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Study Suggests the Number-Line Concept Is Not Intuitive
An anonymous reader writes "The Yupno people of New Guinea have provided clues to the origins of the number-line concept, and suggest that the familiar concept of time may be cultural as well. From the article: 'Tape measures. Rulers. Graphs. The gas gauge in your car, and the icon on your favorite digital device showing battery power. The number line and its cousins – notations that map numbers onto space and often represent magnitude – are everywhere. Most adults in industrialized societies are so fluent at using the concept, we hardly think about it. We don't stop to wonder: Is it 'natural'? Is it cultural? Now, challenging a mainstream scholarly position that the number-line concept is innate, a study suggests it is learned."
Because I know most males know the number of their line, or at least what they think it is.
I just watched a documentary about this on Netflix, called The Story of 1, starring Terry Jones of Monty Python fame.I think it mentioned the ruler wasn't invented until sometime in ancient egypt.
... and in the DRM, bind them.
Try getting a bunch of 10-year-olds to understand the number line concept and you will find out in approximately 3 seconds that it is not innate.
Did anyone else think about older versions of interpreted BASIC first?
Dark Reflection
I wonder how far this goes! Is the notion of the counting numbers innate? I have heard that monkeys cannot count beyond 4. The way that people figured this out is that if five hunters go into a forest as a group, split up and hide. Then one by one, four hunters leave one at a time. The fifth hunter stays in hiding, the monkeys come out of hunting, and the hunter shoots a monkey. This does not happen when there are less than five hunters initially.
Start by studying the Hopi.
Lacking <sarcasm> tags,
Any measuring cup will tell you a number line can be very intuitive. Stacking objects, filling a container; many everyday tasks are perfect physical examples of a number line.
Rulers are another example, though perhaps a bit less physical or intuitive.
How can I believe you when you tell me what I don't want to hear?
Do you speak (or write) it? Intuitive and instinctual are different words.
> the familiar concept of time may be cultural as well.
So *that's* why in some cultures I can eat a fish before I've caught it.
I don't have the reference to hand but I recall there is a South American tribe which don't have words for left and right as most languages do. There words are equivalent to "Up Valley" and "Down Valley" Similarly, if I recall correctly, there's a Native American language that uses before and behind as an analog for time but the other way around to most languages. Their analogy is that you know the past and you can see what it in front of you so forward = the past. You can't see behind you and you don't know the future so behind = the future
From the main article. ""The Yupno people of New Guinea have provide clues to the origins of the number-line concept," Would it not be better to say. ""The Yupno people of New Guinea have provided clues to the origins of the number-line concept," Just asking a silly question here.
Figuring out what isn't intuitive isn't useful, unless we also know what is. Pie graphs for gas gauges, showing the shrinkage of the tank fractionally? Or a circle in a circle shrinking within the "full" one?
"Also, we document that precise number concepts can exist independently of linear or other metric-driven spatial representations."
But TFA doesn't mention any of them, or what we could change a gas gauge to to be intuitive.
Perhaps one day they can figure out why my mother compulsively fills up once the gauge goes under 1/2, but my sister runs cars to empty on a regular basis, usually filling up only after the "e" is lit, sometimes long after.
Learn to love Alaska
I have taught a number of people to code 2D and 3D games. Both 2D and 3D involve a lot of coordinate axis transformations that are almost universally non-intuitive at first. This is true in 2D despite there being a direct correlation between the data being mapped from the model/simulation to the screen (2D to 2D).
This study finds it is non-intuitive to go from an abstract number or count to a line segment? Sure. What I'd like to see is the "sources of evidence [which] suggest that humans naturally associate numbers with space". They would surprise me.
Insert self-referential sig here.
Once a significant percentage of the population becomes interested in measuring pieces of land for various purposes, people will start associating numbers to lines.
Because the amount of food is proportional to the surface of your land, and then... I personally feel it's quite natural, in this context, to associate numbers to geometrical constructs.
new sig
Show me a culture that doesn't have the concept of ordered sets -- which is all that a "number line" is.
