Sophist Mathematics: The Mathematics of Natural Philosophy

This view of Pythagoreanism finds its way into the doxography of Aetius either because Theophrastus followed the early Academy rather than his teacher Aristotle Burkert a, 66 or because the Theophrastan doxography on Pythagoras was rewritten in the first century BCE under the influence of Neopythagoreanism Diels , ; Zhmud a, The evidence for the early Academy is, however, very limited and some reject the thesis that its members assigned late Platonic metaphysics to Pythagoras Zhmud , — Proclus quotes a passage in which Speusippus assigns to the ancients, who in this context are the Pythagoreans, the One and the Indefinite Dyad.

Some scholars argue that this is not a genuine fragment of Speusippus but rather a later fabrication see Zhmud a, — and for a response Dillon , If the Academy did not assign the One and the Dyad to Pythagoras, however, it becomes less clear how these principles came to be assigned to him. If we step back for a minute and compare the sources for Pythagoras with those available for other early Greek philosophers, the extent of the difficulties inherent in the Pythagorean Question becomes clear.

Since Pythagoras wrote no books, this most fundamental of all sources is denied us. In dealing with Heraclitus, the modern scholar turns with reluctance next to the doxographical tradition, the tradition represented by Aetius in the first century CE, which preserves in handbook form a systematic account of the beliefs of the Greek philosophers on a series of topics having to do with the physical world and its first principles.

Here again the case of Pythagoras is exceptional. Thus, the second standard source for evidence for early Greek philosophy is, in the case of Pythagoras, corrupted. Whatever views Pythagoras might have had are replaced by late Platonic metaphysics in the doxographical tradition. A third source of evidence for early Greek philosophy is regarded with great skepticism by most scholars and, in the case of most early Greek philosophers, used only with great caution.

This is the biographical tradition represented by the Lives of the Philosophers written by Diogenes Laertius. Unfortunately, these two additional lives are written by authors Iamblichus and Porphyry whose goal is explicitly non-historical, and all three of the lives rely heavily on authors in the Neopythagorean tradition, whose goal was to show that all later Greek philosophy, insofar as it was true, had been stolen from Pythagoras. The historian Timaeus of Tauromenium ca. In some cases, the fragments of these early works are clearly identified in the later lives, but in other cases we may suspect that they are the source of a given passage without being able to be certain.

Large problems remain even in the case of these sources. They were all written — years after the death of Pythagoras; given the lack of written evidence for Pythagoras, they are based largely on oral traditions.

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Even among fourth-century authors that had at least some pretensions to historical accuracy and who had access to the best information available, there are widely divergent presentations, simply because such contradictions were endemic to the evidence available in the fourth century. What we can hope to obtain from the evidence presented by Aristotle, Aristoxenus, Dicaearchus, and Timaeus is thus not a picture of Pythagoras that is consistent in all respects but rather a picture that at least defines the main areas of his achievement.

This testimony is extremely limited, about twenty brief references, but this dearth of evidence is not unique to Pythagoras. The pre-Aristotelian testimony for Pythagoras is more extensive than for most other early Greek philosophers and is thus testimony to his fame. In the case of Pythagoras, what is striking is the essential agreement of Plato and Aristotle in their presentation of his significance. Aristotle frequently discusses the philosophy of Pythagoreans, whom he dates to the middle and second half of the fifth century and who posited limiters and unlimiteds as first principles.

Aristotle strikingly may never refer to Pythagoras himself in his extant writings Metaph. In the fragments of his now lost two-book treatise on the Pythagoreans, Aristotle does discuss Pythagoras himself, but the references are all to Pythagoras as a founder of a way of life, who forbade the eating of beans Fr. Zhmud a, — argues that in one place Aristotle also describes Pythagoras as a mathematician Fr. For Aristotle Pythagoras did not belong to the succession of thinkers starting with Thales, who were attempting to explain the basic principles of the natural world, and hence he could not see what sense it made to call a fifth-century thinker like Philolaus, who joined that succession by positing limiters and unlimiteds as first principles, a Pythagorean.

Pythagoras (Stanford Encyclopedia of Philosophy)

Plato is often thought to be heavily indebted to the Pythagoreans, but he is almost as parsimonious in his references to Pythagoras as Aristotle and mentions him only once in his writings. In the Philebus , Plato does describe the philosophy of limiters and unlimiteds, which Aristotle assigns to the so-called Pythagoreans of the fifth century and which is found in the fragments of Philolaus, but like Aristotle he does not ascribe this philosophy to Pythagoras himself.

Scholars, both ancient and modern, under the influence of the later glorification of Pythagoras, have supposed that the Prometheus, whom Plato describes as hurling the system down to men, was Pythagoras e. The fragments of Philolaus show that he was the primary figure of this group. For both Plato and Aristotle, then, Pythagoras is not a part of the cosmological and metaphysical tradition of Presocratic philosophy nor is he closely connected to the metaphysical system presented by fifth-century Pythagoreans like Philolaus; he is instead the founder of a way of life.

