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 Noteworthy
Sophyphile
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Quote:
adjust said:
Quote:
Noteworthy said:
good point, daytripper.
I dont know how any of you are gonna try and make a triangle with 185 degrees, without redefining triangle.
If you don't know how to do it, you should learn how to instead of saying it is impossible. 
go ahead and show...
you have made a pretty big suggestion
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 DieCommie
Registered: 12/11/03
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 Re: Knowledge/ truth/ fact  [Re: Noteworthy]
#9843369 - 02/22/09 11:43 AM (15 years, 1 month ago) |
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In non-euclidean spaces, triangles interior angels do not equal 180 degrees.
Consider the non-euclidean surface of the earth... you start at the equator and travel a quarter of the circumference of the earth and then make a right angle to go the the north pole. Once there you make another right angle and head back to the equator. Then once at the equator you have to turn another right angle to face the way you started. You just traveled on a equilateral triangle, one whose interior angles add up to 270 degrees. The term 'triangle' has not been redefined here... the shape we traveled along did have three angles. The catch is the surface of the earth is non-euclidean.
Turns out that space and time itself may be non-euclidean, indicating that nowhere in nature do the internal angles of a triangle add up to exactly 180. The whole idea of flat, euclidean space with 180 deg. triangles is just a construct we have made up in our minds to describe our experiences. Since our experiences are that of a roughly flat space then euclidean axioms do a great job of modeling our perceptions. The ancients who were not privy to the knowledge we have now often assumed an inherent truth of euclidean space, that somehow it is true outside of their perceptions and outside of the axioms. Fortunately (thanks gauss) we arnt restricted by this fundamentalist view, we know that all math is invented to model reality and we can invent any math we see fit to do just that.
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lIllIIIllIlIIlIlIIllIllIIl
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Re: Knowledge/ truth/ fact [Re: DieCommie]
#9843573 - 02/22/09 12:42 PM (15 years, 1 month ago) |
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zouden
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Re: Knowledge/ truth/ fact [Re: DieCommie]
#9848602 - 02/23/09 05:23 AM (15 years, 1 month ago) |
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Yeah it's really easy, just get a tennis ball and draw a triangle on it. The angle sum won't be 180 degrees.
>Turns out that space and time itself may be non-euclidean, indicating that nowhere in nature do the internal angles of a triangle add up to exactly 180.
Yes, which is why I don't think mathematics was designed to model nature. That's what physics is for. Maths is just maths. A triangle in Euclidean space will always have 180 degrees of angles, no matter who you ask, just the same as 1+1=2, no matter who you ask (the fact that aliens may have different uses for the + and = signs is irrelevant; that's just a translation issue. One plus one always equals two, period).
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TameMe
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Re: Knowledge/ truth/ fact [Re: zouden]
#9848609 - 02/23/09 05:27 AM (15 years, 1 month ago) |
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That's because what you are drawing on the ball is not actually a triangle. A triangle is made of straight edges...not curves. A triangle can be places on a two deminsional plane.
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zouden
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Re: Knowledge/ truth/ fact [Re: TameMe]
#9848614 - 02/23/09 05:32 AM (15 years, 1 month ago) |
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But in the 'space' of the ball, it's got straight edges. That's the whole thing about non-Euclidean geometry. The lines are straight, even though they follow the curves of the ball, because in the ball's space they aren't curved - because they're flat on the surface.
Isn't it awesome?
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You know... I'm not a blind man
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TameMe
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Re: Knowledge/ truth/ fact [Re: zouden]
#9848631 - 02/23/09 05:42 AM (15 years, 1 month ago) |
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I'm not visualizing whatever you are. A triangle exists on a 2-d plane...not on a 3-deminsional surface.
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zouden
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Re: Knowledge/ truth/ fact [Re: TameMe]
#9848654 - 02/23/09 06:01 AM (15 years, 1 month ago) |
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Euclid realised that a plane doesn't have to be flat. It can be curved, yet still two-dimensional.
http://en.wikipedia.org/wiki/Euclidean_geometry
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You know... I'm not a blind man
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TameMe
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Re: Knowledge/ truth/ fact [Re: zouden]
#9848657 - 02/23/09 06:04 AM (15 years, 1 month ago) |
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I didn't find anything that claimed a plane can curve.
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TameMe
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Re: Knowledge/ truth/ fact [Re: zouden]
#9848670 - 02/23/09 06:12 AM (15 years, 1 month ago) |
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I dug further and found stuff on it....
hard for me to grasp though.
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DieCommie
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Re: Knowledge/ truth/ fact [Re: zouden]
#9848968 - 02/23/09 08:33 AM (15 years, 1 month ago) |
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Quote:
zouden said:
One plus one always equals two, period
Not always, only under one particular set of axioms. Just like you qualified the triangle geometry must have euclidean axioms for it to hold, also you must qualify that one plus one is two must have particular axioms for it to hold. Like you said, math doesn't necessarily need to model nature (that what physics is for), you can invent math where one plus one is three, or zero or undefined - whatever you want. (the reason that one plus one equals two seems so inherent and ubiquitous to you is because the set of axioms that yield that relation are actually useful, so it is the most studied)
Edited by Qubit (02/23/09 08:42 AM)
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zouden
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Re: Knowledge/ truth/ fact [Re: DieCommie]
#9850481 - 02/23/09 02:25 PM (15 years, 1 month ago) |
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I disagree. Some axioms are always true. And no, the euclidean geometry is not a condition - a triangle is a three-sided figure, period. It makes no claims about the angle sum.
I don't believe you can come up with a set of axioms in which 1+1+3.
