Books on mathematical logic, foundations, and philosophy

In summary: After reading Hartshorne, Griffiths/Harris, and Liu's Arithmetic curves, and EGA and SGA set to be completely read, I would recommend starting with David Hilbert's Grundlagen der Geometrie.
  • #1
Tukhara
15
2
Hello, all. I am looking for some good books to start becoming invested in mathematical logic, the foundations of the field of mathematics, and also basically in general the philosophical heart of this wide subject which has interested me greatly. Now I have already read Shoenfield and Halmos and I already have books on Set/Category theory but I am specifically looking for something that's somewhat like the bible of mathematical logic. Anyone have suggestions?
 
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  • #2
Tukhara said:
I am specifically looking for something that's somewhat like the bible of mathematical logic
I doubt that such a work exists, as there is no single mathematical logic. What usually is referred to as "mathematical logic" is the first order predicate calculus. But there are others. There are even systems with three possible values of truth. As you explicitly mentioned philosophy, the book from Charles (Karl) Popper "The Logic of Scientific Discovery" could be of interest to you. Other relevant scientists are Bertrand Russell and Kurt Gödel. Also of interest could be historical essays about the first half of the 20th century in general (foundational crises), and especially a reflection on Hilbert's program.

I know this might not answer your question with respect to a standard opus, but it hopefully guides you towards possible directions. The Wikipedia articles usually have lists of links, where one can start with research, e.g. on the original papers.
 
  • #3
I was looking for something that has a lot of material at least suitable enough for one to hold a strong foundation in the subject.

Thanks for your assistance either way.
 
  • #5
It enough to make my brain bleeds but its good. I included something of the contents.
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  • #6
How about this one ?

fromabook.png
 
  • #7
How is any of these related to logic? Except that hopefully it's used.
awholenumber said:
Like this book ?
Algebra on high school level.
BL4CKB0X97 said:
It enough to make my brain bleeds but its good. I included something of the contents.
Physics. Neihter logic nor foundational nor philosophy.
awholenumber said:
How about this one ?
History of a certain aspect of physics. Not even close to what the OP described.
 
  • #10
Mark44 said:
@awholenumber and @BL4CKB0X97, try to keep your suggestions applicable to what the OP is asking for.
I believe that the work, and overview of the history (and an in depth working of said mathmatics)of pythagoreans and euclid would but rather apt to the Ops request. I thought ancient mathematics and how it relates to and birthed modern mathematics would also be pretty foundational. Mathmatics is covered first,then the physics comes en masse. He need not read it all.

The are pages devoted to nothing but equations and proofs along with questions to test understanding.
 
  • #11
BL4CKB0X97 said:
I believe that the work, and overview of the history (and an in depth working of said mathmatics)of pythagoreans and euclid would but rather apt to the Ops request. I thought ancient mathematics and how it relates to and birthed modern mathematics would also be pretty foundational. Mathmatics is covered first,then the physics comes en masse. He need not read it all.

The are pages devoted to nothing but equations and proofs along with questions to test understanding.
This might be true or not, but it has definitely not the least to do with
Tukhara said:
to start becoming invested in mathematical logic, the foundations of the field of mathematics, and also basically in general the philosophical heart
Penrose might be entertaining, but he's certainly not the first choice, when it comes to textbooks, and none if mathematical logic is the purpose.

And by the way: "field" in this context does not refer to the algebraic object called field, as
smodak said:
books but do not entirely deal with logic or fields
would suggest.
 
  • #12
Lol. I am already past Abstract/Commutative Algebra. Anyways if anyone knows of Barwise's handbook of mathematical logic, which I have not read and of course will not be able to read due to the fact it's extremely difficult to find and I don't have the money to purchase available copies. However, is there anyone who knows a book like it?
 
  • #14
Just a question, provided one has read Hartshorne, Griffiths/Harris, and also Liu's Arithmetic curves; and also I have EGA and SGA set to be completely read(too bad they will most likely never be translated); which book do you recommend for Algebraic Geometry after all this or just in general as a complement?
 
  • #15
Tukhara said:
I have EGA and SGA set to be completely read
Not really easy to add something upon these. What you're looking for that Grothendiek and Dieudonné haven't addressed? I know of some interesting papers by V. Strassen (Rank and optimal computation of generic tensors, 1982) and others, e.g. on lower bounds of matrix multiplication which uses methods of algebraic geometry (http://www.math.tamu.edu/~jml/LOsecbnd05-28.pdf), and an interesting essay (free download, https://arxiv.org/pdf/1205.5935.pdf, 92 p.) about geometric algebra (which of course is different from algebraic geometry).

