"Nature always proceeds in the simplest or most economical way" (Aristotle).
"Ockham's razor is the supreme maxim of philosophy" (Bertrand Russell).
"Science can be defined as the art of systematic super-simplification" (Karl Popper).
The Principle of Ockham's Razor
Ockham's razor principle, also called the "principle of parsimony" and "principle of economy of thought", is a thesis elaborated by William of Ockham, a 14th-century English Franciscan friar, philosopher, theologian, writer, politician and scholastic thinker. William of Ockham, along with John Duns Scotus (his teacher) and Thomas Aquinas, are the most prominent philosophical figures of the early Middle Ages.
This principle is expressed by Ockham in several ways:
"Entia non sunt multiplicanda praeter mecessitatem" (Entities must not multiply without necessity).
"Pluralitas non est ponenda sine necessitate" (Plurality should not be postulated without necessity).
"Frustra fit per plura quod potest fieri per panciora" (It is vain to do with much what can be done with less).
Behind these phrases underlies the principle of conceptual or intellectual simplicity, under two aspects:
Always strive to create theories of the greatest possible simplicity, with as few concepts as possible.
In the face of several theories that explain or describe a phenomenon, always choose the simplest one.
There are two interpretations of the origin of the term "razor":
It comes from the fact that Ockham employed his principle sharply and precisely on numerous problems or subjects, "dissecting" them to simplify them.
It comes from the fact that, metaphorically, Ockham "shaved" with a razor the beards of Plato, since with its application he obtained a remarkable ontological simplicity, as opposed to the Platonic ontology (the theory of Ideas or Forms), which was very complex by including in it all kinds of entities. Ockham thus eliminated many unnecessary entities, a form of rejection of Platonism.
Ockham was interested and passionate as a young man about logic, a science he considered fundamental to the advancement of knowledge and understanding of reality. For Ockham, the simple is the logical, and logic must be used to simplify.
Ockham's thought
Although Ockham's thought is subject to many interpretations, it does appear that he applied logic and its famous principle to different subjects in an attempt to simplify and clarify them. To do this, he eliminated many entities or concepts, especially those of the scholastic philosophers, which he considered unnecessary. He also applied logic and his "razor" to separate different concepts or entities to simplify them.
In separation of concepts, he distinguished between:
The necessary and the contingent.
The only truly necessary entity is God; everything else is contingent, including the natural order and the moral order.
Theology and science (or faith and reason).
So-called "natural theology" holds that one can ground the existence and knowledge of God from reason; and that one can prove by reason that God is the first cause of all things. One of its advocates was Thomas Aquinas, with his 5 ways to demonstrate the existence of God, using philosophy (especially that of Aristotle) to build the edifice of theology.
For Ockham, science and human reason will never be able to demonstrate neither the immortality of the soul nor the existence of God. These truths can only be known by revelation and faith. Theology is not a science. The first cause cannot be established by reason. The first cause is metaphysical in character and is incompatible with the knowledge of reason.
Real science and rational science.
Real science is that of real things, of individual things and of experience. Rational science is that based on logic and reason. But both sciences must be harmonized: every explanation must match reason and experience.
The evidence (the real) and the possible (or probable).
Nothing must be assumed without a reason, unless it is self-evident (known through itself) or known by experience or demonstrated by the authority of Holy Scripture.
He also used his "razor" to eliminate (at his discretion) unnecessary entities:
He dispenses with universals.
Ockham is founder of the school of thought called "nominalism", which denies that universal concepts are real. They are only names applied to things. Nor are abstractions and concepts real. The real is only the individual, the singular and the particular. Universals are only representations of particulars.
For Aristotle, "True science is that which focuses on the general and universal." Ockham rejects this. For him, science is the knowledge of particular things. Science is of the general only insofar as the general represents the particular. The general propositions of science are not the object of science. The universal is manifested at the mental level by means of terms and propositions, which are also concrete and singular. The universal is in the mind of the subject, not in the object. If the universal were real it would also be individual. The real (external) is singular; the mental (internal) is universal. Nor are there composite elements; the composite is a concept, something internal.
Ockham believed in abstract entities such as "whiteness," but denied that they were universal. There are as many different whitenesses as there are white things. He believed in immaterial entities such as God and angels, but he did not believe in mathematical entities of any kind (numbers, points, lines, etc.) because they were unnecessary.
Ockham's attitude is anti-metaphysical: he renounces metaphysical knowledge of reality. It is an ontology of the singular, particular and concrete.
Ockham's position on universals contrasts with that of his teacher. For Scotus, universals exist, but immanently in particulars. He thus followed the view of Aristotle, for whom matter and form are inseparable and form a unity called "substance". Matter and form represent, respectively, the particular and the universal of substance. According to Scotus, the real are the particulars, but there is a common nature found in all particulars. The universal is in the object before it is grasped by the understanding.
He dispenses with relations.
Relations are not realities added to things but only mental "intentions". Relations belong to the mental world, not the physical.
It dispenses with intermediate entities between God and his creation.
The ideas in the mind of the Creator are the creatures themselves.
It dispenses with sensible or intelligible "species".
