Can $1+1$ be $1$?

$1+1=2$ is often quoted by laypeople (in mathematics) as an epitome of the absolute truth. Those who know a bit of mathematics know that that is not the case. For example, there is a number system where $1+1=0$. It is denoted by $\mathbb{Z}_2$. $\mathbb{Z}_2$ has only two numbers $0$ and $1$ and it is the smallest field. Here, we don’t mean a field by a vector field in physics but a number system where two numbers can be added, a number can be subtracted from another, a number can be multiplied by another, or a number can be divided by another nonzero number. $\mathbb{Z}_2=\{0,1\}$ is also important in computer science as $0$ and $1$ can be identified with two possible values of a bit and to the boolean values true and false.

The mind of young Thomas Edison, the one who would become one of the greatest American inventors in the future, was full of curiosities and the traditional school education unfortunately could not satisfy the young kid’s endless desire to know. Often, what’s being taught in school didn’t make much sense to his creative mind. One day, his teacher was telling students that $1+1=2$ and this happened to be one of the things that did not make sense to him. Edison told his teacher that he did not understand why it is always true and gave a counter example: “If you put a tiger and a rabbit together in a closed room, soon there will be only tiger left, so $1+1$ can be $1$ also.” Such a creative kid in class can be a nightmare for a teacher but a pleasant nightmare, I might add.

We have seen that $1+1$ is not necessarily $2$, so can there be a number system where $1+1=1$ like young Thomas Edison contended? It turns out the answer is not that simple and it is even confusing as the short answer is yes and no. In order for something to be called a number system, it needs to be a field. Let me first state the formal definition of a field. A field $\mathbb{F}$ is a nonempty set of numbers with two binary operations $+$ and $\cdot$ such that

  1. for $a,b\in\mathbb{F}$, $a+b=b+a$.
  2. for $a,b,c\in\mathbb{F}$, $(a+b)+c=a+(b+c)$.
  3. there exists $0\in\mathbb{F}$ such that $a+0=a$ for all $a\in\mathbb{F}$.
  4. for each $a\in\mathbb{F}$ there exists $-a\in\mathbb{F}$ such that $a+(-a)=0$.
  5. for $a,b\in\mathbb{F}$, $ab=ba$.
  6. for $a,b,c\in\mathbb{F}$, $(ab)c=a(bc)$.
  7. there exists $1\in\mathbb{F}$ such that $a1=a$ for all $a\in\mathbb{F}$.
  8. for each $a\in\mathbb{F}\setminus\{0\}$, there exists $\frac{1}{a}\in\mathbb{F}$ such that $a\frac{1}{a}=1$.
  9. for $a,b,c\in\mathbb{F}$, $a(b+c)=ab+ac$.

The properties 1-4 says $\mathbb{F}$ with $+$ is an abelian group. The properties 5-8 says $\mathbb{F}\setminus\{0\}$ with $\cdot$ is an abelian group. In universal algebra, the identities $0$ and $1$ and inverses $-$ and $/$ are also considered as operations, more specifically the identities are nullary operations and the inverses are unary operations. Since the inverse $/$ (division) is not defined for $0$, a field is not considered as an algebra or an algebraic structure, i.e. a set with operations in universal algebra. Examples of fields include $\mathbb{Q}$, the set of all rational numbers, $\mathbb{R}$, the set of all real numbers, and $\mathbb{C}$, the set of all complex numbers. There are also finite fields $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$ where $p$ is a prime number. Fields are not necessarily sets of numbers. There are also fields called function fields, an example being the set of rational functions of one variable such as $\frac{x^3+2x+1}{x^2-1}$. The definition tells that a field must have at least two numbers $0$ and $1$. So, what happens if a field satisfies $1+1=1$? It implies that the field must have a single element and that $0=1$. This pretty much rules out the possibility of $1+1$ being $1$, right? But it is not. First, allowing $0=1$ does not violate the definition of a field, although in conventional mathematics, it is implicitly assumed that $0\not=1$. The definition says any nonzero element has a multiplicative inverse but it doesn’t say $0$ cannot have a multiplicative inverse (although again it is implicitly assumed that $0$ does not have a multiplicative inverse in conventional mathematics). For a moment, let us allow $0=1$. There is a dire consequence of allowing this: any vector space over this field with one element is $0$-dimensional as ${\mathbf v}=1{\mathbf v}=0{\mathbf v}={\mathbf 0}$! Perhaps, we shouldn’t be allowing $0=1$ which seems to be good-for-nothing and should end discussion on the possibility of the field with one element, right? Not so fast, this is not over yet. While we don’t even know whether it can exist, mathematicians are taking the field with one element pretty serious. They want it so badly that they are willing to bend the definition of a field. It is usually denoted by $\mathbb{F}_1$ or $\mathbb{F}_{\mathrm{un}}$. The subscript “un” is from the French word meaning “one”. The field with one element was first introduced by a Belgian-French mathematician Jacques Tits in 1957 in his paper titled “Sur les analogues algébriques des groupes semi-simples complexes.” One of the axioms of projective geometry states that a line must have at least three points. If one replaces this axiom, while keeping others, by “a line admits only two points”, we obtain a degenerate geometry. Tits conjectured that such degenerate geometry can have equal footing with projective geometry if one were to consider such geometry on a field of characteristic one, i.e. a field satisfying $1=0$. Since the late 1980s, $\mathbb{F}_{\mathrm{un}}$ has gained a lot of attention from major league mathematicians after they realized profound implications of $\mathbb{F}_{\mathrm{un}}$ in number theory (including the Riemann hypothesis), algebraic geometry, and noncommutative geometry. Despite many attempts and proposals, as far as I know, there is no widely accepted notion of $\mathbb{F}_{\mathrm{un}}$, yet. Here is my own take on it. I propose to replace the above conventional definition of a field by the following: A field $\mathbb{F}$ is a nonempty set of numbers with two binary operations $+$ and $\cdot$ such that

