Good introductory book to type theory?Good introductory text book on Matroid Theory?How do we express measurable spaces using type theory?Motivating mathematics(particularly algebraic number theory) through historical problemsAre there textbooks on logic where the references to set theory appear only after the construction of set theory?Alternative or reprint of Carter's “Finite Groups of Lie Type: Conjugacy Classes and Complex Characters”Consequences of foundation/regularity in ordinary mathematics (over ZF–AF)?What's the point of cubical type theory?Formal foundations done properlyHow can the simply typed lambda calculus be Turing-incomplete, yet stronger than second-order logic?Explaining the consistency of PRA and ZF from predicative foundations
Good introductory book to type theory?
Good introductory text book on Matroid Theory?How do we express measurable spaces using type theory?Motivating mathematics(particularly algebraic number theory) through historical problemsAre there textbooks on logic where the references to set theory appear only after the construction of set theory?Alternative or reprint of Carter's “Finite Groups of Lie Type: Conjugacy Classes and Complex Characters”Consequences of foundation/regularity in ordinary mathematics (over ZF–AF)?What's the point of cubical type theory?Formal foundations done properlyHow can the simply typed lambda calculus be Turing-incomplete, yet stronger than second-order logic?Explaining the consistency of PRA and ZF from predicative foundations
6
$begingroup$
I don't know anything about type theory and I would like to learn it.
I'm interested to know how we can found mathematics on it.
So, I would be interested by any text about type theory whose angle is similar to the one of Russel and Whitehead in the Principia, or similar to the one of Bourbaki (for instance).
But, I am also interested by any nice presentation of type theory (without any special focus on foundation of mathematics) that would help me to get the best of it.
Books, texts, articles, links are welcomed.
I am interested by any type theory (Martin-Löf's, homotopic, etc.).
PS:
If it can help, in a way, by "foundation of mathematics", I mean "total formalization of mathematics", as in Bourbaki for instance.
reference-request lo.logic textbook-recommendation foundations type-theory
asked 5 hours ago
ColasColas
1313
New contributor
Colas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
|
show 1 more comment
6
$begingroup$
I don't know anything about type theory and I would like to learn it.
I'm interested to know how we can found mathematics on it.
So, I would be interested by any text about type theory whose angle is similar to the one of Russel and Whitehead in the Principia, or similar to the one of Bourbaki (for instance).
But, I am also interested by any nice presentation of type theory (without any special focus on foundation of mathematics) that would help me to get the best of it.
Books, texts, articles, links are welcomed.
I am interested by any type theory (Martin-Löf's, homotopic, etc.).
PS:
If it can help, in a way, by "foundation of mathematics", I mean "total formalization of mathematics", as in Bourbaki for instance.
reference-request lo.logic textbook-recommendation foundations type-theory
asked 5 hours ago
ColasColas
1313
New contributor
Colas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Can you please explain a bit your mathematical and computer science background, so that we can suggest appropriate material. Have you studied logic, or topology, or have you encountered types (maybe in a programming language)?
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
Also, this should be a wiki question.
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
@AndrejBauer Since I'm looking for a text that would describe how to found mathematics, I'm sure my background is so relevant. When I quickly look at HoTT : Univalent foundations of mathematics, I don't see any explanation of "universes" or "types" or "function types" or what is the syntax of the language used, etc. I have a PhD in maths. I studied Algebraic Geometry, Topology, Algebra (commutative, differential), Categories (but not Toposes), etc.
$endgroup$
– Colas
2 hours ago
$begingroup$
If you are looking for something more precise, perhaps Martin Escardo's lecture notes will help. They are very precise because they are written in Agda, and every detail is checked. But when you say that you want an "explanation", what is that supposed to be? It's a foundational theory, it introduces primitive notions. You can get used to these notions, and analogies with your pre-existing mathematical knowledge can be made, but there can be no precise explanation.
$endgroup$
– Andrej Bauer
1 hour ago
$begingroup$
I am looking for a text where the « semantics » of types would be encoded in a formal syntax. If that makes sense to you.
$endgroup$
– Colas
1 hour ago
|
show 1 more comment
6
6
6
2
$begingroup$
I don't know anything about type theory and I would like to learn it.
I'm interested to know how we can found mathematics on it.
So, I would be interested by any text about type theory whose angle is similar to the one of Russel and Whitehead in the Principia, or similar to the one of Bourbaki (for instance).
But, I am also interested by any nice presentation of type theory (without any special focus on foundation of mathematics) that would help me to get the best of it.
Books, texts, articles, links are welcomed.
I am interested by any type theory (Martin-Löf's, homotopic, etc.).
PS:
If it can help, in a way, by "foundation of mathematics", I mean "total formalization of mathematics", as in Bourbaki for instance.
reference-request lo.logic textbook-recommendation foundations type-theory
asked 5 hours ago
ColasColas
1313
New contributor
Colas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I don't know anything about type theory and I would like to learn it.
I'm interested to know how we can found mathematics on it.