The species −postulated by Aristotle and Thomas Aquinas− are intermediary entities between external reality and internal reality (the conception of external reality). For Ockham there is no species between objects and mind, between things and apprehensions of things. Apprehension is realized by intuition, which connects the internal and the external. What the mind apprehends is what is and how it is. Only through intuition do we acquire knowledge. Intuition is the clear and direct grasp of the individual. The relation between cognizing subject and the thing known is direct, without intermediate entities.
Concrete experience is the only basis of knowledge. All human knowledge is founded on the sensory intuition of concrete particulars. What is beyond the senses (such as the existence of God) can only be unveiled by faith, never by reason.
It dispenses with essences.
Essences are not real. The principle of individuation asserts that each individual has an associated essence. This, in Ockham's view, is an error. There is, then, no principle of individuation: there are individuals, individuals are individuals, and that is all. There are no essences, only individuals. He therefore rejects the distinction between essence and existence.
He dispenses with induction.
According to Aristotle, "Induction is a transit from individual things to universal concepts." Ockham disagreed: one cannot justify a universal conclusion by the mere accumulation of particular facts.
He dispenses with 8 of the 10 Aristotelian categories and is left only with substance and quality.
Ockham has had a great influence on science and philosophy:
He was an independent and original thinker who questioned the old ideas (the "ancient way"), the traditional scholastic conception of Thomas Aquinas and Duns Scotus, bent on uniting faith and reason. He tried to configure a "modern way" by applying logic in a systematic way to make science and philosophy independent of theology.
He was the precursor of modern science, experimental or empiricist science.
He opened the modern way to knowledge by formulating a general method: the principle of conceptual economy, his famous "razor".
His nominalist philosophy produced a radical turn in logico-semantic research. He was a precursor of modern logic and of the formal analysis of semantics.
He was also the driving force behind the thesis of the existence of a mental language, which he attempted to describe in some detail.
Evaluation of Ockham's razor principle by the authors
Ockham's razor principle has been admitted, implicitly or explicitly, by almost all philosophers and scientists. But it has also been qualified (so as not to push simplicity to extreme limits) and, in very few cases, questioned. The principle of Ockham's razor goes back to Aristotle:
Aristotle expounds a maxim: "Natura nihil facit frustra" (Nature does nothing uselessly). It follows that if Nature acts according to the principle of economy, at the mental level we must also follow this principle.
Ptolemy stated, "We consider it a good principle to explain phenomena by the simplest possible hypothesis."
Thomas Aquinas wrote: "If a thing can be adequately done by one, it is superfluous to do it by several; for we observe that nature does not employ two instruments where one is sufficient." He admitted the principle of Ockham's razor, but preferred causal explanations to simple ones.
Guillermo Durande −in French, Guillaume Durand de Saint Pourçain− expounded this principle before Ockham, but it was the Franciscan friar who contributed most to its diffusion, while applying it to problems of different kinds. Durande, Dominican, French scholastic philosopher, called "Doctor Resolutissimus" and also "Doctor Modernus", because of the radical and innovative character of his ideas, inaugurated a new period in scholasticism by distinguishing philosophy (the science of reason) from theology (the doctrine of the spiritual). He maintained that, in the search for truth, the exercise of reason is more important than any human authority. Against the Aristotelian realism of Thomas Aquinas, he defended nominalism and abstraction to apprehend reality. He was also in favor of suppressing numerous concepts of scholasticism, which seemed to him useless or superfluous.
The scholastic philosopher Walter Chatton −Ockham's contemporary and fellow member of the Franciscan order− questioned Ockham's razor and Ockham's own use of it. In response, he provided his own "anti-navaja": "If three things are not sufficient to verify an affirmative proposition about things, a fourth must be added, and so on". If Ockham's razor eliminates the unnecessary, Chatton's anti-navaja adds the necessary.
Nicolas d'Oresme −in French, Nicole d'Oresme− also invoked the law of economy. He was an intellectual genius and probably the most original thinker of the 14th century. Economist, mathematician, physicist, astronomer, philosopher, psychologist and musicologist, he was one of the main founders of modern science. He combated pseudosciences (such as astrology) and speculated on the possibility of other inhabited worlds. He was an advocate of nominalism. He invoked the principle of simplicity by observing that if the Earth moved around its axis, it was more economical than the rotation of the entire immense celestial sphere.
Galileo criticized the incorrect interpretation of Ockham's razor in his "Dialogues on the two highest systems of the world: Ptolemaic and Copernican". In this work, Simplicius' character ironically stated that if one really wanted to use the minimum possible number of entities, the letters of the alphabet would be the only fundamental entities, since all human knowledge could be constructed from them.
For Descartes, "The order of our thoughts must always go from the simplest to the most composite" (Discourse on Method).
Newton, in the third edition of his work Principia Mathematica, included four "Rules for reasoning in philosophy", the first being "We must admit no other causes of natural things than those which are true and sufficient to explain their appearances".
Leibniz stated his "Principle of plenitude", which states that "Whatever is possible to occur, will occur". Leibniz argued that the existence of the best of all possible worlds would genuinely confirm every possibility, and postulated in his Theodicy that the best of all possible worlds would contain all possibilities.
George Berkeley believed that all reality could be explained only in terms of the mind. He invoked Ockham's razor against materialism, claiming that matter was not necessary to his metaphysics, so it could be dispensed with.
Kant, in his "Critique of Pure Reason," takes a stand against Ockham's razor by stating, "The variety of beings should not be foolishly diminished."
The mathematician William Rowan Hamilton, in 1852, was the one who named the English Franciscan philosopher's principle of conceptual economy "Ockham's razor" in his writings. Hamilton expressed this principle as "No more or more onerous causes should be assumed than are necessary to explain the phenomena".
Ernst Mach stated that "Scientists should use the simplest means to arrive at their results and exclude everything not perceived by the senses." This philosophy becomes positivism, the belief that there is no difference between something that exists but is not observable and that which does not exist at all. Mach claimed that molecules were metaphysical in character because they could not be directly detected. This is an example of misapplication of Ockham's razor by eliminating something necessary.
Mathematician Karl Menger formulated his "Law Against Miserliness" which took these two forms: 1) Entities should not be reduced to the point of inadequacy; 2) It is vain to do with less what requires more. Menger showed that sometimes many concepts are united under a single concept, as for example the concept of "variable".
Bertrand Russell applied the principle of economy, especially in logic. The search for conceptual simplicity is a recurring theme in his works. He considered Ockham's razor as a heuristic maxim and also as a metaphysical principle. He always tried to minimize the number of entities in his theories.
In his work "Principia Mathematica" he refers to minimal vocabularies. In a minimal (or essential) vocabulary no term can be defined in terms of others. He also distinguishes between direct knowledge (what it is) and knowledge by description or indirect knowledge (what it is like), which is realized through direct knowledge.
He postulated (along with Wittgenstein) "logical atoms", the elementary constituents of propositions. For Russell, they are essentially the particulars and their predicates, which are also the atoms of our knowledge, the primitive constituents with which we apprehend the world. From the particulars and predicates, and adding to them the different logical connectors, it is possible to form propositions. The simplest proposition is the atomic proposition, consisting only of a particular and a predicate, of the form P(a), which simply says that a particular a has predicate P This proposition refers to a fact, which is the truth criterion of the proposition. The world is constituted of facts.
He was a proponent of neutral monism, the metaphysical view that ultimately reality is unified, neutral; it is neither physical nor mental.
Russell offered a particular version of Ockham's razor: "Whenever possible, replace constructions of known entities by inferences to unknown entities" [Russell, 1924]. The "unknown entities" referred to by Russell are to be interpreted as the simple, direct, undefinable entities, those that we cannot know by reason and can only intuit.
Wittgenstein associated the principle of Ockham's razor with the necessary and with the ontology-epistemology identity:
"If a sign is not necessary it lacks meaning. This is the meaning of Ockham's razor" (Tractatus, 3.328).
"Ockham's razor is naturally not an arbitrary rule, nor a rule justified by its practical success: it says that unnecessary sign units mean nothing" (Tractatus, 5.47321).
"In the proposition it must be possible to distinguish exactly the same thing as in the state of affairs it represents" (Tractatus, 4.04).
"The procedure of induction consists in our assuming the simplest law that can be harmonized with our experiences (Tractatus, 6.363).
For Einstein, it is undeniable that the goal of any theory is simplicity, but that this simplicity has a limit. In 1933 he wrote: "Everything should be made as simple as possible, but no simpler". This principle is often referred to as "Einstein's razor (or blade)".
For Karl Popper −philosopher and theorist of science−, the preference for a simple theory is not due to practical or aesthetic considerations. It is justified because a simple theory is more falsifiable than a complex theory, since the simple covers a greater empirical content, i.e., it applies to more cases.
The philosopher David Kellogg Lewis −considered one of the most important analytic philosophers of the twentieth century− is the proponent of the so-called "modal realism", a theory that holds that there are infinite possible worlds totally incommunicable with each other, without any link neither spatial, nor temporal or causal, and with the same ontological level.
This theory has been criticized because it clearly violates the principle of Ockham's razor: "Entities must not multiply without necessity".
Paul Grice −philosopher of language, known for his contributions to the theory of meaning, and originator of modern pragmatic theory− established a principle he called "modified Ockham's razor": "Meanings should not be multiplied beyond necessity". In a sentence that can have several meanings, one should always choose the simplest meaning.
Isaac Asimov, in "Nightfall" (a short story) writes: "We must make use of the sword of Thargola! The principle of parsimony. It was first enunciated by the medieval philosopher Thargola, who would express it as follows: 'We must cut with the sword every hypothesis that is not strictly necessary'". So we have 3 cutting instruments: Ockham's razor, Einstein's razor and Thargola's sword.
Ocam's razor and related concepts
Because of its general nature, the principle of Ockham's razor is related to several concepts that are also general:
Simplicity and complexity.
Simplicity resides in the deep. The complex always appears at the superficial level. The complexity of all that exists is only apparent, for the complex is obtained by combination of the simple. One cannot understand complexity without first understanding simplicity. Complexity is based on simplicity. Simplicity is universal, it affects everything because it is the common essence of all that exists.
"Primitive components must be taken as simple as possible if order and clarity are to be produced" (Frege).
"All great notions must be simple" (Goethe).
"In character, in manner, in style, in all things, the supreme excellence is simplicity" (Henry Wadsworth Longfellow).
"The aspects of things most important to us are hidden by their simplicity and ordinariness" (Wittgenstein).
"To analyze things one must start from the simplest to discover the laws of the complex" (Francis Bacon).
"The great ideas of complexity are simple" (Jorge Wagensberg).
"The great ideas of complexity are simple" (Jorge Wagensberg).
"The central task of a natural science is to make the amazing trivial: to show that complexity, properly viewed, is only a mask for simplicity; to find hidden pattern in apparent chaos" (Herbert Simon).
"The role of knowledge is to explain the visible complex by the invisible simple" (Jean Perrin).
The economy.
Ockham's razor is a particular case of the universal law of economy, operating at the internal (psychic) and external (physical) level.
Probability.
A simple theory is preferred to a complicated theory because of its greater probability. There have even been attempts to derive Ockham's razor principle from probability theory.
Logic and rational.
Ockham's razor is a decision logic, for it is about selecting the simplest alternative. For Ockham, the simple is the logical.
Intuition.
The simple appeals to intuition, to the awareness of the generic and the universal, to the mode of synthetic consciousness of the right hemisphere of the brain.
Common sense.
Common sense is of an integrative type, since it is associated with other characteristics. It appeals to the simplest, the most probable, the most understandable and intelligible, to conscience, logic and intuition.
"Common sense is the instinct of truth" (Max Jacob).
Philosophy.
The principle of conceptual economy provides a profound philosophy that unifies science and humanism.
"Ockham's razor is the supreme maxim of philosophy" (Bertrand Russell).
The general or universal.
The simpler a theory is, the more general or universal it is, the more domains it encompasses.
The aesthetic.
A simple theory is more aesthetic than a complicated theory. A fundamental aesthetic criterion is symmetry.
"Beauty lies in simplicity" (Einstein).
"Beauty lies in simplicity" (Einstein).
"The ideas that govern the world are as beautiful and as simple as possible" (Leizniz).
"I have deep faith that the principle of the universe is beautiful and simple" (Einstein).
"It is my opinion that everything must be based on a simple idea. And it is my opinion that this idea, once we have finally discovered it, will be so irresistible, so beautiful, that we will say to one another how it could have been otherwise" (John Archibald Wheeler).
Conscience.
The concept of the simple is subjective. There is no objective criterion for determining the degree of simplicity of a theory. Simplicity is more than a universal archetype: it is the common property of all primary archetypes. It connects the depths of ourselves with the depths of nature.
"In the deepest depth lies the simplest" (Ken Wilber).
The information.
A simple theory requires less information than a complex theory.
Energy.
Ockham's razor follows the principle of least effort. The principle of economy should be viewed in a general sense: as economy of effort (or law of least effort), both physically and mentally. It is about obtaining the maximum result with the minimum effort. It is a principle that we can abbreviate as "min-max", which unites the extremes of minimum and maximum. In this sense it is a principle of consciousness, as it unites the opposites.
Truth.
The simplest theory is the truest, the one that comes closest to the truth. The simplest explanations are truer than the complex ones.
For example, Copernicus' heliocentric theory is simpler than Ptolemy's geocentric model, which requires complicated trajectories of the planets. Both theories are correct, but one is much simpler than the other. Greater simplicity implies better understanding, less effort, greater awareness and greater truth. Complex theories require greater intellectual effort, obscure awareness, and depart from truth.
"Truth is always to be found in simplicity, and not in the multiplicity and confusion of things" (Newton).
"Nature is pleased with simplicity and is not clothed in the finery of superfluous things" (Newton).
"Simplex sigillum veri" (Simplicity is the seal of truth). Physics auditorium, University of Götinga.
"What is true is simple" (René Mey).
"You can recognize truth by its beauty and simplicity" (Richard Feynman).
"The language of truth must be simple and without artifice" (Seneca).
"Truth is always expressed with the utmost simplicity" (Paul Twitchell).
"The truth is simple. Why doesn't everyone know it. Because it is too simple. Because we have our preconceived notions about what truth is or what it should be" (Harold Klemp).
Mathematics.
The ultimate simplicity is abstract and has to be expressed in mathematical terms.
"Nature cannot be complicated. One must seek the source of truth in mathematical simplicity" (Einstein).
"All the effects of Nature are only the mathematical consequences of a small number of immutable laws" (Laplace).
"Nature is the realization of the simplest mathematical ideas" (Einstein).
Necessity.
Simplicity is what is necessary, what cannot be dispensed with. This is Wittgenstein's point of view, as we have mentioned above.
"When a truth is necessary, its reason can be found by analysis, resolving it into simpler ideas and truths until we arrive at those which are primitive" (Leibniz).
Abstraction.
Deep reality is simple and abstract. And the higher the level of abstraction, the greater the simplicity. At the limit are the primary archetypes, which are supreme simplicity and abstraction.
