4 ms·

Many valued logic arises quite naturally in category theory. If you take usual set theory and change the sets to fibred sets you can easily get a subobject clas

Many valued logic arises quite naturally in category theory. If you take usual set theory and change the sets to fibred sets you can easily get a subobject classifier with, e.g., four truth values. However, these don't tell us much about "truth" in the social sense. Actually, my impression of what category theory has brought to mathematics is that structure itself seems more fundamental than logic; it is as though logic arises because of structure too (and not just the converse of logic leading to structure).

jlg23 7y ago
> However, these don't tell us much about "truth" in the social sense. Which truth? Yours, mine, what we agree upon or what really happened? (Paraphrasing Rousseau)
- mikorym 7y ago
  I mean that it does not supercede the usual logic that we use to argue about everyday things.
phaedrus 7y ago
I like this way of looking at things. I've been (as a studied amateur) reading and thinking about this topic for a long time. In some implementations of logic programming, disjunction is analogous to iteration or sequence (do A then do B) and conjunction is analogous to nested loops or map (map all A with all B). If structure is more fundamental than logic, it's not mysterious why the procedural interpretation of logic programs comes out the way it does - it follows from the structure underpinning the logic.
- mikorym 7y ago
  Mac Lane started out in logic before category theory was invented and I think this had a positive influence on his pedagogical style in writing, for example, textbooks. Does what you say have anything to do with probability theory (e.g.: the set space B given that A is true)? There are of course the basic adjunctions that apply to quantifiers and connectives (e.g.: the adjoints to the preimage). Is iteration and nesting as you mention similar to this?
anongraddebt 7y ago
There are some semi-tenuous, but nevertheless interesting connections between this metamathematical line of thought and contemporary work on "grounding" in metaphysics. I've long held the suspicion that there is some critical unargued for assumption in many of the relevant academic circles, here, that if reality 'bottoms out' somehow it has to bottom out in a monadic fashion. I've often wondered, why can't it be structure all the way down? ---- I should add that many/most grounding theorists likely hold some sort of monadic view. I only mention grounding because it is seemingly closer to a 'structure all the way down' view than the predominant views within analytical philosiphy on what is fundamental.
- mikorym 7y ago
  I am not familiar with this, do you have links to the basics of what grounding theory is?
  - anongraddebt 7y ago
    My pleasure. The first three are entries in the Stanford Encyclopedia of Philosophy. I suggest reading the first link (entry titled "Fundamentality") first, as that will help provide context. The second link (entry titled "Metaphysical Grounding") provides a somewhat dated, but solid overview of the literature and is what you should read right after "Fundamentality". I say somewhat dated only because the entry is from 2014, and 'grounding' sort of took off around 2009 and has accelerated past 2014. The third link (entry titled "Ontological Dependence") can be skipped, but if you want additional conceptual and historical context, then I suggest blitz reading it. The rest of the links are to books or papers/articles which I think you might find interesting. The very last link is to a rather obscure paper (I randomly found it a year ago) from 1997 that is not on grounding, but is related in some way to the 'structure all the way down' perspective I suggested. The papers' argument is incomplete, attempts to do too many things at once, and presents a rather naive and elementary framework for a 'structure all the way down' view. However, it is (in my opinion) a rare gem. It raises a number of important questions and issues that have (to my knowledge) not yet been addressed by anyone within academic philosophy, or other relevant disciplines for that matter. These questions and issues go unaddressed because the monadic assumption(s) is, seemingly, so deep seated and fundamental to most people. The way I see it, either 'things'/'substances'/'elements' are primary and 'relations' are derivative/secondary, or they are on equal footing, or 'relations' are primary and 'substances' are derivative/secondary. There are only three possibilities. I'm inclined to believe either the second or third are true. If you put a gun to my head and made me choose one, I'd say the third is true. Stanford Encyclopedia of Philosophy: https://plato.stanford.edu/entries/fundamentality/ https://plato.stanford.edu/entries/fundamentality/ https://plato.stanford.edu/entries/grounding/ https://plato.stanford.edu/entries/grounding/ https://plato.stanford.edu/entries/dependence-ontological/ https://plato.stanford.edu/entries/dependence-ontological/ Papers: https://www.google.com/url?sa=t&source=web&rct=j&url=https://pdfs.semanticscholar.org/bbb7/d5b6d877d0fd4667c4744d9947fdabbd2d20.pdf&ved=2ahUKEwjYr-K4huriAhULna0KHayvBy0QFjAAegQIARAB&usg=AOvVaw17sV23uYkumL6hCv1wF9NI https://www.google.com/url?sa=t&source=web&rct=j&url=https:/... https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.kellytrogdon.org/ewExternalFiles/GME-8.pdf&ved=2ahUKEwjL_4j7huriAhVEIqwKHVIZCU8QFjAAegQIAxAB&usg=AOvVaw1Bkw7osz_MZSHG3k8hBTnu https://www.google.com/url?sa=t&source=web&rct=j&url=http://... https://www.academia.edu/34059306/Can_grounding_characterize_fundamentality_Analysis_forthcoming_ https://www.academia.edu/34059306/Can_grounding_characterize... https://www.google.com/url?sa=t&source=web&rct=j&url=https://individual.utoronto.ca/jmwilson/Wilson-The-Unity-and-Priority-Arguments-for-Grounding.pdf&ved=2ahUKEwj42fKlieriAhUHLKwKHUxODIAQFjAAegQIAxAB&usg=AOvVaw0MqmazJDMja8MiFfNRcYKo https://www.google.com/url?sa=t&source=web&rct=j&url=https:/... https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.jonathanschaffer.org/groundcause.pdf&ved=2ahUKEwi32pTPieriAhVFeKwKHVRmCUUQFjAAegQIBRAB&usg=AOvVaw0fxaU0bQ-vQw2JGK5UIOLP https://www.google.com/url?sa=t&source=web&rct=j&url=http://... Books: https://www.amazon.com/gp/aw/d/B00A8ICBWO/ref=tmm_kin_title_0?ie=UTF8&qid=1560553506&sr=8-1 https://www.amazon.com/gp/aw/d/B00A8ICBWO/ref=tmm_kin_title_... https://www.amazon.com/gp/aw/d/B0725YKR29/ref=tmm_kin_title_0?ie=UTF8&qid=1560553583&sr=8-1 https://www.amazon.com/gp/aw/d/B0725YKR29/ref=tmm_kin_title_... Obscure Paper: https://www.google.com/url?sa=t&source=web&rct=j&url=http://wellnowwhat.net/transfers/The%2520Mathematical%2520Structure%2520of%2520the%2520World%2520-%2520Dipert.pdf&ved=2ahUKEwiors_wkOriAhVLRqwKHZj7BJcQFjAAegQICBAB&usg=AOvVaw1DLO15hmR32DozOzefGvEr&cshid=1560555269461 https://www.google.com/url?sa=t&source=web&rct=j&url=http://...

jlg23 7y ago

> However, these don't tell us much about "truth" in the social sense. Which truth? Yours, mine, what we agree upon or what really happened? (Paraphrasing Rousseau)

mikorym 7y ago
I mean that it does not supercede the usual logic that we use to argue about everyday things.

phaedrus 7y ago

I like this way of looking at things. I've been (as a studied amateur) reading and thinking about this topic for a long time. In some implementations of logic programming, disjunction is analogous to iteration or sequence (do A then do B) and conjunction is analogous to nested loops or map (map all A with all B). If structure is more fundamental than logic, it's not mysterious why the procedural interpretation of logic programs comes out the way it does - it follows from the structure underpinning the logic.

mikorym 7y ago
Mac Lane started out in logic before category theory was invented and I think this had a positive influence on his pedagogical style in writing, for example, textbooks. Does what you say have anything to do with probability theory (e.g.: the set space B given that A is true)? There are of course the basic adjunctions that apply to quantifiers and connectives (e.g.: the adjoints to the preimage). Is iteration and nesting as you mention similar to this?

anongraddebt 7y ago

There are some semi-tenuous, but nevertheless interesting connections between this metamathematical line of thought and contemporary work on "grounding" in metaphysics. I've long held the suspicion that there is some critical unargued for assumption in many of the relevant academic circles, here, that if reality 'bottoms out' somehow it has to bottom out in a monadic fashion. I've often wondered, why can't it be structure all the way down? ---- I should add that many/most grounding theorists likely hold some sort of monadic view. I only mention grounding because it is seemingly closer to a 'structure all the way down' view than the predominant views within analytical philosiphy on what is fundamental.

mikorym 7y ago
I am not familiar with this, do you have links to the basics of what grounding theory is?
- anongraddebt 7y ago
  My pleasure. The first three are entries in the Stanford Encyclopedia of Philosophy. I suggest reading the first link (entry titled "Fundamentality") first, as that will help provide context. The second link (entry titled "Metaphysical Grounding") provides a somewhat dated, but solid overview of the literature and is what you should read right after "Fundamentality". I say somewhat dated only because the entry is from 2014, and 'grounding' sort of took off around 2009 and has accelerated past 2014. The third link (entry titled "Ontological Dependence") can be skipped, but if you want additional conceptual and historical context, then I suggest blitz reading it. The rest of the links are to books or papers/articles which I think you might find interesting. The very last link is to a rather obscure paper (I randomly found it a year ago) from 1997 that is not on grounding, but is related in some way to the 'structure all the way down' perspective I suggested. The papers' argument is incomplete, attempts to do too many things at once, and presents a rather naive and elementary framework for a 'structure all the way down' view. However, it is (in my opinion) a rare gem. It raises a number of important questions and issues that have (to my knowledge) not yet been addressed by anyone within academic philosophy, or other relevant disciplines for that matter. These questions and issues go unaddressed because the monadic assumption(s) is, seemingly, so deep seated and fundamental to most people. The way I see it, either 'things'/'substances'/'elements' are primary and 'relations' are derivative/secondary, or they are on equal footing, or 'relations' are primary and 'substances' are derivative/secondary. There are only three possibilities. I'm inclined to believe either the second or third are true. If you put a gun to my head and made me choose one, I'd say the third is true. Stanford Encyclopedia of Philosophy: https://plato.stanford.edu/entries/fundamentality/ https://plato.stanford.edu/entries/fundamentality/ https://plato.stanford.edu/entries/grounding/ https://plato.stanford.edu/entries/grounding/ https://plato.stanford.edu/entries/dependence-ontological/ https://plato.stanford.edu/entries/dependence-ontological/ Papers: https://www.google.com/url?sa=t&source=web&rct=j&url=https://pdfs.semanticscholar.org/bbb7/d5b6d877d0fd4667c4744d9947fdabbd2d20.pdf&ved=2ahUKEwjYr-K4huriAhULna0KHayvBy0QFjAAegQIARAB&usg=AOvVaw17sV23uYkumL6hCv1wF9NI https://www.google.com/url?sa=t&source=web&rct=j&url=https:/... https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.kellytrogdon.org/ewExternalFiles/GME-8.pdf&ved=2ahUKEwjL_4j7huriAhVEIqwKHVIZCU8QFjAAegQIAxAB&usg=AOvVaw1Bkw7osz_MZSHG3k8hBTnu https://www.google.com/url?sa=t&source=web&rct=j&url=http://... https://www.academia.edu/34059306/Can_grounding_characterize_fundamentality_Analysis_forthcoming_ https://www.academia.edu/34059306/Can_grounding_characterize... https://www.google.com/url?sa=t&source=web&rct=j&url=https://individual.utoronto.ca/jmwilson/Wilson-The-Unity-and-Priority-Arguments-for-Grounding.pdf&ved=2ahUKEwj42fKlieriAhUHLKwKHUxODIAQFjAAegQIAxAB&usg=AOvVaw0MqmazJDMja8MiFfNRcYKo https://www.google.com/url?sa=t&source=web&rct=j&url=https:/... https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.jonathanschaffer.org/groundcause.pdf&ved=2ahUKEwi32pTPieriAhVFeKwKHVRmCUUQFjAAegQIBRAB&usg=AOvVaw0fxaU0bQ-vQw2JGK5UIOLP https://www.google.com/url?sa=t&source=web&rct=j&url=http://... Books: https://www.amazon.com/gp/aw/d/B00A8ICBWO/ref=tmm_kin_title_0?ie=UTF8&qid=1560553506&sr=8-1 https://www.amazon.com/gp/aw/d/B00A8ICBWO/ref=tmm_kin_title_... https://www.amazon.com/gp/aw/d/B0725YKR29/ref=tmm_kin_title_0?ie=UTF8&qid=1560553583&sr=8-1 https://www.amazon.com/gp/aw/d/B0725YKR29/ref=tmm_kin_title_... Obscure Paper: https://www.google.com/url?sa=t&source=web&rct=j&url=http://wellnowwhat.net/transfers/The%2520Mathematical%2520Structure%2520of%2520the%2520World%2520-%2520Dipert.pdf&ved=2ahUKEwiors_wkOriAhVLRqwKHZj7BJcQFjAAegQICBAB&usg=AOvVaw1DLO15hmR32DozOzefGvEr&cshid=1560555269461 https://www.google.com/url?sa=t&source=web&rct=j&url=http://...