And, no, I don't mean the fancy mathematical formalism. I mean things like narratives, directions from A to B, etc.
Lacking <sarcasm> tags,
... because I use complex numbers for everything, you insensitive clod. Don't you have any feelings for the one dimensionally-challenged?
now we need to go OSS in diesel cars
What does it matter if it's intuitive? English (and any other language, though possibly not language in the abstract) is learned, and it works just fine.
The same subject has been covered in "Here's looking to Euclid". It describes tests done on an Amazon tribe to see how they visually interpret numbers. Unlike most modern adults who visualize number spaced linearly, they visualized them spaced logarithmically. Their reasoning was that the intervals between numbers start (relatively) large and become smaller as the numbers get larger. i.e. from 1 to 2 it's a 100% increase but from 2 to 3 it's only a 33% increase and so on.
I imagine that a thickness gauge (which is what is *really* intuitive in the measuring-cup example) or a color-gauge would be more intuitive. The critical point here is that thicker is "more" and thinner is "less". Even with colors you can have "more red" or "less red". Numbers are a higher-form thought process. When dealing with a line system, your general intention is to gauge this same "more or less" comparions, but is abstracted through numbers which is based on a complex thought process of reading and comprehension.
Can they measure how much time of my life was wasted reading this stupid article? Thanks for the information about nothing...
"Mathematics... is largely taught dogmatically, as objective fact, black and white, right/wrong," Nunez said. "But our work shows that there are meaningful human ideas in math, ingenious solutions and designs that have been mediated by writing and notational devices... Perhaps we should think about bringing the human saga to the subject – instead of continuing to treat it romantically, as the 'universal language' it's not. " ummm ok? someone didn't do well in algebra.
whether anyone gets it or no one gets it, math is inherent and intrinsic in numbers.. 1 + 1 is 2 and 3+3 is 6, regardless of what words and symbols we use to describe it or what we know about it or don't.
Obviously people don't automatically know it by design. Math, like any language, evolves and develops over time. Similar to researchers in other fields, mathematicians research and make discoveries about math and share their results, and those results spread into the collective human body of knowledge. It started out as nothing, then something really basic, and developed over... well, as long as there has been humanity. Math is a pure language because its rules exist on their own, by nature, not by human convention. Applications of the laws of math and numbers, such are rulers and gauges, are human convention - of course they aren't universally known abstracts. Nor is the concept of measuring. I'd bet tho, that given the right experiment, they would find that the people of Papua New Guinea do indeed have notions of amount, and measure.
What next? Are they going to report that chemistry isn't universal because the people of Papua New Guinea don't have any concept of it?
In earlier research, Nunez found that the Aymara of the Andes seem to do the reverse, placing the past in front and the future behind.
I've worked for a number of PHBs who seemed content with the future sneaking up behind them and smacking them in the back of the skull.
Have gnu, will travel.
I'm tired of all the BS splitting hairs over whether something is "intuitive" or not.
Intuitive just means: given your current knowledge, does the thing make immediate sense? If so, it's intuitive.
Very few things are innate. You learn the rest, even if you don't realize that you learned it somewhere. Maybe you saw someone else do it, or maybe you deduced it from other things you already knew. But once you learned the concept, if you can always apply it without thinking, then it's intuitive.
Think of it like mathematical axioms/proofs. You start with a few axioms. As you learn, you gain some new theorems that are so elegant to use that you end up using them as frequently as axioms. These special theorems are what we call "intuitive" facts; they allow us to quickly and "intuitively" deduce things that would be difficult to prove all the way from axioms.
p.s. Intuition is also based on guesswork. For example, suppose you guessed the right theorem(s) to complete the proof; if you guessed right the first time (or 'quickly enough'), then you'll describe the proof as intuitive. If you had to try N different things and finally get an "aha" to use unexpected Theorem XYZ, then it's not intuitive.
