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_nDgN8C&oi=fnd&pg=PR8&dq=godel+on+visualization+of+mathematics&ots=RreUUO2Es&sig=9cIBd4TwevyOj9krMkoyzLztF5w#v=onepage&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), 4346. http://www.ams.org/notices/199901/revcasselman.pdf Dave L. Renfro 
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. 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 ratsense of spatial area. Our ratsense 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 "ratsense" correct rather than mathematically correct. The bottom line is that ratsense is not mathematical reasoning nor is it a possible substitute for mathematical reasoning. Visualizations appeal to ratsense. 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" <[hidden email]> wrote: Joe Niederberger wrote (on 21 June 2013): 
In reply to this post by Dave L. Renfro
>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
Reading a mathematical picture for information is no different in some regards from other kinds of reading (say, reading a trail that the bad guys went down.) There's the information conveyed, and the reader's ability to discern it. The picture itself though, does indeed convey absolutely specific information if you are up to the task of seeing it. I've already proven this with my e^pi example. Google for solutions and you will not find that particular *proof* (the proof I explained after showing the picture, that references the mean value theorem.) At least you will not see it very often (I've looked,) because most people will not have *that* picture in mind when finding other solutions (there are many.) If the *reasoning* stood apart, as you claim, many people should have found that proof because its very simple. That the mean value theorem is a highly *visually* intuitive fact, makes its connection with that particular picture absolutely compelling. I doubt anyone has ever found that proof without seeing the picture first, if only in their mind. I think your argument breaks down completely when you attempt to debunk visual thinking in mathematics by bringing up cases of incorrect conclusions drawn from a picture. But that proves nothing, incorrect conclusions can be drawn from any sort of communication. Cheers, Joe N 
On Jul 10, 2013, at 4:40 PM, Joe Niederberger <[hidden email]> wrote: That the mean value theorem is a highly *visually* intuitive fact, To whom Joe? I have seen NO class of students get anything like the MVT from a picture, UNLESS it is explained to them. Connectedness (a term I will use for what you feel for that curve) is an intuitive ratsense fact. Connectedness is NOT the MVT. I am not saying that you cannot reason and reflect about connectedness, and maybe even come up with the MVT, but that is not the ratsense you are talking about. That requires all of the elements of reasoning that any other domain does. Bob Hansen

In reply to this post by Joe Niederberger
On Jul 10, 2013, at 4:40 PM, Joe Niederberger <[hidden email]> wrote: That the mean value theorem is a highly *visually* intuitive fact, makes its connection with that particular picture absolutely compelling. You are confusing our concrete senses (what I will hereafter call our "ratsense" in praise of Dehaene) with mathematical sense. That ratsense gets in our way more than it helps, when it comes to higher level thinking. It's getting in the way right now.:) Yes, you can connect the MVT to a picture, IF you are familiar with the MVT. Let me ask you this. If I plot f(x) = x^2, that visualization has a distinctive signature to it. That quadratic acceleration of slope, etc. But how is that "mathematical"? Without any explanation or further mathematical understanding, it isn't even a mapping. Not even an intuitive mapping. It is just a curve with a distinctive signature, but not so distinctive as not to be confused with several other curves. You say that the connection of the MVT to that picture is compelling. I agree, but I claim that this feeling of "compelling" is exactly what I warned against. Teachers think that there is something much more there than there actually is because it is so damn "compelling". In this context, compelling means that you know the MVT and it is just a damn neat feeling when you can THEN see it through your ratsense. We all have that feeling. Every time we learn or discover something new in mathematics the first thing we try to do is see it through our ratsense. We love our ratsense. We trust our ratsense. That phenomena doesn't work in reverse. Not without a great deal of reasoning. Which requires a different sense. Bob Hansen

In reply to this post by Joe Niederberger
