Showing posts with label math. Show all posts
Showing posts with label math. Show all posts

Sep 9, 2020

1 + 2 + 3 + ⋯ + ∞ = -1/12?

The Ramanujan Summation:
 1 + 2 + 3 + ⋯ + ∞ = -1/12?

Mark Dodds

“What on earth are you talking about? There’s no way that’s true!” — My mom

This is what my mom said to me when I told her about this little mathematical anomaly. And it is just that, an anomaly. After all, it defies basic logic. How could adding positive numbers equal not only a negative, but a negative fraction? What the frac?
Before I begin: It has been pointed out to me that when I talk about sum’s in this article, it is not in the traditional sense of the word. This is because all the series I deal with naturally do not tend to a specific number, so we talk about a different type of sums, namely Cesàro Summations. For anyone interested in the mathematics, Cesàro summations assign values to some infinite sums that do not converge in the usual sense. “The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series” — Wikipedia. I also want to say that throughout this article I deal with the concept of countable infinity, a different type of infinity that deals with a infinite set of numbers, but one where if given enough time you could count to any number in the set. It allows me to use some of the regular properties of mathematics like commutativity in my equations (which is an axiom I use throughout the article).

Srinivasa Ramanujan (1887–1920) was an Indian mathematician
For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12. Yup, -0.08333333333.

Don’t believe me? Keep reading to find out how I prove this, by proving two equally crazy claims:
    1.    1–1+1–1+1–1 ⋯ = 1/2
    2.    1–2+3–4+5–6⋯ = 1/4
First off, the bread and butter. This is where the real magic happens, in fact the other two proofs aren’t possible without this.
I start with a series, A, which is equal to 1–1+1–1+1–1 repeated an infinite number of times. I’ll write it as such:
A = 1–1+1–1+1–1⋯
Then I do a neat little trick. I take away A from 1
So far so good? Now here is where the wizardry happens. If I simplify the right side of the equation, I get something very peculiar:
Look familiar? In case you missed it, thats A. Yes, there on that right side of the equation, is the series we started off with. So I can substitute A for that right side, do a bit of high school algebra and boom!
1-A =A
1 = 2A
1/2 = A
This little beauty is Grandi’s series, called such after the Italian mathematician, philosopher, and priest Guido Grandi. That’s really everything this series has, and while it is my personal favourite, there isn’t a cool history or discovery story behind this. However, it does open the door to proving a lot of interesting things, including a very important equation for quantum mechanics and even string theory. But more on that later. For now, we move onto proving #2: 1–2+3–4+5–6⋯ = 1/4.
We start the same way as above, letting the series B =1–2+3–4+5–6⋯. Then we can start to play around with it. This time, instead of subtracting B from 1, we are going to subtract it from A. Mathematically, we get this:
A-B = (1–1+1–1+1–1⋯) — (1–2+3–4+5–6⋯)
A-B = (1–1+1–1+1–1⋯) — 1+2–3+4–5+6⋯
Then we shuffle the terms around a little bit, and we see another interesting pattern emerge.
A-B = (1–1) + (–1+2) +(1–3) + (–1+4) + (1–5) + (–1+6)⋯
A-B = 0+1–2+3–4+5⋯
Once again, we get the series we started off with, and from before, we know that A = 1/2, so we use some more basic algebra and prove our second mind blowing fact of today.
A-B = B
A = 2B
1/2 = 2B
1/4 = B
And voila! This equation does not have a fancy name, since it has proven by many mathematicians over the years while simultaneously being labeled a paradoxical equation. Nevertheless, it sparked a debate amongst academics at the time, and even helped extend Euler’s research in the Basel Problem and lead towards important mathematical functions like the Reimann Zeta function.
Now for the icing on the cake, the one you’ve been waiting for, the big cheese. Once again we start by letting the series C = 1+2+3+4+5+6⋯, and you may have been able to guess it, we are going to subtract C from B.
B-C = (1–2+3–4+5–6⋯)-(1+2+3+4+5+6⋯)
Because math is still awesome, we are going to rearrange the order of some of the numbers in here so we get something that looks familiar, but probably wont be what you are suspecting.
B-C = (1-2+3-4+5-6⋯)-1-2-3-4-5-6⋯
B-C = (1-1) + (-2-2) + (3-3) + (-4-4) + (5-5) + (-6-6) ⋯
B-C = 0-4+0-8+0-12⋯
B-C = -4-8-12⋯
Not what you were expecting right? Well hold on to your socks, because I have one last trick up my sleeve that is going to make it all worth it. If you notice, all the terms on the right side are multiples of -4, so we can pull out that constant factor, and lo n’ behold, we get what we started with.
B-C = -4(1+2+3)⋯
B-C = -4C
B = -3C
And since we have a value for B=1/4, we simply put that value in and we get our magical result:
1/4 = -3C
1/-12 = C or C = -1/12
Now, why this is important. Well for starters, it is used in string theory. Not the Stephen Hawking version unfortunately, but actually in the original version of string theory (called Bosonic String Theory). Now unfortunately Bosonic string theory has been somewhat outmoded by the current area of interest, called supersymmetric string theory, but the original theory still has its uses in understanding superstrings, which are integral parts of the aforementioned updated string theory.
The Ramanujan Summation also has had a big impact in the area of general physics, specifically in the solution to the phenomenon know as the Casimir Effect. Hendrik Casimir predicted that given two uncharged conductive plates placed in a vacuum, there exists an attractive force between these plates due to the presence of virtual particles bread by quantum fluctuations. In Casimir’s solution, he uses the very sum we just proved to model the amount of energy between the plates. And there is the reason why this value is so important.
So there you have it, the Ramanujan summation, that was discovered in the early 1900’s, which is still making an impact almost 100 years on in many different branches of physics, and can still win a bet against people who are none the wiser.

P.S. If you are still interested and want to read more, here is a conversation with two physicists trying to explain this crazy equation and their views on it’s usefulness and validity. It’s nice and short, and very interesting.

May 1, 2018

Ludwig Wittgenstein
Lewis Carroll

Math and Humor
Recall Ludwig Wittgenstein's remark that a serious work in philosophy could be written that consisted entirely of jokes. He meant, of course, that "getting" certain jokes is possible if, and only if, one understands the relevant philosophical point. Let us now examine some of this "philosophical humor." George Pitcher (1966) has demonstrated some very interesting similarities between the philosophical writings of Wittgenstein himself and the work of Lewis Carroll. Both were concerned with nonsense, logical confusion, and language, although, as Pitcher notes, Wittgenstein was tortured by these things whereas Carroll was (at least in his writings) delighted by them. Pitcher cites many passages in Alice in Wonderland and Through the Looking Glass as illustrating the type of joke Wittgenstein probably had in mind when he made the comment referred to above.
The following excerpts are representative of the many in Lewis Carroll that concern topics that Wittgenstein wrote about and that demonstrate a purposeful confusion of the logic of the situation.
1. She [Alicel ate a little bit, and said anxiously to herself, "Which way? Which way?" holding her hand on the top of her head to feel which way it was growing, and she was quite surprised to find that she remained the same size. [Alice in Wonderland, p.10]
2. "That is not said right," said the Caterpillar.
“Not quite right. I'm afraid," said Alice timidly. "Some of the words have got altered."
"It is wrong from beginning to end," said the Caterpillar decidedly, and there was silence for some minutes. [Alice in Wonderland, p.471
3. "Then you should say what you mean," the March Hare went on.
"I do," Alice hastily replied; "at least-at least I mean what I say-that's the same thing, you know." "Not the same thing a bit!" said the Hatter. "Why, you might just as well say that 'I see what I eat' is the same thing as 'I eat what I see'!" [Alice in Wonderland, pp. 68-69]
4. "Would you-be good enough," Alice panted out, after running a little further, "to stop a minute just to get one's breath again?" "I'm good enough," the King said, "only I'm not strong enough. You see, a minute goes by so fearfully quick. You might as well try to stop a Bandersnatch!" [Through the Looking Glass, pp. 242-4]
5. "It's very good jam," said the Queen.
"Well, I don't want any to-day, at any rate."
"You couldn't have it if you did want it," the Queen said. "The rule is jam to-morrow and jam yesterday but never jam to-day."
"It must come sometimes to 'jam to-day,'" Alice objected.
"No, it can't," said the Queen. "It's jam every other day; to-day isn't any other day, you know."
"I don't understand you," said Alice. "It's dreadfully confusing." [Through the Looking Glass, p.206]
What do these examples have in common? As noted, they all betray some confusion about the logic of certain notions. One does not lay one's hand on top of one's head to see if one is growing taller or shorter (unless only one's neck is growing). One cannot recite a poem incorrectly "from beginning to end," since then one cannot be said to be even reciting that poem. (Wittgenstein was very concerned with criteria for establishing identity and similarity.) In the third quotation the Mad Hatter is presupposing the total independence of meaning and saying, an assumption that Wittgenstein shows leads to much misunderstanding. The fourth passage confuses the grammar of the word time with that of a word like train, and the fifth illustrates that the word today, despite some similarities, does not function as a date. Both these latter points were also discussed by Wittgenstein.
Wittgenstein explains that "When words in our ordinary language have prima facie analogous grammars we are inclined to try to interpret them analogously; i.e. we try to make the analogy hold throughout." in this way we "misunderstand . . . the grammar of our expressions." These linguistic misunderstandings can be, as I have mentioned, either sources of delight or sources of torture depending on one's personality, mood, or intentions. Wittgenstein was concerned (tortured even) by the fact that a person does not talk about having a pain in his shoe even though he may have a pain in his foot and his foot is in his shoe. Carroll, had he thought of it, probably would have written of shoes so full of pain that they had to be hospitalized.
Open any book on analytic philosophy and you will find clarifying distinctions that, if utilized differently, could be the source of humor. The following pairs of phrases serve as examples of what I mean, "Going on to infinity" versus "going on to Milwaukee"; "honesty compels me" versus "my mother compels me"; "the present king of France is hairy" versus "the present president of the United States is hairy"; "an alleged murderer" versus "a vicious murderer"; "Have you stopped beating your wife?" versus "Have you voted for Kosnowski yet?" "before the world began" versus "before the game began." The first phrase in each case shares the same grammar as the second phrase, yet the logic (in a broad sense) of the two is quite different.
In fact, much of Wittgenstein and modern analytic philosophy in general has been concerned with unmisunderstanding (getting clear about) the logic and (surface) grammar of problematic terms (e.g., time, mind, rule, action, pain, reference) as well as with explicating and clarifying phrases such as the ones in the previous paragraph. Analytic philosophy can in a sense even be called linguistic therapy, and philosophers like Wittgenstein, Ryle, and Austin have devoted much effort and analysis to curing some of these linguistic diseases. Pitcher comments that Alice is a victim of the characters in her mad world of nonsense just as the philosopher is the victim of the nonsense he unknowingly utters. Wittgenstein (1956) writes, "The philosopher is the man who has to cure himself of many sicknesses of the understanding before he can
arrive at the notions of a sound human understanding. If in the midst of life we are in death, so in sanity we are surrounded by madness." In humor the anxiety induced by these misunderstandings as well as by more traditional philosophical concerns (God, death, choice) finds its release in laughter. (Compare Woody Allen and Kierkegaard, say, or the "humor" of Samuel Beckett.)

