Phoenix Rising
|
Just FYI... Brother Michael this is not exactly your question but.. *sigh* I'ma gone reveal another website I visit... it's mainly white... but cerebral! Just the way I like it! anywhoo...
{My name used to be Sophia19... then I changed it to Khalliqa.... Ummm... also.. I did a search on the web... I am not some sister from guyana.... nor any other woman on the net with the name Sophia19... I never posted a picture under the name Sophia19! Which is part of the reason I changed to Khalliqa ... some of my other net names were common... } moving on...
I titled the thread "Math Neophite"
I am challenged there constantly... but I LOVE IT! I learn a lot...
Here is the dialogue:
06-12-2006, 03:40 AM
Sophia19
Phoenix Rising Join Date: May 2006
Posts: 32
Math Neophite...
--------------------------------------------------------------------------------
Math is my weakest subject.... next to physics it is one of my greatest interests........
go figure.....
question.....
from 1 to 2 .......
there are numbers infinitum in between......
does this not make the numbers 1 and 2 arbitrary? abstractions of the mind? illusory? foreign and unnatural ways to "measure"? does this not mean that reality is adrift? or non measureable? in truth?
or am I reaching? overthinking? not making sense?
honestly, this is something that has not made sense to me..... and whenever I asked a teacher.... was told "not to worry about all that right now"...
sigh....
Sophia
Sophia19
View Public Profile
Send a private message to Sophia19
Find all posts by Sophia19
#2 06-12-2006, 06:17 PM
Jihkl
evil and a heathen
Forum Curator Join Date: Mar 2006
Posts: 68
Re: Math Neophite...
--------------------------------------------------------------------------------
Yes. Or no. It depends on whether you want to believe that numbers describe some fundamental underlying structure to reality, or whether they are just constructions. Arguments have been made for both.
Why do you think that the existence of an infinity of real numbers implies the latter?
Jihkl
View Public Profile
Send a private message to Jihkl
Find all posts by Jihkl
#3 06-17-2006, 09:13 PM
Vinny T
Der Meistersinger von Nürnburg Join Date: Mar 2006
Location: Louisiana
Posts: 44
Re: Math Neophite...
--------------------------------------------------------------------------------
Quote:
Originally Posted by Sophia19
from 1 to 2 .......
there are numbers infinitum in between......
does this not make the numbers 1 and 2 arbitrary? abstractions of the mind? illusory? foreign and unnatural ways to "measure"? does this not mean that reality is adrift? or non measureable? in truth?
Sophia
1 and 2 are measures of a certain thing. How many apples do you hold in your hand? I have 1 apple. If you have 2 apples there arn't infinitly many in between. Basicly, numbering somthing is a finite measurable thing.
1 and 2 on the number line have infinitely man in between : 1 1.0000000 ... 001 1.999 ... 999 2. So that being said I don't think that math is illusory. The numbers exist, and some function will yeild them.
Reality isn't a concrete thing. You can measure reality. How real is this table I am sitting at? Impossible question. It is either real or it isn't.
Math is somthing we find in nature. If an apple dropped from the same spot a thousand times, takes the same amount of time each time it is dropped, then how could we be making that up. It is somthing natural and we have given it a name and made assumptions, which are proven and then practiced.
If it exists in nature then it is real. in short is what I believe!
__________________
Die Könnigin Kommpt!
Vinny T
View Public Profile
Send a private message to Vinny T
Find all posts by Vinny T
#6 06-24-2006, 07:18 AM
Vinny T
Der Meistersinger von Nürnburg Join Date: Mar 2006
Location: Louisiana
Posts: 44
Re: Math Neophite...
--------------------------------------------------------------------------------
Quote:
Originally Posted by Sophia19
Does it? Or does the apple land in different spots too small for you to measure? Perhaps the apple lands one millimeter away from the previous spot..
Sophia
You can't say "one millimeter" when trying to disprove or argue what i said, your using a measure, and by the virtue that you measured that perceptable change means that is real!
Nature as a whole. There are not different natures. Nature is our surroundings, our FINITE surroundings. We an measure the distance from a planet, it isn't infinite. However there are infinitely many points in that distance at which a point can be, but there comes a time when rationality and matematical sence need to converge. Yes there are infinitely many points on a surface, but it has a finite surface area. (this last thing was an example)
__________________
Die Könnigin Kommpt!
