thinking logarithmically
October 10, 2012 12:58 PM Subscribe
7 is spacial to me.
posted by weapons-grade pandemonium at 1:03 PM on October 10, 2012 [12 favorites]
posted by weapons-grade pandemonium at 1:03 PM on October 10, 2012 [12 favorites]
6 is much closer to 5 because 7 8 9.
posted by perhapses at 1:06 PM on October 10, 2012 [15 favorites]
posted by perhapses at 1:06 PM on October 10, 2012 [15 favorites]
The Weber–Fechner law combines two different laws. Some authors use the term to mean Weber's law, and others Fechner's law. Fechner himself added confusion to the literature by calling his own law Weber's law.
I can't vouch for the rest of it, but the opening bit of the linked wikipedia article is adorable.
posted by stupidsexyFlanders at 1:09 PM on October 10, 2012 [12 favorites]
I can't vouch for the rest of it, but the opening bit of the linked wikipedia article is adorable.
posted by stupidsexyFlanders at 1:09 PM on October 10, 2012 [12 favorites]
Canine ate seven sick five year olds.
posted by spectrevsrector at 1:10 PM on October 10, 2012 [4 favorites]
posted by spectrevsrector at 1:10 PM on October 10, 2012 [4 favorites]
Some of the knock-on effects of the naturally logarhithmic conceptual models our brains use really suggest that engineers might make fewer gross errors if they did rough calculations on slide rules.
posted by seanmpuckett at 1:10 PM on October 10, 2012 [1 favorite]
posted by seanmpuckett at 1:10 PM on October 10, 2012 [1 favorite]
4.5
posted by blackfly at 1:14 PM on October 10, 2012 [2 favorites]
posted by blackfly at 1:14 PM on October 10, 2012 [2 favorites]
Trick question. Numbers do not have proximity to each other, as they are not spacial.
Uh... I'm pretty sure we've got a well-defined notion of distance. Several, even. Or was this some joke I'm not clever enough to understand?
posted by hoyland at 1:15 PM on October 10, 2012 [1 favorite]
Uh... I'm pretty sure we've got a well-defined notion of distance. Several, even. Or was this some joke I'm not clever enough to understand?
posted by hoyland at 1:15 PM on October 10, 2012 [1 favorite]
This makes sense if you've every adjusted the volume on an older stereo.
posted by 2bucksplus at 1:16 PM on October 10, 2012 [3 favorites]
posted by 2bucksplus at 1:16 PM on October 10, 2012 [3 favorites]
4.5
How many layers of irony are contained here
posted by MangyCarface at 1:16 PM on October 10, 2012 [2 favorites]
How many layers of irony are contained here
posted by MangyCarface at 1:16 PM on October 10, 2012 [2 favorites]
My favorite example of this was (extremely poorly explained) in the wikipedia page:
"Hipparchus ranked the stars he could see in terms of their brightness, with 1 representing the brightest down to 6 representing the faintest, though now the scale has been extended beyond these limits; an increase in 5 magnitudes corresponds to a decrease in brightness by a factor of 100."
The idea is that Hipparchus classified the starts into 6 groups, with class 1 being the brightest and 6 being the darkest. He made his classifications so that they would fall out into even groups, so there was an even difference in brightness between say class 5 and class 6; it was intended to be linear. But if you actually look at the number of photons which hit your eye for the different classes, you find it's logarithmic, and a class 6 is 100x dimmer than a class 1, not 6x. It's also true with the moon, which thousands upon thousands of times brighter in the night sky than anything else, much brighter than it looks to us. Fun little trivia fact I always use to water down the "Do you think there's aliens?" conversations that pervade stargazing.
posted by Buckt at 1:18 PM on October 10, 2012 [4 favorites]
"Hipparchus ranked the stars he could see in terms of their brightness, with 1 representing the brightest down to 6 representing the faintest, though now the scale has been extended beyond these limits; an increase in 5 magnitudes corresponds to a decrease in brightness by a factor of 100."
The idea is that Hipparchus classified the starts into 6 groups, with class 1 being the brightest and 6 being the darkest. He made his classifications so that they would fall out into even groups, so there was an even difference in brightness between say class 5 and class 6; it was intended to be linear. But if you actually look at the number of photons which hit your eye for the different classes, you find it's logarithmic, and a class 6 is 100x dimmer than a class 1, not 6x. It's also true with the moon, which thousands upon thousands of times brighter in the night sky than anything else, much brighter than it looks to us. Fun little trivia fact I always use to water down the "Do you think there's aliens?" conversations that pervade stargazing.
posted by Buckt at 1:18 PM on October 10, 2012 [4 favorites]
The use of computer memory as a metaphor for human memory, plus the evolutionary rationale seem a little fishy to me - is my natural suspicion of science news reports getting a little over sensitive?
That's suspicion of findings being sexed up, misleading metaphors being shoehorned in and so on, not suspicion as in 'science is a left wing atheist plot'.
posted by spectrevsrector at 1:20 PM on October 10, 2012 [1 favorite]
That's suspicion of findings being sexed up, misleading metaphors being shoehorned in and so on, not suspicion as in 'science is a left wing atheist plot'.
posted by spectrevsrector at 1:20 PM on October 10, 2012 [1 favorite]
How many layers of irony are contained here
potentially none.
posted by goethean at 1:20 PM on October 10, 2012
potentially none.
posted by goethean at 1:20 PM on October 10, 2012
The 3/5 debate was the subject of this interesting radiolab bit.
posted by Lutoslawski at 1:28 PM on October 10, 2012 [2 favorites]
posted by Lutoslawski at 1:28 PM on October 10, 2012 [2 favorites]
jinx, mike.
posted by Lutoslawski at 1:28 PM on October 10, 2012 [1 favorite]
posted by Lutoslawski at 1:28 PM on October 10, 2012 [1 favorite]
I just asked a small child 'what number is in the middle of 1 and 9?'
Her response was, 'I don't have my watch.'
Which is a perfectly valid answer, as far as I'm concerned.
posted by bilabial at 1:28 PM on October 10, 2012 [14 favorites]
Her response was, 'I don't have my watch.'
Which is a perfectly valid answer, as far as I'm concerned.
posted by bilabial at 1:28 PM on October 10, 2012 [14 favorites]
Uh, wouldn't it be 5? It's not halfway between 0 and 9; it's halfway between 1 and 9.
(Also, every single time I've tried to clarify a math concept in my head by reading a wikipedia page, I have ended up hopelessly in the weeds of pedantically accurate but useless to a layperson descriptions.)
posted by klangklangston at 1:29 PM on October 10, 2012 [15 favorites]
(Also, every single time I've tried to clarify a math concept in my head by reading a wikipedia page, I have ended up hopelessly in the weeds of pedantically accurate but useless to a layperson descriptions.)
posted by klangklangston at 1:29 PM on October 10, 2012 [15 favorites]
is my natural suspicion of science news reports getting a little over sensitive?
I don't think so. I'm not a psychologist, but the abstract of the article (available freely here and I am sure there is a full-text reprint available somewhere) states that "[t]heoretical results and experimental verification for perception of sound intensity are both presented" -- the article, the empirical law and the results are all to do with responses to stimuli though the lede is buried in the press office piece.
But, as usual, when university press offices get hold of something it gets fiddled with. A cask-strength single malt journal article ends up watered down, blended, chillfiltered and served with ice until it loses all taste and interest.
posted by Talkie Toaster at 1:30 PM on October 10, 2012
I don't think so. I'm not a psychologist, but the abstract of the article (available freely here and I am sure there is a full-text reprint available somewhere) states that "[t]heoretical results and experimental verification for perception of sound intensity are both presented" -- the article, the empirical law and the results are all to do with responses to stimuli though the lede is buried in the press office piece.
But, as usual, when university press offices get hold of something it gets fiddled with. A cask-strength single malt journal article ends up watered down, blended, chillfiltered and served with ice until it loses all taste and interest.
posted by Talkie Toaster at 1:30 PM on October 10, 2012
This part is confusing me:
The STIR researchers demonstrate that if you're trying to minimize relative error, using a logarithmic scale is the best approach under two different conditions: One is if you're trying to store your representations of the outside world in memory; the other is if sensory stimuli in the outside world happen to fall into particular statistical patterns.
If you're trying to store data in memory, a logarithmic scale is optimal if there's any chance of error in either storage or retrieval, or if you need to compress the data so that it takes up less space. The researchers believe that one of these conditions probably pertains — there's evidence in the psychological literature for both — but they're not committed to either. They do feel, however, that the pressures of memory storage probably explain the natural human instinct to represent numbers logarithmically.
The second condition of the first paragraph makes sense to me. Is the first condition, and then the following paragraph, talking about human memory in your brain? Or computer memory? A couple paragraphs before they were talking about compressing mp3s and jpgs, so if it's computer memory, how does that explain the natural human instinct?
Also is there a study or something for the "Kids will tell you 3" in the radio lab? I can't listen to it at the moment but it seems like there might be something published about it they could've linked.
Cool post!
posted by DynamiteToast at 1:32 PM on October 10, 2012
The STIR researchers demonstrate that if you're trying to minimize relative error, using a logarithmic scale is the best approach under two different conditions: One is if you're trying to store your representations of the outside world in memory; the other is if sensory stimuli in the outside world happen to fall into particular statistical patterns.
If you're trying to store data in memory, a logarithmic scale is optimal if there's any chance of error in either storage or retrieval, or if you need to compress the data so that it takes up less space. The researchers believe that one of these conditions probably pertains — there's evidence in the psychological literature for both — but they're not committed to either. They do feel, however, that the pressures of memory storage probably explain the natural human instinct to represent numbers logarithmically.
The second condition of the first paragraph makes sense to me. Is the first condition, and then the following paragraph, talking about human memory in your brain? Or computer memory? A couple paragraphs before they were talking about compressing mp3s and jpgs, so if it's computer memory, how does that explain the natural human instinct?
Also is there a study or something for the "Kids will tell you 3" in the radio lab? I can't listen to it at the moment but it seems like there might be something published about it they could've linked.
Cool post!
posted by DynamiteToast at 1:32 PM on October 10, 2012
The 3/5 debate was the subject of this interesting radiolab bit.
Historically, Americans haven't been real good with the number 3/5.
posted by Benny Andajetz at 1:34 PM on October 10, 2012 [20 favorites]
Historically, Americans haven't been real good with the number 3/5.
posted by Benny Andajetz at 1:34 PM on October 10, 2012 [20 favorites]
Oh yay, it's been far too long since I've had an opportunity to remember the profound confusion and horror of having my asshole 6th grade math teacher shout mercilessly at me for concluding it was two months from January to March.
Protip: roaring "It's either one or three!" at an otherwise mathematically-inclined 11-year-old without further explanation is not a great pedagogical tool. Also: Xanax or another line of work, perhaps?
posted by psoas at 1:35 PM on October 10, 2012
Protip: roaring "It's either one or three!" at an otherwise mathematically-inclined 11-year-old without further explanation is not a great pedagogical tool. Also: Xanax or another line of work, perhaps?
posted by psoas at 1:35 PM on October 10, 2012
I just asked a small child 'what number is in the middle of 1 and 9?'
Her response was, 'I don't have my watch.'
It's cool, I checked mine. The answer is "eleven."
posted by nebulawindphone at 1:36 PM on October 10, 2012 [20 favorites]
Her response was, 'I don't have my watch.'
It's cool, I checked mine. The answer is "eleven."
posted by nebulawindphone at 1:36 PM on October 10, 2012 [20 favorites]
Is the first condition, and then the following paragraph, talking about human memory in your brain? Or computer memory?
I get the impression that it's talking about an information-theoretic approach, which doesn't consider the differences between human memory and computer memory to be relevant. If you're dealing with certain kinds of noise or are trying to compress certain kinds of data, storing the data logarithmically is better. The fact that you happen to be storing the information with neurons or electrical flows in NAND gates or magnetic domains on a hard drive doesn't affect this.
posted by a snickering nuthatch at 1:41 PM on October 10, 2012
I get the impression that it's talking about an information-theoretic approach, which doesn't consider the differences between human memory and computer memory to be relevant. If you're dealing with certain kinds of noise or are trying to compress certain kinds of data, storing the data logarithmically is better. The fact that you happen to be storing the information with neurons or electrical flows in NAND gates or magnetic domains on a hard drive doesn't affect this.
posted by a snickering nuthatch at 1:41 PM on October 10, 2012
me: “Trick question. Numbers do not have proximity to each other, as they are not spacial.”
hoyland: “Uh... I'm pretty sure we've got a well-defined notion of distance. Several, even. Or was this some joke I'm not clever enough to understand?”
I'm being a bit whimsical, but I've been thinking about this since it came up over on Hacker News, and I really believe it's not really a meaningful question; or rather, it's a question that relies on a sort of weak analogy to make sense, and it would be an obvious question if it were only stated correctly.
The problem is that most human beings today are indoctrinated from youth with Dedekind's hackneyed ideas about numbers; so when someone mentions numbers to them, they generally don't think about numbers – they picture a line with notches on it. But that removes most of the actual meaning from numbers, rendering them confusing and obscuring many important things about them.
Numbers really aren't spacial; they're an expression of multitude. And I really do believe that imposing spacial analogies on numbers muddies the waters and serves only to confuse us about their true nature.
posted by koeselitz at 1:41 PM on October 10, 2012 [5 favorites]
hoyland: “Uh... I'm pretty sure we've got a well-defined notion of distance. Several, even. Or was this some joke I'm not clever enough to understand?”
I'm being a bit whimsical, but I've been thinking about this since it came up over on Hacker News, and I really believe it's not really a meaningful question; or rather, it's a question that relies on a sort of weak analogy to make sense, and it would be an obvious question if it were only stated correctly.
The problem is that most human beings today are indoctrinated from youth with Dedekind's hackneyed ideas about numbers; so when someone mentions numbers to them, they generally don't think about numbers – they picture a line with notches on it. But that removes most of the actual meaning from numbers, rendering them confusing and obscuring many important things about them.
Numbers really aren't spacial; they're an expression of multitude. And I really do believe that imposing spacial analogies on numbers muddies the waters and serves only to confuse us about their true nature.
posted by koeselitz at 1:41 PM on October 10, 2012 [5 favorites]
The Weber–Fechner law combines two different laws. Some authors use the term to mean Weber's law, and others Fechner's law. Fechner himself added confusion to the literature by calling his own law Weber's law.I can't vouch for the rest of it, but the opening bit of the linked wikipedia article is adorable.
Every time I read Wikipedia, every time, I get more inspiration for the What Came After Lexicon game. (WCA on the Grey.)
posted by BrashTech at 1:44 PM on October 10, 2012 [2 favorites]
3 holds for me if I'm considering quantity and effort.
Imagine one pie. Got it? Easy.
Imagining over 5 pies gets harder to visualize, and the larger numbers are equally hard, because you have to divide them into two separate smaller groups.
3 is in halfway to the largest easily visualized number of pies, without grouping them.
posted by StickyCarpet at 1:44 PM on October 10, 2012
Imagine one pie. Got it? Easy.
Imagining over 5 pies gets harder to visualize, and the larger numbers are equally hard, because you have to divide them into two separate smaller groups.
3 is in halfway to the largest easily visualized number of pies, without grouping them.
posted by StickyCarpet at 1:44 PM on October 10, 2012
But that removes most of the actual meaning from numbers, rendering them confusing and obscuring many important things about them.
Such as? (Genuine question.)
In any case, I'm not seeing why you need to look at the real-number line to get some sort of distance metric. Is it somehow wrong or incoherent to treat subtraction over the integers as a kind of distance?
posted by nebulawindphone at 1:47 PM on October 10, 2012 [1 favorite]
Such as? (Genuine question.)
In any case, I'm not seeing why you need to look at the real-number line to get some sort of distance metric. Is it somehow wrong or incoherent to treat subtraction over the integers as a kind of distance?
posted by nebulawindphone at 1:47 PM on October 10, 2012 [1 favorite]
Monkeymoo: "Why is 6 scared of 7?"
Because 7 didn't read the previous comments?
posted by Bonzai at 1:48 PM on October 10, 2012 [19 favorites]
Because 7 didn't read the previous comments?
posted by Bonzai at 1:48 PM on October 10, 2012 [19 favorites]
In a couple of months, I'm gonna read this thread and yell "TIMING!" Just thought I'd give y'all a heads-up.
posted by nebulawindphone at 1:49 PM on October 10, 2012 [4 favorites]
posted by nebulawindphone at 1:49 PM on October 10, 2012 [4 favorites]
I get the impression that it's talking about an information-theoretic approach, which doesn't consider the differences between human memory and computer memory to be relevant. If you're dealing with certain kinds of noise or are trying to compress certain kinds of data, storing the data logarithmically is better. The fact that you happen to be storing the information with neurons or electrical flows in NAND gates or magnetic domains on a hard drive doesn't affect this.
