I admit it: I’m a math neanderthal. I just don’t, in so many ways, understand The Number Line, that most basic of mathematical instructional devices.
It’s come up because I’m reading Everything and More: A Compact History of Infinity, by David Foster Wallace. Actually, that’s not reall true. I’ve stopped reading it because I don’t understand, which is one the basic principles explained at the beginning of the book. And I’ve re-read the chapter a number of times, and really, I just don’t get it.
Now, I understand the concept of putting numbers in a line to show their relationship to each other (2 comes after 1, etc). I understand that you can theoretically put and infinite number of numbers between any other two numbers (actually, I start to wig out at that thought, but onwards). What I don’t get is when the idea of ‘space’ is related to the number line. Suddenly, this abstract metaphor that I was quite happy with takes on concrete, measurable properties and muddles it up for me. Why is it important how much space on the number line the set of real numbers occupies, vs, say, the set of irrational numbers (and I may well be mixing terminology here, but onwards). Maybe I don’t get this odd crux of applying physical properties to this completely abstract idea.
And worse, it gets me to thinking about things like Zeno’s (or Xeno? Or someone else entirely?) paradox about how you can never actually go anywhere, because to get anywhere, you have to cross 1/2 that distance, and to cross that 1/2, you have to get 1/2way, etc. and so on and so forth, ad infitum. And I’m generally ok with that, because (and this maybe completely wrong) this paradox assumes that there can be a ‘moment’ in time to worry about travelling within. But I’ve always just kind of assumed our relationship with points of time was also a metaphor, like the number line. There aren’t really any points, we just need to pretend there are, so we can do physics and so on. Which, then, of course, gets me doubting the very nature of physics of which so much seems dependant on the idea of points of time. But because you can always fit an infinite (if infinitely small) amount of time between any two points on the timeline, then… and I run full circle back into that old paradox with points of time.
And there are perfectly reasonably explanations for why we can, in fact, travel anywhere, and I even feel like I understood them at the time, but of course, these have flitted through my head like so much ephemera, and I’m left starting at this abstraction called a number line and worrying about the validity of it’s very existence all over again.
And, if you didn’t have 0 (zero), which, if I’m right in recalling, the Greeks didn’t, how on earth do you have a number line? Like, where do you start it? And how do you do basic arithmetic? Like I want to move 3 spots to the left of 3 on the integer-only number line. Where does that put me? Do I fall off the edge of the metaphor? etc.
I think mathematics would quickly drive me absolutely nutters, where I ever to have studied it. It may, in fact, be why I was never terribly good at it: it’s very foundations clearly bother me. And don’t get me started on the random utlization of base 10 for math. Or why every other base essentially seems to be a corruption of it, the way we use them. Or other meetings of language & math.
Ok. back to work now.
Zeno’s Paradox hasn’t really been answered with total satisfaction yet, but lots of good work has been done by people like Cantor and Russell in the past while (as in hundred or so years).
One thing that you have to realize though is that even though there are infinitely number of numbers between any two numbers, that does not mean that their total distance is infinite. In fact you can easily see that the distance between between 1 and 2 is 1, even though you have infinite number of numbers in between.
A good starting point is to look at something called an ‘infinite series’. A good one is the geometric series. Here you add up all the terms like this: 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … which, if you add up all the terms will be equal to 2. You can even do this on a calculator if you want, just start adding up the terms and watch as the number gets closer and closer to 2, and you can get it as arbitrarly close to 2 as you want, just go further.
The cool thing about this is that it shows how infinity can be finite. So back to Zeno. The paradox says that if a tortoise runs out ahead in a race, and then Achilles tries to catch up he can’t because he must go half the distance, then half again, etc… But you can add up all those halfs and get finite number which cooresponds to a finite length of time.
Zeno’s Paradox hasn’t really been answered with total satisfaction yet, but lots of good work has been done by people like Cantor and Russell in the past while (as in hundred or so years).
One thing that you have to realize though is that even though there are infinitely number of numbers between any two numbers, that does not mean that their total distance is infinite. In fact you can easily see that the distance between between 1 and 2 is 1, even though you have infinite number of numbers in between.
A good starting point is to look at something called an ‘infinite series’. A good one is the geometric series. Here you add up all the terms like this: 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … which, if you add up all the terms will be equal to 2. You can even do this on a calculator if you want, just start adding up the terms and watch as the number gets closer and closer to 2, and you can get it as arbitrarly close to 2 as you want, just go further.
The cool thing about this is that it shows how infinity can be finite. So back to Zeno. The paradox says that if a tortoise runs out ahead in a race, and then Achilles tries to catch up he can’t because he must go half the distance, then half again, etc… But you can add up all those halfs and get finite number which cooresponds to a finite length of time.
(this is too fun, good way to wake up in the morning)
What problem do you have with base 10? Are you on the base 12 camp then (12 having 3 primes as divisors while 10 only has 2). We do have 10 fingers so you can count easily using both hands, although with 12 you could do the 3-knuckle thing and with using your thumb as a pointer count using only one hand. Of course we have to remember to Lagrange who suggested base 11 durring the revolution as every fraction would have the same denominator (and thereby making the metric system people realize that they were being silly in changing the base from 10).