So, I would be interested by any text about type theory whose angle is similar to the one of Russel and Whitehead in the Principia, or similar to the one of Bourbaki (for instance).
But, I am also interested by any nice presentation of type theory (without any special focus on foundation of mathematics) that would help me to get the best of it.
Books, texts, articles, links are welcomed.
I am interested by any type theory (Martin-Löf's, homotopic, etc.).
PS:
If it can help, in a way, by "foundation of mathematics", I mean "total formalization of mathematics", as in Bourbaki for instance.
reference-request lo.logic textbook-recommendation foundations type-theory
reference-request lo.logic textbook-recommendation foundations type-theory
asked 5 hours ago
ColasColas
1313
New contributor
Colas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 5 hours ago
ColasColas
1313
New contributor
Colas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 hours ago
Colas
asked 5 hours ago
ColasColas
1313
New contributor
Colas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 5 hours ago
ColasColas
1313
asked 5 hours ago
ColasColas
1313
1313
New contributor
Colas is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
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Check out our Code of Conduct.
$begingroup$
Can you please explain a bit your mathematical and computer science background, so that we can suggest appropriate material. Have you studied logic, or topology, or have you encountered types (maybe in a programming language)?
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
Also, this should be a wiki question.
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
@AndrejBauer Since I'm looking for a text that would describe how to found mathematics, I'm sure my background is so relevant. When I quickly look at HoTT : Univalent foundations of mathematics, I don't see any explanation of "universes" or "types" or "function types" or what is the syntax of the language used, etc. I have a PhD in maths. I studied Algebraic Geometry, Topology, Algebra (commutative, differential), Categories (but not Toposes), etc.
$endgroup$
– Colas
2 hours ago
$begingroup$
If you are looking for something more precise, perhaps Martin Escardo's lecture notes will help. They are very precise because they are written in Agda, and every detail is checked. But when you say that you want an "explanation", what is that supposed to be? It's a foundational theory, it introduces primitive notions. You can get used to these notions, and analogies with your pre-existing mathematical knowledge can be made, but there can be no precise explanation.
$endgroup$
– Andrej Bauer
1 hour ago
$begingroup$
I am looking for a text where the « semantics » of types would be encoded in a formal syntax. If that makes sense to you.
$endgroup$
– Colas
1 hour ago
|
show 1 more comment
$begingroup$
Can you please explain a bit your mathematical and computer science background, so that we can suggest appropriate material. Have you studied logic, or topology, or have you encountered types (maybe in a programming language)?
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
Also, this should be a wiki question.
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
@AndrejBauer Since I'm looking for a text that would describe how to found mathematics, I'm sure my background is so relevant. When I quickly look at HoTT : Univalent foundations of mathematics, I don't see any explanation of "universes" or "types" or "function types" or what is the syntax of the language used, etc. I have a PhD in maths. I studied Algebraic Geometry, Topology, Algebra (commutative, differential), Categories (but not Toposes), etc.
$endgroup$
– Colas
2 hours ago
$begingroup$
If you are looking for something more precise, perhaps Martin Escardo's lecture notes will help. They are very precise because they are written in Agda, and every detail is checked. But when you say that you want an "explanation", what is that supposed to be? It's a foundational theory, it introduces primitive notions. You can get used to these notions, and analogies with your pre-existing mathematical knowledge can be made, but there can be no precise explanation.
$endgroup$
– Andrej Bauer
1 hour ago
$begingroup$
I am looking for a text where the « semantics » of types would be encoded in a formal syntax. If that makes sense to you.
$endgroup$
– Colas
1 hour ago
$begingroup$
Can you please explain a bit your mathematical and computer science background, so that we can suggest appropriate material. Have you studied logic, or topology, or have you encountered types (maybe in a programming language)?
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
Can you please explain a bit your mathematical and computer science background, so that we can suggest appropriate material. Have you studied logic, or topology, or have you encountered types (maybe in a programming language)?
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
Also, this should be a wiki question.
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
Also, this should be a wiki question.
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
@AndrejBauer Since I'm looking for a text that would describe how to found mathematics, I'm sure my background is so relevant. When I quickly look at HoTT : Univalent foundations of mathematics, I don't see any explanation of "universes" or "types" or "function types" or what is the syntax of the language used, etc. I have a PhD in maths. I studied Algebraic Geometry, Topology, Algebra (commutative, differential), Categories (but not Toposes), etc.
$endgroup$
– Colas
2 hours ago
$begingroup$
@AndrejBauer Since I'm looking for a text that would describe how to found mathematics, I'm sure my background is so relevant. When I quickly look at HoTT : Univalent foundations of mathematics, I don't see any explanation of "universes" or "types" or "function types" or what is the syntax of the language used, etc. I have a PhD in maths. I studied Algebraic Geometry, Topology, Algebra (commutative, differential), Categories (but not Toposes), etc.