Ok so in that case, the confusion for me is coming from this:
If you're trying to store data in memory, a logarithmic scale is optimal if there's any chance of error in either storage or retrieval, or if you need to compress the data so that it takes up less space. The researchers believe that one of these conditions probably pertains — there's evidence in the psychological literature for both — but they're not committed to either.
How do those relate to human memory? I guess I'm just having trouble applying it to human memory, is the "evidence in the psychological literature" linked to? Because that'd probably explain it best.
posted by DynamiteToast at 1:54 PM on October 10, 2012
Ok so in that case, the confusion for me is coming from this:
If you're trying to store data in memory, a logarithmic scale is optimal if there's any chance of error in either storage or retrieval, or if you need to compress the data so that it takes up less space. The researchers believe that one of these conditions probably pertains — there's evidence in the psychological literature for both — but they're not committed to either.
How do those relate to human memory? I guess I'm just having trouble applying it to human memory, is the "evidence in the psychological literature" linked to? Because that'd probably explain it best.
posted by DynamiteToast at 1:54 PM on October 10, 2012
THERE! ARE! FOUR! LIGHTS!
wait, what was the question again?
posted by Halloween Jack at 1:58 PM on October 10, 2012 [5 favorites]
wait, what was the question again?
posted by Halloween Jack at 1:58 PM on October 10, 2012 [5 favorites]
Biology For Photographers - Why Is The Aperture Scale Logarithmic?
Logtime: The Subjective Scale Of Life
posted by the man of twists and turns at 2:00 PM on October 10, 2012 [8 favorites]
Logtime: The Subjective Scale Of Life
Time scarcer? Years getting shorter? Want an explanation? Logtime is the cognitive hypothesis that our age is our basis for estimating time intervals, resulting in a perceived shrinking of our years as we grow older. A simple mathematical analysis shows that our time perception should be logarithmic, giving us a subjective scale of life very different from that of the calendar. Our perception of aging seems to follow the same (Weber-Fechner) law as our perception of physical stimuli.Log Or Linear? - Distinct intuitions of the number scale in Western and Amazonian indigene cultures
The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention, or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages the Mundurucu mapped symbolic and non-symbolic numbers onto a logarithmic scale, while Western adults used a linear mapping with small or symbolic numbers, and a logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. Thus, the mapping of numbers onto space is a universal intuition, and this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.Now, you're thinking with logarithms!
posted by the man of twists and turns at 2:00 PM on October 10, 2012 [8 favorites]
Anyone who does not answer "five" is wrong. Math is not ruled by public opinion polls.
How's this for "theorizing": Give those kids and those "people living in some traditional societies" (whatever the hell that means) nine objects numbered 1 through 9 and placed in order, and tell them to point to the middle one; I bet everything I own that the majority will point to the thing with the 5 on it.
posted by Sys Rq at 2:04 PM on October 10, 2012 [7 favorites]
How's this for "theorizing": Give those kids and those "people living in some traditional societies" (whatever the hell that means) nine objects numbered 1 through 9 and placed in order, and tell them to point to the middle one; I bet everything I own that the majority will point to the thing with the 5 on it.
posted by Sys Rq at 2:04 PM on October 10, 2012 [7 favorites]
Nice belt!
posted by jessamyn at 2:07 PM on October 10, 2012 [3 favorites]
posted by jessamyn at 2:07 PM on October 10, 2012 [3 favorites]
> Trick question. Numbers do not have proximity to each other, as they are not spacial.
True! However vanilla is exactly halfway between chocolate and strawberry.
posted by jfuller at 2:08 PM on October 10, 2012 [5 favorites]
True! However vanilla is exactly halfway between chocolate and strawberry.
posted by jfuller at 2:08 PM on October 10, 2012 [5 favorites]
Okay fine.
We'll split this 9,000 dollars in half. You get the first three thousand, and I'll take the second half.
posted by mule98J at 2:09 PM on October 10, 2012 [5 favorites]
We'll split this 9,000 dollars in half. You get the first three thousand, and I'll take the second half.
posted by mule98J at 2:09 PM on October 10, 2012 [5 favorites]
My abstract dyscalculia suggests that the proper answer is Wassily Kandinsky.
posted by cmyk at 2:13 PM on October 10, 2012 [5 favorites]
posted by cmyk at 2:13 PM on October 10, 2012 [5 favorites]
This does remind me a bit of high school chemistry class, where we were introduced to bizarre rounding methods, including one where if there was a .5, you looked at the next digit for guidance. So you round .54 and smaller down, and .56 up. With .55, you look to the next digit. We were never able to convince the teacher this lead to a systemic bias, with .555555 (repeating) being the midpoint instead of .55. Possibly because we didn't have the mathematical background of countable and uncountable infinities.
Fortunately, this nonsense didn't appear in AP Chemistry.
posted by pwnguin at 2:16 PM on October 10, 2012
Fortunately, this nonsense didn't appear in AP Chemistry.
posted by pwnguin at 2:16 PM on October 10, 2012
koeselitz: "when someone mentions numbers to them, they generally don't think about numbers – they picture a line with notches on it. [...] Numbers really aren't spacial ..."First of all, it's spatial, with a t. Please. It kills my inner geographer to see that other abomination of orthography.
Second, of course numbers are spatial. They are distributed in a one-dimensional space (or in a straight line in n-dimensional space, depending on how many drinks you've had on that particular day). The number line with "notches" is just a map to help you not get lost.
posted by brokkr at 2:16 PM on October 10, 2012 [3 favorites]
Five. Don't make damn simple questions complicated! It doesn't matter than neuroscientists tell you, it is FIVE, because it is the same number of integers away from one and nine.
And spatialness IS a useful way to understand numbers, because they exist along a continuum. The "notched number line" is analogous to the sequences of values, gives us a way to represent fractions, and makes obvious concepts of relative size. We use graphs to depict imaginary numbers, and that's just the same thing but with two axises. What the heck is wrong with that?
posted by JHarris at 2:18 PM on October 10, 2012 [1 favorite]
And spatialness IS a useful way to understand numbers, because they exist along a continuum. The "notched number line" is analogous to the sequences of values, gives us a way to represent fractions, and makes obvious concepts of relative size. We use graphs to depict imaginary numbers, and that's just the same thing but with two axises. What the heck is wrong with that?
posted by JHarris at 2:18 PM on October 10, 2012 [1 favorite]
We'll split this 9,000 dollars in half. You get the first three thousand, and I'll take the second half.
That's different. Set aside $1000 first, and then the analogy works. (But, yes, you'd still end up with more than $3000.)
posted by Sys Rq at 2:19 PM on October 10, 2012 [1 favorite]
That's different. Set aside $1000 first, and then the analogy works. (But, yes, you'd still end up with more than $3000.)
posted by Sys Rq at 2:19 PM on October 10, 2012 [1 favorite]
a line with notches on it
Kids Misperceptions About Numbers And How They Fix Them
for a certain value of 'correct.'
Addition Is Useless Multiplication Is King: Channeling Our Inner Logarithm
posted by the man of twists and turns at 2:19 PM on October 10, 2012
Kids Misperceptions About Numbers And How They Fix Them
for a certain value of 'correct.'
Addition Is Useless Multiplication Is King: Channeling Our Inner Logarithm
posted by the man of twists and turns at 2:19 PM on October 10, 2012
Hmmm, but if you show them nine groups of things
If you lay out the American paper currency,
Interesting suggestion.
posted by fantabulous timewaster at 2:19 PM on October 10, 2012 [4 favorites]
odo you still expect everyone to pick the group of five? I'm not sure.
oo
ooo
ooo
o
ooo
oo
ooo
ooo
ooo
ooo
o
ooo
ooo
oo
ooo
ooo
ooo
If you lay out the American paper currency,
$1 $2 $5 $10 $20 $50 $100then the ten is "in the middle"; the fifty feels "far from the middle."
Interesting suggestion.
posted by fantabulous timewaster at 2:19 PM on October 10, 2012 [4 favorites]
do you still expect everyone to pick the group of five? I'm not sure.
You can make any obvious fact difficult if you depict it in a confusing manner. (You can also make obscure facts obvious if you illustrate them in a clear manner. Like along a number line.)
posted by JHarris at 2:22 PM on October 10, 2012 [3 favorites]
You can make any obvious fact difficult if you depict it in a confusing manner. (You can also make obscure facts obvious if you illustrate them in a clear manner. Like along a number line.)
posted by JHarris at 2:22 PM on October 10, 2012 [3 favorites]
me: “But that removes most of the actual meaning from numbers, rendering them confusing and obscuring many important things about them.”
nebulawindphone: “Such as? (Genuine question.) In any case, I'm not seeing why you need to look at the real-number line to get some sort of distance metric. Is it somehow wrong or incoherent to treat subtraction over the integers as a kind of distance?””
Well – chiefly, I think it obscures the priority of numbers as we actually know them, a priority which is more fundamental to the being of numbers than the general mechanical expressions that were the basis of Dedekind's ideas.
We don't start by experiencing zero, or ten, or two thousand four hundred and twenty six. We start by experiencing one. The experience of a thing as one is wrapped up ineluctably in the experience of a thing as existing; we cannot say that a thing is without saying a thing is one, and we cannot say that a thing is one without saying that it is.
By implication, our confrontation of oneness necessarily brings us in contact with nothingness. If a thing is, all around the edges of that thing are its negative delineations; that is, it is surrounded by things that are not the thing. So zero is a natural result of oneness becoming an object of our cognition.
But clearly the things a oneness might be surrounded by aren't necessarily nothingness; they have distinctness. The next fundamental principle we encounter is that onenesses can have multiplicity; that the many are a possibility. This is where numbers are born: from the paired apprehensions that reality can cohere in oneness and yet can partake of multiplicity.
In fact, as I see it, this is the beginning and the end of numbers: the interplay between oneness and multiplicity. However, necessarily in the physical world "numbers" begin to be identified not with actual numbers but with equal unitary lengths. Which is why we talk about floating-point numbers or fractionary numbers like 4.23 and 32/7ths. But I think it's clear that these are another thing entirely from our fundamental experience of numbers. What is 32/7ths? It is thirty two chunks, seven of which can be combined to create some other chunk which has been defined as a unit. But numbers don't inherently carry size as a quality; three partakes of one in the sense that it carries three units within it, and a certain size is not a necessary characteristic of a unit. So when we talk about a number like 32/7ths, we've already added a whole lot more to the conception of what numbers can be.
All of these things are forgotten and ignored when we consider numbers as a featureless line on which -1, -0.5, 0, 0.5, 1, 1.5, etc appear. In fact, if we wanted to make a line one which numbers actually occur by priority in existence as I've outlined above (although this would still obscure some of the nature of numbers) it would make more sense to put One first, then Zero, then Many, then Fractions.
brokkr: “First of all, it's spatial, with a t. Please. It kills my inner geographer to see that other abomination of orthography.”
Even if that kind of linguistic prescriptivism were warranted, it is an alternate spelling. And from the looks of things it's a common enough one.
“Second, of course numbers are spatial. They are distributed in a one-dimensional space (or in a straight line in n-dimensional space, depending on how many drinks you've had on that particular day). The number line with "notches" is just a map to help you not get lost.”
I'm afraid I can't make much sense of what you've said here. Where are the numbers distributed in one-dimensional space?
posted by koeselitz at 2:25 PM on October 10, 2012 [2 favorites]
nebulawindphone: “Such as? (Genuine question.) In any case, I'm not seeing why you need to look at the real-number line to get some sort of distance metric. Is it somehow wrong or incoherent to treat subtraction over the integers as a kind of distance?””
Well – chiefly, I think it obscures the priority of numbers as we actually know them, a priority which is more fundamental to the being of numbers than the general mechanical expressions that were the basis of Dedekind's ideas.
We don't start by experiencing zero, or ten, or two thousand four hundred and twenty six. We start by experiencing one. The experience of a thing as one is wrapped up ineluctably in the experience of a thing as existing; we cannot say that a thing is without saying a thing is one, and we cannot say that a thing is one without saying that it is.
By implication, our confrontation of oneness necessarily brings us in contact with nothingness. If a thing is, all around the edges of that thing are its negative delineations; that is, it is surrounded by things that are not the thing. So zero is a natural result of oneness becoming an object of our cognition.
But clearly the things a oneness might be surrounded by aren't necessarily nothingness; they have distinctness. The next fundamental principle we encounter is that onenesses can have multiplicity; that the many are a possibility. This is where numbers are born: from the paired apprehensions that reality can cohere in oneness and yet can partake of multiplicity.
In fact, as I see it, this is the beginning and the end of numbers: the interplay between oneness and multiplicity. However, necessarily in the physical world "numbers" begin to be identified not with actual numbers but with equal unitary lengths. Which is why we talk about floating-point numbers or fractionary numbers like 4.23 and 32/7ths. But I think it's clear that these are another thing entirely from our fundamental experience of numbers. What is 32/7ths? It is thirty two chunks, seven of which can be combined to create some other chunk which has been defined as a unit. But numbers don't inherently carry size as a quality; three partakes of one in the sense that it carries three units within it, and a certain size is not a necessary characteristic of a unit. So when we talk about a number like 32/7ths, we've already added a whole lot more to the conception of what numbers can be.
All of these things are forgotten and ignored when we consider numbers as a featureless line on which -1, -0.5, 0, 0.5, 1, 1.5, etc appear. In fact, if we wanted to make a line one which numbers actually occur by priority in existence as I've outlined above (although this would still obscure some of the nature of numbers) it would make more sense to put One first, then Zero, then Many, then Fractions.
brokkr: “First of all, it's spatial, with a t. Please. It kills my inner geographer to see that other abomination of orthography.”
Even if that kind of linguistic prescriptivism were warranted, it is an alternate spelling. And from the looks of things it's a common enough one.
“Second, of course numbers are spatial. They are distributed in a one-dimensional space (or in a straight line in n-dimensional space, depending on how many drinks you've had on that particular day). The number line with "notches" is just a map to help you not get lost.”
I'm afraid I can't make much sense of what you've said here. Where are the numbers distributed in one-dimensional space?
posted by koeselitz at 2:25 PM on October 10, 2012 [2 favorites]
Hey dudes, mathematical psychology is psychology, not math. Nobody's trying to prove that 3 is "actually" halfway between 1 and 9 and mess up all your nice spreadsheets
posted by theodolite at 2:27 PM on October 10, 2012 [3 favorites]
posted by theodolite at 2:27 PM on October 10, 2012 [3 favorites]
It's cool, I checked mine. The answer is "eleven."That's ridiculous. It's not even funny.
posted by fullerine at 2:28 PM on October 10, 2012 [5 favorites]
Here's the thing: integers and real numbers are different things. They appear to overlap but they (the sets) behave totally differently. Koeselitz, you're complaining about people treating -1, -0.5, 0, etc as real numbers and losing some of their integer-ness. But the set of integers still behaves the same way it did before someone* came along and stuck the real numbers in between them.
*dedekind, who was most assuredly not hackneyed, and in fact developed one of the most creative definitions in all of mathematics
posted by milestogo at 2:31 PM on October 10, 2012 [1 favorite]
*dedekind, who was most assuredly not hackneyed, and in fact developed one of the most creative definitions in all of mathematics
posted by milestogo at 2:31 PM on October 10, 2012 [1 favorite]
Well, then it really just depends on what you mean by "behaves." (Also, I didn't call Dedekind hackneyed; he is most definitely fun to read, and very thoughtful. I just think his ideas are hackneyed.)
posted by koeselitz at 2:32 PM on October 10, 2012
posted by koeselitz at 2:32 PM on October 10, 2012
If you're trying to store data in memory, a logarithmic scale is optimal if there's any chance of error in either storage or retrieval, or if you need to compress the data so that it takes up less space. The researchers believe that one of these conditions probably pertains — there's evidence in the psychological literature for both — but they're not committed to either.How do those relate to human memory? I guess I'm just having trouble applying it to human memory, is the "evidence in the psychological literature" linked to? Because that'd probably explain it best.
Storage and retrieval to and from memory are processes (not necessarily distinct processes) that information in humans undergoes. When something is committed to memory, it may be subject to error- simplifications, misperceptions, etc. When something is retrieved from memory, it is subject to interference from other related memories and the filter of our preconceived notions.
