0

= 0 + 0 + 0 + . . .

= (1 - 1) + (1 - 1) + (1 - 1) + . . .

= 1 - 1 + 1 - 1 + 1 - 1 + . . .

= 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + . . .

= 1 + 0 + 0 + 0 + . . . .

= 1

So we have begun with zero and built one out of it. As he said: "something has been created out of nothing."

Regardless of what you think of the conclusion he came to, is there a problem with the "proof"

**mathematically**? If so, where is the problem?

## 18 comments:

Yes, there's a problem with the proof! He conveniently leaves off one of the -1s from a par of 1 - 1s. His final answer should look something more like 1 + 0 + 0 + 0 - 1 = 0.

David

Hmm . . . interesting . . .

Since the 3 dots at the end of each line mean "continue this way forever," are you saying infinity is an even number and when you get to the END of it there will be a -1 left as the LAST number in the infinite list?

Thanks for your comment and valient attempt, but this isn't a problem.

Other thoughts on this? Or do we let the proof stand?

You are way beyond my mathematical abilities!

Oh, Tony, I don't believe that for a minute!

This mathematical reasoning IS quite a conundrum, though!

It's hard to argue with. After all, 0 IS equal to a bunch of zeros added together, and 1-1 does equal zero, so we can substitute it, and so on.

And yet . . .

. . . are we really ready to swallow that 0=1?

But we MUST if we can't find a flaw in the reasoning!

Hmm . . . what to do?

If you don't believe that, then you don't know me as well as you think!

I don't know if you remember, but I asked you about this a couple of years ago and used it as an introduction for a sermon. I believe it was on the Trinity talking about God's math.

Okay. This is getting philosophical, and I grew up with math as something very logical and ... safe. The proof of Ubaldi certainly made my hair bristle. All of me felt there was something wrong here. What is it?

I may be in over my head (!), but I have to try my protest out. Here goes:

As you say, the dots in line 2 signify infinite continuation. And how knows what’s at the end of that, since apparently there’s no end. I don’t remember if the Learned think infinity goes in a circle or just runs ever on and on. I believe the last. And here’s my hope: If we don’t know what’s at the end or rather know there’s no end, then how can we suddenly (in line 5) place a “1+” at the beginning where before there was a 0, or (1-1)=0? Such a +1 might very well be at the end, but no mere human can see or know that end.

If, as I feel, there is a sort of chronology to infinity, suddenly placing something in the beginning that actually comes after(maybe only at the very end) seems … problematic. Are we to take the following line as truth then:

Guesswork = proof

?

COMMENTS (WITH EXPLANATION TO FOLLOW):

Thanks for chiming in! I love your comments, Nethe - and I appreciate the feelings you share about math. Thank you for diving in even if you felt you were getting in over your head. I respect your bravery!

First I'll address some of your comments, and then I'll explain what's up.

Oh, and, by the way, it shows good intuition that your hair would bristle over this. Clearly 1 and 0 cannot be equal; otherwise everything in math falls apart.

But how do we show Ubaldi's reasoning is wrong? It is not easily dismissed, and in his time the means for debunking it did not yet exist, so HE himself was "in over his head" as you put it.

As to the 1 at the front or replacing 0 by 1 - 1, well, 1 - 1 IS zero, and math does legally allow me to substitute something that is of equal value. So, writing 0 as 1 - 1 is fully legal, always.

I have changed the notation in going from line 4 to line 5, but all the things that were there already are still there, and nothing new has been added. I've rewritten "minus" as "plus the opposite," which is also perfectly legal. For instance 5 - 2 is the same as 5 + -2, just as 1 - 1 is the same as 1 + -1, so that change is always OK.

What I did then is to move the parentheses down one space to the right, but the one at the front is the same one that's already there also in lines 3 and 4, so I haven't added it as a new element.

EXPLANATION:

There IS a problem, and it has to do with using the parentheses to regroup.

We actually do this all the time, though, with no problem (in other circumstances).

For instance:

(1 + 2) + 3

can be rewritten as

1 + (2 + 3)

The regrouping DOES make a difference, because we have to "do" what's in parentheses first, right? So the first statement above becomes 3+3 which is 6, and the second one becomes 1+5 (different statement), which is also 6. We just get two different routes to get to the same thing.

