Why can't I divide by zero.
I know it's undefined, but why is it undefined - whay isn't it 1 or 0. Who made this up anyway?
- gaffa
Printable View
Why can't I divide by zero.
I know it's undefined, but why is it undefined - whay isn't it 1 or 0. Who made this up anyway?
- gaffa
Since numerical division is roughly defined as how many times can p be split into q sized pieces with anything left over for p/q any non-zero number can be split into an infinite number of zeroes leaving the original number. Therefore 1/0 is an infinite quantity. That cannot be expressed except notationally. So you CAN divide by zero - you just cannot express the answer and you give your FPU a headache. Remember that zero is not a number - it is a concept and not one that the Romans were aware of! Zero and its uses were first defined by Arab mathematicians from whom we get most of our mathematical notation in use today (and thank goodness - you try multiply XLIV by CDLVIII without using our number system...)
Cheers,
P.
Hmm, x/0 is not infinity, although when you get close to 0 from the right resp left to 0 for the function 1/x, you get close to -infinity resp +infinity.
lim x->0- 1/x=-infinity
lim x->0+ 1/x=infinity
As paul said you can't fill it with any amount of 0's since adding up 0's won't accumulate. Therefore there's no value for 1/0.
the graph for 1/x is a hyperbola with an asymptote y=0, which mean the function 1/x won't cross that line.
Yeah,
I sort of knew why - I justcouldn't express it properly to my friend who wanted to know. Thanks guys
- gaffa
kedaman - I agree that it is asymptotic but infinity is a perfectly valid conclusion and I would take issue with you when you say the result is not infinite - infinity is again a concept. The asymptote can be used to make the general conclusion that 1/0 is infinite since the approach from small values of x for 0+x tends to infinity.
Come on back.
Cheers,
P.
hmm, that's not really correct, 1/x for positive x that goes toward 0 goes toward infinity only means the right limiting value is infinity, not even that there actually is a limiting value at all, since the left limiting value is negative.
Kedaman, I'm gonna have to agree with Paul here, 1/0 is infinity. The approach you take with the limit of 1/x as x tends to zero would prove that I would have thought. Using the same idea you get clearly defined derivitives of functions when you differentiate, and similarly with integration. As far as I remember, the derivation of integration rules is done using limits and strips of trapezia.
What do you mean by the left limiting value and right limiting value?
Harry, you sure agree x=0 is a extreme value for 1/x. by left limiting value i mean what value 1/x approaches when x approaches 0 from the left, which is negative infinity. the right limiting value is positive infinity. Why do you put positive infinity in favour then?
Ked
I think what you've missed is that infinity isn't really a number, it's just a symbol which means an undefined result. The only time we distinguish between +ve and -ve infinity is when we use the expression "The limit of F(x) as x tends to infinity" or "The limit of F(x) as x tends to - infinity"
But strictly speaking these expressions are mathematical slang, what we mean by "The limit of F(x) as x tends to infinity is q" is
For any real number d > 0 there exists a real number Y such that if a real number X > Y then |F(x) - q| < d
For calculating limits of functions such as Sin and Cos we may have to redefine this slightly but we still avoid the mention of +ve and -ve infinity in our definition.
When you think of the left and right limits of 1/0 consider this
1/0 = exp(ln(1/0)) = exp(-ln(0)) = exp(-(-inf)) = +ve infinity
This is technicly a proof by contradiction, (to stop your argument that it doesn't count because I've used +ve and -ve infinity to proove they don't exist) If the 2 infinities are distinct then it can be shown that they are indistinct and hence there is a contradiction, hence they cannot be distinct.
The reason that +ve and -ve infinity are used in the notation of limits is that it's far easier than writing down the formal definition of what you are doing, Everyone understands the notation Lim(x-> -inf) F(x) If they want a formal definition they can go to Cambridge, Harvard, The Royal Society, or wherever the argument over the precice definition happens to be taking place and get a big page full of symbols that define it. +/- infinity aren't numbers, they're ways of defining 2 particular properties of a function
I know that it's not a real number, but i can't accept is equal to "undefined", if a number is real, it's definded with positive infinity as right limit resp negative as left, you can't say that with "positive undefined"
This subject is being made too complicated.
Division is defined as the inverse of multiplication.The above indicates a serious problem with division by zero. Considering ordinary real numbers, there is no possible value for Quotient. End of discussion.
- If Quotient * X = 1, then Quotient = 1 / X
- Now if X = 0, then Quotient * 0 = 1
- What value can you assign to Quotient in this case?
All the talk about infinity is more related to philosophy than mathematics. As far as I know, infinity is dealt with in set theory, which is more a branch of logic than a branch of mathematics.
I have not critically analyzed the posts to this thread, mainly because I do consider myself enough of an expert to avoid errors on this subject.
The above having been said, I would advise others to not trust anyone who posts here (including me) on this subject without verifying via a serious mathematical text.
For example, a very knowledgeable and intelligent individual once posted the statement that 1^infinity = 2.71828... (or the number usually referred to as "e"). This happens to be erroneous, which I only realized after doing some analysis. Due to my respect for the individual, I accepted his post until I had occasion to discuss it with a friend of mine who suggested that we try to verify the statement. This error has to do with the following very subtle concept.One must be very careful when replacing an expression with the limit of that expression. If anybody is interested, I can expand on this.
- Limit(1 + 1 / N ) approaches one as N grows without bound.
- Limit[(1 + 1 / N)^N] approaches e (2.71828...) as N grows without bound.
- It is invalid to claim that Limit(1 + 1 / N)^N equals Limit[Limit(1 + 1 / N)]^N as N grows without bound.
I wonder about the references to positive and negative infinity and left/right limits. It is not clear to me that transfinite numbers have signs. In complex analysis, infinity is a circle.
I do not think mathematicians make statements like 1 / 0 = infinity, instead they say 1 / X grows without bound as X approaches zero. Oddly enough, they sometimes assign a value to zero divided by zero, with one not necessarily being the value assigned. I do not remember any mathematical text which allows using a value like 1 / 0 as a real number in an equation.
As far as I know, mathematicians avoid treating infinity as a real number. The only literature I know of that deals with the concept uses the term "Transfinite Numbers" to refer to the cardinal number associated with certain sets. Once you open that can of worms, you are forced to accept more than one transfinite number. For example: The set containing all of the integers is smaller than the set containing all real numbers. In "slang" terms, there is more than one infinity.
I do not remember any literature which allows transfinite numbers to be used as equivalent to real numbers in analytical expressions.
I am not sure that you can find a reference which allows you to claim which transfinite number corresponds to Limit(1/X) as X approaches zero. Is it the transfinite number associated with the set of all integers, or is it the transfinite number associated with the set of all real numbers? If somebody can cite an authority for this, I will be interested to know of it. I would not be surprised if no such citation can be found. I will not be surprised if there is a citation. If there is a citation, I am guessing that it is the transfinite number associated with the set of all reals.
1^infinity. (an admission of error.)
If you remember when I posted this I claimed that I had read the result in a maths book and then couldn't find the page again.
Well I've found the page and the source of the error.
1^infinity is known as an indeterminate form. this means that if F(x) = G(x)^H(x) and Lim(x->c, G(x)) = 1 and Lim(x->c, H(x)) is infinite we still do not know the answer. (as opposed to axioms 1/infinity = 0, infinity*infinity = infinity etc, ie for any F(x) = G(x)/H(x) and Lim(x->c, G(x)) = 1 and Lim(x->c, H(x)) is infinite we know that Lim(x->c, F(x)) = 0
However, the book didn't offer much explanation of this, and merely gave an example of how to find a particular limit of a fumction which was in this form. specificly
Lim(x->0,(1+x)^cot(x))
As I misenterpreted this example as a result in itself I quoted said result, and you picked me up on it, As I had read it in a maths book I assumed it to be correct.
I'm pretty sure transfinite numbers have nothing to do with limits. I've forgotten the name of this axiom, but it's one of the basic axioms of the real number line
for real numbers x and y exactly one of the following is true
x > y
x = y
x < y
I believe this axiom holds for the transfinite numbers.
ie the cardinality of the real numbers is greater than the cardinality of the integers, the cardinality of the rationals is equal to the cardinality of the integers. etc
I don't think this axiom applies to infinities arrived at by limits, It would be wrong to say that exactly one of these three is true
lim(x->0, 1/x) > lim(x->pi/2, tan(x) )
lim(x->0, 1/x) = lim(x->pi/2, tan(x) )
lim(x->0, 1/x) < lim(x->pi/2, tan(x) )
Even if we can say that the axiom applies to the limits it still doesn't mean that we can assign a transfinite number to 1/0, The 2 fields are so different that we can't just assume there's a nice link between them. Especially as the transfinite numbers deal with the extension of natrual numbers to infinity, rather than real or complex values.
Sam, Some time ago I swore a solemn oath to myself that I would not get involved in discussions of transfinite numbers, infinity, 1/X as X approaches zero, et cetera, but here I am again. I hope my solemn oath was not taken seriously by any of Lovecraft's gods.
First I repeat a statement I made in a previous post. I would advise others to not trust anyone who posts here (including me) on this subject without verifying via a serious mathematical text.
Second, I think that 1/0 = infinity and similar statements are "mathematical slang" avoided by serious mathematicians. A serious text is likely to say something likeSerious texts avoid using infinity in mathematical expressions.Quote:
1 / X grows without bound as X approaches zero. Or Limit(1 / X) = 0 as X grows without bound.
Your following post is quite a mouthful[quote]1^infinity is known as an indeterminate form. this means that if F(x) = G(x)^H(x) and Lim(x->c, G(x)) = 1 and Lim(x->c, H(x)) is infinite we still do not know the answer. (as opposed to axioms 1/infinity = 0, infinity*infinity = infinity etc, ie for any F(x) = G(x)/H(x) and Lim(x->c, G(x)) = 1 and Lim(x->c, H(x)) is infinite we know that Lim(x->c, F(x)) =0[/code]The above confuses me. I am not sure I really understand it. Could you give me the name of the book and page reference? I would like to see the context surrounding these statements, and be able to better understand what the author is saying. I have access to a large public library and some local university libraries.
The first sentence from the above quote is 1 ^ infinity is known as an indeterminate form.. I cannot imagine what the author is trying to say by this statement. If he saying that Limit(1 ^ N ) as N grows without bound is indeterminate, he is dead wrong. If he is not saying that, I do not understand what he is saying. Note: 1^1 = 1, 1^2 = 1, 1^3 = 1...1^50000 = 1... Limit(1 ^ N) is not indeterminate: This limit is one. Nothing magic happens some where along the way as N gets larger and larger. It is not explicitly stated in the proof, but Limit(1 ^ N ) = 1 is used in the proof that that Limit(1 + 1 / N ) ^ N = 2.71828..(Or Limit = e).
The second sentence in the above quote is not applicable to the first sentence which makes no statements about limits. I am not sure what the second sentences is trying to say. It seems to merely reinforce the notion that you do all the algebra first, getting simple terms, and then determine the limits of the individual terms. You do not determine limits of complicated terms and them do algebra on the limits.
The third sentences seems to be a complicated (and informal) way of saying that dividing a finite function by a function that grows without bound results in a quotient of zero in the limit.
If I could read the book myself, the above might be clearer. Obviously, you cannot post an entire book, and out of context the above seems confusing to me.
Guv,
I do apologise, the statement of mine you quoted wasn't in the book at all, it was someting I added that the book was missing.
basicly the passage in the book that I got the result was describing a method of calculating limits of functions in indeterminate form. However the book did not define the term indeterminate form, which lead me to confuse an example of a limit in the form 1^infinity with a general result for all limits in the form 1^infinity.