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DieCommie
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Re: Knowledge/ truth/ fact [Re: zouden]
#9852566 - 02/23/09 08:19 PM (15 years, 1 month ago) |
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What do you mean some axioms are always true? An axiom is assumed to be true because you can never prove it... if you could then that would be by appealing to a different set of axioms. Yes, euclidean geometry (that is, euclid's 5 axioms) is a necessary condition for the theorem that internal angles must equal 180. Similarly, there are a set of axioms (peano) that are necessary for the theorems of simple addition to hold.
Looks to me like you have faith in some kind of metaphysical truth of inherent properties of math - appealing to some kind of axiom-less mathematical structure. I think many people have tried to find it, and many do believe in it. But again, that is faith and I guess some people have it and some people dont. I have no faith in the existence of such a framework.
edit - I didn't mean faith in a derogatory manner, sorry if it came off that way. I have meet mathematicians who believe what you do and others who believe what I do. However, when I press the former group on their beliefs they appeal to faith/intuition in the end.
Edited by Qubit (02/23/09 08:46 PM)
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Sventington
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Re: Knowledge/ truth/ fact [Re: DieCommie]
#9852963 - 02/23/09 09:24 PM (15 years, 1 month ago) |
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How on earth could 1 + 1 = anything other than 2? I mean if I have one apple, and one more apple....how can it be that I have anything other than two apples? It doesn't seem to make any sense.
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zannennagara
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Re: Knowledge/ truth/ fact [Re: zouden]
#9853079 - 02/23/09 09:39 PM (15 years, 1 month ago) |
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Quote:
zouden said:
I don't believe you can come up with a set of axioms in which 1+1+3.
Maybe this is a stupid example, but we can imagine a system of counting objects in which 1, as an integer for a single unit, is really a rounded approximation that represents, say, an orange maybe 1.4 times as large as an average orange. Adding two units of this 140% orange would give us two units of individual orange, but the real whole, 2.8 average oranges, could be rounded to three - the real quantity becomes not only unit but unit content.
Or one orange is twice as massive as another: added together they could be either three (small) oranges or 1.5 (large) oranges.
But these integers aren't accurate measurements, of course; we can use measurements of mass/weight/volume and use our same mathematics, but then we realize that the only accurate integers are the ones that refer to our axiomatic and arbitrary standardized measurements - this much space, this much time - which are likely to be very different for different experiences of the universe that we are (currently) incapable of imagining.
Some form of unit conversion may always be possible, so in that sense you're right that our 1+1 always equals our 2, but it is possible to conceive of numbers systems that would appear nonsensical on the surface until so translated.
How about pantheistic math, where everything is 1? Or nihilistic math, where everything is 0? Ha. Ha.
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zouden
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But those are just number systems. 1+1 still equals 2 in those systems, even if they use different symbols to represent it. In your example where one orange weighs twice as much as another, then it's a different case - that's 1+2=3. Same axiom.
Qubit: I see what you're saying, but I'm not convinced that that other axioms exist (for trivial cases like this). Aren't you essentially arguing that I can't be certain that these axioms are the only ones "because there might be others"?
The wikipedia article on Peano's axioms indicates (as far as I can understand) that they are considered to be consistent, even after Godel's Incompleteness Theorems were published. That agrees with what I mentioned earlier - Godel's theorems don't apply to trivial cases. Godel found that there are mathematical proofs which aren't complete proofs because they rely on axioms that can't be proved - but basic arithmetic is not one of them.
If that's right, then it is safe to say that there are some things which are absolute truths, and 1+1=2 is one of them.
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I know... that just the smallest
part of the world belongs to me
You know... I'm not a blind man
but truth is the hardest thing to see
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Mr. Mushrooms
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Re: Knowledge/ truth/ fact [Re: Ratci]
#9854013 - 02/24/09 01:09 AM (15 years, 1 month ago) |
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Quote:
Ratci said:
If a piece of knowledge must be true to be considered knowledge, how can anyone prove that they know what they know?
Some things do not require proof or evidence. They are inherently known and axiomatic.
Few things have that certitude.
Lakefinger's comment on ontology was spot on.
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lIllIIIllIlIIlIlIIllIllIIl
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Re: Knowledge/ truth/ fact [Re: zouden]
#9854060 - 02/24/09 01:32 AM (15 years, 1 month ago) |
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Quote:
zouden said:
If that's right, then it is safe to say that there are some things which are absolute truths, and 1+1=2 is one of them.
In a way you are right because when you say "1+1=2" it is implied that you are using integers or real numbers. However you can make a new set of numbers where addition is neither commutative nor associative, and in that case simple arithmetic would give a different result. The only reason people resist this idea is because they are thinking in terms of the already established sets of numbers, which are useful is solving real problems.
As I said earlier in some cases it is useful that 1+1=1-1=2, especially in binary calculation for data integrity checks such as CRC checksums.
It is still mathematics but the form of addition used does not follow the same axioms as you would when adding apples.
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zouden
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>As I said earlier in some cases it is useful that 1+1=1-1=2, especially in binary calculation for data integrity checks such as CRC checksums.
Can you explain this further?
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I know... that just the smallest
part of the world belongs to me
You know... I'm not a blind man
but truth is the hardest thing to see
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lIllIIIllIlIIlIlIIllIllIIl
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Re: Knowledge/ truth/ fact [Re: zouden]
#9854093 - 02/24/09 01:48 AM (15 years, 1 month ago) |
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I don't remember exactly the details, but by making it like that the calculation is more efficient and better able to find errors. But I tried looking at wikipedia and it didn't say anything about it. I guess you would have to find some university webpage that talks about that to get into the details of how it is calculated.
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