-
Categories 5 Limits and colimits
This lecture is part of an online course on category theory.
We define limits and colimits of functors, and show how various constructions (products, kernels, inverse limits, and so on) are special cases of this. We also describe how adoint functors preserve limits or colimits.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51F9XZ_Ka4bLnQoxTdMx0AL
published: 24 Sep 2021
-
What are...limits?
Goal.
Explaining basic concepts of category theory in an intuitive way.
This time.
What are...limits? Or: Universal diagrams.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
The distinction between “large classes” and “small classes (sets)” turns out is crucial for many categorical considerations, but somehow makes the language more cumbersome without too much gain imho. So I will strategically ignore all set-theoretical issues.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-ct-2022.html
I used this amazing blog for the exposition of this video.
https://www.math3ma.com/blog/limits-and-colimits-part-3
Limit.
https://en.wikipedia.o...
published: 15 Jan 2022
-
Limits and Colimits
We give the definition of limits and colimits.
published: 09 Jan 2017
-
General limits and colimits 1
Idea of a limit of an arbitrary diagram in a category: cones and universal cones
published: 03 Sep 2008
-
Category Theory II 1.2: Limits
Limits, definition
published: 23 Feb 2017
-
Category Theory For Beginners: Limits
In this video we define what a limit is, by thinking of it as a generalization of a categorical product. We also define equalizers, terminal objects, initial objects and pullbacks, in terms of limits. We also define the dual notions of colimits, co-equalizers and colimits. We also prove that every limits corresponds to a universal morphism.
published: 17 Jun 2019
-
Cone and Cocone, limit and colimit, inverse limit and direct limit in Category
Cone and Cocone, limit and colimit, inverse limit and direct limit in Category
In this video
I explain the definition of:
what is Cone and Cocone, limit and colimit, inverse limit and direct limit in Category
More category theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZtI1njDjtXt_0Hkqbn9RVvi
Instagram account: https://www.instagram.com/mathphyen/
Subscribe to my channel
more video lists:
################
Probability and discrete mathematics:
######################################################################
Probability interview questions:
https://www.youtube.com/watch?v=qphhG_1rWf8&list=PLg9w7tItBlZtI1eM6znSYfyDEaqQtUx5b
Probability theory:
https://www.youtube.com/watch?v=8s-nyRLBsDo&list=PLg9w7tItBlZsvciKHBx8lXXnCtQ7Ou4cV
Complex analysis:
https://www...
published: 18 Aug 2021
-
Section 2.1 - Introduction to Limits - Categories & Sheaves
We give a "modern" definition of the inductive and projective limits, and explain how they capture the ideas we learn in a class on abstract algebra.
If you like category and/or sheaf theory, and can acquire a copy of the book by Kashiwara and Schapira, join our discord: https://discord.gg/vWFhUBjVJu
published: 13 Sep 2020
-
A Reassessment of Gödel’s Doctrine: The Necessity of Infinity (Patrick Ryan, Chapman University)
The Orange County Inland Empire (OCIE) Seminar series in History and Philosophy of Mathematics takes place at Chapman University as its main host, and is co-organized together with researchers from UC Riverside, CSU San Bernardino, and Pitzer College. It also occasionally integrates the Chapman University D.Sc. program in Math, Philosophy and Physics as its Graduate Colloquium.
The seminars are held in hybrid format on the Chapman University campus in the Keck Center, home of Schmid College of Science and Technology, or on Zoom. On November 1, 2024, Patrick Ryan presented his talk, "A Reassessment of Gödel’s Doctrine: The Necessity of Infinity".
ABSTRACT:
In his landmark 1931 paper, Gödel demonstrated the existence of finitary statements that required infinitary resources to prove them....
published: 04 Nov 2024
-
The First Real Application of Category Theory #SoME3
this is a video about category theory... for #SoME3
haha jk this is just algebraic topology
btw i forgot the ray is supposed to go the other way (f(x) to x) im just dumb af pls dont crucify me
published: 17 Aug 2023
31:10
Categories 5 Limits and colimits
This lecture is part of an online course on category theory.