  1. $\mathbb{F}$ with $+$ is a abelian group.
  2. $\mathbb{F}$, possibly excluding an element, with $\cdot$ is an abelian group.
  3. for $a,b,c\in\mathbb{F}$, $a(b+c)=ab+ac$.

In this definition, there is no specific mentioning of particular additive and multiplicative identities, only that they exist. This definition then may allow the existence of a field with one element $\{1\}$ without requiring that $1=0$. I have not examined this idea much further yet, so whether this proposal can be something useful remains to be seen.

Thus far, $\mathbb{F}_{\mathrm{un}}$ is a phantom object in mathematics and mathematicians may be, after all, chasing a phantom. But throughout the history of mathematics, we have had phantom objects which turned out to be extremely important entities not only in mathematics but also in physics. Examples include 0, negative integers, irrational numbers, the imaginary number $i$ (complex numbers), quaternions, non-Euclidean geometry, and the summability of divergent series such as $1+2+3+\cdots=-\frac{1}{12}$ (read here for details if you are curious). One day, we may eventually understand what $\mathbb{F}_{\mathrm{un}}$ is and if and when that happens, it will take up its rightful place in mathematics.

Returning to the original question, can $1+1$ be $1$? Je ne sais pas. (I don’t know.) But those major league mathematicians who are way smarter than I am desperately want that to be true. Perhaps time will tell.

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Could the Great Deluge in the Bible have really happened?

According to Genesis 6:9-9:17, “Noah was a righteous man and walked with God. Seeing that the earth was corrupt and filled with violence, God instructed Noah to build an Ark in which he, his sons, and their wives, together with male and female of all living creatures, would be saved from the waters. Noah entered the Ark in his six hundredth year, and on the 17th day of the second month of that year the fountains of the Great Deep burst apart and the floodgates of heaven broke open and rain fell for forty days and forty nights until the highest mountains were covered to a depth of 15 cubits, and all earth-based life perished except Noah and those with him in the Ark.”

The Bible is not a history book, so perhaps we should take such a story as a myth rather than a record of an actual historical event. Interestingly though, the Bible is not the only place where you can find such a story of the Great Deluge. The Bible is not the oldest source of the story either. A pretty similar story can be found in the Epic of Gilgamesh, the Sumerian Epic poems about Gilgamesh, king of Uruk (an ancient city of Sumer), which predates the Torah (Jewish Bible). Similar Great Deluge stories also appear in many other cultures that include Indian, Chinese, Greek, Norse, Polynesian, and Mayan mythologies. It makes you wonder why. Could it be that such a global scale cataclysmic event might actually have happened?

