Re[2]: 20th Century Mathematics

John W Nageley <jw_nageley@pnlg.pnl.gov>
Date: Tue, 25 May 93 10:37 PST
From: John W Nageley <jw_nageley@pnlg.pnl.gov>
Subject: Re[2]: 20th Century Mathematics
To: sowa@turing.pacss.binghamton.edu
Cc: interlingua@ISI.EDU
Message-id: <2F8EE715865F20E47D@pnlg.pnl.gov>
X-Ccmail-Date: 5/25/93 10:23AM
X-Ccmail-*Cc: Morgan L Yim at ~PNL53
X-Envelope-To: interlingua@isi.edu



          And thank you for your note and help getting the message
          out.

          For my part, I agree that enough has been said.  It is
          important that it be said.  That much is true.  The adage
          that we will repeat the mistakes of the past if we don't
          learn from them very much applies here, I think.  That is
          why I so strongly emphasize a thorough understanding of the
          assumptions underlying all the structures we are now trying
          to construct.  Cantor, as I have stated, began with what I
          consider to be invalid assumptions; on those assumptions he
          then built his rather imposing but now obviously
          insubstantial structure.  It can be rather imposing to
          behold and fun to play on but....  I think enough has been
          said.

          Now, where do we go from here?

          For one thing, I am now developing the conceptual foundation
          for a piece of software under an in-house grant from the
          research laboratory for which I work.  The experimental
          element in this work is the use of an approach I
          have been developing for many years.  This approach, it
          seems, circumvents many of the problems one encounters in
          applying any form of traditional set theory to problems in
          logic and structure. (Have you ever heard of anyone who
          claims to have developed a piece of software using set
          theory in its design?)  The first deliverable is a prototype
          of the software, due by the end of September.  Based on this
          work, I hope to publish not only the design of the software
          but the approach being used in designing it.

          For other areas that need attention, as Godel points out, we
          need to return to the fundamentals.  One such fundamental is
          the real number line.  I have an article in one stage or
          another on this problem but won't be returning to it until
          after the software project is complete.  In any case, it
          should be proven that the concept of the real number line
          is invalid.  (One factor to consider in disproving this
          concept is comparable to disproving that unicorns exist:  in
          both cases, the fact that we have not seen one does not mean
          that one cannot exist.  But there should be a way around
          this factor.)

          So, learning from history--we have all done our homework--we
          can now move forward in developing our conceptual
          structures, assured that we will not make the same mistakes
          again; or can we?  What Kant has to say about this is,
          again, a warning:  we must examine the foundations on which
          we build our structures--before building them.  If we don't
          future generations will have to struggle to survive with
          even more flimsy, ineffectual structures.

          And I hope to make it to Quebec.  Look forward to meeting
          you.  Morgan Yim loaned me a copy of your "Conceptual
          Structures" which I have enjoyed reading.