We define limits and colimits of functors, and show how various constructions (products, kernels, ...
This lecture is part of an online course on category theory.
We define limits and colimits of functors, and show how various constructions (products, kernels, inverse limits, and so on) are special cases of this. We also describe how adoint functors preserve limits or colimits.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51F9XZ_Ka4bLnQoxTdMx0AL
https://wn.com/Categories_5_Limits_And_Colimits
This lecture is part of an online course on category theory.
We define limits and colimits of functors, and show how various constructions (products, kernels, inverse limits, and so on) are special cases of this. We also describe how adoint functors preserve limits or colimits.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51F9XZ_Ka4bLnQoxTdMx0AL
- published: 24 Sep 2021
- views: 15847
10:59
What are...limits?
Goal.
Explaining basic concepts of category theory in an intuitive way.
This time.
What are...limits? Or: Universal diagrams.
Disclaimer.
Nobody is perfect...
Goal.
Explaining basic concepts of category theory in an intuitive way.
This time.
What are...limits? Or: Universal diagrams.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
The distinction between “large classes” and “small classes (sets)” turns out is crucial for many categorical considerations, but somehow makes the language more cumbersome without too much gain imho. So I will strategically ignore all set-theoretical issues.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-ct-2022.html
I used this amazing blog for the exposition of this video.
https://www.math3ma.com/blog/limits-and-colimits-part-3
Limit.
https://en.wikipedia.org/wiki/Limit_(category_theory)
https://ncatlab.org/nlab/show/limit
Diagram.
https://en.wikipedia.org/wiki/Diagram_(category_theory)
https://ncatlab.org/nlab/show/diagram
Universal properties.
https://en.wikipedia.org/wiki/Universal_property
https://ncatlab.org/nlab/show/universal+construction
https://ioc.ee/~amar/notes/ct2019_lecture3.pdf
Examples.
https://en.wikipedia.org/wiki/Initial_and_terminal_objects
https://en.wikipedia.org/wiki/Product_(category_theory)
https://en.wikipedia.org/wiki/Coproduct
https://en.wikipedia.org/wiki/Pushout_(category_theory)
https://en.wikipedia.org/wiki/Pullback_(category_theory)
Pictures used.
https://uploads-ssl.webflow.com/5b1d427ae0c922e912eda447/5b5918988f598b49ec11c950_diagram.jpg
https://uploads-ssl.webflow.com/5b1d427ae0c922e912eda447/5b59120d2f3fe63dbcdb06aa_limit.jpg
https://uploads-ssl.webflow.com/5b1d427ae0c922e912eda447/5bf9e443e1003d775ea8c511_product.jpg
https://uploads-ssl.webflow.com/5b1d427ae0c922e912eda447/5bf9715029b48216bfc1a3de_pullback.jpg
https://en.wikipedia.org/wiki/Limit_(category_theory)#/media/File:Functor_cone_(extended).svg
https://en.wikipedia.org/wiki/Limit_(category_theory)#/media/File:Functor_co-cone_(extended).svg
https://uploads-ssl.webflow.com/5b1d427ae0c922e912eda447/5c45f5c4900aab24be28fe32_chart.jpg
Some books I am using (I sometimes steal some pictures from there).
https://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematician
https://www.cambridge.org/core/books/an-introduction-to-category-theory/38C6B02892C2FE7408F52975756AC88D
http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf
https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf
https://math.jhu.edu/~eriehl/context.pdf
https://github.com/hmemcpy/milewski-ctfp-pdf
Nlab.
https://ncatlab.org/nlab/show/HomePage
TheCatsters.
https://www.youtube.com/channel/UC5Y9H2KDRHZZTWZJtlH4VbA
Mathematica.
https://wildcatsformma.wordpress.com/
#categorytheory
#categoricalalgebra
#mathematics
https://wn.com/What_Are...Limits
Goal.
Explaining basic concepts of category theory in an intuitive way.
This time.
What are...limits? Or: Universal diagrams.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
The distinction between “large classes” and “small classes (sets)” turns out is crucial for many categorical considerations, but somehow makes the language more cumbersome without too much gain imho. So I will strategically ignore all set-theoretical issues.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-ct-2022.html
I used this amazing blog for the exposition of this video.
https://www.math3ma.com/blog/limits-and-colimits-part-3
Limit.
https://en.wikipedia.org/wiki/Limit_(category_theory)
https://ncatlab.org/nlab/show/limit
Diagram.
https://en.wikipedia.org/wiki/Diagram_(category_theory)
https://ncatlab.org/nlab/show/diagram
Universal properties.
https://en.wikipedia.org/wiki/Universal_property
https://ncatlab.org/nlab/show/universal+construction
https://ioc.ee/~amar/notes/ct2019_lecture3.pdf
Examples.
https://en.wikipedia.org/wiki/Initial_and_terminal_objects
https://en.wikipedia.org/wiki/Product_(category_theory)
https://en.wikipedia.org/wiki/Coproduct
https://en.wikipedia.org/wiki/Pushout_(category_theory)
https://en.wikipedia.org/wiki/Pullback_(category_theory)
Pictures used.
https://uploads-ssl.webflow.com/5b1d427ae0c922e912eda447/5b5918988f598b49ec11c950_diagram.jpg
https://uploads-ssl.webflow.com/5b1d427ae0c922e912eda447/5b59120d2f3fe63dbcdb06aa_limit.jpg
https://uploads-ssl.webflow.com/5b1d427ae0c922e912eda447/5bf9e443e1003d775ea8c511_product.jpg
https://uploads-ssl.webflow.com/5b1d427ae0c922e912eda447/5bf9715029b48216bfc1a3de_pullback.jpg
https://en.wikipedia.org/wiki/Limit_(category_theory)#/media/File:Functor_cone_(extended).svg
https://en.wikipedia.org/wiki/Limit_(category_theory)#/media/File:Functor_co-cone_(extended).svg
https://uploads-ssl.webflow.com/5b1d427ae0c922e912eda447/5c45f5c4900aab24be28fe32_chart.jpg
Some books I am using (I sometimes steal some pictures from there).
https://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematician
https://www.cambridge.org/core/books/an-introduction-to-category-theory/38C6B02892C2FE7408F52975756AC88D
http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf
https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf
https://math.jhu.edu/~eriehl/context.pdf
https://github.com/hmemcpy/milewski-ctfp-pdf
Nlab.
https://ncatlab.org/nlab/show/HomePage
TheCatsters.
https://www.youtube.com/channel/UC5Y9H2KDRHZZTWZJtlH4VbA
Mathematica.
https://wildcatsformma.wordpress.com/
#categorytheory
#categoricalalgebra
#mathematics
- published: 15 Jan 2022
- views: 1524
3:35
Limits and Colimits
We give the definition of limits and colimits.
We give the definition of limits and colimits.
https://wn.com/Limits_And_Colimits
We give the definition of limits and colimits.
- published: 09 Jan 2017
- views: 1866
8:09
General limits and colimits 1
Idea of a limit of an arbitrary diagram in a category: cones and universal cones
Idea of a limit of an arbitrary diagram in a category: cones and universal cones
https://wn.com/General_Limits_And_Colimits_1
Idea of a limit of an arbitrary diagram in a category: cones and universal cones
- published: 03 Sep 2008
- views: 14606
1:23:34
Category Theory For Beginners: Limits
In this video we define what a limit is, by thinking of it as a generalization of a categorical product. We also define equalizers, terminal objects, initial ob...
