Why it would be great if more people laugh from this nerdy meme
It took me about a minute and a half, but I got it, and it led to a very interesting feeling of finding gold where I was only looking for copper. In the meme, a bad math student is trying to solve (5/12) — (6/12), which we know is -1/12. In the second scene, we see a clever advanced student who tells the first that the answer is the same for the Gauss sum.
Legend has it that Carl Friedrich Gauss, one of the greatest mathematicians of all time, was born in 1777 in Brunswick, Germany. Even as a child, Gauss showed extraordinary mathematical talent. The famous anecdote goes that when Gauss was just a young schoolboy, his teacher tasked the class with adding up all the numbers from 1 to 100 as a way to keep them occupied. The teacher expected the task to take some time. However, Gauss, in a stroke of genius, quickly found a shortcut.
Instead of tediously adding up the numbers one by one, Gauss realized that he could pair the numbers in a clever way. He observed that if he paired the first and last numbers (1 + 100), the second and second-to-last numbers (2 + 99), the third and third-to-last numbers (3 + 98), and so on, each pair would sum up to 101. Since there were 100 numbers in total, he would have 50 such pairs. Therefore, to find the sum of all the numbers from 1 to 100, he simply needed to multiply 101 by 50, which equals 5050.
This realization allowed Gauss to swiftly provide the correct answer, much to the amazement of his teacher and classmates. Gauss’s ability to recognize and exploit the pattern in the sequence of numbers demonstrated his remarkable mathematical insight, even at a young age. This story showcases not only Gauss’s extraordinary mathematical talent but also his ability to think creatively and find elegant solutions to complex problems. The formula Gauss discovered is now known as the arithmetic series formula and can be used to find the sum of an arithmetic series with a given first term, last term, and number of terms.
Of course, if, as in the meme, the Gauss sum is taken from 1 to infinity, any reader with some common sense would realize that (1 + 2 + 3 + …+ …) will grow without limit and therefore the Gauss sum tends to infinity.
In the meme, we have that (5/12) — (6/12), which we know is -1/12, is then identified as infinity: \((5/12) — (6/12) = \infty\), which is the result from the Gauss sum the second student suggested as a solution.
So, what is wrong?
a) (5/12) — (6/12)\) is not equal to \(-1/12
b) Gauss sum is not equal to infinity
c) the solution to the problem is not the Gauss sum
Well, it is clear for any reader with common sense that evidently the answer is c): the solution to the problem is not the Gauss sum. And that is right, unless you are an Indian touched by the gods called Ramanujan.
An astrophysicist friend of mine who studied at the Cavendish Laboratory in Cambridge told me he once viewed the original math notebooks by Ramanujan, which are exhibited in the Trinity College Wren Library alongside Newton’s “Philosophiæ Naturalis Principia Mathematica.” That is how important the work of Ramanujan is considered, who worked at Cambridge from 1914 until his death (at 32 years old) in 1921.
Srinivasa Ramanujan was a self-taught mathematical genius born on December 22, 1887, in Erode, a town in present-day Tamil Nadu, India. From a young age, Ramanujan displayed an exceptional aptitude for mathematics, often solving complex problems without any formal training.
Despite facing numerous challenges, including financial hardship and lack of access to formal education, Ramanujan’s talent eventually caught the attention of mathematicians in India. In 1913, at the age of 26, Ramanujan secured a clerical position at the Madras Port Trust, where he continued to pursue mathematics in his spare time.
Ramanujan’s breakthrough came when he began corresponding with G. H. Hardy, a prominent mathematician at the University of Cambridge in England. In 1913, Ramanujan sent a letter to Hardy containing a list of mathematical results and conjectures. Impressed by the depth and originality of Ramanujan’s work, Hardy invited him to Cambridge to collaborate.
In April 1914, Ramanujan traveled to England, where he began working with Hardy at Trinity College, Cambridge. Despite facing significant cultural and personal challenges, Ramanujan’s collaboration with Hardy proved fruitful, and he made groundbreaking contributions to several areas of mathematics, including number theory, infinite series, and continued fractions.
Ramanujan’s work was characterized by its brilliance and originality, often involving innovative approaches and profound insights. He had an intuitive grasp of mathematical concepts that allowed him to discover numerous theorems and identities that had eluded other mathematicians.
During his time in Cambridge, Ramanujan published several papers detailing his findings, many of which revolutionized the field of mathematics. His work on partition theory, mock theta functions, and modular forms, among other topics, remains influential to this day.
