GS Chandy
Posts:
7,980
From:
Hyderabad, Mumbai/Bangalore, India
Registered:
9/29/05


Re: A question about straight lines
Posted:
Jan 29, 2014 6:54 AM


Neighbor posted Jan 27, 2014 3:06 AM (http://mathforum.org/kb/message.jspa?messageID=9372856): > I should very much like to know whether a > straightline has infinite points on it or a finite > number of points on it? > For those who answer I'd like to have an explanation > of what they say too. > On trying to put myself in "Neighbor's" frame of mind and reading through with some care (without, however, formally modeling ANYthing), all the responses to date to his/her request above, I observe that there is a HUGE amount of learning ahead for 'Neighbor'! (assuming, of course, that he/she is not just posing the question to 'make us think' somewhat more deeply on the issue[s] raised):
I quote just two of the responses: > Joe Niederberger suggested Jan 29, 2014 4:20 AM (http://mathforum.org/kb/message.jspa?messageID=9373965): >...You could read up on it and think about it for awhile. > In brief, (assuming you seriously wish to get your question answered to your full satisfaction), you may need to read up and think about things like: ======   Point; line; ....   Line  segment  ray...   Infinite ~ Finite ...   Boundaries, limits, ...   Understanding of math and physical reality in Euclid's times ~ understanding of math and physical reality in   'Usable' math ~ 'abstract' math (what is the relationship between the two?  how much of each should we do at each point of our respective 'learning curves')...   Euclid (flaws in; extensions of) ...   Abstract Geometry ~ physical geometry (i.e. geometry in the real world)   Finite sets ~ Infinite sets   Open sets ~ closed sets   Naive Set Theory ~ Axiomatic (Formal) Set Theory   Pointset topology (as a part of 'general topology') ... ... ... (AND much else besides!!)   Language; linguistics ...   Foundations of mathematics ~ Using math in real life (How to teach math to beginners? How do 'experts' learn math?)   Philosophy ~ the real world ... ... ...
(NOTE: "~" does NOT mean "Versus" but it does try to imply there is much that is common between the two sides AND much that is different). ====== Lou Talman had pointed out (Jan 29, 2014 5:04 AM, http://mathforum.org/kb/message.jspa?messageID=9373995) that there are some serious flaws in Euclid's definitions, postulates, propositions [as we perceive them and the nature of math today]: > > It has already been pointed out, but it needs to be > said again: > > Euclid's system of geometry doesn't live up to his > claims. > > His "definitions" are seriously flawed, for one > thing. For another, not > all of Euclid's propositions can be proven from the > Postulates, Axioms, > "Common Notions", and Definitions that he > giveseven though Euclid > asserts that he gives such proofs. Moreover, > propositions Euclid did not > intend, and which seem to contradict what he probably > intended, *can* be > "proved" from what he gives using logic he probably > would've admitted > (because he admitted it elsewhere). Any attempt to > reconcile Euclid's > "straight lines" with modern conceptions of straight > lines by means of > what he says is therefore doomed to failure. From > the getgo. > > This fact and the fact that both Euclid's conception > and the modern > conception of straight lines are mental constructs, > and not physical > realitiesand so also irreconcilable with physical > reality, probably lie > at the root of Neighbor's difficulties dealing with > the explanations that > have been given here. > The underlying issue is, of course, that we do have a great deal to do to bridge a whole lot of *'gaps'* in our common and mutual understanding of math, science, philosophy, language, ... ...
In particular, I observe that there are often sizable gaps within various disciplines, in particular between the various disciplines in *abstract* and the same disciplines as they are used in daily life (*usable disciplines*). We do need to try to bridge these gaps [not all such gaps are readily 'bridgeable', so to speak).
I have often suggested that some study of 'systems' could be most helpful indeed to help us bridge at least some of these gaps. In particular, I'd recommend the approach pioneered by the late John N. Warfield, in which he suggested that the powerful ideas in *abstract systems science* would become a lot better understand all round (and also more usable in our daily lives) through understanding just how the factors in various systems may be related to each other and to the 'higher level goals' of the system under consideration. More information about Warfield's seminal contributions to systems science is available at http://www.jnwarfield.com and at the "John N. Warfield Collection" held at the library of George Mason University (Fairfax, VAUSA), where Warfield was Professor Emeritus  see http://ead.lib.virginia.edu/vivaxtf/view?docId=gmu/vifgm00008.xml).
Some developments [which I call the "One Page Management System" (OPMS)] of Warfield's insights into systems now enable individuals and users (at any level, highschool upwards; no background in systems; modeling or math required) to choose any 'Mission' of current interest and to construct, from available good ideas, an effective 'Action Plan' to accomplish the Mission. For instance, 'Neighbor' could most effectively arrive at an Action Plan to help him/her resolve the "question about straight lines" to his/her *reasonable* satisfaction. The responses provided here by various responders to the question, while most useful, actually are responding to their own understanding of the question and of their own understanding of the 'Euclidean system of geometry' (both of which may not necessarily correspond completely with the underlying question[s] as posed by 'Neighbor').
Brief information about the OPMS is available at the attachments to my message heading the thread "Democracy: how to achieve it?") see: http://mathforum.org/kb/thread.jspa?threadID=2419536
Above all, it would always be most useful to keep in mind Hamlet's saying:
"There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy".   Hamlet (1.5.1678), Hamlet to Horatio
GSC

