Reading assignment 1: pages 1-7 Mathematical Reality
At the moment, I’m envisioning that this group will try to read a few pages of Measurement this book every week. This will likely mean we won’t finish the book in the fastest fashion, but that isn’t really the goal of this conversation anyway.
To help guide our conversation, I, along with the help of a few other math teachers (Michael Perhsan, Shawn Cornally, and Paul Solomon), have put together some reading questions which will hopefully serve as good starters for conversations in the comments of this post, but please do not feel constrained by these questions—we would much prefer that you ask your own questions than simply respond to ours.
If our conversations become too complex to be handled by the simple commenting system here, we can always move them to a different forum.
Introduction: Reality and Imagination
- What does Lockhart mean by mathematical reality? How does he see this world as different from physical reality? Do you buy it? Where do you think the author is going with this distinction?
- Some people say that there are perfect circles, while others say that everything in the world is imperfectly round. Which side do you agree with? Why?
- What do you think of when you hear the term “mathematical reality”? What experiences have you had exploring mathematical reality?
Developing Mathematical Arguments
Here’s a quick video tutorial to show how easy it is to get started with Geogebra:
- How can this problem be tweaked? Is this a narrow version, or part of some larger truth about all shapes?
- Is this true for other shapes? What happens when you draw the diagonals and connect the midpoints of a rectangle—do they meet at a point? Does this hold true for any 4-sided object? Feel free to explore n-sided shapes—pentagons, hexagons, etc.
- Lockhart mentions that the best problems are you own. In that vein, what’s a mathematical problem that you have some interest in?
1A median is a line from the vertex of a triangle to the midpoint of the opposite side.
The Beauty of Math 
On a fundamental level, I agree, the physical world is decidedly NOT what we’re talking about when we do math. Nonetheless, I use it all the time for reference points and ideas. After all, a major part of me (my body) lives here and interacts with physical objects. I tend to blur the two worlds as I like, but when it comes to getting something right – finding some truth – I must resort to the imaginary, pure world in my head.
I have a 5th grade student who, whenever I draw a circle or line or angle or anything, says, “THAT’s not an angle!” Of course he’s right, and thank god he gets it, but I think it’s alright to let the world’s interact. In fact, they must. Am I right?
I love where Paul says something to the extent of “I’m in the physical world, and the mathematical world is in me.” That’s a lovely kind of symmetry. It also feels very real. As a mathematician, I make up all sorts of things and concepts to play around with, but I can only share them with another math nerd to the extent that I can get our imaginations to interact. It’s really quite different from saying, “here feel this dolphin. Smooth, eh? It’s more like, “imagine this 4-dimensional manifold. Smooth eh?”
Finally, Paul says the mathematical Jungle “haunts your waking dreams.” I definitely agree there. The other day one of the 7th graders in my mathematical art class showed me a pattern she came up with, and I stayed up till 2am playing with it that night. But I don’t dare tell her my research. She gets to play with her toy first.
But thinking about the physical world often requires the same sort of use of the imagination. If I want to share my experience of the function of a protein then I need to provide some sort of model, visualization, or aid to help someone else share my thought. That requires imagination, but it’s undeniably part of describing and experiencing physical reality.
Absolutely. My point exactly. They have to interact and can’t function in isolation.
I don’t like how Lockhart uses the word imaginary to describe math.
1. Most imaginary things don’t exist. For instance, unicorns. Is Lockhart saying that numbers and equations are imaginary in the same way that Godzilla is imaginary?
2. Remember the last time mathematicians called their own stuff “imaginary”? And we got stuck with imaginary numbers and lots of people thinking that some (like numbers) are real, and other stuff is just made up.
But I don’t think that we should take Lockhart too seriously when he talks about this stuff. He writes, “this is personal,” and it’s most charitable to read him as describing his uncritical experience of mathematical reality as something not unlike his imaginary worlds of possibilities.
Is that similar to my experience of mathematical reality? Not really. My experience of thinking about math is a lot like my experience of thinking about anything. Thinking about physics, biology, writing, teaching or learning feels similar to the way I feel when I think about math. I don’t know if I’m just wrong about my experience, or maybe that says something about the way that I interact with math.
Does anybody else share my feeling? Is thinking about non-math radically different for you than thinking about math? Or do you find the experience more-or-less similar?