$endgroup$
– Colas
2 hours ago
$begingroup$
If you are looking for something more precise, perhaps Martin Escardo's lecture notes will help. They are very precise because they are written in Agda, and every detail is checked. But when you say that you want an "explanation", what is that supposed to be? It's a foundational theory, it introduces primitive notions. You can get used to these notions, and analogies with your pre-existing mathematical knowledge can be made, but there can be no precise explanation.
$endgroup$
– Andrej Bauer
1 hour ago
$begingroup$
If you are looking for something more precise, perhaps Martin Escardo's lecture notes will help. They are very precise because they are written in Agda, and every detail is checked. But when you say that you want an "explanation", what is that supposed to be? It's a foundational theory, it introduces primitive notions. You can get used to these notions, and analogies with your pre-existing mathematical knowledge can be made, but there can be no precise explanation.
$endgroup$
– Andrej Bauer
1 hour ago
$begingroup$
I am looking for a text where the « semantics » of types would be encoded in a formal syntax. If that makes sense to you.
$endgroup$
– Colas
1 hour ago
$begingroup$
I am looking for a text where the « semantics » of types would be encoded in a formal syntax. If that makes sense to you.
$endgroup$
– Colas
1 hour ago
|
show 1 more comment
3 Answers
3
active
oldest
votes
8
$begingroup$
I am far from being an expert. I will make a few suggestions.
Per Martin-Löf. Intuitionistic type theory. (Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis, 1984
T. Streicher (1991), Semantics of Type Theory: Correctness, Completeness, and Independence Results, Birkhäuser Boston.
Andre Joyal. Notes on Clans and Tribes.
Michael Shulman. Homotopy type theory: the logic of space.
Also, may I suggest the Naive Type Theory written by Thorsten Altenkirch?
answered 4 hours ago
Ivan Di LibertiIvan Di Liberti
1,7871719
$endgroup$
4
$begingroup$
Streicher and Joyal are not introductory. Martin-Löf is a bit like trying to learn Christianity by reading the Bible.
$endgroup$
– Andrej Bauer
4 hours ago
$begingroup$
Does one of these references emphasize on foundations of mathematics?
$endgroup$
– Colas
4 hours ago
1
$begingroup$
@Colas: The only foundational narrative that I know about is in a beautiful paper by Shulman: sciencedirect.com/science/article/pii/…
$endgroup$
– Ivan Di Liberti
2 hours ago
$begingroup$
@IvanDiLiberti Yes, that looks much more like what I'm looking for. Thanks
$endgroup$
– Colas
2 hours ago
add a comment |
6
$begingroup$
Here are some resources:
UniMath school teaching materials, and in particular:
Spartan type theory, an introduction to type theory (slides)
Introduction to Univalent Foundations of Mathematics with Agda by Martín Escardó.
Unialent Foundations programme: Homotopy Type Theory: Univalent Foundations of Mathematics
- Bengt Nordström, Kent Petersson, and Jan M. Smith: Programming in Martin-Löf's Type Theory
answered 3 hours ago
Andrej BauerAndrej Bauer
31.6k480171
$endgroup$
2
$begingroup$
During my first experience with Type Theory, I had quite some troubles to understand your option 2, and in fact, I truly needed to get my hands on the option number 1 in my answer to understand something. I am not trying to defend my answer, the point I want to make is that different backgrounds and sensitivities might react very differently with our suggestions.
$endgroup$
– Ivan Di Liberti
3 hours ago
2
$begingroup$
@IvanDiLiberti: I am well aware of the fact that type theory is difficult to learn and that not everyone likes the same approach. A lot depends on the background of the person who is learning. Actually, OP should explain their background a bit.
$endgroup$
– Andrej Bauer
3 hours ago
add a comment |
2
$begingroup$
It seems that the HoTT book and Vladimir Voevodsky’s program for Univalent Foundations of Mathematics is made for you !
You will find everything from here:
https://homotopytypetheory.org/
answered 5 hours ago
L. GardeL. Garde
872
$endgroup$
1
$begingroup$
Learning univalent foundations so you can understand Russell and Whitehead? Are you mad?
$endgroup$
– Nik Weaver
5 hours ago
1
$begingroup$
@NikWeaver The original question is pretty vague - it's certainly not clear that the OP's intention is to understand Principia. Given that the OP seems to just want to learn something about any kind of type theoretic foundation for mathematics, steering them away from Russell and Whitehead toward something more modern seems wise!
$endgroup$
– Alex Kruckman
5 hours ago
1
$begingroup$
@Colas HoTT is way way way harder to learn than elementary type theory a la Russell and Whitehead. The suggestions in the other answer look fine.
$endgroup$
– Nik Weaver
4 hours ago
2
$begingroup$
@NikWeaver Although elementary type theory might be easy, "a la Russell and Whitehead" can be misleading. The type theory in Principia is a ramified type theory, which needs (or at least uses) a reducibility axiom to undo unwanted effects of the ramification.
$endgroup$
– Andreas Blass
3 hours ago
1
$begingroup$
When people mention type theory, what typically goes through my head is an extension of the simply typed lambda calculus. It is true that Russell stratified objects into types but this is a very different flavour to the utility of types in lambda calculus. For one, lambda calculus can be seen as a programming language, which leads to the whole subject of Curry-Howard correspondence about a duality between proofs and programs. This is the flavour of type theory I think Andrej is talking about. Other than the idea of typing, Russell's types have very little to do with this idea.
$endgroup$
– Ali Caglayan
1 hour ago
|
show 10 more comments
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
8
$begingroup$
I am far from being an expert. I will make a few suggestions.
Per Martin-Löf. Intuitionistic type theory. (Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis, 1984
T. Streicher (1991), Semantics of Type Theory: Correctness, Completeness, and Independence Results, Birkhäuser Boston.
Andre Joyal. Notes on Clans and Tribes.
Michael Shulman. Homotopy type theory: the logic of space.
Also, may I suggest the Naive Type Theory written by Thorsten Altenkirch?
answered 4 hours ago
Ivan Di LibertiIvan Di Liberti
1,7871719
$endgroup$
4
$begingroup$
Streicher and Joyal are not introductory. Martin-Löf is a bit like trying to learn Christianity by reading the Bible.
$endgroup$
– Andrej Bauer
4 hours ago
$begingroup$
Does one of these references emphasize on foundations of mathematics?
$endgroup$
– Colas
4 hours ago
1
$begingroup$
@Colas: The only foundational narrative that I know about is in a beautiful paper by Shulman: sciencedirect.com/science/article/pii/…
$endgroup$
– Ivan Di Liberti
2 hours ago
$begingroup$
@IvanDiLiberti Yes, that looks much more like what I'm looking for. Thanks
$endgroup$
– Colas
2 hours ago
add a comment |
8
$begingroup$
I am far from being an expert. I will make a few suggestions.
Per Martin-Löf. Intuitionistic type theory. (Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis, 1984
T. Streicher (1991), Semantics of Type Theory: Correctness, Completeness, and Independence Results, Birkhäuser Boston.
Andre Joyal. Notes on Clans and Tribes.
Michael Shulman. Homotopy type theory: the logic of space.
Also, may I suggest the Naive Type Theory written by Thorsten Altenkirch?
answered 4 hours ago
Ivan Di LibertiIvan Di Liberti
1,7871719
$endgroup$
4
$begingroup$
Streicher and Joyal are not introductory. Martin-Löf is a bit like trying to learn Christianity by reading the Bible.
$endgroup$
– Andrej Bauer
4 hours ago
$begingroup$
Does one of these references emphasize on foundations of mathematics?
$endgroup$
– Colas
4 hours ago
1
$begingroup$
@Colas: The only foundational narrative that I know about is in a beautiful paper by Shulman: sciencedirect.com/science/article/pii/…
$endgroup$
– Ivan Di Liberti
2 hours ago
$begingroup$
@IvanDiLiberti Yes, that looks much more like what I'm looking for. Thanks
$endgroup$
– Colas
2 hours ago
add a comment |
8
8
8
$begingroup$
I am far from being an expert. I will make a few suggestions.
Per Martin-Löf. Intuitionistic type theory. (Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis, 1984
T. Streicher (1991), Semantics of Type Theory: Correctness, Completeness, and Independence Results, Birkhäuser Boston.
Andre Joyal. Notes on Clans and Tribes.
Michael Shulman. Homotopy type theory: the logic of space.
Also, may I suggest the Naive Type Theory written by Thorsten Altenkirch?
answered 4 hours ago
Ivan Di LibertiIvan Di Liberti
1,7871719
$endgroup$
I am far from being an expert. I will make a few suggestions.
Per Martin-Löf. Intuitionistic type theory. (Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis, 1984
T. Streicher (1991), Semantics of Type Theory: Correctness, Completeness, and Independence Results, Birkhäuser Boston.
Andre Joyal. Notes on Clans and Tribes.
Michael Shulman. Homotopy type theory: the logic of space.
Also, may I suggest the Naive Type Theory written by Thorsten Altenkirch?
answered 4 hours ago
Ivan Di LibertiIvan Di Liberti
1,7871719
edited 4 hours ago
answered 4 hours ago
Ivan Di LibertiIvan Di Liberti
1,7871719
answered 4 hours ago
Ivan Di LibertiIvan Di Liberti
1,7871719
answered 4 hours ago
Ivan Di LibertiIvan Di Liberti
1,7871719
1,7871719
4
$begingroup$
Streicher and Joyal are not introductory. Martin-Löf is a bit like trying to learn Christianity by reading the Bible.
$endgroup$
– Andrej Bauer
4 hours ago
$begingroup$
Does one of these references emphasize on foundations of mathematics?
$endgroup$
– Colas
4 hours ago
1
$begingroup$
@Colas: The only foundational narrative that I know about is in a beautiful paper by Shulman: sciencedirect.com/science/article/pii/…
$endgroup$
– Ivan Di Liberti
2 hours ago
$begingroup$
@IvanDiLiberti Yes, that looks much more like what I'm looking for. Thanks
$endgroup$
– Colas
2 hours ago
add a comment |
4
$begingroup$
Streicher and Joyal are not introductory. Martin-Löf is a bit like trying to learn Christianity by reading the Bible.