The number line represents the continuum, that is the real numbers. The debates about the foundations of mathematics are old and ongoing, but most hold that the natural numbers {1, 2, 3, ...} are intuitive and axiomatic and prove the rest formally. Heck, Kroneckers well-known quote, "God made the natural numbers; all else is the work of man" is saying exactly that.
In the original task, people are shown a line and are asked to place numbers onto the line according to their size, with "1" going on the left endpoint and "10" (or sometimes "100" or "1000") going on the right endpoint.
Go to a class of college students in america, ask them to mark 10, 1 million, and 1 billion on a line, and 99% of them will draw 1 million closer to 1 billion. Usually a lot closer.
I read the article, and it wasn't clear to me what these people have discovered. Maybe I'll have to read the actual study. Or maybe anthropologists are better at understanding primitive cultures than their own.
"First they came for the slanderers and i said nothing."
I don't know why this result is surprising. I thought it was generally understand that counting (there are 10 sheep) and measurement (this fence is 10 feet long) were distinct concepts. The point of the number line is to establish a relationship between the two concepts.
Come to think of it, it should be obvious that a number line relates two distinct concepts, just from the form they usually take. A number line, with its regularly spaced markings perpendicular to the main line, has a form similar to that of a line graph, which shows a relationship between two distinct variables.
Oddly enough, I was telling my girlfriend just tonight that I'm not very visual, and tend to approach concepts best through symbols (numbers, words, etc.) I've always found graphical representations of math more-or-less useless (although they are cool sometimes) and prefer my math without the diagrams. She told me that I'm deeply weird. :)
"He who would learn astronomy, and other recondite arts, let him go elsewhere. " -- John Calvin, commenting on Genesis 1
Neither is reading. Human beings evolved to see "in the round" and not in focused linear scans. When we were children, both my sister and I went through periods when we were just learning to write where we wrote everything "exactly" backwards, like a mirror image. And, it wasn't all the time. We both outgrew it very quickly, but I'm sure it's been studied by some -ologist out there.
I swear to God...I swear to God! That is NOT how you treat your human!
Seriously, I can't be the only one who read the title and thought: Numberwang! m
In the immortal words of Socrates, who said; 'I drank what?'
Wrong question: "Here is 1, and here is 10. Where is the number 7
Yunpo: so, I have to classify, and determine, to which of the two numbers, number 7 is nearer?
California: so, I have to put 10 consecutively-equidistant points on the line, and count up to the 7-th point.
No, go ask Yunpo the right question: "Put 10 consecutive equidistant points, and count up to the 7-th point."
http://alexbellos.com/wp-content/uploads/2010/04/maths.pdf
Log scale = intuitive (ratios - there's twice as many of those as these)
Numberline != intuitive (counting, ordering etc.)
This is nothing new. http://verizonmath.blogspot.com/2006/12/verizon-doesnt-know-dollars-from-cents.html
Logical or not, the number line is equivalent to a finite list of axioms (field axioms, look 'em up, maybe with some stuff I forget atm). When we accept the truth of those axioms, all at once, then we begin studying 'the number line'.
Personally, studying unintuitive concepts via the language of mathematics interests me. That's how mathematics allows you to expand the list of things that you find intuitive. First, only the abstract language of mathematics describes some logical object. The logical object itself may or may not be 'intuitive' from the outset. Eventually, after studying a logical concept via math for a time, I can eventually gain some intuition concerning the object. I've done this with the real and complex number systems (separately), partial and ordinary differential equations, vector/inner product spaces, mathematical knots, and etc.
PS If you're looking for a way to study calculus or the real numbers in a "more intuitive sense", I suggest you look up the hyperreal number system.
PS: I don't reply to ACs.
Kids are taught calculus in most schools without informing about the existence of continuum hypothesis :-(
"Over the past thirty years I have been writing, speaking, and consulting about technology-driven trends that are coming but difficult for most people to see. Back in 2000, I wrote about one such trend that would hit about now, and here we are, on the brink of experiencing a technology that will provide new opportunities for IT to add strategic value and competitive advantage: ultra-Intelligent electronic agents."
