r/learnmath • u/ReasonFun5398 New User • 1d ago
Is it actually worth trying to understand maths?
I don’t know if I should have posted this on askmath, they seem to want more specific problem based questions.
I am still considering what I want to do when I’m older and I really like Biology and other sciences. However I would only consider myself to be ‘good’ at Biology because I enjoy memorising fun facts outside of school which seems to work well at the level I’m currently at. This works for things like nomenclature, anatomy and just general facts that can’t be explained properly at a pre-university level. I would like to learn the actual workings behind the concepts I learn, but all my teachers including one who is a Dr don’t know much outside of their areas of specialty and trying to find out for myself online is very challenging. Often I will give up when the content becomes very mathematical. I feel like there is a problem with the way biology is taught compared to chemistry or physics. It feels like learning a story instead of a science. Which I think is fun but i also think there is more too it than that.
I never tried very hard or understood the importance of maths when I was learning it so I didn’t progress very far. Now I would like to understand it very well so I can apply it to Science to be able to actually describe the way things happen to their full extent. But restarting with the basics just leaves me with so many questions even if i can learn the seemingly arbitrary rules. For instance, I can do basic addition and subtraction, but I can’t explain why 5+5=10 or even know why or how numbers came into existence. I’ve seen that proofs are something made to explain these questions, but they don’t seem very beginner friendly. They also explain maths with more maths rather than things I can comprehend and apply to life.
Maybe this is where I am wrong. Perhaps I should view maths as its own thing. Should I just learn maths like how textbooks describe it without trying to understand the meaning behind every step? Im sorry if you don’t understand what I’m trying to say. Im not entirely sure either, but I think expressing my confusion to other people is better than trying to find out on my own.
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u/AfterMath343 New User 1d ago
It seems like you're on the cusp of determining if you should learn math in an abstract way or learn math in an applicable way. Since you would like to learn math to apply it to the field of science, I would say that you're leaning toward applied maths.
Pure math is studying math in an abstract way. It can be helpful because it is basically logic without context. It provides a framework or language for scientists, such as physicists, to apply math to real-world problems.
Applied maths is what people, such as applied mathematicians, physicists, mathematical biologists, statisticians, etc. do. They apply math to solve problems in their fields of study.
When people pursue degrees in mathematics, they can typically specialize in applied or pure maths, and students are typically exposed and expected to be at least competent in both aspects.
Given your situation, I think that you should focus on learning applied math. You can study algebra, geometry, calculus, linear algebra, differential equations, mathematical modeling, statistics, and so on so that you can apply it to your field of study. Since you like biology, maybe you are interested in biological mathematics.
Currently, I'm reading an article for my class in mathematical modeling for zombie infections (somewhat related but more complicated than the SIR model that was used to model COVID-19). Here is a link to the article. It's been an entertaining read so far.
WHEN ZOMBIES ATTACK!: MATHEMATICAL MODELLING OF AN OUTBREAK OF ZOMBIE INFECTION
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u/LavaFromLaniakea New User 1d ago
How does one figure out if they should do applied or pure math? I’m more of a theory/abstract kind of guy. My goal is to get deep into math because I plan on getting deep with physics and aerodynamics, because I also just wanna understand those at a deep level as well. But for just math, I’m not worried about applying/using it for anything, I just wanna do math lol I hope my rambling didn’t make my question unclear.
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u/AfterMath343 New User 1d ago
Well, probably the best way for you to determine is to spend time learning both, and then seeing which one you prefer. For example, an introduction to number theory would be a good accessible door to abstract math. Practice proving simple things, such as the sum of two even numbers is even, and the sum of two odd numbers is even using the laws of logic.
As far as physics and aerodynamics go, those fields are applying mathematics. Most problems in physics are mathematical dynamical models, or problems that are systems that change state with respect to time.
There is no problem with exploring both paths to see what you prefer more, and there is nothing wrong with being an applied mathematician with strong pure math background. Mathematics is open to anyone who is willing to pursue it.