In this video we define what a limit is, by thinking of it as a generalization of a categorical product. We also define equalizers, terminal objects, initial objects and pullbacks, in terms of limits. We also define the dual notions of colimits, co-equalizers and colimits. We also prove that every limits corresponds to a universal morphism.
https://wn.com/Category_Theory_For_Beginners_Limits
In this video we define what a limit is, by thinking of it as a generalization of a categorical product. We also define equalizers, terminal objects, initial objects and pullbacks, in terms of limits. We also define the dual notions of colimits, co-equalizers and colimits. We also prove that every limits corresponds to a universal morphism.
- published: 17 Jun 2019
- views: 9701
18:17
Cone and Cocone, limit and colimit, inverse limit and direct limit in Category
Cone and Cocone, limit and colimit, inverse limit and direct limit in Category
In this video
I explain the definition of:
what is Cone and Cocone, limit and col...
Cone and Cocone, limit and colimit, inverse limit and direct limit in Category
In this video
I explain the definition of:
what is Cone and Cocone, limit and colimit, inverse limit and direct limit in Category
More category theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZtI1njDjtXt_0Hkqbn9RVvi
Instagram account: https://www.instagram.com/mathphyen/
Subscribe to my channel
more video lists:
################
Probability and discrete mathematics:
######################################################################
Probability interview questions:
https://www.youtube.com/watch?v=qphhG_1rWf8&list=PLg9w7tItBlZtI1eM6znSYfyDEaqQtUx5b
Probability theory:
https://www.youtube.com/watch?v=8s-nyRLBsDo&list=PLg9w7tItBlZsvciKHBx8lXXnCtQ7Ou4cV
Complex analysis:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsYfYG6dUISItqsSRSXaX4B
Combinatorics:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvQt0YNfCqcULzrPme2ai7K
additive combinatorics:
https://www.youtube.com/playlist?list=PLg9w7tItBlZseUrmXgf_2yc2NI_obVUhB
Calculus and advanced mathematics (abstract algebra):
######################################################################
Real analysis
https://www.youtube.com/playlist?list=PLg9w7tItBlZumEJ4DacAKgstCvDHKtY63
Reduction formula in Calculus:
https://www.youtube.com/playlist?list=PLg9w7tItBlZubmhgkHoHvje1uOASnkcke
Integrals of trigonometric functions:
https://www.youtube.com/playlist?list=PLg9w7tItBlZuwEER2YvFV06ri6i1Cj3hh
(algebraic geometry)
https://www.youtube.com/playlist?list=PLg9w7tItBlZssQ4Smzgp_bk2rEszKaW0B
Linear algebra Proof:
https://www.youtube.com/playlist?list=PLg9w7tItBlZueH6LMeYzzwPMUlIaRhCZo
Ring theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZs9w4wrNAuv4oJdrt-sIWUP
Abstract Algebra:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsyZ0Ri0IDPYEwDxDXjCqB0
Group theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZu4Qkxz2WSk4eYZmDAC8ZY0
Galois theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsepuNJ3tOnG238M03KrKaG
number theory and others:
###########
Riemann Zeta functions:
https://www.youtube.com/playlist?list=PLg9w7tItBlZu4ncqL53iPd1GmwKq8yYoM
Math related to pi:
https://www.youtube.com/playlist?list=PLg9w7tItBlZuzFft3Y5h1LRaNvAv7ruKe
Number theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvGgPYrFxfGWaQgQBrLOOws
Set Theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsDqBUL-CIHcVVBs9EODL_J
High school math & AP calculus:
