
   >>   
   >> Actually, part that is needed here is ancient, due to Eudoksos.  Namely,   
   >> real numebers a and b are equal if and only if comparing them with   
   >> rational numbers gives the same result.  In other words, a is different   
   >> from b if and only if there is a rational number c between a and b,   
   >> but not equal to either a or b.  When computing something this   
   >> principle is clumsy, but for proofs it works quit well.  Thanks   
   >> to this ancients we able compute (and prove) a bunch of transcendental   
   >> equalities.  Theory was finished by Dededking and Cantor who proved   
   >> that number produced by limiting process exist (earlier this was   
   >> consider true "by faith" or by invoking geometric intuition).   
   >   
   > I think this needs some sorting. Speaking about "real numbers" in   
   > context of Eudoksos gives a wrong impression. The ancient Greeks   
   > expressed their mathematics in geometric properties and relations   
   > of such entities. (Algebra, Real Numbers, etc., came much later.)   
      
   This is misunderstanding.  What we consider as "analysis" now was   
   long considerd as part of geometry.  Couchy was called "geometer".   
   Even Goursat, who wrote his analysis book after discoveries by   
   Cantor works frequently in geometric setting.   
      
   Already ancients knew that geometric constructions allow calculation.   
   And that geometric concepts are applicable more generally than to   
   concrete geometrical objects.  It is probably underappreciated   
   that Greek mathematics (or more precisely part that we know,   
   there are plausible claims that vast majority of ancient   
   scientific work was lost) was quite rigorous.  European math   
   attained that level of rigour only at the end of 19-th century.   
      
   Greek notation for numbers was clumsy and probably our reasonings   
   with decimal places would be foreign to Greeks.  But that this   
   sub-thread started with request for a proof.  For that Greeks   
   had what is needed.   
      
   > If you read modern articles you may find references to real numbers   
   > in context of ancient Greek mathematics, but these are only used to   
   > explain that old knowledge with modern methods.   
      
   You miss the point: the words used differed and probably human   
   intuitions differed.  But concepts are general and independent   
   of words used to express them.  This does not differ much   
   from translating mathematical text written in a foreign landuage.   
      
   --   
                                 Waldek Hebisch   
      
   --- SoupGate-Win32 v1.05   
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