“What exactly do mathematicians do and how can we understand it better without needing a math PhD?”
Have you ever wondered what high-level mathematicians actually do? It can be a mystery to many of us who are not familiar with the world of advanced math. Let’s dig into this topic and demystify it without having to go through the rigorous process of obtaining a math PhD.
Understanding the Role of Mathematicians
– What are mathematicians involved in on a day-to-day basis?
– How do they contribute to various fields such as science, technology, and economics?
Demystifying Mathematics for Lay People
– Exploring the importance of math in our everyday lives
– Breaking down complex mathematical concepts into simpler terms
By shedding light on what mathematicians do, we can gain a better understanding of their role in society and appreciate the beauty of mathematics. Let’s uncover the world of high-level math together. #Mathematics #Mathematicians #UnderstandingMath #DemystifyingMath #LayPeople #MathPhD #EverydayMath #HighLevelMath
They work on problems no one has solved yet. For example prime numbers are very important to us, in fact your bank probably uses prime numbers to verify your identity, but we still don’t know whether there are infinitely many primes that are exactly 2 apart, such as 3 and 5, or 17 and 19.
Just like medical doctors there are several different disciplines of high level math. Some of them are more abstract than others. It would be hard to truly describe them all in a simple manner. However the broadest generalization I can make is high level mathematicians use complex math equations and expressions to describe both things that exist physically and things that exist in theory alone.
An example would be, One of the most abstract fields of mathmetics is “number theory” or looking for patterns and constants in numbers. Someone working in number theory might be looking to see if they can find a definable pattern in when primes occur (so far it has been more or less impossible to put an equation to when a prime number occurs).
Now you may ask, “why work on something so abstract and purely theoretical” well sometimes that work becomes used to describe something real. For instance for hundreds of years mathematicians worked on a problem they found in the founding document of math “the elements” by Euclid. One part of it seemed to mostly apply, but their intuition told them something was wrong. Generations worked on this problem without being able to prove Euclid wrong. Eventually they realized the issue. Euclid was describing geometry on a perfectly flat surface. If we curve that surface and create spherical and hyperbolic geometry the assumption Euclid made was wrong, and our Intuition was right. Later we learned we can apply that geometry to how gravity warps space and time. Thus the theoretical came to describe reality.
Solve problems, that’s the essence of it. Some of them can be stated simply, like the Collatz Conjecture (iterate a function: on even numbers, divide by 2; on odd numbers, multiply by 3 then add 1; for any positive integer starting point, does it eventually reach the loop 4-2–1-4-2-1-etc.?), and some of them require more advanced knowledge, like the Reimann hypothesis (do all the non-trivial zeros of the analytic extended Reimann function satisfy Re(z)=-1/2?).
It might not be apparent why these problems are important, but their applications can be hidden in the real world and not known for years or decades or centuries. Fermat’s Little Theorem, for example, is why encryption on your computer works. Or, finding solutions of the Navier-Stokes equation is useful for fluid dynamics, which affects engineering of planes, cars, etc. On the flip side, we might never know if there’s a practical use for the Goldbach Conjecture or the Twin Primes Conjecture, but even if there isn’t there’s still the pursuit of knowledge, applying those methods to other problems.
Very broadly, you can classify mathematicians as either applied or theoretical.
Applied mathematicians generally start with real-world problems – like determining the optimal shape of an airplane wing, or predicting the path of a hurricane. They start with real-world measurements and observations, look at how those differ from what the existing math predicts, and help come up with better ways to model the real world using math. Sometimes those new models involve new equations or formulas that can’t be solved using existing techniques, so they figure out techniques to solve them.
Theoretical mathematicians generally start with interesting questions – things we don’t understand about math, even if we’re not quite sure if they’re going to be useful or not. One good way to do that is to generalize a concept. For example, take the factorial function n! = n x (n-1) x … x 2 x 1, for example 5! (“5 factorial”) is 5 x 4 x 3 x 2 x 1. It makes sense to take 5! or 29!, but you can’t take 2.7! – but why not? Some mathematicians wondered whether it was possible to generalize factorial to work for any number, not just whole numbers. It started with just curiosity but now their solution (the gamma function) is quite useful in solving some real-world problems.
Sometimes applied math doesn’t lead to new discoveries. Sometimes theoretical math doesn’t have real-world applications. And that’s okay. Also, the line between applied and theoretical isn’t that clear. There are many mathematicians who do some of both, or work on things that are somewhere in-between.
Whether applied or theoretical, essentially all mathematicians try to come up with new theorems with proofs. Basically they come up with a new mathematical solution to a problem that wasn’t solvable before, and they write a proof that their answer is correct. They publish these in journals and present their findings at conferences. Then other mathematicians can build on their solutions to ask new questions and find new answers. So the total knowledge we have in mathematics keeps growing.