References to Pythagoras by Xenophanes ca. For the details of his life we have to rely on fourth-century sources such as Aristoxenus, Dicaearchus and Timaeus of Tauromenium. There is a great deal of controversy about his origin and early life, but there is agreement that he grew up on the island of Samos, near the birthplace of Greek philosophy, Miletus, on the coast of Asia Minor. There are a number of reports that he traveled widely in the Near East while living on Samos, e. To some extent reports of these trips are an attempt to claim the ancient wisdom of the east for Pythagoras and some scholars totally reject them Zhmud , 83—91 , but relatively early sources such as Herodotus II.

Aristoxenus says that he left Samos at the age of forty, when the tyranny of Polycrates, who came to power ca. This chronology would suggest that he was born ca. He then emigrated to the Greek city of Croton in southern Italy ca. There are a variety of stories about his death, but the most reliable evidence Aristoxenus and Dicaearchus suggests that violence directed against Pythagoras and his followers in Croton ca. There is little else about his life of which we can be confident. The evidence suggests that Pythagoras did not write any books. No source contemporaneous with Pythagoras or in the first two hundred years after his death, including Plato, Aristotle and their immediate successors in the Academy and Lyceum, quotes from a work by Pythagoras or gives any indication that any works written by him were in existence.

Several later sources explicitly assert that Pythagoras wrote nothing e. This fragment shows only that Pythagoras read the writings of others, however, and says nothing about him writing something of his own. The second of these is a Sacred Discourse , which some have wanted to trace back to Pythagoras himself.

The idea that Pythagoras wrote such a Sacred Discourse seems to arise from a misreading of the early evidence. Herodotus says that the Pythagoreans agreed with the Egyptians in not allowing the dead to be buried in wool and then asserts that there is a sacred discourse about this II. For an interesting but ultimately unconvincing attempt to argue that the historical Pythagoras did write books, see Riedweg , 42—43 and the response by Huffman a, — One of the manifestations of the attempt to glorify Pythagoras in the later tradition is the report that he, in fact, invented the word philosophy.

This story goes back to the early Academy, since it is first found in Heraclides of Pontus Cicero, Tusc. Moreover, the story depends on a conception of a philosopher as having no knowledge but being situated between ignorance and knowledge and striving for knowledge.

Such a conception is thoroughly Platonic, however see, e. For a recent attempt to defend at least the partial accuracy of the story, see Riedweg Even if he did not invent the word, what can we say about the philosophy of Pythagoras? For the reasons given in 1. The Pythagorean Question and 2. There is general agreement as to what the pre-Aristotelian evidence is, although there are differences in interpretation of it. It is crucial to decide this question before developing a picture of the philosophy of Pythagoras since chapter 19, if it is by Dicaearchus, is our earliest summary of Pythagorean philosophy.

Porphyry is very reliable about quoting his sources. He explicitly cites Dicaearchus at the beginning of Chapter 18 and names Nicomachus as his source at the beginning of chapter The material in chapter 19 follows seamlessly on chapter Thus, the onus is on anyone who would claim that Porphyry changes sources before the explicit change at the beginning of chapter Wehrli gives no reason for not including chapter 19 and the great majority of scholars accept it as being based on Dicaearchus see the references in Burkert a, , n. Zhmud a, following Philip , argues that the passage cannot derive from Dicaearchus, since it presents immortality of the soul with approval, whereas Dicaearchus did not accept its immortality.

However, the passage merely reports that Pythagoras introduced the notion of the immortality of the soul without expressing approval or disapproval. Zhmud lists other features of the chapter that he regards as unparalleled in fourth-century sources a, but, since the evidence is so fragmentary, such arguments from silence can carry little weight.

In the face of the Pythagorean question and the problems that arise even regarding the early sources, it is reasonable to wonder if we can say anything about Pythagoras. A minimalist might argue that the early evidence only allows us to conclude that Pythagoras was a historical figure who achieved fame for his wisdom but that it is impossible to determine in what that wisdom consisted. We might say that he was interested in the fate of the soul and taught a way of life, but we can say nothing precise about the nature of that life or what he taught about the soul Lloyd There is some reason to believe, however, that something more than this can be said.

The earliest evidence makes clear that above all Pythagoras was known as an expert on the fate of our soul after death. Herodotus tells the story of the Thracian Zalmoxis, who taught his countrymen that they would never die but instead go to a place where they would eternally possess all good things IV. Among the Greeks the tradition arose that this Zalmoxis was the slave of Pythagoras. Ion of Chios 5 th c. Although Xenophanes clearly finds the idea ridiculous, the fragment shows that Pythagoras believed in metempsychosis or reincarnation, according to which human souls were reborn into other animals after death.

The Sophists (Ancient Greek)

According to Herodotus, the Egyptians believed that the soul was reborn as every sort of animal before returning to human form after 3, years. Without naming names, he reports that some Greeks both earlier and later adopted this doctrine; this seems very likely to be a reference to Pythagoras earlier and perhaps Empedocles later. Many doubt that Herodotus is right to assign metempsychosis to the Egyptians, since none of the other evidence we have for Egyptian beliefs supports his claim, but it is nonetheless clear that we cannot assume that Pythagoras accepted the details of the view Herodotus ascribes to them.

Similarly both Empedocles see Inwood , 55—68 and Plato e. Did he think that we ever escape the cycle of reincarnations? We simply do not know. The fragment of Ion quoted above may suggest that the soul could have a pleasant existence after death between reincarnations or even escape the cycle of reincarnation altogether, but the evidence is too weak to be confident in such a conclusion. In the fourth century several authors report that Pythagoras remembered his previous human incarnations, but the accounts do not agree on the details.

Dicaearchus Aulus Gellius IV. Dicaearchus continues the tradition of savage satire begun by Xenophanes, when he suggests that Pythagoras was the beautiful prostitute, Alco, in another incarnation Huffman b, — It is not clear how Pythagoras conceived of the nature of the transmigrating soul but a few tentative conjectures can be made Huffman Transmigration does not require that the soul be immortal; it could go through several incarnations before perishing.

It has often been assumed that the transmigrating soul is immaterial, but Philolaus seems to have a materialistic conception of soul and he may be following Pythagoras. Similarly, it is doubtful that Pythagoras thought of the transmigrating soul as a comprehensive soul that includes all psychic faculties. His ability to recognize something distinctive of his friend in the puppy if this is not pushing the evidence of a joke too far and to remember his own previous incarnations show that personal identity was preserved through incarnations.

Thus, it would appear that what is shared with animals and which led Pythagoras to suppose that they had special kinship with human beings Dicaearchus in Porphyry, VP 19 is not intellect, as some have supposed Sorabji , 78 and but rather the ability to feel emotions such as pleasure and pain. There are significant points of contact between the Greek religious movement known as Orphism and Pythagoreanism, but the evidence for Orphism is at least as problematic as that for Pythagoras and often complicates rather than clarifies our understanding of Pythagoras Betegh ; Burkert a, ff.

There is some evidence that the Orphics also believed in metempsychosis and considerable debate has arisen as to whether they borrowed the doctrine from Pythagoras Burkert a, ; Bremmer , 24 or he borrowed it from them Zhmud a, — Dicaearchus says that Pythagoras was the first to introduce metempsychosis into Greece Porphyry VP Moreover, while Orphism presents a heavily moralized version of metempsychosis in accordance with which we are born again for punishment in this life so that our body is the prison of the soul while it undergoes punishment, it is not clear that the same was true in Pythagoreanism.

It may be that rebirths in a series of animals and people were seen as a natural cycle of the soul Zhmud a, — One would expect that the Pythagorean way of life was connected to metempsychosis, which would in turn suggest that a certain reincarnation is a reward or punishment for following or not following the principles set out in that way of life.

However, there is no unambiguous evidence connecting the Pythagorean way of life with metempsychosis. It is crucial to recognize that most Greeks followed Homer in believing that the soul was an insubstantial shade, which lived a shadowy existence in the underworld after death, an existence so bleak that Achilles famously asserts that he would rather be the lowest mortal on earth than king of the dead Homer, Odyssey XI.

The doctrine of transmigration thus seems to have been extended to include the idea that we and indeed the whole world will be reborn into lives that are exactly the same as those we are living and have already lived. Aristotle emphasized his superhuman nature in the following ways: Kingsley argues that the visit of Abaris is the key to understanding the identity and significance of Pythagoras. Abaris was a shaman from Mongolia part of what the Greeks called Hyperborea , who recognized Pythagoras as an incarnation of Apollo. The stillness of ecstacy practiced by Abaris and handed on to Pythagoras is the foundation of all civilization.

Whether or not one accepts this account of Pythagoras and his relation to Abaris, there is a clear parallel for some of the remarkable abilities of Pythagoras in the later figure of Empedocles, who promises to teach his pupils to control the winds and bring the dead back to life Fr. There are recognizable traces of this tradition about Pythagoras even in the pre-Aristotelian evidence, and his wonder-working clearly evoked diametrically opposed reactions. Similarly Pythagoras may have claimed authority for his teachings concerning the fate of our soul on the basis of his remarkable abilities and experiences, and there is some evidence that he too claimed to have journeyed to the underworld and that this journey may have been transferred from Pythagoras to Zalmoxis Burkert a, ff.

The testimony of both Plato R. It is plausible to assume that many features of this way of life were designed to insure the best possible future reincarnations, but it is important to remember that nothing in the early evidence connects the way of life to reincarnation in any specific fashion. One of the clearest strands in the early evidence for Pythagoras is his expertise in religious ritual. Herodotus gives an example: It is not surprising that Pythagoras, as an expert on the fate of the soul after death.

A significant part of the Pythagorean way of life thus consisted in the proper observance of religious ritual. The earliest source to quote acusmata is Aristotle, in the fragments of his now lost treatise on the Pythagoreans. It is not always possible to be certain which of the acusmata quoted in the later tradition go back to Aristotle and which of the ones that do go back to Pythagoras. Thus the acusmata advise Pythagoreans to pour libations to the gods from the ear i. A number of these practices can be paralleled in Greek mystery religions of the day Burkert a, Gorgias is also credited with other orations and encomia and a technical treatise on rhetoric titled At the Right Moment in Time.

The biographical details surrounding Antiphon the sophist c. However, since the publication of fragments from his On Truth in the early twentieth century he has been regarded as a major representative of the sophistic movement. On Truth , which features a range of positions and counterpositions on the relationship between nature and convention see section 3a below , is sometimes considered an important text in the history of political thought because of its alleged advocacy of egalitarianism:.

Those born of illustrious fathers we respect and honour, whereas those who come from an undistinguished house we neither respect nor honour. In this we behave like barbarians towards one another. For by nature we all equally, both barbarians and Greeks, have an entirely similar origin: Whether this statement should be taken as expressing the actual views of Antiphon, or rather as part of an antilogical presentation of opposing views on justice remains an open question, as does whether such a position rules out the identification of Antiphon the sophist with the oligarchical Antiphon of Rhamnus.

The exact dates for Hippias of Elis are unknown, but scholars generally assume that he lived during the same period as Protagoras. Hippias is best known for his polymathy DK 86A His areas of expertise seem to have included astronomy, grammar, history, mathematics, music, poetry, prose, rhetoric, painting and sculpture. Like Gorgias and Prodicus, he served as an ambassador for his home city. His work as a historian, which included compiling lists of Olympic victors, was invaluable to Thucydides and subsequent historians as it allowed for a more precise dating of past events.

In mathematics he is attributed with the discovery of a curve — the quadratrix — used to trisect an angle. It is hard to make much sense of this alleged doctrine on the basis of available evidence. As suggested above, Plato depicts Hippias as philosophically shallow and unable to keep up with Socrates in dialectical discussion. Prodicus of Ceos , who lived during roughly the same period as Protagoras and Hippias, is best known for his subtle distinctions between the meanings of words. He is thought to have written a treatise titled On the Correctness of Names.

Prodicus spoke up next: There is a distinction here. We ought to listen impartially but not divide our attention equally: More should go to the wiser speaker and less to the more unlearned … In this way our meeting would take a most attractive turn, for you, the speakers, would then most surely earn the respect, rather than the praise, of those listening to you.

For respect is guilelessly inherent in the souls of listeners, but praise is all too often merely a deceitful verbal expression. Socrates, although perhaps with some degree of irony, was fond of calling himself a pupil of Prodicus Protagoras , a; Meno , 96d. Thrasymachus was a well-known rhetorician in Athens in the latter part of the fifth century B.

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He is depicted as brash and aggressive, with views on the nature of justice that will be examined in section 3a. The distinction between physis nature and nomos custom, law, convention was a central theme in Greek thought in the second half of the fifth century B. Before turning to sophistic considerations of these concepts and the distinction between them, it is worth sketching the meaning of the Greek terms. Some of the Ionian thinkers now referred to as presocratics, including Thales and Heraclitus, used the term physis for reality as a whole, or at least its underlying material constituents, referring to the investigation of nature in this context as historia inquiry rather than philosophy.

The term nomos refers to a wide range of normative concepts extending from customs and conventions to positive law. Nonetheless, increased travel, as exemplified by the histories of Herodotus, led to a greater understanding of the wide array of customs, conventions and laws among communities in the ancient world. This recognition sets up the possibility of a dichotomy between what is unchanging and according to nature and what is merely a product of arbitrary human convention.

The dichotomy between physis and nomos seems to have been something of a commonplace of sophistic thought and was appealed to by Protagoras and Hippias among others. Antiphon applies the distinction to notions of justice and injustice, arguing that the majority of things which are considered just according to nomos are in direct conflict with nature and hence not truly or naturally just DK 87 A His account of the relation between physis and nomos nonetheless owes a debt to sophistic thought. Callicles argues that conventional justice is a kind of slave morality imposed by the many to constrain the desires of the superior few.

What is just according to nature, by contrast, is seen by observing animals in nature and relations between political communities where it can be seen that the strong prevail over the weak. Callicles himself takes this argument in the direction of a vulgar sensual hedonism motivated by the desire to have more than others pleonexia , but sensual hedonism as such does not seem to be a necessary consequence of his account of natural justice.

Like Callicles, Thrasymachus accuses Socrates of deliberate deception in his arguments, particularly in the claim the art of justice consists in a ruler looking after their subjects. Justice in conventional terms is simply a naive concern for the advantage of another. From another more natural perspective, justice is the rule of the stronger, insofar as rulers establish laws which persuade the multitude that it is just for them to obey what is to the advantage of the ruling few. Our condition improved when Zeus bestowed us with shame and justice; these enabled us to develop the skill of politics and hence civilized communal relations and virtue.

A human being is the measure of all things, of those things that are, that they are, and of those things that are not, that they are not DK, 80B1. On this reading we can regard Protagoras as asserting that if the wind, for example, feels or seems cold to me and feels or seems warm to you, then the wind is cold for me and is warm for you. All three interpretations are live options, with i perhaps the least plausible. Whatever in any particular city is considered just and admirable is just and admirable in that city, for so long as the convention remains in place c. One difficulty this passage raises is that while Protagoras asserted that all beliefs are equally true, he also maintained that some are superior to others because they are more subjectively fulfilling for those who hold them.

