Metaphysical naturalism

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https://en.wikipedia.org/wiki/Metaphysical_naturalism 

Metaphysical naturalism (also called ontological naturalism, philosophical naturalism and antisupernaturalism) is a philosophical worldview which holds that there is nothing but natural elements, principles, and relations of the kind studied by the natural sciences. Methodological naturalism is a philosophical basis for science, for which metaphysical naturalism provides only one possible ontological foundation. Broadly, the corresponding theological perspective is religious naturalism or spiritual naturalism. More specifically, metaphysical naturalism rejects the supernatural concepts and explanations that are part of many religions.

Definition

In Carl Sagan’s words: "The Cosmos is all that is or ever was or ever will be."

According to Arthur C. Danto, naturalism, in recent usage, is a species of philosophical monism according to which whatever exists or happens is natural in the sense of being susceptible to explanation through methods which, although paradigmatically exemplified in the natural sciences, are continuous from domain to domain of objects and events. Hence, naturalism is polemically defined as repudiating the view that there exists or could exist any entities which lie, in principle, beyond the scope of scientific explanation.

Regarding the vagueness of the general term "naturalism", David Papineau traces the current usage to philosophers in early 20th century America such as John Dewey, Ernest Nagel, Sidney Hook, and Roy Wood Sellars: "So understood, 'naturalism' is not a particularly informative term as applied to contemporary philosophers. The great majority of contemporary philosophers would happily accept naturalism as just characterized—that is, they would both reject 'supernatural' entities, and allow that science is a possible route (if not necessarily the only one) to important truths about the 'human spirit'." Papineau remarks that philosophers widely regard naturalism as a "positive" term, and "few active philosophers nowadays are happy to announce themselves as 'non-naturalists'", while noting that "philosophers concerned with religion tend to be less enthusiastic about 'naturalism'" and that despite an "inevitable" divergence due to its popularity, if more narrowly construed, (to the chagrin of John McDowell, David Chalmers and Jennifer Hornsby, for example), those not so disqualified remain nonetheless content "to set the bar for 'naturalism' higher."

Philosopher and theologian Alvin Plantinga, a well-known critic of naturalism in general, comments: "Naturalism is presumably not a religion. In one very important respect, however, it resembles religion: it can be said to perform the cognitive function of a religion. There is that range of deep human questions to which a religion typically provides an answer ... Like a typical religion, naturalism gives a set of answers to these and similar questions".

Science and naturalism

Main article: Uniformitarianism

Metaphysical naturalism is the philosophical basis of science as described by Kate and Vitaly (2000). "There are certain philosophical assumptions made at the base of the scientific method – namely, 1) that reality is objective and consistent, 2) that humans have the capacity to perceive reality accurately, and that 3) rational explanations exist for elements of the real world. These assumptions are the basis of naturalism, the philosophy on which science is grounded. Philosophy is at least implicitly at the core of every decision we make or position we take, it is obvious that correct philosophy is a necessity for scientific inquiry to take place." Steven Schafersman, agrees that methodological naturalism is "the adoption or assumption of philosophical naturalism within scientific method with or without fully accepting or believing it ... science is not metaphysical and does not depend on the ultimate truth of any metaphysics for its success, but methodological naturalism must be adopted as a strategy or working hypothesis for science to succeed. We may therefore be agnostic about the ultimate truth of naturalism, but must nevertheless adopt it and investigate nature as if nature is all that there is."

Various associated beliefs

Contemporary naturalists possess a wide diversity of beliefs within metaphysical naturalism. Most metaphysical naturalists have adopted some form of materialism or physicalism.

Natural sciences

According to metaphysical naturalism, if nature is all there is, the Big Bang, the formation of the Solar System, abiogenesis, and the processes involved in evolution would all be natural phenomena without supernatural influences.

The mind is a natural phenomenon

Metaphysical naturalists do not believe in a soul or spirit, nor in ghosts, and when explaining what constitutes the mind they rarely appeal to substance dualism. If one's mind, or rather one's identity and existence as a person, is entirely the product of natural processes, three conclusions follow according to W. T. Stace. Cognitive sciences are able to provide accounts of how cultural and psychological phenomena, such as religion, morality, language, and more, evolved through natural processes. Consciousness itself would also be susceptible to the same evolutionary principles that select other traits.

Utility of intelligence and reason

Metaphysical naturalists hold that intelligence is the refinement and improvement of naturally evolved faculties. Naturalists believe anyone who wishes to have more beliefs that are true than are false should seek to perfect and consistently employ their reason in testing and forming beliefs. Empirical methods (especially those of proven use in the sciences) are unsurpassed for discovering the facts of reality, while methods of pure reason alone can securely discover logical errors.

View on the soul

According to metaphysical naturalism, immateriality being unprocedural and unembodiable, is not differentiable from nothingness. The immaterial nothingness of the soul, being a non-ontic state, is not compartmentalizable nor attributable to different persons and different memories, it is non-operational and it (nothingness) cannot be manifested in different states in order it represents information.

Arguments for metaphysical naturalism

Argument from physical minds

In his critique of mind–body dualism, Paul Churchland writes that it is always the case that the mental substance and/or properties of the person are significantly changed or compromised via brain damage. If the mind were a completely separate substance from the brain, how could it be possible that every single time the brain is injured, the mind is also injured? Indeed, it is very frequently the case that one can even predict and explain the kind of mental or psychological deterioration or change that human beings will undergo when specific parts of their brains are damaged. So the question for the dualist to try to confront is how can all of this be explained if the mind is a separate and immaterial substance from, or if its properties are ontologically independent of, the brain.