There are some great unsolved problems in mathematics. Many of them are easy to state but despite the work of thousands or even millions of brilliant people, no solution has been found yet. Some of these questions are just curiosities, some of them would potentially unlock all sorts of real technological innovations if they could be solved. However, most mathematicians spend most of their time on less ambitious problems. A lot of mathematicians try to focus their career on an area – often an obscure one – that has lots of interesting questions and few answers so far, maximizing their chances they’ll be able to find a lot of answers.
The Man Who Knew Infinity: [https://www.youtube.com/watch?v=npcmIC-I7Ec](https://www.youtube.com/watch?v=npcmIC-I7Ec)
An excellent movie about mathematicians. Though the story is a bout Srinivasa Ramanujan, I found the character of G.H. Hardy as played by Jeremy Irons to be remarkable.
Mr. Hardy was one of the greatest English “pure” mathematicians who said “I have never done anything “useful”. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world”.
Of course, his many discoveries have had wide application in the physical sciences over time.
[https://en.wikipedia.org/wiki/G._H._Hardy](https://en.wikipedia.org/wiki/G._H._Hardy)
[https://en.wikipedia.org/wiki/Srinivasa_Ramanujan](https://en.wikipedia.org/wiki/Srinivasa_Ramanujan)
From what I’ve seen as a uni student,teaching math to undergrads and doing their own research on what they’re into math wise.Publishing papers is important too in the academic world.All the math profs I’ve seen are stressed out weirdos though tbh.
A somewhat long read, but great and classic on this topic:
[https://www.mimuw.edu.pl/~pawelst/rzut_oka/Zajecia_dla_MISH_2011-12/Lektury_files/LockhartsLament.pdf](https://www.mimuw.edu.pl/~pawelst/rzut_oka/Zajecia_dla_MISH_2011-12/Lektury_files/LockhartsLament.pdf)
At my previous company, we had a resident mathematician. Absolutely brilliant guy who specialized in developing physics simulations for us. He ended up solving and modeling a very niche physics problem in 3D that had previously only been done in 2D. This allowed us to design a system with optimal parameters along all 3 axes, something that would have been impossible to brute force by experimentation. It’s been a few years but I haven’t seen any papers on the topic so as far as I know it’s still a trade secret that I have to be purposefully vague about.
The best example I can give comes from VMware, a software company
The trade show blurb was “We have a room full of guys with pony tails that do math all day so you don’t have too”
During development of the software they ran into a series of insurmountable mathematical problems. Without thorough analysis the software developers would just have to guess what to do.
The math involved was so complicated that they needed a team of professionals with Doctorate degrees working on it for months to figure it out.
They hired a team of professional mathematicians to analyze statistical models and optimize how the software handled a multitude of different problems. They created new equations and algorithms to program into the software to analyze the data and make processing more efficient.
What did that translate to in the real world?
Significantly improved performance in the software and the ability to handle much larger workloads.
Some mathematics majors (and physics majors) actually end up working in the financial industry. With their ability to understand complex equations and systems of equations, they are good at calculating risks and developing derivative trading.
Source: my physics majoring roommate in college who now works on Wall Street
I work in IT have a degree in applied mathematics but also one in telecommunications systems management and work mostly as a Sys Admin but also i make models based on trends to help push for more allocation of resources etc.
You know how you can say everything with any language that doesn’t make sense?
Math is a language that has some rules which, if you stick to them, everything you say automatically makes sense.
Using this method can reveal solutions to lots of real-life problems.
A walk through my work day:
– Walk to work with coffee and a book or paper that I’m interested in. Sometimes I bring a newspaper instead to do the puzzles. (I love Sudoku and Kubok.)
– Check emails and spend maybe a half hour responding to anything relatively important.
– Attend various meetings or seminars with other mathematicians. Meetings are boring and usually do not help me directly. Seminars are fun but also frustrating. Math is hard and people are rarely good at communicating it.
– Spend some time grading. Arguably the worst part of teaching responsibilities.
– Prep for and teach any lessons. Usually things like calculus, abstract algebra, or graph theory.
– In what little free time remains, spend some time doing the thing I actually got into mathematics for: Thinking about neat problems. This usually involves reading carefully through papers and references, piecing together missing arguments, drawing diagrams, and trying to come up with new approaches to difficult problems.
– Go home, feed and walk the dog, and watch some TV with my family.
The specifics of my actual research are in topology and set theory. I spend a lot of time thinking about infinity and how it impacts various notions of closeness.
Mathematicians start by distilling a real life problem into its most fundamental bits, which sometimes is numbers, but often isn’t! (graphs, geometry, topology, for instance)
Then they take this abstracted form and study its properties, discovering and proving theorems. Every once in a while, a theorem is found that bridges two completely different areas of math, allowing you to use all of their theorems for your subject matter “for free”, as in not having to come up with them yourself.