https://www.youtube.com/playlist?list=PLg9w7tItBlZv165XaBIyj03q3WSMMUqDo
Solving polynomial equation:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsheN8dz8LLPy99AaLb7_-7
Engineering Mathematics:
###########
Laplace transformation:
https://www.youtube.com/playlist?list=PLg9w7tItBlZty21VgNOrLTKg3BM1sFmFE
differential equation:
https://www.youtube.com/playlist?list=PLg9w7tItBlZukQ1yQXKXPo4ZN2yMHwvf1
0xab3acdec0919be9d429baf2190de2e2a4a384c7e
Legendre polynomial:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvG3Ju8YUxEct1tawePe_ZA
Complex analysis:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsYfYG6dUISItqsSRSXaX4B
Gamma Function:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvKhQHLzaULPNtZH7mYNIf9
Theoretical Physics:
################################
Quantum Mechanics:
https://www.youtube.com/playlist?list=PLg9w7tItBlZuP5Wv2jMbeTd7uc9ZXYS6m
Quantum Field Theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvvsWqUz6uR85ktsrcv5Y4r
Electrodynamics:
https://www.youtube.com/playlist?list=PLg9w7tItBlZu6oIdAtRaT-Wtjkg3_dJcQ
Quantum Information:
https://www.youtube.com/playlist?list=PLg9w7tItBlZs2JgNL6qxtPSlnWAFkqoJG
Quantum Algorithms:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvyCUpJAskKqpeLkKhrtQio
https://wn.com/Cone_And_Cocone,_Limit_And_Colimit,_Inverse_Limit_And_Direct_Limit_In_Category
Cone and Cocone, limit and colimit, inverse limit and direct limit in Category
In this video
I explain the definition of:
what is Cone and Cocone, limit and colimit, inverse limit and direct limit in Category
More category theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZtI1njDjtXt_0Hkqbn9RVvi
Instagram account: https://www.instagram.com/mathphyen/
Subscribe to my channel
more video lists:
################
Probability and discrete mathematics:
######################################################################
Probability interview questions:
https://www.youtube.com/watch?v=qphhG_1rWf8&list=PLg9w7tItBlZtI1eM6znSYfyDEaqQtUx5b
Probability theory:
https://www.youtube.com/watch?v=8s-nyRLBsDo&list=PLg9w7tItBlZsvciKHBx8lXXnCtQ7Ou4cV
Complex analysis:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsYfYG6dUISItqsSRSXaX4B
Combinatorics:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvQt0YNfCqcULzrPme2ai7K
additive combinatorics:
https://www.youtube.com/playlist?list=PLg9w7tItBlZseUrmXgf_2yc2NI_obVUhB
Calculus and advanced mathematics (abstract algebra):
######################################################################
Real analysis
https://www.youtube.com/playlist?list=PLg9w7tItBlZumEJ4DacAKgstCvDHKtY63
Reduction formula in Calculus:
https://www.youtube.com/playlist?list=PLg9w7tItBlZubmhgkHoHvje1uOASnkcke
Integrals of trigonometric functions:
https://www.youtube.com/playlist?list=PLg9w7tItBlZuwEER2YvFV06ri6i1Cj3hh
(algebraic geometry)
https://www.youtube.com/playlist?list=PLg9w7tItBlZssQ4Smzgp_bk2rEszKaW0B
Linear algebra Proof:
https://www.youtube.com/playlist?list=PLg9w7tItBlZueH6LMeYzzwPMUlIaRhCZo
Ring theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZs9w4wrNAuv4oJdrt-sIWUP
Abstract Algebra:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsyZ0Ri0IDPYEwDxDXjCqB0
Group theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZu4Qkxz2WSk4eYZmDAC8ZY0
Galois theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsepuNJ3tOnG238M03KrKaG
number theory and others:
###########
Riemann Zeta functions:
https://www.youtube.com/playlist?list=PLg9w7tItBlZu4ncqL53iPd1GmwKq8yYoM
