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bible vs quran essay COMPARING THE BIBLE AND THE QUR'AN. (How to do it accurately.) By Samuel Green. Relationship! The Bible and the Qur'an are often compared to each other when Christians and drawing essays, Muslims are discussing issues. Relationship Essay! They are typically compared in the following ways: Preservation of text. Formation of canon. Teaching about conclusions essays, Jesus, women, warfare, etc. Scientific accuracy. Contradictions. This type of comparison is something that I myself have done in relationship, the past. However, it is actually inaccurate and misleading to burning faulkner, compare the Bible to the Qur'an.
There are two reasons for this. Reason 1. The Context of the relationship Bible and Qur'an. To Complete! The Qur'an revolves and relationship, evolves around Muhammad's life. Muhammad recited the words of the Qur'an in response to various situations in on artists, his life, but what these situations were is not recorded in the Qur'an. That is, the Qur'an does not provide its own context or chronology.
Knowing the correct context and essay, chronology is essential to understand the Qur'an. But to to complete, know this context and essay, chronology you must go outside of the Qur'an to the Islamic traditions - books like the Hadith or Sira literature. These books provide the context for the Qur'an. The Islamic scholar Habib Ur Rahman Azami clearly states the Qur'an's dependence upon the Hadith and essays on fictional characters, Sira. (I)t is almost impossible to understand or explain the meaning of a large number of relationship essay Qur'anic verses if the Traditions are rejected as useless and inauthentic. Burning Faulkner! (Habib Ur Rahman Azami, The Sunnah in Islam , pp. Essay! 29-31.) The Bible, however, is very different. It provides its own context and chronology. Its revelation begins with creation and tells the story of God, the world, his people, the conclusions essays prophets and the Messiah, right through to the new creation - the age of the resurrection. Therefore, when the relationship Bible gives various commands or announces the gospel it does so within a context that it itself has provided. Thus, to understand the Bible you only need the Bible. The Bible is self-sufficient but the conclusions Qur'an is not.
Therefore a proper comparison must include these other essential Islamic books. Reason 2. Practices and relationship, Beliefs. The Qur'an does not contain most of the barn basic practices or many of the beliefs of Islam. The Sunnah (the example of Muhammad) is the relationship crucial complement to essays on artists, the Koran; so much so, that there are in essay, fact isolated instances where, in fact, the on artists Sunnah appears to prevail over the Koran as, for example, when the Koran refers to three daily prayers (24:58, 11:116, 17:78-79, 20:130, 30:17-18), but the relationship essay Sunnah sets five. On the on artists other hand, there are cases from the earliest days of Islam of essay universal practices which appear to contradict express Sunnah. Moreover, the Koran does not make explicit all of its commands; not even all those which are fundamental.
Thus it enjoins prayer, but not how it is to morrison essay, be performed: the form of canonical prayer (salah) is relationship essay based entirely on Sunnah. Conclusions Essays! (Cyril Glass, Sunnah, The Concise Encyclopedia of Islam , pp. Essay! 381-382) (T)he obligatory injunction to establish Salah (regular, formal worship) was revealed in essays on lies me, the Qur’an as were some of the elements of Salah (like Qiyam, Ruku`, Sujud and relationship, Qira’ah). But the to complete dissertation actual manner of offering Salah and essay, the order in on future technology, which the various acts connected with it were to be performed, were not described in the Qur’an. . Essay! Similarly the Hajj (pilgrimage to Mecca) was prescribed as a religious duty in the Qur'an but its method and on artists, formalities were not defined. The Prophet showed the correct way by relationship essay, performing the Hajj himself. My Teacher Told! (Habib Ur Rahman Azami, pp. 10-11) Narrated Al-Irbad ibn Sariyah as-Sulami: . They gathered and the Prophet (peace_be_upon_him) led them in essay, prayer, stood up and morrison analysis essay, said: Does any of you, while reclining on his couch, imagine that Allah has prohibited only that which is to be found in relationship, this Qur'an? By Allah, I have preached, commanded and dissertation, prohibited various matters as numerous as that which is found in the Qur'an, or more numerous. . (Abu-Dawud: bk. 19, no. Relationship Essay! 3044, Hasan) Islamic Shariah is complete only essays technology, with recourse to both the Qur'an and the Sunnah . (Habib Ur Rahman Azami, p. 5) How, when and what to pray, what to relationship essay, do on Hajj, circumcision, the signs of the hour, the essays on lies my teacher told me story of relationship essay Hagar, in fact most of the drawing conclusions essential Islamic practices and essay, beliefs come from the Sunnah (practices) of Muhammad. The Sunnah is essential to Islam but it does not come from the Qur'an but the books of the toni morrison beloved analysis Hadith and Sira. Again, this is not the case with the Bible.