Finally, you can use those theorems and apply them back to a real world problem, which lets you shortcut an absurd amount of manual work (often an impossibly large amount of work) to get to a solution.
The only reason why math research sometimes feels useless is that the uses are found on average some hundreds of years after the discoveries, so no one is alive to say “I told you so”.
Most of engineering uses stuff that Euler, Laplace and co. invented in the 1700s, and Einstein’s general theory of relativity is a relatively (ha) simple application of algebraic geometry, which he learned from his mathematician friend Marcel Grossmann. Nevermind number theory being useful in cryptography being completely unimaginable to the mathematicians who invented it hundreds of years before computers.
It is my opinion that if more people studied advanced mathematics, we would invent and discover amazing applications faster. But unfortunately it’s mostly left to professional mathematicians only.
Math is the art of making up rules and deducing implications of such rules. A lot of what we know and use today was invented at some point (like numbers base ten, addition, the zero etc.), so we have a good foundation for most things. Then mathematicians use these rules to derive facts that were not known before.
Others already write how this is important, so I’ll briefly talk about how they do it. Mostly, they sit down and read scientific publications by other mathematicians and then they try to apply ideas and techniques to the question at hand. Often this work is done by hand and on paper or on a blackboard.
Basically, you write all the information and rules down that appear to be relevant for the current step and then you try to deduce the next step. Usually, there is an open question like, “does this expression hold true?” and then the mathematician has an intuition whether they believe it to be true and they try to find a chain of arguments that proofs the answer correct.
In some areas of math, people heavily rely on computers as well. Some questions can be answered by a computer and mathematicians write programs to do so. Other questions can be answered under restricted assumptions and mathematicians write programs for those cases as well. Often the next step is to generalize these computational results by hand.
I could go on and on, but I’ll stop it here and reply to questions if there happen to be any.
Mathematicians take observations of the physical world and ’abstract’ or ‘generalize’ them into theorems. Here theorems make use of a limited number of axiom (basic ideas that are self-evidently true) and specific methods of providing that the axioms require the theorem to be correct. The number of basic axioms in mathematics is very small. About a dozen if I remember correctly. They ways of proving theorems is also small. About 4 methods. As theorems are proved, more abstract (and often more complex) theorems are proved from earlier theories.
You can think of this as having one pig and someone gives you another. When abstracted, this becomes 1+1=2. Multiplication is simply abstraction of addition. 3×4=4+4+4… or adding up 3 4s. And on it goes, becoming more abstract with each new idea/theory.
(This is what I do as a Mathematical physicist…)
They invent new languages and concepts in those languages.
They create procedures for solving problems using these languages.
Newton, Euler, and Pythagorus are all great mathematicians.
They help us put words and symbols (sometimes called numbers) towards abstract thought, and give us a way to perform computations on these numbers.
Unless this is a trick question…. Math?
Most people think mathematics is about numbers. They’re wrong. Mathematics is about what you can say about logical systems.
For example, if you have a set of strings of letters, can you chose valid strings so that if someone mixes an order or puts the wrong letter, you’d be able to tell and correct this? A mathematician working in codes would be able to figure this out.
Another example, if you are playing connect 4, will the first person always win playing perfectly? Or what is the best move for this? A mathematician working on games or search might be able to help you on this.
That’s not to say math doesn’t have numbers. Numbers are extremely expressive for logical systems. But just like how literature isn’t about the alphabet, math isn’t about the numbers. And when you have a question about a logical system, a mathematician can help you answer it.
Mathematicians do mathematics, just like scientists and engineers do. The difference between them is that scientists and engineers tend to push the applications, and be driven by them. Mathematicians are more focused on discovering what mathematics makes possible.
For example, there is a notion called an operator algebra that is a convenient way to describe things that happen in quantum mechanics. A physicist would use this notion to develop physical theories or to predict physical behaviors. A mathematician would tend to focus on things like, given an operator algebra, maybe with some extra conditions, what are the possibilities for what it actually is, and what are the ways that we can get information about its structure?
Some, although not all, of these explorations turn out to be useful, and some would be difficult to approach if the applications were the only driver. For example, one tool that is used to understand changes of symmetry in a physical system (also known as “phase changes”) is representation theory, but in order for this tool to be useful, a tool that was easy to understand (namely characters of abelian groups), had to be generalized to several increasingly more complicated structures (to arbitrary groups, and then to C^* algebras, Lie algebras, or some other notion depending on application). The physical application would be very difficult to make headway on without some mathematical groundwork already being laid.
The fraction of work done by mathematicians that is useful to science allows us to take larger leaps from what is known to what we would like to understand than would otherwise be possible.
Well, professional mathematicians work ridiculously hard every day through the most mind-numbingly difficult problems, while also writing lecture notes for their courses, delivering lectures and marking student’s work. They also have to constantly apply for funding by convincing people that what they do has value.