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User:Egm6322.s09.bit.gk/r2

Quasilinear Partial differential equation
For a PDE of order n,i.e,highest order derivative are of order n ,coefficients of the nth

order derivative are functions of $$(x,y,u,u_x,u_y,...,\frac{\partial^m u }{\partial x^p\partial x^q},...)$$

Where,

$$(x,y,u,u_x,u_y,...,\frac{\partial^m u }{\partial x^p\partial x^q},...)$$

is the mth order derivative

and

$$p+q=m$$

The general solution of a Quasilinear partial differential equation is given by

$$A(x,y,u)\frac{\partial u }{\partial x}+B(x,y,u)\frac{\partial u }{\partial y}=C(x,y,u) $$

The quasilinear PDE is non linear PDE but linear in $$u_{x}$$ and $$ u_{y}$$

For Example a PDE linear wrt 2nd order derivative but still non linear in general can be given as

$$Div\left(\kappa.grad u\right)+f(x,y,u_x,u_y)=0$$

where $$\kappa=\kappa \left(x,y\right)$$

For example

$$Div\left(\kappa.grad u\right)+ax^2+by+\sqrt u_x+(u_x)^4+2(u_y)^2=0$$

Proof of linearity of a particular differential operator
We Have,

$$Div V :=\kappa.Grad u$$

Therefore exapnding it,assuming 2-D

$$Div V=\frac{\partial }{\partial x}\left(\kappa_{11}\frac{\partial u}{\partial x}\right)+\frac{\partial }{\partial x}\left(\kappa_{12}\frac{\partial u}{\partial y}\right)+\frac{\partial }{\partial y}\left(\kappa_{21}\frac{\partial u}{\partial x}\right)+\frac{\partial }{\partial y}\left(\kappa_{22}\frac{\partial u}{\partial y}\right)$$

To prove that the above Divergence is linear ,we have to show,

Let,

$$D\left( \right):$$ be an operator,such that $$L\left(\ u\right)$$ is linear with respect to $$u$$ if,

$$ D\left(\alpha u+\beta v\right)=\alpha D\left(\ u\right)+\beta D\left(\ v\right)$$

Therefore, in the present problem, we have,

$$D=\frac{\partial }{\partial x}\left(\kappa_{11}\frac{\partial }{\partial x}\right)+\frac{\partial }{\partial x}\left(\kappa_{12}\frac{\partial }{\partial y}\right)+\frac{\partial }{\partial y}\left(\kappa_{21}\frac{\partial }{\partial x}\right)+\frac{\partial }{\partial y}\left(\kappa_{22}\frac{\partial}{\partial y}\right)$$

Let $$\kappa=\kappa(x,y)$$

$$\Rightarrow D= \left(\kappa_{11x}\frac{\partial }{\partial x}+\kappa_{11}\frac{\partial^2  }{\partial x^2}\right)

+\left(\kappa_{12x}\frac{\partial }{\partial y}+\kappa_{12}\frac{\partial^2  }{\partial x\partial y}\right)$$

$$+\left(\kappa_{21y}\frac{\partial }{\partial x}+\kappa_{21}\frac{\partial^2  }{\partial x\partial y}\right)

+\left(\kappa_{22y}\frac{\partial }{\partial y}+\kappa_{22}\frac{\partial^2  }{\partial y^2}\right)$$

$$\therefore$$

$$\Rightarrow D(\alpha u+\beta v)= \left(\kappa_{11x}\frac{\partial (\alpha u+\beta v) }{\partial x}+\kappa_{11}\frac{\partial^2 (\alpha u+\beta v) }{\partial x^2}\right)

+\left(\kappa_{12x}\frac{\partial (\alpha u+\beta v) }{\partial y}+\kappa_{12}\frac{\partial^2 (\alpha u+\beta v) }{\partial x\partial y}\right)$$

 * $$+\left(\kappa_{21y}\frac{\partial (\alpha u+\beta v) }{\partial x}+\kappa_{21}\frac{\partial^2 (\alpha u+\beta v) }{\partial x\partial y}\right)

+\left(\kappa_{22y}\frac{\partial (\alpha u+\beta v) }{\partial y}+\kappa_{22}\frac{\partial^2 (\alpha u+\beta v) }{\partial y^2}\right)$$

$$\Rightarrow D(\alpha u+\beta v)= \left(\kappa_{11x}\left[\alpha\frac{\partial ( u) }{\partial x}+\beta\frac{\partial ( v)}{\partial x}\right]+\kappa_{11}\left[\alpha\frac{\partial^2 ( u)}{\partial x^2}+\beta\frac{\partial^2 (v) }{\partial x^2}\right]\right)$$

$$+ \left(\kappa_{12x}\left[\alpha\frac{\partial ( u) }{\partial y}+\beta\frac{\partial ( v)}{\partial y}\right]+\kappa_{12}\left[\alpha\frac{\partial^2 ( u)}{\partial x\partial y}+\beta\frac{\partial^2 (v) }{\partial x\partial y}\right]\right)$$

$$+ \left(\kappa_{21x}\left[\alpha\frac{\partial ( u) }{\partial x}+\beta\frac{\partial ( v)}{\partial x}\right]+\kappa_{21}\left[\alpha\frac{\partial^2 ( u)}{\partial x\partial y}+\beta\frac{\partial^2 (v) }{\partial x\partial y}\right]\right)$$

$$+ \left(\kappa_{22y}\left[\alpha\frac{\partial ( u) }{\partial y}+\beta\frac{\partial ( v)}{\partial y}\right]+\kappa_{22}\left[\alpha\frac{\partial^2 ( u)}{\partial y^2}+\beta\frac{\partial^2 (v) }{\partial y^2}\right]\right)$$

Therefore by rearranging we have,

$$\Rightarrow D(\alpha u +\beta v)= \alpha\left[\left(\kappa_{11x}\frac{\partial (u) }{\partial x}+\kappa_{11}\frac{\partial^2 (u) }{\partial x^2}\right) +\left(\kappa_{12x}\frac{\partial (u) }{\partial y}+\kappa_{12}\frac{\partial^2 (u) }{\partial x\partial y}\right)\right]$$

$$+\alpha\left[\left(\kappa_{21y}\frac{\partial (u) }{\partial x}+\kappa_{21}\frac{\partial^2 (u) }{\partial x\partial y}\right) +\left(\kappa_{22y}\frac{\partial (u) }{\partial y}+\kappa_{22}\frac{\partial^2 (u) }{\partial y^2}\right)\right] $$

$$+\beta\left[\left(\kappa_{11x}\frac{\partial (v) }{\partial x}+\kappa_{11}\frac{\partial^2 (v) }{\partial x^2}\right) +\left(\kappa_{12x}\frac{\partial (v) }{\partial y}+\kappa_{12}\frac{\partial^2 (v) }{\partial x\partial y}\right)\right]$$

$$+\beta\left[\left(\kappa_{21y}\frac{\partial (v) }{\partial x}+\kappa_{21}\frac{\partial^2 (v) }{\partial x\partial y}\right) +\left(\kappa_{22y}\frac{\partial (v) }{\partial y}+\kappa_{22}\frac{\partial^2 (v) }{\partial y^2}\right)\right] $$

$$\therefore$$

$$ D\left(\alpha u+\beta v\right)=\alpha D\left(\ u\right)+\beta D\left(\ v\right)$$

$$\Rightarrow$$ the operator $$ D\left(\right)$$ is Linear

Additive and Homogeneity of PDE

$$\mathfrak{L}(0)=0$$

Image of 0 under $$\mathfrak{L}(.)$$ is zero "0" if $$\mathfrak{L}(.)$$ is linear

 * Additive: The additivity of an operator is defined as follows,

$$\forall u ,v:\omega\rightarrow \R$$

In the present case if $$\alpha$$=1 and $$\beta$$=1, we can clearly check the additive property of the operator

$$ L\left( u+ v\right)= L\left(\ u\right)+ L\left(\ v\right)$$

 * Homogeneous:The Homogeneity of an operator is defined as follows,

$$\forall u ,v:\omega\rightarrow \R$$

$$ L\left( \alpha u\right)=\alpha L\left(\ u\right)$$

If either $$\alpha$$=0 or $$\beta$$ =0, we notice the operator is homogeneous.

The adjoining figure in essence demonstrates that a zero intercept indicates homogeneity.

Proof of Equivalency of linearity
Note: Equivalency $$A\equiv B$$

"if and only of" $$\equiv$$ "equivalent to" $$\equiv$$ "necessary and sufficient condition

$$A\Rightarrow B$$ A is a sufficient condition for B

$$A\Leftarrow B$$ B is a sufficient condition for A

A is a necessary condition for B

$$\begin{bmatrix}

A\Rightarrow B\end{bmatrix}\Leftrightarrow \begin{bmatrix}

\mathbf{B}\Rightarrow \mathbf{A}\end{bmatrix}$$

Where: B is non B or negation of B

and A is non A or negation of A

1) $$(\Rightarrow) $$ (sufficient condition)

2) $$(\Leftarrow) $$ (necessary condition)

$$\R^m$$,$$\R^n$$ are species of vectors (tensors,column matrix). Div maps, vector fields

vector fields (vector-value function) into a scalar function.

In other words, domain and range of div(.) are function spaces

General Form
The General form of a Second-order linear PDE in two independant variable is given as ,

$$A(x,y)\frac{\partial^2u }{\partial x^2}+2B(x,y)\frac{\partial^2u }{\partial x\partial y}+C(x,y)\frac{\partial^2u }{\partial y^2}+D(x,y)\frac{\partial u }{\partial x}+E(x,y)\frac{\partial u }{\partial y}+F(x,y)u+G(x,y)=0 $$

The coefficients $$A$$,$$B$$,$$C$$ are constant
With the assumption that in the general form of the Second order PDE, the coefficients $$A$$, $$B$$,$$C$$

are constants and the remaining coefficients $$D$$, $$E$$, $$F$$,

$$G$$ are still functions of $$(x,y)$$, we can write the general equation  in

the matrix form as follows

$$\left[{\begin{matrix} \frac{\partial }{\partial x} &  \frac{\partial }{\partial x}  \\ \end{matrix}}\right]\left[{\begin{matrix} A & B \\ B & C \\ \end{matrix}}\right]\left[{\begin{matrix}

\frac{\partial u }{\partial x}   \\

\frac{\partial u}{\partial y}    \\ \end{matrix}}\right] +\left[{\begin{matrix} D &  E  \\ \end{matrix}}\right]\left[{\begin{matrix}

\frac{\partial u }{\partial x}   \\

\frac{\partial u}{\partial y}    \\ \end{matrix}}\right]+F(x,y)u+G(x,y)=0 $$

Let,

$$\alpha=\left[{\begin{matrix} \frac{\partial }{\partial x} &  \frac{\partial }{\partial x}  \\ \end{matrix}}\right]\left[{\begin{matrix} A & B \\ B & C \\ \end{matrix}}\right]\left[{\begin{matrix}

\frac{\partial u }{\partial x}   \\

\frac{\partial u}{\partial y}    \\ \end{matrix}}\right]$$

and let,

$$\beta=\left[{\begin{matrix} D &  E  \\ \end{matrix}}\right]\left[{\begin{matrix}

\frac{\partial u}{\partial x}   \\

\frac{\partial u}{\partial y}    \\ \end{matrix}}\right]+F(x,y)u+G(x,y)$$

Expanding $$\alpha$$,

$$\alpha=\left[{\begin{matrix} \frac{\partial }{\partial x} &  \frac{\partial }{\partial x}  \\ \end{matrix}}\right]

\left[{\begin{matrix} A\frac{\partial u }{\partial x}+B\frac{\partial u}{\partial y}   \\

B\frac{\partial u }{\partial x}+A\frac{\partial u}{\partial y}   \\ \end{matrix}}\right]$$

$$\Rightarrow \alpha=(A\frac{\partial^2 u }{\partial x^2}+A_x\frac{\partial u}{\partial x})+(B\frac{\partial^2 u }{\partial x\partial y}+B_x\frac{\partial u}{\partial y})+(B\frac{\partial^2 u }{\partial x\partial y}+B_y\frac{\partial u}{\partial x})+(C\frac{\partial^2 u }{\partial^2 y}+C_y\frac{\partial u}{\partial y})$$

Expanding $$\beta$$,

$$\Rightarrow \beta=D(x,y)\frac{\partial u }{\partial x}+E(x,y)\frac{\partial u }{\partial y}+F(x,y)u+G(x,y)$$

If $$A, B, C$$ where to be a constant and we make the assumption that$$\frac{\partial }{\partial x\partial y}=\frac{\partial }{\partial y\partial x}$$ then in $$\alpha $$ the $$A_x$$,    $$B_y$$ and $$C_y$$ terms would be zero.

$$\therefore$$

$$\alpha=(A\frac{\partial^2 u }{\partial x^2})+(B\frac{\partial^2 u }{\partial x\partial y})+(B\frac{\partial^2 u }{\partial x\partial y})+(C\frac{\partial^2 u }{\partial^2 y})$$

Adding $$\alpha$$ and $$\beta$$, we have the gen general form of the Second order PDE,

$$\Rightarrow \alpha+\beta=A(x,y)\frac{\partial^2u }{\partial x^2}+2B(x,y)\frac{\partial^2u }{\partial x\partial y}+C(x,y)\frac{\partial^2u }{\partial y^2}+D(x,y)\frac{\partial u }{\partial x}+E(x,y)\frac{\partial u }{\partial y}+F(x,y)u+G(x,y)=0$$

The coefficients $$A$$,$$B$$,$$C$$ are constant are functions of $$(x,y)$$
If we have have to incorporate the coefficients into the partial differential equation,the general form of the Second order PDE would be

$$\frac{\partial^2A(x,y)u }{\partial x^2}+\frac{\partial^2B(x,y)u }{\partial x\partial y}+\frac{\partial^2C(x,y)u }{\partial y^2}+\frac{\partial D(x,y)u }{\partial x}+\frac{\partial E(x,y)u }{\partial y}+F(x,y)u+G(x,y)=0$$

Expanding this we have ,

$$[A_x\frac{\partial^2u }{\partial x^2}+A(x,y)\frac{\partial u}{\partial x}]+[2B(x,y)\frac{\partial^2u }{\partial x\partial y}+B_y\frac{\partial u}{\partial x}+B_x\frac{\partial u}{\partial y}]+[C(x,y)\frac{\partial^2u }{\partial y^2}+C_y\frac{\partial u}{\partial y}]$$

$$+D(x,y)\frac{\partial u }{\partial x}+E(x,y)\frac{\partial u }{\partial y}+F(x,y)u+G(x,y)=0 $$

Rearranging the terms ,we have

$$A(x,y)\frac{\partial^2u }{\partial x^2}+2B(x,y)\frac{\partial^2u }{\partial x\partial y}+C(x,y)\frac{\partial^2u }{\partial y^2}+(D+A_x+B_y)\frac{\partial u }{\partial x}+(E+B_x+C_y)\frac{\partial u }{\partial y}+F(x,y)u+G(x,y)=0$$

Let,

$$\overline D=D+A_x+B_y$$

and

$$\overline E=E+B_x+C_y$$

Rotation of Coordinate axes

Rotation of coordinate axes is an example of linear coordinate translation.

Following is general representation of linear coordinate translation.

$$\begin{pmatrix} x\\

y\end{pmatrix} = V_{2\times2}\begin{pmatrix} \overline x\\

\overline y\end{pmatrix} $$

Example of linear map for $$\R^n$$ (in a n dimension case)

$$A:\R^n \to \R^n$$

$$x \mapsto y=Ax$$

$$A \in R^{nxn} $$ nxn matrix

In this case:

$$\mathfrak{D}(Domain)=\R^n$$

$$\mathfrak{R}(Range)=\R^n$$

Analyzing a more general where m≠n:

$$A:\R^n \to \R^n$$

$$x \mapsto y=Ax$$

$$A \in R^{nxn}$$

where: n=row and m=columns

y=Ax

Now consider:

y=Ax

where: y=nx1 matrix

Ax=nx1

b=nx1

$$M: \R^m \mapsto \R^n$$

$$x \mapsto y=Ax+b$$

Clearly:

$$M(0)=b\neq 0\Rightarrow M(.)$$

The $$\neq$$ signal means that $$M(.)$$ is not homogenic

$$M(.)$$ is not a linear map, affine map

Example: Rotation followed by translation

$$y=Rx+b$$

m=n=2

$$\begin{Bmatrix} y_1\\y_2

\end{Bmatrix}=\begin{bmatrix} R_{11} &R_{12} \\ R_{21} &R_{22} \end{bmatrix}\begin{Bmatrix} x_1\\x_2

\end{Bmatrix}+\begin{Bmatrix} b_1\\b_2

\end{Bmatrix}$$

Coordinates:

$$x=\phi (\bar{x},\bar{y})$$

$$y=\psi (\bar{x},\bar{y})$$

Linear coordinate transformation

Let consider:

$$u=(x,y)= u(\phi (\bar{x},\bar{y}),y(\psi (\bar{x},\bar{y}))$$

$$u=(x,y)= u(\bar{x},\bar{y})$$ * abuse of notation by using "u"

$$u=(x,y)= \bar{u}(\bar{x},\bar{y})$$ * this is more rigorous notation

Example:

Let: $$u(x)=ax+b$$

Consider: $$x=x(\bar{x})=sin\bar{x}$$ where x $$\to $$ general function or $$(\phi (\bar{x}))$$

$$u(x)=u(\phi (\bar{x}))=a sin \bar{x}+b$$

$$u(x)=u(\bar{x})=\bar{u}$$

where:

$$u(\bar{x})$$ is an abuse of notation

and $$\bar{u}(\bar{x})$$ is a more rigorous notation

$$u_x(x,y)=\frac{\partial u}{\partial x\,}(x,y)=u_x(\phi (\bar{x},\bar{y}),y(\psi (\bar{x},\bar{y}))$$

$$=\frac{\partial \bar{u}}{\partial x\,}(\bar{x},\bar{y})=\frac{\partial \bar{u}}{\partial \bar{x}\,}\frac{\partial \bar{x}}{\partial x\,}+\frac{\partial \bar{u}}{\partial \bar{y}\,}\frac{\partial \bar{y}}{\partial x\,}$$

where:

$$\frac{\partial \bar{u}}{\partial \bar{x}\,}\frac{\partial \bar{x}}{\partial x\,}=\bar{u}_\bar{x}$$

$$\frac{\partial \bar{u}}{\partial \bar{y}\,}\frac{\partial \bar{y}}{\partial x\,}=\bar{u}_\bar{y}$$

Define:

$$\bar{x}=\bar{x}(x,y)=\phi (x,y)$$

$$\bar{y}=\bar{y}(x,y)=\psi (x,y)$$

$$u_y(x,y)=\frac{\partial \bar{u}}{\partial y\,}(\bar{x},\bar{y})=\bar{u}_{\bar{x}}\frac{\partial \bar{x}}{\partial y\,}+\bar{u}_{\bar{y}}\frac{\partial \bar{y}}{\partial y\,}$$

In a matrix form:

$$=\partial \bar{x}(u)=\begin{bmatrix}\frac{\partial \bar{x}}{\partial x\,} & \frac{\partial \bar{y}}{\partial x\,}\end{bmatrix} \begin{bmatrix}\partial \bar{x} \\ \partial \bar{y}\end{bmatrix}(\bar{u})$$

$$\begin{Bmatrix}\partial x \\\partial y\end{Bmatrix}(u)=\begin{bmatrix}\frac{\partial \bar{x}}{\partial x} & \frac{\partial \bar{y}}{\partial x}\\ \frac{\partial \bar{x}}{\partial y} & \frac{\partial \bar{y}}{\partial y}\end{bmatrix}\begin{Bmatrix}\partial \bar{x} \\\partial \bar{y}\end{Bmatrix}(\bar {u})$$

$$\begin{bmatrix}\frac{\partial \bar{x}}{\partial x} & \frac{\partial \bar{y}}{\partial x}\\ \frac{\partial \bar{x}}{\partial y} & \frac{\partial \bar{y}}{\partial y}\end{bmatrix}$$ This matrix is known as the transpose of Jacobian matrix: $$\underline{J}^{T}$$ (sometimes defined as Jacobian matrix)

Note: Easier and more general

$$(x_1,...,x_n)\to (\bar{x}_1,...,\bar{x}_n)$$

Ind. notation: $$\bar{x}_i=\bar{x}_i(x_1,...,x_n)$$

$$\mathbf{J}_{nxn}=\begin{bmatrix}\partial \bar{x}_i\\ \partial \bar{x}_j\end{bmatrix}_{nxn}$$

where:

i= row index

j=column index

= References = Egm6322.s09.bit.sahin 20:50, 6 February 2009 (UTC) Egm6322.s09.bit.gk 20:51, 6 February 2009 (UTC) Egm6322.s09.bit.la 20:52, 6 February 2009 (UTC)

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Ethics/Nonkilling/Mathematics

The appeal for nonviolence

 * This Course is based mainly on "A Nonkilling Mathematics?", chapter prepared by Professor Ubiratan D’Ambrosio (Universidade Estadual de Campinas, Unicamp, Brazil) for Toward a Nonkilling Paradigm (Honolulu: Center for Global Nonkilling, 2009). The Course is part of the Interdisciplinary Program on Nonkilling Studies at the School of Nonkilling Studies.

Nonviolence is the magnificent scenario we are struggling for. I want to envisage a road that makes Lawrence believe in man.

Political scientist Glenn D. Paige published, in 2002, a pioneering book on Nonkilling Global Political Science, featuring a very provocative and basic chapter entitled Is A Nonkilling Society Possible? In it Paige says

A document elaborated by an international group of scientists, convened by the National Spanish National Commission for UNESCO, in Seville, Spain, in 1986 and adopted by UNESCO, became known as the Seville Statement on Violence. In the last paragraph, it claims that

In the 8th World Summit of Nobel Peace Laureates, conveyed in Rome in 2007, participants produced the Charter for a World without Violence, which states:

I agree with the Seville Statement on Violence in accepting that I am also responsible for inventing peace and, as invited in the Charter for a World without Violence, I join Glenn D. Paige in committing myself to the enormous task of participating in the efforts to create a World society in which there is no killing of humans and no threats to kill.

The great challenge, which I face in writing this chapter, is how, as a mathematician and mathematics educators, to act to fulfill this commitment. How to go beyond the humanitarian dream? I believe an academic quest of the nature and history of mathematics may be helpful. This will be the tone of this chapter.

Introduction
As Peace Educator Leah Wells once said, “Violence comes from fear, fear comes from incomprehension, incomprehension comes from ignorance … we eliminate ignorance with education”. To recognize, to respect and not to fear different values is the way to eliminate violence.

Education is a practice present in every culturally identified group. The major aims of education are to convey to new generations the shared knowledge and behavior and supporting values of the group, and, at the same time, to stimulate and favor progress.

Let us consider groups of individuals who share modes and styles of knowledge and behavior, supported by a system of values, which were generated and accumulated throughout a common past. This characterizes a culture. Thus, a culturally identified group, be it a professional guild, a family, a community, a nation, share sets of modes and styles of knowledge and behavior and values, ingrained in traditions, which support knowledge and behavior. Knowledge, behavior and values which come from the past, justify present behavior and, at the same time, entice and make it possible the advancement of knowledge. Inevitably, the supporting values also go through permanent revision. This is the essence of progress. The phenomenon of globalization leads us to consider a much larger group, indeed the total group of the human kind. This leads to envisage a universal culture. The major challenge is to recognize shared knowledge and behavior and supporting values for this total group, that is, for the human kind. This asks for universal and transcultural knowledge, behavior and values. Examples of transcultural and universal knowledge are mathematics and the sciences in general. Modern, euphemistically called civilized, behavior, as expressed in manners, in dressing, in the appropriation of technology, particularly the media, is advancing worldwide as universal behavior. A strong piece of resistance is, as it has historically been, the systems of values. Education has been focusing on knowledge, behavior and values of culturally identified groups and on past struggles for keeping the identity of the group. The violent facet of the struggles has dominated the historical narratives within education. If we accept the initial premises that action in the present reflects the past, it is undeniable that education has been favoring violence. The historical narratives are impregnated with hostilities and atrocities, and emphasize moments of success or failure. Although the moments of temporary success are, sometimes, marked by efforts to build up new styles and modes of knowing, behaving and accepting different values, these efforts have not been deserving attention in history education.

Every human being experiences biological, physical, social, psychological, spiritual needs and also wants. A road to peace is to achieve a balance between needs and wants. Education for peace must consider the realms of inner peace, social peace and environmental peace, paving the way to military peace. These four are intimately related. To achieve peace between human beings, we must understand how man is integrated in nature and we must respect the equilibrium that exists in nature. This means, to be in peace with the environment. Taking advantage of natural resources to accumulate wealth to a few, which is perpetrated at a structural level of the economy, generates social injustices which cause violence.

In this chapter I will discuss mathematics, the earliest and most recognized universal system of knowledge. As it has been said by historian Mary Lefkowitz “the evolution of general mathematical theories from those basics [mathematics of Egyptians, Sumerians and others] is the real basis of Western thought (emphasis added)”. History shows that Mathematical ideas have been expropriated by the Arts, Religions, Sciences and, in modern civilization, by the technological, industrial, military, economic and political complexes. Mathematics and mathematicians benefitted, and continue to draw resources, from these complexes, relying on them for the material bases of its continuing progress. I will also discuss the origins of mathematics and how a universal set of values, essential for peace, is intrinsic to mathematics.

I raise many issues, leaving most of then unanswered. This text is an introduction to a large and ambitious program of looking into mathematics as the real basis of civilizations, hence into the relations of mathematics with the arts, religions, sciences, economics, politics and architecture and urban life, hence with peace.

To achieve peace is essential for the survival of civilization. We are a threatened species. When I refer to peace, I am concerned with peace in its several dimensions: inner peace, social peace, environmental peace and, of course, military peace. Violations of peace, in all these dimensions, permeate the history of the world.

Violations of peace, in all dimensions, are frequently shown in the media and are dramatized in the arts. Recently, the Academy of Motion Picture Arts and Sciences recognized the violation of inner peace in American society by granting an Oscar to the movie American Beauty, which denounced this situation. Research institutions such as The World Watch Institute and many NGOs systematically denounce violations of Social Peace and Environmental Peace.

Violations of Military Peace, that is, the insane practice of war, are a recurrent theme of the artistic, religious and scientific discourses. The impact produced by Picasso’s Guernica synthesizes dramatic visualizations of the horror of wars in literature, music, photography and the plastic arts. Appeals to sanity and to stop war are frequent. The exhibit “Thermonuclear Garden”, installed by Sheila Pinkel in several cities of the USA from 1982-1992, is an example of appeal to the American people to protest against production and export of weapons. Ecumenical meetings all over the world call for forgiveness and tolerance, love and harmony. And scientists lead the call for a stop in the insanity of war. Most pungent is the appeal of Albert Einstein and Bertrand Russell in the Pugwash Manifest, 1955: “We appeal, as human beings, to human beings: remember your humanity, and forget the rest.”

The Pugwash Movement or Pugwash Conferences on Science and World Affairs, which was awarded Nobel Price for Peace in 1995, has the motto “Thinking in a new way.” Indeed, to go beyond wishful thinking and inspiring discourses, some innovative action is need.

I have a utopia: a world in peace! We need utopias in the sense given by Karl Mannheim, which sees utopia as the substratum of will. And will guides our actions. Mannheim says:

Global responsibility
This paper deals basically with the global responsibility of Mathematicians and Mathematics Educators. The guiding question is “How do we fulfill, as Mathematicians and Mathematics Educators, our commitments to mankind?”

To be highly provocative, I invite people to reflect about the embarrassing fact that people who have attained a high level of cultural development, particularly excellence in Mathematics, have performed the most despicable human behavior in recent times. Let me make it very clear that this is not an insinuation of an intrinsic malignity of Mathematics. But it is clear that Mathematics has been an instrumental companion in the historical events that we all deplore. Let me make very clear that I see Mathematics playing an important role in achieving the high humanitarian ideals of a new civilization with equity, justice and dignity for the entire human species, without distinction of race, gender, beliefs and creeds, nationalities and cultures. But this depends on the way we understand how deeply related are Mathematics and human behavior. Mathematicians, Historians of Mathematics and Mathematics Educators rarely consider these questions.

It is undeniable that Mathematics is well integrated into the technological, industrial, military, economic and political systems of the present world. Indeed, Mathematics has been relying on these systems for the material bases of its continuing progress. We may say that Mathematics is intrinsic to today’s culture. So, we are led to examine History of Mathematics as related to World History.

In order to appreciate the real significance and importance of Mathematics in different cultures and in different times, it has to be viewed through what might be termed “cultural lens”. It is hoped that this approach will illuminate many areas of mathematical thought and indicate new directions of research. As a result, we may better understand the implications of mathematical research, its contents and its pedagogical methodologies, for the achievement of peace in its several dimensions: military peace, environmental peace, social peace and inner peace. This is essential for building up a civilization that rejects inequity, arrogance and bigotry, which are the behaviors which initiate and support violence. Paradoxically, the intense rejection of these behaviors sometimes are, themselves, arguments favoring violence.

As a mathematician proposing strict nonviolence, it is for me very difficult to understand why and how the recognized pacifist Albert Einstein sent to President Franklin Delano Roosevelt, in August 2nd 1939, the decisive letter to build an atomic bomb, that killed thousands of Japanese civilians, families, elder and children and deflagrated the Cold War. In his letter, Einstein says

The USA was then neutral. Only after the Japanese attack on Pearl Harbor, in December 7th, 1941, the USA declared war on Japan and Hitler, drawn by his alliance with Japan, declared war on the USA. But the atomic bomb project was well under way.

This is supported by the concept of being prepared for a just war. The argument is that the destruction and killing of civilians is necessary, although regretted. This same argument is as old as civilization, and it is employed nowadays.

Can the argument of just war be supported? In the name of what? The maxim “For the winners and just, medals and paradise, for the losers and wicked, scaffolds and hell” seems to be universally accepted. The concept of bellum iustum is as old as mankind. Laurens Winkel synthesizes it well:

The prevailing attitude
It is not sufficient to say, as it is common in our profession — indeed, in every profession — that we are fulfilling our commitment and responsibility to mankind “By doing good Mathematics” or “By being a good Mathematics teacher” But doing good mathematics should be complemented with the question “What will be done with the Mathematics I am helping to develop?” and a good mathematics teacher must always be asking “How will my students perform? Will they be conscious of their moral commitment in their professional life?” Our responsibilities include the uses society makes of our intellectual production and what is the influence we have in the behavior of future generations.

It is naïve or sarcastic to say, as G. H. Hardy has said, that

Indeed, the theory of numbers is a fascinating subject, even for children in early schooling. But what bothers me is that the most attractive jobs for specialists in the theory of numbers are offered by the Department of Defense. It is one of the most important resources for military purposes.

The possibility of final extinction of civilization in Earth is real. Not only through war. We are now witnessing an environmental crisis, mounting social crises in just about every country and, above all, the recurring threat of another World War. I can not accept that it is normal to solve regional conflicts by military means and that isolated wars can be tolerated. Mainly as retaliation, which produce a chain of retaliatory actions, inevitably chastising innocents, although conveniently used as human shields, thus serving as a very efficient argument for cooptation. Although isolated, the violence and violation of human dignity going on in these conflicts are abhorrent. It is perturbing that discourses of obvious pacifists let it open the way for necessary wars and just wars. Even in Tao Te Ching, #31, we read

History has shown us that regional and limited conflicts eventually lead to larger involvement of nations. The escalation is the door to World War. Even more alarming, because it is a subtle violation of peace, is the lack of inner peace of individuals, leading to drugs, nihilism and violence.

To survive as a species we have to achieve peace, in its several dimensions: Inner Peace, Social Peace, Environmental Peace and Military Peace. This means peace with dignity. In a correspondence to Albert Einstein, Sigmund Freud said

We all, particularly mathematicians, have a responsibility in finding these ways.

As it was said above, and I repeat, Mathematics is well integrated into the technological, industrial, military, economic and political systems and mathematicians have been relying on these systems for the advancement of their professional career and for material reward.

Rare, but exemplary, is the attitude of Derek Smith, who, in 1992, was working in speech recognition for Texas Instruments. When he learned that the results of his work was playing a role in the control systems of an antiradar missile developed by the Pentagon, he decided to quit his job and joined, thanks to his expertise, a research group to model the immune system recognition of influenza viruses. (Science, April 18, 2008, pp. 310–311).

Cooperative subservience is not restricted to specialists in Science and Technology. They are found in Economics, in Communication, even in Philosophy—indeed in all fields of academic specialties and professions. It is extremely difficult to avoid. The cooptation strategies are subtle and, sometimes, intimidate. Ideological and even academic zealots play a fundamental role on this.

If, as Mathematicians and Mathematics Educators, we try to answer the challenge of Sigmund Freud to Albert Einstein, it is natural for us to reflect about our personal role in putting an end to and avoiding future wars. According to Freud

The issues are essentially political. There has been reluctance among mathematicians, to a certain extent among scientists in general, to recognize the symbiotic development of mathematical ideas and models of society. Mathematics has grown parallel to the elaboration of what we call Modern Civilization. Historians amply recognize this. Particularly explicit on this is Mary Lefkowitz, as quoted in Note 1 above, in recognizing that mathematics is universal.

We can not disregard the fact that the most universal problem—that is, survival with dignity—must have much to do with the most universal mode of thought—that is, mathematics. I believe that to find the relation between these two universals is an inescapable companion to the claim of the universality of mathematics.

Our commitment implies to assume a broad view of the world and of mankind in general. This is possible through a reflection about the future and a broad perception of the state of the world, which is disturbing. It is a general feeling that human behavior has not been ethical. Particularly mathematicians and mathematics educators have not been explicit about a comprehensive ethics guiding their practices. An ethics of responsibility is needed. But, given the universality of mathematics and of its effects, this ethics must go beyond professional codes of behavior and professional ethics, such as the Hippocratic Oath.

It is natural to express the discontentment with the state of the world by chastising Science and Technology, which are recognized as the embodiment of modern society. Science and Technology are thus blamed for the malaise of humanity. Mathematics is, obviously, directly affected by this criticism.

The challenges and counter-challenges we are witnessing reflect a defensive posture that is growing to contain the wave of discontent. For generations and generations, access to facts have been controlled by moral and material instruments, among them norms and codes, language and literacy, all organized in systems such as religions, sciences and technology. Reminding the ideological zealots of Senator Joseph McCarthy era, academic mobbing is a powerful control instrument. Paradoxically, the same instruments, which were fragmentarily constructed to preserve the prevailing order, have become so complex that they are no longer effective and became increasingly permeable. An old Spanish refrain says “Cría cuervos y ellos te comerán los ojos” [“Raise crows and they will peck your eyes out”]. The creature escapes the control of the creator. The fall in disgrace of Senator McCarthy, as well as metaphors like Adam, Frankenstein, Hal of 2001, the androids of Blade Runner, all point into this direction. Our hope is that a new thinking in Science, mainly in Mathematics, will be able to go through the control mechanisms.

The reaction to the challenge
To raise these questions is sometimes interpreted as opening doors to anti-science and irrationality. In his recent book, Carl Sagan cautions about the lure of new directions in inquiry. In his denouncement of the “new Dark Age of irrationality”, Sagan says:

It is misleading to denounce discontentment as such. Indeed, these conflicting postures have led to the so-called “Science War”. Research done by Sociologists of Science have been more focused on the relations of Science and Society. But the new field of Social Studies of Science has been chastised. Alan Sokal draw much attention to the theme in a hoax published in one of the cherished journal of the postmodern critics.

The polemic thus started is not different from those focusing afro-centrism and feminism. These polemics reveal that the discussion of scientific knowledge by postmodern critics reveal the real issue of the subordination of Science, which is a political one, and that goes much beyond the national arenas. Ideological labels are commonly subtly used to justify fundamentalism in the defense of the prevailing academic order. This is very well illustrated by the fact that Sokal’s hoax was used, a few weeks after its publication, by Brazilian Congressman Roberto Campos to support his political rightist harangue. A few days later, Alan Sokal published a reply to Congressman Campos, in the same influential Brazilian newspaper, explicitly chastising Campos as a rightist and declaring himself as a leftist. Another example is the fact that the television debate between candidates Clinton and Dole, on October 6, 1996, revealed an insistence of Senator Dole on using the word “liberal” as a form of attacking the policies of President Clinton. There is a danger that these polemics result in the deviation of the main objective, which is to “condemn injustices and inequities of the capitalist system and try to eliminate or, at least, minimize them”, using the same words of Alan Sokal, which contradicts his posture in deflagrating a total Science War. To challenge scientific, religious, socio-political and historical knowledge does not mean retrogress. It has always been a coherent response to the state of society and it can be understood if we look into the full cycle of knowledge in a historical perspective, of course freeing ourselves of the epistemological biases that are adopted to justify the prevailing socio-political and economical order. The essence of these biases is the argument that Science is an object of knowledge of a different nature, in the realm of the ratioïd. This is particularly strong when we refer to Mathematics. Metaphorically, Mathematics is manichaestic. Its foundations rely on very strict dichotomies.

Knowledge is generated by individuals and by groups, is intellectually and socially organized, and is diffused. The full cycle of the generation, organization and diffusion of knowledge intertwines with needs, myths, metaphors, and interests. The human species develops, like other animal species, strategies of hierarchical power. Intrinsic to hierarchical power is the control of knowledge.

In the discussion about the current state of the World, it is not so important to claim that although the Egyptian, the Sumerian and other civilizations were ahead of the Greek, the contribution to build up general mathematical theories was indisputably Greek. It is irrelevant, though largely accepted, that the medieval scholars received Euclid through the Arabs. What is very relevant is the fact that Mathematics as it is recognized today in the academia, developed parallel to Western thought (philosophical, religious, political, economical, artistic and, indeed, every sector of culture). It would lead to a redundant boredom to give examples justifying this assertion. Indeed, Mathematics and Western Civilization belong to each other.

When we question the current social, economical and political order, we are essentially questioning the righteousness of Western Civilization in face of a real threat to its continuation, how is it possible to avoid questioning its pillars, Science and Mathematics? How can discussions about these pillars be closed to non-scientists and non-mathematicians? The resource to arguments of authoritative competence leads to intimidation and to passionate arguments, as discussed above about the ideological zealots. How can we reach the new by refusing, discouraging, rejecting, denying the new? Indeed, a subtle instrument of denial is discouragement through intimidation. Language plays an important role is this process, as every schoolteacher knows. Particularly in Mathematics, the use of a formal language, inherent to academic Mathematics, has been a major instrument in deterring critics.

The organization of this language is the realm of epistemology. Epistemologies and histories, the same as norms, differ from group to group, from society to society, and are incorporated in what is called culture. The crux is the dynamical process of encounters of cultures and the resulting mutual expositions, which underlie the construction and reconstruction of knowledge and the maintenance, substitution, dissolution and modification of epistemologies and norms. When authority dominates this process, as it was in the colonial process and equally characterizes conservative schools, the outcome is predictable: contest. The problem thus resides with authority and the denial of participation in the dynamics of this process.

Social and political scientist Marcus G. Raskin and physicist Herbert J. Bernstein, in their analysis of the linkage between the generation of knowledge and political directions, claim that

The nature of Mathematics
The criticism inherent in reestablishing the lost connection of the sciences, technology and human values is causing unavoidable conflicts. This is particularly true with Mathematics, in which the acknowledgement of human attributes is conspicuously absent in its discourse.

This has not been so in the course of history. Mathematics, the same as the other sciences, used to be impregnated with religious, as well as social and political considerations. Current Epistemology and History, and above all the educational framework, were constructed to justify the prevailing socio-political and economical order, in which we recognize different “theories of science”.

The theories of science largely fail to recognize that the generation of knowledge is the result of a complexity of sensorial, intuitive, emotional and rational factors. We are “informed” by these factors and process the information in a way as yet unknown. We need more understanding on how the human mind functions. A holistic approach to knowledge, going from reality to action, owes much to artificial intelligence, biology and sociobiology.

Let us now turn to the question of political power. There are indicators that students spend less time studying or doing homework and that they are bored in class. There is no point in putting the blame on youth, claiming that the current generation is uninterested in learning and intellectually “lost”. Maybe we should look into the blamers. The problem does not reside in youth, but in the older generation, in family, in schools, in the institutions in general. Chiefs of staff are ready to justify sending troops, of young age, even teenagers to the battlefield. I know of no decision taken by a young chief of staff to engage in a war and sending the older generation to the battlefield!

As Fred M. Hechinger (1992: 206) puts it,

The real problems facing education are political, essentially the result of an unequal distribution of material and cultural goods, intrinsic to modern economy. There is no need to elaborate on these issues. I suggest a few sources where we find discussion of property, production and global issues in modern society.

Some readers will claim that this has not much to do with the relations among Violence, Mathematics and Mathematics Education. I claim they have everything to do. This relation has been avoided in the discussions about the state of the world and Mathematics and Mathematics Education have been absent of the critical views on the main issues. The cultural consumerism, practiced both in schools and in the academia, has been efficient in trimming processes and focusing only in results. Mathematics and History of Mathematics are delivered as frozen systems of knowledge, conforming to the status quo. A frequent argument, when one calls for a broader view, is “this belongs to another discipline, not to mathematics classes”.

Exceptions are notable. We have to mention the activities of the research group on “Political Dimensions of Mathematics Education/PDME” and also the movements “critical mathematics” and ethnomathematics.

There have been few writings about values attached to Mathematics and even less about the moral quality of our action. To search for a correlation between the current state of civilization and mathematics has been uncommon among mathematics educators. Particularly the political component, which was so well studied by Paulo Freire, Michael Apple, Henry Giroux and others with respect to education in general, seem to have drawn little attention of Mathematics Educators.

To a great extent, the polemics around the postmodern discourse of sociologists of science is a reflection of the ideology intrinsic to words. Indeed, language has been the main instrument in denying free inquiry. There is an implicit intimidating instrument in the language of academia and society in general. One must be reminded that of the major confrontations of the sixties, particularly the Civil Rights Movement, the demonstrations against the Vietnam War and the student movements of 1968, probably the first of such contestations of the established order was the Free Speech Movement, initiated by Lenny Bruce.

Human mind is a complex of emotional, intuitive, sensorial, rational perceptions, involving all at the same time. Maybe we have been emphasizing too much the rational perception and denying, rejecting and repressing the others. Indeed, there is a general feeling that, as math teacher, one has to teach “serious math”, that is, objective reason, and to stimulate rational thinking among the students. It is not uncommon to see a child punished for being “too happy” in the classroom. And we always know of teachers saying to a boy “Stop crying. Men do not cry!” Is it possible to build knowledge dissociating the rational from the sensorial, the intuitive and the emotional?

It comes to my mind the case of a school teacher who asked children to draw a color picture of a tree seen through the window of a classroom. Jane came with a tree painted red. The teacher corrected the child, even suggested to the parents that Jane might have a vision problem! A few days later the teacher was sitting in the same place as Jane was, at the same time of the day, and the Sun was in the same position. The teacher saw the tree red. Many say that this example is misleading, since it does not deal with objective reason.

I see the multi-dimensionality in building up knowledge as a very important aspect of the History of Mathematics, which has been practically ignored. And, of course, this is very important in learning.

There has been a resurgence of interest in the intuitive, sensorial (hands-on projects) and affective aspects in Mathematics Education. We must go beyond education and question the discipline itself. What is the role of emotions in Mathematics? When Gustave Flaubert wrote “Mathematics: the one who dries up the heart”, what did he have in his mind?

The usual reaction to these comments is: “But this is natural, since Mathematics is the quintessence of rationalism.” Indeed. But much of the polemics going on relate to the prevailing acceptance of the superiority of rationality over other manifestations of human behavior. This was one of the main concerns of the mathematician-writer Robert Musil in his masterpiece The Man Without Qualities. Commenting on scientists and engineers, the main character Ulrich says,

Musil’s oeuvre anticipates the intellectual framework of Nazi Germany, in which he identifies the incapacity to tolerate pluralism. Indeed, much of the reactions against irrationalism are mixed with a latent emotional incapability of accepting the different. The denial of access to knowledge is a strategy for the exclusion of the different.

The threat of extinction is a fact. Paraphrasing Martin Luther King Jr. in his 1963 speech, the change to nonviolence instead of violence is, indeed, a decision between nonexistence and nonviolence. Do we prefer nonexistence to eradicating violence?

As human beings, we can not relinquish our duty to cooperate with each other, with respect and solidarity, for the preservation of the natural and cultural patrimony. This is the essence of an ethical behavior of respect for the other, who is different in many natural and cultural aspects; solidarity with the other; cooperation with the other. This is a sure road to quality of life and dignity for the entire mankind.

Our main goal is nonkilling. Otherwise, we are in the road of extinction. I am simple in my proposal—we need ethics; and didactic in my style—every individual, whether the sophisticated intellectual or the common man, has a responsibility and should find the means to direct his energies to socially constructive goals.

This is an unusual piece on Mathematics and Mathematics Education, many will say. But if we accept, very clearly and unequivocally, that our professional commitments are subordinated to a more vital commitment to nonviolence, it is absolutely necessary to understand how and why mathematics became such a central instrument, both intellectual and material, in human knowledge and behavior.

The essence of being human: survival and transcendence
Peace, in all its dimensions, depends on an ethical posture not only on human behavior, but also in the production of knowledge. Current systems of knowledge give to the prevailing social, economical and political order a character of normality. Both the religions and the sciences have advanced in a process of dismantling, reassembling and creating systems of knowledge with the undeniable purpose of giving a sense of normality to prevailing human individual and social behavior.

The fundamental problem in this capability is the relation between brain and mind. It is possible to know much about the human body, its anatomy and physiology, to know much about neurons and yet know nothing why we like or dislike, love or hate. This gives rise to the modern theories of consciousness, which claims to be the last frontier of scientific research.

Through a sophisticated communication system and other organic specificities, man tries to probe beyond the span of one’s existence, before birth and after death. Here we find the origins of myths, traditions, religions, cults, arts and sciences. Essentially, this is a search for explanations, for understanding, which go together with the search for predictions. One explains in order to anticipate. Thus builds up systems of explanations (beliefs) and of behavior (norms, precepts). These are the common grounds of religions and sciences, until nowadays.

The drive towards survival is intrinsic to life. But the incursion into the mysteries beyond birth and death, which are equivalent to the search for past and future, seem to be typical of the human species. This is transcendence. The symbiotic drives towards survival and transcendence constitute the essence of being human.

The analysis of this symbiotic drive is focused in three elements, the individual, the other(s), organized as a society, and nature, plus the three relations between them. Metaphorically, complex life may be represented by a triangle, emphasizing that the six elements are in mutual solidarity. The image of a triangle to relate basic components of the model is very convenient. I owe the idea for this triangle (the primordial triangle) as well as for the other two (the enhanced triangle and the humanness triangle) to a paper of Antti Eskola (1989). A mathematical triangle ceases to be by the removal of any of the six elements. The same occurs with the life of an individual. It terminates with the removal of any of the six elements. Life ceases by the suppression of any of the three vertices or the interruption of the relation between them. The following image of the primordial triangle is very convenient.

INDIVIDUAL -- NATURE \                     /                                             \                    /                                              \                  /                                               \                /                                                \              /                                                 \            /                                                  \          /                                                   \        /                                                    \      /                                                     \    /                                                      \  /                                                       \/                                                      OTHER(s) (SOCIETY)

In species with developed neocortex, which we might call superior living species, the pulsion of survival, of the individual and of the species, and gregariousness, are genetically programmed. Reflexes, part of this programming, are usually identified as instinct.

The relations (sides) generate individual and social behavior. The triangle metaphor, meaning the indissolubility of the six elements, is resolved by the principles of physiology and ecology. Basically, the relation between individual and nature is responsible for nurturing, the relation of the individual and the other of opposite sex for mating and continuity of the species. Gregariousness is responsible for individuals organizing themselves in groups and herds, and hierarchy develops, most probably as an evolutionary strategy. The group, thus organized as society, relates to nature aiming at general equilibrium, following basic principles of ecology. Thus, the primordial triangle keeps its integrity. The rupture of each of the six elements eventually causes the extinction of a species. Individual and social behaviors are actions taken “here” and “now”.

Individuals of the human species, differently than other species with neocortexes, are provided with will, that subordinates instinct. Every individual has the ability to generalize and to decide actions that go beyond survival, thus transcending survival. Individuals acquire the sense of before/now/after and here/there. Individual and social behavior transcend here and now. Thanks to will, individuals develop preferences in nurture and in mating. They protect themselves and their kin and they plan ahead and provide. Physiological and ecological principles are not enough. Humans have to go beyond them and the relations (sides) and increment the primordial triangle by creating intermediacies. Between individual and nature, humans create instruments ; language intermediates individual and the others; the relation between groups/society and nature is intermediated by production. In the process of recognizing the potential of these intermediacies, humans acquire an enlarged perception of nature. It becomes what is generally understood as reality, comprising natural, cultural and social environments. The primordial triangle becomes an enhanced triangle:

instruments INDIVIDUAL -- REALITY \                     /                                             \                    /                                              \                  /                                               \                /                                                \              /                                        language \            / production \         /                                                   \        /                                                    \      /                                                     \    /                                                      \  /                                                       \/                                                      OTHER(s) (SOCIETY)

The three intermediacies are, clearly, related. Instruments, both material and intellectual, are shared through language and decisive in the production system. The distinguishing feature of language is that it goes beyond mere communication and is responsible for the formation of new concepts. Language becomes essential in forming thought and determining personality features. It is the roots of emotions, preferences and wants, which determines the enhanced relations of the individual and the other(s). Language is also essential in the definition and distribution of tasks, necessary for organizing systems of production. Thus, the intermediacies have also a form of solidarity which synthesizes what is called culture. Culture may be thus metaphorically expressed as a triangle, which I call the humanness triangle:

instruments INDIVIDUAL .. REALITY \        .  .         /                                             \       .    .       /                                              \     .      .     /                                               \   .        .   /                                                \ .          . /                                        language .............. production \         /                                                   \        /                                                    \      /                                                     \    /                                                      \  /                                                       \/                                                      OTHER(s) (SOCIETY)

Human life is, thus, synthesized as the pursuit of the satisfaction of the pulsions of survival and transcendence. It is a mistake to claim, as many mathematicians do, that this refers to other forms of knowledge and that Mathematics has little to do with these pursues. A holistic view of History of Mathematics traces the origins of mathematics in pursuing the satisfaction of these two pulsions.

Engaging in survival, humans develop the means to work with the most immediate environment, which supplies air, water, food, necessary for nurturing, and with the other of opposite sex, necessary for procreation. These strategies, common to all superior living species, are absolutely necessary for the survival of the individuals and of the species. They generate modes of behavior and individual and collective knowledge, including communication, which is a complex of actions, utilizing bodily resources, aiming at influencing the action of others. In the species homo, behavior and knowledge include instruments, production and a sophisticated form of communication, which uses, as its means, language, as well as codes and symbols.

In the search of transcendence, the species homo develops the perception of past, present and future and their linkage, the explanations of and the creation of myths and mysteries to explain facts and phenomena encountered in their natural and imaginary environment. These are mentifacts incorporated to the individual memory and retrievable only by the individual who generated them. Material representations of the real, which we generally call artifacts, are organized as language, arts and techniques. Artifacts are observable and interpreted by others. In this process, codes and symbols are created. Shared mentifacts, through artifacts, have been called sociofacts by biologist Julian Sorell Huxley (1887-1975), who also introduced the words artifacts and mentifacts. The concept of culture for Huxley contemplates artifacts+mentifacts+sociofacts.

Explanations of the origins and the creation of myths and mysteries lead to the will to know the future [divinatory arts]. Examples of these arts are astrology, the oracles, logic, the I Ching, numerology and the sciences, in general, through which we may know what will happen—before it happens! The strategy of divinatory arts is deterministic.

Divinatory arts are based on mathematical concepts and ideas: observing, comparing, classifying, ordering, measuring, quantifying, inferring. Indeed these concepts and ideas are present in all the steps of the search for survival and transcendence.

As every form of knowledge, mathematical artifacts, in the form of practices and tools, and mentifacts, in the forms of aims or objectives, concepts and ideas, are first generated by individuals trying to cope and to deal with the natural and social environment, to resolve situations and problems, and to explain and understand facts and phenomena. These ad hoc artifacts and mentifacts are individually organized and are transmitted to other(s) and shared. They attain objectives, they serve, they are useful, they become methods, which are shared and acquired by the other(s), by society. They are part of the sociofacts of the group. How are they transmitted and shared? These are the basic questions when we ask for the origins of mathematics. Was the transmission and sharing through observation, mimicry? Eventually, using language. But when? This is historically unknown. We have indications of the emergence of mathematical ideas thanks to artifacts, as it will be discussed later in this paper.

We have no idea when language was used in this socialization. Indeed, the origin of language was an academic “forbidden” theme about one hundred years ago. When language occurred, most probably systems of codes and symbols and specific words were created to design mathematical objects and ideas. This is major research subject for oral cultures. With the appearance of graphic registry, like cave drawings and bone carving, we have more elements to understand the development of mathematical concepts and ideas. The progress of mathematics through history, in different cultural environments, is a central issue to understand the nature of mathematics. In a recent book, Ladislav Kvasz (2008) discusses the historicity of linguistic tools as a major factor in the development of mathematics. We may infer that, socially, this factor, which isolates mathematics from consideration of those that are outside the restricted circle of professional mathematicians, is a form of censorship. This kind of obstacle to critical views on the advances of mathematics, of its purpose and appropriation for interest, sometimes, non-acceptable, was already discussed above. Research that can not be disclosed, is euphemistically identified, in academic circles, as “classified” research, not as “confidential” research. This was clearly illustrated in the movie A Brilliant Mind (2001), directed by Ron Howard, a fiction based in the real life of John Nash.

To share mathematics advances with the general population requires demystifying mathematics language. In an emblematic phrase, David Hilbert (1862-1943), probably the most eminent mathematician of the 20th century, said, in the major conference of the 2nd International Congress of Mathematicians:

Demystifying mathematical language may open the way to a new form of mathematical education, with more space for critical analyses of mathematical development.

The threat of extinction
The only possibility of escaping the threat of extinction of civilization is to attain peace in its broadest sense, in all its dimensions, that is inner peace, social peace, environmental peace and military peace. I see peace not as the inexistence of conflicts, since, as discussed in the beginning of this paper, every human being experiences different biological, physical, social, psychological, spiritual needs and also different wants. Since the individual and the other are different, conflicts are to be expected. The crucial point is to resolve the conflicts without violence. Violence ranges from evident confrontation and aggression and the resource of prepotency, but also in more subtle forms of arrogance and bigotry, intolerance and fanaticism.

The only road to peace is through conflict resolution, based on a global understanding of the life phenomenon and intermediacies created by man, which implies the acknowledgement of differences in the inter- and intra-cultural dialogue.

A primordial ethics recognizes the mutual essentiality of the three vertices and three sides of the primordial triangle and aims at the preservation of its integrity and survival with dignity. This primordial ethics is synthesized in the box:

Mathematics in General Education
I repeat what I said above. Many will say that this is an unusual piece on Mathematics and Mathematics Education. Without denying the fundamental importance of nonviolence, they claim that the role of a mathematician and of mathematics educators is to act, seriously and with competence to attain the specific objectives of the discipline.

But this competence, without a firm ethical commitment, may be directed to reproachable consequences. Particularly, to military innovation. An unsustainable argument of the neutrality of analytical treatment is a resource to support reproachable actions. The seduction of mathematics is responsible for “promoted tricks in technique and the assimilation of dogma at the expense of considered thought” (Hodgson; Screpanti, in Keir, 2006:22).

This is coherent with what some philosophers of science claim. There is, indeed, a seduction in mathematics. Based on remarks of Thomas Reissinger, Sanford L. Segal says:

This attitude does not differ from what other professionals say of their responsibility vis-à-vis their discipline. But if we do accept, very clearly and unequivocally, that our commitment to mankind is much more important that our commitment to the discipline and to their objectives, we can not, passively, relinquish our action and give this responsibility to other educational constituencies.

Our professional commitments must be subordinated to a global ethics, such as the primordial ethics, proposed above. Otherwise, it will be impossible to engage in a deeper reflection about our role as mathematicians and mathematics educators.

It is an undeniable right of every human being to share all the cultural and natural goods needed to material survival and to intellectual enhancement. This is the essence of the Universal Declaration of Human Rights (1948), to which every nation is committed. The educational strand of this important profession of faith in the future of mankind is the World Declaration on Education for All (1990; see Haggis, Fordham and Windham, eds. 1992), to which 155 countries are committed. Of course, there are many difficulties in implementing the resolutions contained in the document. But as yet this is the best instrument available that may lead to a planetary civilization, with peace and dignity for the entire mankind.

The crux is to understand how Mathematics and Mathematics Education can be directed as a response to these principles.

I see my role as an Educator and as a teacher of my specific discipline, Mathematics, as complementary instruments to move toward my utopia of a world in peace.

In order to make good use of these instruments, I must master them, but I also need to have a critical view of their potentialities and of the risk involved in misusing them. Of course, this has everything to do with ethics.

I believe most mathematicians and mathematics educators share these views. No doubt they are authentically concerned with nonviolence, quality of life and dignity for mankind. But sometimes the relations between concerns and professional practice are not clear. Particularly in Mathematics, there is a general acceptance that if we do Mathematics well, thus instilling attitudes of rigor, precision and correctness in the students behavior, we are fulfilling our broad responsibilities. Undeniably true. But this is not enough. This must be subordinated to a much broader attitude towards life and towards how mathematics can be used for good or for bad.

The first issue is to understand how Mathematics, as a knowledge system, emerges as a result of the search of survival and transcendence.

My proposal to achieve this understanding is to discuss the elements of the primordial and the enhanced triangles. Then to proceed into the knowledge and behavior acquired in the search of survival and transcendence. Mathematics, as manifest in the techniques of observing, comparing, classifying, ordering, measuring, quantifying, inferring, is inherent to these searches.

The curriculum I propose below is organized in two steps. The two steps must be integrally covered, but the level of exposition and the required complementary reading is absolutely flexible. I have been developing this curriculum in courses for both future mathematicians and teachers. It is very frequent to have, among my students, individuals coming from other specialties. It is the teacher responsibility to adapt the exposition to the level of the students. It has been possible to develop the curriculum in elementary classes.

The Proposed Curriculum
I have been using an image of the evolution of the species which is very convenient, since it allows for talking about the emergence of the basic ideas of mathematics, particularly observing, comparing, classifying, ordering, measuring, quantifying, inferring. There is much to be explored in this image. Particularly, the autonomy of the individual, which is symbolic represented by its erect posture.

It is very important to pay attention to the various phases of human evolution. Bipedism, the first differential from apes, allowed the new species to move using two feet and to discover other things to do with the idle hands (equilibrium is the mathematical manifestation in such step); among these discoveries: stone tools, for which the mathematical concept of comparison of dimension, rendering the tool appropriate for the designed use, became necessary; the invention of the spear, later developed into arrows and bows, which required the identification of a target in a distant complexity and the development of the mathematical concepts of distance, direction and force (nowadays characterized as a vector, with has magnitude and direction). In this phase, there is a good motivation for philosophical reflections about the autonomy of the individual, well exemplified by the possession of a sword in medieval times, and about the generation of a sense of accuracy through mental discipline, as seen in archery. Next phase, leading to history and modern human behavior, is the invention of agriculture, and the necessary consequence of coordinated labor, hence hierarchy and power of a different nature (not deriving from physical strength), and of property. It is appropriate, in this phase, to discuss the roots of the capitalistic system. Next phase is the development of industry, due to the invention of non-animal power. A reflection about the mathematics involved in this invention is very appropriate. Again, it is the appropriate moment for socio-political reflections on the condition of the new character of being a worker and the emergence of modern capitalism. The next phase, humans-with-media, represents the dominating presence of informatics in all sectors of the modern world.

The figure reflects a very relevant fact: the ascent of man to individual autonomy, through bipedism, stone tools and culminating with the spear and derivates, arrow, bow and sword. The symbolic status of possessing a sword in medieval times is most relevant for reflections about autonomy. In a sense, with the emergence of agriculture, individual autonomy was lost. The attachment to the small group of family and tribe was subordinated to an increasingly complex social structure. Agriculture brought the end of nomadism, the concept of property and collective labor and the development of astronomy, a very important moment in the development of mathematics. Industry paved the way to modern capitalism. The age of informatics requires new concepts of privacy. Every one of these phases marked the emergence of new directions for mathematics. Each of these steps demands a deeper discussion of the primordial ethics, which is the most important pedagogical practice leading to nonviolence and peace.

Final remarks
In this curriculum proposal, the right moment for a discussion about the search of survival and the search of transcendence is the move from Step 1 to Step 2. This discussion shall emphasize the nature of mathematics as an instrument to deal with the human pulsions of survival and transcendence. This is the moment to elaborate on examples of the relation between Mathematics and religion, Mathematics and tool making, Mathematics and art.

It is of fundamental importance to stress the fact that breaking the primordial triangle implies nonexistence. The enhanced triangle does not change this. The only reason for the enhanced triangle is to make it possible to keep the integrity of the primordial triangle.

Again, this is a discussion of how essential is the behavior according to the primordial ethics for avoiding total destruction of civilization. Paraphrasing Martin Luther King Jr. it is either adherence to the primordial ethics or nonexistence.

How about a nonkilling mathematics? This is an ill-posed question. Mathematics is in the realm of ideas and, as such, is abstract. For reasons not explained in human nature, its results, methods and language may be appropriated, but do not master, as it was made very explicit by eminent physicist Eugene Wigner in a classic paper:

Regrettably, Mathematics is practiced, both in its pure and applied forms, and also presented as a cold and austere sequence of formal steps. In a figurative, somewhat imprecise way, we might say that it emphasizes syntax over semantics. I believe this is responsible for the easy cooptation of mathematicians, as well as of other educated individuals, to put mathematical results, methods and language at the service of material and ideological wants and needs. We might identify this facility to coopt mathematics, a cold and austere sequence of formal steps, as prone to be a killing mathematics. On the contrary, a practice and presentation of mathematics, critically and historically grounded, as proposed in my model of curriculum above, emphasizing semantics over syntax, may resist cooptation and be prone to be used for humanitarian and dignifying purposes. This might be a nonkilling mathematics.

title: Ethics/Nonkilling/Mathematics


text:
User:Kylnso/documentary-dfvp212

kJLpuO

DUE DATES
WEEK 6 (FEB 16) - 1st draft of script (individual & group) WEEK 7 (FEB 23) - Treatment (individual & group) WEEK 8 (MAR 2)- Production Book: prod sched, crew list, equipment list, final script - VTR LOGS (individual & group) WEEK 9 (MAR 9)- EDL'S (individual & group) WEEK 11 (MAR 23)- individual project & group project

CREW ASSIGNMENTS
- Research History - Create 5-10 questions for segments 2, 3, & 4 - Contact Managers / clubs - Shoot neighborhood b-roll - Contact 2nd club - Shoot interview with neighbor store - Find music - Shoot Lusty Lady Interviews - Research Positive aspect - Create 5-10 questions for segments 2, 3, & 4 - Record VTR log - Create interview schedule with neighborhood stores/managers - Shoot interview with neighbor store - Contact manager and set up interview dates - 1st draft of script - Shoot Lusty Lady Interview - Research Psychological aspect - Create 5-10 questions for segments 2, 3, & 4 - Capture footage - Search for stock footage - Rough cut 1 - Rough cut 2 - Final cut - Research Negative aspect - Create 5-10 questions for segments 2, 3, & 4 - Contact any "dancers" - Shoot neighborhood b-roll - Create interview schedule with ex dancer girls - 2nd draft of script - Transcribe
 * DREW (DIRECTOR)
 * KAYLEN (WRITER)
 * NATHAN (EDITOR)
 * KYLE (WRITER)

RESEARCH/RESOURCES PACKET
http://www.foxnews.com/video-search/m/20251986/strippers_suit.htm http://www.miaminewtimes.com/2008-07-24/news/day-strippers/ http://en.wikipedia.org/wiki/Striptease http://www.historylink.org/index.cfm?DisplayPage=pf_output.cfm&file_id=8150 http://en.wikipedia.org/wiki/Exotic_dance http://www.43things.com/things/view/31045/be-a-stripper http://www.geocities.com/alysabethc/why.html http://www.albany.edu/scj/jcjpc/vol10is1/bernard.pdf http://en.wikipedia.org/wiki/Lusty_Lady

EQUIPMENT LIST
- Panasonic DVX 100 / Tripod - Sony HVR-Z1U / Tripod - HVR Battery - CGP Battery - Power Strip - Extension Cord - Lowel VIP Kit - Gloves - Reflectors - Gels - A/C Power - Fishpole - Sennheiser Shotgun Mic - XLR Cable - Tote Bag

VTR LOG
http://imaginarydigital.com/downloads/forms/VTR-Log-Sheet.doc http://www.nku.edu/~strobelc/archive/vtrlog.doc

SCRIPT
...

TREATMENT
This film will be a ten minute documentary on exotic dancers with a realistic and in-depth look at the profession, including its roots in Seattle. This film would be exploring various variables of the girls and their profession such as, different ages, different experiences, those who just started and those who have been in it for more than 20 years. We would look into the job itself, how the girls get into it, how their hours occur, and much more. Also we would go deeper and reveal if they choose to hide the job as a secret or if they allow their family and friends to know about it.

Filming will be structured through a narrated script with interviews of exotic dancers, their manager(s), and neighboring businesses.

OUTLINE

 * INTRO

- Dancing / stripping itself = Where / When did it start? - Clubs in Seattle (lust ladies) = When did the first one show up? = How quickly they spread? - Evolution over time = Change with business = Change with girls
 * SEGMENT 1:HISTORY

- Identity / daily activities = work hours - Job process = pay / tips = Training or initiation = Applications / hiring process - Why the job? = options - How they feel about the job?
 * SEGMENT 2: JOB / GIRLS

- Public life = Separation between work and normal life - Do others know? = Parents (mom/dad) = Married / boyfriend or girlfriend = kids? - truth or lies
 * SEGMENT 3: FAMILY / FRIENDS

- Effect on business = customer complaints = loitering - Positives or negatives or neutral = more or less people - Conflicts?
 * SEGMENT 4: NEIGHBORING BUSINESSES / PUBLIC

 * TAG / CONCLUSION

Manager
MANAGER – (HISTORY) - How long its been in business - Problems in the past
 * State your name, job title, location, and how long you’ve been there
 * Tell us about the history of the Lusty Lady

MANAGER - How many rooms - Types of rooms - Stages - Ages, looks, height, personality - Any trials before being hired - Interviews - Any age limit, or circumstances? - Find another job? Family reasons? Other reasons?
 * State your name, job title
 * Describe The Lusty lady’s building / layout (what to expect there)
 * How many girls are employed there?
 * Are there any certain qualifications to work there?
 * Is there any type of hiring process?
 * Do any of them receive some kind of training or classes?
 * Who manages or assigns the days or times the girls work?
 * How are they paid?
 * Only hourly, only by tips, or by hourly and tips?
 * Do girls have to pay for their stage time or are they scheduled?
 * How long do you keep your dancers/girls?
 * If you have girls that leave this job, do you know any reasons why?
 * How is the competition like with other businesses in the adult industry?
 * How has the economy affected the business?
 * How long do you think The Lusty Lady will last here in Seattle?

Self / Job

 * State your name, age, and job title and describe it in your own words
 * Do you enjoy your job?
 * How long have you been working?
 * Why did you choose this job?
 * How were you introduced into this type of work?
 * How did you feel before your first day?
 * Do you remmber your 1st day? (describe what you remember)
 * What are your work hours now or days now?
 * Does it affect your personal life?

- Why? Why not? - (if yes) How do they feel about it? - (if no) Why don’t they know? - (if yes) Do they know about your job? - (yes) How do they feel about it? (no)Why don’t they know?
 * Do you have to hide or keep your job a secret?
 * Do your parents know about your job?
 * Are you in a relationship? (boyfriend/girlfriend/husband)
 * Do any of your friends know?

 * What do you like about your job?
 * What do you not like about your job?
 * How long do you plan on staying?
 * Do you have any strange or funny customer stories that you can share?

Family / Friends / Kids
Yes - Are they ok with it? - Are you comfortable that they know? No - Why don't they know? - Do you want them to know? - Do they know? - how do they feel about it? - Do they know about the job? - Do you lie about the job?
 * Do (they) know about your job?
 * Are you involved? (Married, BF, GF)
 * Any conflict with the people who know about your job?
 * Does anyone visit you at your job?
 * Do you have kids?

Neighboring Stores

 * your name and your business?
 * what are your business hours?
 * how long have you had your business? (how long have you worked here?)
 * How do you feel about a strip club next to your store?
 * Does it affect your business negatively?
 * Does it affect your business positively?
 * Is it just neutral to your store?
 * Do you ever hear customers talk about the strip club next to your store?
 * Do you ever have customers that complain about it?
 * How is the area/street at night compared to the day?
 * Do the clubs create more people/solicitation in the streets?
 * Have you been in there?

PRE-PRODUCTION
- Concept Meeting - Research - Create Outline - Review outline - Contact "dancers" / clubs - Research - Review outline - Contact "dancers" / clubs - Research - Collect and combine research via E-mail - Create interview questions - Finalize Outline - Start script (intro, body, conclusion) - Create treatment - contact neighboring stores - Review script - Review treatment - Review interview questions - Pre interview with "dancers" - Pre interview with neighboring businesses - Review script - Review treatment - Review interview questions - Pre interview with "dancers" - Setup interview dates
 * JAN 12
 * JAN 19
 * JAN 26
 * JAN 28 (wed)
 * FEB 2
 * FEB 4 (wed)
 * FEB 6 (fri)
 * FEB 7 (sat)
 * FEB 9

PRODUCTION
- Shoot b-roll of neighborhood - Interview Neighboring store (Rob - Crumpet Shop) - Interview Manger at Lusty Lady - Interview "dancer" 1 - Collect b-roll of "dancer" 1 - Interview "dancer" 2 - Collect b-roll of "dancer" 2 - Interview "dancer" 3 - Collect b-roll of "dancer" 3 - Interview "dancer" 4 - Collect b-roll of "dancer" 4 - Interview "dancer's" family / friends - Collect b-roll for family / friends - Interview "dancer's" family / friends - Collect b-roll for family / friends - Intro 1 minute - Tag / Conclusion 1 minute - Extra shooting
 * FEB 9
 * FEB 13 (fri)
 * FEB 17 (tues)
 * FEB 19 (thur)
 * FEB 21 (sat)
 * FEB 23
 * MAR 2

POST PRODUCTION
- Capture footage - Lower third - Edit - 1st rough cut - Voice Over - Music - Review Edit - 2nd rough cut - Review Edit - 3rd rough cut - Review Edit - Final cut - Turn in project
 * FEB 16
 * MAR 9
 * MAR 14 (sat)
 * MAR 16
 * MAR 21 (sat)
 * MAR 23

title: User:Kylnso/documentary-dfvp212


text:
Portal:Greysmith Institute/languages division/Tab1

 * style="padding:0.8em; background-color:gold; border:solid 3px #003366; border-bottom:solid 3px gold; text-align:center" width="30%" | Languages
 * style="background-color:#AACCFF; border-bottom:3px solid #003366" width="3%" |


text:
Nonkilling Human Biology/Glossary

Definition of terms for Nonkilling Human Biology.

Multidisciplinary research is the most effective approach for human biocultural studies, which provide a holistic and comprehensive understanding of human affairs. This approach necessitates explaining specific terms and their related concepts for two purposes: to communicate with a non-specialist audience and to avoid semantic misunderstanding when terminology is not clear in both the media and specialist literature.

Below is a glossary for those terms and concepts whose particular usage in this work needs to be explained. A glossary is normally offered at the end of an essay, but we think that an initial list of short definitions prepares the reader to better integrate special terms when later met in a complex text. The asterisks indicate terms listed in the glossary; in the chapter this is done only once and for the first time. This version of the original glossary includes references.

Aggression
A specific behaviour aiming at damaging or destroying a living being (plant or animal), normally for alimentary purpose. Hunting and gathering involve aggression. Sexual competition among males is normally a display of fitness to improve the choice to be made by females. Note that, unlike violence*, aggression is not specifically intended to damage or destroy other individuals of the same species. In the literature, aggression, aggressiveness* and violence* are normally used as synonyms, causing in this way much misunderstandings and conceptual confusion.

Aggressiveness
It represents a congenital predisposition* to acquire aggressive behaviour after birth. Therefore it does not represent information for a specific behaviour. In the literature, aggression*, aggressiveness and violence* are normally used as synonyms, causing in this way much misunderstandings and conceptual confusion.

Behavioural predispositions
In humans they are congenital predispositions*, not congenital information for specific behaviour. A behavioural predisposition only set the level of postnatal stimulation necessary to channel an individual toward a certain category of behaviours. A specific behaviour within that category will then be defined by postnatal models. Therefore behavioural predispositions do not contribute to the definition of social behaviour in any shape or form (what the individual will actually do and how). For this reason the current idea of a 50-50 contribution of genetic information and postnatal learning to define specific social behaviour is invalid. It would be like adding apples and pears. In humans social behaviour is defined only by postnatal social models and personal experiences.

Biocultural evolution
Parallel evolution of behavioural predispositions* and specific behavioural models acquired by children and youngsters after birth from their cultural context. As a consequence of biocultural evolution, human behavioural predispositions* are common to all human beings without being specific behaviours (we have no instincts*, a part from the few ones of babies). Specific social behaviours (aggression* and violence* included) are not congenitally or genetically defined and differs in different cultures. In biocultural evolution, natural selection acts on both prenatal and postnatal information to make them change in harmony and slowly. In the late Neolithic this harmony was lost, as purely cultural changes occurred fast and in absence of corresponding changes in congenital predispositions. As a consequence we are still born with a brain suiting a hunter-gathering culture or, to be precise, for living in a nonviolent culture.

Biocultural studies
All academic disciplines have to do with human beings, some only marginally (e.g. Astronomy, Physics, Chemistry) most of them very directly (e.g. Medicine, Sociology, Psychology, Economy). It is becoming increasingly clear that human affairs cannot be studied without a multidisciplinary approach: Sociology, Political science, History and Philosophy are not enough to understand what happened in the past, causal mechanisms and what would be better for us in the future. We cannot continue exploring human beings only partially with individual disciplines or, worst, sub-disciplinary specialisations. For the purpose of this work – answering the question of whether we are congenital killers – we suggest that both so-called scientific disciplines and so-called humanities should be involved, in particular biomedical sciences and cultural studies are necessary, hence the term biocultural studies.

Conflict
In the general literature this term is marred by semantic vagueness. The whole field of studies dealing with conflicts and conflict management is also affected by confused terminology. For clarity and effectiveness, we suggest the following terminology. When two persons or communities are confronted with a difference of opinion or interests, but no conflict has yet materialised, they are facing a conflict of interests* (not a conflict). If the conflict of interests* is dealt with in a violent way (the stronger one will prevail) they will face a conflict, often a social confrontation, or a physical fight or a war. If the conflict of interests* is dealt with in a nonviolent way (dialogue, consultation, formulation of win-win solutions, etc.) a conflict will be prevented. This terminology, and its related concepts, allow a more sophisticated discussion and, importantly, include the concept of prevention. The strategies of conflict resolution or conflict management normally deal, instead, with situations after the conflict has occurred.

Conflict of interests
A conflict of interests is a social situation that represents a potential conflict*, which will occur if nothing is done to prevent it. See Conflict.*

Congenital characteristics
Congenital literally means ‘born with.’ It refers to both genetic characteristics (specific DNA sequences) and conditions experienced by the foetus in uterus. The popular literature often uses the term ‘genetic’ to mean congenital. For example, mental deficiency due to malnutrition of the pregnant mother is congenital, not genetic. Importantly, uterine congenital characteristics are not inherited by the next generation, while genetic characteristics are. For the themes discussed in this paper the important genetic/uterine congenital characteristics are behavioural predispositions.*

Cultural violence
See violence.* A special case of structural violence* that affects the way a person thinks. Indoctrination, misleading political propaganda and commercial advertising are among the many forms of cultural violence.

Deep culture
Important neuronal connections are formed in our brain during the first 5-6 years of life under the defining influence of post-natal experiences and non-verbal behavioural models. They define social values by establishing logical connections (cell contacts) between cultural situations and their consequences. At a later age memories of this experience settle within sub-cortical regions and escape consciousness to become the so-called deep culture operating at a subconscious level. Deep culture keeps influencing conscious behaviour throughout life via reciprocal connections with the cerebral cortex, but escaping the awareness of the subject. In this way we are convinced of carrying congenital 'drives' or 'impulses' that are beyond our control (the imaginary instincts*), while we are just influenced by very early postnatal inputs.

Direct violence
See violence.* Aggressive behaviour displayed by a person against another person; it can be verbal or physical (wounding, torturing and killing). Intentional killing is the extreme form of man-to-man violence* and is typical of (historical) human beings. As this is not practised by animals, one needs a term (violence*) different from aggression. In this sense, animals are aggressive not violent, and hunting is aggression against other species, not violence.*

Functional potentialities
They are functions that are eventually displayed by adult humans, but are in our congenital (genetic) developmental program only as potentialities (incomplete structures), because they need a strong postnatal input in order to complete their development as conceived by our biocultural evolution.* Examples: bipedal gait, speech and language, hand dexterity, social behaviour, etc. For example, a child can be born perfectly normal (larynx and brain regions ready to work) but the neural connections for speech must be constructed under post-natal stimulation (hearing adult speaking) otherwise the child will say nothing. Conclusion: one is not born a human being; one becomes one after birth if the necessary models are available.

Human beings
They are individuals of the species Homo sapiens who emerged in Eastern Africa about 100,000 years ago. Therefore the study of human nature should not be limited to historical humans who lived in the last few thousand years. On the other hand, other species of the genus Homo or other Hominids or Primates are not ‘our ancestors’, as often said, because they belong to other species and survived through different adaptive strategies; we only share a common ancestry with them, as we do with all other species of animals at more or less ancient times.

Human nature
Human nature is the set of characteristics that distinguish human beings from other species of animals, other primates in particular. Zoologists have no hesitation in defining the nature of all animal species, by describing body shape, geographical region(s) inhabited, diet, and specific behaviour. In the case of humans, it is relatively easy to define human nature in terms of physical features such as gene sequences and body shape (anatomy), but there is still debate about which aspects of human behavior should be described as human nature and which behaviors should be classified as cultural artifacts. It has been proposed that prehistoric and contemporary hunter-gatherers (their cultural practices and spirituality included) could be a better model for human nature than are most cultures from recent and current times. Giorgi's position is that recent (since the development of human cultures that are not limited to the hunter-gather pattern) modifications to human culture have been invented and are "purely cultural". If so, then claims about human nature that emphasize cultural practices that have developed during historical times and that have so far dominated sociological analyses and political proposals are misleading. If there are human cultural universals then exactly what are they? This issue can be explored at Nonkilling Human Biology/Human nature.

Human instincts
Terms used by Sigmund Freud (together with 'drives' and impulses') to indicate specific behaviours that are built in our congenital (prenatal) characteristics. In fact this is the way zoologists and animal ethologists use the term instinct: a specific behaviour that expresses itself even in absence of specific postnatal input. As animals evolved from Fish to Herbivores, Carnivores and Primates, the repertoire of instincts gradually decreased because postnatal acquisition* and learning* turned out more advantageous than congenital behaviour to adapt to a changing environment. Human beings* have only a few instincts, all associated with babies in the first year of life: searching the nipple, suckling, orient its senses toward mother, clasping objects passing in front of their visual field and swimming around the first year of age. All other sensory-motor functions are acquired* in the first few years of life under social guidance and learned* throughout life. Therefore social behaviour, violence* and killing included, are not instincts; Freud did not benefit of modern scientific advances.

Neurological imperatives
The human nervous system makes possible the learning and development of cultural elements such as technologies that allow humans to adapt to a wide range of physical or social environments. However, fossil and genetic evidence indicates that Homo sapiens and chimpanzees evolved by divergence from a common ancestor and that modern humans arose in Africa. Due to our evolutionary origins, are humans healthier and happier in a tropical/subtropical environment? Humans lived in a relatively small communities until recently. Are people more at ease (healthier) in a social environment where members know each other and display solidarity toward each other? Are chronic depression and other health problems made more likely when people live in high-density populations or competitive and violent societies? We can define "neurological imperatives" in terms of environmental conditions that favor healthy behavior and societies because they are a good fit with biological needs that arise from how the human nervous system evolved.

Nonviolence
It represents a mental attitude and behavioural strategies that favours consultation and negotiation in order to set in place win-win solutions of conflicts of interests.* Nonviolent solutions are not passive or appeasing; they require action, courage and intelligence. It does not take much intelligence to resolve a conflict of interests with violent strategies, that is, with a conflict, as we have done in the last 8,000 years or so.

Peace studies
They are multidisciplinary studies that aim at understanding the causes of violence* and war* in order to prevent (not just reduce) them and propose possible nonviolent solutions of conflicts of interests.* At the moment the theoretical bases of peace studies are weak, because academics and intellectuals still avoid dealing with the issue of human nature and the origins of human behaviour, nonviolence and violence.*

Postnatal acquisition
The completion of functional potentialities* in parallel with the definition of neural circuitries during the first 5-6 years of life. After this initial period of basic neural construction, one begins postnatal learning* of new skills. Erect posture, hand dexterity, and speech, for example, are acquired, and not learned, as one normally says. The general literature wrongly uses ‘acquiring’ and ‘learning’ as synonyms.

Postnatal learning
Information added to the memory bank of the brain after all functional potentialities* have been acquired. In fact, the function of transforming short-term memory into long-term memory (learning) is itself one of these functional potentialities.*

Religion
Religion emerged after the production of food (agricultural and pastoral economies), as a superstructure of pre-existing spirituality.* Religion carried the following novelties: a priestly authority, the concept of god(s), rituals, and moral instructions. The traditional collaboration with civil authorities has often embroiled religion in structural violence* and caused a loss of spirituality.*

Spirituality
Spirituality is a functional potentiality* typical of human beings, who are concerned with important metaphysical questions: the origins of natural features, the origins of humans, the relationship among humans and between humans and nature, and fundamental questions about life and death. The human cerebral cortex has an area in the frontal lobe that becomes particularly active during meditation and mental concentration on metaphysical issues. Prehistoric cultures and contemporary hunter-gatherers demonstrated sophisticated forms of metaphysical association with elements of nature and with other human beings. They did not consider themselves masters of nature (rather guardians of it) nor masters of other people (but as equals in society). The particular way this functional potentiality* was expressed was through their particular culture (just as language was), not separate instructions provided by clergy and political movements.

Structural violence
Structural violence is the source of all forms of violence.* According to Johan Galtung, it is the sum of those ideas and institutions that limit the development of the human potentialities of each individual. Falling ill of a preventable disease, lacking education, being deprived of love or cultural identity, are among the many examples of structural violence.

Violence
Violence is a term to be used only for human beings. It represents intentional oppression, wounding and killing directed toward other human beings, therefore member of the same species. Arguably other species do not display violence, only aggression*. In the literature, aggression*, aggressiveness* and violence are normally used as synonyms, causing in this way much misunderstandings and conceptual confusion. We cannot discuss about nonviolence* without a good definition of violence, especially of structural violence.*

War
It is a form of direct violence* against a perceived “enemy” involving a sophisticated social organisation, a culture accepting or admiring armed forces and weapons, and a dominant minority that has a vested interest in staging war, while not fighting in it.

title: Nonkilling Human Biology/Glossary


text: User:Viktormedina

title: User:Viktormedina


text:
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title: User talk:Hillgentleman/20090204232312