Proofy math versus applied math

In summary: I have seen rigorous mathematical proofs in physics, but they are usually used to show the flexibility of the mathematical tools that are being used. For example, the proof for the Riemann hypothesis uses rigorous mathematical techniques to show that certain properties of the zeta function are consistent with the hypothesis.In summary, mathematics can be found in many different places in the physical world. It can be found in the formulas that are used to solve problems in physics, or in the definitions of concepts that are used in physics. However, it is not always necessary to understand how these mathematical tools work on a molecular level to be able to understand and use them.
  • #1
gibberingmouther
120
15
i guess a lot of physics could be considered applied math, though the more you know about how nature works, including how more abstract subjects in math work, probably the better will be your overall understanding.

so there are problems of the type you have in a physics textbook - these are usually word problems, or problems to show you know how to perform the calculations required in physics subjects like engineering. often the textbook will show how to derive the formulas used, or as i do, you could look on the internet.

an abstract book in, say, calculus (or analysis? don't know much about that) will have strict logical definitions, like the confusing definition for what a limit is, and there may be lots of proofs. the book may challenge the reader to do some of his own proofs.

i seem to be able to understand my college level physics textbook, as far as the end of the 8th chapter. but i was curious, i took calculus a couple years ago, and today i decided to review some of the theorems. I've done proofs before in a class called discrete math, and i seemed to be okay at doing them, but they were almost formulaic and I'm not good at doing proofs in other contexts. i can be effusive when it comes to writing about some subjects, but math is not really one of them. it's like the text might say "prove this" and my mind would not really have much to say on the matter unless it's something that I've seen done before in a similar context.

Andrew Wiles wrote a 129 page proof for Fermat's last theorem (according to the Wikipedia). i mean that boggles my mind. physics isn't easy i guess but is there anything in physics that is comparable to that, in terms of difficulty? math is all about logic, while it seems like physics is more concerned with just using the mathematical models that can be built with the toolset provided by mathematicians. do i have this right? I'm not sure what my question is, i just want to know more about how deep you can delve in these subjects, and how they relate to each other.
 
Physics news on Phys.org
  • #2
gibberingmouther said:
i'm not sure what my question is,..
Honestly? Me neither.
... i just want to know more about how deep you can delve in these subjects,..
Which? Math or physics?
... and how they relate to each other.
If you take a closer look on quantum field theory, you will discover a lot of pretty pure math. Good theoretical physicists are usually also good mathematicians. But they use math on a different level than mathematicians do. E.g. coordinates are all over the place in physics and the average mathematician hates them. Physicist use math as a doctor uses a drug. It's not necessary to understand how they function on a molecular level, that's what pharmacologists do. It is similar here. A sequence ##(a_n)_{n\in \mathbb{N}}## converges to a limit ##L## if for every ##\varepsilon > 0## there is a number ##N_\varepsilon## such that ##|a_n-L| < \varepsilon## for all ##n > N_\varepsilon##. That's the mathematical point of view. In physics on the other hand, it often (not always) is sufficient to know: The bigger the index ##n## gets, the closer the sequence ##a_n## gets to the limit ##L##. It is basically the same thing, and if it comes down to a rigorous paper, the mathematical definition will be needed. Until then, the description is as good. Apart from that, the daily scientific business is not what Andrew Wiles did, but instead a steady communication and dialogues, even across the disciplines, especially between mathematicians and physicists. And although Wiles solved Fermat's last theorem mainly on his own, he, too, had colleagues he talked to - and with whom he repaired his first incomplete version. It is therefore difficult to draw lines where in real life there aren't any.
 
  • Like
Likes gibberingmouther
  • #3
thank you fresh_42, that was some of the additional information i was looking for. i figured it would work something like that, but it's hard to get insight into how academia/industry works when you're just a student.
 
  • #4
Until now all the math that I have seen being used in physics is just taking a rigorous mathematical idea and applying it loosely to the physical scene to describe it. This is what calculus is in mechanics, rates of change happen to explain well the mechanics of nature. However, I have not yet seen (myself in MY education) any actual rigorous mathematical proof using the rigor I saw in my Analysis and Algebra courses. It was disappointing as I was expecting those in physics for some reason. But it seems as if it isn't that beneficial to what I have seen. No one in physics cares if the function of the position of a particle is continuous pointwise or uniformly continuous. They only care that it is continuous enough to make sense and allow predictions.
I think that this is a glaring difference between math and physics. Most people think that physics is just math with some meaning in it. But it is not that. It is better described as a science where math has been proven to be stupidly efficient and helpful, but it does not care about math. Physics exited without math and it still could, but it would not be as powerful. And that tells you why proofs such as those are not existent in physics. Most of the time the proof that a physicist needs for some mathematical issue was already done and if it wasn't and he made it, it would be as if a butcher is sharpening his knife, making it more useful and powerful, but that does not mean that his trade is based on sharpening knives.

Disclaimer: This is sometimes offensive to some mathematicians, and it makes them feel as if their math is just a tool. It is not! its way more, but explaining that will take another essay :biggrin:
 
  • Like
Likes gibberingmouther

1. What is the difference between proofy math and applied math?

Proofy math, also known as pure math, focuses on the theoretical and abstract aspects of mathematics, often proving theorems and developing new mathematical concepts. Applied math, on the other hand, uses mathematical principles to solve real-world problems and make predictions.

2. Which type of math is more useful in the real world?

Both proofy math and applied math have their own uses in the real world. Proofy math is important for developing new mathematical theories and understanding the underlying principles of math. Applied math, on the other hand, is more directly applicable to solving real-world problems and making predictions.

3. Do proofy math and applied math require different skill sets?

While both types of math require a strong foundation in mathematical concepts and critical thinking skills, they do require different skill sets. Proofy math often requires more abstract thinking and problem-solving abilities, while applied math may require more practical skills such as programming and data analysis.

4. Can you give an example of proofy math versus applied math?

An example of proofy math would be the study of group theory, which focuses on the abstract properties of mathematical groups. An example of applied math would be using differential equations to model the spread of a disease in a population.

5. Which type of math is more challenging?

This is a subjective question and can vary from person to person. Some may find proofy math more challenging due to its abstract nature and the need for strong critical thinking skills. Others may find applied math more challenging because it requires a strong understanding of mathematical concepts and their applications in the real world.

Similar threads

  • STEM Academic Advising
Replies
5
Views
987
Replies
2
Views
632
  • STEM Academic Advising
2
Replies
60
Views
3K
Replies
22
Views
816
  • STEM Academic Advising
Replies
9
Views
1K
  • STEM Academic Advising
Replies
10
Views
1K
  • STEM Academic Advising
Replies
9
Views
2K
  • STEM Academic Advising
Replies
20
Views
3K
  • STEM Academic Advising
Replies
23
Views
3K
  • STEM Academic Advising
Replies
11
Views
2K
Back
Top