Put A Fence Around It

“Put a Fence Around It” is a way of thinking about Mania and Depression while you are manic or depressed. Consider the following analogy. Suppose you have a dog and a fenced yard. You leave your dog in the yard while you go to work. When you get back from work what can you expect? You can expect your dog to still be contained within the confines of your fence. However, it is uncertain what your dog did during those hours you were gone. Within this bounded region there exists uncertainty. Your yard is like your moods and your mind is like your dog. Mania and depression are bounded events in time despite their encompassing chaotic behavior. Each mood definitely has a beginning and an end regardless of the uncertain behavior of your mind within that mood. However, to help you gain certainty over the duration of your moods you must “fence your yard.”

 An imaginary “fence” can be constructed around your mood by actively predicting its duration while you are experiencing the mood. It's very easy while you are depressed or manic to forget that these things have beginnings and ends with or without your conscious prediction. These feelings as I’m sure you already know can be all consuming. You might feel like light trying to escape the event horizon of a black hole. There's no turning back despite travelling at the greatest possible speed in precisely the opposite direction. The gravity of these feelings becomes so immense that time itself becomes warped. A minute can feel like an hour and hours can feel like minutes.

When I came up with “Put a Fence Around It” I created a running tally of my moods on an Xcel spread sheet. However, I decided to add the mood called centered. I call it centered because it's somewhere in-between mania and depression. Centered turns out to be my dominant state of mind at least in terms of the amount of time that I experience it. However, the other two moods are so much more powerful than being centered. So they must be weighted in terms of their contributions. The amount of energy required to be either depressed or manic is greater per unit of time than being centered.

Prescription Thoughts for Manic-Depression

Cell-phones used to have only one function. They were hardwired to operate in one and only one way. Now cell-phones are computers. They serve multiple purposes and can even be repurposed by downloading new software that expands their functionality. Cell-phones or any computing machine with these kinds of capabilities are called universal machines. They can simulate any computer with a fixed program. In an abstract sense, both you and I are universal machines. As part of our input we accept data that describes the operation of other minds. The knowledge that you will encounter here is precisely of this nature. It is a collection of ways of thinking that you can “download” that will make up the difference that prescription medication alone cannot make. I have never been able to by force of will improve my mental health. There were times in my life when I believed I could. I had a very strong will before my diagnosis. It subsequently was irrevocably shattered.

Bipolar disorder used to be called manic-depression. Mania and depression are fairly easy to distinguish between because they are essentially polar opposites of each other. Thoughts and moods have a distribution or a shape spread out over time. Some ideas or memories I’ve discovered are "stored" or only occurred in conjunction with specific states of mind or moods. I have manic thoughts or depressed thoughts and mostly won’t remember those thoughts until I am depressed again or manic. It used to be the case that I would make changes to my soul frequently but then I couldn’t remember the work that I had done until I found myself in the same state of mind. Even after years of dedication to my prescription regiment I was still trying to pick up the pieces of my mind and restore continuity to my soul. It's humbling to realize that when you go to sleep tonight you won't be picking up where you left off today.

However, my confidence was restored when I discovered that I have a saving grace. The universe has blessed me and you with something very valuable and especially given our circumstance i.e. that we suffer from mental illness. Intelligence has some kind of protective effect when it comes to bipolar disorder. As IQ increases the likely hood of someone who is bipolar needing hospitalization decreases. My experience has shown me that I can't fight my illness and win, nor can I be proactive and defeat it with strength of spirit. It seems my only option is to outsmart it! But how does one handle fixing a broken mind with the very mind that is broken? It’s not easy but it can be done. The main idea here is to use intelligence not force and the problems you face will begin to reveal their secrets.

Prescription medication takes center stage in most of the literature on bipolar disorder. What I realized I was lacking was medicine in the form of pure thought. There are no manuals written specifically for my thoughts, no words to show me how to live with this illness in my mind. From the date of my diagnosis until the day that I began working on prescription thoughts not my psychiatrist, religious leaders, friends, others with bipolar disorder, or online sources and books had any ideas to share regarding what I need to do with my thinking in order to restore continuity to my soul and gain dominance over my illness. Since no one had produced medicine in the form of words for me, I decided to do it myself.

The Product Rule (Calculus)

Assume u = f(x) and v = g(x) are both positive differentiable functions.  Then the product uv can be interpreted as the area of a rectangle. If x changes by an amount △x, then the corresponding changes in u and v are 

△u = f(x + △x) - f(x) and △v = g(x + △x) - g(x)

And the value of the product (u + △u)(v + △v) can be interpreted as the area of the largest rectangle above. The change in the rectangle then is 

△(uv) = (u + △u)(v + △v) - uv = u△v + v△u + △u△v

Or in other words the sum of the three shaded regions above. Now dividing by △x, yields

△(uv)/△x = u(△v/△x) + v(△u/△x) + △u(△v/△x)

Now take the limit of △(uv)/△x as △x ⟶ 0 and you get the derivative of uv:

d(uv)/dx = u lim△x⟶0(△v/△x) + v lim△x⟶0(△u/△x) +  lim△x⟶0(△u) lim△x⟶0(△v/△x)

d(uv)/dx = u(dv/dx) + v (du/dx) +  0 (dv/dx)

d(uv)/dx = u(dv/dx) + v (du/dx)

Calculus

A function is a rule that assigns each element in a set D to exactly one element in a set R.

If f is an even function, then f(-x) = f(x) for all x. If f is an odd function, then f(-x) = -f(x) for all x. f(x) = x^2 is an even function because f(-x) = (-x)^2 = (-x)(-x) = x^2 = f(x). f(x) = x^3 is an odd function because f(-x) = (-x)^3 = (-x)(-x)(-x) = (-x)x^2 = -(x^3) = -f(x). Odd functions are symmetric with respect to the origin. Even functions are symmetric with respect to the y-axis.

A function is called increasing on an interval I if for all x1 < x2 in I, f(x1) < f(x2). And a function is called decreasing on an interval I if for all x1 < x2 in I, f(x1) > f(x2).

Linear Algebra (5)

Ax = 0 is a homogeneous system of linear equations. 0 represents the zero vector. Every homogeneous system of linear equations has at least one solution which is the trivial solution or in other words the zero vector. A homogeneous system of linear equations with a nontrivial solution can be thought of as a set of vectors that when summed together start at the origin and end at the origin. Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.

Linear Algebra (4)

A vector y is said to be a linear combination of a set of vectors v1, v2, ... , vn if there exists scalars c1, c2, ... , cn, such that y = c1(v1) + c2(v2) + ... + cn(vn). span{v1, v2, ... , vn} is the set of all linear combinations of v1, v2, ... , vn. A linear combination can be viewed as a product of a matrix and a vector. The vector is the set of scalars and the matrix the set of vectors. The product of a matrix A and a vector x is written Ax. Ax = b has a solution if and only if b is a linear combination of the columns of A.

                                  

Ax = [ a1 , a2 , ... , an ]x =  x1(a1) + x2(a2) + ... + xn(an)

So, linear systems can be viewed in three different ways. As a matrix equation, a vector equation, or as a set of linear equations in an augmented matrix.

A very important matrix to understand is the identity matrix. It is any square matrix with 1's along the diagonal and zeros elsewhere. 

1   0   0   0   0   0
0   1   0   0   0   0
0   0   1   0   0   0
0   0   0   1   0   0
0   0   0   0   1   0
0   0   0   0   0   1

is a 6 x 6 identity matrix.

If A is an m x n matrix, u and v are vectors in ℝ^n, and c is a scalar, then A(u + v) = Au + Av and A(cu) = c(Au).


                                         

Linear Algebra (3)

A matrix with only one column is called a vector.

         1
V =  2
         3

V can also be expressed as V = [ 1 , 2 , 3 ]. Essentially, a vector is an ordered set of numbers. The set of all vectors with three entries is denoted by  ℝ^3. Given two vectors u and v in ℝ^3, their sum u + v is a vector w obtained by adding the corresponding entries of u and v. For example, if u = [ 1 , 2 , 3 ] and v = [ 2 , 3 , 4 ], then w = u + v = [ 1 , 2 , 3 ] + [ 2 , 3 , 4 ] = [ 1 + 2 , 2 + 3 , 3 + 4 ] = [ 3 , 5 , 7 ].

Given a vector u and a real number c, the scalar multiple of u by c is the vector c(u) obtained my multiplying each entry in u by c. The number c in c(u) is called a scalar.

Geometric Descriptions of ℝ^2
The following is a graph of the set of all vectors in ℤ^2 which is a subset of ℝ^2 such that each vector is not a scalar multiple of any other vector in the set and their components are between -8 to 8. The vectors that are in the set are marked with a purple dot. 

Vector addition geometrically amounts to the construction of a parallelogram.

Here are some algebraic properties of ℝ^n:

For all u, v, and w in ℝ^n and scalars c and d

1.) u + v = v + u

2.) (u + v) + w = u + (v + w)

3.) u + 0 = u

4.) u + (-u) = 0

5.) c(u + v) = cu + cv

6.) (c + d)u = cu + du

7.) c(du) = (cd)u

8.) 1u = u

Linear Algebra (2)

In a matrix a leading entry is the leftmost nonzero entry in a row. Consider the following matrix.

1   3   4   5   6
0   1   3   5   7
0   0   1   2   3

Rows 1, 2, and 3 have leading entries at columns 1, 2, and 3 respectively. A matrix is in row echelon form if all nonzero rows are above any rows of all zeros, each leading entry of a row is in a column to the right of the leading entry of the row above it, and all entries in a column below a leading entry are zeros. 

If a matrix is in row echelon form and each nonzero row has a leading entry of 1 and each leading 1 is the only nonzero entry in its column, then the matrix is said to be in reduced row echelon form. For example,

1   0   0   5   6
0   1   0   5   7
0   0   1   2   3

A pivot position in a matrix is a location that corresponds to a leading 1 in the reduced echelon form of the matrix. A pivot column is a column that contains a pivot position.

The Row Reduction Algorithm

Step 1
Begin with the leftmost nonzero. This is a pivot column. The pivot position is at the top.

Step 2
Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position.

Step 3
Use row replacement operations to create zeros in all positions below the pivot.

Step 4
Repeat steps 1 - 3 to the submatrix that remains until there are no more nonzero rows to modify.

Step 5
Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, use a scaling operation to make it 1.

Linear Algebra (1)

Any system of linear equations can be transformed into a row equivalent system of equations or in other words one with the same solution set by using elementary row operations.

Elementary Row Operations
1.) Interchange two rows.
2.) Scale a row by a non-zero constant.
3.) Add a multiple of row to another row.

Consider the following matrix

1  0  1
0  1  1

Here, x = 1 and y = 1. If the rows are switched then the solution remains the same. If I scale a row, say row two by a constant c, then cy = c which means that cy/c = c/c but c/c = 1. So, y = 1; the solution remains the same. Now if I add row 1 to row 2 and replace row 1 with the result, then

1  1  2
0 1  1

Row 1 is now x + y = 2. So, this line has a y-intercept of 2 and a slope of -1 but still intersects row 2 at (1,1) because 1 + 1 = 2; a solution to the equation. In fact, no matter the multiple c of row 2 when added to row 1, which results in x + cy = c + 1 will always ensure that (1,1) is a solution since (1) + c(1) = c + 1.

Linear Algebra (0)

A linear equation is an equation that can be written in the form 

a1x1 + a2x2 + ... + anxn = b

where a1,a2,...,an and b are real or complex numbers. A linear system is a collection of one or more linear equations involving the same variables. A solution of a system is a list s1, s2, ..., sn of numbers that make each equation a true statement when the values of s1, s2, ... , sn are substituted for x1, x2, ... , xn, respectively. The set of all possible solutions is called the solution set of the linear system. Two linear systems are equivalent if they have the same solution set.

A system of linear equations has

1.) no solution, or

2.) exactly one solution, or

3.) infinitely many solutions.

A system of linear equations is said to be consistent if it has either one solution or infinitely many solutions; a system is inconsistent if it has no solution.

Matrix Notation

The equations 

x    -   2y +  z  =  0

          2y -  8z =  8

-4x + 5y + 9z = -9

can be compactly recorded in the following ways. First, by using a coefficient matrix, and by using an augmented matrix.

  1   -2    1

  0    2   -8

-4     5    9

  1   -2    1    0
  0    2   -8    8
-4     5    9   -9

About Proofs

Definition: A sequent is an expression (A ⊢ C) where C is a statement called the conclusion of the sequent and A is a set of statements called the assumptions of the sequent. (A ⊢ C) is read 'A entals C' and means that there is a proof whose conclusion is C and whose undischarged assumptions are all in the set A. (⊢ C) means there is a proof of C relying on no undischarged assumptions.

Sequent Rule (∧I): If (A ⊢ C) and (B ⊢ D), then [(A ⋃ B) ⊢ (C ∧ D)].

Sequent Rule (∧E): If  [A ⊢ (B ∧ C)], then (A ⊢ B) and (A ⊢ C).

Sequent Rule (⇒I): If [A ⋃ {b} ⊢ C], then [A ⊢ (b ⇒ C)].

Sequent Rule (⇒E): If (A ⊢ C) and [B ⊢ (C ⇒ D)], then [(A ⋃ B) ⊢ D]

Just Kidding

I found a more appropriate textbook than Hurley's. It's title is Mathematical Logic and it is written by Ian Chriswell. 

Examples of Proofs with Quantifiers

∀x(Ax ⇒ Bx), ∀x(Bx ⇒ Cx) ⊢ ∀x(Ax ⇒ Cx)
1.) ∀x(Ax ⇒ Bx)                   Hyp
2.) Ax ⇒ Bx                          UI
3.) ∀x(Bx ⇒ Cx)                   Hyp
4.) Bx ⇒ Cx                          UI
5.) Ax ⇒ Cx                          Th5
6.) ∀x(Ax ⇒ Cx)                   UG

∀x(Bx ⇒ Cx), ∃x(Ax ∧ Bx) ⊢ ∃x(Ax ∧ Cx)
1.) ∃x(Ax ∧ Bx)                        Hyp
2.) ∀x(Bx ⇒ Cx)                      Hyp
3.) Ac ∧ Bc                               EI
4.) (Ac ∧ Bc) ⇒ Bc                  Th28
5.) (Ac ∧ Bc) ⇒ Ac                   Th29
6.) Bc                                        mp 3,4
7.) Ac                                        mp 3,5
8.) Bc ⇒ Cc                               UI
9.) Cc                                        mp 6,8
10.) Ac ⇒ [ Cc ⇒ (Ac ∧ Cc)]    Th30
11.) Cc ⇒ (Ac ∧ Cc)                  mp 7,10
12.) (Ac ∧ Cc)                          mp 9,11
13.) ∃x(Ax ∧ Cx)                      EG

Rules of Inference for Quantifiers

Universal Instantiation (UI)

The operation of deleting the universal quantifier and replacing every variable bound by that quantifier with a constant or a variable. 

Universal Generalization (UG)

The operation of adding a universal quantifier to a formula possessing at least one occurrence of a variable to be quantified. This rule cannot be applied to constants.

Existential Instantiation (EI)

The operation of deleting the existential quantifier and replacing in each occurrence of the variable with a constant. The existential name or constant to be used must be a new name that has not occurred in any previous lines in the proof.

Existential Generalization (EG)

The operation of adding a existential quantifier to a formula possessing either a constant or a variable and replacing it with a new quantified variable.

Predicate Logic Translations (2)

19.) Whoever is a socialite is vain: ∀x(Sx ⇒ Vx)
20.) Any caring mother is vigilant and nurturing: ∀y[(Cy ∧ My) ⇒ (Vy ∧ Ny)]
21.) Terrorists are neither rational nor empathic: ∀z[Tx ⇒ (¬Rx ∧ ¬Ex)
22.) Nobody consumed by jealousy is happy: ¬∃y(Cy ∧ Hy)
23.) Everything is imaginable: ∀w(Iw)
24.) Ghosts do not exist: ¬∃z(Gz)
25.) A thoroughbred is a horse: ∀y(Ty ⇒ Hy)
26.) A thoroughbred won the race: ∃z(Tz ∧ Wz)
27.) Not all mushrooms are edible: ∃x(Mx ∧ ¬Ex)
28.) Not any horse chestnuts are edible: ∀w(Hw ⇒ ¬Ew)
29.) A few guests arrived late: ∃x(Gx ∧ Ax)
30.) None but gentlemen prefer blondes: ∀x(Px ⇒ Gx)
31.) A few cities are neither safe nor beautiful: ∃w[(Cw ∧ ¬Sw) ∧ ¬Bw]
32.) There are no circular triangles: ¬∃z(Cz ∧ Tz)
33.) Snakes are harmless unless they have fangs: ∀x[(Sx ∧ Fx) ⇒ ¬Hx]
34.) Some dogs bite if and only if they are teased: ∃y[(Dy ∧ By) ⇔ Ty]
35.) An airliner is safe if and only if it is properly maintained: ∀z[(Az ∧ Sz) ⇔ Pz]

Predicate Logic Translations (1)

1.) Elaine is a chemist: Ce
2.) Nancy is not a sales clerk: ¬Sn
3.) Neither Wordsworth nor Shelley was Irish: ¬Iw ∧ ¬Is
4.) Rachel is either a journalist or a newscaster: (Jr ∨ Nr) ∧ ¬(Jr ∧ Nr)
5.) Intel designs a faster chip only if Micron does: Dm ⇒ Di
6.) Belgium and France subsidize the arts only if Austria or Germany expand museum holdings: (Ea ∨ Eg) ⇒ (Sb ∧ Sf)
7.) All maples are trees: ∀x(Mx ⇒ Tx)
8.) Some grapes are sour: ∃x(Gx ∧ Sx)
9.) No novels are biographies: ¬∃y(Ny ∧ By)
10.) Some holidays are not relaxing: ∃z(Hz ∧ ¬Rz)
11.) If Gertrude is correct, then the Taj Mahal is made of marble: Cg ⇒ Mt
12.) Gertrude is not correct only if the Taj Mahal is made of granite: Gt ⇒ ¬Cg
13.) Tigers exist: ∃z(Tz)
14.) Anything that leads to violence is wrong: ∀x(Lx ⇒ Wx)
15.) There are pornographic art works: ∃u(Pu ∧ Au)
16.) Not every smile is genuine: ¬∀x(Sx ⇒ Gx)
17.) Every penguin loves ice: ∀z(Pz ⇒ Lz)
18.) There is trouble in River City: ∃u(Tu ∧ Ru) 

Textbook Found

I finally found a textbook that I can trust. I've been through quite a few. The predicate logic presented in this textbook is user friendly and more for philosophy but is just as valid as any mathematical presentation. So, I'm going to go with my new textbook and trust it. Besides it's from Patrick J. Hurley. He's been writing textbooks for decades.

Set Theory and Symfluence (0)

The symbol ' ∈ ' has a measurable effect on the mind. 

Exercise 1
Look at the symbol ' ∈ ' and in your mind's voice say ' is an element of '. Repeat!

New possibilities now exist within your mind. For example:  A ∈ B.

Exercise 2
Look at the symbol ' ¬ ' and in your mind's voice say 'It is not the case that'. Repeat!

Abstractions such as '¬ A ∈ B' and ' X ∈ Y ' select very specific thoughts for your conscious experience. They're symfluence or symbol-influence finds its origin in the English language and is strengthened with each repetition of exercises 1 and 2.

Exercise 3

Look at the symbol ' ∧ ' and in your mind's voice say 'and'. Repeat!

It is not the case that the English language has provided you with all of the patterns of thought necessary to understand set theory. Thoughts such as ' A ∈ B ∧ B ∈ C ' are perfectly well understood now without any further training. However, making all of the abstractions of set theory available to your conscious experience will require a new training method.