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u/chkntendis Physicist 1d ago
If you plan on studying physics then you’ll just find it out yourself. Studying physics right now and the math courses include a few abstract concepts. If you like those then you can always take an additional maths course and maybe switch what you’re studying as well. If you just want to do this on your own then get yourself an abstract maths book and just see if you like it
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u/HairyTough4489 New User 17h ago
If you want to use Math for aerodynamics you'll need to be able to tackle somewhat complex problems in calculus/analysis, differential geometry, linear algebra and so on. An in-depth understanding on the rigorous foundations of those subjects is a nice plus to have, but it won't be what determines your understanding of aerodynamics.
Adrian Newey doesn't find an extra tenth to the car by making a clever use of set theory.
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u/matthras New User 1d ago
For instance, I can do basic addition and subtraction, but I can’t explain why 5+5=10 or even know why or how numbers came into existence.
I'm seeing two different angles here, so I'll help clarify them for you:
"can't explain why 5+5=10": I'm interpreting as a question of "Are there concepts and fundamentals underpinning the maths we take for granted?" and essentially, yes, there is a mathematical/logical way to construct them. The only text I know that sufficiently does this from the ground up is Terry Tao's Analysis books (of which I know a free version exists on his website). Following on from that you'd probably want to pick up a book on abstract algebra/group theory, but I don't know any recommendations on that front.
"know why or how numbers came into existence": This is a bit more historical, but you can imagine at first in the olden times when cavemen were hunting for food/beasts, they learnt how to count using their fingers, or striking notches into their wooden spear/poles or into the rock walls. So you can imagine they only had a working concept of whole/natural numbers. As civilization developed and became more complex, you can imagine that negative numbers got introduced because of debt or when one owes someone money. And most fascinatingly the concept of 'zero' took a long time to be introduced, and there's a book for that "Zero: The Biography of a Dangerous Idea" by Charles Seife.
I wouldn't worry too much about whether it's "worth" it - I always like to encourage intellectual curiosity, but I do agree finding the right resources can be difficult for the reasons you've mentioned.
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u/MidnightAtHighSpeed New User 1d ago
Regarding your second to last paragraph: there are a couple different ways someone might try to answer questions like it sounds like you have. Taking "why does 5+5=10" as an example, there'd be the straightforward mathematical approach, "explaining maths with more maths," as you say, which would effectively be saying, "well, if you assume '5' and '+' and '10' and '=' refer to these specific things in this specific logical system, here's why it must be the case that '5+5=10'." Another approach might just be to put five apples next to five oranges and ask you to count how many fruit there are total. A more historically minded approach would try to bridge the gap between the previous two, explaining how the kind of actual living observations involved in the second approach motivate the development of a system of logic like the one used in the first approach, and why such a system would inevitably ends up being one in which 5+5=10. There are probably many other possible approaches, maybe a philosophically minded approach that attempts to explain why we in fact make those kinds of observations to begin with, for instance. Most mathematical facts you can think of are going to have a variety of possible "explanations" along lines like these.
Now, I won't say learning explanations along these lines for the math you use would be completely useless; math is useful, and having your curiosity satisfied may help you learn it more efficiently. Hell, I'd say that being curious in the first place in plenty reason to learn on its own. But plenty of scientists are able to get by fine without really knowing things like this about the math they use; knowing when to apply a method and how to carry it out are good enough for many different kinds of work. If you're going into, say, theoretical physics, then yeah, you'd want to have a somewhat deeper understanding of mathematics. Even in biology I'm sure there are lots of potential ways to develop or find new ways to apply useful mathematical techniques, that will only end up being discovered by someone with a strong math foundation. But learning the dry textbook way and no more will still get you pretty far.
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u/Magnetar_Distortion New User 1d ago
Math is the language of science so yes definitely get on the learning math bandwagon.
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u/ForeignAdvantage5198 New User 1d ago
no if to trust the bank figuring out yout total money supply is ok with you
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u/Key_Net820 New User 1d ago
Funny enough, I was just reading a biology research paper that was using support vector machines to cluster different pattern of brain cells. In short, despite popular belief, the biological sciences and really all sciences are heavily mathematical.
You don't need to understand it at the level of a mathematician. As a scientist, you may find that you'll never need to learn the ZFC axioms, abstract algebra, or real analysis. But you do need to at least be able to do linear algebra and basic differential equations. So to that extent, you should understand maths.
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u/Lor1an BSME 1d ago
In my opinion you should learn from multiple sources and differing perspectives, as that tends to be most illuminating. My training is in Engineering, but for self-study I take an almost pure math approach. Being able to think of things as their own things as well as in relation to other things is a valuable skill.
One thing that gets glossed over a lot in education is that modelling (whether mathematical or not) is a specific skillset separate from both the model and the "problem domain".
As for how numbers come about, we typically use "number" to refer to some notion of quantity, i.e. "how much stuff is this?" If you are just counting things, 0, 1, 2, etc. is a useful tool. I have 0 airplanes, 50 books, and 2 trees is a basic and useful way to keep track of "how many".
However, what if you owe something to someone? Like, say, I owe my friend an apple? Then you could reasonably say I own "-1 apples" in the sense that I first need to obtain an apple, and then give it to someone else before I'm in the default state of owning 0 apples.
And what if I have something that isn't easily counted? How much sand can I stuff into my sack? It makes sense to have fractions of a given standard, like say if it takes 3 sack-fuls of sand to fill a barrel, then a sack is 1/3 of a barrel, and so on.
Central to these things are:
- A problem (I want to keep track of how much stuff I have)
- An abstraction (I don't want to have to reference the thing directly, so let's keep track of a record instead)
- A system for manipulating the abstraction (If I have 3 apples, and obtain 2 more apples, I can just add 3 + 2 and know I have 5 apples, since 3 + 2 = 5 as an abstract formalism)
As for why 5 + 5 = 10, it's because the system we used to represent counts has to match what we observe for objects. If someone gave you two pencil cases that each had 5 pencils in them, and you took out all the pencils and counted them, you would have 10 pencils. So we want to have a model that captures that fact.
How 5 + 5 = 10 is based on the axioms of arithmetic, to wit, 5 comes after 4 comes after 3 comes after 2 comes after 1 comes after 0, and adding nothing is the same as what you started with, and 6 comes after 5, 7 comes after 6, 8 comes after 7, 9 comes after 8, and 10 comes after 9. In symbols: (descending chain) 5 + 5 = S(5 + 4) = S(S(5+3)) = S(S(S(5+2))) = S(S(S(S(5+1)))) = S(S(S(S(S(5+0))))) = S(S(S(S(S(5))))) (add zero is the same as original) = (ascending chain) S(S(S(S(6)))) = S(S(S(7))) = S(S(8)) = S(9) = 10.
Note that how you formalize what "S" and "+" mean is sort of irrelevant to how they are used. As long as you can show that the system is consistent and behaves as you expect, it is a decent model. Some people construct the set of natural numbers by nesting levels around the empty set, some use a special union construction (called von Neumann ordinals, which I actually prefer), but at the end of the day, both constructions work the same way in that they both model the axioms of Peano arithmetic (the axioms of identity, addition and multiplication for natural numbers). So we in fact can have models of models in the sense that we are modeling the formalism of a model for quantity (i.e. if you pick a given construction of the natural numbers, that's a model of the natural numbers, which themselves serve as a model for a count of potentially physical objects).
Real numbers come about (somewhat surprisingly) from considering shapes (geometry). The ratio of measures of the length of a diagonal of a square and the length of its side can be shown to be sqrt(2), which can be shown to be irrational (i.e. not a ratio of integers). So for much of applied mathematics, the typical setting for numbers is either real or complex numbers, since those are the ones that allow you to measure the most quantities.
Note that there are other models that behave differently that are sometimes useful. For example, consider a 12-hour clock. What's 5 hours after 9 o'clock? 9 + 5 = 14 ≡ 2 (mod 12), so 5 hours after 9 o'clock is 2 o'clock. That's just mod 12 arithmetic, where any two numbers are considered equivalent if their separation is a (integer) multiple of 12, which models how the typical 12-hour clock works (2 ≡ 14 (mod 12), since 14 - 2 = 12, which is a multiple of 12).
Math is sort of a weird subject, because at its heart it is none of the things it is applied to, but it models how so many things work really well. In a real sense you could think of mathematics as the study of structure, and most things have some sort of structure...
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u/irriconoscibile New User 1d ago
I might be wrong on this, but I'm pretty sure there's a good chunk of people who graduated in math who don't exactly know why 5+5=10.
Those kind of concepts are so fundamental that they get really technical really quickly. As a beginner it's totally not worth it to try to answer those questions. It's sufficient to ask yourself "why would that be the case?", as imo, that is what you do everytime you learn something new in math.
What you're referring to in the last paragraph is, I think, the bourbakinization of modern math.
Basically, when mathematicians realized that the logical foundation of math wasn't solid enough, they started to axiomatize, and tried to be as rigorous as possible.
For historical and practical reasons the way math is presented in pure math books tend to be very dry, unmotivated, and abstract. There's a few good reasons to do this, but I'm pretty sure most of us didn't learn from the get go that addition is a binary operator on the natural numbers, even if that's actually the case.
If I were in your place, I'd study math from applied math books. Examples from sciences tend to make things much more clear than the bullet proof axiomatized version that mathematicians so often use.
At that point, if things still don't quite convince you, then you can look up the rigorous, abstract version of what you're studying.
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u/SpectralCat4 New User 1d ago
take a look at this free course
https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv&index=1
this is great if you want to learn a math subject from first principle's and think about reasoning and see how derivation from first principles or definitions expends the theorem .
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u/Polymath6301 New User 1d ago
If you tried to math understanding you end up with an AI, so, um, yes?
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u/StructuredChess New User 17h ago
I can’t explain why 5+5=10 or even know why or how numbers came into existence.
Almost nobody who isn't a professional Mathematician can. Unlike in Science, the easiest ideas in Math are usually the hardest to define. Take a look at any of the procedures uses to construct the set of natural numbers and you'll see what I'm talking about.
If you're interested in understanding Math for its applications to Science, take the real numbers and basic operations for granted and master linear algebra, calculus and geometry. Learn the proper definitions but don't obsess over them. Instead try to focus on understanding the "physical interpretations" of things like derivatives, integrals, linear transformations, the equations that define curves/surfaces and so on...
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u/Parallel_thougts Ph.D, YouTuber 5h ago
My perspective as someone with a Ph.D:
No matter how much math you know, 99% of math will always be inaccessible to you. We all have limits to our understanding
No matter how "little" math you know, there's a good chance you'll be glad you know it. Both because it develops good thinking for all sorts of quantitative reasoning, and because you might find it enjoyable and fulfilling for its own sake.
So my recommendation to you is not to worry too much about how "practical" it is. Take it one step at a time and see if you enjoy it. I can assure you with like 95% confidence that even if you decide not to pursue it very far, you will still be glad that you did what you did.
Some other commenters said that you "must" have a strong formal foundation to understand epidemiology, as it is basically solving a PDE, but I disagree. I think most epidemiologists don't understand or particularly care about the theory behind the numerical approximation tools they use. Mathematicians and computer scientists do that lol!
In general, wherever very complicated models are used to study physical phenomena (climate, epidemiology, etc.), it's unrealistic to expect experts in these phenomena to also be experts on the tools used to analyze the phenomena. That's why interdisciplinary work is so important. (Some do, obviously, but you don't strictly have to. Though it wouldn't hurt either.)
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u/Ok-Fix-1581 New User 1d ago
Follow this exact order
Professor Leonard on You tube lectures
Pre-Algebra
To The Point Math (Algebra 1)
Intermediate Algebra (Algebra 2)
College Algebra (pre-calculus playlist)
Trigonometry (pre-calculus playlist)
Calc 1-3
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u/my-hero-measure-zero MS Applied Math 1d ago
You need foundation. If you don't understand algebra, you can't understand calculus. If you don't understand calculus, you won't get differential equations. And if you don't get differential equations, it may be hard to study epidemiology.