"It is said that one of the signs of the advancement of a civilization is its ability to communicate large amounts of information by encoding it in the smallest possible space, as an abstract symbol" (Freddy Silva).
Compression and compression.
A simple theory implies greater compression and understanding.
"To understand is to compress" (Gregory Chaitin).
"To understand is to compress" (Gregory Chaitin).
"The shorter the theory, the better we understand what it explains" (Gregory Chaitin).
"I love everything brief and find that, in general, the longer a paper is, the less there is in it" (Gödel).
"A theory is formulated as a simple principle or set of principles, expressed in a relatively short message" (Murray Gell-Mann).
"The simplest questions can push the limits of human knowledge" (Richard Feynman).
Freedom.
The simple provides greater freedom. The complex restricts and limits.
Practicality (or pragmatism).
A simple theory is more practical, easier to apply than a complex theory.
"Everything that is complex is not useful. Everything that is useful is simple" (Mikhail Kalashnikov).
Creativity.
Simplicity allows more things to be related, which facilitates creativity.
"Simplification facilitates discovery" (Gödel).
"In the future, as in the past, the great ideas will be the simplifying ideas" (André Weil).
"The most fundamental and important discoveries come from very simple concepts" (David Lister Grimsby).
"By great idea I mean a simple concept of immense scope, a seminal idea that grows into a great oak tree whose branches would be its many applications" (Peter Atkins).
Perfection (or the ideal).
A simple theory tends toward or approaches the ideal and perfect.
The universal paradigm.
The principle of economy is a unified way of contemplating reality. Virtually all disciplines of knowledge have benefited from this principle, which can be considered a universal paradigm.
The heuristic.
Ockham's razor is a heuristic principle for decision making. It is also applicable during the process of searching for a solution to a problem, always thinking as simply as possible. Therefore, the principle of conceptual economy is a universal heuristic, as it can be applied in all kinds of situations.
Heuristics is a practical or informal art, technique, or procedure for solving problems. It involves intuition, synthesis and creativity. Heuristic ability is a characteristic trait of humans. The term "heuristic" comes from Greek and means "to find, invent or discover". It shares etymology with "eureka". The popularization of this concept is due to George Pólya, with his book "How to solve it".
The transcendental or spiritual.
Ockham's razor is even considered a transcendental or spiritual principle.
"When the solution is simple, God is answering" (Einstein).
"God always takes the simplest path" (Einstein).
"God has chosen the most perfect world, the only one that is at the same time the simplest in hypothesis and the richest in phenomena" (Leibniz).
The Principle of Economy in Nature
Nature follows the principle of economy, makes use of as few resources as possible and always uses the simplest laws.
It always follows the easiest, the shortest, the shortest way.
It works effortlessly, without resistance, harmoniously.
Always seeks balance and stability.
It is not analytical; it is synthetic, spontaneous, holistic and intuitive.
It is not linear and uniform; it compresses. The consumption of space is the minimum possible. The universe itself is compressed.
It tends to the simplest and most harmonious forms, such as symmetrical, spherical and spiral shapes.
Some quotes:
"Nature always proceeds from the simplest or most economical form" (Aristotle).
"Nature expresses itself in a simple way" (Sesha).
"Many scientists believe as fervently as I do in the principle that all of nature is governed by mathematical laws of great simplicity and elegance." (C.A.R. Hoare).
"If the law is simple, then it is a genuine law of nature" (Leibniz).
"Any fundamental description of the universe must be simple" (Max Tegmark).
"Nature usually opts for the simplest and most economical solution" (Marius Livy).
"Nature is simple, and does not indulge in superfluous causes for things" (Newton).
"Nature uses as little as possible of everything" (Kepler).
"Nature is pleased with simplicity" (Newton).
For example:
It manifests itself in the way sunflowers face the sun; in the way water flows following the path of least resistance; in the way a lightning bolt follows toward the earth; in how seagulls take advantage of air currents to expend the least amount of energy; etc.
Animals explore all possibilities at every moment, at an intuitive level, and choose the most efficient and simplest action.
Soap bubbles adopt the spherical shape, occupying the minimum surface area.
The logarithmic spiral of natural shapes, using the golden ratio, which provides maximum compression.
The human brain is the expression of maximum physical compression.
The human mind follows the universal principle of economy by minimizing all effort in performing any task. For example, a new information or concept is acquired through associations of already existing concepts.
The Principle of Economy in Science
The principle of conceptual economy is a basic principle of the scientific method. It is the foundation of so-called "methodological reductionism", which also includes ontological and epistemological features.
Science always advances in the direction of simplicity and unification of concepts. Science prefers the simplest explanation that is consistent with the experimental data available at any given time. But this simplest explanation may later be rejected when new data become available.
An illustrative example is that of planetary orbits. Copernicus postulated that the orbits were circular, the simplest hypothesis. Kepler, with more data, deduced that the orbits were elliptical, with the Sun at one of their foci, which led him to deduce his famous 3 laws. But, in any case, Copernicus' theory was a good approximation to the truth.
Some quotes on science and simplicity:
"Science may be defined as the art of systematic supersimplification" (Karl Popper).
"Science is the simplest and most compressed form of understanding reality" (Jorge Wagensberg).
"The purpose of science is to seek the simplest explanation of complex facts" (Alfred North Whitehead).
"Most of the fundamental ideas of science are essentially simple, and can, as a rule, be expressed in language understandable to all" (Einstein).
"The simpler a theory is, the greater its value" (Bertrand Russell).
"We seek the simplest possible scheme of thought that can bring together all observable facts" (Einstein).
"The best things in science are both beautiful and simple" (John Gribbin).
"Science is really the pursuit of simplicity." (Claude A. Villee).
"The role of science is to produce economy of thought, as the machine produces economy of labor" (Henri Poincaré).
Some examples of the tendency and triumph of simplicity or the principle of economy in the history of science are:
Physics
The simple heliocentric theory of Copernicus, as opposed to the complicated trajectories of the planets required by Ptolemy's geocentric model. Galileo, in adhering to the Copernican theory, also advocated the simpler hypothesis of the heavens.
Newton's theory of gravitation, which made it possible to explain by the same physical laws the motion of terrestrial and celestial bodies.
All physical theory is built on only 3 concepts: mass, length and time. All physical laws are expressed by combination of these quantities (or physical dimensions).
Attempts have been made to derive known laws from simplicity criteria. The paradigmatic example is the theorem of Emmy Noether (1915) −considered one of the most beautiful and profound in science−, which merged symmetry and conservation as two facets of the same property: to each abstract symmetry in a physical system corresponds a conservation law and vice versa. Symmetry is nature's way of achieving maximum economy of resources and maximum simplicity. Symmetry is a key feature of quantum phenomena, superstring theory and relativity theory. Heisenberg said: "In the beginning was symmetry".
In modern physics, the principle of economy is formalized as "the principle of least action" (PLA). Action is defined in physics as the product of energy times time. The time evolution of a physical system occurs in such a way that the action is the minimum possible. The action is defined as the integral between t1 and t2 ∫(Ec − Ep)dt, where t1 and t2 are the initial and final times, and Ec and Ep are the kinetic and potential energies at instant tEc+Ep remains constant along its trajectory, by the principle of conservation of energy.
The first formulation of the PLA is due to Pierre-Louis Moreau de Maupertuis, in 1744. "Nature is economic in all her actions". "Nature always works by employing the least effort or energy to achieve a given end".
Pierre Louis Maupertuis mathematically formalized the principle of least effort under the name "law of economy of nature" (Lex Parsimoniae). In his "Essay on Cosmology" he states, "This is the principle so wise, so worthy of the Supreme Being: in whatever change takes place in nature, the sum of action expended in that change will be the smallest possible".
Euler also studied it the same year as Maupertuis. But it seems that Leibniz had arrived at the same result years earlier.
Fermat discovered that the laws of reflection and refraction of light follow the minimum time path. A ray of light always selects the path from an initial point to its destination in such a way that the time spent is the minimum possible.
The PLA is a universal physical principle, a "deep principle", the most important principle in physics. It applies to classical mechanics, electromagnetic theory, thermodynamics, fluid mechanics, general relativity theory, quantum mechanics, etc. Maupertuis considered it a universal principle, beyond even physics, used by the Supreme Being in his creations.
The PLA led to the development of the Lagrangian formulation of classical mechanics, introduced by Joseph-Louis de Lagrange in 1788. It reformulated classical physics, giving rise to analytical mechanics. A physical system is described by a set of variables or parameters that define the configuration space. The state of the system at an instant t is given by the values of the variables at that instant. The system evolves over time and describes a trajectory in the configuration space. The function describing the action at instant t is a function of the variables, their derivatives and time, and is called the "lagrangian". The trajectory of a physical system is obtained by minimizing the action, which is the integral of the lagrangian in time. The Lagrangian (or Euler-Lagrange) equations of motion are thus obtained.
The advantages of the Lagrangian formulation are:
It considerably simplifies the resolution of physical problems. From the deep level of general or universal principles, everything is simplified. In particular, it makes it possible to obtain simply and directly the equations of motion of a given physical system.
It is independent of the coordinate system used. It is invariant under coordinate transformations. And it is applicable to non-inertial reference systems, unlike Newton's equations of motion.
No force appears explicitly in the equations of motion. This issue is crucial in the theory of general relativity.
It allows to derive Newton's equations of motion and those of the theory of general relativity. It also allows to derive the Schrödinger equation of quantum mechanics by considering all possible trajectories of a quantum entity. This was discovered by Feynman in 1948.
The first law of thermodynamics or law of conservation of energy is an example of the law of economy: "In a closed system, energy is neither created nor destroyed; it is only transformed".
The second law of thermodynamics (or Carnot-Clausius principle) is also an example of the law of economy: the entropy of a closed (or isolated) system grows or remains constant. Entropy measures the degree of disorder of a system, so this law says that closed systems tend to disorder. Disorder is simpler than order. Therefore, closed systems seek maximum simplicity, homogeneity and non-differentiation, which is where they find equilibrium. The increase in entropy indicates the direction of the arrow of time. Order implies more information than disorder. Therefore, nature evolves towards states of less information, to states of maximum simplicity, of maximum homogeneity, of lower energy, with the maximum number of common properties.
