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Noke Lieu
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relative maths
« on: Nov 9th, 2008, 11:21pm » |
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Once more, slipping into the ambiguous world of meta-maths...
I've been thinking a bit recently (which makes a nice change).
I assume most of us would agree that maths isn't "in the bag" stuff- it's all about the processing, the finding out how and when and why to apply different patterns and observations to different scenarios to yeild the information you're after.
1+1=2 is recall for almost everyone- there's no processing, it's just out and out memory use (and the same is true for 2+2=4 or 5+5=10).
Yet, at some point, we did have to process, we did have to use maths for 7+5=12 (typically counting on fingers) before we could dedicate it to memory. (From there, we make the choice of storing it as a stand alone fact, or learning how to intertwine it in our mental scaffolding so that it can be used in other sitations.) {that's heading off topic, sorry}
Or, for example, fractions. Understanding what they mean takes a huge amount of time and effort for most students. Or understanding Pythagoras.
Yet once understood, they're easy tools to use- and they stop being "Maths" and start being a tool for you to use when doing maths- figuring out etc...
This leads me to think that there something akin to "relative maths" in that the amount of processing required differs for one person to another for the same task- sort of what is "maths: for you might not be "maths" for me (though round here, it's WAY more likely the other way round...)
Huhn. Not quite sure how to wrap this up- gotta soon though...
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towr
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Re: relative maths
« Reply #1 on: Nov 10th, 2008, 12:23am » |
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Well, yes; people process problems differently. e.g. Some use insertion sort, some use merge sort.
I wouldn't say the 'tools' aren't part of maths though.
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Noke Lieu
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Re: relative maths
« Reply #2 on: Nov 10th, 2008, 4:43pm » |
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on Nov 10th, 2008, 12:23am, towr wrote:
I wouldn't say the 'tools' aren't part of maths though. |
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I've been wrestling with the difference between "calculation" and "maths".
Sure calculation is a subset of maths, so it's part of maths.
However, I'm leaning towards these quasi-definitions
"Maths" ...is the set of tools.
"doing Maths"... is applying/using/developing the tools.
"Maths lessons"... typically train you in the existence and use of the tools.
So is 1+1=2 Maths?
I'd answer No. It's a bunch of symbols that we use to help us convey the maths message, but the maths exists in your head- not in the symbols. The fact that you recall the fact 1+1=2 prevents you from actually actually calculating it, hence stopping any processing.
Yet 15+38=53, though it's essentially the same thing (as in adding two numbers together) does require processing. Choosing how to process it takes an understanding of Maths.
The deeper your understanding of Maths, the more elegant your manner of processing is likely to be.
Actually churning through the calculation though, that doesn't require deep understanding. It requires blindly following rules. Otherwise calculators couldn't do it.
on a related tangent...
Is the Mona Lisa "art"? Or is it an artifact of the process of art? For that matter, is the McDonalds logo "art"?
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Re: relative maths
« Reply #3 on: Nov 11th, 2008, 1:20am » |
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on Nov 10th, 2008, 4:43pm, Noke Lieu wrote:| Actually churning through the calculation though, that doesn't require deep understanding. It requires blindly following rules. Otherwise calculators couldn't do it. |
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It depends on the calculation. There are many calculations a calculator can't do.
You can't really separate the process from the selection (sometimes creation) of tools for doing the process; it's part of it, and may require some creativity that goes beyond current calculator's means. (For example, clever substitutions for solving difficult integrals.)
Quote:on a related tangent...
Is the Mona Lisa "art"? Or is it an artifact of the process of art? For that matter, is the McDonalds logo "art"? |
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I think art is the result, not the process (except in as far as it exists in the result). The McDonald's logo fails on several common criteria for art, like unicity.
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0.999...
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Re: relative maths
« Reply #4 on: Nov 11th, 2008, 3:40pm » |
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Quote:| I think art is the result, not the process (except in as far as it exists in the result). The McDonald's logo fails on several common criteria for art, like unicity. |
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What is this art you speak of?