This does remind me a bit of high school chemistry class, where we were introduced to bizarre rounding methods, including one where if there was a .5, you looked at the next digit for guidance. So you round .54 and smaller down, and .56 up. With .55, you look to the next digit. We were never able to convince the teacher this lead to a systemic bias, with .555555 (repeating) being the midpoint instead of .55. Possibly because we didn't have the mathematical background of countable and uncountable infinities.
Fortunately, this nonsense didn't appear in AP Chemistry.
They should have been rounding by looking at whether the subsequent number was odd or even, rather than whether it was >5.
posted by a snickering nuthatch at 2:36 PM on October 10, 2012 [1 favorite]
But humans are biologically incapable of thinking logarithmically, that's why the Singularity is going to happen!
posted by ckape at 2:38 PM on October 10, 2012
posted by ckape at 2:38 PM on October 10, 2012
It probably says a lot about me that I couldn't even begin to make sense of this FPP until the PetaPixel article about aperture. /photodork
posted by Doleful Creature at 2:38 PM on October 10, 2012
posted by Doleful Creature at 2:38 PM on October 10, 2012
They should have been rounding by looking at whether the subsequent number was odd or even, rather than whether it was >5.
They should have been rounding up if it was greater than or equal to five.
posted by Sys Rq at 2:41 PM on October 10, 2012
They should have been rounding up if it was greater than or equal to five.
posted by Sys Rq at 2:41 PM on October 10, 2012
... of course numbers are spatial. They are distributed in a one-dimensional space (or in a straight line in n-dimensional space, depending on how many drinks you've had on that particular day). The number line with "notches" is just a map to help you not get lost.
You've got an awful lot riding on the word "distributed" there.
something something Zeno something something
posted by solotoro at 2:45 PM on October 10, 2012 [1 favorite]
You've got an awful lot riding on the word "distributed" there.
something something Zeno something something
posted by solotoro at 2:45 PM on October 10, 2012 [1 favorite]
No, 4 is the distance from the middle to the ends.
posted by Pruitt-Igoe at 2:49 PM on October 10, 2012
posted by Pruitt-Igoe at 2:49 PM on October 10, 2012
2 is the loneliest number so I presume it the furthest away from all other numbers.
posted by Hairy Lobster at 2:49 PM on October 10, 2012
posted by Hairy Lobster at 2:49 PM on October 10, 2012
They should have been rounding up if it was greater than or equal to five.
Derp, you are correct.
posted by a snickering nuthatch at 2:51 PM on October 10, 2012
Derp, you are correct.
posted by a snickering nuthatch at 2:51 PM on October 10, 2012
Historically, Americans haven't been real good with the number 3/5.
posted by Benny Andajetz
I know. My drunk uncle buck is fastidious about how much he drinks.
posted by clavdivs at 2:52 PM on October 10, 2012
posted by Benny Andajetz
I know. My drunk uncle buck is fastidious about how much he drinks.
posted by clavdivs at 2:52 PM on October 10, 2012
All of these things are forgotten and ignored when we consider numbers as a featureless line on which -1, -0.5, 0, 0.5, 1, 1.5, etc appear. In fact, if we wanted to make a line one which numbers actually occur by priority in existence as I've outlined above (although this would still obscure some of the nature of numbers) it would make more sense to put One first, then Zero, then Many, then Fractions.
I'm sorry, but I think if anyone is "obscuring many important things about [numbers]" here it has to be you. Numbers are abstractions that are not necessarily bound to the concepts you've tried to link them to here, whatever their historical lineage be, and your prioritization of those concepts feels a little ad hoc. Anyway, the number line follows very neatly from the total ordering of the integers, a notion that is pretty central to how we use numbers and that your proposed number line throws out completely. The typical left-to-right orientation of the number line is admittedly arbitrary, but the orientation isn't an essential component of the idea anyway. I'm sure you're putting all this forward in earnest, but honestly, calling such a simple, useful and well-founded construct "hackneyed" would read to me as trolling if I didn't know better.
posted by invitapriore at 3:05 PM on October 10, 2012 [1 favorite]
I'm sorry, but I think if anyone is "obscuring many important things about [numbers]" here it has to be you. Numbers are abstractions that are not necessarily bound to the concepts you've tried to link them to here, whatever their historical lineage be, and your prioritization of those concepts feels a little ad hoc. Anyway, the number line follows very neatly from the total ordering of the integers, a notion that is pretty central to how we use numbers and that your proposed number line throws out completely. The typical left-to-right orientation of the number line is admittedly arbitrary, but the orientation isn't an essential component of the idea anyway. I'm sure you're putting all this forward in earnest, but honestly, calling such a simple, useful and well-founded construct "hackneyed" would read to me as trolling if I didn't know better.
posted by invitapriore at 3:05 PM on October 10, 2012 [1 favorite]
Will these kids give you square root of any number you ask for (X ^ 1/2)? If you asked for the middle between 1 and 100, would they say 50 or 10? Or maybe children who are educated-stupid enough to count above 9 do not exhibit this behaviour.
posted by Pruitt-Igoe at 3:06 PM on October 10, 2012
posted by Pruitt-Igoe at 3:06 PM on October 10, 2012
This does remind me a bit of high school chemistry class, where we were introduced to bizarre rounding methods, including one where if there was a .5, you looked at the next digit for guidance. So you round .54 and smaller down, and .56 up. With .55, you look to the next digit. We were never able to convince the teacher this lead to a systemic bias, with .555555 (repeating) being the midpoint instead of .55. Possibly because we didn't have the mathematical background of countable and uncountable infinities.
Something got lost in translation then. For rounding to the integer (unity) place, anything with .0-.4, no matter what comes after the tenths, is rounded down. Anything with .6-.9, no matter what comes after the tenths, is rounded up. Anything with .5 ...
So 2.500000000000001 rounds to 3, but 2.5 rounds to 2. Whereas 3.500000000000001 and 3.5 both round to 4.
posted by solotoro at 3:07 PM on October 10, 2012 [1 favorite]
Something got lost in translation then. For rounding to the integer (unity) place, anything with .0-.4, no matter what comes after the tenths, is rounded down. Anything with .6-.9, no matter what comes after the tenths, is rounded up. Anything with .5 ...
- If there is any non-zero digit after the tenths (the 5), round up.This gives a bias for even numbers, but no bias for a skewed midpoint among many numbers.
- If the tenths place is the final digit or all the subsequent digits are zero, then the number rounds up if the ones place is odd and down if it is even.
So 2.500000000000001 rounds to 3, but 2.5 rounds to 2. Whereas 3.500000000000001 and 3.5 both round to 4.
posted by solotoro at 3:07 PM on October 10, 2012 [1 favorite]
Dear People Who Believe that the Number Line Is an Unassailable Truth:
Your conception of the way numbers and arithmetic work is absolutely good enoughTM and will never fail you in your day to day life unless you suddenly accidentally become pure mathematics professors. It is much the same as how Newtonian physics works perfectly fine for everything a normal person ever needs to worry about.
That said, though number theory looks opaque, masturbatory, and just plain wrong from the outside, it is actually quite important and well-established. You don't need to learn set theory, but it's probably best if you refrain from talking down to people who have.
Signed,
Collective Past and Present Students of MAT 301
posted by 256 at 3:19 PM on October 10, 2012 [10 favorites]
Your conception of the way numbers and arithmetic work is absolutely good enoughTM and will never fail you in your day to day life unless you suddenly accidentally become pure mathematics professors. It is much the same as how Newtonian physics works perfectly fine for everything a normal person ever needs to worry about.
That said, though number theory looks opaque, masturbatory, and just plain wrong from the outside, it is actually quite important and well-established. You don't need to learn set theory, but it's probably best if you refrain from talking down to people who have.
Signed,
Collective Past and Present Students of MAT 301
posted by 256 at 3:19 PM on October 10, 2012 [10 favorites]
it was two months from January to March
What's especially hilarious about this is it is really something of an ill-formed question and there are (at least) three possible correct answers:
1 - if you you mean 'how many months *strictly between* January and March' (ie, from the end of January to the beginning of March)
2 - if you mean 'how many months from a certain date in January to the same date in March'
3 - if you mean 'how many months from the start of January to the end of March'
And that isn't even getting into "Do you mean January and March in the same year?" which is where the really asshole HS students would be going with this . . .
posted by flug at 3:31 PM on October 10, 2012 [1 favorite]
What's especially hilarious about this is it is really something of an ill-formed question and there are (at least) three possible correct answers:
1 - if you you mean 'how many months *strictly between* January and March' (ie, from the end of January to the beginning of March)
2 - if you mean 'how many months from a certain date in January to the same date in March'
3 - if you mean 'how many months from the start of January to the end of March'
And that isn't even getting into "Do you mean January and March in the same year?" which is where the really asshole HS students would be going with this . . .
posted by flug at 3:31 PM on October 10, 2012 [1 favorite]
We don't start by experiencing zero, or ten, or two thousand four hundred and twenty six. We start by experiencing one.
We don't "experience" any number. They're abstract concepts we invented to describe the world. But that invention came from direct observation of quantity.
Hey dudes, mathematical psychology is psychology, not math. Nobody's trying to prove that 3 is "actually" halfway between 1 and 9 and mess up all your nice spreadsheets
Read the text of the FPP. That's fairly provocative. And the text of the article, at the front at least, seems to imply that notions of "halfway" should be based on a logarithmic, not a linear, scale. It doesn't say anything that sounds like "speaking psychologically."
You've got an awful lot riding on the word "distributed" there.
something something Zeno something something
I find Zeno's paradox greatly overrated. If you split the distance you travel in half over and over you'll never get to the end of the road -- unless you split the time you took to cover that distance in half as well, which makes sense. Thus the infinitely decreasing distances you cover are matched by infinitely decreasing amounts of time. But we don't experience time like that, to us time is linear; we just cover the remaining distance and skip over that infinity of steps.
Signed,
Collective Past and Present Students of MAT 301
The number line isn't "truth," it's a description. Just as valid a description as anything you learned in 301, they're just used for different purposes. You don't own the numbers.
posted by JHarris at 3:33 PM on October 10, 2012 [5 favorites]
We don't "experience" any number. They're abstract concepts we invented to describe the world. But that invention came from direct observation of quantity.
Hey dudes, mathematical psychology is psychology, not math. Nobody's trying to prove that 3 is "actually" halfway between 1 and 9 and mess up all your nice spreadsheets
Read the text of the FPP. That's fairly provocative. And the text of the article, at the front at least, seems to imply that notions of "halfway" should be based on a logarithmic, not a linear, scale. It doesn't say anything that sounds like "speaking psychologically."
You've got an awful lot riding on the word "distributed" there.
something something Zeno something something
I find Zeno's paradox greatly overrated. If you split the distance you travel in half over and over you'll never get to the end of the road -- unless you split the time you took to cover that distance in half as well, which makes sense. Thus the infinitely decreasing distances you cover are matched by infinitely decreasing amounts of time. But we don't experience time like that, to us time is linear; we just cover the remaining distance and skip over that infinity of steps.
Signed,
Collective Past and Present Students of MAT 301
The number line isn't "truth," it's a description. Just as valid a description as anything you learned in 301, they're just used for different purposes. You don't own the numbers.
posted by JHarris at 3:33 PM on October 10, 2012 [5 favorites]
My fourteen dimensional calendar counts flirteen blornsaries between the fleen of February and the negative twenty-first of January, assuming an annual intra-leap fortnight occurs every 76.4 kiloweeks following each inverse semi-month.
posted by blue_beetle at 3:37 PM on October 10, 2012 [1 favorite]
posted by blue_beetle at 3:37 PM on October 10, 2012 [1 favorite]
Of course my calendar weighs 19 tons and requires a team of 400 to operate, so it's not always accurate.
posted by blue_beetle at 3:38 PM on October 10, 2012
posted by blue_beetle at 3:38 PM on October 10, 2012
The number line isn't "truth," it's a description. Just as valid a description as anything you learned in 301, they're just used for different purposes. You don't own the numbers.
Y'know, I already regret phrasing that in such a snarky way. But, truly, the number line model and post-Principia Mathematica number theory are not equally valid representations. It really is like saying that Newtonian physics and relativity are equally valid. They aren't. They are functionally the same as approximations in most cases, but one contains the other. And just as Newtonian physics will lead you astray when talking about things at the limits of our observable reality (like the Plank length or the speed of light), the number line will lead you astray when talking about things at the limits of mathematics (like what it means for one number to be "between" two numbers).
posted by 256 at 3:47 PM on October 10, 2012 [1 favorite]
Y'know, I already regret phrasing that in such a snarky way. But, truly, the number line model and post-Principia Mathematica number theory are not equally valid representations. It really is like saying that Newtonian physics and relativity are equally valid. They aren't. They are functionally the same as approximations in most cases, but one contains the other. And just as Newtonian physics will lead you astray when talking about things at the limits of our observable reality (like the Plank length or the speed of light), the number line will lead you astray when talking about things at the limits of mathematics (like what it means for one number to be "between" two numbers).
posted by 256 at 3:47 PM on October 10, 2012 [1 favorite]
me: “We don't start by experiencing zero, or ten, or two thousand four hundred and twenty six. We start by experiencing one.”
JHarris: “We don't ‘experience’ any number. They're abstract concepts we invented to describe the world. But that invention came from direct observation of quantity.”
Hm. I felt odd about the word "experience," but I used it because "observation" seems distinctly wrong to me. I mean, although they are different, you could also say that we "observe" color or "observe" shape, but that isn't strictly true either, is it? We've never "observed" anything that didn't have color or shape; and it would be hard to say anyone had observed the color yellow itself or squareness itself. I think it's fair to say that we don't "observe" number, color, or shape in the way we "observe," say, the principle of gravity or the conservation of energy; that is, number, color, and shape aren't things we discovered through observation, they're things without with there would be no observation at all. But I am not sure what exactly you meant by the word.
posted by koeselitz at 3:50 PM on October 10, 2012
JHarris: “We don't ‘experience’ any number. They're abstract concepts we invented to describe the world. But that invention came from direct observation of quantity.”
Hm. I felt odd about the word "experience," but I used it because "observation" seems distinctly wrong to me. I mean, although they are different, you could also say that we "observe" color or "observe" shape, but that isn't strictly true either, is it? We've never "observed" anything that didn't have color or shape; and it would be hard to say anyone had observed the color yellow itself or squareness itself. I think it's fair to say that we don't "observe" number, color, or shape in the way we "observe," say, the principle of gravity or the conservation of energy; that is, number, color, and shape aren't things we discovered through observation, they're things without with there would be no observation at all. But I am not sure what exactly you meant by the word.
posted by koeselitz at 3:50 PM on October 10, 2012
Your conception of the way numbers and arithmetic work is absolutely good enoughTM
Interestingly, some people believe that the concept of ordinal numbers (ie, first, second, third, fourth, etc) arose earlier than and independently of the idea of numbers as a quantity (1,2,3,4 etc).
(One reason they believe that is that in many languages, as in English, the first few--and more to the point, most commonly used--ordinals and cardinals have very different roots. 4->4th 5->5th 6->6th, and so on, but how does "one" relate to "first and "two" to "second"? The answer is, they don't.)
I'm not pointing that out to get into a linguistic argument, but merely to illustrate the idea that, as 256 points out, numbers really are used in dramatically different ways. We usually wallpaper over this in daily life. Yes, 1 as a natural number, 1 as a rational number, 1 as a decimal number, 1 as a real number (and so on) are all sorta the same, and can be used interchangeably much of the time. And yet, those are all fundamentally different concepts at heart, though we've designed certain correspondences into them.
And back to ordinal vs cardinal: The idea of "one" thing and the "first" thing really are quite different and distinct. Think about it.
posted by flug at 3:51 PM on October 10, 2012 [2 favorites]
Interestingly, some people believe that the concept of ordinal numbers (ie, first, second, third, fourth, etc) arose earlier than and independently of the idea of numbers as a quantity (1,2,3,4 etc).
(One reason they believe that is that in many languages, as in English, the first few--and more to the point, most commonly used--ordinals and cardinals have very different roots. 4->4th 5->5th 6->6th, and so on, but how does "one" relate to "first and "two" to "second"? The answer is, they don't.)
I'm not pointing that out to get into a linguistic argument, but merely to illustrate the idea that, as 256 points out, numbers really are used in dramatically different ways. We usually wallpaper over this in daily life. Yes, 1 as a natural number, 1 as a rational number, 1 as a decimal number, 1 as a real number (and so on) are all sorta the same, and can be used interchangeably much of the time. And yet, those are all fundamentally different concepts at heart, though we've designed certain correspondences into them.