I stopped reading the article after this first paragraph about awesome the author thinks he is, and his claim of IT consulting, predicting tech trends since at least 1982.
If anything were, there would be no need to teach it.
The concept may not be innate, and is learned, but it can still be natural. The symbol for a full/empty battery does not resemble how a real battery drains. It represents how a cup or container looks when it's full, half-empty or empty. I say that's quite natural.
as well as number form and personification. Numbers - depending on if they are simply numbers or dates - have a specific "geography", color, and personality.
46 & 2
I once took a course in "Math philosophy" (a simple introduction course, with e.g. Gödel numbers, introduction to infinity, and things like that), and at the end of that course we were asked to write about something. I decided to ask friends about how they viewed numbers. To my surprise, everyone had pretty much their own unique way. I think I asked about 10 people. Some viewed numbers as colors ("the number 2 is of course blue" or something along that line), some viewed the numbers as on a traditional line, one guy thought of the numbers as being in a circle and you took one out as you wanted to use it and then had to put it back. Not everyone included the number zero (or negative numbers) in their explanation. My self, I see the natural numbers on a line, but the line has "angles" at the numbers 10 and 20. Perhaps this is because in my native language, the spoken words for 10..19 are not constructed in the same simple manner as 30..39, 40..49, and so on.
Many things that we consider intuitive, are not. For instance, if you give most children some numbered cards, and tell them to put them in order, they will lay them out from left to right, and intuitively assume that anyone else would do the same. But an Arabic person would probably lay them out from right to left, and a Chinese child would likely lay them out from top to bottom. An Australian aborigine child would most likely lay them out from east to west.
The continuum hypothesis has nothing to do with the calculus taught in high school, or this article for that matter. I like talking about math, though, so for those who don't know, here's a non-technical explanation of the continuum hypothesis.
(1) "smaller" sets: A set is a particular group of objects (numbers, fruit, names, your Toyota, etc.). Given two sets, we say the first is smaller than the second if (a) we can assign to each object in the first a corresponding, unique object in the second, yet (b) we cannot assign to each object in the second a unique object in the first. For instance, {1, 2, 3} is smaller than {apple, 1, Karen, Jim} since we can pair up 1 with apple, 2 with 1, and 3 with Karen, but any attempt to pair in the other direction will necessarily fail by using one of the numbers 1, 2, or 3 twice, breaking uniqueness. The good part of this definition is that it also makes sense for infinite sets, which is where the "just count it" definition breaks down.
(2) An unintuitive example: The natural numbers are just the numbers 1, 2, 3, 4, 5, .... They form an infinite set. The integers are just the numbers ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, .... The integers contain the naturals so one might expect the naturals to be smaller than the integers. This is *not* the case, since I can pair each integer with a unique natural as follows:
0 with 1 ...
1 with 2
-1 with 3
2 with 4
-2 with 5
Thus (b) cannot be satisfied since the pairing attempt does not always fail.
(3) Real numbers: The real numbers are fractions like 4/5 together with the irrationals like pi or Sqrt(2). These also form an infinite set and again contain the naturals. However, they are "large" enough so that the naturals are smaller than the reals in the sense above; this is the content of Cantor's Diagonal Argument.
(4) The Continuum Hypothesis states that there is *no* set S where both (a) the naturals are smaller than S and (b) S is smaller than the reals. Intuitively, there is no "size" strictly between the naturals and the reals.
It turns out the Continuum Hypothesis is neither provable nor falsifiable under standard but technical assumptions; one can assume either its truth or falsehood without creating contradictions. It's probably the most famous example of an "undecidable" statement. In my experience, though it doesn't come up much among non-set theorists, there's no general consensus on which path to take. I suppose I lean towards accepting the generalized continuum hypothesis since it collapses down the hierarchy in an appealing way while also giving the axiom of choice as a bonus (which I actually "believe").
