In my opinion, these are my 5 favorite math things… From Zero to Infinity (literally), here are the most intriguing topics involving math that tickle my fancy the most. Information about the Reimann Hypothesis.
The Top 5 Mathematical Concepts (in my opinion)
5. The Concept of Infinity
∞
If -1 is the input for the Reimann Zeta Function, the answer becomes +1 + 2 + 3 + 4… all the way up to infinity, but the output ends up converging into -1/12. This leads some mathematicians to conclude that infinity = -1/12.
If you didn’t read that above statement because it sounds like gibberish, I get it. But I will say, as an English major and a casual math enthusiast, I’m not as concerned with getting a right answer to a single problem as a more professional math person might be. But I do like to think about what mathematics represents. What math has concluded, not only about the actual universe through physics, but also about all possible universes, is what attracts my mind. Things that have been proven in the realm of math (a proof, not a conjecture or theory or inference) are the purest forms of truth. This is why you’ll find the Fibonacci sequence (golden ratio) is ubiquitous throughout nature. Pi also goes on forever…
I could go on forever about infinity, but wanted to highlight two interesting characteristics of the concept that can appear counterintuitive. The first is that ∞ = -1/12 and involves the Reimann Zeta Function and I decided to just write that out above with little explanation for fear I’d start rambling on…
The second interesting attribute of infinity is that are multiple types of infinities, in the sense that some types of infinity are bigger than others. Again, without getting into the weeds, this can be illustrated beautifully through square numbers. Every whole number can be squared by multiplying it by itself (0 x 0 = 0 , 1 X 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16, etc.) The products (2, 4, 9, 16…) are known as square numbers. Now think about how there are many more non-square numbers than there are square numbers, but every whole number can be squared. The infinity of squares is larger than the infinity of whole numbers, and yet both are considered infinite.
Recommended Reading: ‘Humble Pi: When Math Goes Wrong in the Real World’ by Matt Parker (2020)
4. The Concept of Zero
ᴓ
Zero, as a number, is fun. It represents nothing. It’s the set with nothing in it but still remains a set. It is emptiness. The beginning in the end. Poetically, zero is fantastic. It is neither prime nor composite, but it is even, algebraic, natural, and real… but it is the only number that is both real an imaginary simultaneously And in some sense, it is the point where math breaks. Multiplication is fine, anything multiplied by 0 = 0. X + 0 = X an X - 0 = X but with division… math completely breaks. There isn’t an answer with anything except zero itself because nothing divided by nothing is still nothing. Anything else divided by nothing? You can’t even do it. It doesn’t make sense. To me, this is such a glaring indicator of where math is limited in signifying reality.
Back to the Reimann hypothesis… zero plays a crucial role here. There are trivial zeros and non-trivial zeros (which we’ll get to shortly) but trivial zeros are pretty much all the negative even numbers on the real number line. If the imaginary part of complex number is 0, the real part will be a negative even number like with (-8 , 0). When the “real” part of a complex number is 0, it is considered nontrivial and the imaginary piece will always be between 0 and 1…
Somewhat related to this discussion of complex number is the Mandelbrot Set, which is a product of chaos theory, stating initial conditions wildly change the outcome.
Recommended Reading: ‘Chaos: Making a New Science’ by James Gleick (2008)
3. The Incompleteness Theorem
G
F
Math can rarely be called “gangsta” but Kurt Gödel did one of the most gangsta things I’ve ever heard about. I’ll try to be brief. A bunch of pompous mathematicians around World War I were trying to create a system of math that could stand on its own. As in – it would be complete. Principia Mathematica was published in 1912 and tried to construct a series of symbols and axioms that would be free of paradox and contradiction. If it had been successful, the book would’ve had astounding implications for science, physics, and numerous other facets of the universe. But around 1932, Kurt Gödel found one beautiful contradiction amongst the hundreds of pages of weird symbols and numbers, a language that looks more like computer code than anything else. And in doing so, defiantly proved that mathematics can never be “complete.” There will always be a self-referential paradox to arise that will require a mind to decipher it or nothing makes sense. Essentially, it brings us to my favorite paradox, Russel’s. (Note Bertrand Russel worked on the Principia Mathematica and while his endeavor was bit pompous in hindsight, he was definitely a great mind.)
Russel’s paradox is basically: as soon as X belongs in one group, that property itself places X in an opposite group, which in turn qualifies X for the first group, and this goes on forever. X never “settles” in a group. This is often illustrated with a barber who only cuts hair for people who do not cut their own hair... so does the barber cut his own hair? As soon as he does, he doesn’t. As soon as he doesn’t, he does…
Recommended Reading: “I Am A Strange Loop” by Douglas R. Hofstadter (2007)
2. The Monster
ᴹ
I am still reading the book on this one, but what I garner is fascinating. All mathematical objects can be broken down in set theory as series of groups. All ‘finite simple’ groups have been classified, and all of these belong to either one of 18 “countably infinite” families or one of 26 “sporadic groups” – which have no systematic pattern. 20 of these 26 groups are known as the “monster” group and within these 20 is the “monster” itself – an insanely inconceivable thing (the largest sporadic simple group) in math that requires at least 196,883 dimensions before it can “act non-trivially.”
While opaque and abstract, if you can appreciate the fact that math represents some reflection of truth or reality, then what does it mean that such a “monster” can mathematically exist? The idea of 196,883 dimensions struggles to materialize in one’s thoughts as picturing a reality beyond 3 or 4 dimensions is difficult to construct.
Recommended Reading: “Symmetry and the Monster” by Mark Ronan (2006)
1. Prime Numbers
There are two things that I like to think about when it comes to prime numbers.
Even though there are “real” numbers in math that means something else, I find the primes to be the most real. No matter what symbol you give them, they remain indivisible, being only the number One or itself. It will not divide evenly in any other way (we know zero isn’t an option) so it is not a construct of several numbers but stands with it’s own… identity. Whether you call a 7 a 7 (base-10) or 111 (binary), it is still prime within that base. You can work in hexadecimal where 971 is written as 3CB and it’s still prime.
The second involves the Reimann Hypothesis, infinity, and non-trivial zeros (neatly tying this whole list together). The famous hypothesis states that all prime numbers lie on the “critical” line, meaning their imaginary part is exactly ½ (remember we already know this have to be between 0 and 1) and their real part is a “non-trivial” zero. Every prime number we know about, and we’ve discovered them all up to the number 10^30 (a 10 with 30 zeros: 1,000,000,000,000,000,000,000,000,000,000) and every single one has been found to be on the critical line. But this is what is great about math. We know there are an infinite number of primes thanks to the great mathematician Euler and there has to be an absolute proof of the Reimann hypothesis before we can rule out the possibility that somewhere on the critical line, which is infinite, primes might start to stray from this line or act sporadically.
Recommended Reading: “Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics”
by John Derbyshire (2004)