$endgroup$
– Andrej Bauer
4 hours ago
$begingroup$
Does one of these references emphasize on foundations of mathematics?
$endgroup$
– Colas
4 hours ago
1
$begingroup$
@Colas: The only foundational narrative that I know about is in a beautiful paper by Shulman: sciencedirect.com/science/article/pii/…
$endgroup$
– Ivan Di Liberti
2 hours ago
$begingroup$
@IvanDiLiberti Yes, that looks much more like what I'm looking for. Thanks
$endgroup$
– Colas
2 hours ago
4
4
$begingroup$
Streicher and Joyal are not introductory. Martin-Löf is a bit like trying to learn Christianity by reading the Bible.
$endgroup$
– Andrej Bauer
4 hours ago
$begingroup$
Streicher and Joyal are not introductory. Martin-Löf is a bit like trying to learn Christianity by reading the Bible.
$endgroup$
– Andrej Bauer
4 hours ago
$begingroup$
Does one of these references emphasize on foundations of mathematics?
$endgroup$
– Colas
4 hours ago
$begingroup$
Does one of these references emphasize on foundations of mathematics?
$endgroup$
– Colas
4 hours ago
1
1
$begingroup$
@Colas: The only foundational narrative that I know about is in a beautiful paper by Shulman: sciencedirect.com/science/article/pii/…
$endgroup$
– Ivan Di Liberti
2 hours ago
$begingroup$
@Colas: The only foundational narrative that I know about is in a beautiful paper by Shulman: sciencedirect.com/science/article/pii/…
$endgroup$
– Ivan Di Liberti
2 hours ago
$begingroup$
@IvanDiLiberti Yes, that looks much more like what I'm looking for. Thanks
$endgroup$
– Colas
2 hours ago
$begingroup$
@IvanDiLiberti Yes, that looks much more like what I'm looking for. Thanks
$endgroup$
– Colas
2 hours ago
add a comment |
6
$begingroup$
Here are some resources:
UniMath school teaching materials, and in particular:
Spartan type theory, an introduction to type theory (slides)
Introduction to Univalent Foundations of Mathematics with Agda by Martín Escardó.
Unialent Foundations programme: Homotopy Type Theory: Univalent Foundations of Mathematics
- Bengt Nordström, Kent Petersson, and Jan M. Smith: Programming in Martin-Löf's Type Theory
answered 3 hours ago
Andrej BauerAndrej Bauer
31.6k480171
$endgroup$
2
$begingroup$
During my first experience with Type Theory, I had quite some troubles to understand your option 2, and in fact, I truly needed to get my hands on the option number 1 in my answer to understand something. I am not trying to defend my answer, the point I want to make is that different backgrounds and sensitivities might react very differently with our suggestions.
$endgroup$
– Ivan Di Liberti
3 hours ago
2
$begingroup$
@IvanDiLiberti: I am well aware of the fact that type theory is difficult to learn and that not everyone likes the same approach. A lot depends on the background of the person who is learning. Actually, OP should explain their background a bit.
$endgroup$
– Andrej Bauer
3 hours ago
add a comment |
6
$begingroup$
Here are some resources:
UniMath school teaching materials, and in particular:
Spartan type theory, an introduction to type theory (slides)
Introduction to Univalent Foundations of Mathematics with Agda by Martín Escardó.
Unialent Foundations programme: Homotopy Type Theory: Univalent Foundations of Mathematics
- Bengt Nordström, Kent Petersson, and Jan M. Smith: Programming in Martin-Löf's Type Theory
answered 3 hours ago
Andrej BauerAndrej Bauer
31.6k480171
$endgroup$
2
$begingroup$
During my first experience with Type Theory, I had quite some troubles to understand your option 2, and in fact, I truly needed to get my hands on the option number 1 in my answer to understand something. I am not trying to defend my answer, the point I want to make is that different backgrounds and sensitivities might react very differently with our suggestions.
$endgroup$
– Ivan Di Liberti
3 hours ago
2
$begingroup$
@IvanDiLiberti: I am well aware of the fact that type theory is difficult to learn and that not everyone likes the same approach. A lot depends on the background of the person who is learning. Actually, OP should explain their background a bit.
$endgroup$
– Andrej Bauer
3 hours ago
add a comment |
6
6
6
$begingroup$
Here are some resources:
UniMath school teaching materials, and in particular:
Spartan type theory, an introduction to type theory (slides)
Introduction to Univalent Foundations of Mathematics with Agda by Martín Escardó.
Unialent Foundations programme: Homotopy Type Theory: Univalent Foundations of Mathematics
- Bengt Nordström, Kent Petersson, and Jan M. Smith: Programming in Martin-Löf's Type Theory
answered 3 hours ago
Andrej BauerAndrej Bauer
31.6k480171
$endgroup$
Here are some resources:
UniMath school teaching materials, and in particular:
Spartan type theory, an introduction to type theory (slides)
Introduction to Univalent Foundations of Mathematics with Agda by Martín Escardó.