In Lingala (Kingshasa area in Congo), they only have one word which both means "yesterday" and "tomorrow". Basically things happen today or they happen not-today. This kind of makes sense in a climate that has no cold and hot season, and where it is useless (or even a very bad idea) to do typical northern stuff like plan way ahead, conserve food or make warm clothes. Most pre-Columbus south american indians saw time as a strictly circular thing, with everything always comming back.
10 ?"Hello World" life was simple then
It also contains an error: Peano defined 2 as { {}, {{}} } = {0,1}. 3 is 2 U {2} = { 2, 1, 0 }. Larger numbers are defined inductively as (n+1) := n U {n}.
You can tell it was supposed to be the Peano construction (and not something else) because the GP defined zero as the empty set and 2 as {0,1}. The error was to also define 2 as {{{}}}, which is clearly not equivalent to {0,1} (since the former set has cardinality 1 and the latter has cardinality 2).
This is an incredibly common mistake even for math undergrads and good evidence that set theory really isn't very intuitive. There's a reason New Math failed.
I guess that a line only makes sense once you get to fractions. When you work with Integers, it's more intuitive to think of distinct objects.
One has only to look at how the data was collected and the independent review of that data, to see that there are major flaws in the study.
The Emperor has tutored many people in set theory and seen that exact error many times. You made the mistake because the idea of computing the union of sets of sets (as opposed to sets of "objects" (ur-elements)) is very non-intuitive, and requires a significant leap of abstract thinking.
The point of number line is not to teach numbers but to show the analogy between numbers and distances/segments, connecting geometry and arithmetics. And for that purpose, it's perfect.
Well, numbers are abstract. I'm not sure how a number line representation, which can take real shape would be an intuitive extension of an artificial concept. It isn't. Actually, it's the other way around, I would think. The number lines help us understand numbers and it's numbers that aren't intuitive.
I'm not sure if they've fixed it yet, but the defaults for line charts in MS Excel were insanely set to have equal spacing between data points on one axis no matter what values they have.
Thus you could have an axis that looked like:
1 4 7 8 14 35
IMHO that sort of defeats the purpose of a line graph. I can userstand linear or log scales but a random changing scale is pointless.
The point of number line is not to teach numbers but to show the analogy between numbers and distances/segments, connecting geometry and arithmetics. And for that purpose, it's perfect.
But it is pretty good at demonstrating why addition and subtraction are inverse operations, and getting started on negative numbers.
Well, your unintuitive example is not that unintuitive because you still can mentally imagine how the association works (basically, take first the positive, then the corresponding negative number - it's intuitively clear that you get all of them this way). It gets really unintuitive if you go to the rational numbers which are dense.
And of course once you are there, with a dense set of numbers, it is completely unintuitive that you can still put numbers "in between". And even more numbers than you already have.
Usability experts have responded to this study by stating that non-intuitive interfaces to reality are to be frowned upon and should be replaced by better, more intuitive ones. The researchers, however, responded by taking a Mac to New Guinea and showed beyond any doubt that state-of-the-art GUIs are not intuitive at all.
It gets really unintuitive if you go to the rational numbers which are dense.
Actually I disagree. There is a simple way to visualize the countability of the rationals, which is a variation on the following (I believe standard) enumeration that I won't formalize: ...
1/2
1/3, 2/3
1/4, 2/4, 3/4
1/5, 2/5, 3/5, 4/5
I realize numbers are repeated but it's simple conceptually to cut out the duplicates and, if you want, to repeat the process in each interval [n, n+1]. The picture corresponding to the version I've written is of course equally spaced points with ever finer spacing, and for any rational number with a given denominator that number will be hit at that denominator's stage.
By "unintuitive" I meant that, for finite sets, proper subsets are smaller in the sense above, whereas infinite sets do not possess this familiar property. Of course it's subjective :).
And of course once you are there, with a dense set of numbers, it is completely unintuitive that you can still put numbers "in between". And even more numbers than you already have.
