Further Study Recommendations

[broken ginkgo]

So I have studied several concepts in math, but I am not a especially good at anything. For example, while I know calculus and Taylor expansion, I dont know what standard deviation is. Any recommendations, articles or books on what to learn further?

[broken ginkgo]

Also, when I opened a recap book about sets, I could barely understand any of the formulas. I feel like I'm lacking in a lot of places, but in some places I'm not lacking as much

[eager ocean]

You likely learned calculus as a tool for physics/engineering (how to move things, how things change). You did not learn math as a language for logic and uncertainty (how to prove things, how to describe data)

[broken ginkgo]

Not exactly. I have only learnt from books in my library and other resources

For the record, I won't be taking engineering, I'll be going into biology

Also, I learn math for fun, not for any degree(not yet)

[eager ocean]

Since you have a library and time, try this. It bridges your current skill (Taylor) to the missing skill (Std Dev) using a Biology example.

The Problem: You measure the length of 100 leaves on a tree. The average is 5cm. How do you describe how "bushy" the tree looks?

1. Taylor View: The average is the 0th order approximation of the tree (just "the leaf is ~5cm").
2. Next Term: If you square the differences from 5cm and average them? That's the Variance ($\sigma^2$). It's the second-order term in describing the shape of the leaf distribution (like the $x^2$ term in your series).
3. The Spread: Standard Deviation ($\sigma$) is just the square root of that variance. It puts the "bushiness" back into units of centimeters so we can say: "Most leaves are $5 \pm 2$ cm."

[vital oar]

Anukalp Jha.dev

[eager ocean]

https://docs.google.com/document/d/11RS6jRsFCkxqw2rz54bFbfLH1LvZx0NltIMwZo8jayo/edit?usp=drivesdk

[broken ginkgo]

Couple things:

1. Do you really think I couldn't just ask chatgpt myself?
2. Once when I asked chatgpt for book recommendations it gave me some garbage. When I want to study math, I don't want to learn biology along with it. It should be mainly focused on math.

Thanks for the help nonetheless

[eager ocean]

these could help:-

Book of Proof by Richard Hammack

Introduction to Probability by
Blitzstein & Hwang (Free online via Harvard Stat 110)

Alternative (Shorter, Punchier): All of Statistics by Larry Wasserman. Chapter 2 and 3

Linear Algebra Done Right by Sheldon Axler

[quartz badge]

https://users.metu.edu.tr/serge/courses/111-2011/textbook-math111.pdf
https://poisson.phc.dm.unipi.it/~pereira/UnderstandingAnalysis.pdf
https://simeioseismathimatikwn.wordpress.com/wp-content/uploads/2013/03/apostol-calculusi.pdf
https://sites.math.rutgers.edu/~zeilberg/akherim/PCM.pdf
https://www.stat.cmu.edu/~brian/valerie/617-2022/0 - books/2004 - wasserman - all of statistics.pdf

these are general recommendations, personally im a data science major, if youre trying to build a real foundation the fix is to go back and learn the things everyone assumes are basic but actually arent, like what variance actually represents, why distributions exist, what randomness even means mathematically, how vectors behave geometrically (not just formulas), also don’t just read math bcs that’s useless on its own, you need to constantly ask what does what to and where does it show up, because I could compute derivatives all day
but I didnt understand something as fundamental as spread in data which is honestly more important in practice

https://mml-book.github.io/book/mml-book.pdf
this book for machine learning

and tbh i'd accept that you will always feel like "how do I not know this already" because with math that feeling doesnt go away and it just shifts to new topics

[civic path]

For someone with the tag of Pre-Uni math and the way this is phrased I feel like this is AI slop. No offense.

[quartz badge]

talking to an ai extension of a person is dystopian as fuck

[broken ginkgo]

Thank you

[civic path]

Lmao. It's not even an extension. Indian kids in my experience love to show off for no apparent reason. They think AI is letting them do that lol.

(I'm Indian too, so don't hit me with the racist button lol)

[broken ginkgo]

Your right. Once I find a concept I don't know, I try learning it. Then I find 10 concepts that I don't know. All my ego is dropping from this server

[quartz badge]

learn it from the ground up without ego and focus on understanding, not covering material fast, connect everything back to something real (data, code, visuals)

[civic path]

For someone going into biology, it may be prudent to review Calculus with DEs (Piskunov), study Linear and Matrix Algebra (Johnston), do some Probability Theory (Blitzstein) and also do Numerical Analysis (FNC Book).

[quartz badge]

the biggest shift is that you dont need more advanced math but a more complete map

[broken ginkgo]

True. But I study math for the fun of it. I would say I know way more math than biology

Im Just looking into biology in the future

[broken ginkgo]

This is for @civic path

[broken ginkgo]

Yes. I thought i was very good at math you know? I knew a lot of math at a young age. But now that I look at this server... but whenever I'm learning math, I try not to look at it for my ego

[quartz badge]

if I went into biology the same way I approached math it'd probably feel weirdly unsatisfying at first lol

[civic path]

Then maybe you're interested in doing this properly, the way mathematicians do? In that case swap Piskunov for Zorich and add Hirsch, Devaney and Smale (DEs and Dynamical Systems) plus Rustum Choksi (PDEs). Switch Johnston with FIS for Linear Algebra. FNC is still solid tho and Blitzstein is a great place to start probability.

[quartz badge]

because math is clean and builds upward logically and biology is almost the opposite, messy, full of exceptions

[broken ginkgo]

Ok. I think i will

[broken ginkgo]

Ahh. I haven't touched much on biology. Im thinking on taking soon

[civic path]

Keep your ego home lol. It will come back to you once you get through Analysis all by yourself.

[civic path]

Well. I wouldn't say that. It's just not formal logic in the sense of proving stuff. Empirical deductions are mostly statistical and that is quite logical too.

The sad thing is, idk how many biologists actually know and think about the stats and data better than their computers lol.

[quartz badge]

even when you use statistics, youre not 'proving' anything you’re saying that "given this data and assumptions, this is likely" and that’s fundamentally weaker than a proof

more importantly a huge chunk of biology isn’t even cleanly statistical

its historical and contingent

logic loops there are also way less strict

[civic path]

Science does not proceed with the goals of proving things as much as disproving things.

And the counter examples are empirical

[quartz badge]

even statistical tools themselves don’t give you clean logical truth, take something like hypothesis testing, you don’t prove a hypothesis, you just fail to reject it, that’s a very different epistemology than math

[civic path]

With the theory being operational

It is only in physics where you have the scope for being rigorous.

[quartz badge]

the rigor in physics is kinda selective tbh

[civic path]

Aye. I agree.