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.
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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. This point has been recognised by recent poststructuralist thinkers such as Jacques Derrida and Jean Francois-Lyotard in the context of their project to place in question central presuppositions of the Western philosophical tradition deriving from Plato.
Lyotard views the sophists as in possession of unique insight into the sense in which discourses about what is just cannot transcend the realm of opinion and pragmatic language games , The prospects for establishing a clear methodological divide between philosophy and sophistry are poor. Apart from the considerations mentioned in section 1, it would be misleading to say that the sophists were unconcerned with truth or genuine theoretical investigation and Socrates is clearly guilty of fallacious reasoning in many of the Platonic dialogues. This in large part explains why contemporary scholarship on the distinction between philosophy and sophistry has tended to focus on a difference in moral character.
For Plato, at least, these two aspects of the sophistic education tell us something about the persona of the sophist as the embodiment of a distinctive attitude towards knowledge. The fact that the sophists taught for profit may not seem objectionable to modern readers; most present-day university professors would be reluctant to teach pro bono.
It is clearly a major issue for Plato, however. Plato can barely mention the sophists without contemptuous reference to the mercenary aspect of their trade: It is significant that students in the Academy, arguably the first higher education institution, were not required to pay fees. This is only part of the story, however. A good starting point is to consider the etymology of the term philosophia as suggested by the Phaedrus and Symposium.
Similarly, in the Symposium , Socrates refers to an exception to his ignorance. The philosopher is someone who strives after wisdom — a friend or lover of wisdom — not someone who possesses wisdom as a finished product, as the sophists claimed to do and as their name suggests.
The sophists, according to Plato, considered knowledge to be a ready-made product that could be sold without discrimination to all comers. The Theages , a Socratic dialogue whose authorship some scholars have disputed, but which expresses sentiments consistent with other Platonic dialogues, makes this point with particular clarity.
The farmer Demodokos has brought his son, Theages, who is desirous of wisdom, to Socrates. As Socrates questions his potential pupil regarding what sort of wisdom he seeks, it becomes evident that Theages seeks power in the city and influence over other men. Since Theages is looking for political wisdom, Socrates refers him to the statesmen and the sophists.
Disavowing his ability to compete with the expertise of Gorgias and Prodicus in this respect, Socrates nonetheless admits his knowledge of the erotic things, a subject about which he claims to know more than any man who has come before or indeed any of those to come Theages , b. In response to the suggestion that he study with a sophist, Theages reveals his intention to become a pupil of Socrates.
Perhaps reluctant to take on an unpromising pupil, Socrates insists that he must follow the commands of his daimonion , which will determine whether those associating with him are capable of making any progress Theages , c. The dialogue ends with an agreement that all parties make trial of the daimonion to see whether it permits of the association. Whereas the sophists accept pupils indiscriminately, provided they have the money to pay, Socrates is oriented by his desire to cultivate the beautiful and the good in promising natures.
In short, the difference between Socrates and his sophistic contemporaries, as Xenophon suggests, is the difference between a lover and a prostitute.
By contrast, Protagoras and Gorgias are shown, in the dialogues that bear their names, as vulnerable to the conventional opinions of the paying fathers of their pupils, a weakness contributing to their refutation. The sophists are thus characterised by Plato as subordinating the pursuit of truth to worldly success, in a way that perhaps calls to mind the activities of contemporary advertising executives or management consultants.
In the Sophist , Plato says that dialectic — division and collection according to kinds — is the knowledge possessed by the free man or philosopher Sophist , c. Here Plato reintroduces the difference between true and false rhetoric, alluded to in the Phaedrus , according to which the former presupposes the capacity to see the one in the many Phaedrus , b. The philosopher, then, considers rational speech as oriented by a genuine understanding of being or nature. The sophist, by contrast, is said by Plato to occupy the realm of falsity, exploiting the difficulty of dialectic by producing discursive semblances, or phantasms, of true being Sophist , c.
In response to Socratic questioning, Gorgias asserts that rhetoric is an all-comprehending power that holds under itself all of the other activities and occupations Gorgias , a. He later claims that it is concerned with the greatest good for man, namely those speeches that allow one to attain freedom and rule over others, especially, but not exclusively, in political settings d.
