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See above. I am trying to re-teach myself mathematics in a different manner than is formally taught (i.e., set theory, number theory, mathematical logic, abstract algebra, discrete math and then precalculus (college algebra and analytical geometry). These are the pillars to upper level mathematics correct?

Git Gud
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  • The missing parentheses makes it confusing. – Git Gud Mar 22 '14 at 20:55
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    It's certainly possible, but there's generally a level of mathematical maturity required to approach abstract algebra or the other (usual undergraduate) upper-division courses. Lacking the experience that precalc and a few semesters of calc gives might hinder you quite a bit. –  Mar 22 '14 at 20:56
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    Honestly I think it would be quite possible because the material from pre-calc isn't fundamental for learning abstract algebra but agree with the above poster that it's about the experience of the math classes. – EgoKilla Mar 22 '14 at 21:07
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    My middle school started abstract algebra in the second half of eighth grade. It was very elementary abstract algebra, but it was the real thing. The answer to your question is certainly yes. – MJD Mar 25 '14 at 22:04
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    Pre-calculus is a must for abstract algebra. In my first two-semester introductory course (covering groups, rings, fields, Galois theory), algebraic manipulations abound and at least some exposure to linear algebra, complex analysis and real analysis presumed. For example, the cyclotomic roots. How else can you understand the field extensions and Galois groups? Even still, this progression doesn't make any sense: number theory uses complex analysis and abstract algebra. – Christopher K Apr 05 '14 at 00:42
  • In principle you could start with abstract algebra having only a very elementary background in math, except that course materials for learning it that way don't really exist. – Nate C-K May 31 '16 at 13:12

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You could learn mathematics any way you want, but the way of mathematics, as formally taught, is elegant and (most of the time) displays the beauty of the subject. As T. Bongers has said, one requires some mathematical maturity in order to fully appreciate the beauty of mathematics.

Abstract algebra is the study of groups, rings, fields, modules, vector spaces, and algebras. I agree with EgoKilla that precalculus is not fundamental to learning abstract algebra - however, it gives one the foundation for learning the subject in detail, and appreciating its mathematical beauty.

You need to know the following to learn abstract algebra:

  1. Set Theory
  2. Logic
  3. Proof techniques
  4. Functions and Relations
  5. Induction
  6. Cardinal numbers
  7. Number theory

Most universities also need Calculus along with this. You could learn these in whichever order you want, but you must make sure to cover all these topics in order to fully understand abstract algebra.

I recommend reading the responses here - A Book for abstract Algebra.

Just as a note, this is the usual way of going about learning mathematics:

Start with precalculus, then go to single variable calculus. (I would like to note here that I am a self-learner, and I skipped almost the whole of precalculus.) After single-variable, go to multivariable calculus. Then learn differential equations, and then proceed on to linear algebra. This should set a sound basis for real and then complex analysis. Go on to functional analysis, topology, differential geometry, and then whatever else interests you.

I hope this helps.

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  • Thank you for the help. I tried the traditional way in college and it didn't work as well for me. I am trying to learn math in the way that I described with the hope that I will be able to tackle Calculus and other higher level topics with ease. – Creator Mar 22 '14 at 21:54
  • I am also adding a philosophy/history component to help with my thinking. – Creator Mar 22 '14 at 21:56
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    I kinda disagree. I think you learn about the numbered items in an abstract algebra class rather than before one. – JP McCarthy Mar 22 '14 at 22:34
  • @JpMcCarthy In abstract algebra, you learn groups, rings, fields, modules, vector spaces, and algebras. It depends on how you study the course - this would determine if one learned the numbered items in abstract algebra or not. Also, if one does learn the numbered items in abstract algebra, then this basically means that an understanding of high-school math is sufficient to learn abstract algebra! –  Mar 22 '14 at 22:40
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You can certainly learn abstract algebra before you learn calculus (I did this myself), but you (almost certainly) won't be able to learn it if you aren't comfortable with high school algebra (which, I guess, is a sizable component of what is called pre-calculus). A basic facility with factorization, algebraic manipulations (like modifying two sides of an equation by applying the same process to both sides), exponential notation, and complex numbers (among various algebra topics of this sort) is more-or-less essential to successfully learning abstract algebra.

Matt E
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If anyone is looking for a great mathematics series, get used copies of University of Chicago Mathematics School Program or get the series by Dolciani for high school mathematics. The UCMSP teaches precalculus and discrete math simultaneously. I am not sure where the abstract algebra (AA) and number theory (NT) segments go but AA precedes NT. I will implement before pre-calculus and after "Algebra 2/Geometry". Another way is to place NT before AA but after the precalculus/discrete mathematics sequence.

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