107 cancer papers retracted due to peer review fraud

There was a report at ars Technica that the journal Tumor Biology is retracting staggering 107 research papers due to fraudulent peer review process. It appears that such practice has been an on-going business. According to the report, last year 58 papers were retracted from 7 different journals (the report does not specify what they are but I suspect mostly biology journals) and 25 of them came from Tumor Biology. What happened was that when authors submitted their manuscripts for review the editorial office asked the authors to recommend reviewers (the editors perhaps assumed that the authors would suggest best possible reviewers with no conflict of interest who would give a fair and professional review). Apparently the process was abused by some authors and they recommended people they know personally who would give their papers a favorable review regardless of their academic value. I would say those journals are also responsible for the mayhem as they basically let it happen. If people think that scientists would always conduct their research and academic activities with honesty, pride and integrity, they would be flatly naive and wrong. Research publication is directly tied to tenure, promotion and also grants (especially in biological sciences grants come with a big money). Never think that academia would never spoil. Scientists are also men with many flaws. Unfortunately for some often these things weigh more than their pride and academic integrity.

Theoretical physics journals have the same kind of review process but there is little room for such fraudulent practice as it is relatively easier for any third party experts to verify the results and academic merits of a paper in theoretical physics. While theoretical physics research often comes with hypes of all levels, it is relatively much more honest area compared with certain experimental sciences. Without any bias, mathematics is certainly the most honest area. If you are not honest about what you do, you really cannot be a good mathematician. Also there is no room for cooking up your results as everything is readily verifiable by experts. In mathematics, most common malpractices are abusing citation and plagiarism. But such malpractices have never been a big issue in mathematics community as no serious mathematicians would even think of committing them. Whoever commit such things could/would never be regarded as a mathematician.

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Are We Alone in the Universe After All?

In the movie Contact (which is based on the novel with the same title by Carl Sagan), Ellie Arroway (played by Jodie Foster) was speaking to a group of children: “I’ll tell you one thing about the universe, though. The universe is a pretty big place. It’s bigger than anything anyone has ever dreamed of before. So if it’s just us… seems like an awful waste of space. Right?” For us scientists, perhaps I should say most of us, it is almost like faith that there should be an intelligent life other than us somewhere out there in the universe. In 1961, an astronomer Frank Drake proposed the Drake equation (which is not really a mathematical equation but a probabilistic argument) $$N=R_\ast\cdot f_p\cdot n_e\cdot f_\ell\cdot f_i\cdot f_c\cdot L,$$ where

  1.  $N$ is the number of active, communicative extraterrestrial civilizations.
  2. $R_\ast$ is the average rate of star formation in our galaxy.
  3. $f_p$ is the fraction of formed stars that have planets.
  4. $n_e$ is the average number of planets per star that has planets.
  5. $f_\ell$ is the fraction of those planets that actually develop life.
  6. $f_i$ is the fraction of planets bearing life on which intelligent, civilized life has developed.
  7. $f_c$ is the fraction of these civilizations that have developed communications, i.e., technologies that release detectable signs into space.
  8. $L$ is the length of time over which such civilizations release detectable signals.

The original estimate for $N$ given by Frank Drake in 1961 is somewhere between 1000 and 100,000,000 civilizations in the Milky Way galaxy alone. (See here for more details on Drake’s educated guesses on the above quantities 1-8.) I remember back in the 80’s the Drake equation was a great talking point for astronomers to argue why we should support SETI (Search for Extraterrestrial Intelligence).

But then here comes a piece of discouraging news, via RT, “a new study has shown that Earth truly is one of a kind with nothing coming close to life on this planet. Astronomers in Sweden used computer simulations to model the known universe from existing data and then applied to the laws of physics to advance the model 13.8 billion years. Out of the 700 quintillion potential planets, none resembled Earth, meaning we may be very alone after all.” Click here for full story and also here for a related story. Full results of the research can be seen here. Another piece of discouraging news, also via RT, says astronomers examined 93 nearby galaxies for signs of advanced alien life such as the production of waste heat, but found none. Those were the most promising galaxies as they were emitting the largest amounts of heat or mid-infrared emission. Disappointingly the emissions from the majority of observed galaxies could be explained by astrophysical processes, such as dust being generated and heated by a massive star formation. Click here for full story. The Netherlands-based team led by the National Institute for Radio Astronomy (ASTRON) used advanced telescopes to identify the sources of the excessive radiation in hopes of determining if they exhibited signs of advanced alien civilization life. Click here to read ASTRON’s press release.