To better convey my point, I remember when I was ten years of age or so that I had painted with water colors a mountain of sorts with the sky above it. This mountain was composed of five or six colors which all happened to go well together. The sky a light blue. With the back of the paint brush I was trying to write in cursive; it didn't turn out at all like cursive and appeared to be a number of random scratchings. My father, who is an accomplished artist, had thought extremely highly of this painting.
Did I do art?
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| « Last Edit: Nov 11th, 2008, 3:41pm by 0.999... » |
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Noke Lieu
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Re: relative maths
« Reply #5 on: Nov 11th, 2008, 4:18pm » |
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on Nov 11th, 2008, 1:20am, towr wrote:
It depends on the calculation. There are many calculations a calculator can't do.
You can't really separate the process from the selection (sometimes creation) of tools for doing the process; it's part of it, and may require some creativity that goes beyond current calculator's means. (For example, clever substitutions for solving difficult integrals.)
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That's a nice point.
Thining of cooking, though.
Imagine preparing a meal from scratch. That's definately cooking.
Heating a tin of prepared food though... does that count as cooking? Or scooping out icecream. They have the same steps, and the same outcomes (in that food is now prepared).
Following prepared instructions isn't "maths" though it might take basic maths skills. Chosing how to apply instructions to situations is "maths".
(speaking from the shallower end of the maths pool though)
Doing the clever substitution takes deep understanding and insight of Maths, doing the calculation as is doesn't- it takes a degree of compentecy with maths knowledge.
on Nov 11th, 2008, 1:20am, towr wrote:
I think art is the result, not the process (except in as far as it exists in the result). The McDonald's logo fails on several common criteria for art, like unicity. |
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Surely 'artworks' are the result of art? That's a point of pedantry, though.
Direct copies of the painting don't require art to make them. Interpretations of the painting, tweaks of it etc do require art, irrespective of the medium.
As for the unicity of the M sign... I see your point. However, there is only one of them, just that it's represented in millions of places.
Otherwise copies of the Mona Lisa remove the unicity.
Ahhh- you say- the original is still unique.
So is the original M sign. Or the 1000034th one. The fact that they're not distinguishable is a curious turn though. If the copy of the Mona Lisa is indistinguishable...?
The Eiffel Tower, I'd wager is a mix of art and technology and function (and heritage and...)
Little minature replicas of it though... Possibly the art was in making the mould?
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| « Last Edit: Nov 11th, 2008, 4:28pm by Noke Lieu » |
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Re: relative maths
« Reply #6 on: Nov 12th, 2008, 1:09am » |
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on Nov 11th, 2008, 3:40pm, 0.999... wrote:| What is this art you speak of? |
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You can say that about pretty much any concept.
What is a chair? Is a box you use for sitting on a chair? What if you start using it as table? What if you upholster it and sell it as chair?
Or what about games? Try finding a definition that encompasses everything from playing dice to chess to football, yet excludes non-games involving balls, dice, cards etc.
It's not for lack of examples, or for lack of characteristics you can name; they're simply fuzzy concepts. If pressed to delineate them people will draw different lines, and will often not even be happy with the result themselves.
Quote:I don't see why not. As long as it possesses sufficient aesthetic characteristics.
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Re: relative maths
« Reply #7 on: Nov 12th, 2008, 1:58am » |
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on Nov 11th, 2008, 4:18pm, Noke Lieu wrote:That's a nice point.
Thinking of cooking, though.
Imagine preparing a meal from scratch. That's definitely cooking.
Heating a tin of prepared food though... does that count as cooking? Or scooping out icecream. They have the same steps, and the same outcomes (in that food is now prepared). |
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Just because you didn't do (much of) the cooking doesn't mean it isn't cooked food.