Protagoras thus seems to want it both ways, insofar as he removes an objective criterion of truth while also asserting that some subjective states are better than others. His appeal to better and worse beliefs could, however, be taken to refer to the persuasiveness and pleasure induced by certain beliefs and speeches rather than their objective truth. The other major source for sophistic relativism is the Dissoi Logoi , an undated and anonymous example of Protagorean antilogic. In the Dissoi Logoi we find competing arguments on five theses, including whether the good and the bad are the same or different, and a series of examples of the relativity of different cultural practices and laws.

Overall the Dissoi Logoi can be taken to uphold not only the relativity of truth but also what Barney , 89 has called the variability thesis: Understandably given their educational program, the sophists placed great emphasis upon the power of speech logos. Logos is a notoriously difficult term to translate and can refer to thought and that about which we speak and think as well as rational speech or language. The sophists were interested in particular with the role of human discourse in the shaping of reality.

Rhetoric was the centrepiece of the curriculum, but literary interpretation of the work of poets was also a staple of sophistic education. The extant fragments attributed to the historical Gorgias indicate not only scepticism towards essential being and our epistemic access to this putative realm, but an assertion of the omnipotence of persuasive logos to make the natural and practical world conform to human desires. The elimination of the criterion refers to the rejection of a standard that would enable us to distinguish clearly between knowledge and opinion about being and nature. Whereas Protagoras asserted that man is the measure of all things, Gorgias concentrated upon the status of truth about being and nature as a discursive construction.

About the Nonexistent or on Nature transgresses the injunction of Parmenides that one cannot say of what is that it is not. Employing a series of conditional arguments in the manner of Zeno, Gorgias asserts that nothing exists, that if it did exist it could not be apprehended, and if it was apprehended it could not be articulated in logos. The elaborate parody displays the paradoxical character of attempts to disclose the true nature of beings through logos:. For that by which we reveal is logos , but logos is not substances and existing things.

Therefore we do not reveal existing things to our comrades, but logos , which is something other than substances DK, 82B3. Even if knowledge of beings was possible, its transmission in logos would always be distorted by the rift between substances and our apprehension and communication of them. Gorgias also suggests, even more provocatively, that insofar as speech is the medium by which humans articulate their experience of the world, logos is not evocative of the external, but rather the external is what reveals logos. An understanding of logos about nature as constitutive rather than descriptive here supports the assertion of the omnipotence of rhetorical expertise.

If humans had knowledge of the past, present or future they would not be compelled to adopt unpredictable opinion as their counsellor. The endless contention of astronomers, politicians and philosophers is taken to demonstrate that no logos is definitive. Human ignorance about non-existent truth can thus be exploited by rhetorical persuasion insofar as humans desire the illusion of certainty imparted by the spoken word:.

The effect of logos upon the condition of the soul is comparable to the power of drugs over the nature of bodies. For just as different drugs dispel different secretions from the body, and some bring an end to disease and others to life, so also in the case of logoi , some distress, others delight, some cause fear, others make hearers bold, and some drug and bewitch the soul with a kind of evil persuasion DK, 82B All who have persuaded people, Gorgias says, do so by moulding a false logos.

While other forms of power require force, logos makes all its willing slave. The distinction between philosophy and sophistry is in itself a difficult philosophical problem. This closing section examines the attempt of Plato to establish a clear line of demarcation between philosophy and sophistry.

It was Plato who first clearly and consistently refers to the activity of philosophia and much of what he has to say is best understood in terms of an explicit or implicit contrast with the rival schools of the sophists and Isocrates who also claimed the title philosophia for his rhetorical educational program.

The related questions as to what a sophist is and how we can distinguish the philosopher from the sophist were taken very seriously by Plato. He also acknowledges the difficulty inherent in the pursuit of these questions and it is perhaps revealing that the dialogue dedicated to the task, Sophist , culminates in a discussion about the being of non-being. It can thus be argued that the search for the sophist and distinction between philosophy and sophistry are not only central themes in the Platonic dialogues, but constitutive of the very idea and practice of philosophy, at least in its original sense as articulated by Plato.

The second deals with the problem of whether mathematical explanations occur within mathematics itself. Accordingly, this entry surveys the contributions to both areas, it shows their relevance to the history of philosophy, mathematics, and science, it articulates their connection, and points to the philosophical pay-offs to be expected by deepening our understanding of the topic.

Mathematics plays a central role in our scientific picture of the world. How the connection between mathematics and the world is to be accounted for remains one of the most challenging problems in philosophy of science, philosophy of mathematics, and general philosophy. A very important aspect of this problem is that of accounting for the explanatory role mathematics seems to play in the account of physical phenomena.

Consider the following example from evolutionary biology introduced in Baker and discussed extensively in the philosophical literature. It turns out that three species of such cicadas:. Several questions have been raised about this specific type of life cycle but one of them is why such periods are prime.

The mathematical component of the explanation complements the biological claim by pointing out that prime periods minimize intersection. When we move to physics, it becomes even more difficult—given the highly mathematized nature of the subject—to distinguish between the mathematical and the physical components of an explanation.

Consider the following example. Mark the faces of a tennis racket with R for rough and S for smooth. Hold the tennis racket horizontally by its handle with face S facing up. Let y be the intermediate principal axis. This is the vertical axis perpendicular to the handle and passing through the center of mass of the racquet. Toss the racket attempting to make it rotate about the y axis.

Catch the racket by its handle after one full rotation. The surprising observation is that the R face will almost always be up one would expect S to be up. In other words, the racket makes a half twist about its handle. There is no question that we are explaining a physical regularity but mathematics enters here both in the modeling of the phenomenon and in the explanatory account by means of the classical dynamics of a rotating tennis racket.

Another simple example, in which a geometrical fact seems to do much of the explaining, has been offered by Peter Lipton:. While not everyone agrees as to the role of mathematics in the above explanations, or whether they are explanations, it is clear that one of the reasons why philosophers are especially interested in such explanations is that they appear to be counterexamples to the claim that all explanations in the natural sciences must be causal.

The dominant accounts of scientific explanation see Salmon , Cartwright , Woodward , Strevens , and Woodward for an overview in the natural sciences have been causal accounts: And since many of the non-causal explanations that have been discussed in the literature are mathematical explanations, this area of philosophy of science is fueled by concerns related to the nature of scientific explanation. Importantly related is the issue of mathematical modeling and of whether an explanatory modeling of a scientific phenomenon needs to provide a veridical representation of the salient causes or of a causal mechanism in the target phenomenon.

Many philosophers who defend the causal theory of explanation also defend a veridical causal conception of modeling but many challenges have been raised against the latter conception. The topic interacts also with the problem of characterizing the role of idealization in science. As we shall see, alleged counterexamples to the causal theory of explanation as well as to the causal theory of modeling come from areas as diverse as mathematics, physics, biology, and economics. Having established that mathematics seems to play an important role in giving explanations in the natural sciences, we now move to a few historical remarks on how this problem has emerged in the history of philosophy and science.

Does mathematics help explain the physical world or does it actually hinder a grasp of the physical mechanisms that explain the how and why of natural phenomena? It is not possible here to treat this topic in its full complexity but a few remarks will help the reader appreciate the historical importance of the question. The causes [aitia] in question are the four Aristotelian causes: But how do we obtain scientific knowledge? Scientific knowledge is obtained through demonstration.

However, not all logically cogent proofs provide us with the kind of demonstration that yields scientific knowledge. Although both are logically cogent only the latter mirror the causal structure of the phenomena under investigation, and thus provide us with knowledge. There is a relation of subordination between these mixed sciences and areas of pure mathematics. For instance, harmonics is subordinated to arithmetic and optics to geometry.

Aristotle is in no doubt that there are mathematical explanations of physical phenomena:. However, the topic of whether mathematics could give explanations of natural phenomena was one on which there was disagreement. As the domains to which mathematics could be applied grew, so also did the resistance to it. One source of tension consisted in trying to reconcile the Aristotelian conception of pure mathematics, as abstracting from matter and motion, with the fact that both physics natural philosophy and the mixed sciences are all conversant about natural phenomena and thus dependent on matter and motion.

For instance, an important debate in the Renaissance, known as the Quaestio de Certitudine Mathematicarum, focused in large part on whether mathematics could play the explanatory role assigned to it by Aristotle. By the time we reach the seventeenth century and the Newtonian revolution in physics, the problem reappears in the context of a change of criteria of explanation and intelligibility. This has been beautifully described in an article by Y. What Gingras describes, among other things, is how the mathematical treatment of force espoused by Newton and his followers—a treatment that ignored the mechanisms that could explain why and how this force operated—became an accepted standard for explanation during the eighteenth century.

He granted that the proposition connected mathematically the inverse square law to the ellipticity of the course of the planets. In Newton, there is none of this kind. Rather, the aim of the above was to prepare the ground for showing how in contemporary discussions in philosophy of science, to which we now turn, we are still confronted with such issues.

There are two major areas in which the discussion of whether mathematics can play an explanatory role in science makes itself felt. The first concerns issues of modeling and idealization in science and more generally, as pointed out in section 1, the nature of scientific explanation. The second, concerns the nominalism-platonism debate. By and large the former area is a major concern to philosophers of science. The latter is the preoccupation of those philosophers of mathematics primarily concerned with issues of existence of mathematical entities.

There are however non-trivial intersections between the two areas. These are surveyed in the next two sub-sections. One of the major theses of the book is that unification and explanation often pull in different directions and come apart contrary to what is claimed by unification theories of explanation. In short, the mathematical formalism facilitates unification but does not help us explain the how and why of physical phenomena.

These methods proceed by ignoring many details, even of a causal nature, about the phenomenon being analyzed one speaks of minimal models in this connection. But despite this fact, nay, in virtue of it, one arrives at correct explanations of the phenomena. Batterman offers examples from, among other areas, statistical mechanics, thermodynamics, optics, hydrodynamics, and quantum mechanics. In addition to his book, he developed his position in several articles , , a, b, See also Pincock b for a detailed working out of the mathematical explanation of the rainbow that extends this approach.