Apr 29, 2018

The algebra of Alice

Alice’s Adventures in Wonderland by Lewis Carroll continues to attract new readers ever since it was told to three sisters on a summer afternoon during a boat ride on the Thames. The apparently whimsical fairy tale charmed its listeners on its first telling but the story was expanded by Carroll into the Alice of today. On the 152nd anniversary of the classic’s publication on November 26, 1865, as a Christmas release in England, let’s consider the book as a mathematical puzzle.
Lewis Carroll in the preface to the work ‘All in the Golden Afternoon’, claimed to have invented the story on demand from Alice Liddell, and her two sisters, daughters of an Oxford don – Carroll himself taught mathematics at Oxford – during the boat ride. However, the profusion of mathematical puzzles, logical paradoxes and innuendoes throughout the body of the text tell a different story. While there is no doubt about the fact that it was created for, and to be told to children and young adults, what 21st century readers read today is a cleverly crafted tale to poke fun at the mathematics in Carroll’s time and its practitioners.
Carroll, a nom de plume of Charles Lutwidge Dodgson, a mathematics tutor at the Christ Church College in Oxford, was actually not a front-ranking mathematician. He swore by Elements, the famous geometry text by Euclid. Carroll waged a long battle with his peers who were revolutionising Victorian mathematics. Projective geometry, imaginary numbers, quaternion were turning the old-world of algebra and geometry upside down. Mathematics was no longer tied to the ground insofar as it was becoming more abstract, and logic that appealed to Carroll and his ilk could not be used to demystify the new avatar. Carroll was a Euclidean geometry orthodox who did throw the gauntlet at the new kids on the mathematics block but lost out. These were the times when Alice Liddell asked the young mathematics tutor to tell a story.

Close to a decade and a half later, in 1879, Carroll, under his real name, published Euclid and his Modern Rivals. Written in the form of a play, it was Carroll’s way of telling the world that Euclid’s Elements is the best textbook for teaching geometry. Carroll’s introduction lays out his purpose and why he went about it the way he did. His words on writing for a non-scientific audience still sound particularly relevant. “It is presented in a dramatic form,” writes Charles Dodgson in the introduction, “partly because it seemed a better way of exhibiting in alteration the arguments on the two sides of the question; partly that I feel myself at liberty to treat it in a rather lighter style than would have suited an essay, and thus to make it a little less tedious and little more acceptable to unscientific readers.” Not many now are even aware of this curious publication but this can be seen as an extension of Carroll’s thought process that started with Alice’s Adventures in Wonderland.

There is, however, no direct evidence that Carroll actually planned such a tale. Martin Gardner notes is his book, The Annotated Alice, the definitive edition, that Reverend Robinson Duckworth, who accompanied Carroll and the Liddell sisters on the boat ride, says in his account of the trip: “…when three Miss Liddells were our passengers, and the story was actually composed and spoken over my shoulder for the benefit of Alice Liddell…I remember turning round and saying, “Dodgson, is this an extempore romance of yours?” And he replied, “Yes, I’m inventing as we go along.”” That story, on the insistence of Alice, was turned into a manuscript and presented to her by the Oxford mathematician.
By now, the content of the story is presented in disguised form with the use of riddles, apparently meaningless poems, puzzles, puns, and a lot more that is ostensibly nonsense. Carroll was surely not the first to use such devices.
Several examples of puns and riddles are found in nursery rhymes, and folk tales for children. The mastery of Carroll over this kind of recreational mathematics and logic takes Alice’s Adventures in Wonderland to a different league – it is not without reason that the story continues to inspire mathematical puzzles and word-game designers even today.

Raymond S Smullyan wrote a delightful little book titled Alice in Puzzle-Land: a Carrollian Tale for Children Under Eighty in which Alice and her friends return for another trip through Wonderland and the Looking-Glass. The book has 88 engaging puzzles, paradoxes, and logic problems. Smullyan’s characters speak and behave like the Carroll creations, and their puzzles abound in typical Carrollian word-play, logic problems, and dark philosophical paradoxes.
The rich tapestry of puzzles and paradoxes in Alice’s Adventures in Wonderland was a lifelong fascination for Carroll that in some way brought his ‘fairy tales’ closer to Austrian-British philosopher Ludwig Wittgenstein. In his 1965 essay “Wittgenstein, Nonsense, and Lewis Carroll,” philosopher George Pitcher’s talks about striking similarities between the philosophical writings of Wittgenstein and the children’s stories of Carroll. According to Pitcher, both were concerned with nonsense and language puzzles. While Wittgenstein was tortured by these things, Carroll appeared to be delighted by them.
Reverend Dodgson had a playful approach to mathematics that he imported into the Alice stories. He was known to use little puzzles in his lessons to make mathematics class more engaging. For instance, here is one of his classics (many versions of this puzzle now can be found all over the web): A cup contains 50 spoonfuls of brandy, and another contains 50 spoonfuls of water. A spoonful of brandy is taken from the first cup and mixed into the second cup. Then a spoonful of the mixture is taken from the second cup and mixed into the first. Is there more or less brandy in the second cup than there is water in the first cup? (If you are scratching your head for an answer, it is equal.)