And back to ordinal vs cardinal: The idea of "one" thing and the "first" thing really are quite different and distinct. Think about it.
posted by flug at 3:51 PM on October 10, 2012 [2 favorites]
It really is like saying that Newtonian physics and relativity are equally valid. They aren't. They are functionally the same as approximations in most cases, but one contains the other. And just as Newtonian physics will lead you astray when talking about things at the limits of our observable reality (like the Plank length or the speed of light), the number line will lead you astray when talking about things at the limits of mathematics (like what it means for one number to be "between" two numbers).
That's not news to me, and I wasn't arguing against it. The point is that no one would call Newtonian physics "hackneyed" unless they felt like being aggressively contrarian, because it's extremely useful when you're dealing with the subset of all phenomena that it describes.
posted by invitapriore at 3:53 PM on October 10, 2012
That's not news to me, and I wasn't arguing against it. The point is that no one would call Newtonian physics "hackneyed" unless they felt like being aggressively contrarian, because it's extremely useful when you're dealing with the subset of all phenomena that it describes.
posted by invitapriore at 3:53 PM on October 10, 2012
But – well, I'll say it: I have a hard time seeing how Dedekind's number line is "extremely useful." Can you explain that, please? As far as I can tell, it did not birth whole new realms of modern mathematics; most of modern mathematics seems to stand utterly distinct from it, and was not helped or hindered by the number line. The only place the number line has really been extremely popular is in grade-school classrooms.
posted by koeselitz at 4:14 PM on October 10, 2012
posted by koeselitz at 4:14 PM on October 10, 2012
koeselitz: But that removes most of the actual meaning from numbers, rendering them confusing and obscuring many important things about them.
256: the number line will lead you astray when talking about things at the limits of mathematics (like what it means for one number to be "between" two numbers).
Are you guys alluding to p-adic numbers or to something else? Are there any other completions of ℚ?
posted by Nioate at 4:55 PM on October 10, 2012
256: the number line will lead you astray when talking about things at the limits of mathematics (like what it means for one number to be "between" two numbers).
Are you guys alluding to p-adic numbers or to something else? Are there any other completions of ℚ?
posted by Nioate at 4:55 PM on October 10, 2012
ℚ
Such language! Ya know, there are sometimes days when I regret making that unicode post....
But, truly, the number line model and post-Principia Mathematica number theory are not equally valid representations. It really is like saying that Newtonian physics and relativity are equally valid. They aren't.
That is a false analogy, because Newtonian physics and relativity are meaningless without the natural world they describe, while numbers, while initially derived from the natural world, have been used to explain a plethora of other things. While it doesn't matter where those values come from, it is useful to remember how they were invented. And when people come up to me and say that thinking of numbers as existing along a continuum is wrong wrong wrong I have to cock my head funny.
I am no populist; I'm not taking any position that "since most people use it this way, they're right." And I may well be wrong, but I'd appreciate it if you'd explain why, instead of just taking the position and assuming authority.
But – well, I'll say it: I have a hard time seeing how Dedekind's number line is "extremely useful."
They're extremely useful in teaching. For starters, the regular marks on the line are integers. You can use that to show children, pictorally, how the integers follow each other in sequence, giving them a second way to understand the concept than quantity. Then you can point to the beginning of the line and say "zero," which is hard to do by showing a handful of apples because yoinks there's nothing there, and you can even extend the line beyond zero to get to negative numbers, which is some seriously brain-twisting shit to a third-grader. When you do fractions, you make more marks, between the integers, their positions relative to the distance between the numbers. I suspect you aren't going to introduce them to the concept with set theory nearly so easily.
When you get to higher math this system may well be less useful, but it doesn't stop me from noticing that graphs are still used a lot in math, and those are just a two-dimensional version of the same thing. I am not a math major (humanities), but doesn't calculus depend on the concept of a continuous number line?
posted by JHarris at 5:51 PM on October 10, 2012 [2 favorites]
Such language! Ya know, there are sometimes days when I regret making that unicode post....
But, truly, the number line model and post-Principia Mathematica number theory are not equally valid representations. It really is like saying that Newtonian physics and relativity are equally valid. They aren't.
That is a false analogy, because Newtonian physics and relativity are meaningless without the natural world they describe, while numbers, while initially derived from the natural world, have been used to explain a plethora of other things. While it doesn't matter where those values come from, it is useful to remember how they were invented. And when people come up to me and say that thinking of numbers as existing along a continuum is wrong wrong wrong I have to cock my head funny.
I am no populist; I'm not taking any position that "since most people use it this way, they're right." And I may well be wrong, but I'd appreciate it if you'd explain why, instead of just taking the position and assuming authority.
But – well, I'll say it: I have a hard time seeing how Dedekind's number line is "extremely useful."
They're extremely useful in teaching. For starters, the regular marks on the line are integers. You can use that to show children, pictorally, how the integers follow each other in sequence, giving them a second way to understand the concept than quantity. Then you can point to the beginning of the line and say "zero," which is hard to do by showing a handful of apples because yoinks there's nothing there, and you can even extend the line beyond zero to get to negative numbers, which is some seriously brain-twisting shit to a third-grader. When you do fractions, you make more marks, between the integers, their positions relative to the distance between the numbers. I suspect you aren't going to introduce them to the concept with set theory nearly so easily.
When you get to higher math this system may well be less useful, but it doesn't stop me from noticing that graphs are still used a lot in math, and those are just a two-dimensional version of the same thing. I am not a math major (humanities), but doesn't calculus depend on the concept of a continuous number line?
posted by JHarris at 5:51 PM on October 10, 2012 [2 favorites]
I suspect you could present any number of mathematical questions to small children and they'd answer "3".
posted by CheeseDigestsAll at 6:28 PM on October 10, 2012 [1 favorite]
posted by CheeseDigestsAll at 6:28 PM on October 10, 2012 [1 favorite]
JHarris: “They're extremely useful in teaching. For starters, the regular marks on the line are integers. You can use that to show children, pictorally, how the integers follow each other in sequence, giving them a second way to understand the concept than quantity. Then you can point to the beginning of the line and say "zero," which is hard to do by showing a handful of apples because yoinks there's nothing there, and you can even extend the line beyond zero to get to negative numbers, which is some seriously brain-twisting shit to a third-grader. When you do fractions, you make more marks, between the integers, their positions relative to the distance between the numbers. I suspect you aren't going to introduce them to the concept with set theory nearly so easily.”
This is my biggest problem with the number line, actually. I think that, not only does it introduce bad ways of thinking about numbers that are hard to fix later, but it makes it more difficult for children to understand something they already actually really understand better than one might think (since numbers are fundamental to the way human beings experience the world). And the number line instantly confuses the entire subject of geometry, which should be the starting point in mathematics education anyway.
I have a lot of problems with the way math is taught, frankly.
“When you get to higher math this system may well be less useful, but it doesn't stop me from noticing that graphs are still used a lot in math, and those are just a two-dimensional version of the same thing. I am not a math major (humanities), but doesn't calculus depend on the concept of a continuous number line?”
The calculus absolutely does not depend on the concept of a continuous number line, no. The idea of a number line was introduced by Richard Dedekind in the late nineteenth century, hundreds of years after Liebniz and Newton. In fact, calculus really doesn't even rely on algebra; and this is another problem I have with mathematics education – calculus isn't taught geometrically, even though it makes a whole lot more sense if you start that way. When I was in school, they just slapped dy/dx in front of me and said, "always replace this with that, and you'll get the answer." I'd say "why? What the heck am I calculating?" and they'd say "the derivative, silly. Just do the math." And since I was good at calculating, I thought that was fine.
It wasn't until I actually read Newton's Principia in college that the light went on and I realized: we're figuring rate of change, which is the same thing as the slope at a point on a curve! That seemed so simple, so obvious, that I was both vexed at myself for going through two years of calculus without realizing it and vexed at my teachers for not thinking I needed to know that.
But mathematics isn't taught as an exercise in understanding now. Mathematics is taught as a means to an end and a machine you plug numbers into to get the answer you're supposed to get.
posted by koeselitz at 6:46 PM on October 10, 2012 [1 favorite]
This is my biggest problem with the number line, actually. I think that, not only does it introduce bad ways of thinking about numbers that are hard to fix later, but it makes it more difficult for children to understand something they already actually really understand better than one might think (since numbers are fundamental to the way human beings experience the world). And the number line instantly confuses the entire subject of geometry, which should be the starting point in mathematics education anyway.
I have a lot of problems with the way math is taught, frankly.
“When you get to higher math this system may well be less useful, but it doesn't stop me from noticing that graphs are still used a lot in math, and those are just a two-dimensional version of the same thing. I am not a math major (humanities), but doesn't calculus depend on the concept of a continuous number line?”
The calculus absolutely does not depend on the concept of a continuous number line, no. The idea of a number line was introduced by Richard Dedekind in the late nineteenth century, hundreds of years after Liebniz and Newton. In fact, calculus really doesn't even rely on algebra; and this is another problem I have with mathematics education – calculus isn't taught geometrically, even though it makes a whole lot more sense if you start that way. When I was in school, they just slapped dy/dx in front of me and said, "always replace this with that, and you'll get the answer." I'd say "why? What the heck am I calculating?" and they'd say "the derivative, silly. Just do the math." And since I was good at calculating, I thought that was fine.
It wasn't until I actually read Newton's Principia in college that the light went on and I realized: we're figuring rate of change, which is the same thing as the slope at a point on a curve! That seemed so simple, so obvious, that I was both vexed at myself for going through two years of calculus without realizing it and vexed at my teachers for not thinking I needed to know that.
But mathematics isn't taught as an exercise in understanding now. Mathematics is taught as a means to an end and a machine you plug numbers into to get the answer you're supposed to get.
posted by koeselitz at 6:46 PM on October 10, 2012 [1 favorite]
Anyone who does not answer "five" is wrong.
In fact, if you restrict yourself to a scale over the positive numbers defined only by the magnitude of your observed numbers, the perceptual midpoint is indisputably 3.
What is the ideal perceptual code for samples take from a distribution? Surely, it is the cumulative distribution function. For example, if you want to put student grades on a scale that evenly and maximally separates everyone, the ideal scale is to give quantiles. Otherwise, there might be clumping in some parts of the scale. Thus, the ideal perceptual scale is the quantile of a given distribution.
Now, suppose that a hat is filled with slips of paper with positive integers and you draw a couple (say, 1 and 9) and record their average magnitude. How do you imagine the positive integers in the hat are distributed? The minimum assumption you can make is to assume the maximum entropy distribution, which is a geometric distribution with the given mean.
If you imagine that everyone naturally is using such a geometric distribution with which to perceptually encode numbers, then it follows that the perceptual encoding function is the cumulative distribution function of the geometric distribution, which is logarithmic. This function evaluated at three has a value exactly between its values at one and nine.
If you measure more than the average magnitude of the numbers in the hat, for example, you also measure their variance, then you get the familiar normal distribution, and the perceptual midpoint is the mean.
posted by esprit de l'escalier at 7:00 PM on October 10, 2012
In fact, if you restrict yourself to a scale over the positive numbers defined only by the magnitude of your observed numbers, the perceptual midpoint is indisputably 3.
What is the ideal perceptual code for samples take from a distribution? Surely, it is the cumulative distribution function. For example, if you want to put student grades on a scale that evenly and maximally separates everyone, the ideal scale is to give quantiles. Otherwise, there might be clumping in some parts of the scale. Thus, the ideal perceptual scale is the quantile of a given distribution.
Now, suppose that a hat is filled with slips of paper with positive integers and you draw a couple (say, 1 and 9) and record their average magnitude. How do you imagine the positive integers in the hat are distributed? The minimum assumption you can make is to assume the maximum entropy distribution, which is a geometric distribution with the given mean.
If you imagine that everyone naturally is using such a geometric distribution with which to perceptually encode numbers, then it follows that the perceptual encoding function is the cumulative distribution function of the geometric distribution, which is logarithmic. This function evaluated at three has a value exactly between its values at one and nine.
If you measure more than the average magnitude of the numbers in the hat, for example, you also measure their variance, then you get the familiar normal distribution, and the perceptual midpoint is the mean.
posted by esprit de l'escalier at 7:00 PM on October 10, 2012
The Weber–Fechner law combines two different laws. Some authors use the term to mean Weber's law, and others Fechner's law.
To which I feel compelled to add, Hitler. It stands to reason.
posted by philip-random at 7:04 PM on October 10, 2012 [1 favorite]
To which I feel compelled to add, Hitler. It stands to reason.
posted by philip-random at 7:04 PM on October 10, 2012 [1 favorite]
The calculus absolutely does not depend on the concept of a continuous number line, no. The idea of a number line was introduced by Richard Dedekind in the late nineteenth century, hundreds of years after Liebniz and Newton.
Um, yes, but... one big reason that Dedekind and others were doing what they were doing was to fix conceptual problems in calculus. Calculus could be used, but there were serious problems in the foundations, problems which were eventually resolved by (among other things) adopting a certain formal model for the intuitive notion of "a continuous line", a model we now call "the real numbers".
Anyway, yes, I completely agree that it was a big step to merge the notion of "number" and "[geometric] magnitude". (The Greeks considered them distinct, for example — that famous story about the Pythagoreans' discovery of irrational numbers (as we would say it today) was, in their terms, more like a discovery that numbers (as they understood them) were not enough to describe magnitudes.) But you seem to be saying that it was a big step backwards, not forwards. That is astonishing. If your concern is with solely with pedagogy, that you're concerned students no longer understand that it is a big step to merge number and magnitude, then I see your point. But I can't believe that you really see no value in being able to do arithmetic with magnitudes.
It wasn't until I actually read Newton's Principia in college that the light went on and I realized: we're figuring rate of change, which is the same thing as the slope at a point on a curve! That seemed so simple, so obvious, that I was both vexed at myself for going through two years of calculus without realizing it and vexed at my teachers for not thinking I needed to know that.
Yeah, if your calculus teachers didn't tell you that derivatives are about rates of change and slopes of tangent lines, then (a) they were complete incompetents and (b) they support no inferences about typical calculus education. Every introductory calculus textbook explains these things. (There was really no need to break out Principia.)
posted by stebulus at 7:09 PM on October 10, 2012 [3 favorites]
Um, yes, but... one big reason that Dedekind and others were doing what they were doing was to fix conceptual problems in calculus. Calculus could be used, but there were serious problems in the foundations, problems which were eventually resolved by (among other things) adopting a certain formal model for the intuitive notion of "a continuous line", a model we now call "the real numbers".
Anyway, yes, I completely agree that it was a big step to merge the notion of "number" and "[geometric] magnitude". (The Greeks considered them distinct, for example — that famous story about the Pythagoreans' discovery of irrational numbers (as we would say it today) was, in their terms, more like a discovery that numbers (as they understood them) were not enough to describe magnitudes.) But you seem to be saying that it was a big step backwards, not forwards. That is astonishing. If your concern is with solely with pedagogy, that you're concerned students no longer understand that it is a big step to merge number and magnitude, then I see your point. But I can't believe that you really see no value in being able to do arithmetic with magnitudes.
It wasn't until I actually read Newton's Principia in college that the light went on and I realized: we're figuring rate of change, which is the same thing as the slope at a point on a curve! That seemed so simple, so obvious, that I was both vexed at myself for going through two years of calculus without realizing it and vexed at my teachers for not thinking I needed to know that.
Yeah, if your calculus teachers didn't tell you that derivatives are about rates of change and slopes of tangent lines, then (a) they were complete incompetents and (b) they support no inferences about typical calculus education. Every introductory calculus textbook explains these things. (There was really no need to break out Principia.)
posted by stebulus at 7:09 PM on October 10, 2012 [3 favorites]
What about this old chestnut: 4+4=9 for very large values of 4 (I like that one because it gives me a nice logarithmic sense of four fading imperceptibly into 4.5...)
posted by sneebler at 7:19 PM on October 10, 2012
posted by sneebler at 7:19 PM on October 10, 2012
stebulus: “Every introductory calculus textbook explains these things.”