Math related to pi:
https://www.youtube.com/playlist?list=PLg9w7tItBlZuzFft3Y5h1LRaNvAv7ruKe
Number theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvGgPYrFxfGWaQgQBrLOOws
Set Theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsDqBUL-CIHcVVBs9EODL_J
High school math & AP calculus:
https://www.youtube.com/playlist?list=PLg9w7tItBlZv165XaBIyj03q3WSMMUqDo
Solving polynomial equation:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsheN8dz8LLPy99AaLb7_-7
Engineering Mathematics:
###########
Laplace transformation:
https://www.youtube.com/playlist?list=PLg9w7tItBlZty21VgNOrLTKg3BM1sFmFE
differential equation:
https://www.youtube.com/playlist?list=PLg9w7tItBlZukQ1yQXKXPo4ZN2yMHwvf1
0xab3acdec0919be9d429baf2190de2e2a4a384c7e
Legendre polynomial:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvG3Ju8YUxEct1tawePe_ZA
Complex analysis:
https://www.youtube.com/playlist?list=PLg9w7tItBlZsYfYG6dUISItqsSRSXaX4B
Gamma Function:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvKhQHLzaULPNtZH7mYNIf9
Theoretical Physics:
################################
Quantum Mechanics:
https://www.youtube.com/playlist?list=PLg9w7tItBlZuP5Wv2jMbeTd7uc9ZXYS6m
Quantum Field Theory:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvvsWqUz6uR85ktsrcv5Y4r
Electrodynamics:
https://www.youtube.com/playlist?list=PLg9w7tItBlZu6oIdAtRaT-Wtjkg3_dJcQ
Quantum Information:
https://www.youtube.com/playlist?list=PLg9w7tItBlZs2JgNL6qxtPSlnWAFkqoJG
Quantum Algorithms:
https://www.youtube.com/playlist?list=PLg9w7tItBlZvyCUpJAskKqpeLkKhrtQio
- published: 18 Aug 2021
- views: 466
31:21
Section 2.1 - Introduction to Limits - Categories & Sheaves
We give a "modern" definition of the inductive and projective limits, and explain how they capture the ideas we learn in a class on abstract algebra.
If you l...
We give a "modern" definition of the inductive and projective limits, and explain how they capture the ideas we learn in a class on abstract algebra.
If you like category and/or sheaf theory, and can acquire a copy of the book by Kashiwara and Schapira, join our discord: https://discord.gg/vWFhUBjVJu
https://wn.com/Section_2.1_Introduction_To_Limits_Categories_Sheaves
We give a "modern" definition of the inductive and projective limits, and explain how they capture the ideas we learn in a class on abstract algebra.
If you like category and/or sheaf theory, and can acquire a copy of the book by Kashiwara and Schapira, join our discord: https://discord.gg/vWFhUBjVJu
- published: 13 Sep 2020
- views: 129
1:55:46
A Reassessment of Gödel’s Doctrine: The Necessity of Infinity (Patrick Ryan, Chapman University)
The Orange County Inland Empire (OCIE) Seminar series in History and Philosophy of Mathematics takes place at Chapman University as its main host, and is co-org...
The Orange County Inland Empire (OCIE) Seminar series in History and Philosophy of Mathematics takes place at Chapman University as its main host, and is co-organized together with researchers from UC Riverside, CSU San Bernardino, and Pitzer College. It also occasionally integrates the Chapman University D.Sc. program in Math, Philosophy and Physics as its Graduate Colloquium.
The seminars are held in hybrid format on the Chapman University campus in the Keck Center, home of Schmid College of Science and Technology, or on Zoom. On November 1, 2024, Patrick Ryan presented his talk, "A Reassessment of Gödel’s Doctrine: The Necessity of Infinity".
ABSTRACT:
In his landmark 1931 paper, Gödel demonstrated the existence of finitary statements that required infinitary resources to prove them. This led him to postulate what Solomon Feferman called Gödel’s Doctrine, namely, that “the unlimited transfinite iteration of the powerset operation is necessary to account for finitary mathematics.” This claim garnered further support over the course of the 20th century because of the production of various other “finitary independence” results. Nonetheless, proof theoretic work by Feferman and others showed that these finitary results could be proved using relatively weak systems, e.g., predicatively justifiable systems, thereby challenging Gödel’s Doctrine. In this talk, I would like to argue that, though the technical results of Feferman and others are unimpeachable, their philosophical significance is overstated. That is, even if Gödel’s Doctrine is dubious when we understand "necessary" to mean "proof theoretically necessary," it can be vindicated when we think of other senses in which strong infinitary resources might be necessary for mathematics. This is done by investigating a fascinating collection of finitary statements that possess multiple proofs employing both infinitary and finitary resources. I consider how an analysis of such results can inform debates in the philosophy of mathematics, especially discussions of purity, content, and explanation. In particular, if a finitary theorem τ has a perfectly cogent, finitary proof, why then provide an infinitary proof of τ , a proof involving principles of an ostensibly different sort? What is gained? Do such infinitary proofs play an explanatory role? Is there then a sense in which infinity is necessary? I conclude by indicating some promising directions for future research.
https://wn.com/A_Reassessment_Of_Gödel’S_Doctrine_The_Necessity_Of_Infinity_(Patrick_Ryan,_Chapman_University)
The Orange County Inland Empire (OCIE) Seminar series in History and Philosophy of Mathematics takes place at Chapman University as its main host, and is co-organized together with researchers from UC Riverside, CSU San Bernardino, and Pitzer College. It also occasionally integrates the Chapman University D.Sc. program in Math, Philosophy and Physics as its Graduate Colloquium.
The seminars are held in hybrid format on the Chapman University campus in the Keck Center, home of Schmid College of Science and Technology, or on Zoom. On November 1, 2024, Patrick Ryan presented his talk, "A Reassessment of Gödel’s Doctrine: The Necessity of Infinity".
ABSTRACT:
In his landmark 1931 paper, Gödel demonstrated the existence of finitary statements that required infinitary resources to prove them. This led him to postulate what Solomon Feferman called Gödel’s Doctrine, namely, that “the unlimited transfinite iteration of the powerset operation is necessary to account for finitary mathematics.” This claim garnered further support over the course of the 20th century because of the production of various other “finitary independence” results. Nonetheless, proof theoretic work by Feferman and others showed that these finitary results could be proved using relatively weak systems, e.g., predicatively justifiable systems, thereby challenging Gödel’s Doctrine. In this talk, I would like to argue that, though the technical results of Feferman and others are unimpeachable, their philosophical significance is overstated. That is, even if Gödel’s Doctrine is dubious when we understand "necessary" to mean "proof theoretically necessary," it can be vindicated when we think of other senses in which strong infinitary resources might be necessary for mathematics. This is done by investigating a fascinating collection of finitary statements that possess multiple proofs employing both infinitary and finitary resources. I consider how an analysis of such results can inform debates in the philosophy of mathematics, especially discussions of purity, content, and explanation. In particular, if a finitary theorem τ has a perfectly cogent, finitary proof, why then provide an infinitary proof of τ , a proof involving principles of an ostensibly different sort? What is gained? Do such infinitary proofs play an explanatory role? Is there then a sense in which infinity is necessary? I conclude by indicating some promising directions for future research.
- published: 04 Nov 2024
- views: 100
12:45
The First Real Application of Category Theory #SoME3
this is a video about category theory... for #SoME3
haha jk this is just algebraic topology
btw i forgot the ray is supposed to go the other way (f(x) to...
this is a video about category theory... for #SoME3
haha jk this is just algebraic topology
btw i forgot the ray is supposed to go the other way (f(x) to x) im just dumb af pls dont crucify me
https://wn.com/The_First_Real_Application_Of_Category_Theory_Some3
this is a video about category theory... for #SoME3
haha jk this is just algebraic topology
btw i forgot the ray is supposed to go the other way (f(x) to x) im just dumb af pls dont crucify me
- published: 17 Aug 2023
- views: 93574