# HOW MINDS THEORIZE: I. Opposites

Theories are made—made by us—in our struggle to come to terms with the given reality. In the universe of mathematical discourse, Lawvere’s Functorial Semantics objectifies abstraction as a functor. Here we construct examples of the abstraction functor, which can inform the scientific discourse on mind—as it reaches for generals.

The abstraction functor

*S*^{Top} → T

(where the domain category *S*^{Top} is the category of models [in the category of sets ** S**] of the codomain category T of theories) and its ‘opposite’ interpretation functor

T → *S*^{Top}

are the focus of our study (Sets for Mathematics, pp. 154 – 5, 248 – 50). Beginning with simple cases such as T = **1**, a category with one morphism, we slowly unravel the mathematical content of the opposite pair—abstraction, interpretation—of functors.

The category **1** with one morphism has one object and the definition of CATEGORY forces the one morphism to be the identity morphism on the one object. The opposite category **1**^{op} has same objects as **1** but with arrows reversed (Sets for Mathematics, page 25). Since reversing the identity morphism results in the same identity morphism, we have **1**^{op} = **1**.

The category

*S*^{Top}

with T = **1** is the category

*S*^{1}^{op}

of functors

**1**^{op} → *S*

from the category **1**^{op} (with exactly one identity morphism) to the category of sets ** S**.

A functor is an assignment of objects and maps in the codomain category to the objects and maps, respectively, of the domain category in a way respectful of the domain, codomain, composition, and identity of the domain category. So an assignment of a set and its identity function in ** S** to the only object and the only identity morphism, respectively, of

**1**

^{op}constitutes a functor

X: **1**^{op} → *S*

Denoting the only object of **1**^{op} with

•

and its identity morphism with

• — 1_{•} —> •

we find

X (•) = X (a set in ** S**)

X (1_{•}) = 1_{X} (identity function on X)

So we say that the functors

X: **1**^{op} → *S*

correspond to sets in ** S**.

*f*: X → Y

from a functor

X: **1**^{op} → *S*

to a functor

Y: **1**^{op} → *S*

is an assignment of a function

X (•) — *f*_{•} —> Y (•)

to the only object (•) in **1**^{op }subject to the condition that the square

X (•) — *f*_{•} —> Y (•)

^ ^

| |

X (1_{•}) Y (1_{•})

| |

X (•) — *f*_{•} —> Y (•)

assigned to the only map (1_{•}) in **1**^{op} commutes.

Upon substituting

X (•) = X (a set in ** S**)

X (1_{•}) = 1_{X} (identity function on X)

Y (•) = Y

Y (1_{•}) = 1_{Y}

we find that the natural transformations

*f*: X → Y

correspond to functions in ** S**. So we say that the category

*S*^{1}^{op}

whose objects and morphisms are functors

X: **1**^{op} → *S*

and natural transformations

*f*: X → Y

respectively, corresponds to the category of sets ** S**. With this identification, we can study the opposition between abstraction

*S*^{Top} → T

and interpretation

T → *S*^{Top}

With T = **1** and *S*^{Top} = S, we have an adjoint situation

A: ** S** →

**1**

I: **1** → *S*

Since there is only one object (•) in the category **1**, all sets of ** S** are mapped to that only object and all functions (in

**) are mapped to the only morphism (1**

*S*_{•}) in the category

**1**and that is the only functor (A) from the category of sets

**to the category**

*S***1**. In contrast to this situation, there are many functors from

**1**to the category of sets; each functor corresponding to a set (in

**). Out of these, there’s one**

*S*I: **1** → *S*

whose value at the only object (•) in the category **1** is the terminal set **1** (= {•}) in the category of sets ** S**.

I (•) = **1** (in ** S**)

I (1_{•}) = 1_{1}

and this interpretation functor

I: **1** → *S*

is the right adjoint to the abstraction functor

A: ** S** →

**1**

Some more work is required to show that I is right adjoint to A, and I need a break ;) More importantly, you are probably not feelin the connection between all this (assuming I didn’t screw-up the math big-time) and your mind busy theorizing. Well, you will once we get to somewhat more involved categories such as

T = **2** (= [0 → 1])

T = G (= [D = •, A = • → •, *s*, *t*: D → A])

embodying correspondingly richer theories. In the meantime, you may want to check out the Child’s problem (Conceptual Mathematics, page 96) and a simple and disturbingly delightful (deep) illustration of how an abstract structure arises from a particular example (Conceptual Mathematics, page 150).

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