First, how dare you insinuate that unicorns don’t exist!
I think that even though Bio, Physics, Chem, English, Writing, etc. are very grounded in the real world, they are imaginary in that we use our minds to create models, and to discern facts that aren’t necessarily directly stated. For example, what would English be without inferences? Where would physics be without models of situations in ideal worlds? I think that when Lockhart says imaginary, he means that we use our minds to augment our sensory experiences, not that we are making something up completely unconstrained by perceptions of reality.
But that means that mathematical reality isn’t really any different than biological reality, right? So Lockhart is wrong?
We know that there is no such realizable or observable length as 1/10^30000, so in what sense is that number real? I’m not all to interested in how real math is or not, but the things I consider are mental constructs that may or not represent my experience. I’m buying in.
Disregarding your position, Paul, I find your argument interesting. But who says that length is the only way to empirically ground numbers?
This is an interesting line of thought. I’m open to hearing more.
I will say that if we call 10^30000 feet a wun, then 1/10^30000 wuns is an actual length, namely 1 foot.
Here’s another thought: So much of what we talk about with number is built on the lie of things being the same when they are not. When we split a cake in 6 pieces, it’s quite impossible for us to do it equitably. We just imagine the scenario of equal division.
And countless other examples.
The word imaginary jumped out at me, just as it did for Michael. “Imaginary” and “imagine” bring to mind something that exists only in my my mental world. So here’s my question: Was 7 a prime number 4 million years ago? Everyone I ask this simple question of immediately answers “Yes!” and I do, too. I think. Maybe. Not so sure, now that I think some more.
If pure mathematical objects (the perfect circle, the idea of primeness etc.) exist outside of humankind’s thinking about them, then there is a Platonic world of math concepts that many insist exists. That’s weird for me, doggone it, but I might agree.
I used to be very dismissive of the concept of a Platonic world. As a physicist, I figured I was far too grounded in reality to fall for that silliness. But the more I’ve thought about how we know what we know, our reliance on models of reality to describe reality and our inherent reliance on part of the physical world (our brains) to comprehend the physical world (a lot of bootstrapping we’ve got going on… can this even WORK?), I’m starting to understand why thoughtful mathematicians think in terms of a Platonic world. Well, that and the question I asked above in my first paragraph.
Mark: You might find this interesting.
http://plato.stanford.edu/entries/mathphil-indis/
But there are some things that probably also exist only because of a person or people thinking about them. This image of a symmetric 11-Venn diagram comes to mind.
I’m not so sure that this structure existed before a mathematician thought of it.
But where should we draw the line on these things? Seven has always been a prime, 2^(43112609)-1 (largest known prime)? Numbers like that don’t exist in nature, only in our minds. Are they still pure forms?
I’m still waiting for the book (arghh!). Regarding Q1 above in the ‘Developing Mathematical Arguments’ section, you may be interested in having a look at this quick video on an interesting new function on geogebra. I saw it about 3 months ago and wasn’t quite sure what to make of it (more of a negative reaction than positive). I wonder what other people’s thought are on this?
Dan,
I find this fascinating. My first question is how in the world do you program geogebra to be able to do something like this. But as a teaching tool, I’m not nearly as enthusiastic—it seems to rob the students of any ability to devise the argument for themselves.
We share exactly the same feelings about this John. Whilst I think that it’s a pretty cool function, I don’t have a clue how you would program geogebra to do this (I think I’ll ask the geogebra forum via twitter if anyone can explain this simply). As a teaching and learning tool it completely takes any enjoyment, logical thought and creativity out of mathematics.
It’s a lovely demonstration, but I wouldn’t say it robs the kids of the argument. I don’t think it’s an argument or proof at all. It’s evidence perhaps, but it doesn’t account for all triangles in the way we need it to.
I also don’t have the book yet. When is the end of this assignment? Today?
No rush. I’ll probably post reading questions once per week.
Ok, sounds good!
I am really enjoying Lockhart’s book!
I love the discussion of physical reality vs. mathematical reality. It reminds me of Tegmark’s “Mathematical Universe Hypothesis” (bit.ly/apauvL), and I have blogged (http://bit.ly/SMN8X3) a little about Brian Greene’s wonderful introduction to possibly mathematical multiverses in “The Hidden Reality”
The philosophical question of the physical reality of internal experiences (or the nature of physical reality at all) is so rich and going-down-a-rabbit-hole intense that I am not sure how to respond to the “circle” or the “mathematical reality” question. I really like what Mark, Paul and Michael have posted about this, and I will add that Bertrand Russell has a good discussion of the nature of mathematical and internal truths that would be wonderful to discuss at some point as well (bit.ly/Sgbqwn)!