Unialent Foundations programme: Homotopy Type Theory: Univalent Foundations of Mathematics
- Bengt Nordström, Kent Petersson, and Jan M. Smith: Programming in Martin-Löf's Type Theory
answered 3 hours ago
Andrej BauerAndrej Bauer
31.6k480171
answered 3 hours ago
Andrej BauerAndrej Bauer
31.6k480171
answered 3 hours ago
Andrej BauerAndrej Bauer
31.6k480171
answered 3 hours ago
Andrej BauerAndrej Bauer
31.6k480171
31.6k480171
2
$begingroup$
During my first experience with Type Theory, I had quite some troubles to understand your option 2, and in fact, I truly needed to get my hands on the option number 1 in my answer to understand something. I am not trying to defend my answer, the point I want to make is that different backgrounds and sensitivities might react very differently with our suggestions.
$endgroup$
– Ivan Di Liberti
3 hours ago
2
$begingroup$
@IvanDiLiberti: I am well aware of the fact that type theory is difficult to learn and that not everyone likes the same approach. A lot depends on the background of the person who is learning. Actually, OP should explain their background a bit.
$endgroup$
– Andrej Bauer
3 hours ago
add a comment |
2
$begingroup$
During my first experience with Type Theory, I had quite some troubles to understand your option 2, and in fact, I truly needed to get my hands on the option number 1 in my answer to understand something. I am not trying to defend my answer, the point I want to make is that different backgrounds and sensitivities might react very differently with our suggestions.
$endgroup$
– Ivan Di Liberti
3 hours ago
2
$begingroup$
@IvanDiLiberti: I am well aware of the fact that type theory is difficult to learn and that not everyone likes the same approach. A lot depends on the background of the person who is learning. Actually, OP should explain their background a bit.
$endgroup$
– Andrej Bauer
3 hours ago
2
2
$begingroup$
During my first experience with Type Theory, I had quite some troubles to understand your option 2, and in fact, I truly needed to get my hands on the option number 1 in my answer to understand something. I am not trying to defend my answer, the point I want to make is that different backgrounds and sensitivities might react very differently with our suggestions.
$endgroup$
– Ivan Di Liberti
3 hours ago
$begingroup$
During my first experience with Type Theory, I had quite some troubles to understand your option 2, and in fact, I truly needed to get my hands on the option number 1 in my answer to understand something. I am not trying to defend my answer, the point I want to make is that different backgrounds and sensitivities might react very differently with our suggestions.
$endgroup$
– Ivan Di Liberti
3 hours ago
2
2
$begingroup$
@IvanDiLiberti: I am well aware of the fact that type theory is difficult to learn and that not everyone likes the same approach. A lot depends on the background of the person who is learning. Actually, OP should explain their background a bit.
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
@IvanDiLiberti: I am well aware of the fact that type theory is difficult to learn and that not everyone likes the same approach. A lot depends on the background of the person who is learning. Actually, OP should explain their background a bit.
$endgroup$
– Andrej Bauer
3 hours ago
add a comment |
2
$begingroup$
It seems that the HoTT book and Vladimir Voevodsky’s program for Univalent Foundations of Mathematics is made for you !
You will find everything from here:
https://homotopytypetheory.org/
answered 5 hours ago
L. GardeL. Garde
872
$endgroup$
1
$begingroup$
Learning univalent foundations so you can understand Russell and Whitehead? Are you mad?
$endgroup$
– Nik Weaver
5 hours ago
1
$begingroup$
@NikWeaver The original question is pretty vague - it's certainly not clear that the OP's intention is to understand Principia. Given that the OP seems to just want to learn something about any kind of type theoretic foundation for mathematics, steering them away from Russell and Whitehead toward something more modern seems wise!
$endgroup$
– Alex Kruckman
5 hours ago
1
$begingroup$
@Colas HoTT is way way way harder to learn than elementary type theory a la Russell and Whitehead. The suggestions in the other answer look fine.
$endgroup$
– Nik Weaver
4 hours ago
2
$begingroup$
@NikWeaver Although elementary type theory might be easy, "a la Russell and Whitehead" can be misleading. The type theory in Principia is a ramified type theory, which needs (or at least uses) a reducibility axiom to undo unwanted effects of the ramification.
$endgroup$
– Andreas Blass
3 hours ago
1
$begingroup$
When people mention type theory, what typically goes through my head is an extension of the simply typed lambda calculus. It is true that Russell stratified objects into types but this is a very different flavour to the utility of types in lambda calculus. For one, lambda calculus can be seen as a programming language, which leads to the whole subject of Curry-Howard correspondence about a duality between proofs and programs. This is the flavour of type theory I think Andrej is talking about. Other than the idea of typing, Russell's types have very little to do with this idea.