This is flatly not true. I don't think I've ever seen any calculus textbook that explains derivatives geometrically. They all use algebra, assuming that that's easiest, because (they figure) the 70% of people who didn't understand quite get algebra should be left behind anyway.
posted by koeselitz at 7:30 PM on October 10, 2012
This is flatly not true. I don't think I've ever seen any calculus textbook that explains derivatives geometrically. They all use algebra, assuming that that's easiest, because (they figure) the 70% of people who didn't understand quite get algebra should be left behind anyway.
posted by koeselitz at 7:30 PM on October 10, 2012
Really? Your calculus book & teachers didn't go over derivatives using limits of measuring slopes of curves? And that integration is a method of measuring the area under the curve? Both my high school and college calculus classes did it that way.
posted by fings at 7:46 PM on October 10, 2012 [1 favorite]
posted by fings at 7:46 PM on October 10, 2012 [1 favorite]
So you don't think geometry is vastly de-emphasized in favor of algebraic notation in mathematics classrooms today?
posted by koeselitz at 8:03 PM on October 10, 2012
posted by koeselitz at 8:03 PM on October 10, 2012
I was explained what calculus was geometrically, but then told how to solve it (so to speak) algebraically. But that was my prof, who was excellent; I barely ever looked at the unwieldy textbook, except for homework.
posted by Casuistry at 8:04 PM on October 10, 2012
posted by Casuistry at 8:04 PM on October 10, 2012
(And apparently you have no problem with the fact that it's maybe 5% of us in the civilized world who were lucky enough to finally get even intermediate mathematics? As opposed to the general literacy rate? Mathematics education is failing, miserably, and the effects and indications are all around us.)
posted by koeselitz at 8:04 PM on October 10, 2012
posted by koeselitz at 8:04 PM on October 10, 2012
I don't think I've ever seen any calculus textbook that explains derivatives geometrically. They all use algebra, assuming that that's easiest,
Geometry with coordinates is still geometry. Explaining geometrically, then moving to a coordinate representation for calculations, is still explaining geometrically. And so is an explanation that maintains a coordinate-endowed geometric picture and fluidly moves back and forth between geometric considerations and algebraic manipulations of coordinates, developing a calculational apparatus simultaneously with the geometric insight.
I love synthetic geometry, but it's more than a little fundamentalist to insist that it's the only true geometry. (And like other kinds of fundamentalism, it requires jettisoning hundreds of years' worth of progress.)
because (they figure) the 70% of people who didn't understand quite get algebra should be left behind anyway.
This is an unfounded, and quite nasty, attribution of motive. I teach calculus, and I know lots of people who teach calculus. It is a rare instructor that has such contempt for their students.
I get that you had some bad experiences in your mathematical education, and that you feel raw about it, but I'd appreciate it if you tried to find generous attributions of motive. (And I think you'll find that that's a good policy in general, whatever you might think of my preferences in particular.) I also get that you might be too pissed to do that — I often am, though not right at this moment — and if so, perhaps we should simply drop the matter.
posted by stebulus at 8:05 PM on October 10, 2012 [1 favorite]
Geometry with coordinates is still geometry. Explaining geometrically, then moving to a coordinate representation for calculations, is still explaining geometrically. And so is an explanation that maintains a coordinate-endowed geometric picture and fluidly moves back and forth between geometric considerations and algebraic manipulations of coordinates, developing a calculational apparatus simultaneously with the geometric insight.
I love synthetic geometry, but it's more than a little fundamentalist to insist that it's the only true geometry. (And like other kinds of fundamentalism, it requires jettisoning hundreds of years' worth of progress.)
because (they figure) the 70% of people who didn't understand quite get algebra should be left behind anyway.
This is an unfounded, and quite nasty, attribution of motive. I teach calculus, and I know lots of people who teach calculus. It is a rare instructor that has such contempt for their students.
I get that you had some bad experiences in your mathematical education, and that you feel raw about it, but I'd appreciate it if you tried to find generous attributions of motive. (And I think you'll find that that's a good policy in general, whatever you might think of my preferences in particular.) I also get that you might be too pissed to do that — I often am, though not right at this moment — and if so, perhaps we should simply drop the matter.
posted by stebulus at 8:05 PM on October 10, 2012 [1 favorite]
I'm not really pissed off, although I probably could dial back a little. But I do think that algebra is relied on far too heavily, and geometry is left behind far too quickly, in mathematics education today. I'm sorry if some of my comments here have seemed like personal attacks – I truly didn't mean them as such, although now it's easy to see why you might take them that way.
I also appreciate that I might be generalizing my own pedagogical problems a bit too much. I will try to stay aware of that.
stebulus: “Geometry with coordinates is still geometry. Explaining geometrically, then moving to a coordinate representation for calculations, is still explaining geometrically. And so is an explanation that maintains a coordinate-endowed geometric picture and fluidly moves back and forth between geometric considerations and algebraic manipulations of coordinates, developing a calculational apparatus simultaneously with the geometric insight.”
I agree, and I appreciate that the whole point of algebra is that it a symbolic language intended to be convertible with geometry. But I believe that that isn't as intuitive as those who are familiar with mathematics might think it is. I think this might be the fundamental issue that people struggle with in math – again, I accept that might be a rash statement, but it makes sense to me: it is difficult for people to get to the point where algebra and geometry are completely convertible.
“Yeah, if your calculus teachers didn't tell you that derivatives are about rates of change and slopes of tangent lines, then (a) they were complete incompetents and (b) they support no inferences about typical calculus education. Every introductory calculus textbook explains these things. (There was really no need to break out Principia.)”
I happen to really enjoy Principia, and think it's an elegant and coherent explanation when taken in the right doses.
Regardless of that: "rate of change" was never once mentioned in my calculus class. At the beginning, "area under a curve" was mentioned, but I had no idea why "area under a curve" could possibly be significant. It made no sense to me; it seemed like there was this whole realm of mathematics bent on solving apparently unimportant problems that were not applicable to life at all. I think I must have had a pretty bad teacher, too, and I know I spent more time reading the book than listening to her.
Newton really helped. What also helped was seeing calculus apply to so much in the world – Maxwell, Einstein, and the rest. Knowing that this was a tool for measuring all of these things was very useful, and I guess if (when?) I get to the point where I can teach, I'm hoping to emphasize that these things have real practical applications in the world, rather than drilling numbers and getting people to be proficient at solving equations.
But that is rather far afield, I grant. And it might only be my experience. Also, education seems like something that's widely diverse, so that quality changes rapidly from classroom to classroom. So at this point I should make no claims that my experience is universal. Anyway, that's where I'm coming from.
(Also, mathematics itself didn't make much sense until I read Apollonius, but that's an entirely different discussion. I am weird in having gone to a school where we read such things.)
posted by koeselitz at 8:17 PM on October 10, 2012 [1 favorite]
I also appreciate that I might be generalizing my own pedagogical problems a bit too much. I will try to stay aware of that.
stebulus: “Geometry with coordinates is still geometry. Explaining geometrically, then moving to a coordinate representation for calculations, is still explaining geometrically. And so is an explanation that maintains a coordinate-endowed geometric picture and fluidly moves back and forth between geometric considerations and algebraic manipulations of coordinates, developing a calculational apparatus simultaneously with the geometric insight.”
I agree, and I appreciate that the whole point of algebra is that it a symbolic language intended to be convertible with geometry. But I believe that that isn't as intuitive as those who are familiar with mathematics might think it is. I think this might be the fundamental issue that people struggle with in math – again, I accept that might be a rash statement, but it makes sense to me: it is difficult for people to get to the point where algebra and geometry are completely convertible.
“Yeah, if your calculus teachers didn't tell you that derivatives are about rates of change and slopes of tangent lines, then (a) they were complete incompetents and (b) they support no inferences about typical calculus education. Every introductory calculus textbook explains these things. (There was really no need to break out Principia.)”
I happen to really enjoy Principia, and think it's an elegant and coherent explanation when taken in the right doses.
Regardless of that: "rate of change" was never once mentioned in my calculus class. At the beginning, "area under a curve" was mentioned, but I had no idea why "area under a curve" could possibly be significant. It made no sense to me; it seemed like there was this whole realm of mathematics bent on solving apparently unimportant problems that were not applicable to life at all. I think I must have had a pretty bad teacher, too, and I know I spent more time reading the book than listening to her.
Newton really helped. What also helped was seeing calculus apply to so much in the world – Maxwell, Einstein, and the rest. Knowing that this was a tool for measuring all of these things was very useful, and I guess if (when?) I get to the point where I can teach, I'm hoping to emphasize that these things have real practical applications in the world, rather than drilling numbers and getting people to be proficient at solving equations.
But that is rather far afield, I grant. And it might only be my experience. Also, education seems like something that's widely diverse, so that quality changes rapidly from classroom to classroom. So at this point I should make no claims that my experience is universal. Anyway, that's where I'm coming from.
(Also, mathematics itself didn't make much sense until I read Apollonius, but that's an entirely different discussion. I am weird in having gone to a school where we read such things.)
posted by koeselitz at 8:17 PM on October 10, 2012 [1 favorite]
The calculus absolutely does not depend on the concept of a continuous number line, no.
Actually, it does. Being continuous is a topological property (loosely: how subsets (eg. intervals) of the set of real numbers are related to each other), and the topology of the space in which the domains of functions that you do calculus on live affects which results are true. If you used some fractal set, such as the Cantor middle third set, instead of the real numbers, or if you used a different topology on the real numbers (eg. a discrete topology), then some statements would remain true, but some would be false and some different statements would be true. (Such changes among other generalizations lead to the sub-field of math known as analysis.)
Mathematicians have many different ways to think about numbers, all of which are useful and valid in some contexts, and less useful in other contexts. I've read that one of the difficulties that many students have when they first encounter algebra is that it's not made clear to them that a new, different way of thinking about numbers is being introduced and used. Algebra, in middle school or high school, being the first class where students see a little bit more abstraction in much of North America. I'm told that in France, they use geometry as this transition step class? I don't have any opinion on which one is better, only having experience with the one, but I suspect that if, say, US schools switched the order of algebra and geometry, we'd end up with lots of folks turned off to math by their geometry classes because some other abstraction or new and different way of thinking about some concept that students thought they already understood fully was not made clear in their geometry classes. That is, it seems to me that, apart from issues of curriculum design/ordering, there are also issues in classroom pedagogy that contribute to the difficulty that so many students have with basic algebra.
Calculus is another step where we introduce yet another different, and in some ways more abstract, way of thinking about numbers (and functions!), and developing the agility to work with these different interpretations - to go back and forth between them, and to choose which is best suited for a given context - is not something that calculus students are explicitly taught, in my experience. (But, koeselitz, I am shocked and appalled that the expression "rate of change" was not mentioned in your calculus class! The idea of a rate of change and why a derivative is a rate of change may not be well explained always, but I think it is quite unusual for it not to be mentioned at all.)
posted by eviemath at 8:22 PM on October 10, 2012 [3 favorites]
Actually, it does. Being continuous is a topological property (loosely: how subsets (eg. intervals) of the set of real numbers are related to each other), and the topology of the space in which the domains of functions that you do calculus on live affects which results are true. If you used some fractal set, such as the Cantor middle third set, instead of the real numbers, or if you used a different topology on the real numbers (eg. a discrete topology), then some statements would remain true, but some would be false and some different statements would be true. (Such changes among other generalizations lead to the sub-field of math known as analysis.)
Mathematicians have many different ways to think about numbers, all of which are useful and valid in some contexts, and less useful in other contexts. I've read that one of the difficulties that many students have when they first encounter algebra is that it's not made clear to them that a new, different way of thinking about numbers is being introduced and used. Algebra, in middle school or high school, being the first class where students see a little bit more abstraction in much of North America. I'm told that in France, they use geometry as this transition step class? I don't have any opinion on which one is better, only having experience with the one, but I suspect that if, say, US schools switched the order of algebra and geometry, we'd end up with lots of folks turned off to math by their geometry classes because some other abstraction or new and different way of thinking about some concept that students thought they already understood fully was not made clear in their geometry classes. That is, it seems to me that, apart from issues of curriculum design/ordering, there are also issues in classroom pedagogy that contribute to the difficulty that so many students have with basic algebra.
Calculus is another step where we introduce yet another different, and in some ways more abstract, way of thinking about numbers (and functions!), and developing the agility to work with these different interpretations - to go back and forth between them, and to choose which is best suited for a given context - is not something that calculus students are explicitly taught, in my experience. (But, koeselitz, I am shocked and appalled that the expression "rate of change" was not mentioned in your calculus class! The idea of a rate of change and why a derivative is a rate of change may not be well explained always, but I think it is quite unusual for it not to be mentioned at all.)
posted by eviemath at 8:22 PM on October 10, 2012 [3 favorites]
It is kind of interesting to wonder whether there are helpful pedagogical clues about mathematics education in the study this post is about.
posted by koeselitz at 8:30 PM on October 10, 2012
posted by koeselitz at 8:30 PM on October 10, 2012
stebulus: “Every introductory calculus textbook explains these things.”
koeselitz: This is flatly not true. I don't think I've ever seen any calculus textbook that explains derivatives geometrically.
I think there's a little confusion in the thread about what people mean by "geometrically". Newton made some pretty much completely geometric argument to "prove" some of his theorems about calculus, iirc. (Technically, calculus studies properties of functions (such as rate of change), and functions are an algebraic concept, so you can't get away from the algebra entirely.) I've certainly seen this done. Giving a geometric proof is not quite the same thing as giving a geometric motivation, but then going on to develop the results using more analytic/algebraic techniques.
While the geometric motivation is pretty key, there are some good reasons why calculus classes typically don't bother with the geometric style proofs, including: (i) students typically do know more algebra than geometry when they enter calculus (the not insignificant proportion of students who never quite "got" algebra in high school or whenever don't show up in calculus classes) -- some of my students have neven ever taken a high school geometry course, for example (there's a feedback loop between the college prep math courses offered in high schools and the way that classes like calculus are taught, of course); and (ii) Newton's proofs were wrong. (Or, if you prefer, incomplete.) They give a good intuition, but fail for more complicated functions that one encounters in the real world where not everything is smooth and continuous. It took mathematicians one or two hundred years of revisions to sort out the gaps in Newton and Leibniz' first arguments for why their theorems held and to develop the epsilon-delta definitions of limits and related calculus concepts that allowed for correct proofs (and that also have a nice geometric intuition behind them, but contain some very deep ideas that many of the best mathematical minds in earlier centuries overlooked, so in North America we generally save that for a real analysis course that math majors typically take in their third year, after completing the whole calculus sequence, and that non-math majors typically avoid like the plague even if they were considering taking other higher level math courses.)
posted by eviemath at 8:46 PM on October 10, 2012
koeselitz: This is flatly not true. I don't think I've ever seen any calculus textbook that explains derivatives geometrically.
I think there's a little confusion in the thread about what people mean by "geometrically". Newton made some pretty much completely geometric argument to "prove" some of his theorems about calculus, iirc. (Technically, calculus studies properties of functions (such as rate of change), and functions are an algebraic concept, so you can't get away from the algebra entirely.) I've certainly seen this done. Giving a geometric proof is not quite the same thing as giving a geometric motivation, but then going on to develop the results using more analytic/algebraic techniques.
While the geometric motivation is pretty key, there are some good reasons why calculus classes typically don't bother with the geometric style proofs, including: (i) students typically do know more algebra than geometry when they enter calculus (the not insignificant proportion of students who never quite "got" algebra in high school or whenever don't show up in calculus classes) -- some of my students have neven ever taken a high school geometry course, for example (there's a feedback loop between the college prep math courses offered in high schools and the way that classes like calculus are taught, of course); and (ii) Newton's proofs were wrong. (Or, if you prefer, incomplete.) They give a good intuition, but fail for more complicated functions that one encounters in the real world where not everything is smooth and continuous. It took mathematicians one or two hundred years of revisions to sort out the gaps in Newton and Leibniz' first arguments for why their theorems held and to develop the epsilon-delta definitions of limits and related calculus concepts that allowed for correct proofs (and that also have a nice geometric intuition behind them, but contain some very deep ideas that many of the best mathematical minds in earlier centuries overlooked, so in North America we generally save that for a real analysis course that math majors typically take in their third year, after completing the whole calculus sequence, and that non-math majors typically avoid like the plague even if they were considering taking other higher level math courses.)
posted by eviemath at 8:46 PM on October 10, 2012
From the article in the FPP itself, I would conclude that we need to bring back slide rules as a basic teaching tool! :D
(Also abacuses (more), but that's unrelated to the article.)
posted by eviemath at 8:59 PM on October 10, 2012
(Also abacuses (more), but that's unrelated to the article.)
posted by eviemath at 8:59 PM on October 10, 2012
I am no populist; I'm not taking any position that "since most people use it this way, they're right." And I may well be wrong, but I'd appreciate it if you'd explain why, instead of just taking the position and assuming authority.
The practical applications of pure maths are difficult to describe in a comment (not because people are too dumb to understand, but just because it is too big).
For very starters, we can use pi. The depths to which we can calculate pi exist only because pure maths discovered irrational numbers. We never could have gotten that precision by just making and measuring circles. And yet, our calculated values of pi allow us to engineer things we would not otherwise have been able to engineer.
In a more abstract (and modern sense) number theory has proven directly applicable not only to cryptography (a filed that would not exist as we know it otherwise) but also to pretty much the entirety of physics. I encourage you to check out this link: http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm
posted by 256 at 9:26 PM on October 10, 2012
The practical applications of pure maths are difficult to describe in a comment (not because people are too dumb to understand, but just because it is too big).
For very starters, we can use pi. The depths to which we can calculate pi exist only because pure maths discovered irrational numbers. We never could have gotten that precision by just making and measuring circles. And yet, our calculated values of pi allow us to engineer things we would not otherwise have been able to engineer.
In a more abstract (and modern sense) number theory has proven directly applicable not only to cryptography (a filed that would not exist as we know it otherwise) but also to pretty much the entirety of physics. I encourage you to check out this link: http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm
posted by 256 at 9:26 PM on October 10, 2012
I'm sorry if some of my comments here have seemed like personal attacks – I truly didn't mean them as such, although now it's easy to see why you might take them that way.
Ok, no worries.
helpful pedagogical clues about mathematics education in the study this post is about.
Heh, yes, I guess we're a bit derailed. Maybe we can continue the geometry/algebra/calculus thing in MeMail.
On the question of innate understanding of numbers: My grandmother has an unusual way of thinking about numbers — to her, they are not on a line, but on a path. This path goes up and down, goes around corners, has scenery with trees and whatnot. She was able to understand addition with this mental model, somehow, but multiplication was a disaster. She didn't find out that her mental model of numbers was unusual until it happened to come up in a conversation in college — she and another person started comparing number paths and everybody else was like, "WTF are you two talking about?" Turns out, my grandmother says, that this is a known thing that some kids do, but most unlearn it at some point and forget. But she never did (in part because she never met a teacher who diagnosed the issue), and so in her it persisted into adulthood.
posted by stebulus at 9:33 PM on October 10, 2012 [1 favorite]
Ok, no worries.
helpful pedagogical clues about mathematics education in the study this post is about.
Heh, yes, I guess we're a bit derailed. Maybe we can continue the geometry/algebra/calculus thing in MeMail.
On the question of innate understanding of numbers: My grandmother has an unusual way of thinking about numbers — to her, they are not on a line, but on a path. This path goes up and down, goes around corners, has scenery with trees and whatnot. She was able to understand addition with this mental model, somehow, but multiplication was a disaster. She didn't find out that her mental model of numbers was unusual until it happened to come up in a conversation in college — she and another person started comparing number paths and everybody else was like, "WTF are you two talking about?" Turns out, my grandmother says, that this is a known thing that some kids do, but most unlearn it at some point and forget. But she never did (in part because she never met a teacher who diagnosed the issue), and so in her it persisted into adulthood.
posted by stebulus at 9:33 PM on October 10, 2012 [1 favorite]
Sys Rq: How's this for "theorizing": Give those kids and those "people living in some traditional societies" (whatever the hell that means) nine objects numbered 1 through 9 and placed in order, and tell them to point to the middle one; I bet everything I own that the majority will point to the thing with the 5 on it.
I'm sure you're right. But the study was actually linked upthread, so we can read what they did instead:
The RadioLab episode linked upthread (and in the article in the post) seems to be pretty misleading here. Ten minutes in:
posted by stebulus at 9:38 PM on October 10, 2012 [2 favorites]
I'm sure you're right. But the study was actually linked upthread, so we can read what they did instead:
A total of 33 Mundurucu adults and children were tested individually in a number-space task (figure 1)(24). On each trial, a line segment was displayed on a computer screen, with one dot at left and ten dots at right (or, in a separate block, 10 and 100 dots respectively). Then other numbers were presented in random order, in various forms (sets of dots, sequences of tones, spoken Mundurucu words, spoken Portuguese words). For each number, the participant pointed to a screen location and this response was recorded by a mouse click, without feedback. Only two training trials were presented, with sets of dots whose numerosity corresponded to the ends of the scale (e.g. one and ten). The participants were told that these two stimuli belonged to their respective ends, but that other stimuli could be placed at any location.That's not the procedure you describe, but it is also interesting. (And as the study says, they did it with Westerners too, and were able to get the same behaviour from them under some circumstances.)
The RadioLab episode linked upthread (and in the article in the post) seems to be pretty misleading here. Ten minutes in:
Scientist [Dehaene, one of the authors of the study above]: So, we've done these very funny experiments in the Amazon with people in the Amazon who do not count. Basically, in their culture, they do not have number words beyond five, and they don't recite [?] these numbers. What we found is that these people still think of numbers in logarithmic way. Even the adults. What that means is if you give them a line and on the left you place one object and on the right you place nine objects...I'm not sure what the scientist actually said to the producer in the part of the interview that's been edited out, but based on the actual text of the study, they actually *did not* ask them what number is exactly between one and nine. The minor thing is that it was ten. (Nine may have been chosen for explanatory purposes because it's square and makes the numbers come out nice.) The major thing is that they were just given numbers and asked to place them, and, as the scientist says, "What they put in the middle is 3."
Producer [Miller]: You got that?
Sidekick [?]: Uh huh.
Producer: And he asked them, what number is exactly between one and nine?
Sidekick: Okay.
Producer: So you'd say?
Sidekick: Five.
Producer: Exactly. But:
Scientist: What they put in the middle is 3.
posted by stebulus at 9:38 PM on October 10, 2012 [2 favorites]
they did it with Westerners too
Correction: The study says Americans specifically. The word "Westerners" was mine, and I'm not sure why I made the substitution.
posted by stebulus at 9:43 PM on October 10, 2012
Correction: The study says Americans specifically. The word "Westerners" was mine, and I'm not sure why I made the substitution.
posted by stebulus at 9:43 PM on October 10, 2012
Signed,
Collective Past and Present Students of MAT 301
posted by 256 at 3:19 PM on October 10 [8 favorites −] Favorite added! [!]
I think the real question is, which number is larger: 256 or 100?
posted by Brak at 9:48 PM on October 10, 2012
Collective Past and Present Students of MAT 301
posted by 256 at 3:19 PM on October 10 [8 favorites −] Favorite added! [!]
I think the real question is, which number is larger: 256 or 100?
posted by Brak at 9:48 PM on October 10, 2012
what number is in the middle of 1 and 9?
In the case of Pi it's 415. At first, anyway. Then it's (null), then 6, then 05820, then 6406286208. (Citation)
(To be continued)
posted by Twang at 10:26 PM on October 10, 2012
In the case of Pi it's 415. At first, anyway. Then it's (null), then 6, then 05820, then 6406286208. (Citation)
(To be continued)
posted by Twang at 10:26 PM on October 10, 2012
Hello, there! I am a psychophysicist(-in-training), and while I haven't read the main paper very carefully yet, I thought I would address some background questions that came up.
Weber-Fechner Law; Psychophysical Scaling
To begin, we must understand the idea of the Weber-Fechner law. As stated in the article, it is the notion that the human perceptual system is sensitive to relative portions. Suppose that I had a 5-lb chihuahua and a 50-lb bulldog (do bulldogs get that heavy? I don't know, this is just an example), and that they both gained 5 lbs. Although the amount of gain is the same for both the chihuahua and the bulldog, the change will be much more noticeable on the chihuahua than on the bulldog. Rather, for the weight increase to appear perceptually equivalent, the increase should be stated in percentage change -- so, since the chihuahua underwent a 100% increase (doubling) in weight, the bulldog must also undergo a 100% increase in weight to appear equally obese. This is what the article means by "psychological experiments suggest that multiplying the intensity of some sensory stimuli causes a linear increase in perceived intensity," which, when worded that way, sounds horribly confusing.
Anyway, many psychological dimensions have been found to obey the Weber-Fechner law, including brightness and loudness, although not all do. There is a whole sub-field of psychophysics dedicated to understanding psychophysical scaling -- i.e., how the physical intensity of a stimulus relates to the perceived intensity of a stimulus.
Numerosity Perception
The article opens by talking about numerosity perception -- how do people experience "manyness?" If I briefly showed you a picture of two dots versus one dot, you could readily say that the the two-dot picture has twice as many dots as the one-dot picture without counting. What if I showed you a picture of twenty dots versus ten dots? Would twenty dots appear to be twice as many as ten? Or, would fifteen dots appear to be twice as many as ten? These stimuli are presented only for a few seconds at a time, so you cannot simply count to see how many dots there are. (You're probably wondering if people rely on other cues, such as dot density. Numerosity studies tend to require a lot of control conditions to rule out alternative explanations.) Anyway, psychophysicists use all sorts of rigorous experimental techniques to measure things like the point of subjective equality (PSE), which is taken to mean perceptual equivalence.
The numerosity article mentioned is Dehaene (2003). This is not my area, so if you know more than I do, please correct me. In this experiment, each participant (a 5-year-old child) was presented with a pair of drumbeating sequences. The participant had to indicate (using age-appropriate pointing methods) whether the first drumbeat or the second drumbeat sequence had "more." So, a trial might consist of something like: beat-beat-beat (pause) beat-beat-beat-beat-beat. Then, the correct answer would be the second sequence.
Apparently, five-year-olds are not very good at this task; whereas no adult would ever choose the three-beat sequence over the five-beat sequence, some children will. For the different stimulus conditions (e.g., two-beat vs. three-beat, two-beat vs. four-beat, etc.), the researchers calculated the children's scores, and used them to analyze the PSE. I have to guess a little bit at their analysis, but here are the results. In the two-beat condition, in which the two-beat baseline is compared against higher numbers of beats, children treat two beats as though they are indistinguishable from about four beats (i.e., PSE is about 4); children need four beats or higher to reliably choose this sequence over the two-beat sequence. In the four-beat condition, in which the four-beat baseline is compared against higher numbers of beats, children behave as though four beats are indistinguishable from about eight beats (i.e., PSE is about 8); again, children need eight beats or more to reliably choose this sequence over the four-beat sequence. So, for these children, the minimum requirement for "moreness" increases approximately 2-4-8 (whether geometric or exponential is up for debate), unlike with the 2-3-4 of adults. (Sorry, I am simplifying a lot here, since the PSE is a topic in and of itself, but that's the gist of it.)
The tl;dr:
- Psychophysics is an awesome research area, and I'm happy that I got to ramble about it to people outside the department.
- The main article (Sun et al., 2012) looks awesome, and I will have a careful look. I am still but a novice in this area, but Bayesian optimality is something that my boss specializes in.
- Numerosity perception sounds insane, but it's a legitimate area of research with some pretty cool results! Children have poor number sense.
posted by tickingclock at 11:10 PM on October 10, 2012 [5 favorites]
Weber-Fechner Law; Psychophysical Scaling
To begin, we must understand the idea of the Weber-Fechner law. As stated in the article, it is the notion that the human perceptual system is sensitive to relative portions. Suppose that I had a 5-lb chihuahua and a 50-lb bulldog (do bulldogs get that heavy? I don't know, this is just an example), and that they both gained 5 lbs. Although the amount of gain is the same for both the chihuahua and the bulldog, the change will be much more noticeable on the chihuahua than on the bulldog. Rather, for the weight increase to appear perceptually equivalent, the increase should be stated in percentage change -- so, since the chihuahua underwent a 100% increase (doubling) in weight, the bulldog must also undergo a 100% increase in weight to appear equally obese. This is what the article means by "psychological experiments suggest that multiplying the intensity of some sensory stimuli causes a linear increase in perceived intensity," which, when worded that way, sounds horribly confusing.
Anyway, many psychological dimensions have been found to obey the Weber-Fechner law, including brightness and loudness, although not all do. There is a whole sub-field of psychophysics dedicated to understanding psychophysical scaling -- i.e., how the physical intensity of a stimulus relates to the perceived intensity of a stimulus.
Numerosity Perception
The article opens by talking about numerosity perception -- how do people experience "manyness?" If I briefly showed you a picture of two dots versus one dot, you could readily say that the the two-dot picture has twice as many dots as the one-dot picture without counting. What if I showed you a picture of twenty dots versus ten dots? Would twenty dots appear to be twice as many as ten? Or, would fifteen dots appear to be twice as many as ten? These stimuli are presented only for a few seconds at a time, so you cannot simply count to see how many dots there are. (You're probably wondering if people rely on other cues, such as dot density. Numerosity studies tend to require a lot of control conditions to rule out alternative explanations.) Anyway, psychophysicists use all sorts of rigorous experimental techniques to measure things like the point of subjective equality (PSE), which is taken to mean perceptual equivalence.
The numerosity article mentioned is Dehaene (2003). This is not my area, so if you know more than I do, please correct me. In this experiment, each participant (a 5-year-old child) was presented with a pair of drumbeating sequences. The participant had to indicate (using age-appropriate pointing methods) whether the first drumbeat or the second drumbeat sequence had "more." So, a trial might consist of something like: beat-beat-beat (pause) beat-beat-beat-beat-beat. Then, the correct answer would be the second sequence.
Apparently, five-year-olds are not very good at this task; whereas no adult would ever choose the three-beat sequence over the five-beat sequence, some children will. For the different stimulus conditions (e.g., two-beat vs. three-beat, two-beat vs. four-beat, etc.), the researchers calculated the children's scores, and used them to analyze the PSE. I have to guess a little bit at their analysis, but here are the results. In the two-beat condition, in which the two-beat baseline is compared against higher numbers of beats, children treat two beats as though they are indistinguishable from about four beats (i.e., PSE is about 4); children need four beats or higher to reliably choose this sequence over the two-beat sequence. In the four-beat condition, in which the four-beat baseline is compared against higher numbers of beats, children behave as though four beats are indistinguishable from about eight beats (i.e., PSE is about 8); again, children need eight beats or more to reliably choose this sequence over the four-beat sequence. So, for these children, the minimum requirement for "moreness" increases approximately 2-4-8 (whether geometric or exponential is up for debate), unlike with the 2-3-4 of adults. (Sorry, I am simplifying a lot here, since the PSE is a topic in and of itself, but that's the gist of it.)
The tl;dr:
- Psychophysics is an awesome research area, and I'm happy that I got to ramble about it to people outside the department.
- The main article (Sun et al., 2012) looks awesome, and I will have a careful look. I am still but a novice in this area, but Bayesian optimality is something that my boss specializes in.
- Numerosity perception sounds insane, but it's a legitimate area of research with some pretty cool results! Children have poor number sense.
posted by tickingclock at 11:10 PM on October 10, 2012 [5 favorites]
The practical applications of pure maths are difficult to describe in a comment (not because people are too dumb to understand, but just because it is too big).
Is there a web page that makes your point for you? I'd Google it myself but I'm honestly uncertain how to phrase the search term.
We never could have gotten that precision by just making and measuring circles. And yet, our calculated values of pi allow us to engineer things we would not otherwise have been able to engineer.
But that proves nothing. Relativity depends on irrational numbers, that doesn't mean we need to teach grade-schoolers all about i. In any case aren't there multiple ways to derive π* now? It was long ago that the number was only considered to be a geometric constant. But while π is important, it's not all-important.
I think I'm a bit offended by your tone because too many people are scared to death of math, and your conclusion that they're being taught wrong, but then going on to explain things in a more obtuse way, will not solve this problem. And anyway, I don't think people are necessarily beholden to the first way they come to understand something, people are more flexible than that.
* I can do it too.
posted by JHarris at 12:49 AM on October 11, 2012
Is there a web page that makes your point for you? I'd Google it myself but I'm honestly uncertain how to phrase the search term.
We never could have gotten that precision by just making and measuring circles. And yet, our calculated values of pi allow us to engineer things we would not otherwise have been able to engineer.
But that proves nothing. Relativity depends on irrational numbers, that doesn't mean we need to teach grade-schoolers all about i. In any case aren't there multiple ways to derive π* now? It was long ago that the number was only considered to be a geometric constant. But while π is important, it's not all-important.
I think I'm a bit offended by your tone because too many people are scared to death of math, and your conclusion that they're being taught wrong, but then going on to explain things in a more obtuse way, will not solve this problem. And anyway, I don't think people are necessarily beholden to the first way they come to understand something, people are more flexible than that.
* I can do it too.
posted by JHarris at 12:49 AM on October 11, 2012
Hmmm, but if you show them nine groups of things
Are those beans?
posted by obiwanwasabi at 2:46 AM on October 11, 2012
Are those beans?
posted by obiwanwasabi at 2:46 AM on October 11, 2012
Sorry JHarris. I know my tone started off snarky and patronizing and I really feel bad about that. I've been trying to avoid that in subsequent posts, but I may have failed. I really want to be clear that I don't think it's in any way a failing not to have learned advanced number theory. It's an esoteric discipline and, while it is very interesting, so is Medieval Hungarian history. And our lives are only so long, so we can't learn everything about everything.
