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).
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.
Engineers: let's work through this optimization problem to test and categorize the limits of materials for resistance to heat or vibration forces or fuels for energy density to choose the right materials for a rocket
Physicists: let's work through these multidimensional equations of velocity and distance so we can map the stars and build trajectories for getting from any system to another in the galaxy
Mathematicians: You're lost in a forest and you don't know where the boundaries are or which direction you're facing, let's try to calculate how hard it is to leave
I don't think bugs can be in a set. Bugs are physical and sets arent. Sets don't occupy physical space. I mean they cannot be seen or touched or observed by any experiments so we can conclude that they are not part of our world and bugs need physical space therefore they cannot be part of a set.
Defined don't equals existent I can be wrong but I am rather not part of your set because I am physical so it doesent exist or it contains object that is not me =( When you draw for example square its not really a square its physicala representation of it. And your object in your singleton is actually only mathematical representation of me.
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.