Modern experiments have demonstrated that the relation between brain and mind is much more than simple correlation. By damaging, or manipulating, specific areas of the brain repeatedly under controlled conditions (e.g. in monkeys) and reliably obtaining the same results in measures of mental state and abilities, neuroscientists have shown that the relation between damage to the brain and mental deterioration is likely causal. This conclusion is further supported by data from the effects of neuro-active chemicals (e.g., those affecting neurotransmitters) on mental functions, but also from research on neurostimulation (direct electrical stimulation of the brain, including transcranial magnetic stimulation).

Critics such as Edward Feser and Tyler Burge have described these arguments as "neurobabble", and consider them as flawed or as being compatible with other metaphysical ideas like Thomism. According to the philosopher Stephen Evans:

We did not need neurophysiology to come to know that a person whose head is bashed in with a club quickly loses his or her ability to think or have any conscious processes. Why should we not think of neurophysiological findings as giving us detailed, precise knowledge of something that human beings have always known, or at least could have known, which is that the mind (at least in this mortal life) requires and depends on a functioning brain? We now know a lot more than we used to know about precisely how the mind depends on the body. However, that the mind depends on the body, at least prior to death, is surely not something discovered in the 20th century.

Argument from cognitive biases

In contrast with the argument from reason or evolutionary argument against naturalism, it can be argued that cognitive biases are better explained by natural causes than as the work of God.

Arguments against

Arguments against metaphysical naturalism include the following examples:

Argument from reason

Main article: Argument from reason

Philosophers and theologians such as Victor Reppert, William Hasker, and Alvin Plantinga have developed an argument for dualism dubbed the "argument from reason". They credit C.S. Lewis with first bringing the argument to light in his book Miracles; Lewis called the argument "The Cardinal Difficulty of Naturalism", which was the title of chapter three of Miracles.

The argument postulates that if, as naturalism entails, all of our thoughts are the effect of a physical cause, then we have no reason for assuming that they are also the consequent of a reasonable ground. However, knowledge is apprehended by reasoning from ground to consequent. Therefore, if naturalism were true, there would be no way of knowing it (or anything else), except by a fluke.

Through this logic, the statement "I have reason to believe naturalism is valid" is inconsistent in the same manner as "I never tell the truth." That is, to conclude its truth would eliminate the grounds from which it reaches it. To summarize the argument in the book, Lewis quotes J. B. S. Haldane, who appeals to a similar line of reasoning:

If my mental processes are determined wholly by the motions of atoms in my brain, I have no reason to suppose that my beliefs are true ... and hence I have no reason for supposing my brain to be composed of atoms.

— J. B. S. Haldane, Possible Worlds, page 209

In his essay "Is Theology Poetry?", Lewis himself summarises the argument in a similar fashion when he writes:

If minds are wholly dependent on brains, and brains on biochemistry, and biochemistry (in the long run) on the meaningless flux of the atoms, I cannot understand how the thought of those minds should have any more significance than the sound of the wind in the trees.

— C. S. Lewis, The Weight of Glory and Other Addresses, page 139

But Lewis later agreed with Elizabeth Anscombe's response to his Miracles argument. She showed that an argument could be valid and ground-consequent even if its propositions were generated via physical cause and effect by non-rational factors. Similar to Anscombe, Richard Carrier and John Beversluis have written extensive objections to the argument from reason on the untenability of its first postulate.

Evolutionary argument against naturalism

Main article: Evolutionary argument against naturalism

Notre Dame philosophy of religion professor and Christian apologist Alvin Plantinga argues, in his evolutionary argument against naturalism, that the probability that evolution has produced humans with reliable true beliefs, is low or inscrutable, unless their evolution was guided, for example, by God. According to David Kahan of the University of Glasgow, in order to understand how beliefs are warranted, a justification must be found in the context of supernatural theism, as in Plantinga's epistemology. (See also Supernormal stimuli.)

Plantinga argues that together, naturalism and evolution provide an insurmountable "defeater for the belief that our cognitive faculties are reliable", i.e., a skeptical argument along the lines of Descartes' evil demon or brain in a vat.

Take philosophical naturalism to be the belief that there aren't any supernatural entities—no such person as God, for example, but also no other supernatural entities, and nothing at all like God. My claim was that naturalism and contemporary evolutionary theory are at serious odds with one another—and this despite the fact that the latter is ordinarily thought to be one of the main pillars supporting the edifice of the former. (Of course I am not attacking the theory of evolution, or anything in that neighborhood; I am instead attacking the conjunction of naturalism with the view that human beings have evolved in that way. I see no similar problems with the conjunction of theism and the idea that human beings have evolved in the way contemporary evolutionary science suggests.) More particularly, I argued that the conjunction of naturalism with the belief that we human beings have evolved in conformity with current evolutionary doctrine... is in a certain interesting way self-defeating or self-referentially incoherent.

— Alvin Plantinga, "Introduction" in Naturalism Defeated?: Essays on Plantinga's Evolutionary Argument Against Naturalism

Branden Fitelson of the University of California, Berkeley and Elliott Sober of the University of Wisconsin–Madison argue that Plantinga must show that the combination of evolution and naturalism also defeats the more modest claim that "at least a non-negligible minority of our beliefs are true", and that defects such as cognitive bias are nonetheless consistent with being made in the image of a rational God. Whereas evolutionary science already acknowledges that cognitive processes are unreliable, including the fallibility of the scientific enterprise itself, Plantinga's hyperbolic doubt is no more a defeater for naturalism than it is for theistic metaphysics founded upon a non-deceiving God who designed the human mind: "[neither] can construct a non-question-begging argument that refutes global skepticism." Plantinga's argument has also been criticized by philosopher Daniel Dennett and independent scholar Richard Carrier who argue that a cognitive apparatus for truth-finding can result from natural selection.