The Bible fully declares what God has done to save us and relationship essay, bring glory to himself and how we are to live. It is the dissertation basis for our wisdom and defines our liberty. Therefore to compare the Bible to essay, the Qur'an alone is essays on artists misleading and relationship, inaccurate; we must include these other Islamic books. What is the Bible and why is it complete? The Bible is on fictional complete because it contains the Torah (the law of Moses) the Psalms of David, the books of Solomon, the relationship essay prophetic books of essays technology Isaiah, Jeremiah, Ezekiel, Daniel, Hosea, Joel, Amos, Obadiah, Jonah, Micah, Nahum, Habakkuk, Zephaniah, Haggai, Zechariah and Malachi. Essay! It has the gospel of Jesus from Matthew, Mark, Luke and John, and essays on artists, the writings of the apostles of the Messiah.
The Bible is not one book but a collection of many books from many prophets. Christians believe all the relationship essay prophets and toni morrison beloved analysis essay, make no distinction between them, therefore, they read all of the prophets. Relationship! What Christians believe about God comes from reading all of the prophets. The prophets build on essays each other and are meant to be read together, and together they give us God’s complete message. Essay! This is why the pro euthanasia essay Bible is complete. The Other Essential Books in relationship, Islam.
The Hadith. The word hadith means an account or news about essays on artists, something. A hadith can vary in relationship, length from a sentence to a full page. In Islam the main subject of the Hadith is what Muhammad did and said, that is, the Hadith contain the Sunnah. There are many large collections of Hadith. Prof. On Artists! Masud-ul-Hasan explains main collections: The recognised collection of relationship essay Hadith on the Musannaf pattern are the collections of: Al-Bukhari (d.
870 C.E.) [A collection of 7658 hadiths (ahadith).] Muslim (d. 875 C.E.) [A collection of 7748 hadiths.] Abu Daud (d. 875 C.E.) [A collection of 5276 hadiths.] Al-Tirmizi (d. 892 C.E.) [A collection of 4415 hadiths.] Al-Nasai (d. 915 C.E.) [A collection of 5776 hadiths.] Ibn Maja (d. 886 C.E.) [A collection of 4485 hadiths.] . The collections of Al-Bukhari and essays on lies my teacher told me, Muslim rank high and are known as Al-Sahihain i.e. authentic and essay, authoritative.
The best known collection on the Musnad pattern is the collection of pro euthanasia Ahmad ibn Hanbal (d. 855 C.E.). The Shia works on Hadith, do not merely refer to what the Holy Prophet said or did, they also refer to relationship essay, what the toni morrison beloved analysis Shia Imams said or did. The Shia works on Hadith are the collections of: Muhammad ibn Yakub Al-Kulluni (d. 939 C.E.) Muhammad Al-Hummi (d. 991 C.E.) Tahir Al-Sharif Al-Murtaza (d.
1004 C.E.) Muhammad Al-Tusi (d. 1067 C.E.) (Prof. Masud-ul-Hasan, History of Islam , vol. 1, p. Relationship Essay! 613) There is my teacher also the important collection of the relationship essay Muwatta of Malik. The Hadith are part of the Islamic canon, yet Sunni and essays on artists, Shia disagree dramatically over essay, which hadith to on fictional characters, accept, and some Muslims do not accept the Hadith at relationship, all. This means that within Islam Muslims do not agree about their canon. The Sira. The Sira are the essays on artists biographies of relationship Muhammad's life. They provide the toni morrison beloved essay context and chronology of his life, and thus the essay context and chronology of the toni morrison analysis essay Qur'an. The two oldest Sira are: Muhammad ibn Ishaq (d. 773 C.E.) via ibn Hisham's (d.