@mathwonk once recommended some:
mathwonk said:
Miles Reid's Undergraduate algebraic geometry is a good book, but will not give you this categorical notion of affine products in relation to tensor products. but the first 15 pages of the second link I gave above will, the notes from Charles Siegel at UPenn, based on the terse little book Algebraic varieties, by George Kempf.
https://www.amazon.com/Algebraic-Varieties-Mathematical-Society-Lecture/dp/0521426138/ref=sr_1_1?s=books&ie=UTF8&qid=1449594305&sr=1-1&keywords=george+kempf,+algebraic+varieties

I first learned it from Mumford's "red book" of algebraic varieties.

https://www.amazon.com/Red-Book-Varieties-Schemes-Mathematics/dp/354063293X/ref=sr_1_1?s=books&ie=UTF8&qid=1449594259&sr=1-1&keywords=mumford's+red+book

heres a cheaper used copy, but maybe without the nice little addition of his lectures on curves from Michigan:

http://www.abebooks.com/servlet/SearchResults?an=david+mumford&sts=t&tn=red+book
and @lavinia
lavinia said:
Steenrod's book, The Topology of Fiber Bundles, if you haven't looked at it already, is one of the first that specifically focuses on fiber bundles. I find it difficult going but it covers other examples of bundles than vector bundles e.g. sphere bundles,covering spaces, and principal bundles. These are just as important as vector bundles. Milnor's book is advanced and his chapter on what vector bundles are ,while rigorous ,is also quite dense. Bott and Tu's Differential Forms in Algebraic Topology deals with bundles from the differentiable view point. This book is also advanced but gives a wonderful introduction to the use of calculus in algebraic topology. The book I started with is Singer and Thorpe's Book, lecture Notes on Elementary Topology and Geometry - also in PDF form. It deals with bundles in a way more elementary way but still has the modern viewpoint.

Maybe not exactly what you've been looking for, but judging by whom they have been recommended, worth to keep in mind in any case.
 
  • #16
fresh_42 said:
Not really easy to add something upon these. What you're looking for that Grothendiek and Dieudonné haven't addressed? I know of some interesting papers by V. Strassen (Rank and optimal computation of generic tensors, 1982) and others, e.g. on lower bounds of matrix multiplication which uses methods of algebraic geometry (http://www.math.tamu.edu/~jml/LOsecbnd05-28.pdf), and an interesting essay (free download, https://arxiv.org/pdf/1205.5935.pdf, 92 p.) about geometric algebra (which of course is different from algebraic geometry).

@mathwonk once recommended some:

and @laviniaMaybe not exactly what you've been looking for, but judging by whom they have been recommended, worth to keep in mind in any case.
Well do you anything close to something that's like the English translations of EGA and SGA?
 
  • #17
Tukhara said:
Well do you anything close to something that's like the English translations of EGA and SGA?
Sorry, no. I assume the Bourbaki approach isn't really a bestseller. I'v seen some interesting books on the internet about the historical developments like the correspondence between Grothendiek and Serre, but no conceptional treatment. Leo Corry seems to have written a nice overview of the modern mathematical structural concepts, but this also looks more like a historical treatment:
https://books.google.de/books?id=8G0FCAAAQBAJ&dq=Bourbaki+algebraic+geometry
 
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  • #18
Does this go over similar topics as EGA/SGA?
 
  • #19
Tukhara said:
Does this go over similar topics as EGA/SGA?
I don't know. I only know about EGA/SGA what is written on Wiki. My assessment is based on what I know about the authors (and an old French version of a Bourbaki text about algebraic geometry I found and had a look into). But I don't think that the historical treatments I mentioned above can be compared to it. I only added them as being interesting on their own. To be honest, I cannot really imagine that a book written Bourbakian style would find many readers on the Anglo-American market. As far as I know, it is a fundamentally different tradition. And the books I have are again in another "wrong" language.
 
  • #20
Tukhara said:
Hello, all. I am looking for some good books to start becoming invested in mathematical logic, the foundations of the field of mathematics, and also basically in general the philosophical heart of this wide subject which has interested me greatly. Now I have already read Shoenfield and Halmos and I already have books on Set/Category theory but I am specifically looking for something that's somewhat like the bible of mathematical logic. Anyone have suggestions?
For the philosophical part, maybe reading the book by Morris Kline: Mathematics in Western Culture? I have not looked at it.
 

Related to Books on mathematical logic, foundations, and philosophy

What is mathematical logic?

Mathematical logic is a branch of mathematics that studies the use of formal reasoning and symbols to represent and manipulate mathematical concepts and statements. It is concerned with the study of valid reasoning and proof, and it provides the foundation for many areas of mathematics and computer science.

What are the foundations of mathematics?

The foundations of mathematics are the fundamental principles and concepts upon which all of mathematics is built. These include axioms, definitions, and logical rules that are used to construct and prove mathematical statements. The study of foundations of mathematics is closely related to mathematical logic and philosophy of mathematics.

What is the relationship between mathematical logic and philosophy?

Mathematical logic and philosophy are closely intertwined, as both fields are concerned with the nature of truth, knowledge, and reasoning. Mathematical logic provides the tools and methods for studying and analyzing the logical structure of mathematical statements and arguments, while philosophy addresses broader questions about the nature of reality and our understanding of it.

Why is mathematical logic important?

Mathematical logic is important because it provides a rigorous and formal framework for reasoning and proof in mathematics. It allows us to analyze the structure of mathematical statements and identify which arguments are valid and which are not. It also has practical applications in computer science, artificial intelligence, and other fields.

What are some key topics in books on mathematical logic, foundations, and philosophy?

Some key topics in these books include set theory, propositional and predicate logic, proof theory, computability theory, and philosophical approaches to the foundations of mathematics. Other topics may include non-classical logics, model theory, and the relationship between mathematics and language or reality.

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