[i][Mathematics that is}[;)][/i]

Since it's popular now to bring up "apparent" inconsistencies in certain fields of inquiry (Biblical theology for one), I'd thought I'd continue this with a few inconsistencies or contradictions in other fields.

I'll start with a just few easy ones... Mathematics and Euclidian Geometry:
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#1)
Point: a zero-dimensional figure.
Line Segment: the set of points consisting of two distinct points and all inbetween them.

A line segment has a defined dimension (i.e. exact length)

[u]Based on these definitions[/u], since a single "point" has zero dimensions, how can a set of zero dimension points have a defined dimension?

Does a line segment have width? By definition - no.
Then how do dimensionless points aligned sequentially in a segment create length but no width?

How can many line segments placed next to each other create a plane with defined width if each line segment creating it has no width individually?
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#2)
According to the Pythagorean Theorem, the length of a hypotenuse (C) of a right triangle is equal to the square root of the sum of the other sides (A and B) each squared. (A^2 + B^2 = C^2)

If A and B are exactly equal to one, the length of the hypotenuse is an irrational number (square root of 2).

I can conceive that there exists a real, physically tangible triangle with two sides that  measure exactly 1.000000... meters long, but the hypotenuse cannot be exactly measured.

How can the physical length of this hypotenuse be exact (i.e. it does have an exact beginning point and ending point), yet its measurement proves not to be exact (the measurement has no ending)?

In other words, since this hypotenuse is a line segment with definite beginning and ending points, how can its measurement have no ending?

(this one is not exactly the same as Zeno's paradox so don't use that solution)
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#3) (my favorite)
There is no logical answer to the question, "What is the square root of negative one?"
So Einstein CREATED the imaginary number "i" to fill in the gap in his work in relativity.

Since the imaginary number "i" has no real (i.e. physical or logical) origin, it's a fudge-factor and without it, conventional mathematics or logic can not fill in the gaps in relativity.

Yet "i" is used to link space, time, matter, energy, gravity, and inertia, all to a single unit of conversion.

So why not just call "i" an "invisible spirit" and say that this "mysterious spirit" is the inexplicible, unexplainable, illogical force missing from our understanding of relativity?

Isn't "i" to mathematicians what "God" is to theologians - both are undetectable, untestable, illogical entities simply used to fill in the gaps to problems where no logical answer will suffice?

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Okay, I do welcome insightful explanations to these "apparent" inconsistencies in the way many here delighted in thrashing out the inconsistencies of the Bible.

Once these "apparent" mathematical inconsistencies have been thoroughly hashed out, I'll bring up the apparent inconsistencies and contradictions in Chemistry, Physics and Biology to toss around.

What's wrong with the square root of 2 being irrational?  If that tickles your bum too much you could always measure the sides in inches instead of meters that would probably help hehe.  BTW I like where you are going though.

1,2,3,5,8,9,11,21,47...


Hey you're right! That's really messed up.


Seriously, you are gonna compare the sqaure root of a negative number being represented by "i" in certain "theories of relativity" to the absurdities in the bible?

Furthermore the formulas that use "i" are just that "theories." They are not fact but assumptions used to search for the truth. There are scientific knowns such as E=mc2 which are no longer just theory because we have proven them.

You see in math and science, when something is not known for sure it is assigned a hypothosis, theory or a variable unknown. This is science saying "Gee, we really don't know but lets see if we can find out." Science and to a lesser extent, math are based upon the fact that we don't know something. They are a search for the answer. Much of science is categorized as "current theory" this means our best guess based upon available data. Science freely and readily admits when it doesn't know and when it has been wrong. Because REAL science is simply the search for truth and understanding.

If you presented bible stories as theoretical history and theology, nobody would object. But it is presented as fact. And it never admits when it is wrong.

And now for something completely different:

111,111,111 times 111,111,111 =

12345678987654321 [:D]

I hated math in school and I hate it now....[rolleyes]

"less filling" or was that "tastes great"?

Okay, so you're gonna pass on that one. No problem.

Yes, I agree with your assessment of the intent of science wrt the search for truth.

But what of these "apparent" inconsistencies that are the foundation of Euclidean Geometry, basic measurement and simple mathematics?

Shirley you can't say these are obscure notions that are not "factual"?

science if different from math in that science is based in part on physical measurements, while math is based only on logic.

scientific hypotheses, after being tested in the physical world, are either rejected or accepted as working theory. on the other hand, with mathematical theories, the only test is that of logical consistency.

regarding mathematics, you're not describing logical inconsistencies or contradictions. you're saying that it doesn't make sense to you. there's a difference.

#1)

infinitesimals are infinitely small numbers.

when you add an infinite number of infinitesimals together, you can arrive at a finite number.

particular to your question: a point has an infinitesimal length, and when you add together an infinite number of points, you can arrive at a finite lengthed line-segment.

#2)

there are different ways that numbers can be "exactly known". many numbers are exactly known, though they cannot be expressed as rational numbers.

the square-root of 2 is exactly known. however, it cannot be expressed as m/n, where m and n are integers. so it is not a rational number, but it is still a number.

#3)

there is nothing "imaginary" about i. it was named inappropriately. it has as every bit as much physical and logical reality as 1.

1. Stop calling me Shirley.

2. These items will either be classified as "Math and Science" or "Theoretical Math and Science." You just have to put them in the proper category. If it cannot be proven over and over without exception it is neither math or science.

3. I actually approach the bible in much the same way. Lots of actual history in that book and some decent philosophy as well. But I also have categories such as "highly improbable" and "complete bullshit."

Bzzzzzzzzt!! Nope. Wrong.

A point has ZERO length. Zip, Zilch, Naught, Nothing, Nada - NOT "infinitesimal length" but "0.0000..." length.

0 + 0 = 0 (except with geometric points it seems)

Please cut me a length of string equal the square root of two meters long.

What's that, you can't do that?

But I just constructed a triangle with that exact length side!

I'm sorry but you're going to have to do much better than that "by fiat" statement.

An imaginary number is any number multiplied by [i]i[/i] with [i]i[/i] being the square root of negative one, which in mathematics has no logical answer other than this particular fudge factor which could have just as easily been named "deus ex machina".

[:D]

So does the logical math statement "0+0=0" apply to measuring the length of all geometric entities (lines, planes, angles, curves etc) EXCEPT geometric points?

If so, can someone explain why Euclidean geometry is the exception to the basic rules of addition using zero?

Are the "complete bullshit" in the category of stated facts (e.g. the flood, Red Sea, Passover, Resurrection) or are they in the category of Biblical proclamations (e.g. Mosaic Laws, )

in the theory of calculus (by leibniz and newton) a point has an infinitesimal length, not a length of 0.

but you wish to discuss euclidean points, not newtonian points. you seem to think that euclid writes "0+0=1", but he does not write this. what is the inconsistency/contradiction you find in euclid's theory? where is the logical breakdown?

i agree, it is possible to construct a segment whose length is the square root of 2. if you have been told that such a construction is impossible, then you have been told wrong.

you keep making the fiat statement that 1 is more "logical" than i. i don't accept your fiat statement. why do you think 1 is more logical than i? why do you think i is more of a fudge than 1? they have logically equivalent bases in mathematics.

You would do better to ask a expert in those fields.



They are a case by case basis. For instance the lineage in Genesis, from Adam who existed in the first week of creation, to a known man who lived in a known date was used by the Church to calculate the date of the Earth and Universe.

Even given that long ago many lived to be 250+ years old [rolleyes] the arrived at a approximate date of 10,000 years. But if one can see the light from a star a million light years away (and we can) then we know that the 10,000 years old (approx.) date is complete bullshit.

