OBSERVE MORE

–Conceptual Mathematics, page 317

Before I can ask:

how much more?

more than what?

I need to know what you have in mind when you say ‘observe.’

Fair enough!  Let’s say you have a dynamical system (e.g. a light bulb) with two discrete states

ON, OFF

Let’s further say that the bulb comes equipped with a toggle switch, which, when you press, switches the light bulb to OFF if it’s ON and to ON if it’s OFF.  Summing up: what we have here is a discrete dynamical system with a set of states

X = {ON, OFF}

and a state-transition rule

a: X –> X

a(ON) = OFF and a(OFF) = ON

When our light bulb is ON, we notice that it emits a certain amount (say, 1) of light and not surprisingly no light (i.e. 0) when the bulb is OFF.  This function

f: X –> Y

from the set of states X to a set Y (= {0, 1}) of values is what we think of as a process of observation or measurement (Conceptual Mathematics, pp. 316 – 8).  Then there are sequences of observations, configuration of states et cetera, which we need to get hold of while changing a light bulb, but first: what exactly is a discrete dynamical system?

A set

X

equipped with an endomap

a: X –> X

is what we think of as a discrete dynamical system; with the set X as the set of possible states of the system and with the endomap a: X –> X as the state-transition rule.  Dynamical systems, much like many other mundane things in this material universe, have parts—subdynamical systems—that we can look at, pull apart and maybe even put them back together.

Given

X = {-1, 1}

a: X –> X

with a as ‘squaring’ i.e.

a(-1) = 1 and a(1) = 1

how many parts does the object

(X, a: X –> X)

have?

That depends on the universe we are in!

Oh, no—not again

We can think of the object

(X, a: X –> X)

as an object in the category of graphs, as an object in the category of maps, and as an object in the category of dynamical systems, which is not that unusual if we take a moment to notice that you can think of me as a guy checking out girls, as a person lost in thought, and as a body with little mass—it all depends on the universe of discourse you want to locate me.

Case I. Category of Dynamical Systems

Object

X

^

a |

X

with X = {-1, 1}; a(-1) = 1 and a(1) = 1

Imagine an oval with two dots (with the one on the left denoting -1 and the one to the right denoting 1) and two arrows: one going from -1 to 1 and the other looping back to 1 from 1, and that’s how (the internal diagram of) our object looks like.

A part of our object (X, a) is a monomorphism with the object (X, a) as codomain (Conceptual Mathematics, page 340).  But first: what does a map look like in the category of dynamical systems?  A map f from an object (Y, b: Y –> Y) to the object (X, a: X –> X) in the category of dynamical systems is a set map (function)

f: Y –> X

satisfying

f b = a f

which looks like a [commutative] square

Y –f– > X

^           ^

b |             | a

Y –f– > X

Taking Y = {1}, b: Y –> Y with b(1) = 1

and f(1) = 1

we find that

fb(1) = f(1) = 1

af(1) = a(1) = 1

i.e. fb = af

So the f: (Y, b: Y –> Y) –> (X, a: X –> X) is a part i.e. a subdynamical system of the dynamical system (X, a: X –> X).  (I’m skipping the part about checking to make sure that f: (Y, b: Y –> Y) –> (X, a: X –> X) is indeed a monomorphism.  Also saved for later is what got me started in the first place:

elements never leave subdynamical systems

Let me thank Danilo for his question: What about the case of an element leaving the subsystem?

–Conceptual Mathematics, page 347)

Case II. Category of Maps

Object

X

^

a |

X

with X = {-1, 1}; a(-1) = 1, a(1) = 1

Imagine two ovals each with two dots (the two dots [in both ovals] denoting -1 and 1), along with two arrows: one going from the dot denoting -1 in the bottom oval to the dot denoting 1 in the top oval and the other arrow going from the dot denoting 1 in the bottom oval to the dot denoting 1 in the top oval.  And with that vision we have the internal diagram of the map a: X –> X in sight.

A part of

X

^

a |

X

is (as always) a monomorphism with the object a: X –> X as codomain.  But first (think Big-Bang if this ‘first’ in every other sentence is getting on your nerves): what does a map look like in the category of maps? A map f from an object b: Y –> Z to our object a: X –> X is a pair of maps

f = <f^d, f^c>

f^d: Y –> X

f^c: Z –> X

satisfying

f^c b = a f^d

Taking Y = {-1}, Z = {1}, and b: Y –> Z with b(-1) = 1

and

f^d: Y –> X with f^d(-1) = -1

f^c: Z –> X with f^c(1) = 1

we (after reminding ourselves that a: X –> X with a(-1) = 1 and a(1) = 1) find that

a f^d(-1) = a(-1) = 1

f^c b(-1) = f^c(1) = 1

i.e. f^c b = a f^d

So we say

Z –f^c– > X

^             ^

b |             | a

Y –f^d– > X

is a part of our object

X

^

a |

X

Is this the only part of the map a: X –> X?  (I’m saving, yet again, the part about how the number of parts of a: X –> X varies as one hops from one category to another.)