(this is too fun, good way to wake up in the morning)
What problem do you have with base 10? Are you on the base 12 camp then (12 having 3 primes as divisors while 10 only has 2). We do have 10 fingers so you can count easily using both hands, although with 12 you could do the 3-knuckle thing and with using your thumb as a pointer count using only one hand. Of course we have to remember to Lagrange who suggested base 11 durring the revolution as every fraction would have the same denominator (and thereby making the metric system people realize that they were being silly in changing the base from 10).
But isn’t it just wonderful that the universe confounds our attempts to make it simple?
The fact that there’s just so much going on between 1 and 2 on the number line is such a great example of the fact that you don’t have to come up with aliens or gods or psychic friends to find that the world is a mysterious place.
Note that DFW does include a fantastic foreword that describes quite well the feeling you are having. When you start thinking super abstractly you start to doubt the possibility for anything to actually exist.
Steve, regarding your problem with “space” on the number line, I would suggest that you review your first year calculus and the concept of limit, which I realize DFW doesn’t do a super job of explaining (he shouldn’t have tried, is the problem).
The problem is that if you had math courses that tought you to memorize formulas (instead of figure out what they meant), then this stuff gets totally missed.
Math education generally sucks dead donkey dicks.
Regarding zero, that isn’t such an obstacle. The greeks didn’t want to give a symbol for the absense of something. They didn’t do computations using formulas like we do, they figured things out by constructing geometrical figures. Showing that a line segment has no length is the same as saying there is no line segment, so the concept of a zero symbol wasn’t needed. They would say things like, the value of an angle “vanished”.
The history of math is filled with period where new crazy-seeming symbols are introduced and eventually become accepted. Perhaps one day people will wonder why we have such a hard time with complex numbers? (Complex numbers have the form x + yi, were i is the square root of negative 1 – used all the time n electronics, for example).
Anyway, I hope you keep reading the book, if only to discuss ways that DFW could have improved it.
But isn’t it just wonderful that the universe confounds our attempts to make it simple?
The fact that there’s just so much going on between 1 and 2 on the number line is such a great example of the fact that you don’t have to come up with aliens or gods or psychic friends to find that the world is a mysterious place.
Note that DFW does include a fantastic foreword that describes quite well the feeling you are having. When you start thinking super abstractly you start to doubt the possibility for anything to actually exist.
Steve, regarding your problem with “space” on the number line, I would suggest that you review your first year calculus and the concept of limit, which I realize DFW doesn’t do a super job of explaining (he shouldn’t have tried, is the problem).
The problem is that if you had math courses that tought you to memorize formulas (instead of figure out what they meant), then this stuff gets totally missed.
Math education generally sucks dead donkey dicks.
Regarding zero, that isn’t such an obstacle. The greeks didn’t want to give a symbol for the absense of something. They didn’t do computations using formulas like we do, they figured things out by constructing geometrical figures. Showing that a line segment has no length is the same as saying there is no line segment, so the concept of a zero symbol wasn’t needed. They would say things like, the value of an angle “vanished”.
The history of math is filled with period where new crazy-seeming symbols are introduced and eventually become accepted. Perhaps one day people will wonder why we have such a hard time with complex numbers? (Complex numbers have the form x + yi, were i is the square root of negative 1 – used all the time n electronics, for example).
Anyway, I hope you keep reading the book, if only to discuss ways that DFW could have improved it.
Physics actually fucks me up much more than math.
I read recently that when the universe was small, it was so compact that it was like a volume of gas – an atmosphere – and vibration of stuff in the early universe expanding actually made a sound (!) in this atmosphere, and apparently you can look at the background microwave radiation and see the evidence of that sound.
Or how about this thing where our universe has seven dimensions, some of which are very small (!)
These are just reminders that it’s not just abstract math that leads us to seemingly impossible things. The fabric of the universe is fundamentally outrageous.
Physics actually fucks me up much more than math.
I read recently that when the universe was small, it was so compact that it was like a volume of gas – an atmosphere – and vibration of stuff in the early universe expanding actually made a sound (!) in this atmosphere, and apparently you can look at the background microwave radiation and see the evidence of that sound.
Or how about this thing where our universe has seven dimensions, some of which are very small (!)
These are just reminders that it’s not just abstract math that leads us to seemingly impossible things. The fabric of the universe is fundamentally outrageous.
Here’s a crazy thought….
Vancouver Island has an infinitely long shoreline.
Vancouver Island is a fractal. The more precisely you measure a shoreline, the bigger the value you get. An infinitely precise measurement would be infinitely long.
Here’s a crazy thought….
Vancouver Island has an infinitely long shoreline.
Vancouver Island is a fractal. The more precisely you measure a shoreline, the bigger the value you get. An infinitely precise measurement would be infinitely long.
Well, the coastline isn’t really infinite, although it is quite long (a real fractal doesn’t stop at protons, or quarks, or whatever) compared to what is usually reported.
Well, the coastline isn’t really infinite, although it is quite long (a real fractal doesn’t stop at protons, or quarks, or whatever) compared to what is usually reported.