What is the statement to be proven by induction? Prove it is true for n=1, then prove that if it true for n, it is also true for n+1. You will find this impossible to do.
Seriously?
I suppose that "all math textbooks and all mathematicians ever" prove me wrong.
Clayton -
SD,
Do you have any experience with math outside of your beloved book by Hardy?
I've actually been long intrigued by why some people have such strong objections to the use of the infinite as a distinct object or property of mathematical objects but have no objections to using mathematical objects which are stealth-infinite (e.g. pi, square-root-of-2, geometric diagonals, pretty much any real number).
I think that it goes back to the role of language in communicating abstractions. Steven Pinker points out in several of his online lectures on this subject that it doesn't make any sense to say that someone is buried "under ground" when they are not actually under the ground but in it. This highlights the hardwired geometrical abstractions in the human brain that treat "the ground" as either an extensive 3D substance or as a 2D surface.
I think that for some people, infinity violates their internal abstractions - it just doesn't make any sense to them. But the same was once true of zero, negative numbers and complex numbers, all of which are completely "fictitious" yet mathematicians casually operate on them all the time. Infinity is like zero in this regard - its something we can represent but to which we cannot point to any physical example. But we cannot point to a physical example of a geometric cube or the Mandelbrot set, either. So what? As long as the abstractions work inside our heads, that's all that matters.
The point is that analysis is the wrong tool for investigating methodological questions such as whether it is appropriate to treat infinity as a proper mathematical object, whether it is appropriate to use mathematical induction (which, by the way, analysis does do), and so on.
What is the right tool, in your opinion? What backup do you have for your opinion?
are you telling me that no math equation that uses pi expresses anything more than some kind of least-upper-bound?
Are you shocked?
epi*i merely approaches -1
As fate would have it, there are three least upper bounds in that equation. Pi, e, and one more. Do you know which one? [Hint: it's on the left hand side of the equation, and it's not i].
But to be more precise, the equation you wrote asserts that the only accumalation point [for we are in the complex plane and the concept of greater than does not apply to imaginary numbers] of the sequence symbolized by the right hand side is -1. The left hand side is not a sequence because of the e, or because of the pi, [although these numbers are themselves defined as least upper bounds of sequences omitted in the equation], but because of the problem of defining exponentiation for irrational numbers, which is only resolved by resorting to least upper bounds. See Hardy about this.
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It's easy to refute an argument if you first misrepresent it. William Keizer
This is getting comical.
So what you're saying is that an infinite series can be less than say 1 and greater than 0, and so lies in the open interval.
The problem lies in the phrase "can be".
As a child, we all are taught, or figure out, what the concept of "+" is. We also figure out, or are taught, what a+b+c+..n is, as long as n is a finite number. Formal math will use a recursive definition. So that when we are introduced to a+b+..., with the dots representing an infinity of additions, we think we know what that means also.
But sadly, our intuition is way ahead of our rigorous math. A mathemetician realizes that there is no defintion of an infinite sum which is a generalization of a finite sum. There is no recursive defintion of it, or any other. He has to make one up. Great minds tackled this problem, and after much groping around blindly, finally agreed that the only possible definition of an infinite sum found to date is that of least upper bound.
Sure, the least upper bound is in there somewhere, and yes it is finite. But its only a least upper bound, nothing else. There is no "actual" sum.
Really, guys, I'm not saying anything revolutuonary here. It's standard math.
Clayton: I've actually been long intrigued by why some people have such strong objections to the use of the infinite as a distinct object or property of mathematical objects but have no objections to using mathematical objects which are stealth-infinite (e.g. pi, square-root-of-2, geometric diagonals, pretty much any real number). I think that for some people, infinity violates their internal abstractions - it just doesn't make any sense to them. But the same was once true of zero, negative numbers and complex numbers, all of which are completely "fictitious" yet mathematicians casually operate on them all the time. Infinity is like zero in this regard - its something we can represent but to which we cannot point to any physical example. But we cannot point to a physical example of a geometric cube or the Mandelbrot set, either. So what? As long as the abstractions work inside our heads, that's all that matters.
Infinity is not even a concept. You cannot comprehend what infinity is, just like you cannot comprehend anything else you have never experienced. Zero doesn't exist, infinity doesn't exist, and you are correct in assuming that it is merely the use of language that allows us to categorize such things. Paradoxes do not exist, neither do any numbers. We created them and we can do extraordinary things with them, but that doesn't mean they truly exist.
For instance, "The largest number the human brain can comprehend without counting or guessing is 4. Beyond that most people can identify 5 elements in a group by quickly counting them; everything beyond 5 can only be a guess, unless there is enough time for a count." http://www.es.flinders.edu.au/~mattom/science+society/lecture3.html Of course, I don't think this is accurate for everyone, but it brings up a point: people can't comprehend the actual amounts they deal with in numbers beyond very few. They can simply tell a lot or a little of things. Math is a brilliant invention, but the numbers in it are man made, not natural by any means.
That being said, this next statement might sound hypocritical given my stance of mathematical numbers. Infinity means something endless. If it doesn't end, it keeps going, and therefore it cannot have any sort of quantifiable value because it is constantly changing, which means its value is constantly changing. So you really cannot use infinity as an integer or number of any kind. It is certainly not an integer, and it cannot qualify as a number.
I don't know of a single mathematician in the world who doesn't accept .999..... = 1
I don't know of a single mathematician in the world who doesn't accept
.999..... = 1
I accept it, too. And we all accept it in the same way, as saying that the least upper bound of the LHS is the RHS.
No,
Every single mathematician I know will tell you what you're saying is wrong.
Clayton,
Much as I would love to visit 500 BC, what you write is irrelevant.
Have you ever studied the construction of the real numbers from the integers?
Really? Do you know that in math if
a <= b and b <=a (less than or equal to sign here), then a = b?
Seriously? I suppose that "all math textbooks and all mathematicians ever" prove me wrong.
That link is to finite sums, Clayton. We are talking about infinite sums.
C'mon Freidmanite, there are hundreds of math books available free online. Show me one. One.
As to whether i have ever read anything but Hardy, I chose Hardy because his book is available free online, it is a highly respected classic, as he is a highly respected mathematician, and because, yes, I enjoy the book. [If you want to have a good time, go to page one of the book and do the exercises that he says require no more than the laws of arithmatic to solve.] But Hardy is merely saying what every other book on the subject says.
Really? Do you know that in math if a <= b and b <=a (less than or equal to sign here), then a = b?
And your point is?
Here's a website where actual mathematics professors answer questions ranging from elementary to research-level topics. See if any of your assertions hold up. It's pretty quick too. You'll get an answer in the less than 5 minutes.
http://math.stackexchange.com/