On Jul 10, 2013, at 4:40 PM, Joe Niederberger <[hidden email]> wrote: Reading a mathematical picture for information is no different in some regards from other kinds of reading (say, reading a trail that the bad guys went down.) Would reading a chess board then also be just like other kinds of reading? I don't understand why you think these things are that mechanical. I have worked with a lot of people and almost all of them read very well, and I am sure the others read very well in their native language, until, it's not a question of reading. Then the difference in this "sense" really shows up. Btw, on my way home today, all of this talk of ratsense got me thinking... Given that there have been billions of non human primates over millions and millions of years of evolution and given that cognitive ability in this large non human population would fall in a distribution, then it is reasonable to assume that there would be many extraordinary examples of non human primates. Now, if we assume that there is even just a glimmer of a connection between numerosity (ratsense) and mathematical formulation then it would seem very likely that within this distribution of billions of apes with millions of years of evolution on their side that some of them at least would be quite extraordinary. How extraordinary? I think enough such that if ratsense begets humansense just a little bit, then I would expect some cases of humans somewhere saying "Take your stinking paws off me you damn dirty ape!" Regardless of what theory you believe I still think it is remarkable that this difference between us and them is so absolute and so resolute. It just goes to show you what a difference 3% in DNA makes. Although, you could compare a small airplane and an automobile and find that they too are only 3% different at a materials level. Yet, one flies through the air and the other can only roll around on the ground. I see three layers of evolution here. That of the chimpanzee. That of the human. And that of the reasonable human. The chimpanzee layer is so done (evolved) that it will probably not change significantly at any point in the future, barring of course some alien intervention. Yes, unfortunately that means that we will never experience a human telling an ape to get their stinking paws off of them. The human layer (language, motor skills, emotions, etc) is mostly done, but not entirely blended. The reasonable human layer though seems to have a lot of blending left to do. Bob Hansen

In reply to this post by Dave L. Renfro
Joe N says:
> That the mean value theorem is a highly *visually* intuitive fact, R Hansens says: >To whom Joe? I have seen NO class of students get anything like the MVT from a picture, UNLESS it is explained to them. Lots of people. Probably most. We've already established you're not one. But once again your entire argument doesn't scan. Yes, one must be taught how to read the information, whether its words or curves. Cheers, Joe N 
On Jul 11, 2013, at 12:41 AM, Joe Niederberger <[hidden email]> wrote: We've already established you're not one. But once again your entire argument doesn't scan. Your argument rests on me not being one? Bob Hansen

In reply to this post by Dave L. Renfro
R Hansens says:
>To whom Joe? I have seen NO class of students get anything like the MVT from a picture, UNLESS it is explained to them. Joe N says: >Lots of people. Probably most. We've already established you're not one. But once again your entire argument doesn't scan. Yes, one must be taught how to read the information, whether its words or curves. Perhaps, too glib again. I suspect most people, even non mathematical types, could be taught to see the theorem, if its explained what one is looking for. Given an ordinary continuous everywhere differentiable curve, the tangent can be intuitively and visually conveyed. Once the chord between two points is illustrated, the notion that the mean value theorem could be violated would be seen to be impossible. Naturally, a guide is needed until one becomes a scout on their own. Its even more visually compelling with the intermediate value theorem. Your assertion about classes you've experienced means little. Classes where the mean value theorem are introduced and proved without ever mentioning the visual intuition behind them would be few and far between, probably only classes where it was assumed the visual was already understood from prior exposure. By today's standards these things were only analytically "proved" within the last couple hundred years, or even last hundred, all depending on your standard. They were known for a much longer time. The intuition was visual, not some other "sense". By your argument, these things were not really be known before they could be proved by some modern standard. That's not the case. Cheers, Joe N 
On Jul 11, 2013, at 1:10 AM, Joe Niederberger <[hidden email]> wrote: By your argument, these things were not really be known before they could be proved by some modern standard. No. I am saying that they are not really known before they are known. You are being sloppy with the word "known". Let's take your example of the MVT. If I draw a continuous curve on the board from point A to point B it is obvious that if I trace this curve with my finger or just with my eyes that it must go through every point on it's path. But that is just the reality of the world we live in. What thought or reasoning is involved with this? Where is the logical argument or conclusion at the end of this? You are trying to equate our concrete sense of the world with mathematical reasoning. I say they are two very different things. You are also confusing what I am saying with rigor. I don't care about rigor. I also don't care if it is geometric or algebraic. All I care about is backed with REASON. Appealing to one's ratsense of continuous is not the same thing as appealing to one's abstract sense and theory of continuity. That is my line in the sand. That is my warning to those who become enthralled with seeing mathematics in a picture and forget that they knew the mathematics FIRST. You seem to be insisting that one can have a mathematical visual experience without reasoning. Yet every time I ask you to explain it, out pours the reasoning. Here is a thought experiment for you. Take a person with no exposure whatsoever to mathematics and shown them the continuous curve. And then you take a look at the continuous curve. I guarantee that both of your visual ratsense experiences will be the same. So, I ask you, are you both having a mathematical experience or not? Do you have a line in the sand? Bob Hansen 