Hmm... sorry, I again disagree. Rational numbers have repeating decimal expansions; if this fact is intuitive (I believe it is, though a rigorous proof may not be), then picking a number that doesn't have a repeating decimal expansion (easy to do) means you've picked an irrational, so there are numbers "between" the rationals. I would at least find it quite weird if the set of infinite binary strings and the set of finite binary strings had the same cardinality--I would very much want to know the injection. Since I find the converse of Cantor's conclusion unintuitive, I find Cantor's conclusion intuitive. Again, though, the phrase is subjective.
We don't stop to wonder: Is it 'natural'? Is it cultural?
'Cultural' is natural for us humans, so it is a daft question. A better question would be to ask whether this is something we are most likely to have learned through our early experience - and how. And I think the answer is likely to be that we learn the idea of "moreness" being a continuous thing from observing varying amounts of things - water in a glass etc, or the length of a piece of string; these concepts are clearly learned as and when you learn the words to describe them - ie. it is 'cultural'.
But many - maybe most - animals have the ability to gauge the relative size of things, and some, like the corvids - even seem able to count. Thus that would count as a 'natural' ability, I suppose.
The case with the Yupno seems to be that measurements aren't needed in their culture; one can muse over where that need arises from - it could be a result of trade, perhaps?
... I mean, what next, "walking is not intuitive?"
Humans have to learn most things which we take for granted,
it really is important to give children under seven a good, well-rounded exposure,
possibly even including wrestling wild boar
(jk about the boar).
No I haven't read the article, but a "study" claiming x is not "intuitive", I find highly suspect.
The reason you can't define intuitive, is because we don't know what the word means yet.
Furthermore it has been used to describe multiple mental processes.
I mean many call the iPhone "intuitive", and I'm pretty fucking sure we weren't born with Apple in our DNA.
I read the article pointed to in the summary (which is a summary of the scholarly article). The study authors seem to have confused the idea that finding a single population that behaves this way (not arranging piles of oranges linearly along a line according to the number of oranges in a pile) with determining true innate human behavior. Find another dozen isolated groups, and then maybe. Find groups that have been only recently isolated and it will be more impressive.
Put my fist through my alarm clock with its ding-dong death inside my ear. - The Blackjacks.
timeline is stupid and now we have studies to prove it!
Of course it's learned. We teach it in school, every year, from somewhere around second grade right on up through college. Obviously it's learned.
Is that supposed to have some kind of significance? I don't see it. Virtually everything we know is learned. Arithmetic is learned. Color is learned. Language is learned. Food preferences are learned, including even the ability to tell the difference between food and non-food. The notion that a stove burner is hot and you don't want to put your hand on it is learned.
Cut that out, or I will ship you to Norilsk in a box.
I for one have not stopped caring about the problem of numbers, and I am sure I am not alone. It's not a problem that sees much in the way of publication, probably because there hasn't been that much progress and it's not a study likely to get your Phd. It's the sort of problem that sits on the back burner until some genius comes up with a new insight.
Part of the problem with this thread is that there are different meanings being attached to the symbol "number". The "1" in 1 sheep is probably intuitive, the 1 in {0,1} is probably not, yet both might reasonably be called "numbers". As for the "number line", I think that "things laid out in a line to see how many I've got" is innate, and may even be so for animals such as cats and birds. Naming the thing at any point in that row by the "number" that I count to get there seems to be a level of abstraction which requires "teaching".
nec sorte nec fato
... someone please explain to me (I am not from the US).
"Learned" and "intuitive" don't really seem mutually exclusive in a case like this. You do have to notice that one of these is about two of these, but doing so doesn't necessarily require somebody's help. It is communication about numbers that is cultural, not understanding them.
I'm sitting next to a recently-opened bottle of Gatorade. Because the plastic is transparent, and the liquid not, I can see exactly how full the bottle is.
I just took a sip, and guess what? I watched the level go down! If I did the exact same thing, but in two dimensions, it would look exactly like a gas gauge or battery monitor.
If that's not "intuitive" in your culture, there's something wrong with your culture.
MSIE: The world's most standards-complaint web browser.
Just because "the Yupno people of New Guinea" have a certain inherent trait doesn't mean the rest of us do. Even in a relatively homogenous population, we're not all the same. We are not robots, we are human beings. Lazy scientists like the ones who did this study need to stop drawing broad conclusions from just a few individuals.
If you grew up with the metric system you might not realize that common measurements used to be based on supposedly common items, so you had measurements dealing with what a man could hold with his arms around it, and the length of the King's erect cock or whatever. It's a natural advance to go from measuring things in terms of a fingertip to so many fingertip-units. I imagine it would have started with measuring distance, but it could as easily have been someone figuring it out by volume, this container holds so many of that container. Or this stick rolls over x times when it passes down the side of this object.
"You're right," Fisheye says. "I should have set it on 'whip' or 'chop.'"
I sometimes wonder if negative numbers wasn't something that was invented as part of trade/commerce to help accommodate the concept of indebtedness.
You don't really find negative numbers in physical nature, or at least not as the physical world was known in the BC era. A lake or river may be lower than average, but never has a negative quantity of water. Even subtraction is somehow positive, because if you take 2 apples from a basket of 10, the two you take aren't somehow negative, they still exist in the same form and are counted as a group as 2 apples.
The author of the study co wrote a book with George Lakoff on where math concepts come from, their history etc. Math is not as perfect as I used to think. For instance , we want our systems to be closed under their operations and ideally the inverse of the operations. An example of this would be addition on modulo 5 numbers where 1+5=1. You have the numbers 1-5 and the operation always produces one of them. If you allow for infinite extension, addition is closed on positive integers. But then you want the inverse of the operation, and then that leads to the need for "negative" numbers which are not an intuitive concept. Multiplication then leads to division, rational numbers, irrational numbers (ie square roots ), imaginary numbers-dividing by zero - as you try to expand the ideas to preserve closure more and more bizarre constructs have to be added. Then there's Godel and the inherent flaws in formal systems.
So math is really not as perfect as you might think. The book is interesting and I am not at all doing it justice. Also interesting is David Foster Wallace's math book "Everything and more" I think is the title.
I thought that the part of the article where they talk about the past and future being reference by up and down a hill was very interesting. It just made me think of our idea of the 'fabric' of space-time. Maybe space-time is actually more intuitive than our number line. What if the Natives were given a topological map model (3d bent shape) and asked to arrange the oranges on that?
Perhaps this relative/three-dimensional mathematical model is the key to understanding Warp-Drive.
Think about it.
They don't "exists". They are inventions that we make to try to explain the world. This doesn't make them any less useful, by the way.
But yeah, trying to call them "natural" or even "intuitive" misses the point. Don't confuse the map with the terrain.
Geometry was probably invented 8000 years by surveyors, architects and astronomers. Algebra was invested by the Greeks in classical times. Descartes unified the two in the Coordinate System in the 1500s.
Since this took us so long, I would be surprised this concept is not intuitive to unmathed peoples.
The fact that different cultures perceive math and time different was already demonstrated by Australian Aborigines. In an experiment, they showed a set of 2 matchsticks on the left and a set of 1 matchstick on the right. 1 matchstick was then moved from the left set to the right set. When asked to describe the phenomenon, they cited "There are 2 sets of 2, 2 sets of 3, and 1 3-set-making thing."
"Love heals scars love left." -- Henry Rollins
Intuitively, heavier objects fall faster.
Physicists and mathematicians, who have never been outside in the room with the big blue ceiling, think math is reality, rather than a really good way of describing reality.
Counting is natural. Then came measuring. Then came laying out farming plots and geometry. Already PI and irrationals came into play. Cracks in using a lineal number system to describe a multidimensional reality.
Of course numbers are not natural beyond fingers and toes. Why do you think we needed writing, and symbols first. Why do we need calculators and computers instead of using hardware in our brains? Why do you think we spend 12 years in school.