840 C.E.) recension, Sirat Rasul Allah . Relationship Essay! (English translation 798 pages.) Muhammad ibn Sa'd (d. 852 C.E.), Kitab al-Tabaqat al-Kabir , (English translation, 1097 pages.) Summary. As you can see, there are a lot of other essential books in Islam than just the on future Qur'an, and they are all much bigger than the Qur'an. Both Bukhari and Muslim contain more hadiths than the relationship essay Qur'an has verses. I have been told by essays on lies me, an ex-Muslim Islamic scholar that Islam is 10% Quran and 90% traditions (Hadith and relationship essay, Sira). The Quran is toni morrison beloved essay like the relationship essay frame of a picture. It sets some boundaries, but the on artists details of the picture are provided by relationship, the traditions. A More Accurate Comparison. It should be clear now that to compare the Bible to essays on lies me, the Qur'an is misleading because while the essay Bible is the foundation of Christianity, the Qur'an is essays not the foundation of Islam, rather Islam is founded on relationship essay the Qur'an, Hadith and Sira. If you want an honest and accurate comparison between the books of Christianity and on lies told me, Islam then you must compare the essential books of both religions. 1. Do not assume that the Qur'an and Bible are basically the same.
The Bible contains the writings of many prophets from Moses to Jesus. The Qur'an however, only relationship essay, has what Muhammad, one man, recited. Motivation To Complete Dissertation! 2. If Christians and Muslims want to relationship essay, compare books on conclusions any subject then the Hadith and Sira must be included in this comparison for it to be accurate and meaningful. The Hadith and Sira are an essential part of the Islamic canon. If you see an Islamic leader comparing the essay Bible to essays on fictional characters, just the Qur'an then have the courage to explain why this is relationship inadequate and misleading. Motivation Dissertation! 3. In the media, Islamic leaders, and others, often say that certain practices, like female circumcision, are not authentic Islam because they are not in the Qur'an. Relationship Essay! However, statements like these are either ill-informed or deliberately misleading because authentic Islam is essays on fictional not based on relationship the Qur'an but on the Qur'an and Hadith. It is true that the Qur'an does not mention female circumcision, but neither does it mention male circumcision, and on artists, this does not stop circumcision being an authentic Islamic practice because it comes from the relationship Hadith.
4. On Lies My Teacher Me! I have been told by Muslims that the Qur'an is essay complete because it directs you to on fictional, Muhammad's example. Essay! Yes, the Qur'an does direct Muslims to follow Muhammad's example, but the fact remains that you have to go outside of the Qur'an to obtain this, and Muslims disagree strongly as to essays, where this information is to be found and what is authentic. Essay! 5. On Future! Islam is essay a complex religion because it is based on so many books, each of which have their own method of study and critical issues. Pro Euthanasia! It is relationship essay very difficult, if not impossible, for faulkner most Muslims to essay, be familiar with all these essential books. This means that Muslims must depend upon their leaders to drawing conclusions essays, make the essay complex simple. However, it is possible for a Christian to read and be familiar with the whole Bible.
When we invite you to technology, become a Christian we are only relationship essay, asking you to accept the Bible.  Where the morrison beloved essay hadiths are ordered according to essay, topic. Essays On Future Technology!  Where the hadiths are ordered according to narrator.  Ibn Ishaq, Sirat Rasul Allah , translated as, The Life of Muhammad , (tr. Relationship! A. Guillaume), Karachi: Oxford University Press, 1998.  Ibn Sa'ad, Kitab Al-Tabaqat Al-Kabir , (tr. S Moinul Haq) New Delhi: Kitab Bhavan, 2 volumes, no date. Sulaiman Abu Dawud, Sunan Abu-Dawud (translator: Prof. Ahmad Hasan).
Habib Ur Rahman Azami, The Sunnah in to complete, Islam , U.K.: UK Islamic Academy, 1995. Relationship! Prof. Masud-ul-Hasan, History of drawing Islam , Delhi: Adam Publishers Distributors, 2002. Cyril Glass, The Concise Encyclopedia of Islam , San Francisco: Harper Row, 1989.
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Dec 19, 2017 Relationship essay,
Platonism in the Philosophy of Mathematics. Platonism about mathematics (or mathematical platonism ) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects' perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented. The most important argument for the existence of abstract mathematical objects derives from Gottlob Frege and goes as follows (Frege 1953). The language of mathematics purports to refer to relationship essay and quantify over abstract mathematical objects.
And a great number of mathematical theorems are true. But a sentence cannot be true unless its sub-expressions succeed in essays on fictional characters, doing what they purport to do. So there exist abstract mathematical objects that these expressions refer to and quantify over. Frege's argument notwithstanding, philosophers have developed a variety of objections to mathematical platonism. Thus, abstract mathematical objects are claimed to be epistemologically inaccessible and metaphysically problematic. Mathematical platonism has been among the most hotly debated topics in the philosophy of essay mathematics over motivation to complete, the past few decades. Mathematical platonism can be defined as the conjunction of the relationship essay, following three theses: There are mathematical objects.
Mathematical objects are abstract. Mathematical objects are independent of intelligent agents and their language, thought, and practices. Some representative definitions of ‘mathematical platonism’ are listed in the supplement. and document that the above definition is fairly standard. Platonism in general (as opposed to platonism about barn faulkner, mathematics specifically) is any view that arises from the above three claims by replacing the adjective ‘mathematical’ by any other adjective. The first two claims are tolerably clear for present purposes. Existence can be formalized as ‘∃ x Mx ’, where ‘ Mx ’ abbreviates the predicate ‘ x is relationship a mathematical object’ which is true of all and only the objects studied by pure mathematics, such as numbers, sets, and functions. Abstractness says that every mathematical object is abstract, where an object is said to be abstract just in technology, case it is non-spatiotemporal and (therefore) causally inefficacious. Essay! (For further discussion, see the entry on abstract objects.) Independence is barn less clear than the other two claims. What does it mean to ascribe this sort of independence to an object?
The most obvious gloss is probably the counterfactual conditional that, had there not been any intelligent agents, or had their language, thought, or practices been different, there would still have been mathematical objects. It is doubtful that this gloss will do the work that Independence is supposed to relationship essay do (see Section 5.3); for now, Independence will be left somewhat schematic. Platonism must be distinguished from the view of the historical Plato. Few parties to the contemporary debate about on artists, platonism make strong exegetical claims about relationship, Plato's view, much less defend it. Essays On Lies Told! Although the view which we are calling ‘platonism’ is inspired by Plato's famous theory of abstract and eternal Forms (see the entry on Plato's metaphysics and epistemology), platonism is now defined and essay debated independently of its original historical inspiration. Not only is the essays technology, platonism under discussion not Plato's, platonism as characterized above is a purely metaphysical view: it should be distinguished from other views that have substantive epistemological content. Relationship Essay! Many older characterizations of platonism add strong epistemological claims to the effect that we have some immediate grasp of, or insight into, the realm of on artists abstract objects. (See e.g., Rees 1967.) But it is useful (and nowadays fairly standard) to reserve the term ‘platonism’ for the purely metaphysical view described above. Many philosophers who defend platonism in this purely metaphysical sense would reject the additional epistemological claims. Examples include Quine and other philosophers attracted to the so-called indispensability argument , which seeks to give a broadly empirical defense of mathematical platonism. (See the entry on indispensability arguments in the philosophy of mathematics.)
Finally, the above definition of ‘mathematical platonism’ excludes the claim that all truths of pure mathematics are necessary, although this claim has traditionally been made by relationship essay most platonists. Barn Faulkner! Again, this exclusion is justified by essay the fact that some philosophers who are generally regarded as platonists (for instance, Quine and some adherents of the aforementioned indispensability argument) reject this additional modal claim . 1.2 The philosophical significance of mathematical platonism. Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on essays on lies told, the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the relationship, physical world and includes objects which aren't part of the causal and spatiotemporal order studied by characters the physical sciences. 1 Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to relationship accommodate.
Although these philosophical consequences are not unique to characters mathematical platonism, this particular form of platonism is relationship essay unusually well suited to conclusions support such consequences. For mathematics is a remarkably successful discipline, both in its own right and as a tool for other sciences. 2 Few contemporary analytic philosophers are willing to contradict any of the core claims of relationship essay a discipline whose scientific credentials are as strong as those of mathematics (Lewis 1991, pp. 579). So if philosophical analysis revealed mathematics to have some strange and surprising consequences, it would be unattractive simply to reject mathematics. 3 A form of platonism based on a discipline whose scientific credentials are less impressive than those of mathematics would not be in this fortunate situation. For instance, if theology turns out to have some strange and surprising philosophical consequences, many philosophers would not hesitate to reject the relevant parts of theology. In contemporary philosophy nominalism is typically defined as the view that there are no abstract objects. (In more traditional philosophical usage the word ‘nominalism’ refers instead to the view that there are no universals. See Burgess Rosen 1997, pp. 1325 and the entry on abstract objects.) Let anti-nominalism be the essays characters, negation of nominalism, that is, the claim that there exist abstract objects. Anti-nominalism about relationship essay, mathematics is thus just the conjunction of Existence and Abstractness . Because anti-nominalism leaves out essays on fictional, Independence , it is logically weaker than mathematical platonism.
The philosophical consequences of anti-nominalism are not as strong as those of platonism. Many physicalists would accept non-physical objects provided that these are dependent on or reducible to physical objects. They may for instance accept objects such as corporations, laws, and relationship essay poems, provided that these are suitably dependent or reducible to physical objects. Moreover, there appears to be no mystery about epistemic access to on artists non-physical objects that we have somehow made or ‘constituted’. If corporations, laws, and poems are made or ‘constituted’ by us, presumably we gain knowledge of essay them in the process of making or ‘constituting’ them. Some views in the philosophy of mathematics are anti-nominalist without being platonist. One example are traditional intuitionist views, which affirm the existence of mathematical objects but maintain that these objects depend on or are constituted by mathematicians and their activities. 4 Some further examples of views that are anti-nominalist without being platonist will be discussed in Section 5.2. Truth-value realism is the view that every well-formed mathematical statement has a unique and objective truth-value that is independent of whether it can be known by us and whether it follows logically from our current mathematical theories. The view also holds that most mathematical statements that are deemed to be true are in fact true.
5 Thus truth-value realism is essays technology clearly a metaphysical view. But unlike platonism it is not an ontological view. For although truth-value realism claims that mathematical statements have unique and objective truth-values, it is not committed to the distinctively platonist idea that these truth-values flow from an ontology of mathematical objects. Mathematical platonism clearly motivates truth-value realism by providing an account of how mathematical statements get their truth-values. But the relationship, former view does not entail the latter unless further premises are added. For even if there are mathematical objects, referential and quantificational indeterminacy may deprive mathematical statements of a unique and objective truth-value. Conversely, truth-value realism does not by motivation to complete dissertation itself entail Existence and thus implies neither anti-nominalism nor platonism. For there are various accounts of relationship how mathematical statements can come to possess unique and objective truth-values which do not posit a realm of mathematical objects. 6. In fact, many nominalists endorse truth-value realism, at least about more basic branches of mathematics, such as arithmetic. Nominalists of this type are committed to the slightly odd-sounding view that, although the ordinary mathematical statement.
(1) There are primes numbers between 10 and 20. is true, there are in fact no mathematical objects and thus in particular no numbers. But there is no contradiction here. We must distinguish between the language L M in which mathematicians make their claims and conclusions essays the language L P in which nominalists and other philosophers make theirs. The statement (1) is made in L M . But the nominalist's assertion that (1) is true but that there are no abstract objects is made in relationship essay, L P . The nominalist's assertion is thus perfectly coherent provided that (1) is translated non-homophonically from L M into on fictional, L P . And indeed, when the nominalist claims that the truth-values of sentences of L M are fixed in relationship, a way which doesn't appeal to mathematical objects, it is precisely this sort of essays non-homophonic translation she has in essay, mind. The view mentioned in the previous note provides an example. This shows that for the claim Existence to have its intended effect, it must be expressed in the language L P used by us philosophers. Burning! If the claim was expressed in the language L M used by essay mathematicians, then nominalists could accept the claim while still denying that there are mathematical objects, contrary to the purpose of the claim. A small but important tradition of philosophers urge that the debate about essays characters, platonism should be replaced by, or at relationship essay least transformed into, a debate about truth-value realism. Technology! One reason offered in relationship, support of this view is that the conclusions essays, former debate is hopelessly unclear, while the latter is relationship more tractable (Dummett 1978a, pp. 228232 and Dummett 1991b, pp.
1015). Another reason offered is that the debate about truth-value realism is of greater importance to both philosophy and mathematics than the drawing essays, one about platonism. 7. 1.5 The mathematical significance of platonism. Working realism is the methodological view that mathematics should be practiced as if platonism was true (Bernays 1935, Shapiro 1997, pp. 2127 and 3844). This requires some explanation. Relationship Essay! In debates about the foundations of essays mathematics platonism has often been used to defend certain mathematical methods, such as the following:
Classical first-order (or stronger) languages whose singular terms and quantifiers appear to be referring to and ranging over mathematical objects. (This contrasts with the languages that dominated earlier in the history of mathematics, which relied more heavily on constructive and modal vocabulary.) Classical rather than intuitionistic logic. Non-constructive methods (such as non-constructive existence proofs) and relationship essay non-constructive axioms (such as the Axiom of Choice). Impredicative definitions (that is, definitions that quantify over a totality to which the object being defined would belong). ‘Hilbertian optimism’, that is, the belief that every mathematical problem is in drawing conclusions, principle solvable. Relationship Essay! 8. According to working realism, these and other classical methods are acceptable and available in burning, all mathematical reasoning. But working realism does not take a stand on relationship essay, whether these methods require any philosophical defense, and to complete dissertation if so, whether this defense must be based on platonism. Essay! In short, where platonism is an explicitly philosophical view, working realism is first and foremost a view within mathematics itself about the correct methodology of essays this discipline.
Platonism and working realism are therefore distinct views. However, there may of essay course be logical relations between the two views. Given the origin of working realism, it is not surprising that the view receives strong support from mathematical platonism. Assume that mathematical platonism is true. Then clearly the language of mathematics ought to be as described in (i). Secondly, provided it is legitimate to reason classically about any independently existing part of reality, (ii) would also follow. Motivation To Complete! Thirdly, since platonism ensures that mathematics is discovered rather than invented, there would be no need for mathematicians to restrict themselves to constructive methods and axioms, which establishes (iii).
Fourth, there is a powerful and influential argument due to Gödel (1944) that impredicative definitions are legitimate whenever the objects being defined exist independently of our definitions. (For instance, ‘the tallest boy in the class’ appears unproblematic despite being impredicative.) If this is correct, then (iv) would follow. Relationship! Finally, if mathematics is about some independently existing reality, then every mathematical problem has a unique and determinate answer, which provides at least some motivation for Hilbertian optimism. (See, however, the discussion of plenitudinous platonism in Section 5.1.) The truth of mathematical platonism would therefore have important consequences within mathematics itself. It would justify the classical methods associated with working realism and encourage the search for essays on fictional new axioms to settle questions (such as the Continuum Hypothesis) which are left open by our current mathematical theories. However, working realism does not in any obvious way imply platonism. Although working realism says that we are justified in using the essay, platonistic language of contemporary mathematics, this falls short of platonism in at least two ways.
As the on lies my teacher me, above discussion of truth-value realism showed, the platonistic language of mathematics can be analysed in such a way as to avoid reference to and quantification over mathematical objects. Moreover, even if a face-value analysis of the language of mathematics could be justified, what would follow is anti-nominalism but not yet platonism. An additional argument would be needed for the third component of platonism, namely, Independence . The prospects for relationship essay such an argument are discussed in Section 3.2. 2. The Fregean Argument for Existence. We now describe a template of an argument for the existence of mathematical objects. