Aristotelian teleology and the principle of least action.doc

7/27/2019 Aristotelian teleology and the principle of least action.doc

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The Aristotles teleology and the principle of least action

George Mpantes mathematics teacher

From my book : Selected stories in mathematics and

physics, Lambert Academic Publishing. www.mpantes. gr .

Key words: Aristotelian teleology, least action, Maupertius, Hamilton, many stories,

Feynman, Spinoza, final causes,

Abstract

Final causes,

the principle of least action

Hamilton principle

The PLA in physics

The teleological version of Feynman

Final causes

Aristotle is commonly considered the inventor of teleology, although the precise

term originated in the eighteenth century. But if teleology means the use of ends or goals in

natural science, then Aristotle was rather a critical innovator of

teleological explanation. Aristotles teleological explanation,

does not crucially depend upon the application of

psychological concepts such as desires, beliefs and intentions.

This brings the discussion back to the very puzzling chapter 9

of Physics II, and thence to a consideration of the idea that

absolutely everything in nature is for some purpose.

"A difficulty presents itself: why should not nature work, not for the sake of something, nor because it

is better so, but just as the sky rains, not in order to make the corn grow, but of necessity? If a man's

crop is spoiled on the threshing-floor, the rain did not fall for the sake of this-in order that the crop

might be spoiled-but that result just followed. Why then should it not be the same with the parts in


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nature?...Wherever then all the parts came about just what they would have been if they had come be

for an end, such things survived, being organized spontaneously in a fitting way; whereas those which

grew otherwise perished and continue to perish?... Yet it is impossible that this should be the true

view. For teeth and all other natural things either invariably or normally come about in a given way;

but of not one of the results of chance or spontaneity is this true. It follows that they must be for an

end; and that such things are all due to nature even the champions of the theory which is before us

would agree."FromAristotle's Physics,Book II

But if somebody does not comprehend Aristotles ontology cannot conceive his teleology.

The four causes of Aristotle are allocated: in matter is the necessity, but in form is the final cause(

goal). But the most important aspect of nature is the form, so physicist should study both kinds of

causes, but particularly the final cause. Since the goal is the cause of the matter and not the matter

the cause of goal... Physics B9

The main controversy of the new epoch of mechanics as it is clearly shown

in the contents of its ideas, is directed against the ' scientific ' aspects of theology,

mainly against the concept of final causes, which acted as the "scientific " axioms

of theology and the reaction for the emerging science. So the debate is about

Aristotelianism, but actually the target is theology.

Man, says Spinoza, believes that everything (God, nature, himself) functionsthrough final causes (purposes ) because he does not know the true causes, so he

connects all physical phenomena with himself, that is, as nature guided from God

for the benefit of humans ( anthropocentrism ), who (God) looks like a Superman (

anthropomorphism). But in nature everything happens according to the material

constraint and not freely.Moreover, Spinoza expressly denies that nature is a teleological

system, and claims that final causal/teleological explanations turns nature completely

upside down. For what is really a cause it considers as an effect, and conversely, what is an

effect it considers as a cause (Appendix, I). Now all the prejudices which I intend to mention

here turn on this one point, the widespread belief among men that all things in Nature are

like themselves in acting with an end in view.

"The final causes do not help science and knowledge in general , because they encourage

man to attribute any event in the inscrutable will of GodSpinoza.

Spinoza through his critique of the final causes challenges the belief in

divine providence and miracles, claiming that God never acts against the laws of

nature which He endorsed.
http://classics.mit.edu/Aristotle/physics.2.ii.htmlhttp://classics.mit.edu/Aristotle/physics.2.ii.htmlhttp://classics.mit.edu/Aristotle/physics.2.ii.htmlhttp://classics.mit.edu/Aristotle/physics.2.ii.htmlhttp://classics.mit.edu/Aristotle/physics.2.ii.htmlhttp://classics.mit.edu/Aristotle/physics.2.ii.html


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So the final causes were excluded from God, from nature, and theology ,

which had no cognitive value for Spinoza . The teleological delusion is inherent in

human nature and justified only by the ignorance of the chain of efficient causes

that must produce the human understanding.