In reply to this post by Joe Niederberger
On Jul 11, 2013, at 1:10 AM, Joe Niederberger <[hidden email]> wrote: Perhaps, too glib again. I suspect most people, even non mathematical types, could be taught to see the theorem, if its explained what one is looking for. Given an ordinary continuous everywhere differentiable curve, the tangent can be intuitively and visually conveyed. Once the chord between two points is illustrated, the notion that the mean value theorem could be violated would be seen to be impossible. Naturally, a guide is needed until one becomes a scout on their own. Its even more visually compelling with the intermediate value theorem. So now you are talking something very different. You are talking about using visualizations to teach abstract things. I never complained about that. I wouldn't even try to teach without it. My complaint is when people start thinking that the visualization is the abstract thing and the ratsense behind it is the same as thinking. I was part of an interesting discussion on a physics teaching site having to do with electric circuits. It started out with the the usual reformist idiocy of "traditional teaching is wrong because..." The point was that teachers teach about current and voltage in circuits as if the electrons push each other down the wire like peas in a straw. The reformist view was that this is wrong because after a very very brief amount of time the excess charge in the wire arranges it self in a stationary configuration and it is these stationary excess charges doing the pushing on the electrons in the middle of the wire. I had no choice but to concede that the electrons in a wire are nothing like peas in a straw. It was Zeev that saw through the idiocy of this discussion that by this time had lasted a couple hours at least. After I conceded that in an actual circuit the electrons act nothing like peas in a straw, Zeev asked "But how does that help students learn the relationship between voltage, current and resistance in a circuit?" That is when it hit me. We hold onto these intuitive models dearly, even the physically ridiculous ones, not based on how alike they are to the abstract concepts we are learning but based on how well they help us learn the abstract concepts we are learning. There is always some sort of analogy involved but that by no means implies equivalence. You can really dig deep and learn a lot by listening to reformers. Bob Hansen 
In reply to this post by Dave L. Renfro
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. > > 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). Other useful background: Gene Bellinger's "Mental Model Musings"  http://www.systemsthinking.org/ . GSC ("Still Shoveling!") 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 ratsense of > spatial area. Our ratsense 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 "ratsense" correct rather than > mathematically correct. > > The bottom line is that ratsense is not mathematical > reasoning nor is it a possible substitute for > mathematical reasoning. Visualizations appeal to > ratsense. 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" > <[hidden email]> 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), 4346. > > http://www.ams.org/notices/199901/revcasselman.pdf > > > > Dave L. Renfro 
In reply to this post by Dave L. Renfro
R Hansens syas:
>You are also confusing what I am saying with rigor. I don't care about rigor. I also don't care if it is geometric or algebraic. All I care about is backed with REASON. >Appealing to one's ratsense of continuous is not the same thing as appealing to one's abstract sense and theory of continuity. No, I think you are confusing abstraction with correctness. Rigor is the accepted price of increasing confidence in correctness, but one can be correct with less (Gauss, Euler, Newton...) Rigor today is associated almost exclusively with deductive thought (this, therefore that..etc.) from primitives that are as divorced from everyday "reasoning" as possible. Hence all the angst in trying to "derive" numbers (which most people accept as obvious) from things supposedly even more primitive. But everyday "reasoning" can also be correct, and it can be visual, or visual in part. Pictures can be abstract as well, if I think of an infinitely long, 1 dimensional Euclidean line, that's a very abstract picture. But its still a picture, and I can think about that; I can reason about it, visually. I can think about line segments and compare lengths and distances etc. If I see in my mind a curve