Robert Hansen (RH) posted Jul 10, 2013 10:07 PM: > > I like this quote... > > "A heavy warning used to be given that pictures are > not rigorous; this has never had its bluff called and > has perma- nently frightened its victims into play- > ing for safety." > > I do not claim that a visualization of a proof or of > a mathematical conclusion is less rigorous. In fact, > my claims have nothing to do with rigor at all. I > claim that when you show students a visualization > they will see either the visualization or the > mathematical reasoning and the only way to know which > it is they see is to ask them to explain the > reasoning behind the visualization. > QUOTE (RH, with emphasis from GSC): > I claim that when you show students a visualization > they will see EITHER the visualization or the > mathematical reasoning UNQUOTE (Emphasis GSC's)
UTTERLY WRONG! Means precisely nothing.
The visualisation that those students see "MAY CONTRIBUTE TO" their better understanding of the math reasoning. I.e., "VIA" the visualisation, the students may arrive at a better understanding of the formal mathematics.
Your 'probable proof' below my signature is not a proof at all - nor is it likely ever to become one. (Your 'suspicions' do not constitute the basis of a 'proof').
John Sowa (http://www.jfsowa.com/) in his works on 'Knowledge Representation'; 'Conceptual Graphics'; 'Ontology'; 'systems of classification'; (and etc) has provided valuable initial steps towards 'reconciling' visual and graphical representations of knowledge ('mental models') with the kind of artefacts that RH accepts as 'mathematical reasoning'. (Well worth studying for those who may be interested to explore such issues seriously - as opposed to superficially talking about them).
John Sowa's work provides valuable background for the study of John N. Warfield's insights into systems science (and system design).
The rest of RH's post copied below for ready reference: > > This could probably be easily proven. Show students > two "visually appealing" proofs, one mathematically > correct and the other not, even though it "visually" > looks correct. I suspect that you will find that the > students lacking mathematical reasoning sense will > judge both to be correct while the students that have > the sense will see the flaw. I am not saying that the > first set of students can't be taught mathematics or > will never get the sense. I am only pointing out that > a visualization of a mathematical argument still > requires the same sense of mathematical reasoning to > see it as does a mathematical argument in any other > medium. > > Another example of this is data visualization. Data > visualization appeals to our non mathematical senses. > These are the senses Dehaene writes about and > confuses with mathematics. When I can create a data > visualization I can create it in such a way that it > is mathematically correct but invokes a response > completely at odds with the data being represented. > For example, Richard posted some time ago a graphic > of a bag three times as large as another bag, by > volume (or something to that effect). Not a good > representation to feed to our Dehaene rat-sense of > spatial area. Our rat-sense does not recognize it as > three times as big. It requires actual mathematical > sense to make sense of this visual contradiction. > Obviously, when you visualize data you strive to > avoid this situation. You choose graphical methods > that look "rat-sense" correct rather than > mathematically correct. > > The bottom line is that rat-sense is not mathematical > reasoning nor is it a possible substitute for > mathematical reasoning. Visualizations appeal to > rat-sense. The mathematical argument behind the > visualization can only appeal to the sense of > reasoning. Saying that a visualization is appealing > does not tell me which occurred. > > Bob Hansen > > > On Jul 10, 2013, at 10:15 AM, "Dave L. Renfro" > <email@example.com> wrote: > > > Joe Niederberger wrote (on 21 June 2013): > > > > > http://mathforum.org/kb/message.jspa?messageID=9142053 > > > >> But onward and upward - > >> Here's a nice book, but I won;t be paying $159 for > an book copy! > >> It's a collection and has essays on various sides > if the issue. > >> The entire introduction, which I just read, seesm > promising. > >> > >> "Visualization, Explanation and Reasoning Styles > in Mathematics" > > > > > http://books.google.com/books?hl=en&lr=&id=B5QH_-nDgN8 > C&oi=fnd&pg=PR8&dq=godel+on+visualization+of+mathemati > cs&ots=-RreUUO2Es&sig=9cIBd4TwevyOj9krMkoyzLztF5w#v=on > epage&q=godel%20on%20visualization%20of%20mathematics& > f=false > > > > For those who might be interested, I came across > something this > > morning that fits well into this discussion: > > > > Bill Casselman, Review of "Visual Explanations" by > Edward R. Tufte, > > Notices of the American Mathematical Society 46 #1 > (January 1999), 43-46. > > http://www.ams.org/notices/199901/rev-casselman.pdf > > > > Dave L. Renfro