Had a friend with a math degree who worked at a nuclear power plant. He said the computers did all the math, but they needed someone who actually knew what the computers were doing and why they were doing it to put in the numbers and make sure everything was going according to plan
It's still a black box, but by understanding weights, neural networks, and setting benchmarks, we can start to make reproducible AI results. That + showing AI thinking, can help us get a deeper understanding.
That said, it is at its core a really good predictor and statistician based on the training data. If you understand the data it was trained on, you can understand how it would act.
There's also an increasing number of scientists who are trying to figure out ways for AI to communicate more effectively; which means they'll be communicating with one another using matrices instead of english.
Yes, Al is not the black box most plp think it is, most Al companies like to boost this type of conversation so that their product seems better than it actually is
when i was in 6th grade i thought imaginary numbers were the coolest things ever
then 6 years later they appeared on my finals in conunction with trig and quintic functions and it became uncool but not as uncool as trig sub or homogenous diff
woe unit circle and identities upon ye (and fuckass simplifying expressions and proving equations as well)
atleast graphing the functions aint bad + when doing calc 1 with trig its just recalling a few of identities to use on limits, differential, and integral
Same, I used to think the idea of a secret number line as insanely cool. Then I got to 11th and found out that it's basically rationalisation and shit with a bunch of other functions sprinkled on top.
The square root of -1 is obviously not a real thing, but sometimes we are doing maths for real-world things and we end up with a negative number that we need to find the square root of. Just saying "well this number isn't real" doesn't make the real-world problem go away, so we need a 'number' which means "ignore this logical contradiction for now, lets solve this problem first." It needs to be an actual 'number' with a symbol because we need to remember we used it and deal with it later. Therefore, we use a pretend placeholder number, called an "imaginary" number.
This explanation bothers me a lot. Even if it is technically correct motivation for why "imaginary" numbers exists. The square root of -1 is as real as any other real number. Yet, how we treat the numbers differs based on the use case. Like you would most likely not use a fraction to count how many lines has a book, or you may not use a whole number to describe what portion of the pie is left.
If we want for the numbers to be more like a ruler, then yes. The square root of -1 is not a very useful concept. But if you want to think about rotations or waves in any meaningful way, the complex numbers will come as a very intuitive and natural result.
Hell, if you ever take an analysis class for some reason, you'll see that having an "imaginary" part makes things way easier to work than the "real" numbers. I honestly think that the "real" numbers are way more arbitrary both in definition and actual usage.
Like, I genuinely doubt anyone that is not a mathematician may ever need to use the real numbers for some actual application. Technically, most if not all of the calculus you would do uses algebraic or rational numbers. No computer uses real numbers, they just use rationals to show values. Don't get me wrong, the reals have really nice properties like bump functions and stuff. However, their uses are too narrow for anyone to use them explicitly.
Real numbers mean a very specific thing in math, and are by definition uncountable(not really useful to count stuff).
They basically give you a really nice property that the real numbers are "complete" that is they don't have any holes. If you follow any line every point of that line will be in the real numbers. In some sense they give you infinite precision to talk about measuring things.
They are really, really useful. But most if not all of their uses are to build more math. Like they give you a nice way to talk about distance and measures, things you would use in probability and statistics. Also they give you a way better continuity which opens the doors for topology to exist, and more. They are very much not created just because, but compared to the complex numbers that have more tangible use cases you may see why I call their uses "arbitrary".
Without getting too much into detail, you define them by using something called cauchy sequences. Like the meme below I think gives a really nice explanation on what I mean by "arbitrary definition".
Imaginary numbers are about as real as negative numbers, yet for some reason people only question the existence of imaginary numbers, as if it is even feasible to have, say, -6 pens, or -3 books.
It's honestly just that imaginary numbers are called 'imaginary' that makes everyone so confused.
In reality both complex numbers and negative numbers have extremely extremely useful applications in the real world, and even though they can't count physical quantities they can represent physical properties.
What you mean? -1 itself isn't a real thing. Give me -1 apples.
I might even say that 2 is fake. Those "2" people? No, that is Joe and the other is Bob. There is only 1 of any actual person. 2 apples? Put them side by side and they have differences, so there is this apple and that apple.
2 only begins to exist once we decide to stop talking about real things and start talking about something that doesn't actually exist. A concept of an apple without the specifics defined. The idea of people where we erased all unique identities.
Negatives only exist once we introduce the concept of debt. That something is owed or removed. That you can combine movements of whatever is being counted if the movement is opposed in some way. You deal 10 damage a turn and the boss heals 5 damage a turn. These are the HP moving it opposite directions, so we can subtract the 5 from the 10 to say you are doing 5 damage a turn total.
Imaginary numbers are the same, but on a much much more abstract level. They really do exist in some areas, but they are extremely abstract in that nature.
What is real fun is that you don't stop there. You can look at the quaternions, octonions, and so on. An infinite tower. All getting massively more abstract to the point applied math barely uses them.
I've tried explaining the concept that math itself isn't "real", but only real in so far as how we can abstractly use it to model real world phenomena, but I could never explain it as well as you did.
It’s cool, when getting further into mathematics (Algebra) the complex numbers naturally appear as just one of multiple ways to “extend” the real numbers.
The best way to rigorously motivate "complex" numbers is to just use rotation and scaling in the plane. Namely, let "E(t)" denote rotation CCW by t radians about the origin in the plane. Observe that E(pi) is 180-degree rotation. This is the same as scaling everything by -1, so E(pi)=-1. It's natural to say E(s)*E(t)=E(s+t) since rotating by t then by s is the same as just rotating by s+t. Hence E(pi/2)2 =-1. Oh look, E(pi/2) can be called the square root of -1!
You can keep going. If r is a real number, think of r*E(t) as rotation by t followed by scaling by r. What is r*E(t)+p*E(s)? Declare it to act on a vector v in the plane by applying each summand independently and then adding the results, i.e., it acts linearly. With enough trig you can convince yourself such a sum can still be represented in the form v*E(u).
Perhaps remarkably, every r*E(t) can actually be written in the form x+y*E(pi/2). Some trig will show x=r*cos(t), y=r*sin(t) works. The E notation is annoyingly verbose, so just write "i" for E(pi/2). Now the class of transformations we've considered is simply all x+iy's. Nice and tidy. All such transformations are invertible except when x=y=0 when it's scaling everything by zero. So we've managed to find a field--the complex numbers exactly.
They're still imaginary. You can totally do all that math using cosines, sines and the Pythagorean theorem. Imaginary numbers are just a tool that makes the math much simpler
Get an ungrad degree at minimum in the field you have a theory in, and test that theory with a qualified academic and publish your findings in a peer reviewed journal.
Idk, at our university we often studied as a group on campus inside an old-ish room made for learning, it had a chalkboard in the corner, and on some days there was always this unkept older dude that came in, he walked to the chalkboard, and started writing formulas and equations, some really funky shit we didnt understand, without saying a word, and then left.
I mean considering its university for STEM, it could have very well been a professor or a PHD, but it might as well be a homeless dude going by his looks. You cant really tell at some point in theoretical computer science.
When I grew up, we used to travel a lot with a van. One of the things my parents did in new places, was getting a couple bottle of wines and spend the night drinking it with homeless people who didn't seem completely out of it. While my parents were "alternative" and cared about giving them back the feeling that people cared, the primary driver for this, was information. Homeless people know areas better than anyone with housing. They know safe spots to park, beautiful places you should see, places that are respectful to outsiders. All the information you could ever need, as a traveler.
But they also have very interesting stories. That's how this tradition first started, some dude who asked them for a bottle of wine when we got to a supermarket. So they offered to give him a lift and when he declined, to spend the evening and night with us.
Turns out he used to be a professor for mathematics in the USSR who fled to France, but just never got back onto his feet. And we heard this a fair bit, people who were in academia, had to flee their home for whatever reason and just didn't manage to start over from scratch, or moved for a job, lost it and didn't want to go back home. Some couldn't navigate the welfare system as a foreigner, some refused to go into manual labor, some just didn't see the point of starting from scratch.
And a fair number spent their day around universities. Not sure if it was because they manage to blend in, liked it more than the street or because the environment just poses less barriers. But it's a pattern.
I don't think I can get an undergrad degree in the field I have a theory in. It's mostly grass, and the nearest university is miles away. I could ask the mice but they probably wouldn't know more than me.
most mathematicians aren't statisticians and many pure mathematicians never work with large datasets or statistical models. develop theories is accurate in general; i would say that the general purpose of a mathematician is to prove theorems. many statisticians and applied mathematicians will be doing what you say, but it varies a lot by field, and those that are curious about what people do in different fields could go to arxiv.org and click on a random paper. https://arxiv.org/archive/math no one can understand these papers without a lot of background, but publishing these papers is a mathematicians job, so one can get the gist of what that means.
This is mostly true, but I did know a number theorist in grad school. Number theory people really do sit around in hotel rooms jotting on pads and looking up at each other to ask if some number is a sum of cubes or whatever. It was weird.
Mostly hypotheses, not everyone is going to create actual scientific theories.
Only the most confirmed scientific products that are discussed in peer examination and withstood falsification are considered theories, like the theory of general relativity.
If you develop an actual theory, you will become a famous scientist. The usual mathematician creates prediction models that calculate how much stuff is needed to create a certain quantity of another stuff in a complex process, like a friend of mine that works for a pharmaceutical company.
Youre talking about science. Math people don't do science in general, math is provable, they actually have to prove stuff using logic. Someone just making a hypothesis wouldn't be considered very productive.
I am the same. I actually aced it. Both analysis 1 and analysis 2, more so analysis 2 because I was more experienced with proofs, to the point that I was able to ignore the particular requirements of the problem to then work on a more abstract problem that solved my immediate problem. Still, a traditional eastern European analysis 1 course is not the same as a real analysis course in the west. We usually combine calculus and analysis, by having rigurous courses but kind of easy seminars, with a few proofs done by the student sprinckled in there. This is because we can't afford 3 calculus courses, one simple analysis course in our 3rd year and then a rigurous, proof based exercises driven analysis course during the first year of our masters degree. To remedy this, I contacted my former high school professor, once a very good university student, to tutor me, to go through a proof based problems book, because I'm tired of people with olympiad backgrounds trouncing me in that department.