Quote:| Following prepared instructions isn't "maths" |
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And following a recipe isn't cooking? If you really think cooking lies in thinking up the recipe, and following it is just working through the instructions; then you underestimate how many people follow recipes only to end up with a culinary disaster.
Quote:| Doing the clever substitution takes deep understanding and insight of Maths, doing the calculation as is doesn't- it takes a degree of competency with maths knowledge. |
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What does it mean to "do the calculation as is"?
If you go from 4*6+6*6 to 10*6, have you stopped doing the calculation as is, because you should have gone 24+36? Isn't calculating often making appropriate transformations to get closer to the answer?
And some integrals, for example, cannot be done "as is". To do the calculation you have to do something other than follow the "instructions" of the expression.
Quote:| Surely 'artworks' are the result of art? That's a point of pedantry, though. |
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If you ask "is the mona lisa art", the mona lisa being an object, then I can't take "art" in that sentence to be a process, because an object isn't a process. And I'm assuming you're trying to talk sense 
Here, straight from a dictionary; meaning 2 of art
Quote:| 2. the class of objects subject to aesthetic criteria; works of art collectively, as paintings, sculptures, or drawings: a museum of art; an art collection. |
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Quote:| Direct copies of the painting don't require art to make them. |
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And the result is also not the same; because no matter how good the reproduction it is not the original. You can't copy its unicity, you can't copy its history, you can't (completely) copy people's attachment to it or their appreciation of it.
Quote:As for the unicity of the M sign... I see your point. However, there is only one of them, just that it's represented in millions of places.
Otherwise copies of the Mona Lisa remove the unicity. |
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No it wouldn't. People will only pay millions for the original. No matter how often or how well you copy it, the copy won't be quite the same.
If you find the original McDonalds M, it may well be considered art; but none of the copies will unless they have that something extra.
Quote:| Ahhh- you say- the original is still unique. |
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Wow, you're psychic? 
Quote:| So is the original M sign. Or the 1000034th one. |
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Aesthetic unicity is not quite the same as being physically unique. The original M is the first, and in that sense special; and it represent all what was to come after, much moreso than anything else to come after. If it's in good condition I can certainly imagine an art collector having an interest. The 1000034th is no different from the 1000033rd, however. Sure they are not physically the same M, but that's about where the distinction ends.
Quote:| The fact that they're not distinguishable is a curious turn though. |
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If you can say they are each unique, then you must say they are distinguishable. I mean, they're in different places. 
Quote:| If the copy of the Mona Lisa is indistinguishable...? |
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I would be very surprised if someone could pull that off, considering the location, and the forensic techniques of today. Where will you find 500 year old canvas, for a start?
People invest great effort into assuring what they have is original.
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Noke Lieu
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Re: relative maths
« Reply #8 on: Nov 12th, 2008, 5:14pm » |
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on Nov 12th, 2008, 1:58am, towr wrote:
Just because you didn't do (much of) the cooking doesn't mean it isn't cooked food. |
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Quote:| And following a recipe isn't cooking? If you really think cooking lies in thinking up the recipe, and following it is just working through the instructions; then you underestimate how many people follow recipes only to end up with a culinary disaster. |
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Thinking that you've followed the instructions and ACTUALLY following the instructions are 2 very different things... It's one of the main reasons I got out of working in laboratories... 
Quote:What does it mean to "do the calculation as is"?
If you go from 4*6+6*6 to 10*6, have you stopped doing the calculation as is, because you should have gone 24+36? Isn't calculating often making appropriate transformations to get closer to the answer?
And some integrals, for example, cannot be done "as is". To do the calculation you have to do something other than follow the "instructions" of the expression. |
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You're right about that. I'd suggest that following the information as it's presented, and working things out step by step in a pre-described, unimaginative fashion- so plugging into a calculator 4*6, then 6*6 and adding the results, I'd discribe as unimaginative. Spotting that it's 10*6 does take a bit of imagination. Not much, but it does take actively engaging with the problem.