In this article I will show that physics that rejected the theological

interpretation of final causes, will resets them back in their genuine Aristotelian

form.

The principle of least action.

..The variation principles proposed in physics, are surrounded by a philosophical mysticism that

has delayed an evaluation of their value.. J.W.Leech

The teleology in physics comes back through the principle of least action, a

variation principle with a long history, with mathematical foundations and

continuous verification from experimental physics. It is the principle which denies

the omnipotence of deterministic mechanism.

The idea that the laws of nature must satisfy a purpose, a principle of

simplicity, starts from Aristotleto On Sky " which argues that the movement of the

stars should be circular, because" the line enclosing the circle is the line the smallest

length, and the faster movement t is that in the shortest length. Here's an example

of Aristotelian teleology . The stars move as they move, because achieve the shortest

path . Heron of Alexandriahas shown perhaps the first scientific principle for the

minimum, indicating that the path of the reflected light beam is shorter if the angles

of incidence and reflection are equal.

Fermat, between 1601-1605, studying the refraction of light, showed that

the light reaches its destination by following the fastest path (principle of the least

time). The principle of Fermat is a teleological principle, so it met the opposition of

the mechanistic scholarship. Fermat repelled criticism with a response that restates

later from the school of Copenhagen in the field of quantum mechanics : " Forget the


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deep interpretations. The principle i have expressed agrees with experiment up to seven decimal

places. Newton's laws were introduced because they interpreted what had found Kepler . "Fermat

But the mathematical reasoning, that according to Galileo can not be

separated from things, will show another way.

Fifty years after Newton, it was formulated the principle of least action

( PLA), in first form of the French Maupertius and final of the Irishman William

Hamilton. The first was incomplete and descriptive, the second is based on the

mathematical theory of variations.

Maupertius round in 1744 formulated the principle in its first form, with the

( Aristotelian ) intuition that the perfection of the universe requires an economy in

variations and a contrast of nature itself to unnecessary energy expenditure.Nature

does nothing in vain said Aristotle. The natural movements must be such as to

render some quantity of minimum. This principle has produced by the laws of

Newton. According to this principle, there is a different way of interpreting the

motion of planets which gives exactly the same results as Newtonian 'Principle of

acceleration forces, as we call it . It is the "principle of least action, PLA" that the

general idea is that: planets or bodies which move in space, follow that trajectory

which minimizes something called actionbetween the initial and final point. Action

is something that can be measured for each moving object along a track. Its

dimensions are of energy multiplied by time . The details are not necessary since

we will not do calculations.1

.. Having found that laws of motion extracted ( from the principle of least action ) is exactly

what we observe in nature, we can admire its application to all phenomena, in movement of animals,

in plant growth, in the rotation of the celestial bodies : the impressive spectacle of the universe

becomes larger, so beautiful the more worthy of his Creator ... these laws, so beautiful and so simple

are probably the only that the Creator and Organizer of things has installed in matter , in order to act

in all the phenomena of the visible world .... Maupertius

This alternative way of describing the world by Moreau de Maupertius met

reactions, since claimed that it is a rather moral principle, in which God saves energy

in the entire universe, for a better world. It is a divine design so it is about for a

1

The action has dimensions of energy over time and as a term is a relic of the 17thcentury when they had not clarified the concepts of energy and momentum.


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theological teleology in physics. Moreover, the action, which considered the

Maupertius was not correct and the evidence lacked clarity and precision2.

Almost simultaneously, Euler in his book in 1744 (a method for finding curved lines

enjoying properties of maximum and minimum. Or solution of isoperimetric problems),he lays the

foundations of a calculus of variations a method applicable to all the problems of

extremities. This book is one of the first to present and discuss the physical principle

of least action, indicating a deep connection between the calculus of variations and

physics. He demonstrated that the curve will follow a body during free movement

between two points is the one that will minimize the momentum along the orbit, i.e.

its action. He speaksalso of a divine plan3. Sarcasm anyway abated, because it is not

easy to gibe mathematical results .

Hamiltons principle.

But the idea that linked the new principle with the mechanics of Newton, is

the new form given to it (mechanics ) by Lagrange, the greatest scientist of Europe

of his time." The alternative form of the Newtonian theory which had proposed

Lagrange, and Euler's theorems of calculus of variations, supported the Hamiltons

principle, the principle of least action. Now this principle (PLA) will emerge as an

application of Euler's important theorem of the theory of variations in the motion of

bodies. The math do not exist to save the phenomena ! Assuming the Lagrangian of

the system as a function of Euler 's theorem, the mathematical result of the theorem

gives us the equations of motion of Lagrange i.e. Newtonian mechanics , through the

minimization of the action! This is the principle of Hamilton!

2The biggest enemy was Voltaire whose texts against Maupertius are amonument

of sarcasm. For months the high society laughed with Maupertius who eventually became the

laughing stock of Europe, finding refuge in Switzerland. It is said that Voltaire and Maupertius

had a relationship with the same woman.

3"Since the divine plan is the most perfect in the world, there can be no doubt that all

actions in the universe can be determined by the respective causes through the calculus of

maxima and minima. Leonard Euler1707-1783


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When a dynamic system moves from a position 1 to a new position 2,

within a specified time t2-t1, the path followed is the one that makes the action of

the system minimum4.

But the motion that minimizes the action is the motion satisfying the

Newtonian equation F = mdu/dt. So the two principles seem equivalent. In my

University teacher Hatzidimitriou we read

It is possible to build the edifice of theoretical mechanics on a new principle, the principle of

Hamilton, instead of Newton's laws . That is because Newton's laws are resulting from this principle ,

but conversely , the principle of Hamilton may arise by applying Newton's laws . Principle therefore

does not constitute a new theory of mechanics , but is more elegant than Newton's laws and may also

be applied to other areas outside of theoretical mechanics and mostly in the regions of fields, where

Newton's laws do not have a direct applicability.

Well the principle of Hamilton cannot be described as a fundamental, since

the description is equivalent to the classical description, the Hamilton's principle is

equivalent to the principles of Newton. We have just two answers to describe

situations in classical mechanics.

1. the bodies motion are deterministically described from the initial

conditions, each moment produces the next moment, and the trajectories produced,

satisfy the principle of least action, or

2.motions have holistic existence, the orbits are determined to satisfy the

PLA and happens to be those that yield the system of Newtons forces and

accelerations.

While for Spinoza, the final causes were the anthropomorphic causes, it now

appears inversely that the forces, the gravity or inertia, express the

anthropomorphism .

4 The action of the system during its motion is represented by the integral

2

1

t

t

LdtI with L=T-V.

Strictly expressed the principle Hamilton talks about stationary and not for least action, we will

consider the two terms synonymously. Generally for the minimization of action, Karatheodoris

has shown that action becomes minimum for natural movements whose distance is "small."


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But the principle of Hamilton has something incomprehensible in its

interpretation : It seems to imply that the body n e e d s t o k n o w t h e p a t h

t o f o l l o w and carry out the plan of how to go there, protecting the available

energy (equations of motion). And the question that remains is how bodies manage

to know what is the path which makes the action less than another, so choose only

that, the analogous question that we saw in differential motion of Newton, (how

bodies are held in orbit and at all times they know how to move next). This will be

known from the model of "many stories" of Feynman (below) and will be understood

through the Aristotelian ontology .

Hamilton's principle is a teleological principle5. Let's deepen its meaning: A

projectile is thrown into the air performing a parabolic trajectory until it stops on the

ground. Why its orbit is parabolic ? this parabola is the cause or the result of motion?

Is the result, says the Newtonian approach.The result of the action of two

forces, the inertial and gravitational which pushes the projectile in a parabolic

trajectory from moment to moment. (deterministic interpretation).