These new findings tell us that after all the Earth is a very special place in the universe contrary to Copernican principle. To me personally it is still inconceivable that we human beings on the Earth are the only intelligent life in the universe. If it were true, I would probably feel really lonely and sad though there are some bright sides. We would never have to worry about alien invasions and for us the whole universe is up for grabs. Perhaps then it is our sole and sacred duty to go out, explore, colonize, cultivate, and populate the universe for the sake of our own survival and of the preservation of human civilization.

Update: A related article here which is about a paper on the arXiv titled “Dissolving the Fermi Paradox.” In the paper the authors attempt to resolve the Fermi Paradox by examining highly uncertain parameters of the Drake equation.

Update: Ethan Siegel has his taken on the arXiv paper “Dissolving the Fermi Paradox” here.

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A Notable Quote on Richard Feynman

From Freeman Dyson, “Disturbing the Universe”, Harper & Row, New York, 1979:

“Dick was also a profoundly original scientist. He refused to take anybody’s word for anything. This meant that he was forced to rediscover or reinvent for himself almost the whole of physics. It took him five years of concentrated work to reinvent quantum mechanics. He said that he couldn’t understand the official version of quantum mechanics that was taught in textbooks, and so he had to begin afresh from the beginning. This was a heroic enterprise. He worked harder during those years than anybody else I ever knew. At the end he had a version of quantum mechanics that he could understand.”

This quote is, of course, about Richard Feynman‘s path integral formulation of quantum mechanics.

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$\sum_{n=0}^\infty e^{nix}$ is Cesàro Summable

Back when I was a Ph.D. student, a friend of mine (he was a Ph.D. student in physics studying laser optics) asked me if the series $\sum_{n=0}^\infty e^{nix}$ converges. I vaguely remember that his advisor needed to have a finite value for the infinite sum for whatever reason I don’t remember. At that time, I didn’t know other summability methods and I only knew the conventional definition of infinite sums. The series is a geometric series with $r=e^{ix}$ and since $|e^{ix}|=1$, I bluntly told him that you can’t have a finite value for the series. It diverges!

I don’t know what motivated me but I just thought about the series $\sum_{n=0}^\infty e^{nix}$ and I was wondering if it is Cesàro summable. I was able to show that indeed it is. To my pleasant surprise, the Cesàro sum of $\sum_{n=0}^\infty e^{nix}$ is $\frac{1}{1-e^{ix}}$. I am sure that this is well-known and I am just being ignorant about it. I would appreciate if someone can tell me any reference where the Cesàro sum of $\sum_{n=0}^\infty e^{nix}$ appears.

The $(n+1)$-th partial sum $S_{n+1}$ of $\sum_{n=0}^\infty e^{nix}$ is
\begin{align*}
S_{n+1}&=\sum_{k=0}^n e^{nix}\\
&=1+e^{ix}+e^{2ix}+\cdots+e^{nix}\\
&=\frac{1-e^{(n+1)ix}}{1-e^{ix}}
\end{align*}
provided $x\ne 2m\pi$, $m\in\mathbb{Z}$. Now the sum of the first $n$ partial sums is calculated to be
$$\sum_{k=0}^{n-1}S_k=\frac{1}{1-e^{ix}}\left[(n+1)-\frac{1-e^{(n+1)ix}}{1-e^{ix}}\right].$$
The Cesàro sum of $\sum_{n=0}^\infty e^{nix}$ is
\begin{align*}
\sum_{n=0}^\infty e^{nix}&=\lim_{n\to\infty}\frac{\sum_{k=0}^{n-1}S_k}{n}\\
&=\frac{1}{1-e^{ix}}
\end{align*}
as $\lim_{n\to\infty}\frac{1}{n}\frac{1-e^{(n+1)ix}}{1-e^{ix}}=0$. For $x=2m\pi$ with $m\in\mathbb{Z}$, $\sum_{n=0}^\infty e^{nix}=1+1+1+\cdots$ is not Cesàro summable as its Cesàro sum is \begin{align*}
1+1+1+\cdots&=\lim_{n\to\infty}\frac{\sum_{k=1}^n S_k}{n}\\
&=\lim_{n\to\infty}\frac{\frac{n(n+1)}{2}}{n}\\
&=\infty.
\end{align*}
However, it can be shown that $1+1+1+\cdots=-\frac{1}{2}$ using zeta function regularization. (I also discussed it here.)

The real part of Cesaro sum with n=300

The real part of Cesaro sum with n=300

This figure shows the real part of the Cesàro sum $\sum_{n=0}^\infty e^{nix}$ with $x=\frac{\pi}{6}$ converging to the real part of $\frac{1}{1-e^{ix}}=\frac{1}{2}$.

The imaginary part of Cesaro sum with n=300

The imaginary part of Cesaro sum with n=300

This figure shows the imaginary part of the Cesàro sum $\sum_{n=0}^\infty e^{nix}$ with $x=\frac{\pi}{6}$ converging to the imaginary part of $\frac{1}{1-e^{ix}}=\frac{\sin x}{2(1-\cos x)}$.

Cesaro sum with n=100

Cesaro sum with n=100

Finally this figure shows the Cesàro sum $\sum_{n=0}^\infty e^{nix}$ with $x=\frac{\pi}{6}$ converging to $\frac{1}{1-e^{ix}}$.

Posted in Summability Methods | 5 Comments

$1+2+3+4+\cdots=-\frac{1}{12}$?

No, folks! I am not drunk nor I am pot-headed. Yet, I am about to discuss the crazy identity
$$1+2+3+4+\cdots=-\frac{1}{12}.$$
No, I am not joking either. This is actually pretty serious mathematics and is also pretty serious stuff even to physicists. I promise you that by the time you finish reading this blog article, it will all make sense to you (I hope). So, please bear with me.

The very first thing we learned about numbers in elementary school was how to add two numbers, and that was not bad. But when it came to adding three numbers, things were confusing. Because we didn’t know what $1+2+3$ meant. It could mean $(1+2)+3$ i.e. add 1 and 2 first, and then add the resulting number (which is 3) to 3. Or, it could mean $1+(2+3)$ i.e. add 2 and 3 first and then add 1 to the resulting number (which is 5). It turns out that whichever you do it does not matter. They will all turn out to be the same number 6. In fact, for any real numbers $a$, $b$ and $c$, the following property holds:
$$(a+b)+c=a+(b+c).$$
This property, as we recall, is called the associative law. By the induction process, we know that what is true for three numbers is true for any finitely many numbers. For instance, knowing that the associative law holds for three numbers, we also get
$$(a+b+c)+d=a+(b+c+d)$$
for any real numbers $a,b,c,d$. This means that the sum $a+b+c+d$  can be obtained by any of the following four ways
$$[(a+b)+c]+d,\ [a+(b+c)]+d,\ a+[(b+c)+d],\ a+[b+(c+d)].$$
For multiplication, while complex numbers (2-dimensional numbers) and quaternions (4-dimensional numbers) satisfy the associative law, octonions (8-dimensional numbers) do not. The associative law, on the other hand, does not hold in general when we add infinitely many numbers. For instance, let us consider the Grandi’s series
$$1+(-1)+1+(-1)+\cdots.$$
If we assume that the associative law still holds for this case, we can prove something interesting. First, by the associative law we obtain
\begin{align*}
1+(-1)+1+(-1)+\cdots&=(1-1)+(1-1)+(1-1)+(1-1)+\cdots\\
&=0+0+0+0+\cdots\\
&=0.
\end{align*}
But then again by the associative law we also obtain
\begin{align*}
1+(-1)+1+(-1)+\cdots&=1+(-1+1)+(-1+1)+(-1+1)+\cdots\\
&=1+0+0+0+\cdots\\
&=1.
\end{align*}
Therefore, we just proved that $0=1$. Of course this is bullshit! Now that we know the associative law does not hold in general for infinite sums, our question is what do we mean by adding an infinitely many numbers? That is what do we mean by
$$\sum_{k=1}^\infty a_k=a_1+a_2+a_3+\cdots+a_k+\cdots?$$
Although we think that we can perceive the notion of infinity, our brain can actually process only finitely many things just like computers do. So this is the way we perceive natural numbers. We don’t actually perceive the entire infinitely many natural numbers. We can only count finitely many of them but our mind can convince us that the process continues indefinitely as 2 comes after 1, 3 comes after 2,$\cdots$,  1 million 1 comes after 1 million, and so on so forth. This is the way we perceive infinity. We do not perceive actual infinity but only potential infinity through finite processes. From intuitionism point of view, the actual infinity such as the set $\mathbb{N}$ of all natural numbers is an illusion and it should not be considered as a mathematical object. Finitism even rejects the notion of potential infinity and says that “a mathematical object does not exist unless it can be constructed from the natural numbers in a finite number of steps.” (By the way I am not an intuitionist but a
Platonist.) Back to our previous question. What we really can do is the finite sum
$$s_n=\sum_{k=1}^n a_k=a_1+a_2+a_3+\cdots+a_n,$$
which is called the $n$-th partial sum, but we can define the infinite sum $\displaystyle\sum_{k=1}^\infty a_k$ as the limit of the $n$-th partial sum $s_n$,  $\displaystyle\lim_{n\to\infty}s_n$. If this limit exists as a finite number, we say the infinite sum $\displaystyle\sum_{k=1}^\infty a_k$ exists. If the limit does not exist or becomes $\infty$ or $-\infty$, we say the infinite sum does not exist. The $n$-th partial sum $s_n$ for the Grandi’s series is $1,0,1,0,1,0,\cdots$ so the sequence of partial sums $\{s_n\}$ does not converge, and hence the Grandi’s series does not converge in ordinary sense i.e. the way we learned in calculus. Although this definition of infinite sums appears to be most natural and intuitive, there may be other legitimate ways to define infinite sums. In fact, there are. One of them is Cesàro’s sum. Cesàro’s sum of an infinite series is defined by
$$\lim_{n\to\infty}\frac{\displaystyle\sum_{k=1}^ns_k}{n}$$
i.e. the limit of the arithmetic mean of the first $n$ partial sums as $n\to \infty$. If an infinite series is summable, it is Cesàro summable. But the converse need not be true. A counterexample for the converse is Grandi’s series. The partial sums of Grandi’s series are
$$\frac{1}{1},\frac{1}{2},\frac{2}{3},\frac{2}{4},\frac{3}{5},\frac{3}{6},\frac{4}{7},\frac{4}{8},\cdots$$
and the limit of this sequence is $\frac{1}{2}$ as seen in the following picture.

Cesaro's sum of Grandi's series

Cesaro’s sum of Grandi’s series

There is another summation method called Abel summation which is more powerful than Cesàro’s summation. It uses a different mean called the abelian mean. Let $\{\lambda_n\}_{n=0}^\infty$ be a strictly increasing sequence such that $\lambda_0\geq 0$ and $\displaystyle\lim_{n\to\infty}\lambda_n=\infty$. Let $f(x)=\displaystyle\sum_{n=0}^\infty a_ne^{-\lambda_nx}$. Suppose that $f(x)$ converges for all real numbers $x>0$. Then the abelian mean $A_\lambda$ is defined as
$$A_\lambda(s)=\lim_{x\to 0+}f(x).$$
Now let $\lambda_n=n$. Then we obtain
$$f(x)=\sum_{n=0}^\infty a_ne^{-nx}=\sum_{n=0}^\infty a_nz^n\ (z=e^{-x})$$
and
$$\lim_{x\to 0+}f(x)=\lim_{z\to 1-}\sum_{n=0}^\infty a_nz^n.$$
This limit is called Abel summation. Let us consider the infinite sequence
$$\{a_n\}=\{(-1)^n(n+1):n=0,1,2,\cdots\}=\{1,-2,3,-4,\cdots\}.$$
The series $\displaystyle\sum_{n=0}^\infty a_n$ is not summable nor is Cesàro summable as seen in the following pictures.

Cesaro's sum of 1-2+3-4+...

Cesaro’s sum of 1-2+3-4+…

But it is Abel summable. To see this,
\begin{align*}
\sum_{n=0}^\infty a_nz^n&=1-2z+3z^2-4z^3+\cdots\\
&=\frac{1}{(1+z)^2}
\end{align*}
for $|z|<1$. Thus,
$$\lim_{z\to 1-}\sum_{n=0}^\infty(-1)^n(n+1)z^n=\lim_{z\to 1-}\frac{1}{(1+z)^2}=\frac{1}{4}.$$
This means than
$$1-2+3-4+\cdots=\frac{1}{4}$$
as Abel summation. Abel summation also can calculate Grandi’s series. Let $a_n=(-1)^n$, $n=0,1,2,\cdots$. Then
\begin{align*}
\sum_{n=0}^\infty a_nz^n&=1-z+z^2-z^3+\cdots\\
&=\frac{1}{1+z}
\end{align*}
for $|z|<1$. Hence, we obtain
$$\lim_{z\to 1-}\sum_{n=0}^\infty a_nz^n=\lim_{z\to 1-}\frac{1}{1+z}=\frac{1}{2}$$
as we have seen earlier. From this we see that
$$(1-1+1-1+\cdots)^2=1-2+3-4+\cdots.$$
This identity can be also obtained by the Cauchy product. Cesàro summation or Abel summation can be used to calculate oscillating divergent series to possibly produce a finite answer. However, they cannot produce a finite answer for a series that diverges to $\infty$. For example, $1+2+3+4+\cdots$ is neither Cesàro summable nor Abel summable.

Interestingly Srinivasa Ramanujan (1877-1920) showed in his notebook that
$$1+2+3+4+\cdots=-\frac{1}{12}$$
and his proof is pretty elementary. Let $c=1+2+3+4+\cdots$. Then
$$
\begin{aligned}
c&= 1&+&2+3&+&4+5&+&6&+&\cdots\\
4c&= &+&4  &+&8  &+&12&+&\cdots
\end{aligned}
$$
Subtracting the second identity from the first results
$$-3c=1-2+3-4+5-6+\cdots=\frac{1}{4}$$
and hence we obtain
$$c=-\frac{1}{12}\ \mbox{i.e.}\ 1+2+3+4+\cdots=-\frac{1}{12}.$$
This appears to be a way too elementary and simple proof for the outrageous claim. The problem with this proof is that it is carried out by assuming that $c$ is a finite number. But we don’t know that, do we? In ordinary sense, $c=\infty$ so subtracting $4c$ from $c$ would result $\infty-\infty$ on the left hand side which is undefined, while it would result $1-2+3-4+5-6+\cdots$ on the right hand side. You need to be very careful when you deal with infinite sums and treating them like finite numbers is dangerous as it may result an inconsistent result. While I am not happy with Ramanujan’s proof, what he claimed may still be true. And yes, I am still sober. In fact, there is a much more sophisticated way to show that $1+2+3+4+\cdots=-\frac{1}{12}$. It is called zeta function regularization. Let
$$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\cdots$$
where $s=\sigma+it$ is a complex variable. $\zeta(s)$ converges for all complex numbers $s$ with $\sigma=\mathrm{Re}(s)>1$. $\zeta(s)$ diverges when $\sigma=\mathrm{Re}(s)\leq 1$, in particular when $s=-1$ in which case we obtain
$$\zeta(-1)=1+2+3+4+\cdots.$$
Bernhard Riemann (1826-1866) showed that $\zeta(s)$ can be continued analytically to the punctured plane $\mathbb{C}\setminus\{1\}$. The analytic continuation of $\zeta(s)$ is called the Riemann zeta function. If you are not familiar with analytic continuation, I explained the idea of analytic continuation using a simple example here. The Reimann zeta function is a meromorphic function on $\mathbb{C}$, which is holomorphic everywhere except for a simple pole at $s=1$.
$$
\begin{aligned}
\zeta(s)&=1^{-s}&+&2^{-s}&+&3^{-s}&+&4^{-s}&+&5^{-s}&+&6^{-s}&+&\cdots\\
2\cdot 2^{-s}\zeta(s)&=0&+&2\cdot 2^{-s}&+&0&+&2\cdot 4^{-s}&+&0&+&2\cdot 6^{-s}&+&\cdots
\end{aligned}
$$
Subtracting the second identity from the first results
$$(1-2^{1-s})\zeta(s)=1^{-s}-2^{-s}+3^{-s}-4^{-s}+5^{-s}-6^{-s}+\cdots=\eta(s).$$
The Dirichlet series
$$\eta(s)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+1)^s}=\frac{1}{1^s}-\frac{1}{2^s}+\frac{1}{3^s}-\frac{1}{4^s}+\cdots$$
converges only for complex numbers $s$ with $\mathrm{Re}(s)>0$. However, $\eta(s)$ is Abel summable for any complex number $s$. Hence, it can be analytically continued to the entire complex plane $\mathbb{C}$. The analytic continuation of $\eta(s)$ which is an entire function is called the Dirichlet eta function. The identity $(1-2^{1-s})\zeta(s)=\eta(s)$ still holds when both functions are continued analytically to the punctured plane $\mathbb{C}\setminus\{1\}$. Substituting $s=-1$, we obtain $-3\zeta(-1)=\eta(-1)$. $\eta(-1)=1-2+3-4+5-6+\cdots$ diverges to $\infty$ but it is Abel summable and the Abel sum of the series is $\frac{1}{4}$
$$\eta(-1)=\lim_{z\to 1-}(1-2z+3z^2-4z^3+\cdots)=\lim_{z\to 1-}\frac{1}{(1+z)^2}=\frac{1}{4}$$
as we have seen earlier. $\eta(-1)$ as the analytic continuation of $\eta(s)$ evaluated at $s=-1$ is the Abel sum $\frac{1}{4}$. Therefore, we obtain $\zeta(-1)=-\frac{1}{12}$ i.e. $1+2+3+4+\cdots=-\frac{1}{12}$. So, have we actually proved $1+2+3+4+\cdots=-\frac{1}{12}$ now? The answer is yes and no. No, no, I am not playing with you. Let me explain. What we actually have proved here is $\zeta(-1)=-\frac{1}{12}$ and in fact you also remember that $\zeta(-1)$ is not defined because it diverges. So here $\zeta(-1)$ is not really $1+2+3+4+\cdots$ but the Riemann zeta function i.e. the analytic continuation of $\zeta(s)$ evaluated at $s=-1$. Precisely speaking, it is not that $1+2+3+4+\cdots=-\frac{1}{12}$ but that we can assign the infinite series $1+2+3+4+\cdots$ a unique finite number $-\frac{1}{12}$, which coincides with Ramanujan’s calculation, using the analytic continuation of $\zeta(s)$. As an analogy, in my notes here, $\displaystyle f_1(s)=\sum_{n=0}^\infty(-1)^ns^n$ is not defined at $s=\frac{3}{2}i$ because $f_1\left(\frac{3}{2}i\right)$ diverges to $\infty$, however the analytic continuation $F(s)$ is defined at $s=\frac{3}{2}i$ and that $F\left(\frac{3}{2}i\right)=\frac{4}{13}-\frac{6}{13}i$. Hence, we may assign the divergent series $f_1\left(\frac{3}{2}i\right)$ a unique finite number $\frac{4}{13}-\frac{6}{13}i$.

Giving a finite value to a divergent quantity is not an unusual practice in mathematics. For instance, we can turn the infinite number line $\mathbb{R}=(-\infty,\infty)$ into a unit circle $S^1$ by assigning a finite point to the boundary $\{\pm\infty\}$ of $\mathbb{R}$. Here is one way to do it. The infinite number line $(-\infty,\infty)$ is homeomorphic to the open interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ via the map $f(t)=\arctan t$. This means that $f(t)$ is a homeomorphism i.e it is one-to-one and onto, is continuous and its inverse $f^{-1}(t)$ is also continuous ($f^{-1}(t)=\tan t$). In the eyes of topologists two homeomorphic spaces are the same i.e. they do not distinguish $(-\infty,\infty)$ and $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. The open interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ is then homeomorphic to $(0,2\pi)$ via the homeomorphism $g(s)=2s+\pi$. The open interval $(0,2\pi)$ is then homeomorphic to the unit circle $S^1$ with one point $(1,0)$ removed via the homeomorphism $h(\theta)=(\cos\theta,\sin\theta)$. The composition $(h\circ g\circ f)(t)$ is then a homeomorphism from $(-\infty,\infty)$ to $S^1\setminus\{(1,0)\}$. By adding a point $(1,0)$ to $S^1\setminus\{(1,0)\}$, we obtain a unit circle $S^1$. This process is called one-point compactification in topology. Similarly adding a point $\infty$ (this is called the ideal point) to the infinite plane $\mathbb{R}^2$ and identifying the boundary of $\mathbb{R}^2$ with the ideal point $\infty$ results a unit sphere $S^2$. This whole process can be done via the stereographic projection from the north pole of $S^2$ as seen in the following picture. In fact, the stereographic projection can be also used to show that we can obtain a unit circle $S^1$ by adding a point to the real line $\mathbb{R}$ (and this point is identified with the boundaries $\{\pm\infty\}$ of $\mathbb{R}$).

Stereographic projection from the north pole

Stereographic projection from the north pole

In the study of instantons in physics, an infrared cut-off, the finite-energy condition imposed on Yang-Mills action with $|x|\to\infty$ where $x\in\mathbb{R}^4$, geometrically amounts to the one-point compactification of $\mathbb{R}^4$ which is the 4-sphere $S^4$.

In the strip $0<\mathrm{Re}(s)<1$, the zeta function satisfies the functional equation
$$\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s).$$
One can immediately see that $\zeta(s)=0$ for $s=-2,-4,-6,\cdots$, negative even integers. These are called trivial zeros of the Riemann zeta function and there are indeed nontrivial zeros of the Riemann zeta function as well. There is a famous conjecture, called the Riemann Hypothesis, regarding the nontrivial zeros of the Riemann zeta function, namely the nontrivial zeros $s$ of the Riemann zeta function all have real part $\mathrm{Re}(s)=\frac{1}{2}$. This conjecture still has not been resolved (proved or disproved). It is part of Hilbert’s eighth problem along with the Golbach conjecture in the Hilbert’s 23 unsolved problems and is also one of the Clay Mathematics Institute Millennium Prize Problems. Using the Riemman zeta function we can calculate a finite answer for another divergent series $1+1+1+\cdots$.
\begin{align*}
\zeta(0)&=\lim_{s\to 0}2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)\\
&=\frac{1}{\pi}\lim_{s\to 0}\sin\left(\frac{\pi s}{2}\right)\zeta(1-s)\\
&=\frac{1}{\pi}\lim_{s\to 0}\left(\frac{\pi s}{2}-\frac{\pi^3s^3}{48}+\cdots\right)\left(-\frac{1}{s}+\cdots\right)=-\frac{1}{2}.
\end{align*}
That is, we obtain
$$1+1+1+\cdots=-\frac{1}{2}.$$

This sort of regularization i.e. giving a finite value to a divergent quantity is particularly important to physicists along with renormalization as the divergence of path integrals often appears in quantum field theory even when physically you expect finite values for those integrals. Many physicists are trying to find the resolution for the divergences from mathematics such as regularization. However, I don’t think that the answer is in mathematics. I suspect that the occurrence of divergences in quantum field theory may have originated from its foundation and I believe that the answer can be found by carefully and thoroughly reexamining the way the current quantum field theory is formulated. For one, in quantum field theory particles are treated as mathematical points i.e. there are no inner structures of particles contrary to the nature of actual particles. Second, I also believe that the way the path integral was formulated is incorrect. (I am not saying that the idea of path integral formulation is wrong. I do believe that the idea itself is correct. I just believe that the use of complex numbers in the formulation of path integrals is incorrect. For this, see my blog article here.) I will delve into this issue at some other time when I have a better understanding.

Back to mathematics, so what now? Well, we have two conflicting types of arithmetic. One type of arithmetic, which coincides with our perception of numbers, says that $1+2+3+4+\cdots=\infty$ and the other type of arithmetic says that
$1+2+3+4+\cdots=-\frac{1}{12}$. While this second type of arithmetic appears to be mathematically consistent, it is also against everything we experience about numbers. For instance, we know that if we add any finitely many positive numbers, the result would still be positive. Now the infinite sum $1+2+3+4+\cdots$ is not only finite but also negative! The big question is how these two conflicting types of arithmetic can be consistent with each other and how do we cope with this trouble? The situation may be parallel to what happened in geometry about a couple centuries ago. The parallel postulate, also called Euclid’s fifth postulate (because it is the fifth postulate in Euclid’s Elements) states that, in two-dimensional geometry,

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.” In 1795, John Playfair (1748-1819) showed that Euclid’s parallel postulate can be replaced by the following axiom

At most one line can be drawn through any point not on a given line parallel to the given line in a plane.”

Many mathematicians thought Euclid’s parallel postulate might be proved from the other four postulates of Euclid and attempted to prove it and failed. Among them includes Adrien-Marie Legendre (1752-1833). In the beginning of 19th century, mathematicians realized the possibility of other geometries by negating the parallel postulate. Carl Friedrich Gauß (1777-1855) knew the possibility of non-Euclidean geometry but never publicized his finding, perhaps because he was afraid of criticisms from other mathematicians. There are two different cases with negating the Euclid’s parallel postulate. One is “In a plane, given a point $p$ and a line $\ell$ not passing through $p$, there exist two lines through $p$ which do not meet $\ell$.” Replacing the Euclid’s parallel postulate by this postulate, János Bolyai (1802-1860) and Nikolai Ivanovich Lobachevsky (1792-1856) independently discovered a non-Euclidean geometry, called hyperbolic geometry. In fact, in plane hyperbolic geometry given a point $p$ and a line $\ell$ not passing through $p$, there exist infinitely many lines through $p$ which do not meet $\ell$, i.e. there are infinitely many parallel lines. The other case is “In a plane, given a point $p$ and a line $\ell$ not passing through $p$, all the lines through $p$ meet $\ell$,” i.e. there are no parallel lines. Replacing the Euclid’s parallel postulate by this one, Bernhard Riemann discovered a non-Euclidean geometry called elliptic geometry which is the simplest case of Riemannian geometry. The existence of plane geometries with three different parallel postulates may appear to be a contradiction. Due to Arthur Cayley (1821-95), Eugenio Beltrami (1835-1912), Felix Klein (1849-1925), and Henri Poincaré (1854-1912), et. al. it turns out that non-Euclidean geometries can be modelled within Euclidean geometry, and hence there is no contradiction with having both Euclidean and non-Euclidean geometries and they can all be consistent with each other.

Having a lesson from geometry, one must wonder whether we are in the same situation with the two different types of arithmetic we have now, which appear to be conflicting with each other. We obtained $1+2+3+4+\cdots=-\frac{1}{12}$ within the conventional arithmetic through analytic continuation. To me this appears to be a reminiscence of obtaining models of non-Euclidean geometries within Euclidean geometry. But a suitable mathematical or physical interpretation of this new arithmetic still lacks. I cannot shake off the feeling that we are missing something big here with a chain of questions. What is the meaning of this new arithmetic? What are the implications of this new arithmetic regarding its possible impact on mathematics? Does it indicate that there may be a new type of mathematics we don’t yet know about? If so, what could be possible impact of such new mathematics on physics? I will get to these questions in due course and hopefully by then I will have answers for them.

Posted in Foundations of Mathematics, Summability Methods | 4 Comments