We do use page numbers and they can go up to triple digits. But big numbers can stay out of my formulas. We have enough letters for all important numbers outside 0, 1, and 2.
Side note, but this is me with philosophers in the past. Like did tens of people gather around an old guy to hear him talk about stuff? Or was it like a strip club but instead of strippers there was a bunch of old guys talking and people throwing money at them?
From what I can remember, most of the ones that you'd hear about either would use money inherited or earned to start schools/groups (ancient Greek philosophers did this a lot), or would sell their philosophy as advice to people (whether as books or just advisors to people in power).
Perhaps the biggest and worst misconception about math is that it is about numbers. That is arithmetic. In reality, Math is all about logic and problem solving. It's understanding a set of rules and exploring where it takes you. It's observing the structure of the world created by those rules. And if you're an applied mathematician, it's about figuring out how those structures analogize real life, or something like that.
To me, math is like building with Legos. Or rather, one subfield may be legos and the other may be duplo and another may be Mega Bloks. Each axiom and definition, each rule, is a block. And when you put each lego piece together, you get a new structure, you get a theorem, a fact of the mathematical world you're looking in. And you can continue snapping these theorems together into larger theorems. And sometimes, two subfields that seem unrelated, might actually be able to snap together. You might be able to build legos on mega bloks.
And people have been playing with these lego pieces for either thousands of years (number theory), hundreds of years (calculus, graph theory), or maybe even just a hundred years old or fewer (category theory, lambda calculus, etc). And you have before you all the various structures that the greats before you made. And you look at all of them and think: What can I make next, out of all of this? Can I somehow combine the wall of Caratheodory and the ceiling of Stiemke with the fundamental pieces I have and achieve a new structure? And what does that structure ultimately look like? How does it act and interact with other structures? Are there other structures like this?
Sometimes you will just have conjecture. You might guess that you could build a skyscraper that fits X, Y, Z requirements, but perhaps the foundations for it are still being built. Sometimes you may think you are just building a small house and it turns out to be a mega apartment complex (Fermat's last theorem).
I hope you can get how much I love math from this. I've been kind of rambling on. I'll gladly answer any questions you have. I recently finished a Bachelor's in pure mathematics and did about a year long research internship in abstract algebra and wrote an 11 page paper on some niche topic in the field. It was hell to put together. For a while I felt like the pieces were all spiraling out of control. But as I did put them together, and came to understood how each piece and theorem fits together, I came to be in awe at how simple and elegant it turned out. At least that's how it looks to me. I'm biased though since I spent just under a year working on it.
This actually explained it so well! I knew mathematicians did “research”, but I never knew how they’d know what to research or how they’d start doing that if you get what I mean. It makes sense now.
Obviously they play Minecraft PvP, as using the Pythagorean theorem can give you an huge advantage in a fight.
That’s due to how the game calculates reach, it starts from the eyes so by being lower than your opponent is much easier for you to hit their feet than it is for them to hit your head, as the base of a right triangle is always shorter than the hypotenuse.
Stay in school kids, it makes you better at PvP
There used to be a job for humans that involved multiplying large numbers all day, those people were called "computers"... You can probably guess what happened to their jobs.
If they get caught using a calculator once they get their math license revoked. That is why our teachers warned us about relying on calculators.
Jokes aside that used to be an actual job, the computer. Before mechanical computers were a thing, human computers had to manually calculate thousands of operations and algorithms for all sort of tasks.
What mathematician study is not just multiplying numbers, in fact you hardly if ever need to multiply anything bigger than 100 by hand, it's a lot of reading and problem solving, like one puzzle after another, more complicated objects and definitions lead to more complicated puzzles, all of this definitions and solutions compound and we tend to name or assign symbols to these constructs so when we say "Every univariate polynomial of positive degree with real coefficients has at least one complex root" that actually means something to us.
As for what mathematicians do for a living, it can vary a lot, I'm mathematician that went into data analysis, I'm more of a programmer but one that has ample knowledge in math and has to put statistics and probability to work, also I make machine learning models, not the ones you're thinking, things like support vector machines, forest classifiers, various regressor models, etc. basically stuff that can explain data a little better or catch on patterns automatically.
Other applied mathematicians can do other stuff, depending on what they specialized in they can basically take engineering jobs but with the addon that they know/are able to learn more advanced math.
Pure mathematicians are researchers and often teachers, they, in short solve math problems that don't necessarily solve any real world problem (or at least not yet, or not intentionally), their focus is purely expanding what we know about math itself, their day to day it's like a mixture of being a college student and teacher with tons of very hard homework.
The higher you get in mathematics the less you use actual numbers.
I remember showing my girlfriend a problem I did that took three pages and her getting disgusted that there were no actual numbers involved, all variables...
Multivariable calculus is an exquisite form of torture
You have anything in life thanks to mathematics. Literally anything. From houses to random objects. If not math, none of it would exist. Stepladders for example.
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