Determinism often is taken to mean causal determinism, which in physics is known

as cause-and-effect. It is the concept thatevents within a givenparadigm are bound

bycausality in such a way that any state (of an object or event) is completely determined by

prior states.Causality(also referred to as causation, or cause and effect) is the natural or

worldly agency or efficacy that connects one process (the cause) with another process or

5The science historian J. Gleick (1993) writes: 'it is impossible for a physicist to talk

about the principle of least action without unconsciously give some kind of willpower in the

moving body. The body seems to choose the route. It seems to know all the chances that arepresented. "
https://en.wikipedia.org/wiki/Event_(philosophy)https://en.wikipedia.org/wiki/Paradigmhttps://en.wikipedia.org/wiki/Causalityhttps://en.wikipedia.org/wiki/Causalityhttps://en.wikipedia.org/wiki/Paradigmhttps://en.wikipedia.org/wiki/Event_(philosophy)


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state (theeffect), where the first is partly responsible for the second, and the second is partly

dependent on the first. (Wikipedia)

Is the cause (goal), says the principle of Hamilton, because with the

parabolic trajectory comes true the PLA. (holistic interpretation, where systems are

approached as coherent wholes whose component parts are best understood in

context and in relation to one another and to the whole). The trajectory is inscribed

in motion from the beginning. Hamiltons motion minimizes the integral for the

action, namely the action in whole trip, the entire route is involved in interpretation,

is a holistic law. The holistic interpretation is easy to describe but difficult to

understand, deterministic versa.

This practice is in contrast to a purely analytic tradition (sometimes

calledreductionism)which aims to gain understanding of systems by dividing them

into smaller composing elements and gaining understanding of the system through

understanding their elemental properties. The holism-reductionism dichotomy is

often evident in conflicting interpretations of experimental findings and in setting

priorities for future research. In theholistic approach ofDavid Bohm, objects

constitutes an indivisible whole and there is no scientific evidence to support the

dominant view that the universe consists of a huge, finite number of minute particles,

and offered in its stead a view of undivided wholeness: "ultimately, the entire

universe (with all its 'particles', including those constituting human beings, their

laboratories, observing instruments, etc.) has to be understood as a single undivided

whole, in which analysis into separately and independently existent parts has no

fundamental status

How can we capture this exactly Aristotelian teleology, that brings the

principle of Hamilton, as interpretation of motion ?

This will be possible only by redefining the phenomenon of motion. In

Hamiltons principle we must conceive that motion is not the phenomenon of

change of bodys position, but something deeper. Movement is the phenomenon of

minimizing the action in a change of position. This is associated with the mobiles

Aristotelian entelechy, the being completely of the moving body, its tendency

to "consume the minimum energy" in its motions, as was its tendency described

from the law of inertia in Newtons model. But how should we understand the
https://en.wikipedia.org/wiki/Resulthttps://en.wikipedia.org/wiki/Reductionist_sciencehttps://en.wikipedia.org/wiki/Reductionist_sciencehttps://en.wikipedia.org/wiki/Reductionist_sciencehttps://en.wikipedia.org/wiki/Holomovementhttps://en.wikipedia.org/wiki/Holomovementhttps://en.wikipedia.org/wiki/Holomovementhttps://en.wikipedia.org/wiki/David_Bohmhttps://en.wikipedia.org/wiki/David_Bohmhttps://en.wikipedia.org/wiki/Holomovementhttps://en.wikipedia.org/wiki/Reductionist_sciencehttps://en.wikipedia.org/wiki/Result


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phenomenon? It depends on which principle is fundamental in physics, deterministic

or holistic, and within the framework of classical mechanics we saw that they are

equivalent.

The PLA in physics

But it seems that nature prefers the version of the action. The PLA has invaded in

physics, so that all the physics can be rewritten in terms of easier or minimum. Light

finds the shortest path in an optical system, the trajectory of a body in the

gravitational field is a geodesic, i.e. a route with the smallest " space-time " length .

Thomson 's theorem states that electrically charged particles move between them as

if they had the least energy, yet, we can use the PLA in relativity if we find the right

form for the action, such an easy and successful variation principle has been applied

to general relativity by Taylor and Wheeler for the exploration of black holes .... (

M.Hancova ). Larmor applied the calculus of variations in electromagnetic

phenomena and showed that Maxwell's laws can be derived from a suitable

mathematical expression L of electromagnetic function of action, by PLA .

So what happens? Which principle is fundamental in physics? What is the

phenomenon of motion?

The teleological version of Feynman

While a clear answer to these questions does not exist in the

context of classical physics, this will only be given by quantum physics,

and even from the version developed by Feynman, who brings

Aristotle in modern physics.


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What changed in the Feynman interpretation of quantum mechanics?

He gives an interpretation of the formation of the wave function which describes the

activity of a particle.

He introduces the well-known principle of quantum electrodynamics, that of

many stories in motion.6

But how to determine the probability of a path before this motion? He

accepts all equally possible eventsas

potential Aristotelian realities

from which the entelechy will

push ' in actuality those that will

be compatible with the final cause,

which is the PLA. Right here he shows a similar process! The "many stories" of

Feynman are understood by the "potential" Aristotelian reality of matter. He gives

an example of the known reflection phenomenon of light, illustrated in the figure .

The equality of the angles of incidence and reflection is a minor in his interpretation.

Feynman says that the reflectance of the photon from the S to P through horizontal

mirror, is likely to be at the point M, where the angles of incidence and reflection are

equal,7because it dictates the PLA. The photon has initially infinite possibilities of

route ( the many stories ) . Could be reflected in A , B, C , etc. But this is improbable

to happen. The possibilities of these paths are reduced with a mathematical

process, but we cannot give it a real meaning within the world of empirical reality

without the Aristotelian ontology.

6we will not make any arbitrary rule. The only correct rule-what really happens-it's

much simpler : each photon reaches the detector has the same chance to move (it could

move) in any wayFeynman

7The view of classical physics


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This process is: we add the arrows of the waves of probability, whose

directions round the path of PLA -that is the path of least time SMP or minimum

action-, differ slightly, resulting added enhancers and impose their directions, the

phase in the final stage, the final sum - arrow .. Therefore the orbit SMP is most

probable and this seems to take place . This is a natural and automatic process.

Phases in places like K or B cancel each other with a mathematical process and do

not add anything to the final width8. The random motions of potential reality,

primer these of PLA, in the microcosm. So if the PLA considered as purpose, this

purpose is raised by the nature itself9. Aristotle would say that these many stories

are potential orbits, orbits in potentiality, are the infinite possibilities of

transformation of mattertoformand their sorting is interpreted as entelechy of the

photon. So ... the light does not actually move in a straight line. You could say that " smells

" the neighboring paths and uses a ' core area ' round from the straight path .... Feynman

Now we got the roots of the phenomenon of motion for the entire physics.

Because in the classical limit the quantum effect of PLA , is reduced to PLA of

Hamilton, even in forces of Newton, as shown by the Feynman, through the

formalism of ' many histories '. Classical physics is an application of quantum

mechanics , and not vice versa.

8 In a "good" function, round the minimum there is always possibility of less

dispersion of values. This specifies the path of least action as the most possible motion! In

this way the physicist Feynman expresses the entelechy of the mobile. This is the universe

of thought that Jeans says.

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The purpose in nature resulting from the very nature, not something outside of it, thebodies have the purpose within them ..Aristotle.


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Until some size in world observations, the stories of Hamilton and Newton

seem equivalent descriptions. But in the field of quantum physics, the fundamental

of nature is found in PLA as most probable evolution. Feynman applies PLA in an

infinitesimal part of a path, thus the search of extremity of action in this case is

directly related to the derivative of the potential, viz the force that dictates the

particle how to move. So the PLA produces the concept of conservative forces! The

conservative force e.g gravity serves anthropocentrism, despite what believed

classical physics and Spinoza. Everything is PLA.

.The Newton's laws can not be applied to things that are too small, too fast or too large .

In fact is clear that the fundamental laws can be written in the form of a principle of least

action .... Feynman

" Because the purpose is the cause of matter and not matter the cause of purpose,

Aristotle.

George Mpantes mathematician

Serres Greece

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