Case III. Category of Graphs

An object in the category of graphs is a pair of sets:

a set of arrows X_A

and

a set of dots X_D

along with a pair of functions

s: X_A –> X_D

and

t: X_A –> X_D

assigning to each arrow in X_A its source and target dots, respectively, in X_D.

So in the case of our earlier internal diagram of two dots (denoting -1 and 1) and two arrows (x denoting the one going from -1 to 1 and x’ denoting the one going from 1 to 1) we have

X_A = {x, x’}

X_D = {-1, 1}

and

s(x) = -1, t(x) = 1

s(x’) = 1, t(x’) = 1

Now a subgraph of our graph

X_D

^   ^

s |    | t

X_A

is (as usual) a monomorphism with our object

X_D

^   ^

s |    | t

X_A

as codomain.

A map f in the category of graphs

Y_D                               X_D

^   ^                               ^   ^

s‘ |    | t‘       –f–>          s |    | t

Y_A                               X_A

is a pair of set maps

f_D: Y_D –> X_D

f_A: Y_A –> X_A

i.e.

Y_D           –f_D–>          X_D

^   ^                              ^   ^

s‘ |    | t‘                         s |    | t

Y_A           –f_A–>          X_A

satisfying

f_D s‘ = s f_A

f_D t‘ = t f_A

Taking the domain object

Y_D

^   ^

s‘ |    | t‘

Y_A

as Y_A = {x’}, Y_D = {1} with s‘(x’) = 1 and t‘(x’) = 1

and the graph map as

f_A(x’) = x’

f_D(1) = 1

we find that

s f_A(x’) = s(x’) = 1

f_D s‘(x’) = f_D(1) = 1

i.e.

s f_A = f_D s‘

Moving to the second equation we have to check

t f_A(x’) = t(x’) = 1

f_D t‘(x’) = f_D(1) = 1

i.e.

t f_A = f_D t‘

Now we can say

Y_D                               X_D

^   ^                             ^   ^

s‘ |    | t‘         –f–>       s |    | t

Y_A                               X_A

is a part of our beloved object

X_D

^   ^

s |    | t

X_A

If this—starting with CHAOS, switching to dynamical systems under the pretext of measurement, and then running around three different categories just to count parts of a simple idempotent (ee = e)—is little too disorienting, then you need to visit India and try crossing streets; you’ll soon find blooming buzzing confusion all too soothing (if you make it across alive, of course

Seriously, I’m just making my way to a serious study of the chaotic:

An observable

f: X –> Y

on a dynamical system

X

^

a |

X

is said to be chaotic if the induced map

X                       Y^N

^                           ^

a |         –f*– >        | b

X                       Y^N

 is ‘onto for states’, i.e. if for every possible sequence

y: N –> Y, where N = {0, 1, 2, …}

of future observations there is at least one state x of X for which f*(x) = y.

One interpretation of the chaotic nature of the observable

f: X –> Y

is that (although

X

^

a |

X

is perfectly deterministic) f observes so little about the states that nothing can be predicted about the possible sequences of observation themselves.  Often the ‘remedy’ for this is to observe more.

–Conceptual Mathematics, page 317

In particular, syntax is NOT the adjoint of semantics.  Cratylus, Chomsky, and their 21^st century followers can be refuted by looking soberly at the actual practice of mathematics (wherein the construction of sequences of words and of diagrams is pursued with great care for the purpose of communication.  That syntax is only remotely dependent on the structure of the content that is to be communicated).

Both of the functors

? ———————-> theories ————————–> Large categories

Syntax                                                                                       Semantics

are needed.  The domain category of the first can be chosen in various useful ways: sketches or diagrams of signatures et cetera.

Happy New Year!

Bill

f: A –> B

where domain set A = {cat, dog}, codomain set B = {flower, pet}, and with

f(cat) = pet

f(dog) = pet

Why bother counting parts of what looks like nothing more than an all too obvious labeling?  Parts, where to begin, are profound.  Take, for example, a singleton set

1 = {•}

which has two parts

0: 0 –> 1

1: 1 –> 1

each one of which is a 1-1 function.  Just in case you are concerned about the fact that a function is required to assign to each element in the domain set an element in the codomain set and how on earth can there be a function

0: 0 –> 1

when there is no element in the empty domain set 0?  Yes, that’s exactly the reason there is exactly one function from an empty set to a singleton set and that’s exactly the reason we also say there is exactly one function from an empty set to an empty set (of course there’s no arrow going from the empty domain set; not too long ago these two 1-1 functions with empty set as domain used to give me pounding headache; now I don’t feel ;)  Of course, there is no function from a singleton set to an empty set because there is no element in the empty codomain set to which a function, to earn its existence, can assign the element in the domain singleton set (Conceptual Mathematics, page 30).

Now what about the parts of the function f: A –> B with which started?  Parts are parts of an object.  So we need to build a ‘universe of discourse’ with functions such as f: A –> B as objects.

Welcome to the Category of Set Maps (Conceptual Mathematics, pp. 144 – 5)!  Around here objects are functions

f: A –> B

Maps from an object f: A –> B to an object g: C –> D are pairs of functions

<x, y>: f –> g

or, with little more precision, are commutative squares such as

B –y–> D

^………^

f |………| g

A –x–> C

satisfying

y f = g x

Here are some more cast-and-characters of our category of maps.

Returning to the question that got us started: counting the parts of an object (in the category of set maps)

f: A –> B

The parts of f: A –> B, somewhat analogous to the case of category of sets where parts of a set are 1-1 functions with the set as codomain, are monomorphisms (Conceptual Mathematics, pp. 339 – 42) in the category of set maps with

f: A –> B

as codomain object.  So I have to count the number of monomorphisms with f: A –> B as codomain.  Thanks to an exercise (Sets for Mathematics, page 116) which tells me what to look for—when looking for parts—I know that a part of

f: A –> B

is a pair of 1-1 functions

x: X –> A

y: Y –> B

satisfying

y g = f x

with g: X –> Y.

How many 1-1 functions are there with A = {cat, dog} as codomain set?

x₁: 0 –> A, 0 = {}

x₂: X –> A, X = {cat}

x₃: X’ –> A, X’ = {dog}

x₄: A –> A (identity function)

Along similar lines, there are also four 1-1 functions with B = {flower, pet} as codomain set i.e.

y₁: 0 –> B, 0 = {}

y₂: Y –> B, Y = {flower}

y₃: Y’ –> B, Y’ = {pet}

y₄: B –> B (identity function)

So we have 16 pairs of 1-1 functions with f: A –> B.  Of these 16 pairs

<x₁, y₁>

<x₂, y₁>

<x₃, y₁>

<x₄, y₁>

.

.

.

<x₄, y₄>

how many satisfy

f x_i = y_j g_ij

with g_ij: x_i –> y_j and i, j = 1, 2, 3, 4.

Then there is the question of ‘how many g_ij are there for each i, j?’

Oh well, this’s what happens when an unstoppable force meets an immovable object!

Case I. i = 1, j = 1

x₁: 0 –> A, 0 = {}

y₁: 0 –> B, 0 = {}

g₁₁: 0 –> 0

f: A –> B

Since f x₁ = y₁ g₁₁ (think multiplying with zero as in n × 0 = 0 × 0)

K, we have a part <x₁, y₁> of f: A –> B.

Case II. i = 2, j = 1

x₂: X –> A, X = {cat} and x₂(cat) = cat

y₁: 0 –> B, 0 = {}

Unfortunately there’s no function g₂₁ from a singleton set X to an empty set 0.

So <x₂, y₁> is not a part of f: A –> B.

Case III. i = 1, j = 2

x₁: 0 –> A, 0 = {}

y₂: Y –> B, Y = {flower} and y₂(flower) = flower

g₁₂: 0 –> Y

f: A –> B

Since f x₁ = y₂ g₁₂ (think multiplying with zero as in n × 0 = m × 0)

OK then we got another part <x₁, y₂> of f: A –> B.  This g₁₂: 0 –> Y, thinking of the domain and codomain as ‘present’ and ‘past’ respectively, is how you talk about people who didn’t make it to the present (cf. Two-Stage Variable Sets; Sets for Mathematics, pp. 114 – 9).

Lucky you—I, unlike that bunny that keeps going and going, sleep like 12hrs/day; I kinda see adjoint functors high up on the cloud atlas (Exercise 6.14; Sets for Mathematics, page 119) from where we are at (Exercise 6. 6; Sets for Mathematics, page 116) if I squint just right   Being the sadist that I’m I’ll be back to go through all the cases and correct the mistakes I made here.

Happy hump day!

[…]

Truth and the terminal object:

Concerning your question last month about the role of the terminal object in calculating the truth value object of a topos: for example, you note correctly a similarity between the topos of abstract sets and the topos whose objects are maps of abstract sets; this similarity does not extend to any kind of directed graphs, nor to group actions etc. (as you noted).  I think you have re-discovered the important class of toposes known as ‘localic’.  Recall that an important method of analyzing the inside of objects is the employment of a small subcategory of figure shapes.  The basic requirement on this subcategory in its relation to the whole is that for any two maps f, g: X —> Y, one can conclude that f = g provided fx = gx for every figure x: A —>X with A in the little subcategory.  (The actual structure of the inside of X is revealed by incidence relations a: A’ —> A which relate particular figures by x’ = xa and ramifications thereof.)  For example, in categories of graphs, typically a two-object subcategory suffices.  In general, the truth value object is determined by knowing all the subobjects of all these preferred figure shapes and how these substitute along incidence relations.  Of course, ‘points’ in the narrow sense are just 1-shaped figures, so points of the truth value object correspond to subobjects of 1.  In case the subobjects of 1 are sufficient to distinguish maps in the above sense, the whole topos is called ‘localic’, and this subcategory can be visualized as consisting of various open regions in a topological space.  At the level of the 2-category of set-based toposes, your one example (which has only one region besides the two trivial ones) has a certain universality among all the localic toposes.  Contrary to a wide-spread misconception, the localic toposes are not really typical, but do incorporate one important notion of ‘variable set’.  On the other hand, the other large class of cohesive toposes is essentially disjoint from the localic ones, even though there are many relationships between the two classes.

I hope that the above account is helpful.

Bill Lawvere

————————-

Dear Professor F. William Lawvere,

The relation between terminal objects and truth value objects varies across categories.  For example, in the category of sets one could [exhaustively] characterize the truth value object if we are given the terminal object (cf. the 1-1 correspondence between parts of the terminal object and points of the truth value object), but it’s not the case in the category of graphs.  On the other hand, the relation between truth value object and terminal object in the category of two-stage variable sets seems to be like that in the category of sets.

I am wondering if there’s something about a category that speaks directly to the way truth value object corresponds to the terminal object in the category (please forgive me for this diffuse question).  Naively speaking, I am thinking that studying this correspondence can help in the calculation of the truth value object from the terminal object in various categories.

I’d appreciate very much your clarifications and guidance in my study of truth.

My heart-felt congratulations to you on the occasion of being named Fellow of the American Mathematical Society!

Happy Ugadi! (Happy New Year’s Day in my state of Andhra Pradesh)

Thanking you,

Yours sincerely,

posina

There are a couple of cautionary remarks I would like to make:

For one thing, Conceptual Mathematics is not as such a field of study, but only a hopefully helpful guide for the beginner to initiate the study of space and quantity and the human struggle to understand their relationship, with the Science explicitly isolated and developed by Eilenberg, Mac Lane, Grothendieck, Kan, Isbell, Schanuel, et al . You recognized this by your inclusion of more of my writings, although a fairer representation would certainly cite the writings of others too, and even some actual applications of Category Theory. The title [of the textbook] actually arose because there was an NSF proposal for another book on ‘Discrete Mathematics’, which had been for some years a method of promoting one-sidedness in mathematics education. Steve Schanuel and I said ‘No, we want to teach the basics of methods that apply to both discrete and continuous mathematics and to their relationships’. We chose the word ‘Conceptual’ to express this, although some friends wisely warned us that another common mainstream misuse of ‘conceptual’ in School curricula is to suggest imprecise mysticism as opposed to concrete calculations. Thus, the second thing I want to post as a warning is that there is a trend (fortunately still a minor trend) to use our book, not as science, but as a source of subjective highs. […]

A third point has to do with the dialectics of the application of the category of abstract sets to real situations, for ex. choosing which breakfasts to serve to one’s friends. In connection with that and similar examples in the text, we had some misgivings. We tried to emphasize that there are three steps, the extra-mathematical assignment of names of real things to points in definite sets, the application of composition, products, etc. to those definite sets, and then the reading off of real conclusions. For example, reading off the fibers of a map will tell us things like how many friends received the same kind of breakfast. But since in the extreme case, all singleton sets are uniquely isomorphic, there is no value or information to be derived from displaying an arrow from one labeled singleton set to another, so it is non-mathematical to promote that illusion. Science will take us back to reality.

Thank you again,

best wishes

Bill Lawvere

Sin, for some inexplicable reason, reminds me of conservatives.  As we all know, for each conservative there is a liberal that the conservative was during the previous stage [of being sensible].  So a set of people undergoing the transition from the previous liberal stage of having fun to the present conservative stage of panicking about the possibility of somebody somewhere having fun can be analysed in terms of two sets Conservatives, Liberals and a function

f: Conservatives –> Liberals

specifying for each element x in Conservatives the element in Liberals that x was during the previous liberal stage.  We may note here that not all liberals graduate into conservatives (some stay young forever; Sets for Mathematics, pp. 114 – 9).

Given that the objects of the category of two-stage variable sets (Sets for Mathematics, pp. 114 – 9) are functions

ξ_X: X_U <– X₁

is it correct to think, in view of the fact that the functions ξ_X need not necessarily be onto or 1-1, that the [underlying] connection from a previous stage to the present stage

U –> 1

can not be a function.

If, for example, I take

ξ_X: X_U <– X₁

with X₁ = {mother, child}, X_U = {pregnant woman} and

ξ_X(mother) = pregnant woman

ξ_X(child) = pregnant woman

then it is not clear how one could construe the physical process from the past (with one pregnant woman) to the present (of a mother and a child) as a function.  Having used ‘life’ to illustrate the above case of onto ξ_X, we now use ‘death’ to illustrate the case of 1-1 ξ_X i.e.

ξ_X: X_U <– X₁

with X₁ = {Posina}, X_U = {Subrahmanyam, Rayudu}.  Since my brother is no more, there is only one element ‘Posina’ in the present with

ξ_X(Posina) = Rayudu

reminding ‘Posina’ (in X₁) of the ‘Rayudu’ (in X_U) that he was sometime in the long-gone past.  Once again it is not clear how the process from the past (of two brothers) to the present (with just one left) can be thought of as a function.  All this detour into life-death is simply to say that it seems like there’s a difference between the category of two-stage variable sets (Sets for Mathematics, pp. 114 – 9) and the category of maps (Conceptual Mathematics, pp. 144 – 5), and I have no idea what exactly the difference is!

Setting aside the sadness, here’s an happy study with

X₁ = {dog, cat}

X_U = {kitten, puppy}

and the function

ξ_X: X_U <– X₁

specifying for each element x of the domain set X₁ the element that x was during its playful years (in the codomain set X_U) before it got leashed into adulthood i.e.

ξ_X(cat) = kitten

ξ_X(dog) = puppy

Now when I look at this object

ξ_X: X_U <– X₁

what I see depends on what I attend to: attending to the ‘dog’ (along with the ‘puppy’ that it was) looks like a map with the object ξ_X as codomain object of the map

1₁ –> ξ_X

and if we focus-in a little bit more, then it looks like a commutative square

1 –puppy–> X_U

^                     ^

1₁ |                      | ξ_X

1   –dog–>    X₁

satisfying

puppy 1₁ = ξ_X dog

Here there’s a parallel to the category of sets.  Elements of a set, say,

X = {x₁, x₂}

are in 1-1 correspondence with the functions

x₁: 1 –> X

x₂: 1 –> X

from the terminal object (a singleton set 1 = {•}) of the category of sets.  Similarly elements of an object X of the category of two-stage variable sets are maps to X from the terminal object

1₁: 1 –> 1

Just to be clear, an object in the category of sets is a set.  (Why are categories named after their objects?  It has, my porous memory tells me, something to do with NATURAL, and that’s all I know.)  An object in the category of two-stage variable sets is a function

ξ_X: X_U <– X₁

from a domain set X₁ to a codomain set X_U (see also the category of maps; Conceptual Mathematics, pp. 144 – 5).  Terminal object of the category of two-stage variable sets is the function

1₁: 1 –> 1

from 1 (= {•}) to 1.  In other words, there is exactly one map

f: 1 <– X

from every object

ξ_X: X_U <– X₁

in the category of two-stage variable sets to

1₁: 1 <– 1

i.e. a unique commutative square

1 <–f_U– X_U

^             ^

1₁ |             | ξ_X

1 <–f₁– X₁

(f_U sending every element of X_U to the only element of 1 and f₁ sending every element of X₁ to the only element of 1) satisfying

f_U ξ_X = 1₁ f₁

I almost forgot about composition.

Given a pair of composable maps

X –f–> Y and Y –g–> Z

i.e. two commutative squares

X_U –f_U–> Y_U

^              ^

ξ_X |                | ξ_Y

X₁ –f₁–> Y₁

 and

Y_U –g_U–> Z_U

^              ^

ξ_Y |                | ξ_Z

Y₁ –g₁–> Z₁

satisfying

f_U ξ_X = ξ_Y f₁

g_U ξ_Y = ξ_Z g₁

We say that the composite map

X –f –> Y –g –> Z = X –gf –> Z

is the composite square

X_U –f_U–> Y_U –g_U–> Z_U

^              ^              ^

ξ_X |              | ξ_Y            | ξ_Z

X₁ –f₁–> Y₁ –g₁–> Z₁

i.e. the commutative square

X_U –g_Uf_U–> Z_U

^                 ^

ξ_X |                   | ξ_Z

X₁ –g₁f₁–> Z₁

satisfying

g_U f_U ξ_X = ξ_Z g₁ f₁

But does it?  Let’s see—we are given

f_U ξ_X = ξ_Y f₁ and g_U ξ_Y = ξ_Z g₁

and we have to see if

g_U f_U ξ_X = ξ_Z g₁ f₁

Starting with g_U f_U ξ_X which looks like

X_U –f_U–> Y_U –g_U–> Z_U

^

| ξ_X

X₁

which equals g_U ξ_Y f₁ which in turn looks like

               Y_U –g_U–> Z_U

               ^

                 | ξ_Y

X₁ –f₁–> Y₁

which in turn equals ξ_Z g₁ f₁ which in turn looks like

                               Z_U

                               ^

                                 ξ_Z |

X₁ –f₁–> Y₁ –g₁–> Z₁

All of the above can be summed up in words as follows:

going from X₁ to Z_U via the (up-right) path of g_U after f_U after ξ_X

is same as

going from X₁ to Z_U via the (right-up) path of ξ_Z after g₁ after f₁

I don’t know what you are thinking, but I’m like, after what feels like eons, I finally did one exercise (Exercise 6.4; Sets for Mathematics, page 115); I know it’s not even an atom in the grand scheme of things or is it   Next exercise to flex my muscle is Exercise 14 (Conceptual Mathematics, page 144), which, for really good reasons, I love!

(Slowly but surely we’ll get to adjoint functors one exercise after one exercise in the category of two-stage variable sets!  Sorry about the clumsy diagrams above; it looks like I’m better off going back to attaching word.docs)

Given a couple of sentences made up of few words such as

Posina is in India.

India is in China.

I can translate from the given English into Swedish as

Posina är i Indien.

Indien är i Kina.

Translation, moving away from the given particulars, involves

F_W: Words^E –> Words^S

mapping words in English into words in Swedish such as

Posina |-> Posina

is |-> är

and

F_S: Sentences^E –> Sentences^S

mapping sentences as in

Indien är i Kina.

^

|

India is in China.

In mapping sentences we make sure that subjects go to subjects and objects go to objects; otherwise we might end up translating

India is in China.

as

Kina är i Indien.

and find that China is not too happy with our translation, which is rather serious—so let’s formalize it for all future generations to learn before it’s too late!

I think of a sentence as an arrow

• –> •

with the dot on the left as subject and the dot on the right as object of the sentence i.e.

subject –Sentence–> object

Translation F_S of one sentence

A –p–> B

into another

C –q–> D

should respect the subject / object structure of sentences i.e.

if F_S(p) = q

then [it better be] F_W(Sj(p)) = Sj’(F_S(p))

which says something like:

the subject (Sj) of sentence p is mapped (by F_W) to the subject (Sj’) of the sentence (q) to which the sentence p is mapped by F_S.

So is the case with objects i.e.

F_W(Oj(p)) = Oj’(F_S(p))

Just in case you don’t like equations, here it is in pictures

B –F_W–> D

^                ^

|                 |

p  =F_S=>  q

|                 |

A –F_W–> C

In little bit higher resolution it looks like the commutative square below in the case of subjects

Words^Eng –F_W–> Words^Swe

^                              ^

Sj^E |                                | Sj^S

|                                |

Sentences^Eng –F_S–> Sentences^Swe

and in the case of objects

Words^Eng –F_W–> Words^Swe

^                              ^

Oj^E |                                | Oj^S

|                                |

Sentences^Eng –F_S–> Sentences^Swe

If you are now like—I miss equations

Sj^S(F_S) = F_W(Sj^E)

Oj^S(F_S) = F_W(Oj^E)

or better yet, in plain words, subject / object of a translated sentence is same as the translated subject / object of the sentence translated.

Given

Posina is in India.  India is in China.

I’m tempted to think

Posina is in China.

This thought expressed in arrow-language looks like two arrows

Posina –is in–> India.               India –is in–> China.

composed into one arrow

Posina –is in–> China.

Having had enough of lost-in-translation, we make sure translation (F) preserves composition of sentences i.e.

F(qp) = F(q) F(p)

which says something like:

composite of translated sentences is same as the translation of the composite of sentences

Now all this TRANSLATION is looking lil like FUNCTOR (Conceptual Mathematics, page 167), which is an interpretation of one category into another.

CATEGORY, when one looks at the definition (Conceptual Mathematics, page 21), has two things:

OBJECTS

MAPS

Functor F takes objects to objects and maps to maps i.e.

F_Ob: Objects –> Objects

F_Mp: Maps –> Maps

Objects and maps in a category are related to one another via structural maps:

domain: Maps –> Objects

codomain: Maps –> Objects

identity: Objects –> Maps

composition: Maps³ –> Maps

and functors from one category to another respect this structure of categories i.e.

F_Ob(dom) = dom’(F_Mp)

F_Ob(cod) = cod’(F_Mp)

F_Mp(identity) = identity’(F_Ob)

F_Mp(gf) = F_Mp(g) F_Mp(p)

If you translate from English to Swedish, then I see no reason not to translate from Swedish to English; and in translating from English to Swedish and from Swedish to English i.e.

F: English –> Swedish

G: English <– Swedish

it’s possible to end with what we started with:

India is in China.

^

|

Indien är i Kina.

^

|

India is in China.

which is boring enough to warrant careful examination of pairs of opposed functors

F: A –> B

G: A <– B

There are, among these opposed functors, some special pairs called ADJOINT FUNCTORS, which deserve our special attention.  I though of getting a feel for adjoint functors in terms of

adjoints to inclusions

and this

there is at most one inclusion between two parts

there is at most one coarsening between two partitions

is how close I got to it

Imagine, upon reading the definition of truth value object (Conceptual Mathematics, page 337), finding that the textbook ends then and there—no more spilled-ink to make sense of.  There’s, within this doomsday scenario, enough momentum built, definition-after-definition beginning (where shall we) with terminal objects such as 1 (Conceptual Mathematics, pp. 225 – 9), into the story to keep characters moving in the heads heading home long after the curtain dropped on

true: 1 –> Ω

What if I reverse an arrow

–>

into the arrow

<–

How about an arrow from Ω

X <– Ω

instead of the arrow to Ω

X –> Ω

Reversing an arrow

–>

into the arrow

<–

doesn’t cost whole lot of consciousness and that’s all(?) it takes to get to the definition of 0 (empty set):

there is, for each set X, one function 0 –> X

from the definition of 1 (singleton set):

there is, for each set X, one function 1 <– X

Just as I was about to write-off 1 as bland (after all there is just one function to 1; Conceptual Mathematics, page 302), points i.e. functions from 1

1 –> X

are all over the place spicing up the story (Conceptual Mathematics, page 19, pp. 230 – 5).

With this preamble in place, let’s indulge in some head-scratching: the definition of truth value object is beaming with arrows (waiting to be reversed?) such as

(i)                  a given part true: T –> Ω

(ii)                every part g: U –> X

(iii)               exactly one arrow f: X –> Ω

or, shall we say, loaded with heavyweights such as

PART

Belongs to

Beginning with belongs to, what do we get if we reverse arrows in

x belongs to y

I’d have never guessed

determined by

to be the opposite of

belongs to

Then again it’s not like I suspected the initial object

0

to be the opposite of the terminal object

1

or SUM to be the opposite of PRODUCT (Conceptual Mathematics, page 254, 284).  I have nothing to show even after looking in every gyri and sulci of my big male-brain for the opposite of PART (Sets for Mathematics, pp. 37 – 8).

That’s all the self-indulgent blather my shrink can stand this Sunday

We say

f: A –> Y

is determined by

g: A –> X

whenever

f = pg (Conceptual Mathematics, pp. 45 – 9, 68 – 80, 370 – 1).

Note to self: One of these days you need to stop talking words or at least start drawing diagrams!  Until then I need not highlight the fact that science is preoccupied in every step of its long-winding road to reality with DETERMINATION.  Watching bodies-in-motion one wonders about distances traveled by the moving bodies

f: Moving bodies –> Distances

and thinks of noting down the duration of motion

g: Moving bodies –> Durations

and hopefully finds that there is a function

h: Durations –> Distances

(assigning to each duration of motion the distance traveled by moving bodies in that duration) such that

f = hg

in which case we say

f is determined by g

1 = {•}

a singleton set with one element (•).

Here’s a part

g: 1 –> 1

of 1 i.e. g(•) = (•).

This part g is the inverse image of another part

v: 1 –> W

with W = {true, false} and v (•) = true…

Now this is beginning to read a lot like Ramayana in three sentences, which is all the excuse I need to go back—way back to my very good friend function

f: A –> B

Let’s say we have a function

color: A –> B

with A = {Rayudu} and B = {Black, Brown} telling us the hair color of people in A.  For example

color(Rayudu) = Black

which says something like: the hair color of Rayudu is black.

A recurring nightmare of mine [in mathematics] is inversion—fortunately, for now and here, it’s not that scary; it simply asks: who’s that in A who got black hair?

Rayudu

Rayudu, not unlike any other answer, is boring; it’s how we got to ‘Rayudu’ from ‘Black’ that’s worth a penny or two.

But, first, who’s Rayudu?

Rayudu is a part of the domain set A i.e.

Rayudu: 1 –> A

What, while we are at it, is black?

Black is a part of the codomain set B i.e.

Black: 1 –> B

Going from the color Black to the person who’s color is black is, as far as I can tell, going from a given part

Black: 1 –> B

to a part

Rayudu: 1 –> A

along the function

color: A –> B

The question then is: given

color: A –> B, color(Rayudu) = Black

and

Black: 1 –> B

how did we end up with this

Rayudu: 1 –> A

What’s so special about Rayudu?

It’s a part of A; but so is

0: 0 –> A, 0 = {}

Rayudu is special in that any generalized element of A (Sets for Mathematics, page 16) i.e.

x: X –> A

is in

Rayudu: 1 –> A

if and only if

fx: X –> B

is in

Black: 1 –> B

which is just about all the confusion I can cook up out of

Rayudu: 1 –> A is the inverse image of Black: 1 –> B along f: A –> B

Conceptual Mathematics, page 336

Since we are least interested in Rayudu, his color, or for that matter, any other values of physical variables associated with him, let’s get off this tangent and get back to business pronto.

The part

g: 1 –> 1, g(•) = •

is the inverse image of the part

v: 1 –> W, v(•) = true

but along what?

Even more bluntly, only if I specify a function

f: 1 –> W

and say something like

g is the inverse of v along f

then somebody else can tell me if it’s true or false.

Given that

W = {true, false}

we have two functions from 1 to W

f: 1 –> W, f (•) = true

f’: 1 –> W, f’ (•) = false

Now the question we have staring into our pupils:

Is g the inverse image of v along f or along f’ or along both f and f’?

We can say

g is the inverse image of v along f

because the generalized element

x: 1 –> 1

is in (Conceptual Mathematics, page 335)

g: 1 –> 1

i.e. there exists a p such that

x = gp

and

fx: 1 –> W

is in

v: 1 –> W

i.e. there exists a q such that

fx = vq

 and because

 f’x: 1 –> W

is not in

v: 1 –> W

i.e. because there is exactly one function

f: 1 –> W

along which

g is the inverse image of v

Just in case you feeling bad about the other function

f’: 1 –> W, f’(•) = false

being left out of our story, here’s fairness-in-action: there’s another part of 1

g’: 0 –> 1, 0 = {}

that is the inverse image of v along f’ (but not along f).

That’s enough pillow-fights and here comes the hardcore definition (Lawvere element):

An object W together with a given part v: 1 –> W is called a truth value object (for the category of sets) if and only if for every part g of any set X there is exactly one function f: X –> W for which g is the inverse image of v along f (Conceptual Mathematics, page 337).

Here’s the part I like in the above definition:

for every part g… there is exactly one function f

As much as I just do things, here’s the plan: redo this exercise

1. for D (dot) in the category of graphs
2. for U (= 0 –> 1) in the category of two-stage variable sets (Sets for Mathematics, pp. 114 – 9)
3. for 0: 0 –> 1 in the category of labeled sets

The above 3 cases have something in common with the case of 1 = {•} of the present post which is the one dot (in visual depiction) of the objects

1, D, U, and 0

which means… nothin

1                      1                      1                      (being)

1                      1                      2                      (stasis)

1                      2                      1                      (motion)

1                      2                      2                      (result of motion)

2                      1                      1                      (superposition)

2                      1                      2                      (substitution)

2                      2                      1                      (configuration)

But what do you mean by

things: points or bodies?

places: boundaries or neighborhoods?

times: instants or durations?

Diving into all that feels like simulated-drowning  For now here we go drop •     drip •

MOTION:       1 thing in 2 places at 1 time

The concept of motion as the presence of a thing in one place at one time, in another place at another time describes only the result of motion and not motion.

While I’m paraphrasing Professor F. William Lawvere (see Continuously Variable Sets; Algebraic Geometry = Geometric Logic) I might as well share another observation from the same paper which I found fascinating:

Each material quantity (e.g. 120 lbs) is a quantity of something (lb) and hence has its own particular structure which is in need of mathematical clarification.

(Please don’t expect physicists possessed by God particles to do the janitorial work of clarifying MATTER or MOTION; they are too busy tickling themselves to headline news.)

Distancing myself from the verbal description of the material situations with which we started, I see a big-picture:

constant vs. variable

To get a feel for that all too familiar ever-elusive variable:

I am a different person today than I was yesterday, yet I am still the same person.

Sets for Mathematics, page 16

we begin with an acknowledgment of how challenging it is to grasp CHANGE conceptually (definitely not fun like turning it into a lullaby for the masses and the enlightened alike).

Let’s start with, say, then and now and throw some things in.  Some things that were may not be anymore: people die, vehicles move out of parking spaces, etc.  Some things do survive, though not forever, both yesterday and today.  Standing in the shoes of a thing that’s in today we can see the thing that it was yesterday.  Straightening this bipolar rendering of variation, we have

history: Now –> Then

specifying for each thing x in Now the thing that x was back Then (see Truth Values for Two-Stage Variable Sets; Sets for Mathematics, pp. 114 – 119).

I promise to return or better yet stick with the category of two-stage variable sets at least until I get to its truth value object!