We thus see that the problem of the explanatory role of mathematics in science is intimately related to problems of modeling and idealization in science see Morrison In turn, understanding how modeling and idealization work is an integral part of addressing the question of how mathematics hooks on to reality, i. It would of course be impossible here to describe the major accounts of mathematical applications on offer.

However, the discussion on mathematical explanation in the empirical sciences has fruitfully interacted with some of the proposals on offer. One such proposal is the mapping account of mathematical applications offered by Pincock , c, The mapping account emphasizes the centrality of the existence of a mapping between the mathematical structures and the empirical reality that is the target of the scientific modeling.

Critics have claimed that while there are aspects of the applications of mathematics that are correctly captured by the mapping account, there are also aspects related to mathematical idealization and explanation that it does not manage to capture. Pincock a and replies to some of the criticisms and argues that his account can be extended to handle mathematical idealizations and explanations.

More criticisms are raised in Rizza who argues that examples from social choice theory cannot be handled by mapping accounts such as those defended by Pincock and Bueno and Colyvan. They claim that one such case is the case of lattice gas automaton models of fluid dynamics: Batterman and Rice note that these explanations can be understood using renormalization group techniques. Batterman and Rice conclude:. One should point out that the terminology of minimal model is also used in a different sense in the literature, for instance in Weisberg and Strevens ; in this second use, minimal models provide essential causal details.

Chirimuuta distinguishes B-minimal models of the Batterman type and A-minimal models of the Weisberg-Strevens type. Rice see also focuses on optimality models, which are widely used in physics, biology and economics. He argues that these models explain without tracking any genuine causal structure in the target phenomenon.

Baron b also discusses the example in connection to the role idealizations play in such explanations see also Potochnik While Batterman and Rice claim that explanations that depend on renormalization group techniques are not causal, Reutlinger insists that causal factors are still present. But Reutlinger also admits that non-causal factors are crucial due to the role of mathematical operations. For more on renormalization techniques and mathematical explanation see Batterman b, Reutlinger , a and b, Povich , and Morrison Minimal models are also discussed in Chirimuuta , Lange a, and Povich Pincock and Batterman are criticized in Reutlinger and Andersen on account of relying on the assumption that abstract explanations, defined as those that leave out most of the microphysical causal details of the target phenomenon to be explained, have to be non-causal.

They claim, by contrast, that a abstract explanations often do identify causes and b that they often are causally explanatory. Andersen , which aims to provide a friendly amendment to Lange c, sees complementarity and not opposition between causal and mathematical explanations. As it will be obvious from the previous discussion, the literature on mathematical explanations in the sciences has now grown too large to be described in detail.

But here are some further foci of discussion that should be mentioned, without any attempt at completeness. As we have seen, much attention has been devoted recently to structural explanations in physics, namely explanations of specific phenomena given in terms of mathematical properties of the systems in which they occur. Dorato and Felline b argue that contraction of rods and dilation of clocks can be structurally explained by properties of Minkowski space-time but see also Smart , Colyvan and and Lange a and Dorato and Felline a claim that quantum entanglement is structurally explained by the non-commutativity of certain operators and the uncertainty principle by the properties of the limits of Fourier transforms see also Felline and Dorato Much of the interest in this area has been the analysis of whether, and if so how, such explanations differ from mechanistic explanations see Huneman , Brigandt , Woodward , Jones , Kostic , Huneman , Darrason for case studies and further references.

A characteristic of such explanations is that one makes use of graph-theoretic or topological properties of a system to explain properties of the system such as stability under perturbations. A variety of responses reductionism to causal accounts, pluralism, monism can be developed in reaction to the presence of non-causal scientific explanations see Reutlinger c for a discussion of the options. For instance, much work has been devoted to incorporating non-causal explanations into counterfactual theories of explanation see Baron, Colyvan, and Ripley ; Bokulich ; Kistler ; Saatsi and Pexton ; Pexton ; Pincock a; Rice ; Reutlinger , ; Saatsi ; Jansson and Saatsi ; French and Saatsi ; Woodward A pressing issue has become that of identifying what is distinctively mathematical in mathematical explanations of scientific facts.

Lange c, , has recently put forward a modal theory of explanation, according to which mathematical explanations of scientific facts act as constraints which have modal necessity stronger than causal necessity. As he puts it in Lange c: Distinctively mathematical explanations thus supply a kind of understanding that causal explanations cannot. Baron revamps the deductive-nomological model to account for mathematical explanations of scientific facts by using an informational account of relevance logic to capture the dependence relation between mathematical facts and physics facts.

Whereas the issues treated in section 3. Colyvan and the earlier Colyvan contrasts hard road and easy road nominalism. While the hard road nominalist offers non-mathematical versions of our best scientific theories, the easy road nominalist simply retains the original, mathematical versions of the theories, and qualifies our acceptance of them so as to exclude a commitment to abstract entities. Colyvan argues that easy road nominalists such as Azzouni, Melia and Yablo are unable to account for the explanatory role of mathematics in science, for example, the explanation of the absence of asteroids in the Kirkwood gaps.

He claims that any account of how such explanations are possible that avoids mathematical commitments would require hard road nominalism, that is, showing how one can eliminate reference to mathematical entities. Replies include Bueno , Yablo , Leng , and the rejoinder Colyvan a. Bueno claims that the mathematics used in scientific explanations only plays a descriptive role and not an explanatory role.

