True, but I mainly mean in terms of their attitude towards research and their level of skepticism and critical thinking when presented with new information.
Engineers are always thinking in terms of "how can I make this work?" and scientists are trained to think in terms of "where does this theory/method break?"
This means that in general, engineers are far more likely to assume one positive result is significant, whereas scientists are far more likely to be looking at and poking holes in experiment methodology. This is a generalization, but in my experience, engineers are far more likely to fall for pseudoscience BS. Granted, my experience is mostly in chemistry and chemical engineering, but this idea in general has been a topic of discussion and research in peer-reviewed literature for years.
Similarly, from an engineer's perspective, scientists are a great addition to the working group when you need to find the flaws in the system, but awful when you actually just need something to go into the real world and work 80% of the time ;)
Doing research, I used to work with mathematicians, engineers AND physicists on a daily basis for years. Physicists were the least fun. Most of them seemed to think of themselves as a sort of Jesuits of Science. As in: "I just figured this out, and already it's set in stone, why do you even argue with me?" Mathematicians and engineers were a lot humbler, more down-to-earth. Also, some of them were astonishingly edgy in a very positive way.
There are different kinds of physics researchers and it doesn't look like what physics lessons show in university, which is mostly theory. Most are not theoricians, they work on experiments and analyze results, they design and build instruments similarly to engineers. It seems the main difference is the kind of question they want to answer to: scientific question vs client need.
Mathematician: this is category theory. No, it didn’t have anything to do with categorization, it just helps us understand how spaces can map to each other. Yeah I guess it’s kinda like graph theory or algebra, but not really. We made a category of graphs, and you can use the category of graphs to represent endofunctors on the category of categories.
Also no! The “graphing calculator” is an abomination that should be more rightly called a plotting calculator. But that’s what happens when you let engineers in Texas name something.
I've read a fair few unintuitive mathematical things, but category theory has so far been the worst. Some things are just plain unintuitive and don't catch your attention. Then there are things that are intuitive and really do reel you in. Finally there are things that seem intuitive but become so complex that your comprehension inverses: what you thought you knew feels wrong because of the new things you learned.
The latter has been my experience with category theory.
It's a strange feeling to think you understand what you are reading until you get to the end, but you have given me that feeling. I was like "yeah category that's a word I know. Let's math the hell out of some categories." Then I recognized other words you said, but by the time I was at the end of your post I wasn't sure if I understood anything.
I don't mind feeling dumb. Honestly it helps keep my narcissism in check. I like math because I don't understand all of it even though it should be logical.
Oh. Neat. Like a sort of language interpreter trying to explain an analogy from one language to another? The words might not mean the same when it comes down to the word for word interpretation, but the idea behind them is what matters.
Pure mathematician here - some of us argue "mathematics is a language", others of us argue "language gets in the way of mathematics".
The latter feels much more true; as a species we're absolutely awful when it comes to talking about abstract things. The thing is, those abstract things are often VERY interesting.
It's like making a map and being fascinated with the type of trees rather than the shape of the land, because the types of trees tell us about the climate, soil, and even history of the land.
I would say a important part of my job is to find the appropriate mathematical language to model computer programs. In my experience, using efficient language not only helps us discover more structures and connections between different kinds of program, but also leads to efficient and simplistic real-world implementations.
I would argue, from observing the development of this field, It seems like picking the right mathematical language is essential whether you are interested in theory or practicality.
I am not a mathematician, perhaps you can comment on this. From what I read, I feel like a good amount of the achievement for Grothendieck stems from finding the right language to describe the given problem. The result sometimes will follow like magic, once the correct language is discovered.
See now, I'd argue that the language comes after the mathematics. For example, I walk to work each day; part of walking to work is trying to find the route that lets me lie in the longest.
Now, humans are pretty good at exploring and finding alternative routes between locations, and they also tend to locate the shortest route given enough time.
Trying to explain how this intuitive activity works necessitates the use of graph theory. The graph theory was something our brain had constructed in the background, but it wasn't entirely conscious. Trying to explain this in natural language would take pages, however...
Given a set V of street intersections, and a set E of streets connecting two intersections, and a set W of weights assigned to each E. I can calculate the shortest route by applying one of the pathfinding algorithms (which are expressed in this notation).
This explanation will cover any pathfinding problem, but it's not great at conveying what is a really happening. The language we must use gets in the way of conveying the mathematics that is going on.
We do need a language (telepathy not being on the menu), but that language is a separate entity from the mathematics itself.
There are "mathematical languages", but these are present to describe mathematics. There are mathematical theories of language, but again the language itself is not mathematics - its structure, however, has mathematical properties.
I suppose you could say "fire has the property of being hot, but it isn't hotness itself"? Language is used to communicate mathematics, but it is not mathematics itself.
Now, this is not to discount notational developments in easing communication - that's a great branch as you have to check your new language and its rules match the mathematics it tries to describe. However, again, it's important not to conflate the thing you are describing with the thing you are using to describe it!
So, let's say you write down the words "fire is a chain reaction between carbon and oxygen that produces heat". You've characterised fire yes, but is that sentence itself the fire?
Let's say you write down the equation describing this reaction so you can play with it and manipulate it. Is this fire, or just a convenient way to talk about it?
I'd argue neither of these are fire, and both will never completely describe a fire (though they come damn close).
I'm a phd chemist who does safety work for (mostly) engineers. I get a lot of "but you can do quantum physics, this should be easy".
I always reply that it's just basic maths, anyone who graduated highschool can "do" quantum physics. But I'm convinced all the people who say they can visualize whats going on are just liars. But then, that's also how I feel about FEM, so what do I know.
I think you just have to differentiate whether you want to do mathematically rigorous QM (which gets arbitrarily hard), or just do useful calculations.
Interesting. That’s not how I was taught (different time, different language). A set that has some boundary points not being part of a set is open. Otherwise it is closed. It was binary definition. A 1D-sphere (a circle) was classified as a closed set. No boundary. But I looked in google and now it is different.
Wait are we supposed to be making super precise blueprints? They never build what I draw so I just give rough dimensions on a sketch and specify the important bits
I mean there’s not that much precision needed to pick out the toppings on a cheeseburger. You don’t need to specify the mass of the pickles man we do this all day.
I've taken multiple advanced trigonometry courses and still can't really say what trigonometry is. Mathematics is just the fake thing that made puzzle kids feel smart before chess was invented. Oh wow you can make little symbols and they're a special language only you can speak showing how clever you are. Neat they make a circle I thought I could draw one of those but I need a fucking PhD apparently.
I've ended up using calculus and trig for programming multiple times.
You may be able to draw a circle without math, but teaching a computer to draw a circle requires an understanding of math.
All of machine learning is rooted in linear algebra, rust is a very practical programming language that gains most of its power through category theory.
You don't need to know high level math to be a successful developer, but it can really help in many areas. I can't really think of how to categorize which areas high level math is more or less likely to show up in, which I guess itself kind of supports my point.
Just understanding what a derivative is and what an integral is can help you determine what problems are solvable and what aren't, and let you think ahead about what information you might want to hold onto in your data structures. ( Think about what the +C in this integral represents in the real world, and what data you need to pin that down concretely ).
It is valid to criticize how our current society disproportionately economically reward STEM fields while ignoring social sciences, philosophy, anthropology, etc. and thus often creating these math nerd types who are simultaneously racist or reactionary idiots, but (assuming you're being serious) dismissing math as "fake" only reads as very bitter
I would argue our society disproportionately economically rewards TE fields and S&M (no not like that) get lip service because if they didn't get a mention it would be far too obvious how disconnected economic value is from societal value