That example was with a series that has an end, but we can even do this with an infinite series. For instance:

(1+1/2)+(1/4+1/8)+(1/16+1/32)+...

is the same as

1+(1/2+1/4)+(1/8+1/16)+...

where the dots mean to continue in this way forever.

So then why do we get such a "hair bristling" result with Ubaldi's "proof?"

It has to do with the fact that if you were to sit around for eternity doing this addition problem it would never converge to one answer. Let's just start and see what happens:

step 1: 1=1

step 2: 1-1=0

step 3: 1-1+1=1

step 4: 1-1+1-1=0

step 5: 1-1+1-1+1=1

We really don't have to go any further. As you can see the sum of the whole string is going to go back and forth between 0 and 1.

This is where the problem comes in, and it is rather subtle. As I said, the means to debunk his "proof" did not exist in Ubaldi's time.

In both other examples I used:

1+2+3

and

1 + 1/2 + 1/4 + 1/8 + 1/16 +...

the series converge to one single answer - in the first case 6, and in the second case 2.

Ubaldi's series does not converge, so it does not necessarily give the same sum when regrouped.

(I'm being casual in my explanation, but the proof involved is due to Riemann who lived from 1826 to 1866 - about a century after Ubaldi).

END OF EXPLANATION

Thanks to David, Tony and Nethe for entering into discussion of this - you brave souls! And kudos to others who have thought about it but didn't comment.

If you want to give your brain some good, fun exercise, thinking about the infinite is certainly a good place to do it.

For a gentle introduction, I recommend chapter 9 (entitled "The Abyss") of Robert and Ellen Kaplan's book "The Art of the Infinite." For deeper exploration of the infinite, I recommend "To Infinity and Beyond: A Cultural History of the Infinite" by Eli Maor.

Seems a late addition, but try this on for size: if we are allowed to REARRANGE as well as regroup, we can make this sum equal any number you want!

For example, suppose we want to show that 0 = 5.

Start the same way:

0 = 0 + 0 + 0 + 0 + ...

0 = (1-1) + (1-1 ) + (1-1) + (1-1) + ...

0 = 1 -1 +1 -1 +1 -1 +...

now the fun part - grab FIVE of the positive 1's and move them to the front. You'd think there would be some "extra" negative 1's out at the back end, but NO - there are an infinite number of both positive and negative 1's to play with, so we can always pair up the rest of them! So the sum becomes

0 = 1 + 1 + 1 + 1 + 1 + (1-1) + (1-1) + (1-1) + ...

i.e. 0 = 5 + 0 + 0 + 0 + ...

So 0 = 5

If you want to make the sum equal to negative eleven, just put eleven minus 1's on the front and then pair up the rest!

Good job on the explanation Heidi!

Brian

Cool! I hadn't thought about that in this context, but I should have. I love it!! (I'd seem the sum here given as 1, 0, and 1/2 in the past in cute little puzzles I'd come across having to do with the infinite.)

Here is something for you to chew on:

One question on the Real 2 final was "Prove or give a counterexample: EVERY possible rearrangement of a conditionally convergent series converges."

Was not Cantor spiritually motivated to play infinite games like this?

I know there was a spiritual aspect to his work and that at times (most of the time?) he was strongly supported by Catholic clergy - much more so than other mathematicians, especially his former mentor Leopold Kronecker who famously said, "God created the integers. All the rest is the work of man." Karl Gauss also protested vehemently against such a thing as actual infinity (as opposed to potential infinity). Definitions of actual and potential infinity go back at least as far as Aristotle, and struggles with the concept of infinity go back at least as far as Zeno of Elea. Most mathematicians of the time (and some today) did not take well to Cantor's actual infinity, but mathematician David Hilbert called Cantor's work with the actual infinite a "paradise" from which no one would drive us out.

So, I believe Cantor's work was initially and remained mainly a mathematical pursuit, but that it also touched on the spiritual for him, that he found solace in communications with priests and that he personally felt there was a spiritual aspect to it.

Whatever its impetus, I agree with Hilbert that it is a paradise, and I see Cantor as having one of the greatest mathematical minds of all time - to have been able to make this leap to actual infinity that mathematicians and philosophers before him had dismissed.