As suggested above, in the context of Athenian public life the capacity to persuade was a precondition of political success. Retrieved 22 October Stanford Encyclopedia of Philosophy. Sextus Empiricus gives a direct quotation in Adv. The translation, " Man is the measure In traditional English usage, man referred to hominids. Responses to relativism in Plato, Aristotle. Retrieved 28 February Protagoras' prose treatise about the gods began "Concerning the gods, I have no means of knowing whether they exist or not or of what sort they may be.
Many things prevent knowledge including the obscurity of the subject and the brevity of human life". Sophists of the 5th century BC. Pre-Socratic philosophers by school. Pythagoras Hippasus Philolaus Archytas. Protagoras Gorgias Prodicus Hippias. Xenophanes Pherecydes Hippo Diogenes Alcmaeon. Ancient Greek schools of philosophy. Epicureanism Neoplatonism Neopythagoreanism Pyrrhonism Stoicism.
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Influences Democritus , Parmenides. Milesian Thales Anaximander Anaximenes. At the end of their article, Lyon and Colyvan also review a few possible moves the nominalist can make in response. One such move would deny that mathematical explanations have any bearing on physical explanations and that some bridge principles linking the mathematics to the physical system, are required. Saatsi criticizes Lyon and Colyvan by questioning the conclusiveness of the appeal to phase spaces in bringing about explanations which would allow the deployment of an enhanced indispensability argument.
After proposing a strengthening of the indispensability argument by appealing to attractors in dynamical system theory, he concludes that understanding how explanations work in dynamical system theory undermines the Platonist use of the enhanced indispensability argument.
According to Lyon and Colyvan, a proper account of explanations in science requires an analysis of mathematical explanations in pure mathematics. It is important to add that serious objections have been raised to the idea that mathematical explanations of scientific facts require a mathematical explanation of a mathematical fact as part of the explanation. Why do hive-bee honeycombs have a hexagonal structure? The nature of the question is contrastive: Part of the explanation depends on evolutionary facts.
Bees that use less wax and thus spend less energy have a better chance at evolving via natural selection. Thus, the hexagonal tiling is optimal with respect to dividing the plane into equal areas and minimizing the perimeter. The explanation of the biological fact seems to depend essentially on a mathematical fact. Just like the cicada example, this case has been extensively discussed in the literature. Some have questioned whether the example is a case of genuine explanation.
Lyon sketches a more general theory of mathematical explanations of scientific facts. The distinction is that formulated by Jackson and Pettit between process explanations and program explanations. Such alternative explanations work at a level of granularity indicating all the spatio-temporal coordinates, forces acting on the bodies etc. Following the lead of Jackson and Pettit, Lyon calls the former process explanations and the latter program explanation:.
While Lyon does not claim that all mathematical explanations in science are process program explanations, he argues that the cases that have been at the center of attention in the discussion of mathematical explanations of scientific phenomena are special cases of program explanation, namely those in which the mathematics is indispensable to the programming.
Lyon also defends the thesis that what characterizes mathematical explanations of scientific facts is a certain modal stability see also the discussion of Lange c in the previous subsection which account for their being good explanations, an explanation that does not have to be altered even when the causal history for the occurrence of the event say, the cracking of a glass flask had been different from the actual ones. For further discussion of program and process explanations applied to logical explanations of scientific facts see Baron and Colyvan One of the aspects of the enhanced indispensability argument that has generated much attention is that the argument seems to presuppose a form of inference to the best explanation.
An inference to the best explanation licenses, when one is presented with several alternative explanations of the same phenomenon, the acceptance as true of the best among such explanations. In addition to raising the crucial issue of determining whether a certain argument is a mathematical explanation of a scientific fact, we are also confronted with the further issue of how to rank alternative explanations and of whether the appeal to inference to the best explanation in arguing for scientific realism carries over to an argument for mathematical realism.
Such inferences have been at the center of much discussion in philosophy of science see Lipton In the context of enhanced indispensability arguments, they were also discussed, in addition to several of the articles cited so far, in several articles contained in the special issue of Synthese , n. Molinini uses alternative mathematical explanations of scientific facts in kinematics using a case study from Euler studied in Molinini and special relativity using a case study on the FitzGerald-Lorentz contraction studied in detail in Friend and Molinini to put pressure on the inference to the best explanation underlying the enhanced indispensability argument.