I think I can answer that question. My short answer is highly unlikely. I can prove at least that it couldn’t have happened by raining as described in the Genesis using some scientific data and simple arithmetic. First, we need to find out how much water is in the atmosphere. According to an article by US Geological Survey, there is 12,900 cubic kilometers of water in the atmosphere. (The original source of the data is Igor Shiklomanov’s chapter “World fresh water resources” in Peter H. Gleick (editor), 1993, Water in Crisis: A Guide to the World’s Fresh Water Resources, Oxford University Press, New York.) Since 1 cubic kilometer = $2.642\times 10^{11}$ gallons, 12,900 cubic kilometers = $3.40782\times 10 ^{15}$ gallons, i.e. 3.40782 million billion gallons of water in the atmosphere! That is a lot of water but it counts only 0.001% of total Earth’s water. Let us assume that all that water in the atmosphere will turn into rain. Now, the question is how deep would the rainfall be if it were to cover the entire surface of the Earth like the Great Deluge allegedly did in Genesis? The mean radius of the Earth is 6,371 kilometer, so the total surface area of the Earth is $4\times\pi\times (6,371)^2\approx 510\times 10^6$ square kilometers. Dividing 12,900 cubic kilometers by 510 million square kilometers, we have $2.5294\times 10^{-5}$ kilometer = 0.995827 inch since 1 kilometer = 39370 inches. Of course, the Earth’s shape is not a box, so, rigorously speaking, the calculation is not correct. However, the depth is so small compared with the radius of the Earth, the above calculation won’t make much difference from the accurate one. In fact, the accurate depth $x$ is found by solving the cubic equation $\frac{4\pi}{3}[3(6371)^2x+3(6371)x^2+x^3]=12900$. It has one real solution $0.000025$ kilometer=$0.98425$ inch (and two complex solutions).  The rainfall depth is mere 1 inch so it could never have caused the Great Deluge that would have covered even the highest mountain in the world.

I am not, by the way, claiming that the Great Deluge, or something to that effect could never have happened. All I demonstrated was it could not have happened by raining as described in the Bible. But such a cataclysmic event could possibly have happened by other means. For example, a huge comet or asteroid strike on the ocean would result in a super megatsumani possibly causing a great deluge worldwide. Such a comet or asteroid impact on the ocean will also increase water vapor in the atmosphere that results in an increase in rain significantly worsening the flood.

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Twitter and Math

I opened a Twitter account back in March 2016 but had not been active at all without even a single tweeting until this May 2018 simply because I thought it would be just wasting my time. I was so wrong about that and I wish I started using Twitter much earlier. Here is what I found. Twitter can be a great social media platform for intellectual activities including getting and sharing information,  discussing ideas and opinions on math and getting connected with people in the disciplines of your interests in math. I also recently witnessed an amazing collaborative math work being done on Twitter and that makes me think that Twitter can be also an effective research tool for mathematicians to collaborate with other mathematicians. I was also pleasantly surprised that there are quite a few mathematicians who are tweeting actively and daily tweets from some of them are quite intellectually stimulating. I particularly love inspirational and informative tweets from John Baez and Sam Walters. Reading their tweets and participating in the discussions whenever I can became a great joy of my daily life. For someone who has been academically in isolation for a long time (no I am not in jail if that’s what you think, no I am not living on a remote island either. I do meet people. I emphasized on the word “academically.”) reading their tweets is like discovering an oasis in the desert. It’s so refreshing. There are a couple of problems I find with Twitter though. One is it’s 280-character limit but this one is okay because there is a workaround by making a thread using reply function. The other is that the platform is not convenient for writing math expressions or equations. The best thing I could do is generating equations using LaTex and covert them into image files (jpg, gif, png, etc.). If you are using Windows MathType is a nice tool for that. I used it a long time ago. Now I have ditched Windows for good, I can’t use it. The one I found the most convenient is the web site called latex2png. You type a math expression using LaTex codes there and it compiles and converts your equation to an image in png format. You just need to copy and paste it in your tweet. You can adjust the size of the picture (resolution). In my experience resolution=200-300 appears to be most suitable for a tweet.

Okay, it is now time for me to go back to Twitter.

Update: Besides latex2png, I found some additional online LaTex equation editors: LaTex4technics, Codecogs, HostMath, and iTex2Img. I find latex2png and iTex2Img convenient for Twitter but honestly have not had a chance to examine others with Twitter yet.

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E-mail Issue

I just noticed that I am not getting any e-mail notifications and I am sure neither are you. I ran diagnostics on my mail server and it is working fine without any problem. So I concluded that certain ports (like port 25 and port 587) that are required by mail server are blocked. I opened those ports on my router and the issue still persists. I believe that my ISP is actually blocking those ports in which case it does not matter whether I open them on my router. I am currently working on a workaround such as using an alternative SMTP (like Gmail SMTP) instead of my mail server. I will update you if it works (and I surely hope it does). I am sorry for the inconvenience. In the meantime, if you need an assistance that requires an e-mail notification such as changing your password, just e-mail me your request.

Update: The workaround was successful and the e-mail issue has been resolved.

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A Long-Lost NASA Spacecraft Comes Back Alive

A NASA spacecraft IMAGE (Imager for Magnetopause-to-Aurora Global Exploration) was launched in March 2000 to study Earth’s magnetosphere where charged particles from solar winds trapped by Earth’s magnetic field. In December 2005, it suddenly stopped talking and since then it was thought to be lost in space. Recently a signal from IMAGE was detected by a Canadian amateur astronomer after 13 years of long silence. I cannot imagine what goes on in the minds of former IMAGE project members hearing this news. More details can be read here.

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