Hmmm I find it interesting that the Catholic Church would have an interest in Cantor. From my understanding he was Jewish and felt his mathematics was some sort of mystical connection to Judaism as we see that his symbols are Hebrew (aleph).

Thanks

Kaz

Given his very interesting work and life, there are many intriguing stories about Cantor.

In terms of ancestory (though not in practice), Cantor is most likely Jewish. There is implication in some of the correspondence of his mother that she is of Jewish ancestry, but she was baptized Roman Catholic and converted to Protestantism upon her marriage.

Cantor does (once in writing) refer to his paternal grandparents, who fled to Russia during the time of the Napoleanic Wars as "israelitische," but there is no evidence that they were ever practicing Jews. Cantor's father was educated in a Lutheran School.

It seems that the Jewish ancestory of his father and of his mother each rest upon one comment in one letter.

Mathematician and historian E. T. Bell portrayed the conflict between Cantor and Kronecker as a quarrel between two Jews, but, sadly, Bell's work is often romanticized and flawed.

According to the definitive biography of Cantor by Joseph Dauben, " . . . Cantor was not Jewish. He was born and baptized a Lutheran and was a devout Christian his entire life."

Kaz,

I appreciate your comments and have been exploring your blog: Mathematical Poetry. I am in the process of reading the four articles on the sidebar. I'm enjoying this very much and am quite intrigued.

I see mathematics AS poetry, and I am currently pursuring an interdisciplinary graduate degree titled Mathematics, Language and Cognitive Studies.

I have a passion for mathematics, for poetry, and for the art of Rene Magritte and am eagerly continuing to explore your work.

Very interesting thank you!

Kaz

Hi Heidi,

I am happy that you find my explorations intriguing. As you have probably noticed; I do not think of pure mathematics as poetry.

http://mathematicalpoetry.blogspot.com/2008/02/is-pure-mathematics-poetic_15.html

I think there is a lot of confusion among mathematicians about the aesthetics of math and how it relates to art. While I realize that many people are trying to find mathematical connections in the arts and blending the boundaries I think that too many people have not taken the time to sit down and figure out just what it is that they are blending. Before you can put two ideas together you must first understand both ideas. My desire is that the aesthetic field of mathematics defines itself and breaks away from all this art stuff and become an end unto itself. Mathematics does not need art nor does it need to call itself art. There is university Art curriculums set up to offer PhDs in the field of aesthetics. When will university mathematics departments do the same? When will there become museums devoted strictly to the maths? I know that mathematicians are serious about their work but are they serious about defining a critical path that makes aesthetic judgments on which mathematical concepts are more aesthetically pleasing. I look at the myriad of people generating fractals and ask … what make one fractal better than the other? Colors? Composition? Are these people that generate fractals really artists? Is the aesthetic judgment reserved for the button pushers who generate fractals or should it be delivered to the person that wrote the software? I look at the confusion trying to define an artistic aesthetic and witness someone like George David Birkhoff try to apply the mathematical esthetic to the field of art a fail miserably. I see entirely too much confusion here and I welcome anyone who wants to help sort it out. I would love to hear your ideas on the matter.

Thanks

Kaz

I will need to read more of your writing and ponder this further before giving anything like a lucid answer. I have not taken this to the depth that you have, but I will share my sense - and also share a bit of my background.

Poetry is my deepest passion. I have no classroom training in poetry. I discovered it on my own, so I cannot speak of it in a scholarly fashion. Discovering a new poet whose work I appreciate is more exciting to me than getting my most desired Christmas gift. Among my favorite poets are Cummings, Roethke, Auden, Levertov, Dickinson, Millay, Plath, Nemerov, Eliot and Spanish poets Neruda and Lorca.

I tend to like short lyric verse. I like the slant rhyme of Dickinson, and I love the word-play of Cummings. I've been to some poetry slams recently and find I like those too!

My training is in mathematics.

When you stated in your last comment: "When will there become museums devoted strictly to the maths?" You took the words right out of my mouth. I am currently writing a book on math and poetry - for the general public - and that sentence is almost word for word what I have in my introduction.

I do not see the poetry of mathematics as being in the beauty of fractals. I do not see the mathematical nature of poetry as being in the metric.

I know what it is to write mathematical proof. I know what it is to write poetry (although I do not do it well). Each activity feels the same to me. I have not tried to "explain" poetry mathematically (as Poe did - perhaps disengenuously - of The Raven). I do not "judge" the beauty of mathematics poetically. I have simply found that engaging in these activities FEELS the same to me.

With each I have a message that I am trying to share (or even articulate or find for myself), and in that sharing I am trying to do so with brief, powerful language in an elegant way.

There are many philosophies of mathematics. I am a Platonist (most of my colleagues are horrified by this! It's not in vogue, but too bad.). I believe in the actual infinite. I believe in the Platonic forms. I subscribe to "beauty is truth, truth beauty." (Although I do not see that as an absolute.)

In mathematics and in science if something holds together elegantly, it FEELS true - and often (although not always) proves to be so. Poetry that is aesthetically pleasing (to me) is also poetry that I find compelling, that I find speaks to me - that I find to be "true."

I'm afraid that's all rather vague. I'm throwing those thoughts out quickly at the end of a long day.

It is said that aspiring poets would approach W. H. Auden and ask him for advice. He would then ask them why they wanted to be poets. If the answer was, "Because I have so much I want to say." He would tell them to forget about poetry. If the answer was, "Because I just really like to play with words; I like things like crossword puzzles and other word games." He would encourage them to become poets.

To me, pure mathematics is the same thing. One of my favorite fields is number theory. I'm a bit like G. H. Hardy who loved the fact that number theory was not applicable (well, of course now it is . . .). For me math is about exploring and playing; poetry to some degree is also about exploring and playing (for me) - yet both have a depth to them that grabs my guts.

Sorry that's so rough - just the thoughts off the top of my head.

In terms of the museum, I would like to create that. I teach at the Junior College level, and I often work with students who are at a basic skill level (taking beginning algebra courses). All they have ever seen of math is number crunching and finding x.

I can show them pretty pictures of fractals, but they don't understand how that is math. As much as possible I try to broaden their understanding of what math is. I do use fractals, but I also bring in graph theory (Konigsberg Bridge problem), knot theory, the 4-color map theorem and so on.

I can't draw. I am pathetic as an artist, but I can go to an art museum and enjoy what I see. I am not a professional athelete, but I can go to a pro-sports hall of fame and appreciate what is there. I am not a musician, but I can go to a concert of any kind and enjoy the performance at some level. This seems to be true of nearly every field - but not math. People who struggle with math have no museum they can go to in order to appreciate or enjoy it. My goal is to, in some way, create that.

What is always interesting is how people from diverse fields flow into this basin where the waters of math and the arts mingle. You and I are very different yet we have common ground. I am not a Platonist because I haven’t experienced anything to make me such. I look at x^2 + y^2 =r^2 and visualize a crystal clear image that I know cannot exist as an object. I make a living as an engineer in the aerospace industry and deal daily with the problems between our mathematical models and the physical parts of an aircraft. I have had conversations with a linguist who lived in Papua New Guinea and tells me they had no mathematics for they could not even count. I have never experienced any part of math that I could claim as physical. And on the other hand I yield my mathematical consciousness to the likes of Gödel and other Platonists and wonder what it they are experiencing is. My formal training is in visual art with an emphasis in visual poetics and an interest in art theory. Magritte seems to be where you and I meet. What excites me is learning something new in mathematics and finding a way to use it to express poetics. What excites me in mathematics might possibly bore you. On another note, in the arts there are those that believe that art should not say anything and it should function as more like decoration. I find the decorative approach interesting for discovering the language of art but I find art that begs a question to be more interesting. You mention Auden yet others may mention the didacticism of a Pope or Pound … bottom line is there are so many aesthetic paths to enjoy.

As far as the Math Museum goes; there are those working on it and my bet is that Reza Sarhangi will accomplish it first. See the link in my blog sidebar for “Bridges connections in Art Music and Science”

I feel the most interesting thing you said was, “I do not see the poetry of mathematics as being in the beauty of fractals. I do not see the mathematical nature of poetry as being in the metric.” I would love to read an essay that delves into that statement.

Thanks for your conversation!

Kaz

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