I do think you've misunderstood my point though. I don't think people are being taught wrong, not really. The number line is a fine tool for teaching math and there's no real reason to teach grade schoolers set theory (just like Newtonian kinematics are a fine thing that people should learn, despite being technically "wrong").
The only point I'm really trying to make is that number theory (the sort that shows the lie of the number line) is a real and practical field of research, and so the number line needs to be understood as an analogy that, like all analogies, does break down at some point. But most people don't know this.
I guess I started posting in this thread because I was frustrated to see people shutting down koeselitz from a supposed position of authority when in fact he was the one who had it right (though he certainly could have made his first comment a lot less opaque).
And, you know, maybe I do think that math should be taught a little differently. If the curriculum were up to me, the number line would stay and grade school math would remain largely the same. But then, in high school, in the last required math course, it would be nice if there was at least one unit that talked about number theory, introducing the idea that the question "what is 1 and how does it relate to 2?" is actually a very difficult one that people have only relatively recently started to get a handle on. I mean, we teach high-schoolers Newtonian physics but also let them know that relativity exists. And we teach them the Bohr model of the atom, but also mention that it has been superceded by quantum mechanics. Why don't we do the same with math.
But it's not that important. As I said, the number line gets you everything you need to know if you're not a math/physics professor or a research physicist. But it's not a natural law, and the truth of mathematical claims should not be judged based on whether or not they jibe with the number line.
posted by 256 at 5:37 AM on October 11, 2012 [2 favorites]
I do think you've misunderstood my point though. I don't think people are being taught wrong, not really. The number line is a fine tool for teaching math and there's no real reason to teach grade schoolers set theory (just like Newtonian kinematics are a fine thing that people should learn, despite being technically "wrong").
The only point I'm really trying to make is that number theory (the sort that shows the lie of the number line) is a real and practical field of research, and so the number line needs to be understood as an analogy that, like all analogies, does break down at some point. But most people don't know this.
I guess I started posting in this thread because I was frustrated to see people shutting down koeselitz from a supposed position of authority when in fact he was the one who had it right (though he certainly could have made his first comment a lot less opaque).
And, you know, maybe I do think that math should be taught a little differently. If the curriculum were up to me, the number line would stay and grade school math would remain largely the same. But then, in high school, in the last required math course, it would be nice if there was at least one unit that talked about number theory, introducing the idea that the question "what is 1 and how does it relate to 2?" is actually a very difficult one that people have only relatively recently started to get a handle on. I mean, we teach high-schoolers Newtonian physics but also let them know that relativity exists. And we teach them the Bohr model of the atom, but also mention that it has been superceded by quantum mechanics. Why don't we do the same with math.
But it's not that important. As I said, the number line gets you everything you need to know if you're not a math/physics professor or a research physicist. But it's not a natural law, and the truth of mathematical claims should not be judged based on whether or not they jibe with the number line.
posted by 256 at 5:37 AM on October 11, 2012 [2 favorites]
>>We'll split this 9,000 dollars in half. You get the first three thousand, and I'll take the second half.
>That's different. Set aside $1000 first, and then the analogy works. (But, yes, you'd still end up with more than $3000.)
Um. I don't know what the fuck I was on when I wrote that response. It is hilariously wrong. Please ignore it.
But! Finding the middle number is not a matter of cutting in half; half of nine is four and a half. It's the mean you're looking for: (1+2+3+4+5+6+7+8+9)÷9=5
posted by Sys Rq at 7:59 AM on October 11, 2012
>That's different. Set aside $1000 first, and then the analogy works. (But, yes, you'd still end up with more than $3000.)
Um. I don't know what the fuck I was on when I wrote that response. It is hilariously wrong. Please ignore it.
But! Finding the middle number is not a matter of cutting in half; half of nine is four and a half. It's the mean you're looking for: (1+2+3+4+5+6+7+8+9)÷9=5
posted by Sys Rq at 7:59 AM on October 11, 2012
But this isn't about things like the p-adic number system, or rings with characteristic != 0, or whatever you're thinking of that doesn't jibe with the number line, or about acknowledging that the number line does not have supreme validity when you extend your notion of what a number is beyond the form it takes in grade school arithmetic. I was with koeselitz, actually, until the comment about how the number line is bad because it actually distracts us from the concreteness of numbers:
But clearly the things a oneness might be surrounded by aren't necessarily nothingness; they have distinctness. The next fundamental principle we encounter is that onenesses can have multiplicity; that the many are a possibility. This is where numbers are born: from the paired apprehensions that reality can cohere in oneness and yet can partake of multiplicity.
Which, I mean, if math education were a never-ending exhortation to remember what numbers really mean, number theory (and countless other branches of math that rely on the telescoping abstraction of previously concrete entities) would never exist. So if you both have problems with the number line it's for very opposite reasons, and I'm pretty surprised that you missed that.
Also re "supposed position of authority," see "hackneyed ideas about numbers." That just doesn't seem merited, koeselitz.
posted by invitapriore at 8:02 AM on October 11, 2012
But clearly the things a oneness might be surrounded by aren't necessarily nothingness; they have distinctness. The next fundamental principle we encounter is that onenesses can have multiplicity; that the many are a possibility. This is where numbers are born: from the paired apprehensions that reality can cohere in oneness and yet can partake of multiplicity.
Which, I mean, if math education were a never-ending exhortation to remember what numbers really mean, number theory (and countless other branches of math that rely on the telescoping abstraction of previously concrete entities) would never exist. So if you both have problems with the number line it's for very opposite reasons, and I'm pretty surprised that you missed that.
Also re "supposed position of authority," see "hackneyed ideas about numbers." That just doesn't seem merited, koeselitz.
posted by invitapriore at 8:02 AM on October 11, 2012
Also, am I the only person who understands "number line" to mean "a line we're drawing to represent R", not Z? I have very little idea what the objection to the number line is.
Also re "supposed position of authority," see "hackneyed ideas about numbers." That just doesn't seem merited, koeselitz.
Dedekind cuts were very definitely the first way I saw someone construct the real numbers. My memory is not perfect, but I think I've seen that route presented more times than the Cauchy sequence option.
posted by hoyland at 8:57 AM on October 11, 2012
Also re "supposed position of authority," see "hackneyed ideas about numbers." That just doesn't seem merited, koeselitz.
Dedekind cuts were very definitely the first way I saw someone construct the real numbers. My memory is not perfect, but I think I've seen that route presented more times than the Cauchy sequence option.
posted by hoyland at 8:57 AM on October 11, 2012
The rebuttals to the "five or wrong" thing make me think of 1+1=10. Yeah, sure, in base 2, and that's a valid answer for re-examining frames, but it's not really what is being asked by "What's 1+1" in the vast, vast majority of cases.
posted by klangklangston at 9:05 AM on October 11, 2012
posted by klangklangston at 9:05 AM on October 11, 2012
klangklangston: Except that it is relevant here. Because how we interpret the linked study changes dramatically depending on whether we are:
A. looking at the people who say 3 and trying to figure out what makes them get it wrong in that way;
OR
B. looking at the people who say 3 and trying to understand their different but equally valid and practicable understanding of the relationships between numbers.
In the vast majority (essentially 100%) I would never think of piping up with a "well, actually" when someone said something like "5 is halfway between 1 and 9." But, in this particular case, the people saying "5 or wrong" are essentially arguing for interpretation A above, when I think interpretation B is more valid.
posted by 256 at 9:33 AM on October 11, 2012
A. looking at the people who say 3 and trying to figure out what makes them get it wrong in that way;
OR
B. looking at the people who say 3 and trying to understand their different but equally valid and practicable understanding of the relationships between numbers.
In the vast majority (essentially 100%) I would never think of piping up with a "well, actually" when someone said something like "5 is halfway between 1 and 9." But, in this particular case, the people saying "5 or wrong" are essentially arguing for interpretation A above, when I think interpretation B is more valid.
posted by 256 at 9:33 AM on October 11, 2012
"In the vast majority (essentially 100%) I would never think of piping up with a "well, actually" when someone said something like "5 is halfway between 1 and 9." But, in this particular case, the people saying "5 or wrong" are essentially arguing for interpretation A above, when I think interpretation B is more valid."
And like I said, that's like saying that 1+1=10 is an equally valid and practicable understanding of the relationship between numbers. However, if you're given a multiple choice question with "What number is halfway between 1 and 9," you would be marked wrong for answering "3," just as if you answered 10 for 1+1. While the "assuming a standard frame of reference" is elided, it's still in practice.
posted by klangklangston at 10:18 AM on October 11, 2012
And like I said, that's like saying that 1+1=10 is an equally valid and practicable understanding of the relationship between numbers. However, if you're given a multiple choice question with "What number is halfway between 1 and 9," you would be marked wrong for answering "3," just as if you answered 10 for 1+1. While the "assuming a standard frame of reference" is elided, it's still in practice.
posted by klangklangston at 10:18 AM on October 11, 2012
Except that, by your analogy, this is not a case of me saying "1+1=10." This is a case where people unindoctrinated to the standard frame of reference are saying "1+1=10" and the response is "clearly they don't know how to add."
I'm just putting forth that it's entirely possible they are using base 2.
posted by 256 at 11:21 AM on October 11, 2012 [1 favorite]
I'm just putting forth that it's entirely possible they are using base 2.
posted by 256 at 11:21 AM on October 11, 2012 [1 favorite]
Further to my comment above (i.e., feel free to ignore me having a conversation with myself here:P): so, one of the reasons why math has such nit-picky seeming definitions is because there are a lot of different ways of thinking about concepts such as "what is a number?" that are subtly different, and being able to talk about those subtle differences requires some very careful definitions.
So the idea of counting numbers gets formalized as a totally ordered set: take the natural numbers (1, 2, 3, ...; or throw 0 in there too if you want); any two numbers can be compared, and either one is larger than the other, or they are equal. Ordinals and cardinals are two different ways of thinking about this for regular, finite natural numbers, that work slightly differently when you extend them to concepts of infinity (and each have their uses in different contexts).
What if you want to throw in negative numbers? There's a couple different ways to think of this. The more abstract way that isn't quite as intuitive to many people who do not have a strong math background is to say, well, we have this set (say, the natural numbers or positive integers plus 0, or nonnegative rational numbers if you prefer), and it has some operations defined on it: addition and multiplication. And we can think about identity elements with respect to each of these operations: 0 is the additive identity because 0 + anything = that thing; and 1 is the multiplicative identity because 1 x anything = that thing. In other words, performing the operation with the corresponding identity element leaves the other element fixed. Then you can also think about inverses under an operation: if a is some positive integer, then -a is the additive inverse because a + (-a) = 0 (the additive identity). Likewise, 1/a is the multiplicative inverse because a x (1/a) = 1 (the multiplicative inverse). (In this formalism, subtraction and division are just the inverse operations of addition and multiplication.) So we construct the integers from the natural numbers by "extending" the set with respect to the operation of addition, and we construct the set of rational numbers by extending the set of integers with respect to multiplication. Extending the set of rational numbers to include limits of sequences of rational numbers gives us the set of real numbers. (An extending the set of real numbers to include solutions to any polynomial equations gives us the complex numbers: things of the form a + bi, where a and b are any real numbers, and i is "the square root of -1", i.e. the solution to the equation i^2 = -1, which does not have any real number solutions.) In this framework, the real numbers are a set together with two operations defined on the set that satisfy a certain list of properties.
"The real line" is not well-defined in a mathematical sense, but for me it conjures up another way of thinking about the set of real numbers, that is more geometric/topological, and maybe more natural for many people. Here, we define a "metric" (a distance function) d(x,y), where
(i) the distance between any point and itself is 0: d(x,x)=0;
(ii) the distance between any two points that are not the same is positive: d(x,y) > 0 whenever x is not equal to y;
(iii) the triangle inequality is satisfied: if x, y, and z are three distinct points, then d(x,y) + d(y,z) >= d(x,z).
The standard, Euclidean metric on the real numbers says that if n is any integer, then n and n+1 (or n-1) are a distance exactly 1 apart: d(n, n+1) = 1. In other words, we identify real numbers with points on a line, and we place them in such a way that the integers are all evenly spaced.
(On a side note: once we throw in negative numbers, we actually already have direction as well as magnitude, so thinking of the integers or real numbers in terms of position on a line really does make a lot of sense.)
But there are a lot of different metrics that you can define on the set of real numbers. We could use a logarithmic metric, where d(x,y) = |log_a(x) - log_b(y)|. That is, we place the real numbers on a line based on a logarithmic scale instead of a linear scale.
There are many other ways of thinking about the set of real numbers, depending on what other added structure you focus on. No one interpretation is "best" overall: it depends entirely on the context. One way of interpreting the results in the linked article is that untrained humans naturally use context clues to infer a metric on the set of real numbers when they appear in applications, so that if you set out a bunch of points on a line and ask a small child what number is halfway in between a and b, they'll (roughly) point to the spot that is the midpoint (a_b)/2 in the Euclidean distance/metric. But if you lay out the information differently, like fantabulous timewaster did, then the small child will infer a different metric as being the "natural" one for that context. (Or it may have more to do with how we store information in our brains, like the scientists who are actually experts in this area, unlike me, hypothesized.)
You could also use a metric where numbers after, say, 3 are considered to be very close together, and thus harder to distinguish that the numbers up to 3, whose magnitude can be gleaned with a quick glance.
Note that a metric gives intervals on the real line a width. In the Euclidean metric, the interval from 1 to 9 has width 8 (so half the width is 4, so the midpoint is 1+4, or 9-4, which is 5). In the logarithmic metric above, if we take the logarithm base 3, the width of the interval from 1 to 9 is 2: 3^0 = 1, so log_3(1) = 0; 3^2 = 9, so log_3(9) = 2. Thus the distance between 1 and 9 is 2-0 = 2. In this case, half the width of this interval is 1, so the midpoint is the point that is distance 1 greater than the point 1, and distance 1 less than the point 9. That means that the midpoint, m, has log_3(m) = 1, so m = 3^1 = 3. If you take this idea of different ways of assigning widths to intervals (or some other basic set depending on the topology of the space you're working in, for those reading this for whom this comment makes sense), then you can construct a generalization, called a measure. Roughly speaking, a measure tells you how "mass" is distributed on a set. The Euclidean metric generates something called the Lebesgue measure, where the measure of an interval is the same as it's Euclidean width. You can think of this as mass being distributed evenly along the real line. But maybe you want the center around 0 to "weigh more", like if you were Einstein and thinking about a rubber sheet universe. Then you could use a measure that says that an interval containing or near 0 has a larger measure (more weight, aka the width of the interval according to a corresponding metric is larger) than an interval of the same Euclidean width/Lebesgue measure but far away from 0. The example where the "distance" between numbers bigger than 3 is smaller than the "distance" between numbers less than 3 would correspond to such a measure on the set of real numbers.
(Another aside for folks for whom this may make sense: this is closely related to stuff like probability densities and distributions and stuff that's important for statistics as well as mathematical analysis.)
posted by eviemath at 11:58 AM on October 11, 2012
So the idea of counting numbers gets formalized as a totally ordered set: take the natural numbers (1, 2, 3, ...; or throw 0 in there too if you want); any two numbers can be compared, and either one is larger than the other, or they are equal. Ordinals and cardinals are two different ways of thinking about this for regular, finite natural numbers, that work slightly differently when you extend them to concepts of infinity (and each have their uses in different contexts).
What if you want to throw in negative numbers? There's a couple different ways to think of this. The more abstract way that isn't quite as intuitive to many people who do not have a strong math background is to say, well, we have this set (say, the natural numbers or positive integers plus 0, or nonnegative rational numbers if you prefer), and it has some operations defined on it: addition and multiplication. And we can think about identity elements with respect to each of these operations: 0 is the additive identity because 0 + anything = that thing; and 1 is the multiplicative identity because 1 x anything = that thing. In other words, performing the operation with the corresponding identity element leaves the other element fixed. Then you can also think about inverses under an operation: if a is some positive integer, then -a is the additive inverse because a + (-a) = 0 (the additive identity). Likewise, 1/a is the multiplicative inverse because a x (1/a) = 1 (the multiplicative inverse). (In this formalism, subtraction and division are just the inverse operations of addition and multiplication.) So we construct the integers from the natural numbers by "extending" the set with respect to the operation of addition, and we construct the set of rational numbers by extending the set of integers with respect to multiplication. Extending the set of rational numbers to include limits of sequences of rational numbers gives us the set of real numbers. (An extending the set of real numbers to include solutions to any polynomial equations gives us the complex numbers: things of the form a + bi, where a and b are any real numbers, and i is "the square root of -1", i.e. the solution to the equation i^2 = -1, which does not have any real number solutions.) In this framework, the real numbers are a set together with two operations defined on the set that satisfy a certain list of properties.
"The real line" is not well-defined in a mathematical sense, but for me it conjures up another way of thinking about the set of real numbers, that is more geometric/topological, and maybe more natural for many people. Here, we define a "metric" (a distance function) d(x,y), where
(i) the distance between any point and itself is 0: d(x,x)=0;
(ii) the distance between any two points that are not the same is positive: d(x,y) > 0 whenever x is not equal to y;
(iii) the triangle inequality is satisfied: if x, y, and z are three distinct points, then d(x,y) + d(y,z) >= d(x,z).
The standard, Euclidean metric on the real numbers says that if n is any integer, then n and n+1 (or n-1) are a distance exactly 1 apart: d(n, n+1) = 1. In other words, we identify real numbers with points on a line, and we place them in such a way that the integers are all evenly spaced.
(On a side note: once we throw in negative numbers, we actually already have direction as well as magnitude, so thinking of the integers or real numbers in terms of position on a line really does make a lot of sense.)
But there are a lot of different metrics that you can define on the set of real numbers. We could use a logarithmic metric, where d(x,y) = |log_a(x) - log_b(y)|. That is, we place the real numbers on a line based on a logarithmic scale instead of a linear scale.
There are many other ways of thinking about the set of real numbers, depending on what other added structure you focus on. No one interpretation is "best" overall: it depends entirely on the context. One way of interpreting the results in the linked article is that untrained humans naturally use context clues to infer a metric on the set of real numbers when they appear in applications, so that if you set out a bunch of points on a line and ask a small child what number is halfway in between a and b, they'll (roughly) point to the spot that is the midpoint (a_b)/2 in the Euclidean distance/metric. But if you lay out the information differently, like fantabulous timewaster did, then the small child will infer a different metric as being the "natural" one for that context. (Or it may have more to do with how we store information in our brains, like the scientists who are actually experts in this area, unlike me, hypothesized.)
You could also use a metric where numbers after, say, 3 are considered to be very close together, and thus harder to distinguish that the numbers up to 3, whose magnitude can be gleaned with a quick glance.
Note that a metric gives intervals on the real line a width. In the Euclidean metric, the interval from 1 to 9 has width 8 (so half the width is 4, so the midpoint is 1+4, or 9-4, which is 5). In the logarithmic metric above, if we take the logarithm base 3, the width of the interval from 1 to 9 is 2: 3^0 = 1, so log_3(1) = 0; 3^2 = 9, so log_3(9) = 2. Thus the distance between 1 and 9 is 2-0 = 2. In this case, half the width of this interval is 1, so the midpoint is the point that is distance 1 greater than the point 1, and distance 1 less than the point 9. That means that the midpoint, m, has log_3(m) = 1, so m = 3^1 = 3. If you take this idea of different ways of assigning widths to intervals (or some other basic set depending on the topology of the space you're working in, for those reading this for whom this comment makes sense), then you can construct a generalization, called a measure. Roughly speaking, a measure tells you how "mass" is distributed on a set. The Euclidean metric generates something called the Lebesgue measure, where the measure of an interval is the same as it's Euclidean width. You can think of this as mass being distributed evenly along the real line. But maybe you want the center around 0 to "weigh more", like if you were Einstein and thinking about a rubber sheet universe. Then you could use a measure that says that an interval containing or near 0 has a larger measure (more weight, aka the width of the interval according to a corresponding metric is larger) than an interval of the same Euclidean width/Lebesgue measure but far away from 0. The example where the "distance" between numbers bigger than 3 is smaller than the "distance" between numbers less than 3 would correspond to such a measure on the set of real numbers.
(Another aside for folks for whom this may make sense: this is closely related to stuff like probability densities and distributions and stuff that's important for statistics as well as mathematical analysis.)
posted by eviemath at 11:58 AM on October 11, 2012
"Except that, by your analogy, this is not a case of me saying "1+1=10." This is a case where people unindoctrinated to the standard frame of reference are saying "1+1=10" and the response is "clearly they don't know how to add.""
… but they pretty clearly don't know how to add in base 10, the standard frame of reference. It's interesting that people who don't know the standard frame of reference will use another coherent one, but it's reasonable to assume that without other information, answers that hew to the standard frame of reference are better answers than ones that don't.
posted by klangklangston at 2:00 PM on October 11, 2012
… but they pretty clearly don't know how to add in base 10, the standard frame of reference. It's interesting that people who don't know the standard frame of reference will use another coherent one, but it's reasonable to assume that without other information, answers that hew to the standard frame of reference are better answers than ones that don't.
posted by klangklangston at 2:00 PM on October 11, 2012
Understood 256. I do think you're right when you say that number theory is real and practical. I myself don't know that much about it though. I'll try to learn more.
(And don't apologize too much. I recognize that it's difficult to find the right line, but it is good to express strongly-held opinions in a strong manner.)
posted by JHarris at 3:15 PM on October 11, 2012 [1 favorite]
(And don't apologize too much. I recognize that it's difficult to find the right line, but it is good to express strongly-held opinions in a strong manner.)
posted by JHarris at 3:15 PM on October 11, 2012 [1 favorite]
>Regardless of that: "rate of change" was never once mentioned in my calculus class. At the beginning, "area under a curve" was mentioned<
Likewise for myself. In fact, it was later in Physics that a teacher used the rate of change (to directly address the area under the curve) that the light-bulb was switched on.
Later expeditions into calculus were more, enlightening.
posted by twidget at 3:18 PM on October 11, 2012
Likewise for myself. In fact, it was later in Physics that a teacher used the rate of change (to directly address the area under the curve) that the light-bulb was switched on.
Later expeditions into calculus were more, enlightening.
posted by twidget at 3:18 PM on October 11, 2012
3 holds for me if I'm considering quantity and effort.
Imagine one pie. Got it? Easy.
Imagining over 5 pies gets harder to visualize, and the larger numbers are equally hard, because you have to divide them into two separate smaller groups.
3 is in halfway to the largest easily visualized number of pies, without grouping them.
This seems like more of an issue of subitizing to me.
posted by knile at 12:32 AM on October 12, 2012
Imagine one pie. Got it? Easy.
Imagining over 5 pies gets harder to visualize, and the larger numbers are equally hard, because you have to divide them into two separate smaller groups.
3 is in halfway to the largest easily visualized number of pies, without grouping them.
This seems like more of an issue of subitizing to me.
posted by knile at 12:32 AM on October 12, 2012
> while numbers, while initially derived from the natural world
They're not derived from the natural world, they're derived from our perception of the universe through abstraction. We can only say "I have three apples" because we have the abstraction of apples. Numbers don't actually 'exist' in the natural world they only 'exist' in our perception of the world.
If you think this is too metaphysical, fine. But this topic is about perception and conception and so to have different conceptions of 'halfway' is perfectly fine in mathematics. All that mathematics requires is that you build up your system of abstraction in a logically consistent way.
posted by Pranksome Quaine at 5:31 AM on October 12, 2012
They're not derived from the natural world, they're derived from our perception of the universe through abstraction. We can only say "I have three apples" because we have the abstraction of apples. Numbers don't actually 'exist' in the natural world they only 'exist' in our perception of the world.
If you think this is too metaphysical, fine. But this topic is about perception and conception and so to have different conceptions of 'halfway' is perfectly fine in mathematics. All that mathematics requires is that you build up your system of abstraction in a logically consistent way.
posted by Pranksome Quaine at 5:31 AM on October 12, 2012
i is "the square root of -1", i.e. the solution to the equation i^2 = -1
Probably should be careful with that definition… That equation has two solutions, one of which is arbitrarily chosen to be i.
posted by esprit de l'escalier at 11:59 AM on October 13, 2012
Probably should be careful with that definition… That equation has two solutions, one of which is arbitrarily chosen to be i.
posted by esprit de l'escalier at 11:59 AM on October 13, 2012
Probably should be careful with that definition… That equation has two solutions, one of which is arbitrarily chosen to be i.
What's the other solution?
posted by eviemath at 9:07 PM on October 13, 2012
What's the other solution?
posted by eviemath at 9:07 PM on October 13, 2012
-i.
posted by nebulawindphone at 10:35 PM on October 13, 2012
posted by nebulawindphone at 10:35 PM on October 13, 2012
(Basically, as I understand it, the two solutions to x2 = -1 have the property that if you add them together, you get zero. So if one solution is i, the other one must be -i.)
posted by nebulawindphone at 10:45 PM on October 13, 2012
posted by nebulawindphone at 10:45 PM on October 13, 2012
That's true because the quadratic equation has no linear term: Your equation is x2 + 1 = (x-a)(x-b) = x2 - (a+b)x + ab. So, a = -b, and ab = 1.
A more visual rather than symbolic way to see this is to visualize imaginary numbers on the complex plane as in this wonderful explanation here.
posted by esprit de l'escalier at 11:17 PM on October 13, 2012
A more visual rather than symbolic way to see this is to visualize imaginary numbers on the complex plane as in this wonderful explanation here.
posted by esprit de l'escalier at 11:17 PM on October 13, 2012
[needs to not post while tired]
Yeah, I should have been more precise there.
(But that lets me defensively add a paragraph about more exciting math topics! If other readers are to the point where you're thinking about the complex numbers as the completion of the ring of real numbers, you're probably ready to handle homeomorphisms: mappings from a set to itself, or you can think of a homeomorphism as a mapping between two copies of the same set, that preserves operations of addition and multiplication. For example, you could also switch which solution of x^2 = 1 you call "positive", i.e. switch 1 and -1, and correspondingly switch a and -a for all real numbers a to make addition and multiplication look the same after the mapping (geometrically: reflecting the real axis across 0). Though, given the other interpretations of real numbers, this is in some sense less natural or arbitrary than choice of solution to be i, as esprit points out).
posted by eviemath at 6:24 AM on October 14, 2012
Yeah, I should have been more precise there.
(But that lets me defensively add a paragraph about more exciting math topics! If other readers are to the point where you're thinking about the complex numbers as the completion of the ring of real numbers, you're probably ready to handle homeomorphisms: mappings from a set to itself, or you can think of a homeomorphism as a mapping between two copies of the same set, that preserves operations of addition and multiplication. For example, you could also switch which solution of x^2 = 1 you call "positive", i.e. switch 1 and -1, and correspondingly switch a and -a for all real numbers a to make addition and multiplication look the same after the mapping (geometrically: reflecting the real axis across 0). Though, given the other interpretations of real numbers, this is in some sense less natural or arbitrary than choice of solution to be i, as esprit points out).
posted by eviemath at 6:24 AM on October 14, 2012
That's true because the quadratic equation has no linear term
It's actually true for polynomials other than quadratics--complex roots come in conjugate pairs. (Conjugate the polynomial. Conjugation is a homomorphism, so you're sticking bars over everything individually.)
Well... at a certain point this gets a bit circular. What's complex conjugation? It's the homomorphism defined by sending i to -i. Though you can just define i as 'a root of x^2+1', adjoin it to your field, so there's some -i out there, define conjugation and you're away. Or you could just factor, which is what I felt the need to pedant to begin with.
posted by hoyland at 6:49 AM on October 14, 2012
It's actually true for polynomials other than quadratics--complex roots come in conjugate pairs. (Conjugate the polynomial. Conjugation is a homomorphism, so you're sticking bars over everything individually.)
Well... at a certain point this gets a bit circular. What's complex conjugation? It's the homomorphism defined by sending i to -i. Though you can just define i as 'a root of x^2+1', adjoin it to your field, so there's some -i out there, define conjugation and you're away. Or you could just factor, which is what I felt the need to pedant to begin with.
posted by hoyland at 6:49 AM on October 14, 2012
For example, you could also switch which solution of x^2 = 1 you call "positive", i.e. switch 1 and -1, and correspondingly switch a and -a for all real numbers a to make addition and multiplication look the same after the mapping (geometrically: reflecting the real axis across 0).
(Emphasis mine.)
Addition is invariant under such reflection, but multiplication is not. You can tell the difference between 1 and −1 because only one of them is among the solutions to x²=x.
<digression>(Another way to put it: when defining multiplication for integers, at some point you have to say something like, "The additive group has two generators; pick one of them and call it 1. Now define a binary operation * by 0*n = 0 and (m+1)*n = m*n + n, [etc.]" Of course, in practice you might already have distinguished the generators because you defined N first and then constructed Z. But that really only means you're working with a concrete realization of Z and not with the abstract object which is defined only up to isomorphism. Kind of the same way that R^n has a preferred basis but the abstract vector space for which it is a model does not.)<digression>
Anyway, just as ±1 are identical twins as far as addition is concerned but not as far as multiplication is concerned, so it is possible to impose some additional structure on the complex numbers which *does* distinguish between ±i. I asked a number theorist about this once, and he came up with something fairly nice. I'll see if I can dig it up.
posted by stebulus at 11:43 AM on October 14, 2012 [1 favorite]
(Emphasis mine.)
Addition is invariant under such reflection, but multiplication is not. You can tell the difference between 1 and −1 because only one of them is among the solutions to x²=x.
<digression>(Another way to put it: when defining multiplication for integers, at some point you have to say something like, "The additive group has two generators; pick one of them and call it 1. Now define a binary operation * by 0*n = 0 and (m+1)*n = m*n + n, [etc.]" Of course, in practice you might already have distinguished the generators because you defined N first and then constructed Z. But that really only means you're working with a concrete realization of Z and not with the abstract object which is defined only up to isomorphism. Kind of the same way that R^n has a preferred basis but the abstract vector space for which it is a model does not.)<digression>
Anyway, just as ±1 are identical twins as far as addition is concerned but not as far as multiplication is concerned, so it is possible to impose some additional structure on the complex numbers which *does* distinguish between ±i. I asked a number theorist about this once, and he came up with something fairly nice. I'll see if I can dig it up.
posted by stebulus at 11:43 AM on October 14, 2012 [1 favorite]
I'll see if I can dig it up.
Hooray for filing — I had it under "i", in a folder labelled "i & −i".
The short version: It's a p-adic thing. The field C_p is field-isomorphic to the complex numbers, so we can think of it as C with a weird metric, the p-adic metric, which (at least sometimes) establishes different distances between i and (at least some) integers than between −i and those integers.
(A little more detail for people who know about p-adics: If p≡1 (mod 4) then there are two noncongruent solutions to a²≡−1 (mod p), say a1 and a2 (which are just integers), and these extend to two solutions in C_p, say, b1 and b2. Then the field isomorphism to C maps b1 and b2 to i and −i; but in the p-adic absolute value, |a1-b1|≤1/p (since a1 and b1 are congruent mod p) and |a1-b2|=1 (since a1 and b2 are not). So b1 and b2 are distinguishable by their distances to a1, which is an integer and thus canonically defined.)
So, that's a nice structure that can be imposed on C and which distinguishes i and −i.
posted by stebulus at 12:37 PM on October 14, 2012
Hooray for filing — I had it under "i", in a folder labelled "i & −i".
The short version: It's a p-adic thing. The field C_p is field-isomorphic to the complex numbers, so we can think of it as C with a weird metric, the p-adic metric, which (at least sometimes) establishes different distances between i and (at least some) integers than between −i and those integers.
(A little more detail for people who know about p-adics: If p≡1 (mod 4) then there are two noncongruent solutions to a²≡−1 (mod p), say a1 and a2 (which are just integers), and these extend to two solutions in C_p, say, b1 and b2. Then the field isomorphism to C maps b1 and b2 to i and −i; but in the p-adic absolute value, |a1-b1|≤1/p (since a1 and b1 are congruent mod p) and |a1-b2|=1 (since a1 and b2 are not). So b1 and b2 are distinguishable by their distances to a1, which is an integer and thus canonically defined.)
So, that's a nice structure that can be imposed on C and which distinguishes i and −i.
posted by stebulus at 12:37 PM on October 14, 2012
Addition is invariant under such reflection, but multiplication is not. You can tell the difference between 1 and −1 because only one of them is among the solutions to x²=x.
I understood "look the same after mapping" to mean "is a homomorphism", which is the case.
posted by hoyland at 1:13 PM on October 14, 2012
I understood "look the same after mapping" to mean "is a homomorphism", which is the case.
posted by hoyland at 1:13 PM on October 14, 2012
You mean that sending x to −x is a homomorphism of the ring structure on Z? It doesn't preserve multiplication.
posted by stebulus at 1:38 PM on October 14, 2012
posted by stebulus at 1:38 PM on October 14, 2012
Wait... doesn't it? Oh, duh, it doesn't. That's what I get for not writing things down.
posted by hoyland at 1:40 PM on October 14, 2012 [1 favorite]
posted by hoyland at 1:40 PM on October 14, 2012 [1 favorite]
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