In ‘Politics’, Aristotle makes a clear statement: “it is a mistake to believe that the statesman is the same as. the manager of the household.”[1] This idea goes beyond a simple comparison; it points out key differences between the nature of individuals or households and that of the city. In this essay, I’ll examine and explain two of Aristotle’s arguments on why the individual or household and the city are different by nature.
The first argument is a phylogenetic argument, which states that human associations, or communities, first developed through a “conjugal union”[2] of a male and a female to form a household, which then associated with more households to form a village, and eventually a number of villages joined together to form a city (or polis), which is the “final and perfect association”[3] that has achieved the “height of full self-sufficiency.”[4]The phylogenetic argument shows how a human association evolves from an individual to a city. However, if one only considers this argument, then one might think that the two only differ “according to the number”[5] and that the city is simply an extension of the individual and the family.
Aristotle argues that the phylogenetic argument alone “cannot be accepted as correct,”[6] and provides a second argument, a teleological argument, which states that “the city is prior in the order of nature to the family and the individual”[7], even though it developed later in the process of human society. According to Aristotle, the reason why the city takes precedence over the family and the individual in the natural order of things is that “the whole is necessarily prior to the part.”[8]
Aristotle’s teleological argument can be explained in the following way:
Firstly, Aristotle defines the nature of things as “what each thing is when its growth is complete.”[9] Thus for Aristotle, the nature of a thing is not what the thing is but what it is capable of becoming. Its nature is not a static but a developing conception. For example, an acorn naturally grows into an oak tree, and the resulting tree exists by nature and not by craft. Likewise, in Politics the nature of a state can only be discovered by observation of its development and tendencies. Using this definition, Aristotle makes two assertions about human nature. In the first assertion, he argues that humans are naturally communal animals. In the second, he argues that “in a greater degree than bees or other gregarious animals,”[10] human beings are political animals by nature.
Aristotle argues that humans are communal animals because they have an innate desire to live a social life, even when they do not require mutual support. Additionally, he argues that humans are political animals, not only because of their natural inclination to form partnerships with others but also because of their unique capacity for reason speech. This capacity allows humans to communicate and deliberate about what is “what is advantageous and what is the reverse”[11] to each other, leading to the development and communication of moral concepts such as justice, goodness, and evil. This, in turn, leads to the formation of associations that are absent in other animals. For Aristotle, the “most sovereign and inclusive”[12] form of such an association is the city or political association. Since the capacity for reason and speech eventually leads to the pursuit of the city, Aristotle believes that humans are political animals by nature.
Secondly, Aristotle argues that, according to his definition of nature, all associations, including individuals, families, villages, and cities, exist by nature. However, it is crucial to distinguish between the nature of the family/village and that of the city, as they possess different ends. The city differs from the other, more primitive and pre-political associations because it serves as the “consummation” of earlier associations, achieving full self-sufficiency, whereas the family and village exist by nature “for the satisfaction of daily recurrent need.”[13] Aristotle claims that for any city, “it must devote itself to the end of encouraging goodness”[14]. Thus, the distinction between the family and the city lies in the fact that while the purpose of the family is “merely to sustain life”, the purpose of the city is to enable citizens to “live a good life.” [15] In other words, while the end of the family corresponds to achieving the communal aspect of human nature, the end of the city corresponds to achieving both the communal and political aspects. Thus, one can say that the city is prior to the individual and family by order of nature because the city is aiming at achieving a higher moral end.
Aristotle asserts that the city, as the consummation of all associations, “includes all the rest,” “pursues this aim the most,” and is “directed to the most good.”[16] Consequently, it is reasonable for him to claim that the city represents the whole, while other associations only represent parts of the ideal function of the city. The city achieves its end because it is composed of equal and free citizens, with political leadership exercised by the citizens themselves in rotation. The city provides opportunities for individuals to participate in politics and develop their moral character. Only within the city can individuals fully utilize their political nature, through activities such as deliberation and decision-making, which enable their moral character to be fully realized through rational communication.
Overall, Aristotle believed that humans are naturally social and political animals, distinguishing them from other creatures. However, individuals must reside in a city in order to develop and realize their natural capacities. Without the city, an individual is no more than a beast[17], since they are not self-sufficient. This is similar to how a hand loses its ‘nature’ as a hand when the body is destroyed. Therefore, it can be argued that the city takes priority in the natural order over the family and the individual.
Works Cited
Cahn, S. M. (2015). Political philosophy : the essential texts . New York: Oxford University Press.
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and time 
take the form
is smooth curve, then along the curve, 
has rate of change
as speed. The left-hand side of the PDE can also be interpreted as the derivative of 
, in the direction 
in 
space.
![\displaystyle \frac{d}{dt}[u(x(t),t)]=0 \ \ \ \ \ (6)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdt%7D%5Bu%28x%28t%29%2Ct%29%5D%3D0+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0&c=20201002)
. We solve (5) to get
is an unknown constant. Equation (7) is the equation for the family of parallel characteristic curves of (1). We observe that at 
, we have 
, and hence this constant 
is given, we can always find the corresponding anchor point as
is an arbitrary constant.
for the PDE problem (1). We already know that the initial condition for ODE (5) is 
in place of 
, and the general solution at 
and 
. This effectively changes variables from 
to 
.
, where 
is a curve of anchor points and 
is a given function. Suppose that 
is the solution to (24) subject to the side condition (25). We can think of 
as a two-dimensional surface in the 
space. Denote by 
the curve on the surface 
plane is 
is tangent to the solution surface 
which starts from the initial curve 
in terms of the characteristic variable 
and the anchor point 
:
with no frictional forces and no restoring forces is
, 
, where 
. We assume both tension 
and linear density 
to be constant. Tension 
and linear density has units of 
, so 
has the units of 
, which is also the unit of speed. Thus, 
. The end ends of the vibrating strings are fixed.
and 
separates the variables:
is the separation constant. This implies that
, then 
. So 
. Using results from the heat equation, we know that 
and 
yields trivial solutions. For 
, the general solution is
and we arrive at 
. Consider 
, we arrive at 
. If 
, we get a trivial solution. Therefore, we get a nontrivial solution if and only if
. Thus, for 
, each of the functions
satisfies the following orthogonality relation:
and integrating from 
to 
yields
yields,
yields,
, are called the frequencies. The fundamental mode of the string has a frequency of 
. The 
th overtone is just the 
and we arrive at 
and we arrive at 
. Since 
. Thus, for 
. We use the fact that 
satisfies the following orthogonality relation:
and integrating from 
![{f_n:[0,1]\rightarrow \mathbb{R}}](https://s0.wp.com/latex.php?latex=%7Bf_n%3A%5B0%2C1%5D%5Crightarrow+%5Cmathbb%7BR%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
is defined by 
. If 
, then 
as 
. If 
, then 
as 
is continuous on ![{[0,1]}](https://s0.wp.com/latex.php?latex=%7B%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
but the pointwise limit 
converges uniformly on the closed interval ![{[a,b]}](https://s0.wp.com/latex.php?latex=%7B%5Ba%2Cb%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
if the following two conditions are satisfied: First, the series converges to the sum 
for each 
, there exists a 
such that for all 
the partial sums 
satisfy 
for every 
-Test If for all points 
from a certain 
on, and the series of positive numbers 
converges, then the series 
converges uniformly on 
converges absolutely. Therefore, the series 
, then there exist an 
such that 
implies 
.
,
and 
, and to Fourier series in exponential form if 
.
and 
be two finite sets of real numbers. Then,
is defined at the “corners”. To see this, let 
, we have
-periodic, continuous, and piecewise smooth. Let 
, 
, and 
be the Fourier coefficients of 
, 
, and 
be the corresponding Fourier coefficients of 
, then 
is the sum of the derived series
at which ![{\frac{1}{2}[f'(\theta_-)+f'(\theta_+)]}](https://s0.wp.com/latex.php?latex=%7B%5Cfrac%7B1%7D%7B2%7D%5Bf%27%28%5Ctheta_-%29%2Bf%27%28%5Ctheta_%2B%29%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
.
, 
, 
in this series are 
, 
, and 
respectively. Thus theorem 3.5 follows.
periodic and piecewise continuous, with Fourier coefficients 
. If 
, then for all 
on ![{[-\pi,\pi]}](https://s0.wp.com/latex.php?latex=%7B%5B-%5Cpi%2C%5Cpi%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
:
, the sum of the series on the right of (72) is 
.
, 
is the sum of its Fourier series at every 
, and 
of 
or 
is just the usual formula for the zeroth Fourier coefficient of 
, these arguments can be applied to the function 
rather than 
.
for 
, and by (29) (Bessel’s inequality) applied to 
. This completes the proof.
for all 
. We use the following properties:
to 
and one going from 
to 
.
, 
to 
. We obtain
as 
, 
and 
. Similarly, we obtain the following definition of the Fourier series.
![{[-L,L]}](https://s0.wp.com/latex.php?latex=%7B%5B-L%2CL%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
. The periodic extension of 
, which satisfies for 
![{[0,L]}](https://s0.wp.com/latex.php?latex=%7B%5B0%2CL%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
.
and the odd extension 
of 
? Proving the convergence of the Fourier series is not a simple matter. Firstly, let’s derive an estimate of the Fourier coefficients that will be needed to establish convergence.
![\displaystyle \begin{aligned} \mid f(\theta)-\sum^{N}_{-N}c_ne^{in\theta}. \mid^2 & =( f(\theta)-\sum^{N}_{-N}c_ne^{in\theta})( \overline{f(\theta)}-\sum^{N}_{-N}\overline{c_n}e^{-in\theta}) \\ & = \mid f(\theta) \mid^2 -\sum^{N}_{-N} [f(\theta)\overline{c_n}e^{-in\theta}+\overline{f(\theta)}c_ne^{in\theta}]+\sum^{N}_{-N}c_n\overline{c_n}\\ \end{aligned} \ \ \ \ \ (31)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cbegin%7Baligned%7D+%5Cmid+f%28%5Ctheta%29-%5Csum%5E%7BN%7D_%7B-N%7Dc_ne%5E%7Bin%5Ctheta%7D.+%5Cmid%5E2+%26+%3D%28+f%28%5Ctheta%29-%5Csum%5E%7BN%7D_%7B-N%7Dc_ne%5E%7Bin%5Ctheta%7D%29%28+%5Coverline%7Bf%28%5Ctheta%29%7D-%5Csum%5E%7BN%7D_%7B-N%7D%5Coverline%7Bc_n%7De%5E%7B-in%5Ctheta%7D%29+%5C%5C+%26+%3D+%5Cmid+f%28%5Ctheta%29+%5Cmid%5E2+-%5Csum%5E%7BN%7D_%7B-N%7D+%5Bf%28%5Ctheta%29%5Coverline%7Bc_n%7De%5E%7B-in%5Ctheta%7D%2B%5Coverline%7Bf%28%5Ctheta%29%7Dc_ne%5E%7Bin%5Ctheta%7D%5D%2B%5Csum%5E%7BN%7D_%7B-N%7Dc_n%5Coverline%7Bc_n%7D%5C%5C+%5Cend%7Baligned%7D+%5C+%5C+%5C+%5C+%5C+%2831%29&bg=ffffff&fg=000000&s=0&c=20201002)
, we have
we have
, 
and 
are all convergent. As a consequence, we obtain the following result.
in the case of 
, 
and 
are the 
, we say that a function 
;(ii) at each of the points 
) are one of the exception points 
, then we require only the left-hand (or the right-hand) limit to exist). Thus, we say that 
for later convenience. We obtain the following
. The sum is not affected since it ranges from 
to 
. Now, letting 
, we have
is called the 
to obtain![\displaystyle \begin{aligned} D_N(\phi)&=\frac{1}{2\pi}\sum^{N}_{-N}e^{in\theta}\\ &= \frac{1}{2\pi}(\sum^{N}_{1}[e^{in\theta}+e^{-in\theta}]+e^{i(0)\theta})\\ &= \frac{1}{\pi}\sum^{N}_{1}\mathrm{cos}(n\theta)+\frac{1}{2\pi} \end{aligned} \ \ \ \ \ (45)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cbegin%7Baligned%7D+D_N%28%5Cphi%29%26%3D%5Cfrac%7B1%7D%7B2%5Cpi%7D%5Csum%5E%7BN%7D_%7B-N%7De%5E%7Bin%5Ctheta%7D%5C%5C+%26%3D+%5Cfrac%7B1%7D%7B2%5Cpi%7D%28%5Csum%5E%7BN%7D_%7B1%7D%5Be%5E%7Bin%5Ctheta%7D%2Be%5E%7B-in%5Ctheta%7D%5D%2Be%5E%7Bi%280%29%5Ctheta%7D%29%5C%5C+%26%3D+%5Cfrac%7B1%7D%7B%5Cpi%7D%5Csum%5E%7BN%7D_%7B1%7D%5Cmathrm%7Bcos%7D%28n%5Ctheta%29%2B%5Cfrac%7B1%7D%7B2%5Cpi%7D+%5Cend%7Baligned%7D+%5C+%5C+%5C+%5C+%5C+%2845%29&bg=ffffff&fg=000000&s=0&c=20201002)
, we have
,
, we get
![\displaystyle \begin{aligned} \int^{\pi}_{0}D_N(\theta)d\theta&= \int^{\pi}_{0} \frac{1}{\pi}\sum^{N}_{1}\mathrm{cos}(n\theta)+\frac{1}{2\pi}d\theta\\ &= \bigg[\frac{1}{\pi}\sum^{N}_{1}\frac{1}{n}\mathrm{sin}(n\theta)+\frac{\theta}{2\pi}\bigg]^\pi_0\\ &= \frac{1}{2} \end{aligned} \ \ \ \ \ (51)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cbegin%7Baligned%7D+%5Cint%5E%7B%5Cpi%7D_%7B0%7DD_N%28%5Ctheta%29d%5Ctheta%26%3D+%5Cint%5E%7B%5Cpi%7D_%7B0%7D+%5Cfrac%7B1%7D%7B%5Cpi%7D%5Csum%5E%7BN%7D_%7B1%7D%5Cmathrm%7Bcos%7D%28n%5Ctheta%29%2B%5Cfrac%7B1%7D%7B2%5Cpi%7Dd%5Ctheta%5C%5C+%26%3D+%5Cbigg%5B%5Cfrac%7B1%7D%7B%5Cpi%7D%5Csum%5E%7BN%7D_%7B1%7D%5Cfrac%7B1%7D%7Bn%7D%5Cmathrm%7Bsin%7D%28n%5Ctheta%29%2B%5Cfrac%7B%5Ctheta%7D%7B2%5Cpi%7D%5Cbigg%5D%5E%5Cpi_0%5C%5C+%26%3D+%5Cfrac%7B1%7D%7B2%7D+%5Cend%7Baligned%7D+%5C+%5C+%5C+%5C+%5C+%2851%29&bg=ffffff&fg=000000&s=0&c=20201002)
is defined by (40), then![\displaystyle \lim_{n\rightarrow\infty} S^{f}_N(\theta)=\frac{1}{2}[f(\theta-)+f(\theta+)] \ \ \ \ \ (52)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Clim_%7Bn%5Crightarrow%5Cinfty%7D+S%5E%7Bf%7D_N%28%5Ctheta%29%3D%5Cfrac%7B1%7D%7B2%7D%5Bf%28%5Ctheta-%29%2Bf%28%5Ctheta%2B%29%5D+%5C+%5C+%5C+%5C+%5C+%2852%29&bg=ffffff&fg=000000&s=0&c=20201002)
. In particular, 
for every 
![\displaystyle S^{f}_N(\theta)-\frac{1}{2}[f(\theta-)+f(\theta+)] \ \ \ \ \ (55)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+S%5E%7Bf%7D_N%28%5Ctheta%29-%5Cfrac%7B1%7D%7B2%7D%5Bf%28%5Ctheta-%29%2Bf%28%5Ctheta%2B%29%5D+%5C+%5C+%5C+%5C+%5C+%2855%29&bg=ffffff&fg=000000&s=0&c=20201002)
. By (47) we can write it as
is a well-behaved function except when near 
. However, since 
and 
, by L’Hôpital’s rule,
as 
approaches 
. However, we know that formula (57) is just 
. Thus, equation (55) tends to 
are 2
.
that is defined for 
is called periodic if there is a number 
such that
. A periodic function has the following properties:
, then 
for all 
does not depend on 
, its periodic extension is
. This definition does not specify what the periodic extension is at the endpoints 
. In fact, the function 
and 
both exist but not equal.
is symmetric with respect to the y axis.
is an even function if 
are even functions. The functions sin 
. Let ![{\phi(x)=\frac{1}{2}[\phi(x)+\phi(-x)]}](https://s0.wp.com/latex.php?latex=%7B%5Cphi%28x%29%3D%5Cfrac%7B1%7D%7B2%7D%5B%5Cphi%28x%29%2B%5Cphi%28-x%29%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
and ![{\psi(x)=\frac{1}{2}[\psi(x)-\psi(-x)]}](https://s0.wp.com/latex.php?latex=%7B%5Cpsi%28x%29%3D%5Cfrac%7B1%7D%7B2%7D%5B%5Cpsi%28x%29-%5Cpsi%28-x%29%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
. Then we easily check that 
, that 
is odd. The functions 
is any polynomial, its even part is the sum of its even terms, and its odd part is the sum of its odd terms.
and 
are odd. If 
follows directly from (4) by putting 
. The graph of an even function 
, since the derivative is odd (provided the derivative exists).
: Dirichlet BCs corresponding to the odd extension
: Nuemann BCs corresponding to the even extension
: Periodic BCs corresponding to the periodic extension