Book arrived today and I’ve already had quite an interesting experience proving that the medians of a triangle are concurrent. I proved it using co-ordinate geometry but wasn’t very satisfied with the proof – it appeared, to me, to be sligthly inelegant (“My advice is try not to worry about trying to hold yourself to some impossibly high standard of academic excellence”, pg.10 🙂 ) . Anyhow, I then tried using a vector method and struggled with that. So then I naturally decided to check out a few different proofs on the internet and found one done by Sal Khan at the Khan Academy. Funnily enough it’s exactly the same as the one that I used.
It appears that I have a different idea of elegance in a proof to Mr. Khan! Amazingly, this is something that I’ve never really thought much about before – the idea that everyone has ‘their own idea’ of aesthetic beauty in a proof. Proofs were always either elegant or inelegant – I never considered it from the point of view of someone else before. This is probably down to the fact that I never collaborated on anything during University (I know what you’re thinking. Trust me, I wish I could turn back the clock and do it differently!)
By the way, I noticed that James Tanton proved this theorem using vectors. He did it by initially assuming that the intersection point of each median was one third of the distance of the median from the nearest side. His method, as well as mine and Sal’s, just don’t quite do it for me.
Curios to know what other people think about these methods? Also, thoughts on aesthetic beauty of a proof in general?
Dan – Great thoughts here. Love how personal they are. Thanks also for sharing your proof here. It’s not bad, but I’m eager to read what Paul has to say about it. I haven’t worked it through, myself.
Yay! I finally got the book and read the assignment and can make sense of everything you all are saying. 😀 I absolutely love the way he writes! I had never thought of math like that before. And to be honest, I always hated the proofs before. I thought: Who cares about WHY the theorems work? I just cared THAT they work and then use them to solve the problems, that’s what math is all about anyway. But the way he described it really has me thinking like “Yeah, why do things like the Pythagorean theorem work out sooo perfectly?!” This book should really be used in school before geometry. I think it would make a lot more people interested in math, but maybe that’s just me.
Anyway, now for the questions:
1. According to Lockhart, mathematical reality isn’t real, it’s just imaginary. In the real world, nothing is as perfect as it is in math. I would agree with him about this. You can never get an angle that is exactly 90 degrees or a board that is exactly 2 feet long. Everything is an approximation which is why you have to use significant figures in science (not fun! lol). But in math you can take it out to as many decimals as you want because everything is exact! But just because math is in a perfect world of its own does not mean that it can not penetrate the real world. Math is used all the time in real world situations like in construction. Even though things aren’t perfect in the real world, you can still get them close enough to be able to use the math and get a good enough answer to be able to use.
2. I would agree with the side that says that everything in the real world is imperfectly round because there’s really no way to get everything exactly perfect. It will always be at least one atom off.
3. When I hear the term “mathematical reality” I think of the truths of math, how you can prove certain ideas, unlike in science where things are just based on observation and experience. Like you can never prove that everything is always in motion. An atom might have stopped for a split second 13 years ago…you never know. 😀 But with math, you can prove that a^2 + b^2 = c^2 using numbers and the definitions that determine what a triangle is.
So glad to hear your enthusiasm, Elizabeth. It’s so easy to think math is just about solving the problems the teacher hands you, but Lockhart really wakes us up to the fact that math is something else. Something beautiful.
I think you’re pretty spot on about the connection between mathematical reality and our physical world. I love what you said about an atom off. This is the real sticking point. Especially because atoms do all sorts of weird things that are hard (in some sense impossible) to pin down.
How can we ever really point to something as 1 apple, when the atoms that constitute the apple are in a constant state of change, sometimes leaving the apple, while maybe others return.
The same is true of all humans, but that’s another story.
“but Lockhart really wakes us up to the fact that math is something else. Something beautiful.”
Yeah, he really does. Math isn’t just a tool to use, but it’s also a whole realm to explore.
Here’s a pretty interesting blog post on the the question “Do mathematical objects exist”.
An excerpt: