Classification of Mathematics by 42 Branches

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fresh_42
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I often read questions about our classification scheme that we use on physicsforums.com to sort posts by science fields and subjects, what has to be studied first in order to learn something else, what is a good way through physics or mathematics in self-study or simply about the desire to understand, e.g. general relativity theory or the quantum world, or to understand what is meant when people refer to abstract algebra or topology. Threads with such questions usually provide a few very good answers in posts 2-9. A lot of own experiences come next that rarely reflect the circumstances the thread starter is in, posts 10-31, and finally, a discussion of life, the universe, and everything, posts 32 until someone closes the debate. Any classification is necessarily incomplete and by its nature a compromise among personal flavors, cultures, languages, or historically given practices. So will be mine here, incomplete and driven by personal opinion. Nevertheless, I’ll try my best to...

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  • #2
symbolipoint
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I read slightly more than the first paragraph, and then could only look at the rest. What a mess!

Students just can best follow the course prerequisites. People do not learn how everything is related overnight. When students, usually young and not advanced, comment that they do not see how the different parts of Mathematics fit together, they are right and honest.

Just a small piece of how something is related to something else, Long Division for digited numbers is just like polynomial division; one begins to see this more clearly when he is taught polynomial long division (often in Algebra 1 or Algebra 2). Suddenly he understands Place Value better, even if he thought he understood it earlier.
 
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fresh_42
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I read slightly more than the first paragraph, and then could only look at the rest. What a mess!
In which case you should have read
Any classification is necessarily incomplete and by its nature a compromise among personal flavors, cultures, languages, or historically given practices. So will be mine here, incomplete and driven by personal opinion.

Reference: https://www.physicsforums.com/insights/classification-of-mathematics-by-42-branches/--
It - to some extent - has to be a "mess". Otherwise, you are stuck with questionable statements like "learn mathematics if you want to learn physics." You do not really need Galois theory or even numerical analysis if you want to understand relativity theory.

Your statement is a declaration of bankruptcy because it says at its core that you are unable to answer questions about which mathematical branches are necessary for what. I trust you, that you can't. I do.
 
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  • #4
symbolipoint
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Your statement is a declaration of bankruptcy because it says at its core that you are unable to answer questions about which mathematical branches are necessary for what. I trust you, that you can't. I do.
I do not understand the quoted passage above. I HAVE USED basic algebra in real life. I did know what I was doing when I applied what I applied. ("Basic Algebra" as meaning Arithmetic using variables, in place of plain numbers; occasionally simple systems of equations)
 
  • #5
symbolipoint
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In which case you should have read

It - to some extent - has to be a "mess". Otherwise, you are stuck with questionable statements like "learn mathematics if you want to learn physics." You do not really need Galois theory or even numerical analysis if you want to understand relativity theory.

Your statement is a declaration of bankruptcy because it says at its core that you are unable to answer questions about which mathematical branches are necessary for what. I trust you, that you can't. I do.
Somehow you did not read the second part of the post #2. (Not a very long posting). Meaning there is, understanding how the many parts fit takes time.
 
  • #7
S.G. Janssens
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This is redundant, potentially confusing, and pretentious.

Mathematics has a very well-established subject classification.
It was last revised in 2020 but has been around for decades.

MSC page on zbMATH
MSC page on AMS
 
  • #8
fresh_42
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This is redundant, potentially confusing, and pretentious.

Mathematics has a very well-established subject classification.
It was last revised in 2020 but has been around for decades.

MSC page on zbMATH
MSC page on AMS
It always surprises me again and again when I see Americans (the AMS, not you) who claim to know what's right or wrong and decide for the rest of the world. I am not really a fan of this hegemonic attitude and I don't actually care. This wasn't an AMS article.

I have given a personal - non-American - opinion based on personal experiences and questions I have read here over the years. That's, e.g. the reason for the broad algebra splitting. Nobody ever asked about cryptography or number theory. But what they did ask was: "Do I need group theory to study physics?"

This article is subjective by nature. Otherwise, I would have needed considerably more than the 97 categories of AMS and had to write a book. Even the subset (applications of discrete math, coding, cryptography, algebras) resulted in two paperbacks on my shelf.

Btw., this AMS database looks like the ICD-10 or ICD-11 code. Is that on purpose?
 
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fresh_42
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What is meta-mathematics?
The philosophical subfield "about mathematics". Whether we should accept AC or not is meta-mathematics in my opinion.
 
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  • #10
martinbn
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The part about homological algebra is quite uninformative.
 
  • #11
fresh_42
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The part about homological algebra is quite uninformative.
Thanks for the relative constructive criticism. If you have a better one, then write me or post it here. I can edit my description. Homological algebra is not much asked about on PF so I kept it short. Hilton-Stammbach on the other hand has 330 pages. Maybe I was a bit too bored when I wrote it.
 
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  • #12
fresh_42
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I knew beforehand that this would be a controversial task. Yes, it is incomplete, and, yes, subjective. Functional analysis or mathematical optimization could have been split into many parts, e.g.

So any constructive criticism is actually very appreciated and I will see how I could edit the article accordingly. Complaining about the obvious, well, isn't actually very helpful.
 
  • #13
pbuk
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I knew beforehand that this would be a controversial task. Yes, it is incomplete, and, yes, subjective.
I for one think it is a bold attempt at something potentially worthwhile.

So any constructive criticism is actually very appreciated and I will see how I could edit the article accordingly.
I have a few comments:
  • "Stochastic" and "Algorithmic" are adjectives and cannot be used as nouns in English. I'd probably say "Stochastic Processes" and maybe "Algorithmics", although I'd probably rather have "Computation Theory" and "Computational Methods" separately.
  • Is "Combinatorics" missing or do you not think it should stand alone?
  • Have you considered prior art - for instance the Princeton Companion's high level grouping into Algebra, Number Theory, Geometry, Algebraic Geometry, Analysis, Logic, Combinatorics, Theoretical Computer Science, Probability and Mathematical Physics?
Edit: I have always thought of Analaysis as "the study of limits".
 
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fresh_42
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I for one think it is a bold attempt at something potentially worthwhile.


I have a few comments:
  • "Stochastic" and "Algorithmic" are adjectives and cannot be used as nouns in English. I'd probably say "Stochastic Processes" and maybe "Algorithmics", although I'd probably rather have "Computation Theory" and "Computational Methods" separately.
Too bad. Stochastik is the German word for probability theory, but it also includes statistics. Would Stochastic Sciences do? Same with algorithms.
  • Is "Combinatorics" missing or do you not think it should stand alone?
I thought: the counting part is in probability theory, the field itself in discrete mathematics.
  • Have you considered prior art - for instance the Princeton Companion's high level grouping into Algebra, Number Theory, Geometry, Algebraic Geometry, Analysis, Logic, Combinatorics, Theoretical Computer Science, Probability and Mathematical Physics?
Edit: I have always thought of Analaysis as "the study of limits".
I tried to match the 42. Any taxonomy has its weaknesses, simply because the level of details isn't the same for all and the many overlapping areas.

The theory of graphs is also missing.
 
  • #15
Svein
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42 branches - is this related to the "Answer to the Ultimate Question of Life, the Universe, and Everything"?
 
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  • #16
martinbn
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This may be off topic, but it might be of interested. There is a book by Jean Dieudonne called "A panorama of pure mathematics (as seen by N. Bourbaki)"
 
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  • #17
pbuk
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Too bad. Stochastik is the German word for probability theory, but it also includes statistics.
I didn't realize this, in English we would just call this field "Probability and Statistics", "Stochastic [whatever]" would be is a much narrower field. I see you have tried "Stocasticity": this is also not a word, and if it is it would mean "the extent to which a thing is stochastic" (c.f. "toxic" = harmful, "toxicity" = the extent to which something is harmful", "toxicology" = the study of harmful substances; no there is no such thing as "stochastology").

Same with algorithms.
Harder but if you must I'd go with Algorithmics.

I thought: the counting part is in probability theory, the field itself in discrete mathematics.
Yes, makes sense.
 
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  • #18
fresh_42
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42 branches - is this related to the "Answer to the Ultimate Question of Life, the Universe, and Everything"?
My running gag, yes. I think there is no meaningful number of mathematical subfields. The more one tries to describe one field, the more subfield one finds that would deserve a separate description. Maybe something around 200 would do. E.g. the GTM series has more than 250 books and the overlappings are tractable.
 
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fresh_42
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This may be off topic, but it might be of interested. There is a book by Jean Dieudonne called "A panorama of pure mathematics (as seen by N. Bourbaki)"
I have a book from Jean Dieudonné about the history of mathematics from 1700 - 1900 (roughly). I love this book and often cited it here on PF.
 
  • #20
fresh_42
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I didn't realize this, in English we would just call this field "Probability and Statistics", "Stochastic [whatever]" would be is a much narrower field. I see you have tried "Stocasticity": this is also not a word, and if it is it would mean "the extent to which a thing is stochastic" (c.f. "toxic" = harmful, "toxicity" = the extent to which something is harmful", "toxicology" = the study of harmful substances; no there is no such thing as "stochastology").
I found it on Wikipedia, and "Google translate" confirmed stochasticity for Stochastik. The lecture at the university I took was "Stochastik". What's wrong all of a sudden? You (English) import every nonsense that comes around (just lately saw stein and spiel), but the meaningful imports go unseen?
Harder but if you must I'd go with Algorithmics.
https://en.wikipedia.org/wiki/Algorithmics

Edit: I wasn't happy with stochasticity either. You said that "stochastic" is an adjective only. That was the key: I gave it a noun, "mathematics".
 
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  • #21
pbuk
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I found it on Wikipedia, and "Google translate" confirmed stochasticity for Stochastik.
It's not in Cambridge, Mirriam-Webster or my paper Shorter Oxford.

The lecture at the university I took was "Stochastik". What's wrong all of a sudden?
Nothing at all: if you are talking about direct translation Physik -> physics, Mathematik -> mathematics, Stochastik -> stochastics. But what you learn in a Stochastik lecture we learn in Probability and Statistics see e.g. https://www.imperial.ac.uk/study/ug/courses/mathematics-department/mathematics-bsc/#structure.

You (English) import every nonsense that comes around (just lately saw stein and spiel)
Sounds like an entertaining combination of beer and board games? Yes we have bars for that :biggrin:

https://en.wikipedia.org/wiki/Algorithmics
Yes, but even that (poorly referenced) entry admits "the term algorithmics is rarely used in the English-speaking world".
 
  • #22
fresh_42
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It's not in Cambridge, Mirriam-Webster or my paper Shorter Oxford.


Nothing at all: if you are talking about direct translation Physik -> physics, Mathematik -> mathematics, Stochastik -> stochastics. But what you learn in a Stochastik lecture we learn in Probability and Statistics see e.g. https://www.imperial.ac.uk/study/ug/courses/mathematics-department/mathematics-bsc/#structure.


Sounds like an entertaining combination of beer and board games? Yes we have bars for that :biggrin:

https://en.wikipedia.org/wiki/Algorithmics
Yes, but even that (poorly referenced) entry admits "the term algorithmics is rarely used in the English-speaking world".
I think the use of word endings is what makes it different. Stochast-ik and Algorithm-ik are nouns and therefore names of science, whereas stochast-isch and algorithm-isch adjectives are. The example of stochastic shows that such a distinction cannot be expected in English. And why doesn't my spell-checker complain when I write stochastics? If it is an adjective, how does the "s" make any sense?

I also have a problem distinguishing between basically the same English adjective but two endings, e.g. historic - historical, neurologic - neurological. Where is the **** difference?
 
  • #23
pbuk
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I think the use of word endings is what makes it different. Stochast-ik and Algorithm-ik are nouns and therefore names of science, whereas stochast-isch and algorithm-isch adjectives are. The example of stochastic shows that such a distinction cannot be expected in English. And why doesn't my spell-checker complain when I write stochastics? If it is an adjective, how does the "s" make any sense?
Ah, with these words it is a noun without an "s", adding the "s" forms an adjective. Sometimes.

I also have a problem distinguishing between basically the same English adjective but two endings, e.g. historic - historical, neurologic - neurological. Where is the **** difference?
Oh you don't have a chance; here are more examples:
  • stochastic (adj): determined by a probability distribution, the opposite of deterministic.
  • stochastics (n): the study of stochastic processes.
  • algorithmic (adj): determined by an algorithm.
  • algorithmics (n, rare): the study of algorithms.
  • heuristic (adj): adaptive or non-deterministic e.g. "we can try to optimize an NP-hard problem by heuristic methods".
  • heuristic (n): an adaptive method or strategy e.g. "we use an heuristic to arrange the class timetable" (note the alternative pronunciation: aspiration of the "h" is optional, and if silent it is preceded by "an" rather than "a" c.f. "a hotel" (ay-hoh-tell) vs "an hotel" (an-oh-tell). With hotel this is an affectation, but "an-er-is-tic" is much easier to say than "ay-hure-is-tic" so is more often heard.
  • heuristics (n): decision making using adaptive methods or strategies e.g. "a public search engine may use heuristics to show you advertising from which it earns a profit".
 
  • #24
pbuk
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whereas stochast-isch and algorithm-isch adjectives are.
Now don't start confusing auxilliary verb order with me using, Yoda.
 
  • #25
fresh_42
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Now don't start confusing auxilliary verb order with me using, Yoda.
Sometimes translate I first my thought and forget then the words to rearrange.

But this happens only occasionally and if then in that Yoda style.
That first sentence here above took me three minutes to write it down in a German order. It sounds extremely weird even to me.
 
  • #26
Demystifier
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What is meta-mathematics?
Like meta-physics, but for math. It's also called foundations of mathematics, dealing mostly with topics such as logic and set theory, and is analogous to foundations of physics (see e.g. the forum for quantum foundations). See also https://en.wikipedia.org/wiki/Metamathematics
 

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