Numbers, Letters and Writing only came into being maybe 3000 years ago, while we have been Humans for hundreds of thousands of years. Most of the world got comfortably into the 1500's without reading and writing or calculus or engineering.
I also detect a whiff of social 'science' mindset, that thinks math and science is whatever some one says it is, rather than an attempt to learn about, describe and model the true essence of reality.
And a whiff of 'Science and Math are nothing but a racist plot by the Heterosexual White Christian Males to rule the world". Yep. That's what is out there on the fringes of the interweb. It is part of the vast conspiracy to censor, obscure and dumb down. The battle for your mind goes on.
The linear number system is only one of the many possible numbering systems out there. Maybe some new number system will describe some parts of reality better, make the equations simpler, or lend itself to 'AHA!' moments because of it's elegant expression that allows new insights.
We have the computers now, we can explore all this.
Newton invented calculus to have an easy way to write down his theories.
Computers use base two.
What it really comes down to is whether they understand the concept of scale or not. If they can scratch a map in the ground and use an appropriate scale to indicate days' journey from point A to point B, then they understand scale. If they can tell you how many days from point A to point B but they can't draw a map, they understand the distance but don't understand how to make a scale model of it.
They don't measure lines using arbitrary yet equal units. If you want them to count in equally-sized units, give them a unit that isn't arbitrary. To see if they understand the concept of a number line, tell them that the line is 10 days journey, and make marks for starting it (zero) and finishing it (10 days) and ask them where 7 days is. They already know that days come at equal intervals, and they probably already measure the distance by the number of days it takes to journey it, so the question is whether or not they can translate that to linear distance in a scale model.
Is language innate? I don't know Swahili. We are certainly adept at language. We are adept at walking a straight line between two points. We are pretty good with numbers too. Language numbers and number lines are concepts that can be grasped in preschool but not really innate. We can say our brains are ready to develop along culturally directed lines. If we had more children raised by wolves the difference would be more apparent.
Actually, you can graph irrational numbers - you just need a grid.
Alas, I digress. Actually, one thing which I have trouble with when interpreting is translating numbers in my head between Japanese and English.
In English, we have the following places:
1 - ones place
10 - tens place
100 - one hundreds place
1000 - one thousands place
but there is no ten-thousands place - or to put it another way, there is no word for "ten thousand" or "one hundred thousand", so we have to combine terms. (In English, that is).
Suddenly there is a word again for Million, but not for 10 million or 100 million. Billion has a word again, etc. If you only speak one language, you've probably not ever thought about this, but it's relatively arbitrary.
In Japanese, there are words for 1, 10, 100, 1000, and 10,000 (man). There is no word for 100,000 or 1,000,000, though. So "1 million" in English basically becomes "100 man" in Japanese. Likewise "1 man" becomes "10 thousand".
This is mildly confusing, but not too bad by itself, but when you start talking about money at the same time, where you have to do currency conversions, it gets tricky trying to do two conversions in your head. For example "10,000 dollars" becomes "1 man dollars" becomes whatever.. say "8,000 yen". In fact, I pretty much just memorize the combination of the two so that I don't have to do both conversions in my head simultaneously, but when someone starts talking about a very large amount of money, I still have to stop and think, or sometimes even write it down first.
In the case of US dollars to yen, the easy way to cheat is to consider one cent one yen - but that's getting less and less accurate with the string yen.
"As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality." --Albert Einstein
The number line actually continues until it twists in time in the distance and both sides exchanging positions until the previous arrangement is opposing as positive or the negative state or direction with zero's in between each whole integer or measure.More or less than 1 but infinite or finite.
It matches what surrounds you as the Earth revolves around the sun...each orbit twists an 8.
Old stuff that the disciples of Rome even knew...or the Egyptians and the Greeks.
Now look into the crystalline formations (rocks) and visually see the same atomic structure matches the same twist of the 8.
That's how some people know what the time is.
Number line concept is a compliance artefact. The FACT that most number lines have ranges of standard usage means that they are compelling users to comply to a standard and be automated. OK