I've tried to put together the connections I see between AIME and math, especially category theory. I am not a specialist neither in AIME-talk nor in category theory (an apprentice of some kind in both) and especially for category theory I am greatful for the work of Andrei Rodin (http://philomatica.org/andrei_rodin/)
First off AIME is full of mathematical concepts (networks, crossings, boundaries, part-whole relations, scale, scripts(algorithms)) both geometrical and algebrical I think.
One might think, especially due to the connection with Actor Network Theory and the talk about networks that it could be related to network theory (networks with coloured links to account for the various modes of existence) but I think that this would be a wrong turn. Because it seems to me that somebody working in networks works on a very clear distinction between a subject (the mathematician) and the object (the various kinds of networks). It would be another kind of the bifurcatory move, except that now one would change the "basic building blocks" that are "more real than experienced reality"
Category theory on the other hand, especially in its relation to logic seems to me that it works on similarly precipitous grounds with AIME itself.
So more precisely:
1. One can see in AIME a move similar with axiomatization in mathematics in the way followed by Lawvere (I put it in bold to differentiate it from axiomatization a la Hilbert, a distinction that Rodin' book (Rodin, A. (2014). Axiomatic method and category theory. Berlin: Springer.) makes clear)
In this book Rodin quotes Lawvere's approach to axiomatization
"In my own education I was fortunate to have two teachers who used the term \foundations"
in a common-sense way (rather than in the speculative way of the Bolzano-
Frege-Peano-Russell tradition). [..] The orientation of these works seemed to be
\concentrate the essence of practice and in turn use the result to guide practice". I
propose to apply the tool of categorical logic to further develop that inspiration.
Foundations is derived from applications by uni cation and concentration, in other
words, by the axiomatic method. Applications are guided by foundations which have
been learned through education."
To my understanding there are very similar moves going on in AIME. Latour based on empirical grounds makes something equivalent to axiomatization of the modern experience through the modes of existence and the concepts he introduces with respect to them. But then this "axiomatization" is used to guide "applications". Moreover there is a similar stance of "openness" relative to what practice might bring forth.
Could it not be that mathematicians have a long experience with axiomatization that could help the AIME effort?
2. When I read the account of axiomatization of mathematics a la Hilbert (in Rodin's book) I get a similar feeling with the bifurcatory reductionism that I felt was an anathema for Whitehead (and for Latour). It feels as if there is an effort to throw the lived experience of the mathematician (mathematical intuition, the living experience of the mathematical concepts and their interconnections) out of the window and keep only syntactic rules that then a machine can just perform them.
So axiomatization a la Lawvere and the other axiomatization efforts that follow along this line seem very similar in flavor with the "Axiomatic" move in AIME, through setting/proposing "modes of existence"
3. Both in category theory and in Latour's approach there is a prominence of relations relative to "entities that just exist out there". In category theory, if I understand well, obejcts are finally accounted for through their morphisms. Moreover there is a similar move (if I interpret well the article Rodin, A. (2011). Categories without structures. Philosophia Mathematica, nkq027.)
away from "essences" (which are closer to the notion of "isomorphisms") and towards alteration (which is closer to the notion of homomorphism)
" What matters in categorical mathematics is how mathematical objects and constructions transform into one another, not what (if anything) remains invariant under these transformations"
(In the end of the above mentioned article even the Pythagorean theorem is presented as travelling in time, perhaps a case of [REP] in AIME-ese)
4. In pg 15 in "Rodin, A. (2014). Axiomatic method and category theory. Berlin: Springer." Rodin speaks about the way Kant was perceiving mathematics:
"According to Kant the representation of general concepts by imaginary
individual objects (which Kant for short also describes as \construction of concepts") is the principal
distinctive feature of mathematical thinking, which distinguishes it from a philosophical
speculation."
"for Kant any individual mathematical object (like triangle ABC) always
comes with a speci c rule that one follows constructing this object in one's imagination and
that provides a link between this object and its corresponding concept (the concept of isosceles
triangle in our example)."
(and if understand well such views are worked and developed in the axiomatization tradition supported by Rodin)
I think that there is a similar feel between "mathematical concepts" here and "Beings to Institute" in the pivot table in pgs 488-489 in the book "An inquiry into modes of existence". These "beings" (so it seems to me) are not "just out there" in a similar sense that a triangle is not just out there and it is not just an example of a big bag of things. A triangle represents a certain group of operations to which we are sensitive. Similar I think the "beings to institute" are not just "things out there" but represent the complex set of interactions that we can find ourselves entertaining with them.
5. I feel there is a similarity between the effort to move towards a better understanding of quantum mechanics through category theory and logic (as in "Constructive Identities for Physics" https://dspace.spbu.ru/bitstream/11701/1720/1/Rodin_Pro1.pdf) and the the "quantum-like" feel that thinking in AIME-ese produces some times (for example http://vkollias.blogspot.gr/2016/06/the-quantum-feel-of-modes-of-existence.html)
I know that for many people interpreting QM is like a holy grail but perhaps using similar thought paths towards understanding the experience of reality (as in AIME) may be more fullfilling, or, if somebody goes for QM, a diversion that may help that goal as well.
First off AIME is full of mathematical concepts (networks, crossings, boundaries, part-whole relations, scale, scripts(algorithms)) both geometrical and algebrical I think.
One might think, especially due to the connection with Actor Network Theory and the talk about networks that it could be related to network theory (networks with coloured links to account for the various modes of existence) but I think that this would be a wrong turn. Because it seems to me that somebody working in networks works on a very clear distinction between a subject (the mathematician) and the object (the various kinds of networks). It would be another kind of the bifurcatory move, except that now one would change the "basic building blocks" that are "more real than experienced reality"
Category theory on the other hand, especially in its relation to logic seems to me that it works on similarly precipitous grounds with AIME itself.
So more precisely:
1. One can see in AIME a move similar with axiomatization in mathematics in the way followed by Lawvere (I put it in bold to differentiate it from axiomatization a la Hilbert, a distinction that Rodin' book (Rodin, A. (2014). Axiomatic method and category theory. Berlin: Springer.) makes clear)
In this book Rodin quotes Lawvere's approach to axiomatization
"In my own education I was fortunate to have two teachers who used the term \foundations"
in a common-sense way (rather than in the speculative way of the Bolzano-
Frege-Peano-Russell tradition). [..] The orientation of these works seemed to be
\concentrate the essence of practice and in turn use the result to guide practice". I
propose to apply the tool of categorical logic to further develop that inspiration.
Foundations is derived from applications by uni cation and concentration, in other
words, by the axiomatic method. Applications are guided by foundations which have
been learned through education."
To my understanding there are very similar moves going on in AIME. Latour based on empirical grounds makes something equivalent to axiomatization of the modern experience through the modes of existence and the concepts he introduces with respect to them. But then this "axiomatization" is used to guide "applications". Moreover there is a similar stance of "openness" relative to what practice might bring forth.
Could it not be that mathematicians have a long experience with axiomatization that could help the AIME effort?
2. When I read the account of axiomatization of mathematics a la Hilbert (in Rodin's book) I get a similar feeling with the bifurcatory reductionism that I felt was an anathema for Whitehead (and for Latour). It feels as if there is an effort to throw the lived experience of the mathematician (mathematical intuition, the living experience of the mathematical concepts and their interconnections) out of the window and keep only syntactic rules that then a machine can just perform them.
So axiomatization a la Lawvere and the other axiomatization efforts that follow along this line seem very similar in flavor with the "Axiomatic" move in AIME, through setting/proposing "modes of existence"
3. Both in category theory and in Latour's approach there is a prominence of relations relative to "entities that just exist out there". In category theory, if I understand well, obejcts are finally accounted for through their morphisms. Moreover there is a similar move (if I interpret well the article Rodin, A. (2011). Categories without structures. Philosophia Mathematica, nkq027.)
away from "essences" (which are closer to the notion of "isomorphisms") and towards alteration (which is closer to the notion of homomorphism)
" What matters in categorical mathematics is how mathematical objects and constructions transform into one another, not what (if anything) remains invariant under these transformations"
(In the end of the above mentioned article even the Pythagorean theorem is presented as travelling in time, perhaps a case of [REP] in AIME-ese)
4. In pg 15 in "Rodin, A. (2014). Axiomatic method and category theory. Berlin: Springer." Rodin speaks about the way Kant was perceiving mathematics:
"According to Kant the representation of general concepts by imaginary
individual objects (which Kant for short also describes as \construction of concepts") is the principal
distinctive feature of mathematical thinking, which distinguishes it from a philosophical
speculation."
"for Kant any individual mathematical object (like triangle ABC) always
comes with a speci c rule that one follows constructing this object in one's imagination and
that provides a link between this object and its corresponding concept (the concept of isosceles
triangle in our example)."
(and if understand well such views are worked and developed in the axiomatization tradition supported by Rodin)
I think that there is a similar feel between "mathematical concepts" here and "Beings to Institute" in the pivot table in pgs 488-489 in the book "An inquiry into modes of existence". These "beings" (so it seems to me) are not "just out there" in a similar sense that a triangle is not just out there and it is not just an example of a big bag of things. A triangle represents a certain group of operations to which we are sensitive. Similar I think the "beings to institute" are not just "things out there" but represent the complex set of interactions that we can find ourselves entertaining with them.
5. I feel there is a similarity between the effort to move towards a better understanding of quantum mechanics through category theory and logic (as in "Constructive Identities for Physics" https://dspace.spbu.ru/bitstream/11701/1720/1/Rodin_Pro1.pdf) and the the "quantum-like" feel that thinking in AIME-ese produces some times (for example http://vkollias.blogspot.gr/2016/06/the-quantum-feel-of-modes-of-existence.html)
I know that for many people interpreting QM is like a holy grail but perhaps using similar thought paths towards understanding the experience of reality (as in AIME) may be more fullfilling, or, if somebody goes for QM, a diversion that may help that goal as well.
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