Despite his remarkable achievements, Ramanujan’s time in England was marred by struggles with health issues, loneliness, and cultural isolation. In 1919, due to declining health and personal difficulties, Ramanujan returned to India, where he continued to work on mathematics until his untimely death on April 26, 1920, at the age of 32.
According to Ramanujan himself, many of his mathematical discoveries came to him in dreams, often in a state of semi-consciousness or meditation. He claimed that these insights were communicated to him by a Hindu goddess named Namagiri Thayar, who appeared to him in his dreams and provided him with mathematical formulas and theorems.
Ramanujan described these experiences as mystical and divine, believing that the goddess directly inspired his mathematical genius. He stated that the goddess would appear to him with chalk and a slate, writing mathematical equations and formulas that he would then transcribe upon waking.
One of the most fascinating results by Ramanujan is the meme one, that the Gauss sum, 1+2+3+…+…, is exactly -1/12
Yea I agree… WTF!
I first knew about this in a very interesting video on the topic:
After seeing this video, I had a very bad time obsessing over it. After several months of thinking about it, I felt it was necessary for me to write down some ideas about it, and I did so with two friends of mine.
The idea of the paper (https://researchers.one/articles/19.08.00002) is to think about the result presented in a Numberphile (http://www.numberphile.com/) talk where they claim that 1 + 2 + 3 + …, the Gauss sum, converges to -1/12. In the video, they make two strong statements: one, that Grandi’s Series 1–1 + 1–1 + 1–1 + … “tends” (for a technical discussion see for example “Probability the logic of science” by Jaynnes) to 1/2, and the second, that as bizarre as the -1/12 result for the Gauss sum might appear, because it is connected to Physics (this result is related to the number of dimensions in String Theory), it is plausible. In this work, we argue that these two statements reflect adhesion to a particular probability narrative and a specific scientific philosophical posture. We argue that by doing so, these (Gauss and Grandi series) results and String Theory ultimately might be mathematically correct but are scientifically (in the Galileo-Newton-Einstein tradition) inconsistent (at least). The philosophical implications of this problem are also discussed, focusing on the role of evidence and scientific demarcation.
I want to highlight why I decided to write that article and now this essay: this case is a great opportunity to learn a lot of important things about math, science, and philosophy.
In a December 2014 article in Nature, physicists George Ellis and Joe Silk issued a rallying cry to “Defend the integrity of physics.” These respected scientists brought attention to some serious concerns regarding the credibility of science, particularly stemming from String Theory’s multiverse concept. They criticized the ongoing efforts to justify the failures of String Theory by arguing against the necessity of experimental testing, claiming that a theory’s elegance and explanatory power alone should suffice.
However, not all members of the String Theory community share this viewpoint. Theoretical physicist Sean Carroll of the California Institute of Technology, a prominent advocate for the multiverse, addressed these issues in an article on Edge. Carroll emphasized the importance of science in explaining observable phenomena and developing models that align with empirical data. He cautioned against oversimplifying the process of fitting models to data and stressed the need for nuanced thinking about the scientific method.
Nevertheless, as N. N. Taleb discusses in “The Black Swan: The Impact of the Highly Improbable”, this is a path that has been thought through and discarded ever since the classics. As he states in an interview for Philosophy Now (https://philosophynow.org/issues/69/Nassim_Nicholas_Taleb): “You can discover more easily what’s wrong than what’s right. You can know that not all swans are white more easily and confidently than all swans are white.”
The issue of testing String Theory poses a unique challenge for physics, unlike previous scientific revolutions such as Relativity. While Albert Einstein’s theory of Relativity faced scrutiny, it presented falsifiable hypotheses that could be tested through observation. Einstein himself articulated the conditions under which his theory could be disproven, and these predictions were later confirmed by empirical evidence. As identified by Sabine Hossenfelder, the problem is that String Theory is mathematically correct but unscientific because it has so many parameters that one can always choose a configuration that achieves good agreement with observations.
So, String Theory is NOT EVEN WRONG, the very worst of all possible situations in science.
In our paper, we propose that since the mathematical structure of String Theory is based on a non-logical interpretation of probability, the result is a logically inconsistent mathematical apparatus, which translates into an impossibility of scientific falsification. In this way, String Theory could be mathematically correct yet scientifically invalid. In this sense, and paraphrasing Taleb, we sustain that in order to not be fooled by randomness (probability -> logic), mathematicians should learn more physics, physicists should learn more mathematics, and all of us should learn more (non-naive) philosophy.