Vinny T
View Public Profile
Send a private message to Vinny T
Find all posts by Vinny T
#7 06-29-2006, 09:32 AM
VanMetal
Vannicon Join Date: Mar 2006
Location: Illinois
Posts: 615
Re: Math Neophite...
--------------------------------------------------------------------------------
Quote:
does this not make the numbers 1 and 2 arbitrary? abstractions of the mind? illusory? foreign and unnatural ways to "measure"? does this not mean that reality is adrift? or non measureable? in truth?
Numbers are meaningless alone. Only when attached to a construct do they convey meaning.
I have 2. - Means nothing.
I have 2 (quantity) apples. I have 2 liters of water. I have 2 inches of string. I have 2 pounds of dirt. - All have complete and finite meaning.
So yes, all numbers are arbitrary. Intigers are special because they are the only ones that can convey whole quantities without a predetermined standard unit. A unit quantity is already defined by the object that is being quantified.
1 apple: a total sphere of fruit-meat contained complete by a peel-membrane grown from a particular species group of tree. An incomplete sphere, missing surface material, or inconsistant tree species diqualifies an object from definition (a half eaten apple is no longer an apple), therefore only integers may be obtained by a quantity of this defined object.
Of course this definition could be altered to allow fractions of apples to be allowed in the count, in which case the integer complete concept is lost and so becomes arbitrary once again, like any other unit of measure.
In effect, reality can be measured only when it is defined. An non defined reality is non measurable.
__________________
You are not your stupid web sig.
VanMetal
View Public Profile
Send a private message to VanMetal
Visit VanMetal's homepage!
Find all posts by VanMetal
#8 11-12-2006, 01:36 PM
thermoplyae
m-m-m-m-math Join Date: Nov 2006
Location: Midwest, USA
Posts: 22
modern algebra
--------------------------------------------------------------------------------
From the perspective of a mathematician (rather than a philosopher), the naturals are definitely not arbitrary. Consider this construction of the reals from the naturals:
[Step 0: I've seen the "creation" of the natural numbers used inside of a grass-roots formulation of the lambda calculus, but a lot of other places don't cover it. The idea is that you have an empty set, then the set containing the empty set, then the set containing the set containing the empty set, and so on. We define the integer one to represent the number of recursions (plus one) we have to go through before we find the empty set again. The reason that most texts don't cover this initial step is because it's not terribly interesting mathematically, and it doesn't fit with the rest of the theme of modern algebra.
Result: The natural numbers exist.]
Step 1: An equivalence relation R on a set A is a set of elements A x A that satisfies the following properties:
Reflexivity: For all elements a of A, (a, a) is an element of R.
Symmetry: For any two elements a and b in A, if (a, b) is an element of R, then (b, a) must also be an element of R.
Transitivity: For any three elements a, b, and c in A, if (a, b) is an element of R and (b, c) is an element of R, then (a, c) must also be an element of R.
For examples of equivalence relations, we can consider equivalence, or the set of all elements A x A, or the empty set.
Another important part of this story is that equivalence relations partition the set that they're defined on. That is, examining the elements of A under R yields a set of disjoint subsets that covers A and contains no extra elements. Two elements a and b of A are part of the same equivalence class iff (a, b) is an element of R for whatever R we're working with.
Result: We know what equivalence classes are.
Step two: We consider an equivalence relation on pairs of integers. For any two pairs of integers (a, b) and (c, d), (a, b) ~ (c, d) iff a + d = b + c. We then define addition as [(a, b)] + [(c, d)] = [(a + c, b + d)] and multiplication as [(a, b)] * [(c, d)] = [(ac + bd, ad + bc)], where [a] denotes the equivalence class of A under R. This set of ordered pairs /is/ the set of integers, and for any element (a, b), we define n = | a - b | and we write...
... n for a >= b,
... -n for a < b,
simply as a matter of notational convenience.
Result: We manufactured the integers.
Step three: Let's define another type of set called a "ring." A ring is a set of elements A equipped with two operations + and * that satisfies the following properties:
Let a, b, and c be elements of A.
1) (Closure under addition.) a + b is also an element of A.
2) (Closure under multiplication.) a * b is also an element of A.
3) (Existence of additive identity.) There exists an element 0 in A such that a + 0 = a for all a.
4) (Existence of a multiplicative identity.) There exists an element 1 in A such that a * 1 = 1 * a = a for all a.
5) (Associativity of addition.) (a + b) + c = a + (b + c).
6) (Associtativity of multiplication.) (a * b) * c = a * (b * c).
7) (Commutativity of addition.) a + b = b + a.
8) (Left distribution.) a * (b + c) = a * b + a * c.
9) (Right distribution.) (b + c) * a = b * a + c * a.
10) (Additive inverse.) For any a in A, there exists an element denoted -a such that a + (-a) = 0.
In addition, we'll only be working with commutative rings, and commutative rings satisfy all the above properties, plus one more:
11) (Commutativity of multiplication.) a * b = b * a.
It's worth noting that rings never guarantee the existence of a multiplicative inverse.
As for examples of rings, consider the set of all polynomials with the addition and multiplication operations you've come to expect, or the set of all matrices, again with the addition and multiplication operations you've come to expect.
Result: Now we have a vague idea of what a ring is.
Step four: An ideal I of a ring A is a subset of A that satisfies the following properties:
1) 0 is an element of I.
2) For any two elements a and b in I, a + b must also be in I.
3) For any two elements a in I and x in R, a * x must also be in I. (Note that we're working with commutative rings.)
Let a be an element of A. Then we write (a) = {a * x | x is an element of A} and we say "the ideal generated by a". It's easy to verify that this is an ideal as defined above. Any ideal that can be written as (a) for some a we call a "principal ideal".
Let I be an ideal of A and a be an element of A. Then we write (I, a) = {x * a + i | x is an element of A, i is an element of I} and say "the ideal generated by I and a" or "the ideal generated by joining a to I". It's again easy to verify that this is an ideal.
Let I be an ideal. If i is an element of I and i can be factored as a * b, then if a and b are always also elements of I, then we say that I is a "prime ideal". If I is not equal to the whole ring, and if for any a in A, a not in I we find that (I, a) = A, we say that I is a "maximal ideal".
Result: A whole slew of definitions about ideals.
Step five: Now we'll consider quotient rings. Consider A a ring and I an ideal of A. Then define an equivalence relation on A such that a ~ b iff (a - b) is an element of I. We denote this particular equivalence class by R_I (where the _ typically denotes subscript, too bad vBulletin can't handle true HTML ).
With this equivalence relation in hand, we are now ready to define quotient rings. Taking the same A and I as before, we write A / I to mean the set of equivalence classes of A under R_I. We define a surjection pi(x) (referred to as the projection function) that maps all elements x of A to their equivalence classes in A / I. pi is a "ring homomorphism", where a ring homomorphism is a mapping X from one ring A into another ring A' that satisfies the following properties:
1) (Preserves addition.) X(a+b) = X(a) + X(b).
2) (Preserves multiplication.) X(a*b) =X(a) * X(b).
3) (Preserves the identity.) X(0) = 0.
4) (Preserves the identity.) X(1) = 1.
With addition and multiplication defined appropriately, it's easy to show that pi is in fact a ring homomorphism, and thus A / I, the set of equivalence classes of A under R_I, is also a ring. We call A / I a "quotient ring."
Result: Quotient rings defined.
Step six: Consider the pairs of integers Z x Z. If we define addition and multiplication like so:
(a, b) + (c, d) = (a*d + b*c, b*d)
(a, b) * (c, d) = (a * c, b * d)
we can see that Z x Z forms a ring, which we will now denote as A. Let's define an equivalence relation R on A such that (a, b) ~ (c, d) iff a * d = b * c. If we consider the equivalence classes of A over R (and the appropriate projection function, pi), we find that A / R is also a ring. For any particular (a, b), we customarily write a/b, and we thus recover the familiar notation for the set of rationals.
Result: Production of the rationals.
Step seven: Let's go back and look at our ring of polynomials. For any ring A and any dummy variable x, we denote the ring of all polynomials in x with coefficients elements of A by A[x]. Let's consider only the polynomials with rational coefficients -- Q[x]. Let's then consider the polynomial x^2 - 2. If we investigate the quotient group Q[x]/(x^2-2), we find that we generate a field (we'll just think of it as a ring, since we never discussed fields) where x is "equivalent" to the square root of two, and we can perform all arithmetic operations with that assumption in mind. If we take this ring (denoted k), look at all the polynomials over it, and then adjoin a different polynomial with a real irrational root (say t^3 - 6), then we can adjoin another algebraic irrational to Q by looking at the ring k[t]/(t^3 - 6). Continuing in this m |