$endgroup$
– Ali Caglayan
1 hour ago
|
show 10 more comments
2
$begingroup$
It seems that the HoTT book and Vladimir Voevodsky’s program for Univalent Foundations of Mathematics is made for you !
You will find everything from here:
https://homotopytypetheory.org/
answered 5 hours ago
L. GardeL. Garde
872
$endgroup$
1
$begingroup$
Learning univalent foundations so you can understand Russell and Whitehead? Are you mad?
$endgroup$
– Nik Weaver
5 hours ago
1
$begingroup$
@NikWeaver The original question is pretty vague - it's certainly not clear that the OP's intention is to understand Principia. Given that the OP seems to just want to learn something about any kind of type theoretic foundation for mathematics, steering them away from Russell and Whitehead toward something more modern seems wise!
$endgroup$
– Alex Kruckman
5 hours ago
1
$begingroup$
@Colas HoTT is way way way harder to learn than elementary type theory a la Russell and Whitehead. The suggestions in the other answer look fine.
$endgroup$
– Nik Weaver
4 hours ago
2
$begingroup$
@NikWeaver Although elementary type theory might be easy, "a la Russell and Whitehead" can be misleading. The type theory in Principia is a ramified type theory, which needs (or at least uses) a reducibility axiom to undo unwanted effects of the ramification.
$endgroup$
– Andreas Blass
3 hours ago
1
$begingroup$
When people mention type theory, what typically goes through my head is an extension of the simply typed lambda calculus. It is true that Russell stratified objects into types but this is a very different flavour to the utility of types in lambda calculus. For one, lambda calculus can be seen as a programming language, which leads to the whole subject of Curry-Howard correspondence about a duality between proofs and programs. This is the flavour of type theory I think Andrej is talking about. Other than the idea of typing, Russell's types have very little to do with this idea.
$endgroup$
– Ali Caglayan
1 hour ago
|
show 10 more comments
2
2
2
$begingroup$
It seems that the HoTT book and Vladimir Voevodsky’s program for Univalent Foundations of Mathematics is made for you !
You will find everything from here:
https://homotopytypetheory.org/
answered 5 hours ago
L. GardeL. Garde
872
$endgroup$
It seems that the HoTT book and Vladimir Voevodsky’s program for Univalent Foundations of Mathematics is made for you !
You will find everything from here:
https://homotopytypetheory.org/
answered 5 hours ago
L. GardeL. Garde
872
answered 5 hours ago
L. GardeL. Garde
872
answered 5 hours ago
L. GardeL. Garde
872
answered 5 hours ago
L. GardeL. Garde
872
872
1
$begingroup$
Learning univalent foundations so you can understand Russell and Whitehead? Are you mad?
$endgroup$
– Nik Weaver
5 hours ago
1
$begingroup$
@NikWeaver The original question is pretty vague - it's certainly not clear that the OP's intention is to understand Principia. Given that the OP seems to just want to learn something about any kind of type theoretic foundation for mathematics, steering them away from Russell and Whitehead toward something more modern seems wise!
$endgroup$
– Alex Kruckman
5 hours ago
1
$begingroup$
@Colas HoTT is way way way harder to learn than elementary type theory a la Russell and Whitehead. The suggestions in the other answer look fine.
$endgroup$
– Nik Weaver
4 hours ago
2
$begingroup$
@NikWeaver Although elementary type theory might be easy, "a la Russell and Whitehead" can be misleading. The type theory in Principia is a ramified type theory, which needs (or at least uses) a reducibility axiom to undo unwanted effects of the ramification.
$endgroup$
– Andreas Blass
3 hours ago
1
$begingroup$
When people mention type theory, what typically goes through my head is an extension of the simply typed lambda calculus. It is true that Russell stratified objects into types but this is a very different flavour to the utility of types in lambda calculus. For one, lambda calculus can be seen as a programming language, which leads to the whole subject of Curry-Howard correspondence about a duality between proofs and programs. This is the flavour of type theory I think Andrej is talking about. Other than the idea of typing, Russell's types have very little to do with this idea.
$endgroup$
– Ali Caglayan
1 hour ago
|
show 10 more comments
1
$begingroup$
Learning univalent foundations so you can understand Russell and Whitehead? Are you mad?
$endgroup$
– Nik Weaver
5 hours ago
1
$begingroup$
@NikWeaver The original question is pretty vague - it's certainly not clear that the OP's intention is to understand Principia. Given that the OP seems to just want to learn something about any kind of type theoretic foundation for mathematics, steering them away from Russell and Whitehead toward something more modern seems wise!
$endgroup$
– Alex Kruckman
5 hours ago
1
$begingroup$
@Colas HoTT is way way way harder to learn than elementary type theory a la Russell and Whitehead. The suggestions in the other answer look fine.
$endgroup$
– Nik Weaver
4 hours ago
2
$begingroup$
@NikWeaver Although elementary type theory might be easy, "a la Russell and Whitehead" can be misleading. The type theory in Principia is a ramified type theory, which needs (or at least uses) a reducibility axiom to undo unwanted effects of the ramification.