Argument from first-person perspectives

Christian List argues that the existence of first-person perspectives, i.e., one existing as oneself and not as someone else, refutes physicalism. He argues that since first-personal facts cannot supervene on physical facts, this refutes not only physicalism, but also most forms of dualism that have purely third-personal metaphysics. List also argues that there is a "quadrilemma" for theories of consciousness: that at most three of the following metaphysical claims can be true: "first-person realism", "non-solipsism", "non-fragmentation", and "one world"—and thus at least one of them must be false. He has proposed a model he calls the "many-worlds theory of consciousness" to reconcile the subjective nature of consciousness without lapsing into solipsism. These ideas are related to the vertiginous question proposed by Benj Hellie.

Scientific realism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Scientific_realism 

Scientific realism is the philosophical view that the universe described by science (including both observable and unobservable aspects) exists independently of our perceptions, and that verified scientific theories are at least approximately true descriptions of what is real. Scientific realists typically assert that science, when successful, uncovers true (or approximately true) knowledge about nature, including aspects of reality that are not directly observable.

Within philosophy of science, this view is often an answer to the question "how is the success of science to be explained?" The discussion on the success of science in this context centers primarily on the status of unobservable entities apparently talked about by scientific theories. Generally, those who are scientific realists assert that one can make valid claims about unobservables (viz., that they have the same ontological status) as observables, as opposed to instrumentalism.

In a 2020 PhilPapers Survey 72% of academic philosophers favored scientific realism vs only 15% favoring antirealism.

Main features

Scientific realism involves two basic positions:

• Ideal-theory thesis. It offers a set of criteria for an ideal scientific theory; one that science aims to approximate.

• Convergence thesis. It maintains that, in at least some well-established domains, the theories scientists now accept already approximate that ideal and will do so ever more closely as inquiry progresses.

Realists often adopt this thesis selectively; one may be realist about mature fields such as fundamental physics while remaining agnostic about less-settled areas.

An ideal scientific theory, on the realist view, satisfies three interconnected commitments:

• Semantic: Its central claims are either true or false because they purport to describe the real world.
• Metaphysical: The entities it posits (e.g. electrons, genes) exist objectively and independent of the mind.
• Epistemic: We have good reason to believe many of those claims because the theory's explanatory and predictive success would be improbable if they were not at least approximately true.

Taken together, the first two commitments imply that an ideal theory makes definite assertions about genuinely exists entities, while the third justifies believing a significant subset of those assertions.

Realists therefore argue that science progresses: later theories generally answer more questions or do so more accurately, thus moving closer, though never necessarily reaching, the ideal of a literally true account of nature.

Characteristic claims

The following claims are typical of those held by scientific realists. Due to the wide disagreements over the nature of science's success and the role of realism in its success, a scientific realist would agree with some but not all of the following positions.

• The best scientific theories are at least partially true.
• The best theories do not employ central terms that are non referring expressions.
• To say that a theory is approximately true is sufficient explanation of the degree of its predictive success.
• The approximate truth of a theory is the only explanation of its predictive success.
• Even if a theory employs expressions that do not have a reference, a scientific theory may be approximately true.
• Scientific theories are in a historical process of progress towards a true account of the physical world.
• Scientific theories make genuine, existential claims.
• Theoretical claims of scientific theories should be read literally and are definitively either true or false.
• The degree of the predictive success of a theory is evidence of the referential success of its central terms.
• The goal of science is an account of the physical world that is literally true. Science has been successful because this is the goal that it has been making progress towards.

History

Scientific realism is related to much older philosophical positions including rationalism and metaphysical realism. However, it is a thesis about science developed in the twentieth century. Portraying scientific realism in terms of its ancient, medieval, and early modern cousins is at best misleading.

Scientific realism is developed largely as a reaction to logical positivism. Logical positivism was the first philosophy of science in the twentieth century and the forerunner of scientific realism, holding that a sharp distinction can be drawn between theoretical terms and observational terms, the latter capable of semantic analysis in observational and logical terms.

Logical positivism encountered difficulties with:

• The verificationist theory of meaning—see Hempel (1950).
• Troubles with the analytic-synthetic distinction—see Quine (1950).
• The theory-ladenness of observation—see Hanson (1958) Kuhn (1970) and Quine (1960).
• Difficulties moving from the observationality of terms to observationality of sentences—see Putnam (1962).
• The vagueness of the observational-theoretical distinction—see G. Maxwell (1962).

These difficulties for logical positivism suggest, but do not entail, scientific realism, and led to the development of realism as a philosophy of science.

Realism became the dominant philosophy of science after positivism. Bas van Fraassen in his book The Scientific Image (1980) developed constructive empiricism as an alternative to realism. He argues against scientific realism that scientific theories do not aim for truth about unobservable entities. Responses to van Fraassen have sharpened realist positions and led to some revisions of scientific realism.

Arguments for and against scientific realism

No miracles argument

One of the main arguments for scientific realism centers on the notion that scientific knowledge is progressive in nature, and that it is able to predict phenomena successfully. Many scientific realists (e.g., Ernan McMullin, Richard Boyd) think the operational success of a theory lends credence to the idea that its more unobservable aspects exist, because they were how the theory reasoned its predictions. For example, a scientific realist would argue that science must derive some ontological support for atoms from the outstanding phenomenological success of all the theories using them.

Arguments for scientific realism often appeal to abductive reasoning or "inference to the best explanation" (Lipton, 2004). For instance, one argument commonly used—the "miracle argument" or "no miracles argument"—starts out by observing that scientific theories are highly successful in predicting and explaining a variety of phenomena, often with great accuracy. Thus, it is argued that the best explanation—the only explanation that renders the success of science to not be what Hilary Putnam calls "a miracle"—is the view that our scientific theories (or at least the best ones) provide true descriptions of the world, or approximately so.

Bas van Fraassen replies with an evolutionary analogy: "I claim that the success of current scientific theories is no miracle. It is not even surprising to the scientific (Darwinist) mind. For any scientific theory is born into a life of fierce competition, a jungle red in tooth and claw. Only the successful theories survive—the ones which in fact latched on to actual regularities in nature." (The Scientific Image, 1980)

It has been argued that the no miracles argument commits the base rate fallacy.

Pessimistic induction

Pessimistic induction, one of the main arguments against realism, argues that the history of science contains many theories once regarded as empirically successful but which are now believed to be false. Additionally, the history of science contains many empirically successful theories whose unobservable terms are not believed to genuinely refer. For example, the effluvium theory of static electricity (a theory of the 16th Century physicist William Gilbert) is an empirically successful theory whose central unobservable terms have been replaced by later theories.

Realists reply that replacement of particular realist theories with better ones is to be expected due to the progressive nature of scientific knowledge, and when such replacements occur only superfluous unobservables are dropped. For example, Albert Einstein's theory of special relativity showed that the concept of the luminiferous ether could be dropped because it had contributed nothing to the success of the theories of mechanics and electromagnetism. On the other hand, when theory replacement occurs, a well-supported concept, such as the concept of atoms, is not dropped but is incorporated into the new theory in some form. These replies can lead scientific realists to structural realism.

Constructivist epistemology

Social constructivists might argue that scientific realism is unable to account for the rapid change that occurs in scientific knowledge during periods of scientific revolution. Constructivists may also argue that the success of theories is only a part of the construction.

However, these arguments ignore the fact that many scientists are not realists. During the development of quantum mechanics in the 1920s, the dominant philosophy of science was logical positivism. The alternative realist Bohm interpretation and many-worlds interpretation of quantum mechanics do not make such a revolutionary break with the concepts of classical physics.

Underdetermination problem

Another argument against scientific realism, deriving from the underdetermination problem, is not so historically motivated as these others. It claims that observational data can in principle be explained by multiple theories that are mutually incompatible. Realists might counter by saying that there have been few actual cases of underdetermination in the history of science. Usually the requirement of explaining the data is so exacting that scientists are lucky to find even one theory that fulfills it. Furthermore, if we take the underdetermination argument seriously, it implies that we can know about only what we have directly observed. For example, we could not theorize that dinosaurs once lived based on the fossil evidence because other theories (e.g., that the fossils are clever hoaxes) can account for the same data.

Incompatible models argument

According to the incompatible models argument, in certain cases the existence of diverse models for a single phenomenon can be taken as evidence of anti-realism. One example is due to Margaret Morrison, who worked to show that the shell model and the liquid-drop model give contradictory descriptions of the atomic nucleus, even though both models are predictive.

Nominalism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Nominalism 

William of Ockham

In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are two main versions of nominalism. One denies the existence of universals—that which can be instantiated or exemplified by many particular things (e.g., strength, humanity). The other version specifically denies the existence of abstract objects as such—objects that do not exist in space and time.

Most nominalists have held that only physical particulars in space and time are real, and that universals exist only post res, that is, subsequent to particular things. However, some versions of nominalism hold that some particulars are abstract entities (e.g., numbers), whilst others are concrete entities – entities that do exist in space and time (e.g., pillars, snakes, and bananas). Nominalism is primarily a position on the problem of universals. It is opposed to realist philosophies, such as Platonic realism, which assert that universals do exist over and above particulars, and to the hylomorphic substance theory of Aristotle, which asserts that universals are immanently real within them; however, the name "nominalism" emerged from debates in medieval philosophy with Roscellinus.

The term nominalism stems from the Latin nomen, "name". John Stuart Mill summarised nominalism in his aphorism "there is nothing general except names". In philosophy of law, nominalism finds its application in what is called constitutional nominalism.

History

See also: Anti-realism

Ancient Greek philosophy

Plato was perhaps the first writer in Western philosophy to clearly state a realist, i.e., non-nominalist, position:

... We customarily hypothesize a single form in connection with each of the many things to which we apply the same name. ... For example, there are many beds and tables. ... But there are only two forms of such furniture, one of the bed and one of the table. (Republic 596a–b, trans. Grube)

What about someone who believes in beautiful things, but doesn't believe in the beautiful itself ...? Don't you think he is living in a dream rather than a wakened state? (Republic 476c)

The Platonic universals corresponding to the names "bed" and "beautiful" were the Form of the Bed and the Form of the Beautiful, or the Bed Itself and the Beautiful Itself. Platonic Forms were the first universals posited as such in philosophy.

Our term "universal" is due to the English translation of Aristotle's technical term katholou which he coined specially for the purpose of discussing the problem of universals. Katholou is a contraction of the phrase kata holou, meaning "on the whole".

Aristotle famously rejected certain aspects of Plato's Theory of Forms, but he clearly rejected nominalism as well:

... 'Man', and indeed every general predicate, signifies not an individual, but some quality, or quantity or relation, or something of that sort. (Sophistical Refutations xxii, 178b37, trans. Pickard-Cambridge)

The first philosophers to explicitly describe nominalist arguments were the Stoics, especially Chrysippus.

Medieval philosophy

In medieval philosophy, the French philosopher and theologian Roscellinus (c. 1050 – c. 1125) was an early, prominent proponent of nominalism. Nominalist ideas can be found in the work of Peter Abelard and reached their flowering in William of Ockham, who was the most influential and thorough nominalist. Abelard's and Ockham's version of nominalism is sometimes called conceptualism, which presents itself as a middle way between nominalism and realism, asserting that there is something in common among like individuals, but that it is a concept in the mind, rather than a real entity existing independently of the mind. Ockham argued that only individuals existed and that universals were only mental ways of referring to sets of individuals. "I maintain", he wrote, "that a universal is not something real that exists in a subject ... but that it has a being only as a thought-object in the mind [objectivum in anima]". As a general rule, Ockham argued against assuming any entities that were not necessary for explanations. Accordingly, he wrote, there is no reason to believe that there is an entity called "humanity" that resides inside, say, Socrates, and nothing further is explained by making this claim. This is in accord with the analytical method that has since come to be called Ockham's razor, the principle that the explanation of any phenomenon should make as few assumptions as possible. Critics argue that conceptualist approaches answer only the psychological question of universals. If the same concept is correctly and non-arbitrarily applied to two individuals, there must be some resemblance or shared property between the two individuals that justifies their falling under the same concept and that is just the metaphysical problem that universals were brought in to address, the starting-point of the whole problem (MacLeod & Rubenstein, 2006, §3d). If resemblances between individuals are asserted, conceptualism becomes moderate realism; if they are denied, it collapses into nominalism.

Modern and contemporary philosophy

In modern philosophy, nominalism was revived by Thomas Hobbes and Pierre Gassendi.

In contemporary analytic philosophy, it has been defended by Rudolf Carnap, Nelson Goodman, H. H. Price, and D. C. Williams.

Lately, some scholars have been questioning what kind of influences nominalism might have had in the conception of modernity and contemporaneity. According to Michael Allen Gillespie, nominalism profoundly influences these two periods. Even though modernity and contemporaneity are secular eras, their roots are firmly established in the sacred. Furthermore, "Nominalism turned this world on its head," he argues. "For the nominalists, all real being was individual or particular and universals were thus mere fictions."

Another scholar, Victor Bruno, follows the same line. According to Bruno, nominalism is one of the first signs of rupture in the medieval system. "The dismembering of the particulars, the dangerous attribution to individuals to a status of totalization of possibilities in themselves, all this will unfold in an existential fissure that is both objective and material. The result of this fissure will be the essays to establish the nation state."

Indian philosophy

See also: Difference (philosophy)

Indian philosophy encompasses various realist and nominalist traditions. Certain orthodox Hindu schools defend the realist position, notably Purva Mimamsa, Nyaya and Vaisheshika, maintaining that the referent of the word is both the individual object perceived by the subject of knowledge and the universal class to which the thing belongs. According to Indian realism, both the individual and the universal exist objectively, with the second underlying the former.

Buddhists take the nominalist position, especially those of the Sautrāntika and Yogācāra schools; they were of the opinion that words have as referent not true objects, but only concepts produced in the intellect. These concepts are not real since they do not have efficient existence, that is, causal powers. Words, as linguistic conventions, are useful to thought and discourse, but even so, it should not be accepted that words apprehend reality as it is.

Dignāga formulated a nominalist theory of meaning called apohavada, or theory of exclusions. The theory seeks to explain how it is possible for words to refer to classes of objects even if no such class has an objective existence. Dignāga's thesis is that classes do not refer to positive qualities that their members share in common. On the contrary, universal classes are exclusions (apoha). As such, the "cow" class, for example, is composed of all exclusions common to individual cows: they are all non-horse, non-elephant, etc.

The problem of universals

Nominalism arose in reaction to the problem of universals, specifically accounting for the fact that some things are of the same type. For example, Fluffy and Kitzler are both cats, or, the fact that certain properties are repeatable, such as: the grass, the shirt, and Kermit the Frog are green. One wants to know by virtue of what are Fluffy and Kitzler both cats, and what makes the grass, the shirt, and Kermit green.

The Platonist answer is that all the green things are green in virtue of the existence of a universal: a single abstract thing that, in this case, is a part of all the green things. With respect to the color of the grass, the shirt and Kermit, one of their parts is identical. In this respect, the three parts are literally one. Greenness is repeatable because there is one thing that manifests itself wherever there are green things.

Nominalism denies the existence of universals. The motivation for this flows from several concerns, the first one being where they might exist. Plato famously held, on one interpretation, that there is a realm of abstract forms or universals apart from the physical world (see theory of the forms). Particular physical objects merely exemplify or instantiate the universal. But this raises the question: Where is this universal realm? One possibility is that it is outside space and time. A view sympathetic with this possibility holds that, precisely because some form is immanent in several physical objects, it must also transcend each of those physical objects; in this way, the forms are "transcendent" only insofar as they are "immanent" in many physical objects. In other words, immanence implies transcendence; they are not opposed to one another. (Nor, in this view, would there be a separate "world" or "realm" of forms that is distinct from the physical world, thus shirking much of the worry about where to locate a "universal realm".) However, naturalists assert that nothing is outside of space and time. Some Neoplatonists, such as the pagan philosopher Plotinus and the Christian philosopher Augustine, imply (anticipating conceptualism) that universals are contained within the mind of God. To complicate things, what is the nature of the instantiation or exemplification relation?

Conceptualists hold a position intermediate between nominalism and realism, saying that universals exist only within the mind and have no external or substantial reality.

Moderate realists hold that there is no realm in which universals exist, but rather there universals are located in space and time however they are manifest. Suppose that a universal, for example greenness, is supposed to be a single thing. Nominalists consider it unusual that there could be a single thing that exists in multiple places simultaneously. The realist maintains that all the instances of greenness are held together by the exemplification relation, but that this relation cannot be explained. Additionally, in lexicology there is an argument against color realism, namely the subject of the blue-green distinction. In some languages the equivalent words for blue and green may be colexified (and furthermore there may not be a straightforward translation either – in Japanese "青", which is usually translated as "blue", is sometimes used for words which in English may be considered as "green" (such as green apples).)

Finally, many philosophers prefer simpler ontologies populated with only the bare minimum of types of entities, or as W. V. O. Quine said "They have a taste for 'desert landscapes.'" They try to express everything that they want to explain without using universals such as "catness" or "greenness."

Varieties

There are various forms of nominalism ranging from extreme to almost-realist. One extreme is predicate nominalism, which states that Fluffy and Kitzler, for example, are both cats simply because the predicate 'is a cat' applies to both of them. And this is the case for all similarity of attribute among objects. The main criticism of this view is that it does not provide a sufficient solution to the problem of universals. It fails to provide an account of what makes it the case that a group of things warrant having the same predicate applied to them.

Proponents of resemblance nominalism believe that 'cat' applies to both cats because Fluffy and Kitzler resemble an exemplar cat closely enough to be classed together with it as members of its kind, or that they differ from each other (and other cats) quite less than they differ from other things, and this warrants classing them together. Some resemblance nominalists will concede that the resemblance relation is itself a universal, but is the only universal necessary. Others argue that each resemblance relation is a particular, and is a resemblance relation simply in virtue of its resemblance to other resemblance relations. This generates an infinite regress, but many argue that it is not vicious.

Class nominalism argues that class membership forms the metaphysical backing for property relationships: two particular red balls share a property in that they are both members of classes corresponding to their properties – that of being red and of being balls. A version of class nominalism that sees some classes as "natural classes" is held by Anthony Quinton.

Conceptualism is a philosophical theory that explains universality of particulars as conceptualized frameworks situated within the thinking mind. The conceptualist view approaches the metaphysical concept of universals from a perspective that denies their presence in particulars outside of the mind's perception of them.

Another form of nominalism is trope nominalism. A trope is a particular instance of a property, like the specific greenness of a shirt. One might argue that there is a primitive, objective resemblance relation that holds among like tropes. Another route is to argue that all apparent tropes are constructed out of more primitive tropes and that the most primitive tropes are the entities of complete physics. Primitive trope resemblance may thus be accounted for in terms of causal indiscernibility. Two tropes are exactly resembling if substituting one for the other would make no difference to the events in which they are taking part. Varying degrees of resemblance at the macro level can be explained by varying degrees of resemblance at the micro level, and micro-level resemblance is explained in terms of something no less robustly physical than causal power. David Armstrong, perhaps the most prominent contemporary realist, argues that such a trope-based variant of nominalism has promise, but holds that it is unable to account for the laws of nature in the way his theory of universals can.

Ian Hacking has also argued that much of what is called social constructionism of science in contemporary times is actually motivated by an unstated nominalist metaphysical view. For this reason, he claims, scientists and constructionists tend to "shout past each other".

Mark Hunyadi characterizes the contemporary Western world as a figure of a "libidinal nominalism." He argues that the insistence on the individual will that has emerged in medieval nominalism evolves into a "libidinal nominalism" in which desire and will are conflated.

Mathematical nominalism

A notion that philosophy, especially ontology and the philosophy of mathematics, should abstain from set theory owes much to the writings of Nelson Goodman (see especially Goodman 1940 and 1977), who argued that concrete and abstract entities having no parts, called individuals, exist. Collections of individuals likewise exist, but two collections having the same individuals are the same collection. Goodman was himself drawing heavily on the work of Stanisław Leśniewski, especially his mereology, which was itself a reaction to the paradoxes associated with Cantorian set theory. Leśniewski denied the existence of the empty set and held that any singleton was identical to the individual inside it. Classes corresponding to what are held to be species or genera are concrete sums of their concrete constituting individuals. For example, the class of philosophers is nothing but the sum of all concrete, individual philosophers.

The principle of extensionality in set theory assures us that any matching pair of curly braces enclosing one or more instances of the same individuals denote the same set. Hence {a, b}, {b, a}, {a, b, a, b} are all the same set. For Goodman and other proponents of mathematical nominalism, {a, b} is also identical to {a, {b} }, {b, {a, b} }, and any combination of matching curly braces and one or more instances of a and b, as long as a and b are names of individuals and not of collections of individuals. Goodman, Richard Milton Martin, and Willard Quine all advocated reasoning about collectivities by means of a theory of virtual sets (see especially Quine 1969), one making possible all elementary operations on sets except that the universe of a quantified variable cannot contain any virtual sets.

In the foundations of mathematics, nominalism has come to mean doing mathematics without assuming that sets in the mathematical sense exist. In practice, this means that quantified variables may range over universes of numbers, points, primitive ordered pairs, and other abstract ontological primitives, but not over sets whose members are such individuals. Only a small fraction of the corpus of modern mathematics can be rederived in a nominalistic fashion.

Criticisms

Historical origins of the term

As a category of late medieval thought, the concept of 'nominalism' has been increasingly queried. Traditionally, the fourteenth century has been regarded as the heyday of nominalism, with figures such as John Buridan and William of Ockham viewed as founding figures. However, the concept of 'nominalism' as a movement (generally contrasted with 'realism'), first emerged only in the late fourteenth century, and only gradually became widespread during the fifteenth century. The notion of two distinct ways, a via antiqua, associated with realism, and a via moderna, associated with nominalism, became widespread only in the later fifteenth century – a dispute which eventually dried up in the sixteenth century.

Aware that explicit thinking in terms of a divide between 'nominalism' and 'realism’ emerged only in the fifteenth century, scholars have increasingly questioned whether a fourteenth-century school of nominalism can really be said to have existed. While one might speak of family resemblances between Ockham, Buridan, Marsilius and others, there are also striking differences. More fundamentally, Robert Pasnau has questioned whether any kind of coherent body of thought that could be called 'nominalism' can be discerned in fourteenth century writing. This makes it difficult, it has been argued, to follow the twentieth century narrative which portrayed late scholastic philosophy as a dispute which emerged in the fourteenth century between the via moderna, nominalism, and the via antiqua, realism, with the nominalist ideas of William of Ockham foreshadowing the eventual rejection of scholasticism in the seventeenth century.

Nominalist reconstructions in mathematics

A critique of nominalist reconstructions in mathematics was undertaken by Burgess (1983) and Burgess and Rosen (1997). Burgess distinguished two types of nominalist reconstructions. Thus, hermeneutic nominalism is the hypothesis that science, properly interpreted, already dispenses with mathematical objects (entities) such as numbers and sets. Meanwhile, revolutionary nominalism is the project of replacing current scientific theories by alternatives dispensing with mathematical objects (see Burgess, 1983, p. 96). A recent study extends the Burgessian critique to three nominalistic reconstructions: the reconstruction of analysis by Georg Cantor, Richard Dedekind, and Karl Weierstrass that dispensed with infinitesimals; the constructivist re-reconstruction of Weierstrassian analysis by Errett Bishop that dispensed with the law of excluded middle; and the hermeneutic reconstruction, by Carl Boyer, Judith Grabiner, and others, of Cauchy's foundational contribution to analysis that dispensed with Cauchy's infinitesimals.

Mathematical beauty

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Mathematical_beauty

Mathematical beauty is a type of aesthetic value that is experienced in doing or contemplating mathematics. The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such as G.H. Hardy, have characterized mathematics as an art form that seeks beauty. The logician and philosopher Bertrand Russell made a now-famous statement of this position:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

Beauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understanding beauty in general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstract ideas which can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than (for example) listening to music. Furthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors seem identify mathematical elegance with mathematical beauty; others distinguish elegance as a separate aesthetic value, or as being, for instance, limited to the form mathematical exposition. Beauty itself is often linked to, or thought to be dependent on, the abstractness, purity, simplicity, depth or order of mathematics.

Examples of beautiful mathematics

Results

Starting at e⁰ = 1, travelling at the velocity i relative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an Argand diagram.)

Euler's identity is often given as an example of a beautiful result:

$${\displaystyle {\displaystyle e^{i\pi }+1=0.}}$$

This expression ties together arguably the five most important mathematical constants (e, i, π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of Euler's formula, which the physicist Richard Feynman called "our jewel" and "the most remarkable formula in mathematics".

Another example is Fermat's theorem on sums of two squares, which says that any prime number such that ${\displaystyle p\equiv 1(\mathrm{mod}4)}$ can be written as a sum of two square numbers (for example, ${\displaystyle 5=1^2+2^2}$, ${\displaystyle 13=2^2+3^2}$, ${\displaystyle 37=1^2+6^2}$), which both G.H. Hardy and E.T. Bell thought was a beautiful result.

In a survey in which mathematicians were asked to evaluate 24 theorems for their beauty, the top-rated three theorems were: Euler's equation; Euler's polyhedron formula, which asserts that for a polyhedron with V vertices, E edges, and F faces, ${\displaystyle V-E+F=2}$; and Euclid's theorem that there are infinitely many prime numbers, which was also given by Hardy as an example of a beautiful theorem.

Proofs

An example of "beauty in method"—a simple and elegant visual descriptor of the Pythagorean theorem.

Cantor's diagonal argument, which establishes that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers, has been cited by both mathematicians and philosophers as an example of a beautiful proof.

A proof without words for the sum of odd numbers theorem

Visual proofs, such as the illustrated proof of the Pythagorean theorem, and other proofs without words generally, such as the shown proof that the sum of all positive odd numbers up to 2n − 1 is a perfect square, have been thought beautiful.

The mathematician Paul Erdős spoke of The Book, an imaginary infinite book in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would proclaim it "straight from The Book!". His rhetorical device inspired the creation of Proofs from THE BOOK, a collection of such proofs, including many suggested by Erdős himself.

Objects

In Plato's Timaeus, the five regular convex polyhedra, called the Platonic solids for their role in this dialogue, are called the "most beautiful" ("κάλλιστα") bodies. In the Timaeus, they are described as having been used by the demiurge, or creator-craftsman who builds the cosmos, for the four classical elements plus the heavens, because of their beauty.

Kepler's Platonic solid model of the solar system

In his 1596 book Mysterium Cosmographicum, Johannes Kepler argued that the orbits of the then-known planets in the Solar System have been arranged by God to correspond to a concentric arrangement of the five Platonic solids, each orbit lying on the circumsphere of one polyhedron and the insphere of another. For Kepler, God had wanted to shape the universe according to the five regular solids because of their beauty, and this explained why there were six planets (according to the knowledge of the time).

Petrie projection of ${\displaystyle E_8}$

A more modern example is the exceptional simple Lie group ${\displaystyle E_8}$, which has been called "perhaps the most beautiful structure in all of mathematics".

Scientific theories

The mathematical statements of scientific theories, especially in physics, are sometimes considered to be mathematically beautiful. For example, Roger Penrose thought there was a "special beauty" in Maxwell's equations of electromagnetism: $${\displaystyle \begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{E}& =\frac{\rho }{{\epsilon }_0}\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\\ \mathrm{\nabla }\times \mathbf{E}& =-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\\ \mathrm{\nabla }\times \mathbf{B}& ={\mu }_0\left(\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)\end{array}}$$

Einstein's theory of general relativity has been characterized as a work of art, and, among other aesthetic praise, was described by Paul Dirac as having "great mathematical beauty" and by Penrose as having "supreme mathematical beauty".

(There can be more to the beauty of a scientific theory than just its mathematical statement. For example, whether a theory is visualizable or deterministic might have an influence on whether it is seen as beautiful.)

Properties of beautiful mathematics

Many mathematicians and philosophers who have written about mathematical beauty have tried to identify properties or criteria that are conducive to the perception of beauty in a piece of mathematics. It is debated whether beauty can be clarified or explained by such properties: Paul Erdős thought that it was no more possible to convince someone of the beauty of a piece of mathematics than to convince them of the beauty of Beethoven's Ninth Symphony, if they couldn't see it for themselves.

Results

In his 1940 essay A Mathematician's Apology, G. H. Hardy said that a beautiful result, including its proof, possesses three "purely aesthetic qualities", namely "inevitability", "unexpectedness", and "economy". He particularly excluded enumeration of cases as "one of the duller forms of mathematical argument".

In 1997, Gian-Carlo Rota, disagreed with unexpectedness as a necessary condition for beauty and proposed a counterexample:

A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago [from 1977] the proof of the existence of non-equivalent differentiable structures on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or now.

In contrast, Monastyrsky wrote in 2001:

It is very difficult to find an analogous invention in the past to Milnor's beautiful construction of the different differential structures on the seven-dimensional sphere... The original proof of Milnor was not very constructive, but later E. Briscorn showed that these differential structures can be described in an extremely explicit and beautiful form.

This disagreement illustrates both the subjective nature of mathematical beauty and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them.

Proofs

Besides Hardy's properties of "unexpectedness", "inevitability", "economy", which he applied to proofs as well as results, mathematicians have customarily thought beautiful proofs that are short and simple.

In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, with hundreds of proofs having being published. Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity. In fact, Carl Friedrich Gauss alone had eight different proofs of this theorem, six of which he published.

In contrast, results that are logically correct but involve laborious calculations or consideration of many cases, are not usually considered beautiful, and may be even referred to as ugly or clumsy. For example, Kenneth Appel and Wolfgang Haken's proof of the four color theorem made use of computer checking of over a thousand cases. Philip J. Davis and Reuben Hersh said that when they first heard that about the proof, they hoped it contained a new insight "whose beauty would transform my day", and were disheartened when informed the proof was by case enumeration and computer verification. Paul Erdős said it was "not beautiful" because it gave no insight into why the theorem was true.

Philosophical analysis

Aristotle thought that beauty was found especially in mathematics, writing in the Metaphysics that

those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a very great deal about them; for if they do not expressly mention them, but prove attributes which are their results or their formulae, it is not true to say that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.

In the twentieth century, some philosophers questioned whether there was genuinely beauty in mathematics. The philosopher of science Rom Harré argued that there were no true aesthetic appraisals of mathematics, but only quasi-aesthetic appraisals. Any mathematical success described by an aesthetic term was a second-order success besides understanding and correctness. In contrast, aesthetic appraisal of a work of art was first-order. Harré considered this to be the difference between a quasi-aesthetic and a genuinely aesthetic appraisal.

Nick Zangwill thought that there were no true aesthetic experiences of mathematics and that a proofs or theories could only be metaphorically beautiful. His argument had two bases. First, he thought that aesthetic properties depended on sensory properties, and so abstract entities could not have aesthetic properties. Second, he thought that proofs, theorems, theories, and so on had purposes such as demonstrating correctness or granting understanding, and that any praise of them reflected only how well they achieved their purpose.

Scientific analysis

Information-theory model

In the 1970s, Abraham Moles and Frieder Nake analyzed links between beauty, information processing, and information theory. In the 1990s, Jürgen Schmidhuber formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory: the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e., Kolmogorov complexity) relative to what the observer already knows. Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the first derivative of subjectively perceived beauty: the observer continually tries to improve the predictability and compressibility of the observations by discovering regularities such as repetitions and symmetries and fractal self-similarity. Whenever the observer's learning process (possibly a predictive artificial neural network) leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.

Neural correlates

Brain imaging experiments conducted by Semir Zeki, Michael Atiyah and collaborators show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial orbito-frontal cortex (mOFC) of the brain and that this activity is parametrically related to the declared intensity of beauty. The location of the activity is similar to the location of the activity that correlates with the experience of beauty from other sources, such as visual art or music. Moreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers.

Mathematical beauty and the arts

Main articles: Mathematics and art, Mathematics and music, and Mathematics and architecture

Music

Examples of the use of mathematics in music include the stochastic music of Iannis Xenakis, the Fibonacci sequence in Tool's Lateralus, counterpoint of Johann Sebastian Bach, polyrhythmic structures (as in Igor Stravinsky's The Rite of Spring), the Metric modulation of Elliott Carter, permutation theory in serialism beginning with Arnold Schoenberg, and application of Shepard tones in Karlheinz Stockhausen's Hymnen. They also include the application of Group theory to transformations in music in the theoretical writings of David Lewin.

Visual arts

Diagram from Leon Battista Alberti's 1435 Della Pittura, with pillars in perspective on a grid

Examples of the use of mathematics in the visual arts include applications of chaos theory and fractal geometry to computer-generated art, symmetry studies of Leonardo da Vinci, projective geometries in development of the perspective theory of Renaissance art, grids in Op art, optical geometry in the camera obscura of Giambattista della Porta, and multiple perspective in analytic cubism and futurism.

Sacred geometry is a field of its own, giving rise to countless art forms including some of the best known mystic symbols and religious motifs, and has a particularly rich history in Islamic architecture. It also provides a means of meditation and comtemplation, for example the study of the Kaballah Sefirot (Tree Of Life) and Metatron's Cube; and also the act of drawing itself.

The Dutch graphic designer M. C. Escher created mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity, architecture, visual paradoxes and tessellations.

Some painters and sculptors create work distorted with the mathematical principles of anamorphosis, including South African sculptor Jonty Hurwitz.

Origami, the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the mathematics of paper folding by observing the crease pattern on unfolded origami pieces.

British constructionist artist John Ernest created reliefs and paintings inspired by group theory. A number of other British artists of the constructionist and systems schools of thought also draw on mathematics models and structures as a source of inspiration, including Anthony Hill and Peter Lowe. Computer-generated art is based on mathematical algorithms.