In that famous conversation with the Cheshire Cat, who wants to convince Alice that they both are mad, the feline tells her that she “…must be, or you wouldn’t have come here”, but Alice refuses to believe him and in turn asks how the cat knows that he is mad. The next set of conversations that appears in Chapter IV of the book shows how deep is the logic play in this work. Here Carroll has employed the so-called modus ponens, or affirming the antecedent logic.
“To begin with,” said the Cat, “a dog’s not mad. You grant that?”
“I suppose so,” said Alice.

“Well, then,” the Cat went on, “you see a dog growls when it’s angry, and wags its tail when it’s pleased. Now I growl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad.”

One can read the above dialogue without even realising that one is trapped in a logic web spun by Carroll. Here, the Cheshire Cat’s argument may appear sound but it is invalid. Here is how Carroll constructed the trap.
Suppose P and Q are two sentences; here, P is ‘an animal growls when angry and wags its tail when pleased’ and Q is ‘it is not mad’. Let us see what the cat says: ‘If an animal growls when angry and wags its tail when pleased, it is not mad.’ This means, if sentence P is true, then Q is also true.
‘I growl when pleased, and wag my tail when angry.’
Here the cat is not saying what P says.
‘Therefore, I am mad.’
So if the cat’s statement does not agree with P then how can it say Q is true?
One interesting aspect of Carroll’s work is that in the world of literature, especially literary criticism, a lot of emphasis has been on the psychoanalytic aspects of characters. There have been critiques highlighting Carroll’s own personal psychological and sexuality issues but almost nothing on reading the tale as a mathematical text. In 2009, Melanie Bayley, of the University of Oxford, published an article in the popular science magazine New Scientist titled “Alice’s Adventures in Algebra: Wonderland Solved”.
In the article Bayley says that Carroll added a lot of material to the illustrated manuscript he personally made for Alice before it was sent for publication. It is in these parts that Carroll took on the proponents of new mathematics, ridiculing their methods and questioning their rigour. The Cheshire Cat becoming a grin, according to the Oxford researcher, was Carroll’s way of portraying increasing and damaging abstraction in mathematics. In the Mad Hatter’s tea party, Bayley discovered the writer’s satire on Irish mathematician William Rowan Hamilton’s discovery – the quaternion.
There are other similar discoveries made by the Oxford researcher. In the scene where Alice is troubled by growing taller or shorter and meets the hookah-smoking Caterpillar, the creature tells Alice “keep your temper.” This Alice interprets as keeping cool but here Carroll is using an older meaning of the word ‘temper’ which was used for “the proportion in which qualities are mingled.” Bayley interprets this as the Caterpillar telling Alice irrespective of her body size she should maintain her body in proportion. If that is true, this reflects Carroll’s love of Euclidean geometry. In this geometry, absolute magnitude does not matter, it’s important to know the ratio of one length to another.
For a little more than 155 years after the story was first told to Alice, Lewis Carroll’s bestseller continues to throw new conundrums. No one can be absolutely sure whether Carroll actually plays those devious games with his readers. The reverend who stammered a lot and enjoyed the company of young girls did love his logic and Euclid like a fanatic. He is not remembered for his mathematics but for puzzles, logic games and biting satire. It is therefore not surprising that some of it made its way into his boat-ride story.

Debkumar Mitra