What ramifications will my scientific breakthrough have?

Factual Questions
81

You appear to be stuck on this statement.

It would seem that just one sensible meaning can be given to “… then said result is unreachable”: that the result cannot be reached in a finite time. Is that what you mean?

If so, you need merely consider how it is known that the sum of certain infinite series of positive numbers is finite. For example, if each step takes half as long as the previous one all your conditions are met and yet when we sum the series we find that all the steps are completed in a finite time.

You thus have a specific, verifiable counterexample that disproves what you are claiming.

82

All right, but apart from the engineering, computing, economics, medicine, weather forecasting, and statistics…what has Calculus ever done for us?

People’s Front of Judea FTW!!

83

Splitters!

84

Glad to. Thanks for asking.

First, I thought to show that 1/2 + 1/4 + 1/8 + 1/16 … = 1

But I figured your answer would be something like “No, it gets very close to 1, but it doesn’t actually equal 1.”

Well, there must be a proof for it somewhere, but I can’t remember it, so instead I’ll use the “0.99999…=1”, cited by Commander Keen, because I do know the proof to that:

Step 1: 0.99999… = X. X is a number which might be 1, or it might be slightly less than 1. (Let’s call that difference “Z”, and X is exactly 1-Z.)

Step 2: Multiply both sides by 10, and you get this : 9.99999… = 10 X. 10 X might be exactly 10, or it might be slightly less than 10, in which case it is 10 - 10 Z.

Step 3: Subtract the first from the second, and you get 9 = 9 X. This (“9 X”) might be exactly 9, or it might be slightly less than 9, in which case it is 9 - 9 Z.

Step 4: Let’s look at step 3 more carefully:


 9.99999...
- .99999...
----------
 9.00000...

Step 5: This shows us that of the two possiblities in Step 3, the difference is indeed exactly 9. Alternatively, you can say that it is “9-9Z”, if Z is exactly zero.

Step 6: Divide both sides (of either step 3 or step 5) by 9, and we have proven that 0.99999… is indeed exactly 1, not any less. This is an infinite series with a finite value.

85

Ah! I remember now!


    1/2 + 1/4 + 1/8 + 1/16 ....  <?>  1    then double both sides

1 + 1/2 + 1/4 + 1/8 + 1/16 ....  <?>  2    then subtract the line 1 from line 2

1                                 =   1          QED!
86

OK, let me see if I can express it in words.

The original problem (problems, since a number of less famous paradoxes reduce to the same issue) was that the Greeks knew very well that an infinite number could not be counted. They knew, better than many today, that infinity did not mean “the largest number” but meant literally “endless.” Endless cannot be counted except in an endless amount of time.

If an endless amount couldn’t be counted, then it couldn’t be summed. Yet they were surrounded by sums that seemed endless, like 1/2 + 1/4 + 1/8 …, that in fact seemed to sum.

The Greeks couldn’t get past this conundrum. Those they passed their legacy to also got stuck.

When math started to unstick in the 17th century, people began to look at this question again to see if they had a clever answer for it. Infinitesimals was one proposed solution. What if you postulated a tininess, tinier than any number and so stringable together in no time? This was Newton’s route, and his version of the calculus indeed used infinitesimals to get the answers. Calculus provided you the answer for the area under a curve, to give one classic example. You sliced that area into infinitesimals, added them together, and came out with an answer.

It worked, but it was complicated and ugly. Mathematicians searched for a better approach. (Infinitesimals survive today, slightly changed, as differentials, in the differential calculus.)

Let’s go back to the original problem, they said. The arrow goes 1/2 the distance, and then 1/2 the remaining distance, and 1/2 of the next remainder, etc. What does that mean?

Well, after the second iteration, it’s gone 3/4 of the way. After the third, 7/8 of the way. After the fourth 15/16 of the way. After a very few iterations, the arrow is so close to the target that the difference is no longer measurable. For all practical purposes, therefore, the arrow has reached the target.

That’s not sufficient. We’re really dealing with filling in an idealized mathematical line of length one. 15/16, 31/32, 63/64, 127/128. There’s still a space left to be filled in. That’s essentially where the Greeks stopped.

But let’s not stop. Let’s keep going. How close do we in fact get? It was easy to show that you could get as close as you wanted. Closer than 10[sup]-10[/sup]? Sure. Closer than 10[sup]-10000000000000[/sup]? Sure. Name any finite number you could possibly come up with and it was easy to demonstrate that the sum more than matched it (or the difference was smaller). If - and this is the big step - if you can say that the line is filled in to a degree closer than any number you can possibly name, how can you say that your line is different in length than the line you had previously defined as 1?

They said, well, it’s not any different. If you can’t name a single number that describes that difference then there isn’t any difference.