The existence of alternative mathematical explanations of the same phenomenon, appealing to two different mathematical entities, speaks against the indispensability of any of those entities for providing an explanation of the phenomenon in question. Sereni , a reply to Molinini , offers some important reflections on the issue of the equivalent explanations problem. Indeed, they claim that inference to the best explanation is used differently in arguments for scientific realism from the way it is used in arguments for mathematical realism see also Colyvan , and that this undermines the use of the enhanced indispensability argument as establishing Platonism.
Connected to the evaluation of indispensability arguments in philosophy of mathematics are also a set of articles comparing indispensability arguments in moral theory and philosophy of mathematics contained in the collection Leibowitz and Sinclair see the introduction to the volume by the editors and contributions by Miller, Liggins, Roberts, Leng, Baker, and Enoch. Before concluding this section, let me mention a few additional contributions.
Bangu proposes a challenge for the nominalist by offering a simple example in terms of games between two players and claims that the nominalist, unlike the realist, is unable to account for the common features of certain rearrangements that can only be explained by reference to a common mathematical structure of the rearrangements. Marcus distinguishes epistemic and metaphysical readings of the explanatory indispensability argument and claims that once one disambiguates the two readings, the explanatory indispensability argument is no improvement on the standard Quinean indispensability argument.
Baron argues for the validity of the following conditional: Baron a is an exploratory investigation on the nature of the ontological dependence relating mathematicand and empirical facts in a genuine mathematical explanation of scientific facts.
Sophist Mathematics : The Mathematics of Natural Philosophy
The topic is also connected to the general issue of grounding see Raven for an overview on grounding and Liggins and Plebani for further contributions on indispensability arguments and grounding. Friend and Molinini generalize the question of mathematical explanations of scientific phenomena to two further questions: Much mathematical activity is driven by factors other than justificatory aims such as establishing the truth of a mathematical fact. The phenomenology of the variety of such explanatory activities has been partially investigated in Sandborg , ch. Consider for instance the case of Gregory Brumfiel, a real algebraic geometer.
One of them relies on a decision procedure for a particular axiomatization of the theory of real closed fields. Another method of proof consists in using a so-called transfer principle which allows one to infer the truth of a sentence for all real closed fields from its being true in one real closed field, say the real numbers.
Explanation in Mathematics
Despite the fact that the transfer principle is a very efficient tool, Brumfiel does not make any use of it, and he is very clear about this. Brumfiel prefers a third proof method which aims at giving non-transcendental proofs of purely algebraic results. This does not mean that he restricts himself to just elementary methods; he does use stronger tools but it is crucial that they apply uniformly to all real closed fields. In some cases explanations are sought in a major conceptual recasting of an entire discipline.
In such situations the major conceptual recasting will also produce new proofs but the explanatoriness of the new proofs is derivative on the conceptual recasting. This leads to a more global or holistic picture of explanation than the one based on the focus on individual proofs. Mancosu describes in detail such a global case of explanatory activity from complex analysis; see also Kitcher and Tappenden for additional case studies. Once mathematics and natural science were placed on the same footing, it became possible to apply a unified methodology to both areas. Thus, it made sense to look for explanations in mathematics just as in natural science.
However, this historical reconstruction would be mistaken. Mathematical explanations of mathematical facts have been part of philosophical reflection since Aristotle. Both are logically rigorous but only the latter provide explanations for their results. The rediscovery of Proclus in the Renaissance was to spark a far-reaching debate on the causality of mathematical demonstrations referred to above as the Quaestio de Certitudine Mathematicarum.
The first shot was fired by Alessandro Piccolomini in This led to one of the most interesting epistemological debates of the Renaissance and the seventeenth century. The historical developments have been presented in detail in Mancosu and Mancosu What is more important here is to appreciate that the basic intuition—the contraposition between explanatory and non-explanatory demonstrations—had a long and successful history and influenced both mathematical and philosophical developments well beyond the seventeenth century. For instance Mancosu shows that Bolzano and Cournot, two major philosophers of mathematics in the nineteenth century, construe the central problem of philosophy of mathematics as that of accounting for the distinction between between explanatory and non-explanatory demonstrations.
In the case of Bolzano this takes the form of a theory of Grund ground and Folge consequence. Kitcher was the first to read Bolzano as propounding a theory of mathematical explanations see Betti and Roski for recent contributions. Related to inductivism are Cellucci and , which emphasize the connection between mathematical explanation and discovery. In section 4 it was pointed out that two major forms of the search for explanations in mathematical practice occur at the level of comparison between different proofs of the same result and in the conceptual recasting of major areas.
These two types of explanatory activity lead to two different conceptions of explanation. These conceptions could be characterized as local and global. The point is that in the former case explanatoriness is primarily a local property of proofs whereas in the latter it is a global property of the whole theory or framework and the proofs are judged explanatory on account of their being part of the framework. While these two types of explanatory activity do not exhaust the varieties of mathematical explanations that occur in practice, the contraposition between local and global captures well the major difference between the two major classical accounts of mathematical explanation, those of Steiner and Kitcher more recent accounts will be discussed in section 7.
Before discussing them, it should also be pointed out that other models of scientific explanation can be thought to extend to mathematical explanation. They will be discussed in section 7. Steiner proposed his model of mathematical explanation in a. In developing his own account of explanatory proof in mathematics he discusses—and rejects—a number of initially plausible criteria for explanation, e.
In order to avoid the notorious difficulties in defining the concepts of essence and essential or necessary property, which, moreover, do not seem to be useful in mathematical contexts anyway since all mathematical truths are regarded as necessary, Steiner introduces the concept of characterizing property.
Let me mention as an aside that Kit Fine distinguishes between essential and necessary properties and that perhaps the distinction could be exploited in this context. Hence what distinguishes an explanatory proof from a non-explanatory one is that only the former involves such a characterizing property.
Furthermore, an explanatory proof is generalizable in the following sense. Thus Steiner arrives at two criteria for explanatory proofs, i. They also provided counterexamples to the criteria defended by Steiner. Kitcher is a well known defender of an account of scientific explanation as theoretical unification. Kitcher sees one of the virtues of his viewpoint to be that it can also be applied to explanation in mathematics, unlike other theories of scientific explanation whose central concepts, say causality or laws of nature, do not seem relevant to mathematics.
Kitcher has not devoted any single article to mathematical explanation and thus his position can only be gathered from what he says about mathematics in his major articles on scientific explanation. In Kitcher , he uses unification as the overarching model for explanation both in science and mathematics:. What should one expect from an account of explanation? Kitcher in points out two things. First, a theory of explanation should account for how science advances our understanding of the world.
Secondly, it should help us in evaluating or arbitrating disputes in science. He claims that the covering law model fails on both counts and he proposes that his unification account fares much better. Kitcher found inspiration in Friedman where Friedman put forward the idea that understanding of the world is achieved by science by reducing the number of facts we take as brute:.
Friedman tried to make this intuition more precise by substituting linguistic descriptions in place of an appeal to phenomena and laws. Let us make this a little bit more formal. Let us start with a set K of beliefs assumed to be consistent and deductively closed informally one can think of this as a set of statements endorsed by an ideal scientific community at a specific moment in time; Kitcher , p.
A systematization of K is any set of arguments that derive some sentences in K from other sentences of K. The explanatory store over K , E K , is the best systematization of K Kitcher here makes an idealization by claiming that E K is unique. Corresponding to different systematizations we have different degrees of unification. The highest degree of unification is that given by E K. But according to what criteria can a systematization be judged to be the best?
There are three factors: A general discussion is found in Tappenden but not a detailed analysis. Apart from a single dissenting voice Zelcer , see also the reply Weber and Frans , contributors in this area all accept mathematical explanation of mathematical facts as a datum about mathematical practice that philosophers of mathematics have to investigate.
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Marc Lange a has argued that a whole class of arguments fail to be explanatory, namely arguments by mathematical induction. But if so, one ends up, so the argument concludes, with P 1 part of the explanation for P k and P k part of the explanation for P 1 and this contradicts the non-circularity of mathematical explanations which is one of the premises about mathematical explanations defended by Lange.
By reductio, the original proof by ordinary mathematical induction is not explanatory. Baker b rejects the claim that the proof by ordinary mathematical induction and the proof by upward and downward induction from k are equally explanatory. Cariani see Other Internet Resources also focuses on the explanatory asymmetry between the two types of induction but with considerations different from those of Baker. Lange section 3, chapters 7 to 9 brings together several investigations on mathematical explanation by the same author see Lange a, b, , b, c; in what follows, we mainly reference the book of In chapter 7 of Lange , Lange emphasizes the importance of symmetry and simplicity in mathematical explanations.
The idea is that faced with a result that displays a salient symmetry, the explanation will be the one that will be able to account for the symmetry of the result by exploiting a symmetry in the set-up of the problem. Lange gives many examples drawn from probability, real analysis, number theory, complex numbers, geometry etc.
What explains this symmetry? A non-explanatory proof can be given by algebraic manipulations but this does not reveal the reason for the result which, according to Lange, is the fact that the axioms of complex arithmetic are invariant under substitution of i for - i.
Lange offers similar considerations, using further case studies from geometry and combinatorics, for simplicity: In chapter 8 of his book, Lange discusses the notion of mathematical coincidence see also Lange a, Baker c. Consider the numbers generated by juxtaposing each row, column, or main diagonal with its reverse image. For instance, or Every such number there are sixteen in total is divisible by Is this a coincidence?
Notice that you can check each of the sixteen numbers and prove that 37 is a factor. But this provides no explanation. A theorem by Nummela shows that divisibility by 37 follows from a common feature of the sixteen numbers i. Appealing to this and other examples, Lange analyzes at length what distinguishes coincidences from non-coincidences and argues that non-coincidences admit of an explanatory proof.
The analysis brings into play several interesting features, such as unification, saliency, natural properties in mathematics, fruitfulness etc. Among the claims defended in chapter 8 is that there are some clear cases that show that purity and explanation come apart. The topics of purity, fruitfulness, coincidence, unification, natural properties etc. Lange discusses and contrasts proofs of the theorem given in the context of synthetic geometry, analytic geometry and projective synthetic geometry.
He argues that the projective proof, which obtains the result by projective means from a spatial version of the same result, provides the required explanation which eludes all the other proofs. Noteworthy are the connections Lange establishes between explanation, natural properties, and fruitfulness see also Tappenden a and b.
Another rich discussion of an example of explanation from mathematical practice has been offered in Pincock b. Pincock focuses on the proof of the impossibility of solving a quintic equation by radicals. Several proofs are looked at and the proof given in terms of Galois theory is singled out as explanatory.
Pincock is especially interested in the abstract nature of the explanation, which in his account functions by appeal to an entity that is more abstract than the subject matter of the theorem itself. Another interesting case study is found in Colyvan et al. There the main example is the theorem to the effect that, given any set X , there exists a free group over X The Free Group Theorem. The authors analyze and contrast a constructive proof of the result and a more abstract but non-constructive proof. They compare and contrast the two proofs with known models of scientific explanation and find that the first proof has features in common with the reductive account of explanation while the second is more in line with unificationist accounts.
They conclude that both proofs have explanatory virtues. The discussion ends up highlighting the problem of how to compare and rank the explanatoriness of proofs and claims that explanatory virtues come in degrees. An additional case study concerning compactness is provided in Paseau In addition to the unification model, other models of scientific explanation have been tested with an eye to mathematical explanation.
Frans and Weber propose to apply mechanistic accounts to mathematical explanations where talk of capacities in the former account is rendered in terms of dependencies in the latter account the approach is discussed also in Baracco A number of fruitful conceptual intersections have been investigated in the recent literature, such as explanation and beauty, explanation and purity, explanation and depth, explanation and inter-theoretic reduction, and explanation and style. It is not my intent here to provide an encompassing overview of the literature on mathematical beauty, purity of methods, understanding in mathematics, mathematical style, and mathematical depth.
We simply refer to one or two such background studies and encourage the reader to explore the bibliography of the studies referred to. The most extensive studies connecting mathematical beauty and explanation are Giaquinto , Lange , and Dutilh Novaes, Giaquinto argues against what might be a prima facie persuasive claim, namely that explanatory proofs and beautiful proofs tend to be the same. He claims that there are reasons to doubt that explanatory proofs tend to be beautiful and insufficient evidence for establishing the converse implication.
The notion of purity of method has long been of interest to mathematicians and philosophers for recent contributions see Detlefsen and Arana and Arana and Mancosu Informally, the notion can be explained s follows. For instance, given a statement of elementary number theory one would like its proof to make use of resources that belong to elementary number theory.