That you can't just crunch through some calculations without understanding what it is you are doing is obvious, yet doesn't sit well with me.
It may be that by the time one is expected to be able to do such calculations, one's library of tools is extensive enough that spotting where and when such substitutions are required ... to be frank, my library wrt calculus is in such decline that I can't really remember much.
When I was at school, I sat a pure maths exam of 2 halves- the first half was out and out -do this, do that. The second half involved using clever substitiutions and more insight as to what was going on. I scored very well in the first, terribly in the second.
The first exam was following pre-learned instructions to test accuracy, the second was applying those lessons.
The amount of 'maths' required for the first, I'd offer, was less than that required for the second (hence i did well )
Quote:| If you ask "is the mona lisa art", the mona lisa being an object, then I can't take "art" in that sentence to be a process, because an object isn't a process. And I'm assuming you're trying to talk sense |
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yeah... sometimes...?
Damn rhetoric tricks not working on you... 
Quote:| You can't copy its unicity, you can't copy its history, you can't (completely) copy people's attachment to it or their appreciation of it. |
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No, you can't copy it's unicity- that's the point of unicity. But it's unicity could be removed, or redefined to be insignificant. Imagine that a "copy" was found that showed all the signs of beign a year older that the "real" one. Which becomes the original? Does the original original stay, or become the the original? (reminds me of la citie d'enfants perdues)
Quote:| Wow, you're psychic? |
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I try my best... want to know the sex of your unborn baby?
Quote:| Aesthetic unicity is not quite the same as being physically unique. *snip* Sure they are not physically the same M, but that's about where the distinction ends. |
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Got me there. 
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Re: relative maths
« Reply #9 on: Nov 13th, 2008, 6:27am » |
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on Nov 12th, 2008, 5:14pm, Noke Lieu wrote:| Thinking that you've followed the instructions and ACTUALLY following the instructions are 2 very different things... It's one of the main reasons I got out of working in laboratories... |
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Actually, a large part of the problem is that the instructions don't tell you what you have to do, they're a guide. A recipe may tell you to add a pinch of salt, but it doesn't say how large a pinch is, nor how it may depend on the rest of the recipe.
Following guidelines to the letter is often one of the best ways to sabotage an approach.
Quote:That you can't just crunch through some calculations without understanding what it is you are doing is obvious, yet doesn't sit well with me.
It may be that by the time one is expected to be able to do such calculations, one's library of tools is extensive enough that spotting where and when such substitutions are required ... to be frank, my library wrt calculus is in such decline that I can't really remember much. |
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One way I've heard talked about mathematics is as a kind of story, or soap opera, you have to get to know the actors. "Doing maths" is exploring/elaborating the story, keeping it in accord to canon. Part of it, is of course just recognizing players, another part is inventing new players that fit in the story.
Quote:| No, you can't copy it's unicity- that's the point of unicity. But it's unicity could be removed, or redefined to be insignificant. Imagine that a "copy" was found that showed all the signs of being a year older that the "real" one. Which becomes the original? Does the original original stay, or become the the original? (reminds me of la citie d'enfants perdues) |
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It's hard to tell, really. Suppose the (non)original has played a historical role, while the "copy" just lay somewhere tucked up for perhaps hundreds of years. Art is a story, as much as anything else. And certainly subject to cultural convention, and imagination.
If you buy a ticket to the Louvre to see the Mona Lisa, you don't buy it to simply see the Mona Lisa; there are plenty of posters and photographs that come cheaper and quite probably give you a better view. You buy it to be in the presence of over 500 years of history, among other things; you "buy into the story".
It is however not "just" a story, any more than culture is "just" a fiction. We live it; it becomes significant through us. And while criteria for art may not be universal and absolute, they exist, albeit somewhat vaguely and hand-wavy.
Quote:| I try my best... want to know the sex of your unborn baby? |
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I'd rather know where to find the mother of my unborn child.
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