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.
Clayton -
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.
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.
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 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?
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.
Every single mathematician I know will tell you what you’re saying is wrong.
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.
In fact, I posted this question:
Now let’s just see what them mathematicians say.
So? You challenged me to demonstrate that mathematical induction is a valid deductive principle (despite its confusing name) and the link gives a simple, universally accepted example of the use of mathematical induction to prove an infinite number of true facts from only a base case and an induction step. Then you shift the goal-posts. Nice.
As for 0.999…, please explain where this proof goes wrong:
0.9999… × 10 = 9.999999…
0.9999… × (9+1) = 9.999999…
0.9999… × 9 + 0.99999… × 1 = 9.99999…
0.9999… × 9 = 9.9999…-0.999999 = 9
0.9999… × 9 = 9
0.9999… = 1
*sigh (bracing self)
It is true that once you re-arrange the terms or decompose an infinite sum, it is no longer algebraically identical to its numerical summand. Treating the sum this way leads to all sorts of absurdities. But for naive algebraic manipulation, it is never wrong to treat the infinite sum as algebraically identical to its summand so long as you wrap it in parentheses and do not rearrange its terms in any way. If you want to move to more sophisticated manipulations, then you need to observe the rules of convergence, which analysis has helped us discover.
But to say that a convergent infinite sum that has not been decomposed or rearranged in any way is not identically equal to its numerical summand is ridiculous. No serious mathematician except perhaps a few finitists would ever say such a thing. Contrary to popular belief, analysis does not asterisk the equality between the infinite sum and its summand, it simply analyzes the behavior of the partial sums.
And I did not misquote Hardy - I intentionally inserted the “also” for emphasis. It is you, not me, who is misreading him.
Clayton -
Yes.
Clayton -
You challenged me to demonstrate that mathematical induction is a valid deductive principle…
No. I challenged you to use it to prove 1/2 + 1/4 …= 1. I know about math induction, have used it many times.
…please explain where this proof goes wrong:
It goes wrong in the very first line, because two of the symbols are undefined [i.e. 0.9999… and 9.9999…], unless you go with all mathematicians everywhere and define them as least upper bounds. Then the whole thing becomes a proof about least upper bounds, with the final result being that the l.u.b. of .9999… is 1.
But to say that a convergent infinite sum that has not been decomposed or rearranged in any way is not identically equal to its numerical summand is ridiculous.
I agree, with the proviso [that you deny] that a “convergent infinite sum” means, by definition, a least upper bound. But no more than that. Corrolary: The sqaure 1x1 has not been proven anywhere that it will be filled by dropping in squares of 1/2 , 1/4, etc.
No serious mathematician except perhaps a few finitists would ever say such a thing.
So Hardy wasn’t a serious mathematician. And every mathematician on the subject since the 19th century is also not serious. Clayton, time to put up or shut up. Show me one, just one, book, website, link, something, that says what you do.
Contrary to popular belief, analysis does not asterisk the equality between the infinite sum and its summand, it simply analyzes the behavior of the partial sums.
Contrary to popular belief [as exhibited in this thread], analysis does not define an infinite sum in any other way but as a least upper bound. There is no other infinite sum. The infinite sum is distinct from its summand, because the infinite sum is the limit of the summand [which is an infinite sequence].
The mistake lies in an ignorance of the meaning of a summation sign that has an infinite index. Whenever you see one, it means only one thing, by definition. It means the limit of a certain sequence.
And I did not misquote Hardy - I intentionally inserted the “also”…
Oh. OK. You mean you did not realize that inserting that innocent looking word negates everything he actually wrote.
It is you, not me, who is misreading him.
OK then. I leave it to whoever is reading this to go straight to the master [=Hardy], sit at his feet [=read Chapter 4 and beginning of Chapter 8] and see what he says.
We are done here, Clayton. If we disagree on the meaning of what Hardy actually said, there is no room for further discussion.