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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.