Topos math and dialectics!

[ComradeRed]

All right, I have a fun challenge for all you philosophers out there with too much time on your hands!

So, that should be everyone.

In math, specifically Category Theory, there is a field called Topos Theory. This allows you to play around with axioms of logic, forming new methods of logic, and so forth.

What I am challenging you to do is to put dialectics (either dialectical materialism or Hegelian dialectics, whichever) in the language of topos math!

So, the question now is: is anyone as bored as I am?!

[Athena]

What a great challenge! And it isn't a matter of bored, but as fish swim and birds fly, humans were designed to think. There is no other reason to here than pleasure of thinking. I have some things to take care and need time to work on this work this challenge. Hopefully, others come to your thread better prepared. I will be back.

[ComradeRed]

Here are the "Three Laws" put in formal logic notation, if it helps.

Transformation of Quantity into Quality: {B: ∫ A d^nA ≠ (A^n)/n}
Interpenetration of opposites: {S: ∃x⇒∃~x}
Negation of the negation: ∃x⇒∃~x

Check me on that, I am not too sure on the transformation of quantity into quality. :eek:

[edit] Also, the final product should be resembling proof theory, so dialectics is used as "logic", per se.

[Starboy]

Would "dialectical materialism or Hegelian dialectics, whichever" qualify as a group?

Starboy

[Athena]

I have to confess to explanations of topos Theory being over my head. Gosh this is frustrating! Is what you are talking about similar to this quote from "The Mayan Factor" by Jose Arguelles?

"Looking at the entire baktun as the creative intensification of a particular morphogenetic field as well as the climax of the total wave harmonic conmmonly know as history, the dialectical action of two qualities colors the whole process. The first is the alchemical impetus to transform matter through the interconnected stages inclusive of scientific revolution, industrial revolution, democratic social revolutions, culminating finally in nuclear action. This is what characterizes the overall movement of the twenty katuns constituing the transformation of matter".

[Starboy]

Quote by: Athena "Looking at the entire baktun as the creative intensification of a particular morphogenetic field as well as the climax of the total wave harmonic conmmonly know as history, the dialectical action of two qualities colors the whole process. The first is the alchemical impetus to transform matter through the interconnected stages inclusive of scientific revolution, industrial revolution, democratic social revolutions, culminating finally in nuclear action. This is what characterizes the overall movement of the twenty katuns constituing the transformation of matter".

That is a fine example of gibberish. The "creative intensification of a particular morphogenic field" should be the tipoff. The rest of it just pushes it over the cliff. You gotta just love "alchemical impetus". This is some very refined gibberish. I can only guess what "nuclear action" could be.

Starboy

[ComradeRed]

I must confess, that I too was overwhelmed with topos when I first saw it too. Anyone who understands topos at first glance really doesn't understand topos at all!

Well, in the beginning there are sets. Essentially a set is a set of elements represented by these barckets "{" and "}", the elements are listed in between them (e.g. {1, 3, ..., 6}).

Now, if we had two sets, say {1, 2, 3} and {5, 3, 4}. The first set is set A, the second is set B. If we had a function "f" that maps from A to B, i.e. changing the elements in set A to be identical in value (although not in order) as set B; this is represented by f : A --> B.

In my example, f(A)=A+2; {1+2, 2+2, 3+2} are all equal in value as set B, although not in the same order as set B.

Alternatively, one can construct a set with only a few things known. The notation for building a set is {X : (conditions)}, the set X is such that the conditions are true.

Now, given a set A and a binary operator *, a group is {A, *}. A binary operator is something like addition, subtraction, multiplication, etc.

It's important to know that a Cartesian product is the product of two sets {a, b} and {1, 2, 3} such that it becomes a set of ordered pairs {(a,1),(a,2),(a,3),(b,1),(b,2),(b,3)}. A Cartesian product of two sets A and B is denoted by AxB.

A binary operation is one where * : AxA --> A described by (x, y) |--> x*y.

A Category consists of all sets G endowed with a binary operation satisfying a certain set of axioms.

Now, each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set theoretic mathematics. But one could instead choose to work with many alternate topoi.

It is also possible to encode a logical theory, such as the theory of all groups, in a topos. The individual models of the theory, i.e. the groups in our example, then correspond to functors from the encoding topos to the category of sets that respect the topos structure.

Functors can be thought of as morphisms in the category of small categories.

Suppose C and D are categories. A functor F from C to D is a mapping (or a function) that does two things:
1. it associates to each object X in C an object F(X) in D, and
2. it associates to each morphism f : X → Y in C a morphism F(f) : F(X) → F(Y) in D

So basically a functor translates the elements of one category into those of another.

But what the hell is a "morphism"?

A category C is given by two pieces of data: a class of objects and a class of morphisms.

There are two operations defined on every morphism, the domain (or source) and the codomain (or target).

Morphisms are often depicted as arrows from their domain to their codomain, e.g. if a morphism f has domain X and codomain Y, it is denoted f : X → Y.

That should explain it all: each topos completely defines its own mathematical framework.

[Athena]

I read every word, but I don't understand their meaning. If I could understand what you are saying, I think I could understand this:

"For the Mayan Factor, the qualitative, harmonic function of number is paramount. Thus, while for us the measurement of time is the counting of a sequence of quantitative unites, be they days or minutes, years or hours, for the Maya what we call time is a function of the prinicple of harmonic resonance. Thus, days are actually tones, called kin, represented by corresponding numbers; sequences of days (kin) create harmonic cycles, called vinal, tun , katuns, baktuns, and so forth; and sequences of harmonic cycles taken as larger aggregates describe the harmonic frequencies of calibrations of a larger organic order, so the harmonic pattern of planet Earth in relation to the Sun and galaxy beyond."

I understand math as the ultimate language of the logos, but I can not wrap my mind around the concepts. The words to explain the concepts, mean nothing to me. I would give just about anything for the power of the knowledge of math. Being math illiterate is as bad as being blind or deaf.

At first when I read your challenge, I saw an explanation of government in geometric form, but my stupid mind can not carry through the conceptualization of government in any language by English, and that is so inadequate! Surely we could reason better if we could do it mathematically.

[ComradeRed]

Well, are you at a university? If so, their libraries are terrific resources, especially for good books on topos theory.

You may also want to slowly creep into topos theory by studying set theory, then group theory, then category theory, and finally topos theory.

If you could point out a specific location of trouble, e.g. in category theory, etc. I may be able to help.

But I don't really understand where you got your quote from Mayan whatevers. It has nothing to do with either topos or dialectics.

Nor does government really have anything to do with dialectics.

There are two books I highly recommend:
* F. William Lawvere and Stephen H. Schanuel: Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, Cambridge, 1997. An "introduction to categories for computer scientists, logicians, physicists, linguists, etc." (cited from cover text).

* F. William Lawvere and Robert Rosebrugh: Sets for Mathematics, Cambridge University Press, Cambridge, 2003. Discusses the foundations of mathematics from a categorical perspective. A book "for students who are beginning the study of advanced mathematical subjects".

Good luck!

[Boetie]

What is the point of this mathematics? Will this math foretell the future? Will learning all this be all in vain in the scheme of Beauty and Art?

[Starboy]

Quote by: Boetie What is the point of this mathematics? Will this math foretell the future? Will learning all this be all in vain in the scheme of Beauty and Art?

Boetie, don't worry about it. Mathematics is like art. If you have to ask the point you don't get it. So just run along.

Starboy

[Boetie]

Starboy posts Boetie, don't worry about it. Mathematics is like art. If you have to ask the point you don't get it. So just run along.

Oh a Pythagorean! did anyone tell you that math is like the liars paradox?

[Starboy]

Quote by: Boetie Oh a Pythagorean! did anyone tell you that math is like the liars paradox?

Exploring the paradoxes of mathematics is what mathematicians do. Pure mathematicians really don't care if there is any application at all for their mathematics. Many are very disappointed when an application is found. They do mathematics for their own reasons mostly having to do with the love of exploring completely abstract man made realms.

BTW, I am not a mathamatician. I am a scientist. A very big and significant difference.

Starboy