$endgroup$
– Andreas Blass
3 hours ago
1
$begingroup$
When people mention type theory, what typically goes through my head is an extension of the simply typed lambda calculus. It is true that Russell stratified objects into types but this is a very different flavour to the utility of types in lambda calculus. For one, lambda calculus can be seen as a programming language, which leads to the whole subject of Curry-Howard correspondence about a duality between proofs and programs. This is the flavour of type theory I think Andrej is talking about. Other than the idea of typing, Russell's types have very little to do with this idea.
$endgroup$
– Ali Caglayan
1 hour ago
1
1
$begingroup$
Learning univalent foundations so you can understand Russell and Whitehead? Are you mad?
$endgroup$
– Nik Weaver
5 hours ago
$begingroup$
Learning univalent foundations so you can understand Russell and Whitehead? Are you mad?
$endgroup$
– Nik Weaver
5 hours ago
1
1
$begingroup$
@NikWeaver The original question is pretty vague - it's certainly not clear that the OP's intention is to understand Principia. Given that the OP seems to just want to learn something about any kind of type theoretic foundation for mathematics, steering them away from Russell and Whitehead toward something more modern seems wise!
$endgroup$
– Alex Kruckman
5 hours ago
$begingroup$
@NikWeaver The original question is pretty vague - it's certainly not clear that the OP's intention is to understand Principia. Given that the OP seems to just want to learn something about any kind of type theoretic foundation for mathematics, steering them away from Russell and Whitehead toward something more modern seems wise!
$endgroup$
– Alex Kruckman
5 hours ago
1
1
$begingroup$
@Colas HoTT is way way way harder to learn than elementary type theory a la Russell and Whitehead. The suggestions in the other answer look fine.
$endgroup$
– Nik Weaver
4 hours ago
$begingroup$
@Colas HoTT is way way way harder to learn than elementary type theory a la Russell and Whitehead. The suggestions in the other answer look fine.
$endgroup$
– Nik Weaver
4 hours ago
2
2
$begingroup$
@NikWeaver Although elementary type theory might be easy, "a la Russell and Whitehead" can be misleading. The type theory in Principia is a ramified type theory, which needs (or at least uses) a reducibility axiom to undo unwanted effects of the ramification.
$endgroup$
– Andreas Blass
3 hours ago
$begingroup$
@NikWeaver Although elementary type theory might be easy, "a la Russell and Whitehead" can be misleading. The type theory in Principia is a ramified type theory, which needs (or at least uses) a reducibility axiom to undo unwanted effects of the ramification.
$endgroup$
– Andreas Blass
3 hours ago
1
1
$begingroup$
When people mention type theory, what typically goes through my head is an extension of the simply typed lambda calculus. It is true that Russell stratified objects into types but this is a very different flavour to the utility of types in lambda calculus. For one, lambda calculus can be seen as a programming language, which leads to the whole subject of Curry-Howard correspondence about a duality between proofs and programs. This is the flavour of type theory I think Andrej is talking about. Other than the idea of typing, Russell's types have very little to do with this idea.
$endgroup$
– Ali Caglayan
1 hour ago
$begingroup$
When people mention type theory, what typically goes through my head is an extension of the simply typed lambda calculus. It is true that Russell stratified objects into types but this is a very different flavour to the utility of types in lambda calculus. For one, lambda calculus can be seen as a programming language, which leads to the whole subject of Curry-Howard correspondence about a duality between proofs and programs. This is the flavour of type theory I think Andrej is talking about. Other than the idea of typing, Russell's types have very little to do with this idea.
$endgroup$
– Ali Caglayan
1 hour ago
|
show 10 more comments
Colas is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
Can you please explain a bit your mathematical and computer science background, so that we can suggest appropriate material. Have you studied logic, or topology, or have you encountered types (maybe in a programming language)?
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
Also, this should be a wiki question.
$endgroup$
– Andrej Bauer
3 hours ago
$begingroup$
@AndrejBauer Since I'm looking for a text that would describe how to found mathematics, I'm sure my background is so relevant. When I quickly look at HoTT : Univalent foundations of mathematics, I don't see any explanation of "universes" or "types" or "function types" or what is the syntax of the language used, etc. I have a PhD in maths. I studied Algebraic Geometry, Topology, Algebra (commutative, differential), Categories (but not Toposes), etc.
$endgroup$
– Colas
2 hours ago
$begingroup$
If you are looking for something more precise, perhaps Martin Escardo's lecture notes will help. They are very precise because they are written in Agda, and every detail is checked. But when you say that you want an "explanation", what is that supposed to be? It's a foundational theory, it introduces primitive notions. You can get used to these notions, and analogies with your pre-existing mathematical knowledge can be made, but there can be no precise explanation.
$endgroup$
– Andrej Bauer
1 hour ago
$begingroup$
I am looking for a text where the « semantics » of types would be encoded in a formal syntax. If that makes sense to you.
$endgroup$
– Colas
1 hour ago