that is constantly "falling off" another straight line (the log curve versus a line of constant slope 1/e), I can correctly deduce the distance between those at any x value will increase as x increases, even though that does not constitute a rigorous proof. But neither does it rely on any "abstract" sense of distance that is not itself visual. (I've mentioned before that continuity seems to be the norm in intuition, one need not know the modern definition of continuity to get the above correct. Many mathematicians made correct conclusions before any theory of continuity existed. When you bring up such matters as "the theory of continuity", of course I'm going to think you are confusing proof and rigor with more general reasoning.) Usually, the visual thinking will be mixed with other facts and modes of thought. Since we don't know how thought works, its pretty difficult to say more that that, but its certain that the visual apparatus of our brains is involved. There's quite a lot of thinking involved in vision per se, such as suppression of that uninteresting details and even insertion of details that do not originate on the retina. We may as well call all that "vision"  that "stimulus + processing" is what we experience as vision. And, I doubt I see exactly as a rat does. I've mentioned this before as well, artists and photographers *see* differently than untrained people, and so do visually oriented mathematicians. Cheers, Joe N 
On Jul 11, 2013, at 12:16 PM, Joe Niederberger <[hidden email]> wrote: I can reason about it, visually. Everything else you wrote I agree with, except this line. We both agree that we reason, and I think we both agree that if there is no reasoning then it isn't mathematical. But then you say that you reason visually. And then I ask "Really? What's that like?" You say "It's super duper trooper." I say "No. I mean how does it work?" You say "log(x)" I say "Huh?" You say "The graph of log(x) proves that e^pi > pi^e?" Renfro says "Huh?" I say "You mean like this, with rectangle's and such?" You say "No, dumb ass! If you use algebra to reduce the expression to logarithms and then use calculus to take the derivatives of that and understand that this is the slope of the tangent that touches the graph here, here and there, then it becomes intuitively obvious that e^pi > pi^e" I say "Oh! So you mean reasoning applied to a visualization, not visual reasoning." You say "No! I mean visual reasoning." I say "Can you somehow differentiate that from just reasoning?" Bob Hansen 
On Jul 11, 2013, at 12:45 PM, Robert Hansen <[hidden email]> wrote: I say "Can you somehow differentiate that from just reasoning?" To clarify... A. You reason about what you see.' B. You see in things that which you have reasoned. C. You reason visually. I am great with A and B. I do not follow C at all. Bob Hansen

In reply to this post by Dave L. Renfro
R Hansens says:
A. You reason about what you see. B. You see in things that which you have reasoned. C. You reason visually. I am great with A and B. I do not follow C at all. This is a strawman of your own making. You're hung up on the word "reason" adn what *you* think is involved in it. Reasoning, at least, is a process. So is vision, and likewise envisioning. If one reaches a reasonable conclusion, using in part the visual parts of the brain, then the reasoning is at least in part visual. Your definition of reason would seem to be "everything in the process except visualization,..." (and whatever else is on your personal list that you don't consider "reasoning".) I don't care about that definition. I'd rather consider whatever is involved in the overall process. Cheers, Joe N 
On Jul 11, 2013, at 1:31 PM, Joe Niederberger <[hidden email]> wrote: You're hung up on the word "reason" adn what *you* think is involved in it. Um yeah. That has been the theme of my reflection the last 10 years. I am bit more than hung up on it. Bob Hansen

In reply to this post by Dave L. Renfro
Breaking an egg is not the same thing as baking, so I cannot bake breakingly.
Measuring is not baking, so I cannot bake measuringly. Mixing is not baking, so I cannot bake mixingly. I cannot bake at all, except as makebelieve, if I can't do the above. Cheers, Joe N 
In reply to this post by Robert Hansen
On Thu, 11 Jul 2013 10:45:22 0600, Robert Hansen <[hidden email]> wrote:
On Jul 11, 2013, at 12:16 PM, Joe Niederberger wrote: I can reason about it, visually. Everything else you wrote I agree with, except this line. We both agree that we reason, and I think we both agree that if there is no reasoning then it isn't mathematical. Has it occurred to you that you may lack a sense of visualization that Joe possesses? Louis A. Talman Department of Mathematical and Computer Sciences Metropolitan State University of Denver <http://rowdy.msudenver.edu/~talmanl> 
In reply to this post by Dave L. Renfro
Robert Hansen (RH) posted Jul 11, 2013 11:28 PM:
> > On Jul 11, 2013, at 1:31 PM, Joe Niederberger > <[hidden email]> wrote: > > > You're hung up on the word "reason" adn what *you* > think is involved in it. > > Um yeah. That has been the theme of my reflection the > last 10 years. I am bit more than hung up on it. > GSC ("Still Shoveling Away!") 
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