Not only that, this process is exactly equivalent to the working out of the real number system. What’s the closest number to 1? Is it .99? No. Is it .999999? No. No matter how many 9’s you write down, there is always a larger number of 9’s that defines a number that is closer to 1. An infinite number of 9’s must therefore be exactly equal to 1. This is the problem that Commander Keen cited. (For which we have had a number of threads too large to name.) And of course it is exactly the same problem. The only distinguishing factor is that instead of adding 1/2 + 1/4 + 1/8 … we’re adding .9 + .09 + .009 … No wonder they give the same answer. They’re the same problem.

In today’s math this is part of the theory of limits and is written as an infinite sum. But it goes back to axioms developed by the Greeks. If two things - triangles, lines, whatever - cannot be distinguished from one another in any way, then they are identical to one another.

That an infinite number of items can add to a finite sum is non-intuitive, but nothing in modern math can function without acknowledging its existence. You can try to argue that you can’t do the actual step by step summing in real time, but completely valid mathematical shortcuts exist that do the work for you, get exactly the answers that satisfy everything we know of real problems, and work completely consistently with all other constituents of math. That’s essentially the definition of reality. If you insist on denying this, your answers will not - can not - match the answers we know that reality gives us. All that does is put you back with the Greeks, who were convinced that their theory was right even if it didn’t give the correct answers.

We don’t swing that way. Theory has to agree with practice. Mathematicians understand infinite sums as well as they understand anything in math. It all works, and works perfectly. If you choose to stand outside that perfection you are the one who is diminished. You’re invited in. It’s up to you.

87

Or, alternatively:

Let S[sub]n[/sub] = (1/2)+(1/4)+(1/8)+…+(1/2)[sup]n[/sup]

Then

(1/2) + (1/2)S[sub]n[/sub]
= (1/2) + (1/2)[(1/2)+(1/4)+(1/8)+…+(1/2)[sup]n[/sup]]
= (1/2) + [(1/4)+(1/8)+…+(1/2)[sup]n+1[/sup]]
= (1/2) + (1/4)+(1/8)+…+(1/2)[sup]n[/sup]+(1/2)[sup]n+1[/sup]]
= S[sub]n[/sub] + (1/2)[sup]n+1[/sup]

So

(1/2) + (1/2)*S[sub]n[/sub] = S[sub]n[/sub] + (1/2)[sup]n+1[/sup]

which yields

(1/2)*S[sub]n[/sub] = (1/2) - (1/2)[sup]n+1[/sup]

multiply each side by 2 for

S[sub]n[/sub] = 1 - (1/2)[sup]n[/sup]

You can test this for arbitrary values of n

n = 1, S[sub]n[/sub]=(1/2), 1 - (1/2)[sup]1[/sup] = 1/2

n = 4, S[sub]n[/sub]=(1/2)+(1/4)+(1/8)+(1/16) = (8/16)+(4/16)+(2/16)+(1/16) = (15/16), 1 - (1/2)[sup]4[/sup] = 1 - (1/16) = (15/16)

As n approaches infinity, (1/2)[sup]n[/sup] goes to zero, thus the limit of S[sub]n** is 1 - 0 = 1.

If you have not already been exposed to the concept, you will definitely want to check out the epsilon-delta definition of limits.

88

Epsilon-delta. Brrr. One day I will travel back in time and kill or at least seriously distract Cauchy while he is developing this stuff.

Straggler, the only thing you really need to understand is - all the things people told you here work. Predictions were made based from calculus, and observations matched this predictions. Forces pointed in the right directions, objects accelerated as they should, it all works out. While all this looks a little counterintuitive and might feel wrong, it really works.

89

straggler, I don’t mean this to be patronizing, but have you studied calculus? All of integral calculus, at least, can be described as “summing an infinite amount of things”, and it has an almost limitless number of applications in physics, engineering, etc.

There’s no shame in thinking “Hey, this business of infinities seems like a bunch hooey.” Lots of smart people had their doubts at one point, even after Newton and others had used calculus to solve lots of problems in physics. But then an area of math called “real analysis” was developed, which explains precisely what we really mean when we talk about infinities, and from which one can prove the results of calculus by means of logical deduction.

In short, when mathematicians and such talk about “infinity”, they really mean that they’re talking about things that hold true “no matter how big the numbers get.” We don’t have to worry about whether infinity “really exists” to say “X is true no matter how big the numbers get”.

For example:
Consider the area of a rectangle. I can split that rectangle into two equal pieces, and the area of the big rectangle will be the sum of the areas of the two smaller rectangles. Or I can split the big rectangle into three pieces, and the big rectangle’s area will be the sum of the three pieces. Or I can split it into 4 pieces, or 5 pieces, or N pieces for any integer N. And no matter how big N gets, the sum of the areas of all N pieces adds up to the area of the whole rectangle.