Now here is where someone will offer "Well what is a day to God?" Well if it is not what it says it is, than how can we accept any part literally. "What is covet to God?" Or they will be genius enough to suggest we haven't accurately gauged the speed of light or the dimensions of the universe.

Another example is Dinosaurs existed for millions of years before man. Does this qaulify as a day to God? If so see above. others completely discount that dinosaurs ever existed at all. After all the Bible doesn't mention them. I even had one guy suggest that dinosaur fossils were the work of the devil put in the ground to confuse us and lead us astray. That devil sure is a stinker.

And I don't even want to get started on the Noah story or Jonah and the Whale. If you substituted the part from PINOCCHIO where Geppetto and Pinocchio find themselves in the belly of a whale, where Pinocchio's nose grows so long the whale becomes acutely uncomfortable and disgorges him, do you think anyone would notice?

Very well. Let's take them one at a time.

Given:
a) A point has zero dimensions,
b) A line segment is a set of points consisting of two distinct points and all points in between them.
c) A line segment has a defined length greater than zero.

How can a set of points, each of whose individual length is zero, have a combined length greater than zero?



Edited to add: yes I did understand your original "point" (pun intended) regarding infinitesimally small measurements. But I WAS referring to Euclidean geometric points, hence my hint to avoid repeating the solution to Zeno's paradox.

i agree with you on (a), (b), and (c).

in logic and mathematics, a theory is assumed to be consistent unless a specific logical contradiction is found in the theory. you are pointing to a phenomenon that challenges your intuition (it challenges mine too), but you are not pointing to any contradiction.

to prove that euclid's theory is inconsistent, you have to write out a mathematical proof that ends with the statement "0+0=1". (in thousands of years, no one has succeeded in finding a logical inconsistency in euclid's theory.)

biblical stories, on the other hand, contradict evidence found throughout the world -- as steyraug demonstrated previously.

(incidentally, second to the Bible, the most read book in history is euclid's.)

Yep. I don't for a minute believe the Universe is 6,000 years old. I think that's a result of some folks' literal interpretation of the Genesis story which has both literal AND symbolic meaning.


I see nothing absurd about a lifespan exceeding hundreds of years - [b]IF[/b] there are very few viruses (which are currently thought to cause more than half of all human cancers) or other pathogenic microbes, very few natural toxins, abundant food and year-round mild weather (the "just right" environment of the supposed "Eden").

There is a passage where it says something to the effect that "to God, a day is 1000 years and 1000 years is a day"... meaning time is relative to God and our term "day" is not His term "day".

There are some who still keep to this interpretation of a day/1000year relativity and say that when Jesus said He would be risen up on the "third day" he meant it according to both God's "day" and man's "day".

Accordingly, Jesus DID arise on the "third day" (for man's benefit) and WILL return in the "third millineum" (for God's benefit) thus fulfilling his promise according to both God's "day" and man's "day".

I don't try to figure out the age of the universe from interpretations of "day" or "year" in the Bible, much like I tire of people trying to figure out who "666" is. It's interesting but I don't hang my hat on any of it.

Yeah, that's a bit much.


Seems things really went downhill after Noah. Lifespans shortened real quick, the tower of Babel, wars...
I don't discount a great flood that upset the "world" (or maybe just Noah's little corner of it).
As far as Jonah, is that metaphor or fact? I don't know I wasn't there.


Pinocchio... LOL.

Hence my parenthetical use of "apparent" inconsistencies in my opening post.

I think our intuition is right though. I do believe there is a logical contradiction in Euclidean geometry, the source of which is probably one of definitions.

Maybe I'll go ahead and write it as an authentic proof that ends with "0+0=1".

You heard it here first folks. [;)]




Edited to add:
... actually I was just asking a simple question to which no one has yet given an answer:
[b]How can a set of points, each of whose individual length is zero, have a combined length greater than zero?[/b]

So does paleontology. Whoops... that's another thread.

i've been studying math for years, and i still can't grasp this intuitively. but the theory holds together and has put the space shuttle on the moon, so i believe it.