William Rowan Hamilton generalized the PLA in the form of the "Principle of Stationary Action" (PAE), which states that the process of change of a physical quantity adopts an extreme value (minimum or maximum). The action can be not only minimum, but also maximum; minimum and maximum must be considered in a relative sense and not absolute. This is especially applicable in general relativity.
The PAE is a variational principle. The calculus of variations consists of finding maxima and minima of continuous functions defined in a configuration space. It uses the equation δ∫Ldt = 0, where L is the Lagrangian. The integral is between t1 and t2
According to the Weber-Fechner psychophysical law, our senses are organized to detect relative differences between stimuli, rather than the absolute intensity of a stimulus. Perception does not act in a linear fashion. The linear is not perceived, it does not become conscious. We need qualitative leaps. That is why the Richter scale of earthquakes is logarithmic. All perception implies work and, as such, has to be subjected to the principle of minimum action.
Stephen Hawking credits the discovery of quantum mechanics to Ockham and his penknife. Physics seeks a "theory of everything," a simple unifying theory.
Mathematics
Mathematics has tried to base itself on a few general or universal axioms. The most prominent examples have been Hilbert's universal axiomatic program and the logical program (also axiomatic) of Russell and Whitehead's Principia Mathematica This claim was shown to be impossible when Gödel presented his famous incompleteness theorem for formal axiomatic systems.
Informatics
In the face of the increasing complexity of computer systems, there is the principle called "KISS" (Keep It Small and Simple), also interpreted as "Keep It Simple, Stupid".
Linguistics
In 1995, Chomsky presented his "Minimalist Program", a general conceptual framework for general language theory research, which attempts to explain linguistic phenomena with as few conceptual resources as possible.
Biology
A simple evolutionary algorithm −natural selection− suffices to explain evolution, with no need to resort to supernatural explanations. It is the simplest principle capable of explaining complexity.
Philosophy
The materialist philosophy −which holds that everything is matter− applies, consciously or unconsciously, Ockham's razor. The opposite view also applies it by holding that everything is consciousness, and that matter is manifested consciousness.
The Philosophy of Simplicity
The two types of simplicity
A distinction must be made between conceptual simplicity and definitional simplicity:
Conceptual simplicity is based on the search for the simplest concepts. Here, the principle of Ockham's razor applies: among the different possible alternatives, the simplest must be selected.
Definitional simplicity is to go beyond conceptual simplicity and try to express concepts in terms of some minimal concepts, and to express the rest of the concepts in a definitional way. Paradoxically, definitional simplicity implies greater complexity. When the line of conceptual simplicity is crossed, everything becomes more complex and impractical. Einstein's razor principle applies here: "Make it as simple as possible, but not simpler", which we could translate as "Make it as simple as possible at the conceptual level, but don't make it simpler through the definitional level because you will complicate it and it will be impractical at the human level".
Ockham's razor is a horizontal principle, of selection among several alternatives. Einstein's razor is a vertical principle, of not going beyond the conceptual.
Five significant examples.
Propositional logic.
At the conceptual level we choose the concepts of "negation" and "conjunction" (or alternatively, "negation" and "disjunction"). At the definitional level we can choose the unique operation called "Peirce's arrow" or "Quine's dagger: NOR (Not or) = p↓q = ¬(p∨q) = ¬p∧¬q), from which the rest of the logical operations can be defined, but this definition is complex and unintuitive. We can also choose Sheffer's slash: NAND (Not and) = p|q = ¬(p∧q) = (¬p∨¬q).
Category theory.
Category theory is based on a single generic type concept: morphism, which admits many particular interpretations (function, process, link, link, sequence, rule, etc.). The theory becomes enormously complex.
Principia Mathematica, by Russell and Whitehead.
Russell and Whitehead opted for logicism in wanting to ground mathematics in logic. The result was a failure due to the resulting complexity. Moreover Gödel showed that in his famous 1931 paper ("On formally undecidable sentences in Principia Mathematica and related systems") that it is not possible to ground mathematics from mathematics itself: with a formal axiomatic system. The grounding has to come from a higher level of abstraction.
The Turing machine.
The Turing machine (TM) is a theoretical machine that is based on non-generic and implementer-like concepts, which limits expressiveness. The TM does not follow Einstein's razor principle because it goes beyond conceptual simplicity because programming algorithms (even the most simple ones) with TM primitives becomes very complex and laborious, impractical. Precisely one way to detect whether one has gone beyond conceptual simplicity is to see if the subject gets complicated.
MT has been the foundation and inspiration for the creation of computers, of practical computing, because of its simplicity of implementation. But the theoretical foundation of a science has to be simple and not limited a priori. Therefore, MT cannot be the foundation of theoretical computer science.
Church's lambda calculus.
It reduces everything to functional expressions, functions that return functions as results. This leads to complexity when expressing concepts such as numbers, predicates, logical operations and arithmetic operations. Also functions are not recursive and require an artificial formalism: the fixed-point AND combinator (Curry's combinator).
One must follow the principles of Ockham's razor and Einstein's razor. Apply supreme conceptual simplicity without going towards further abstraction by means of the definitional.
Definitional simplicity is only justified in the case of physical implementations, as in the case of MT. Turing himself applied it using NAND gates for the simulation of neural networks. This topic is described in the paper "Intelligent Machinery, A Heretical Theory", one of the pioneering papers in artificial intelligence, written in 1948, but published long after his death.
Abstraction vs. simplicity
We perceive reality as fragmented, dispersed and complex. But complexity is only apparent, because behind it hides simplicity. What appears complex (at the superficial level) is really the expression or manifestation of the underlying simplicity (at the deep level). And what appears to be separate at a superficial level is actually unified at a deep level.
The essence of reality is that which is common to all, that which provides unity to diversity. This unity is permanent, it never changes, it is outside of space and time, it is the supreme stability. Everything has come out of unity and tends to return to it in search of equilibrium and stability. Unity is the state that grounds everything, the level from which all possibilities can manifest.
To simplify reality the mechanism of abstraction is applied. Abstraction dispenses with the accessory and focuses on the fundamental and generic mechanisms.
Abstraction makes it possible to extract common properties in the various domains. For example, the mathematical concept of group is a powerful abstraction: the common property is the possibility of combining two elements of a set to obtain another element of that same set.
Abstraction allows us to contemplate things in a broader context, with greater awareness and greater simplicity. The essence of reality lies in supreme conceptual simplicity. Abstraction should go no further.
Simplicity in computing
Computers have changed everything. They have introduced a new paradigm, a new way of looking at reality. What computing has taught us is the power of simplicity:
A small number of instructions make it possible to produce (by combinatorics) potentially an infinity of possible programs. Computers were born based on an abstraction: a theoretical machine which today we call "universal Turing machine" that allowed to define and formalize the concept of computation. The key ideas were two: 1) the simplicity of its computational model; 2) the concept of a program stored in memory, which made it possible to make flexible something that could not be flexible: the hardware.
With only two values (0 and 1) we can represent all kinds of contents: numerical data, texts, sounds, videos, graphics, etc. In this sense, boundaries have been broken down, although a digital file needs an external interpretation. A digital file has no semantics, it is just a sequence of zeros and ones. Although semantics is always present because semantics is always at a higher level than syntax and underlies it. Semantics in this case is an entity capable of adopting or manifesting itself in two states, which we symbolize as 0 and 1, but any other pair of symbols could be used.
Thanks to this simplicity, computers have become a universal tool for science, as it allows modeling all kinds of phenomena of nature and the processes of calculation and reasoning.
The paradoxes of simplicity
Paradoxically, the search for the simple is a very complex task because the simple is hidden behind the superficial manifestations, which are apparently complex.
"Simplicity is the ultimate sophistication" (Leonardo da Vinci).
"There is nothing as complex as simplicity" (Kai Krause).
"There is nothing so complex as simplicity" (Kai Krause).
"The aspects of things most important to us are hidden by their simplicity and ordinariness" (Wittgenstein).
"Ironically, even the simplest things can be difficult to understand, because they are abstract" (John Baez).
"The simple can be harder than the complex. You have to work very hard to clear your mind and make things simple" (Steve Jobs).
"The best thing is always the simplest thing, the bad thing is that to be simple you have to think hard" (John Steinbeck).
"To arrive at the simplest truth, as Newton knew and practiced, requires years of contemplation. No activity. No reasoning. No calculation. No busy behavior of any kind. No reading. No effort. No thinking. Simply holding in one's mind what one needs to know" (George Spencer Brown).
"What is read without any effort has always been written with great effort" (Jardiel Poncela).
Another paradoxical aspect of simplicity is that the criterion for determining which is the simplest theory is a question that can be difficult to resolve because simplicity is subjective. There is no general definition of what is simple, and what is also simple. This is justified by the close relationship between simplicity and consciousness. Consciousness cannot be explained and neither can the simple; one must turn to intuition.
"It is not simple to define 'simple'" (Murray Gell-Mann).
However, in some specific contexts or domains there does exist an objective criterion of simple. For example, in the area of computational information, the algorithmic complexity of a piece of information is defined as the length of the shortest program that produces that information. The shorter the program length, the greater the simplicity. Solomonoff is considered the father of the concept of algorithmic complexity, a concept formalized by Andrei Kolmogorov and extended by Gregory Chaitin, all during the 1960s.
The final paradox is that the more we broaden the domain to be considered, the greater the simplicity. A "theory of everything" is simpler than a theory of a particular domain. Expressed in other words: with less you get more. In the limit, a "theory of everything" should be extraordinarily simple. And if a "theory of everything" also includes possible worlds, then supreme simplicity would be achieved.
"Less is more" (Ludwig Mies Van der Rohe).
"To know everything it is necessary to know very little; but to know that very little one must first know a great deal" (Gurdjieff).
"More and more apparently diverse phenomena are explained by fewer and fewer underlying principles" (John C. Taylor).
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