Look at that bolded statement at the end of the previous paragraph. That’s what I really mean if I say “split the rectangle into infinitely many pieces, and the sum of the areas of all infinity pieces is the area of the whole rectangle.” (If I were being more formal, I’d use the phrase “the limit as N goes to infinity”, but often people speak more casually.)

Whether this makes sense or not, here’s my basic point: It’s not dumb to be suspicious of infinity, and to wonder if notions of infinity are really rigorous. But lots of smart people have already wondered just that, and they’ve done the work to put these ideas on a rigorous footing. If you want to argue they’re wrong, it is first necessary to understand their work. That means putting in the time to study calculus and real analysis extensively. Maybe that will convince you that the experts are right. But if it doesn’t, at least you’ll understand the specifics of the arguments you’ll need to refute to prove them wrong.

This is a general truth: If you think you’ve discovered that the world’s experts on physics and math are all wrong about something, you have to at least understand the details of their arguments before you’ll have any hope of convincing them. At a minimum, that means reading some textbooks on the subject.

90

straggler: This instrumentalist point is actually exceedingly important (I wanted to raise it myself earlier but overlooked it, my thanks to in hiding for doing so) and it explains why your theory, even if we could be assured of its truth beyond all doubt, will not soon displace the edifice of physics and engineering already built with the scaffolding of the continuous space concept.

91

Is 0.9999… a real number, that we can treat in a multiplicative manner? 0.9999… looks unquantifiable to me, with no application in reality.

92

That’s it. I’m out of here.

Someone else can point him to all the threads we’ve done on this.

Tell me, why exactly did I waste all that time trying to answer the question of somebody who doesn’t want to learn?

93

Even if space isn’t infinitely divisible in practice, it has to be infinitely divisible conceptually - we could discuss fractions of Planck length, even though we can’t necessarily expect to whip out a micrometer and measure such a distance, or observe a tortoise moving that far.

So it doesn’t matter whether space is or is not physically divisible indefinitely - the breakdown of Xeno’s paradox must happen for some other reason.

94

(portion graped by me for emphasis)

You’re either quite terribly bad at using Google, or perhaps you have cast us as the mice in your magnificent Cat Game.

I just Googled space is not infinitely divisible and got nothing but relevant results.

What on earth could you possibly have typed into Google to have managed to completely avoid finding all relevant discussion on this topic?

95

When mathematicians write “0.9999…” they don’t really mean “Write a 0, a decimal point and an infinite number of nines.” That wouldn’t be a rigorous definition, because of course you can’t write an infinite number of nines.

Instead, “0.9999…” is a shorthand for "The limit of the sequence 0.9, 0.99, 0.999, 0.9999 . . . " We can be even more rigorous by saying “The n-th element of the sequence is given by 1 - 0.1[sup]n[/sup].”

Moreover, the word “limit” has a precise definition. In this case, the limit is a particular number X for which we can ensure that the value of the n-th element of the sequence is as close to X as we want just by making sure n is large enough. That is, if you give me any positive real number E and say “I want you to find me an element of the sequence that is no more than E away from X”, there is guaranteed to be some N where I can say “The n-th element of the sequence is certain to meet your demand so long as n is greater than N.” (That’s a lot of verbiage, but it can all be said much more concisely if I were using symbolic language.)

Here’s the key point: Whenever mathematicians talk about infinity, there is some precise definition of what they actually mean. It may seem like nonsense if you’re just trusting your intuitive understanding of infinity, but if you look at the actual formal definitions it’s all spelled out in a way that is completely legitimate and well-defined.

96

Short version:

Whenever mathematicians talk about “infinitely many”, “infinitely small”, “infinitely large”, etc., that’s a short hand for a much more precise definition.

These definitions refer to concepts that are very useful in real world calculations. They may not be what you think of as infinity, but they’re what mathematicians mean when they say infinity. If you want to argue that infinity is a load of bunk, you have to learn those definitions first (at least if you want to convince anyone).

97

Ditto.

98

Except before he gets there the physics grad student cum drummer from a Phish cover band walks through and sweeps her off to his fluorescent-lid, shag interior van, and the engineer and mathematician end up arguing about Goldbach’s Conjecture.

The o.p. needs to go take a calculus course and learn about convergent series. Augustin Cauchy had this one solved two centuries ago.

Stranger

99

Well, I for one have found straggler’s argument very convincing.

If I may paraphrase:

  1. An infinite number of non-zero steps must sum to infinity. And if anyone says otherwise, I’m not listening.

  2. Therefore space is granular / finitely divisible. This also makes sense because of course continuous space is just crazy talk.

  3. Therefore “scientific breakthrough”.

  4. Therefore God.

100

Not anything close to mainstream, but: