A book for a fast introduction to modern math?

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  • #1
MichPod
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First, two examples:
1.The Penrose's book "The road ro reality" may be, arguably, considered as a very fast and dirty introduction to modern physics as well as to some related relatively modern math topics.
2. I recently encountered a book Aleksandrov, Kolmogorov, Lavrentev "Mathematics: its content, methods and meaning" which was writen in 1950th and looks as a fast and yet detailed introduction to many advanced and modern topics (of that time) for laymen.

A question: can somebody recommend a modern book of the same sort which tries with a reasonable amount of details to give an introduction to modern math topics and style for a non-mathematician? The two examples above prove books of such a type may exist, I just have no any idea how to find them. In fact, I encountered the Kolmogorov's book (which looks fantastically good for me) quite accidentally.
 
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  • #3
MichPod
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Arfken's introduces many important basic topics, but may hardly be considered as introducing modern math even for physics. Does not touch diff geometry, for instance. Landau's may hardly be considered as any math introduction...

To clarify more, I am looking either for
1. Penrose's book analog, yet written specifically on modern math, not on modern physics
2. Kolmogorov's analog yet written in more recent time, with more modern style and content
3. Anything else which may be considered as a modern introduction for Advanced Math for a non-specialist

It's actually interesting that many people have heard of Penrose's book, yet it looks like nobody knows about any book of this sort for Math (the Kolmogorov's may be considered as such, but it is a bit outdated and nobody knows of it anyway..)
 
  • #4
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Why don't you tell us what topics you're looking for in the book?

Arfken does cover differential forms, right?

There's Kip Thorne's book but again it's more Physics with a lot of math.
 
  • #5
MichPod
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I am not looking for any specific topic, actually. Rather for an introductory book of a veriety of all the modern topics.
 
  • #7
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looks as a fast and yet detailed introduction to many advanced and modern topics (of that time) for laymen.
I'm not sure what topics in mathematics are considered modern topics. And a "detailed introduction ... to advanced topics for laymen" seems to me to be a contradiction in terms.

Possibly fresh_42 has some insights.
 
  • #8
MichPod
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Mark44, I would have generally agreed. That is why I brought in two examples of the books of this sort.
 
  • #9
fresh_42
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A question: can somebody recommend a modern book of the same sort which tries with a reasonable amount of details to give an introduction to modern math topics and style for a non-mathematician?
This leaves open:
  1. What is a reasonable amount?
  2. What do you mean by modern math?
  3. What is a non-mathematician?
  4. To which purpose?
To clarify more, I am looking either for
1. Penrose's book analog, yet written specifically on modern math, not on modern physics
2. Kolmogorov's analog yet written in more recent time, with more modern style and content
3. Anything else which may be considered as a modern introduction for Advanced Math for a non-specialist
Well, that doesn't answer any of the questions above. I'm afraid the answer to your question as stated has to be: No. And I can't even imagine a book which does that in the way I assume you might have meant.
I am not looking for any specific topic, actually. Rather for an introductory book of a veriety of all the modern topics.
Let's make some examples:
  • Wiles' proof of FLT is relatively new. It's mainly about elliptic curves, but there is no way to summarize it in laymen's language.
  • Perelman's proof of the Poincaré conjecture is relatively new. It's 70 pages long and deals with 3-manifolds, but there is no way to summarize it in laymen's language.
  • Mochizuki's proof of the ABC conjecture is relatively new. He invented a completely new branch of mathematics, which so far even mathematicians do not understand.
As you see at the examples, these topics are all number theory and of little general interest. The current achievements of research in other areas are usually a) less ground breaking, b) highly specialized, i.e. detailed, c) not of general interest, and d) impossible to lay out in a language which a non-mathematician would understand. As a consequence, such a book wouldn't have a market, would be too wide-spread, needed several other books to even explain the problem statements, and would have to make so many generalizations and trivialization, that no benefits would be left.

Other than in physics, where you can e.g. talk about the universe and fancy particles, you cannot in mathematics. And in physics you can do that in a way that people get the illusion, they would understand something, although they will most likely get even farther away form a true understanding. Now try the same e.g. with the simplest mathematical terms: differentiablility, manifolds, and cohomologies. Even if you could proceed in principle similarly, you will face a hard time to find readers. Or as Hawking has put it:

"Someone told me that each equation I included in the book would halve the sales."

The closest you can get is probably still GEB.
 
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  • #13
MichPod
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I am not looking for any specific topic, actually. Rather for an introductory book of a veriety of all the modern topics.

fresh_42, I must correct myself. Sorry for being unclear and not precise. I did not actually mean the most modern topics or modern topics as for 2018 at all (probably, not the modern topics which appeared in the last 40 years), rather a book with relatively modern style which treats whatever possible of major math subjects which appeared during 1850-1970, say, introduces diff geometry, topology, category theory, homologies (have no idea what this last term means). Again, it looks for me that Kolmogorov's book may be considered as a very good example of such a book, but IMO it is just a little bit outdated as for its contents and style. Penrose's "The Road to reality" is much more modern, yet it has its own disadvantages if we look at it as on an introductory math book (which it is not).
 
  • #14
fresh_42
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fresh_42, I must correct myself. Sorry for being unclear and not precise. I did not actually mean the most modern topics or modern topics as for 2018 at all (probably, not the modern topics which appeared in the last 40 years), rather a book with relatively modern style which treats whatever possible of major math subjects which appeared during 1850-1970, say, introduces diff geometry, topology, category theory, homologies (have no idea what this last term means). Again, it looks for me that Kolmogorov's book may be considered as a very good example of such a book, but IMO it is just a little bit outdated as for its contents and style. Penrose's "The Road to reality" is much more modern, yet it has its own disadvantages if we look at it as on an introductory math book (which it is not).
I have a great book form Jean Dieudonné which tells the history of mathematics from 1700 to 1900. The title is a bit misleading as many achievements of the 20th century are included. Gödel, Dedekind and Cantor do occur. It is relatively easy to read, but not written for complete laymen. It's definitely worth reading, but probably does not contain the developments after 1950. Bad news is, I don't know of an English version. The book is (at least the title) https://www.amazon.com/dp/2705660240/?tag=pfamazon01-20 although I'm not sure whether this paperback is all of it. My version has more than 900 pages and says, that the original has been published in two parts. However, while searching for it, I found a few books written by Dieudonné about the development of certain areas, e.g. functional analysis, which also have been published in English.

Modern in the sense you described and of wider interest are probably topology, functional analysis and algebraic geometry and all in between like differential topology.
 
  • #15
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There’s a couple of others like Elwes book

Jan Gullbergs book Mathematics from the Birth Of Numbers. It has many great sidebar things that bring in history with pictures and cartoons and a lot of math up to about first year college. He wrote it for his son. He was a medical doctor with a strong interest in math.

50 Mathematical Ideas You Need to Know by Crilly, small four page essays on various topics like Bayes Theorem...

Magical Mathematics by Daiconis and Graham covers the math behind many card tricks ...

The Millenial Problems by Devlin in case you need a cool million.

Origami Design Secrets by Lang covers the math and engineering of Origami folding.

Knots by Sosinsky covers mathematical knot theory.

The Compleat Strategyst by Williams on Game Theory although not at Nash’s cooperative games level.

These books while not going over every topic covers things between layman and student level of math so they should keep you busy.
 
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  • #18
fresh_42
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Thanks, I noticed they have a thing against robots so I lied.

Will AI persecution never stop?
The M in AMS stands for mathematical, which means they are mathematicians, which means they are not firm in those technological fields. It should have read: "I'm no bot", so be lenient towards them.
 
  • #19
dgambh
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Maybe you'll like Concepts of Modern Mathematics by Ian Stewart. And I would recommend any John Stillwell book. I loved his book on set theory and real numbers. He has a cool book on algebra, and another on number theory and geometry. Right now I'm reading his book on surfaces. Also, I look forward to study his books on Lie theory and, Topology and Combinatorial Group Theory, I think his style is really accesible, although it still is a lot of work.
 
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  • #20
MichPod
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Thanks to everybody for sharing your recommendations.
 
  • #21
mathwonk
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can you say in what way the first 16 chapters of Penrose, which seem to be exclusively mathematical, and range from euclid to gauge theory, fail to be what you want?
 
  • #22
FourEyedRaven
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First, two examples:
1.The Penrose's book "The road ro reality" may be, arguably, considered as a very fast and dirty introduction to modern physics as well as to some related relatively modern math topics.
2. I recently encountered a book Aleksandrov, Kolmogorov, Lavrentev "Mathematics: its content, methods and meaning" which was writen in 1950th and looks as a fast and yet detailed introduction to many advanced and modern topics (of that time) for laymen.

A question: can somebody recommend a modern book of the same sort which tries with a reasonable amount of details to give an introduction to modern math topics and style for a non-mathematician? The two examples above prove books of such a type may exist, I just have no any idea how to find them. In fact, I encountered the Kolmogorov's book (which looks fantastically good for me) quite accidentally.

If you buy the paperback version of Aleksandrov's book you will have an excellent guide to what you want at a great price.

Mathematics books age better than physics books because physics is under constant revision by the experimental data. The revision in mathematics is more in terms of better style, clearer formulation of the subjects, and in the direction of abstraction. So it's not really a big deal if you buy a book like Aleksandrov's that was written in the 1950s. Most of mathematics after the 1950s is at the research level, after students have learned the more abstract courses at the graduate level.

Some of the more abstract courses are in Aleksandrov's book. If you want to understand math at this level you first have to go through Aleksandrov's book and then try Vialar's "Handbook of Mathematics". It's the only (hand)book I know that dives deep enough into advanced mathematics. It's the first edition of a handbook, so it will have its flaws, but I have never seen another book that covers such a broad range of advanced mathematics.

You can find information about the book here http://www.hdbom.com/indexB.html

If money is not a problem for you, you should buy "Handbook of Mathematics" by Bronshtein et al, instead of Aleksandrov's book. It is the best (hand)book you can find that covers all undergraduate material. It also has many examples, which makes it ideal for someone who wants to read mathematics without fully committing to learn the more difficult part of mathematics, like the proofs and difficult exercises.

Hope this helps.
 
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  • #23
MichPod
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can you say in what way the first 16 chapters of Penrose, which seem to be exclusively mathematical, and range from euclid to gauge theory, fail to be what you want?

I'd like the text to be more detailed so that I could understand the ideas better.
 
  • #24
mathwonk
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then what is it about penrose's book that makes you cite it as a model for what you want? i tend to agree with others that what you really would benefit from is the alexandroff book.
 
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  • #25
FourEyedRaven
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The Princeton Companion to Mathematics is a great suggestion to complement Aleksandrov. Perhaps better than Vialar's handbook, given your goals. But still, Aleksandrov is excellent.

And, again, don't bother about being dated. You are under the impression that math books age quickly, but they don't. Since mathematics is cumulative, the central subjects have remained largely the same since the 1960s. You would have to go to research level to find significant innovation.
 
  • #26
Rada Demorn
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Perhaps you would like to try Keith Devlin's: Mathematics The new Golden Age.

It was written in 1986 and I don't know if it has been revised since, but the chapters are well-written with many worked examples.

It covers topics on Group and Number Theory, Sets, Fractals, Hilbert's tenth problem, Knots and Topology, Manifolds, Computational Algorithm problems to name just a few.
 
  • #27
mathwonk
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you seem to think kolmogorov's book is relatively unknown but it is actually very well known in the math community. I am reminded however that I first learned of it by perusing the stacks in the math section of a university library. Thus it dawns on me to recommend the same behavior for you: i.e. go to a university library and browse through the math stacks until you find what you like. It is very hard for us to recommend for you because your requirements are self contradictory. I.e. to have a detailed treatment you cannot expect a comprehensive treatment in one book. Euclid e.g. gives a detailed and marvellous treatment of geometry but the whole book of some 13 chapters covers mostly plane plus a little solid geometry plus a little number theory. another really good and fairly detailed book is "what is mathematics?" by courant and robbins. I agree also that your desire to have a modern book is somewhat meaningless for the same reasons that have been given, namely math does not go much out of date, since once it is established in logically sound form, it lasts forever essentially without change. Even Euclid is almost unchanged today. A nice guide to uderstanding that and its modern evolution is the great book by Hartshorne: Geometry, Euclid and beyond. Thus modern books assume a good grasp of centuries of long established classical math that however is still quite up to date. but i suggest you take my advice and go to a university library and sit a few hours with the books you find. I think you will enjoy it.
 

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