Yablo distinguishes three degrees of mathematical involvement in physical explanation, the third of which entails the existence of mathematical entities but, he claims, it is hard to establish whether it ever obtains. Leng defends the possibility of accounting for mathematical explanations nominalistically and emphasizes the importance of structural explanations, which according to her do not commit us to mathematical objects but only to physical objects that instantiate a mathematical structure. Much of the discussion in this area, which addresses whether the mathematics present in the formulation of scientific theories or mathematical explanations offered in science is eliminable, is deeply related to so-called indispensability arguments.

There is actually a variety of classical indispensability arguments see Colyvan and Panza and Sereni but the general structure of the argument runs as follows. One begins with the premise that mathematics is indispensable for our best science. But, second premise, we ought to believe our best theories. Thus, we ought to be committed to the kind of entities that our best theories quantify over. In general this is an argument in favor of Platonism, as our best science quantifies over mathematical entities.

There are many ways in which one can attempt to block the argument. However, the key feature related to our discussion is the following. Several versions of the indispensability argument rely on a holistic conception of scientific theories according to which the ontological commitments of the theory is determined by looking at all the existential claims implied by the theory. However, no attention is paid to how the different parts of the theory might be responsible for different posits and to the different roles that the latter might play.

Baker offers a version of the indispensability argument that does not depend on holism. Baker starts from a debate between Colyvan , and Melia , that saw both authors agreeing that the prospects for a successful platonist use of the indispensability argument rests on examples from scientific practice in which the postulation of mathematical objects results in an increase of those theoretical virtues which are provided by the postulation of theoretical entities. Both authors agree that among such theoretical virtues is explanatory power.

Baker believes that such explanations exist but also argues that the cases presented in Colyvan fail to be genuine cases of mathematical explanations of physical phenomena. Recall that the question of interest was why the life cycle periods of such cicadas are prime numbers and that the answer appealed to evolutionary facts and mathematical properties of prime numbers. After the reconstruction of the explanation, Baker concludes that:.

Such explanations give a new twist to the indispensability argument. The argument now runs as follows.

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The argument has not gone unchallenged. Indeed, Leng tries to resist the conclusion by blocking premise b. She accepts a but questions the claim that the role of mathematics in such explanations commits us to the real existence as opposed to a fictional one of the posits. This, she argues, will be granted when one realizes that both Colyvan and Baker infer illegitimately from the existence of the mathematical explanation that the statements grounding the explanation are true.

She claims that mathematical explanations need not have a true explanans and consequently the objects posited by such explanations need not exist see now also Leng , chapter 9. By focusing on inference to the best explanation, Saatsi also resists the implication that scientific realism forces a commitment to mathematical Platonism. A similar objection to any attempt to use mathematical explanations in physics for inferring the existence of the mathematical entities involved in the explanation had already been raised in b by Steiner, who had discarded such arguments with the observation that what needed explanation could not even be described without use of the mathematical language.

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Thus, the existence of mathematical explanations of empirical phenomena could not be used to infer the existence of mathematical entities, for this very existence was presupposed in the description of the fact to be explained. We can only provide here a very short summary of the main positions.

Daly and Langford side with Melia to claim that the role of positing concrete unobservables in science allows the explanation of the behavior of observable facts. By contrast, mathematics has no such explanatory value: In particular, they also make appeal to the existence of mathematical explanations of mathematical facts, for which see below.

Saatsi emphasizes the representational role of mathematics and criticizes the alleged explanatory role of mathematics in the explanation of empirical phenomena such as the honeycomb case see below and the cicadas case. Rizza is also deflationary about the alleged ontological consequence of the role of mathematics in the cicadas case; however he does not deny the centrality of mathematical concepts in constructing explanations. Finally, Pincock , chapter 10, also rejects the cogency of various formulations of the enhanced indispensability argument.

The objection is that inference to the best explanation for novel abstract entities requires that those new entities are needed to explain the phenomenon in question. But in the cases discussed in this debate, there is no explanatory value to adding the natural numbers or the real numbers to our theories, over and above nominalistically acceptable surrogates see Pincock , A challenge against this strategy is offered by Baker b who claims that the entanglement of mathematics in scientific explanation is much deeper than has been realized so far and it extends to mathematical properties that, in addition to having empirical consequences, are consequences of the mathematical theorems appealed to explicitly in the explanations.

The claims rely on an explanation of the periods of magicicadas that improves on the original version given in Baker Baker , still relying on the periodical cicada explanation, argues that mathematics can sometimes reduce the ontological commitment to concrete entities and poses a challenge to nominalism, especially the piecemeal nominalism described above, based on that observation.

Modified versions of the indispensability argument stressing the importance of the indispensability of mathematics for explanations in science were considered, before Baker, by the nominalist Field as a challenge to the platonist use of such arguments. Field , 14—20 accepts the cogency of this type of inference to the best explanation but he argued Field that platonist mathematics could be replaced by a nominalistically acceptable theory that was sufficient for the development of classical mechanics.