In the quoted statement I was trying to define the term indeterminate form.
Unfortunatley I'm one of those mathematicians who relies very heavily on symbols, and trying to define something in words is not something I'm good at. (especially when I'm unsure of the true definition and am just guessing) but I'll try again.
an expression in the form 1^infinity is in indeterminate form, in the same way as an expression in the form ax^2 + bx + c is in polynomial form. ( There are many expressions in the form 1^infinity and there are expressions in indeterminate form that are not in the form 1^infinity.)
Indeterminate form means what it says, just because we know an expression is in the form 1^infinity does not mean we can determine its value, we need to know more about it. (Just because we know an expression is in the form ax^2 + bx + c doesn't mean we know its value, we need to know what a, b and c are.)
The book didn't explain this, it simply showed an expression in indeterminate form specificly lim(x->0, (1+x)^cot(x)) which as it happens is e.
I misinterpreted this to mean any function in the form 1^infinity = e. which by the definition of indeterminate form is obviously erronious.
so that was where the error occurred.
I hope that makes sense. 1^infinity is a form of an expression rather than an expression itself, and not all expressions in that form have the same value.
Sam, now we might be getting somewhere.Can you give me examples of two/three expressions in form 1^infinity? Also a few expressions in indeterminate form which are not in the form 1^infinity?Quote:
There are many expressions in the form 1^infinity and there are expressions in indeterminate form that are not in the form 1^infinity
I think I am beginning to see the light at the end of the tunnel. I hope it is not a train as wide as the tunnel coming toward me.
No problem.
1 ^ infinity
- lim(x->0, (1+x)^cot x) = e
- lim(x->0, cos(x)^(1/x)) = 1
- lim(x->0, tan( ln(-1)/4i ) ^ csch( 1 - (cos(x)^(1/x))) = I have better things to do with the next few months
Other indeterminate forms
- 0/0
- infinity / infinity
- infinity - infinity
- 0 ^ 0
- infinity ^ 0
the key point (according to the book) is that they all have 2 conflicting parts, (eg 0/0 0/x = 0 for all x, x/0 = infinity for all x) so we can't just say ahh, it's in this form it must have this value, like we can 1/0 or 0^infinty.
for example the limit I didn't work out,
lim(x->0, tan( ln(-1)/4i ) ^ csch( 1 - (cos(x)^(1/x)))
it's fairly easy to show that lim(x->0, tan( ln(-1)/4i )) = 1 and that lim(x->0, csch( 1 - (cos(x)^(1/x))) is infinite
but we can't work out the whole expression, however if we changed the ^ between the 2 terms to a /
lim(x->0, tan( ln(-1)/4i ) / csch( 1 - (cos(x)^(1/x)))
that's easy, it's 0 because it's in the form 1/infinity, which isn't an indeterminate form (I suppose it's determinate then) we know that 1/infinty is 0.
An interesting thought: (to me anyway)
We seem to be arguing about the nature of infinity - remember that parallel lines can be defined as lines that will only meet at infinite length - this is clearly nonsensical since parallel lines will never meet, but it illustrates the strange nature of infinity.
If a line is asymptotic to a result the line will never touch that result for any given number, but it is still valid to say that when the number is infinity, the line will touch its asymptote. Hence 1/0 can be said to be infinite since 1/x tends to 0 for large values of x (-ve or +ve x).
The problem is that we are dealing with concepts rather than concrete ideas.
Guv: I take issue with your idea that
Mathematics is a tool - simply a meta-language to describe the real world. Serious mathematicians include those who apply mathematics (such as engineers) as well as professors in ivory towers.Quote:
Second, I think that 1/0 = infinity and similar statements are "mathematical slang" avoided by serious mathematicians.
To derive a result from a self-limiting set is perfectly valid, this does not mean that there will not be different ideas that may appear self contradictory. There is a school of thought that will not allow proof by induction sice the formality of the proof is not rigorous enough, most mathematicians would use induction though, since it is a useful tool.
Maths and philosophy merge since Maths is a langauage to describe ideas and like any language it develops. The main problem is that one cannot be too dogmatic about anything - can one even rely upon 1 + 1 = 2 - that may not hold valid in the heart of a black hole!
Interesting, eh?
P.
PaulW, you are correct about parallel lines never meeting. I do not know the origin of the concept that they meet at infinity. It might be a "slang expression" used instead of saying that they never meet. Perhaps it is equivalent to saying that parallel lines meet when Hell freezes over. As far as I know this is a totally erroneous "common notion" rather than a formal statement about geometry. Stating that 1 / 0 = infinity is a common notion that cannot be called erroneous, but it does not lead to any usable mathematical ideas.
Dealing with these concepts is confusing and counter-intuitive enough without introducing erroneous notions like parallel lines meeting at infinity (see below).
Among other statements, you saidI cannot fault your statement, but I do like to think that mathematics has esthetic value and is more than a tool. By the way, I do not think that 1 / 0 = infinity is a concept used by practical engineers. While above is correct, the professors in ivory towers do their best to make sure that the tools used by the engineers have no hidden flaws.Quote:
Mathematics is a tool - simply a meta-language to describe the real world. Serious mathematicians include those who apply mathematics (such as engineers) as well as professors in ivory towers.
I have been on both sides of the theoretical fence, and have actually used computational techniques to validate numerical methods not considered provable by the academic community, something considered a sin by some of my college professors.
When mathematical analysis (as opposed to modern set theory) deals with infinity, it uses the concept of limits or limiting values.
I repeat what I said about how mathematicians deal with infinity and limits. They really do avoid statements like "1 / 0 = infinity" and "A curve touches its asymptote at infinity." They prefer to make statements like "1 / X grows without bound as X approaches zero" or the curve X * Y = 1 approaches the X-Axis as X grows without bound." It is not considered a punishable blasphemy to say "1 / X = infinity when X equals zero" or the "The curve X * Y = 1 touches the X-Axis when X equals infinity." It is merely considered bad form or sloppy language to be avoided in formal literature.
Claiming that a curve meets its asymptote at infinity has some merit. This is not analogous to claiming that parallel lines meet at infinity.[/B] Consider the straight line Y = 1 and the curve X * Y = 1.Considering the above, the statement that the curve touches the X-Axis at infinity has some justification. There is no justification for stating that the line Y = 1 ever touches the X-Axis.
- For X = 1, 2, 3, 4 . . . one hundred million . . . Y = 1 on the straight line. Nothing seems to change as X grows without bound.
- For X = 1, 2, 3, 4 . . . one hundred million . . . on the curve, Y = 1, 1/2, 1/3, 1/4 . . . .00000000001 . . ..
There are good reason for mathematicians to be nit-picky about language. Sometime in the last half of the 19th century, it was discovered that some erroneous concepts had been introduced into mathematics. They discovered provable paradoxes (an oxymoron made up by me) due to sloppiness in the way theorems were proved. In an effort to eliminate past and future erroneous concepts, they became very fussy about language.
The issue that we have here though is that we are not proving a theorem formally, and today's slang is tomorrow's orthodoxy. The original question was why can't I (sic) divide 1 by 0? With relation to computational methods, the definition of 1/0 as infinite is reasonable.
The definition of parrallel lines meeting at an infinite distance was part of a paper I studied at university and was extremely formal! I couldn't lay my hands on it now, but it struck me at the time. This was to show the paradoxical nature of infinity and how 'laws' break down in extreme situations. This also covered sets of infinite infinities. It was very wierd.
Infinity was certainly not avoided in the maths that I did as an undergraduate, however the concepts were represented notationally and never computed. maybe its 'New Math' that I cannot understand?:(
I'm playing devil's advocate a little. I do not really want to get too deep or entrench my position and I see what you are saying, but I still maintain that 1/0 can be regarded as infinite in all practical senses.
Flame on...
Cheers,
P.
I hate this subject (and am fascinated by it) because of the confusion and paradoxical conclusions which turn up. When I was taking math courses, transfinite numbers and infinity were dealt with extensively as part of Set Theory and logic. In calculus, algebra, and geometry courses, infinity was dealt with occasionally, and not in depth.
On the main topic of this post, I suppose it is not a crime to say that 1 / 0 = infinity. Those who say you cannot divide by zero are perhaps merely saying that you should not base any further computations or logical analysis on the quotient.
The concept of parallel lines meeting at infinity still seems outrageous to me. Can anyone provide an argument or a citation for the idea that parallel lines meeting at infinity? I am familiar with the following two opposing arguments. I will describe the one I do not believe first.
In Complex Analysis, there is a mapping of a sphere to the plane, which seems consistent with the concept of parallel lines meeting at infinity. The idea is to imagine a sphere with its South Pole at the origin of an XY-Coordinate system. A point of the sphere is mapped to the plane by drawing a straight line from the North Pole through the point, and extending the line it until it intersects the plane. The intersection with the plane is the mapped point. All well and good until we try to map the North Pole itself.
Consider mapping circles of constant latitude like the Equator, the Tropic of Cancer, the Arctic Circle, et cetera. Circles closer and closer to the North Pole map to larger and larger circles on the plane. A circle with an extremely small radius, which looks like a point, maps to a circle with an incredibly large radius. In the limiting case, the North Pole seems to map to a circle at infinity on the plane.
The above suggests that all tangents at the North Pole intersect the plane in a circle at infinity. These tangents are parallel to lines in the XY-Plane. This suggests that parallel lines meet at infinity. I do not remember this idea being used to argue that parallel lines meet at infinity, but this is certainly implied.
The above is the only analysis I can remember which suggests that parallel lines meet at infinity. I do not accept this argument, but others might. To me, this is a situation where is not allowable to base conclusions on the limiting condition.
To me a more compelling argument is based on considering the line defined by Y = 1. For every finite value of X, this line is one unit from the X-Axis. I see no justification for assuming that something magic happens at infinity, causing this line to meet the X-Axis. For this to happen, it seems that the line must make a quantum leap (or right angle turn) to the X-Axis at infinity. I would sooner assume that the above mapping is invalid when used to map the North Pole.
Exactly:):)Quote:
For every finite value of X, this line is one unit from the X-Axis
P.
Guv,
Consider for a moment the gometry of things on the surface of a sphere, Radius R, and draw 2 parrallel lines on it, These of course meet at 2 points on the surface of the sphere, these points are pi*R apart.
Now, making R bigger and bigger 2 things happen, 1, the distance between the lines gets larger, and 2 the surface of the shere becomes flatter.
This leads to the conclusion that plane geometry is in fact the limit of spherical geometry as R tends to infinity.
and so the lines meet at pi*R/2 (which is infinite) in either direction.
The other argument I can think of is if you imagine drawing a line perpendicular to one of the lines, (draw a line d perpendicular to line a, call the other parralell line b)
now, forget for a minute that a and b are parralell, call the angle line d makes with line b theta
So the distance between the points wherea crosses d and where a crosses b is (the distance between the points wherea crosses d and where b crosses d ) * Tan(theta)
and when the lines are parrallel theta = pi/2 and this distance = infinity.
I think the big problem here is a question of how to pronounce the term = infinity.
In maths we don't write the word infinity, we use the drunken eight symbol. So it may not be correct to write the word infinity, as maths symbols do not as a rule correspond exactly to English words, if they did we wouldn't need them.
OK, revert back to your various childhoods for a moment and picture yourself in front of the class reciting your times tables. In my classroom there was a mirror on the wall in front and the times tables written out behind me on big posters. So if you stood in the right place and could read mirror writing you could just read them off the posters.
So picture the scene, in the mirror I can see this
and all I had to say wasQuote:
1 x 7 = 7
2 x 7 = 14
3 x 7 = 21
4 x 7 = 28
There is a difference between what I was saying and what I was reading. Because I was translating between the written symbols on the posters, and the spoken word the teacher wanted to hear.Quote:
One times seven is seven.
Two times seven is fourteen.
Three times seven is twenty-one.
Four times seven is twenty-eight.
Now I wasn't wrong (except morally) A translation had to be made and, as I was 6 at the time nobody was fussed that I was saying is instead of equals.
So after this touching tale of scandal and deciet let's look again at the statement 1/0 = drunken-eight.
maybe rather than pronounceing this as "One divided by zero equals infinity" we can say "One divided by zero is infinite." which is literally "One divided by zero is not finite."
That seems a less controversial meaning to the phrase, esspecially when equality vey much implies the Idea that
a = c & b = c => a = b
wheras it would be very wrong to say
1/0 = drunken-eight
ln(0) = drunken-eight
=> l/0 = ln(0)
So I'd say that "1/0 = drunken eight" is a correct statement as long as it's pronounced correctly, and a misprocunciation is a forgivable mistake, even in maths.
Sam, you're teasing me right?Both proofs start with lines that meet, which is what you intend to prove.Quote:
Consider for a moment the gometry of things on the surface of a sphere, Radius R, and draw 2 parrallel lines on it, These of course meet at 2 points on the surface of the sphere, these points are pi*R apart.
The other argument I can think of is if you imagine drawing a line perpendicular to one of the lines, (draw a line d perpendicular to line a, call the other parralell line b)
now, forget for a minute that a and b are parralell, call the angle line d makes with line b theta
So the distance between the points wherea crosses d and where a crosses b is (the distance between the points wherea crosses d and where b crosses d ) * Tan(theta)
PaulW, you stress the word finite in my discussion of the line Y = 1. Does this mean you that expect any relationship which is valid for a finite value of X to be invalid in the limiting case? Perhaps you should talk to Sam about his proof based on spherical geometry. In the limiting case, you can claim that his great circles leap apart and no longer intersect, while claiming that my parallel lines in the plane make a quantum leap toward each other.
- The first proof starts with great circles on a sphere, which are known to meet. By the way, I have never heard anyone refer to parallel lines on a sphere, whose geometry does not allow parallel lines to exist.
- The second starts by assuming that parallel lines meet to form a triangle, so that you can use the tangent function.
I would maintain that the problem is that we are talking about the nature of infinity. If I want I can 'prove' that black is white etc.
I was being jocular about 'finite' but there is a serious point. The rules that apply to finite values may not be applicable in the infinite case. The point that I am trying to make (clumsily, obviously) is that in mathematics there is nothing provable beyond doubt. Do you want to discuss the work of Godel?
This almost becomes the realms of philosophy and from there it is a short trip to theosophy and theology. Happy to go there if you are.
Time for a new argument though, don't you think?
Cheers,
P.
"Theosophy" <--- never heard this word before... is it just a combo of "theology" and "philosophy", or... err... what?
Theososophy (as every Greek scholar will know) comes from Theos (God) and Sophia (Wisdom). It is literally Divine Wisdom and is essentially a search for 'revealed' truths. You are in a bit of trouble if you are an Atheist though. It is a sort of cross between Philosophy (Love of Wisdom for its own sake) and Theology (Knowledge of God and His nature).
Bottom of the class, Harry (except in programming, natch).
Cheers,
P.
PaulW, if you can follow the works of Godel, you are too much for me. When I looked at his work, I decided that it was not worth the effort. I am not sure that I could follow his proofs even if I did spend the effort.
I am sorry to hear that you take his work as a reason to dismiss the notion of proofs and critical analysis. I have a cherished belief that black and white are different. You have disillusioned me.
At least Godel claims that we have a choice between consistency and completeness. I opt for consistency. I never expected that anyone in general would ever know everything, and I am sure that I never will. So the loss of completeness does not seem tragic to me.
I am sorry to hear that we atheists are not capable of dealing with philosophy and theology. Or is it only the combination that is beyond us? Actually, as you define theology, I really have no use for it.
Cheers Paul. Excuse my ignorance of Greek, for some reason at school they thought perhaps French and German would be more useful :rolleyes:
Anyway, it's a good word, I'll try to remember it. Thanks :)
Hey Guv - I didn't say I understood Godel's work, I am just aware of it. However the interesting thing is that you have to make a choice between consistency and completeness. It is a nice echo of Heisenberg and the uncertainty principle. I don't opt for either, I keep as open a mind as poss and try not to be too dogmatic. Actually, as it goes, I think a lot of what you say is spot on (but I couldn't possibly admit it publicly:D)
Black and White - simply different aspects of the same thing - what you see as black may be what I call white - entirely subjective.
As to proofs and critical analysis, depends on your viewpoint. I am quite attracted to the notion that, ultimately, nothing is provable.
Now, finally,
You spoiling for a bout of fisticuffs, or what? The point I am making (as you know) is that Atheists would not have much need for knowledge of God (as He cannot exist for them). I do not define Theology, but its greek roots are the combination of the words for God and Knowledge. So what else could Theology be defined as. Where do you get the idea that Artheists are not capable of dealing with Philosophy and Theosophy? Not what I said at all. I was saying that a search for 'revealed truths' via God would be a bit hard for an Atheist to accept.Quote:
I am sorry to hear that we atheists are not capable of dealing with philosophy and theology. Or is it only the combination that is beyond us? Actually, as you define theology, I really have no use for it.
Don't be so hard on me. I am here for the enjoyment and the argument. You have every right not to like me:( but I reserve the right to like you (and respect your views) anyway.:)
PaulW, sorry! I thought I was being humourous, not argumentative. I am not at all interested in fisticufs.
In my last post, the only comment I intended seriously was the bit about opting for consistency. While I agree that nothing is absolutely provable, I like to think that if my fundamental beliefs are true, then my conclusions based on them are valid. Hence, I would be horrified by the thought that mathematical logic might be inconsistent.
By the way: Do you have some type of equipment which makes fisticuffs possible over the Internet? Do I have to worry about offending people who post at this forum? If I do not have compatible hardware can they attack me physically anyway?
I'm kidding, I'm kidding...
Virtual fisticuffs is much better than the real thing <Paul swings an uppercut at Guv> it doesn't hurt. <Guv avoids the uppercut and replies with a jab> I'm a creature of logic myself but Quantum theory shows me that what may be logically consistent may not be right let alone provable. I guess that is where I come from on the Infinity bit. It's all about defining the parameters.
BTW Have you noticed how the Brits supported my position and you fought a pretty lonely battle? Maybe 1/0 is infinity in the UK only?
Sorry if I came over a bit agressive there - it's easy to lose humour when ther is no inflection...
Cheers,
P.
PS I've enjoyed the argument, haven't you?
1/0 equals a protein infinity according to a math teacher. Do not ask me what this is; I dont know.
It looks like a misheard sound, typo, or bad hand writing, depending on whether you heard the teacher say it in a lecture, saw in typed notes, or read it from the blackboard.Quote:
1/0 equals a protein infinity according to a math teacher. Do not ask me what this is; I dont know.
Today, I skimmed through a book (Infinity and the Mind by Rudy Rucker) which seems to be a thorough discussion of all the concepts developed by mathematicians and philosophers on the subject of infinity.
Oddly enough, it does not seem to discuss zero as a divisor. Since I did not attempt to actually read the book, it is possible that I missed such a discussion, but I do not think so.
Does anybody know of a book that discusses division by zero? This Thread has gotten to me. I have access to a large public library and several university libraries.
The book you mentioned by Rudy Rucker is actually a very good book. I would recommend actually reading it as although it doesn't actually talk about div by zero it covers some really interesting concepts.
Guv stated that 'Provable Paradoxes' were a sympton of sloppy theorem proving. That may be true but Godel's Incompleteness theroem states that every formal system is capable of proving a statement that contradicts itself. Thus 'provable paradoxes' are not merely a result of language problems but an inevitable part of every formal system.
'Infinity and the Mind' goes right into Godel's proof of this fact.
I know this is slightly off the point of the thread but I thought I should mention this.
The problem with opting for consistancy is that if you disallow an (otherwise) valid statement because it contradicts itself, how can you trust any of the axioms that led to it's construction?
If you opt for completeness then you have to accept the fact that contradictions are a part of reality (What I would go for).
Sucked in again. The Godel Theorem and transfinite numbers are two areas that I try to avoid. It is too damn difficult to follow the proofs developed by Godel and Cantor without spending far more time than I am willing to devote to this subject matter.
I read popularizations like Infinity and the Mind, and leave the deep understandings to others. Having declared that I am not an expert in these areas, I will state what I think are the implications of the Godel Theorem
Simonm, I believe that you are misinterpreting the Godel Theorem.
It is my understanding that the Godel Theorem is viewed as proving that there is at least one valid but unprovable statement in any somewhat complex axiomatic system. I think that the nature of the proof forced one and only one of the following two conclusions.In other words: An axiomatic system is inconsistent or incomplete.
- An axiomatic system is inconsistent. Id est: There are provable statements which contradict the postulates.
- There are valid statements which are not provable. Such statements could be added to the axioms, expanding the scope of the system, and introducing further valid but unprovable statements.
I would rather believe that there are limitations on what can be proven than believe that axiomatic logic can lead to inconsistent statements. The axioms themselves are considered valid but not provable. What harm is there in accepting more esoteric statements which are similarly valid but not provable?
I still wish that somebody could provide a citation which discusses division by zero. If there is some analysis of this, I really wonder which transfinite number they choose as the result. Is it the infinity associated with the the set of all integers or the infinity associated with the set of all real numnbers?
Guv, I apprecite you temporarily putting aside your dislike for this subject.
That your preference for accepting the incompleteness of logic over inconsistancy must be rooted in your desire to view the truth as a solid, unambiguous and non-contradictory (albiet, not wholly provable) thing.
But this is only a prefered assumption due to the implications that accepting a (potentially) inconsistant truth would bring. Maybe it means accepting that truth is not absolute but merely a relative and subjective notion.
As for division bu zero, I don't belive you could really say the answer is 'Infinite'. Even an infinite number (of any cardinality) is still reduced to zero when multiplied by zero. What we're asking here is how do we make something out of nothing? Unless you start with something, there's nothing you can do (with multiplication) to get a non zero number. So I would agree that the unswer is undefinable.
Simonm, you are not the first to post opinions like the following.Such opinions really irritate one of my sore spots.Quote:
. Maybe it means accepting that truth is not absolute but merely a relative and subjective notion.
Whenever I see such opinions, I wonder about the person's attitude concerning the following.The above are from headlines in various newspapers sold at super markets. I never buy them, but amuse myself by reading and remembering the more outrageous headlines.
- The skeletons of Adam and Eve were recently discovered in Colorado.
- The Loch Ness monster has been captured.
- An infant has just been discovered wearing clothes from 1912 and floating in a life preserver from the Titanic. Scientists attribute the lack of aging to a Time Warp.
Do the people who consider truth a relative notion accept the truth value of such stories as being being equivalent to the truth value of the notion that Earth orbits Sol in an approximately elliptical orbit? Are the truth values of such stories equivalent to the truth values of various simple mathematical proofs? After all, you cannot absolutely prove or disprove anything.
Would you folks use astrology instead of physics to decide how to build bridges? Are you folks
willing to fly in airplanes and trust bridges not to collapse? After all, if the principles used by designers/engineers are relative notions, why risk your life in such a potentially dangerous vehicle as an airplane? Why not use prayer instead of penicillin to fight gangrene? Why use the accepted Microsoft syntax when you code in VB or C++? Why not use IBM PL/1 syntax? It is just as valid. Do any of you really run your life as though truth is a relative notion and the universe is a gigantic Whim-based Machine?
People who talk about truth being a relative and subjective notion use that concept as an excuse to avoid the use of critical analysis. Politicians, tyrants, and swindlers love you folks.
The use of critical analysis and striving for truth are more important than any other goals. Without them, how can you claim that democratic principles are better than fascist ones? Or perhaps you view Nazi Germany and Idi Amin's government as differences of opinion about how to run a country, rather than inherently evil regimes. How can you claim that an intentional lie is worse than telling what you believe to be the truth? If a person does not get caught, is theft justifiable as a difference of opinion about ownership? If there is no truth, what makes you think you own anything, including your life and liberty? Do you believe in obeying laws merely bacause of the consequences of getting caught breaking them? After all, the laws we pass are merely opinons about what is right.
I have suggested some absurd conclusions due to a failure to consider truth more than a relative notion. Note, however, that more subtle absurdities are bound to creep into your belief system if you accept the notion that nothing is provable and truth is a relative notion.
I take your point.
I think that if one accepts the relativity of truth, it doesn't necessarilly imply that any idea or notion is as equally valid as any other. At the end of the day, I live my life with certain rules and trust that certain things will happen as I have learned to expect them.
The only thing the notion of relative truth forces us to accept is that we can't make any 100% absolute judgements about things. We have to accept a particular level of uncertainty about what we know (even though this level may be negligable for most practical purposes).
I believe a notion of an 'objective' underlying truth as something that is out there and we are merely striving to understand is a way of simplifying the complexity of reality.
My view of truth as a relative notion is not merely an excuse not to be disciminating or a reason to have no morals. I have a strong idea about what is right and wrong but I don't necessarily believe that this should apply to everyone else.
The main problem is that the word 'Truth' itself is not finitely describable so anyone who claims to have an understanding of what truth is are making an intuitive leap to grasp it.
Sometimes, those who claim to know the truth (and believe it is absolute) feel they are therefore justified in imposing it on others. If the Nazi's had believed in the subjectivity of truth, would there have been a world war two?
Thanks for indulging me in this discussion (I know we have strayed far from the path) as you have made clear that this is a 'Sore' subject for you.
Just tell me this, how does the notion of absolute truth rest with quantum mechanic's uncertainty principle? I'm sure you'll be familiar with it but here's a quick reminder: You cannot know both the position and momentum of a sub atomic particle. The more accurately you know one, the less certainty you know about the other.
actually absolutely absolute.
true is a valid statement, and that leds to the next qwestion, are we capable of validating something? I think the relativeness of our measurements is what fails us.
On the other hand i can't find a definition on truth, it's a valid statement and a valid statement is true, so i'm going in circles. Logic to me is a mathematical language and probably just common way to describe our environment, so reality remains clouded
The definition of truth that I have read about (Infinity and the Mind) is something like this:
"This statement is true."
is defined as:
" 'This statement is true', is true"
which is defined as:
" ' 'This statment is true', is true', is true"
And so on (to infinity). In fact you can substitute 'This Statement' with any sentance and the same problem arises.
I don't know how anyone can insist (except a fundamentalist) that there is any such thing as absolute truth when it comes to moral, value and taste judgements.
I think that when most people talk of an absolute truth they are refering to the physical world (Which we can usually verify one way or the other). Intuition leads us strongly to believe that something exists wheather or not we are around to observe it.
The only problem is that the physical world is made from sub-atomic particals that do not have any definite underlying state until they are measured. Measuring them forces them to take on a definite reality but we cannot say that it therefore had a definite reality before measurement (or with any certainty what it will do next).
The overiding problem is this:
Even if there is an absolute truth, we (with our finite minds) can never know it in it's entirety or ever be sure (without any degree of uncertainty) how close we actually are to the truth.
So that is the real issue. Not if there is an 'Underlying' objective and absolute reality or not, but if we are capable of knowing it (without any uncertainty attall).
If you're wrong and truth is relative, then the study of logic and formal systems is still a worthwhile pursute.
Formal systems are just imaginary constructs that we use as a tool to solve problems. If used well they can be very useful but any formal system is only as valid as the assumptions they are based on and we can never be 100% certain of these (or else they wouldn't be assumptions).
Still, we trust in the systematic process of deduction that we call logic to asses the implications of our assumptions being true (i.e. what else must also be true). This tends to work very well most of the time but even this process may fail (if you accept Godel's Theorem).
The essence of what I'm saying is that logic is a worthwhile pursuit in a universe of relative or objective truth.
You said 'Probabilities are a way to mix logics and unknown elements'.
What you say is true in classical physics but not in quantum mechanics. These 'unknown elements' are unknown because there is nothing to be known. If an electron has a 50/50 chance of going one way or the other, it is truly random because there is no determanistic way to infer which way it will travel.
Unknown elements are elements that are not assumed neither conclusions, which makes them unknown. There's no "known" elements. Everything is therefore also truly random and nothing is deterministic.
But if i'm right, a reality exists, logics have a meaning and truth is reality, otherways, there's absolutely no connection.
I don't see the point in relative truth, that's some kind of decision you make because you don't know, if i knew, i would call it truth but since i don't, i call it unknown, not relative truth. If you split up an enough complex boolean function you will find that the result is dependent on each variable, "unknown", but if you don't, you could draw a conclusion "it's random", and "it has a probability of 935 of 1 to return true." It's still not relative truth.
I am not basing my belief on relative truth on the fact that I cannot know everything. I base it on the notion that two people could experience two conflicting but equally valid truths.
The point is with your complex boolean funciton is that you could (if you had the inclination) find out all the unknown variables.
The point of Quantum mechanics is that these 'unknown' variables do not have a definitive state until you measure them. The act of measurement gives them a definite state but then the collective behaviour of the system changes.
The fact that measuring these 'unknowns' changes the behaviour of the system implies that these variables have no actual value until you measure them.
Infact, two identical experiments could be set up by two different scientists. Each tested in the same way and two different but equally valid results might emerge. Hence the relativity of truth.
You might think it unfair of me to base my argument on quantum mechanics (and perhaps it is) but at least it gives a "real world" example of the non-objectivity of truth.
It's your true world, i've seen how you use these words now and i think you assume you have the right to measure reality. Actually giving everyone/anything the right to not to assume but measure reality.
Two identical experiments has to be the same experiment to be identical, since otherways we have other circumstances. I don't think you should just say "infact..." and ignoring all parameters you don't even know how to measure. If you knew, you would proove determinism and fail, so stating facts would contradict you.
My boolean function won't give any result until all variables have been given value, assigning them by assuming would not proove determinism, but would not have any connection with reality, but if i have the inclination i proove determinism by knowing the result, since i don't, i proove indeterminism. Time will still tell you the result, by truely assigning the unknowns according to reality, in both your and my case. This would not be possible if a variable couldn't be calculated. It would flaw logics and turn reality to be meaningless.
Maybe i'm not making sense, but i should. If you don't then you should ask because i have this feeling people never understand my words, i'm bad at expressing myself and if something is missunderstood, the whole thing could be.
OK, I'll admit; Sometimes I struggle to understand what you are trying to say but I believe I have the gist of it (I may be wrong of course).
I'm not to sure about your first paragraph in the last post: "It's your true world, i've seen how you use these words now and i think you assume you have the right to measure reality. Actually giving everyone/anything the right to not to assume but measure reality. "
With regard to the next two paragraphs,
You stated :"Time will still tell you the result, by truely assigning the unknowns according to reality, in both your and my case. "
The above statement seems to me to indicate that you do not have an understanding of quantum principles. If you did, you would not say that you can assign to the unknowns according to reality because the only reality they 'have' is the subjective reality one assigns to them in the act of measurement.
The behavior of subatomic particles do not conform to what we know as classical mechanics. In classical mechanics, Objects have an underlying reality independant of what we might know about them. I stress again that you cannot apply this chain of thought to quantum particles.
So I stand by my statement that two identical experiments (of quantum particles) could yield different results. This is because the probabilities involved represent an underlying randomness that doesn't exist in the classical mechanic's view of the world.
I have to admit too, i do not understand Quantum mechanics at all, and i can't see the connections between subjective realities, it there are such. I slill can't say should i believe in any reality, but i hate the way they assume they can measure reality, so i have to be brainwashed before i can understand.
If Quantum mechanics explains the behavior of subatomic particles, but substitues objective reality with nothing and tell me my measurements is a subjective reality, i don't believe i ever could. What i meant with my first paragraph is just this, that you assume you (or anyone) can measures reality, or redefine reality.
I don't think you can compare two subjective realities, and apply any logics, for contradiction will flaw it.
If Classical physics does not explain something that i measure, i won't care. I'm not relying on it anyway.
If I can explain that an objective reality exists, i'd prefer myself living in one even if i have no idea how it looks like.
After all i'm happy not knowing anything at all.
I understand you now.
I didn't mean to give the impression that quantum scientists measure reality. They don't. They measure properties of sub-atomic particles (in the same way a classical scientist might measure the properties of a macro object).
The difference is that where a classical scientist takes measurements to get a clearer understanding of the 'objective' underlying state (reality) of an entity, In quantum mechanics, a subjective reality emerges from the process of interaction between the object being measured and the measurer.
I know this is hard to accept (and I must admit that I myself find it difficult to grasp) but this really is the weird nature of the quantum universe.
To be honest, I don't think there's any harm in believing reality to be objective, as you also accept that you can never really know what it is. It's those who believe in an objective reality and think they know what it is (Better than everyone else) who are dangerous.
I just can't see how anybody can reconcile the notion of an objective reality with quantum mechanics.
There's certainly something missing in the picture, Quantum mechanics wouldn't explain why we have these measurements at all and QM wouldn't explain why you are alone when you measure you're not.
Thinking about what you said, if QM might just be another way to look at the reality, not that we take it as truth, while truth remains unknown and absolute.
I'm not sure what you mean but I would recommend further reading on the subject (it is very interesting).
At the end of the day, I must admit that I probably do believe in a kind of underlying reality. That is a reality that it meaningless until someone gives it meaning by interacting (observing, measuring etc.) with it. Generally it leas us to 'share' reality with others as we tend to perceive things in a similar way.
As for as Godel's theorem, I don't think you can choose between the inconsistancy and incompleteness of truth; You have to accept both. i.e. There are some things that are true that we can never prove and there are some things that we can prove but contradict other truths. The only way one can accept contradictions in truth is to allow for (at least a degree of) subjectivity.
Strange.
Well, I admit that I'm not as consise with my terminology as I should be and I know I have strayed far from the original point of the thread but, I still stand by my point that we should accept the possibility of contradictory truths.
Why does this have to kick the doors open and lead to the notion that if one 'truth' can be proved true that contradicts andother 'proven' truth, then all versions of truth are equally valid (or invalid). I don't believe it does. It just means that things aren't as black and white as they we would like them to be.
I did read something interesting that drew a distinction between 'weak' objectivity and 'strong' objectivity.
'Weak' objectivity is the belief that any particular scientific experiment can be performed by anybody at any time (as long as they follow the same rules) to achieve the same set of results.
If you believe that these results will then go on to illuminate a consistant and objective truth then this is called 'strong' objectivity. If they don't, it is merely 'weak' objectivity.
The author who made the above distinction said that only a belief in weak objectivity is required for a scientist to do his job. Furthermore, quantum mechanics seems to go against the notion of strong objectivity.
Anyway, as you say, This should really be in another thread (perhaps even in another forum) so I'll let the matter rest (unless a new thread is started up).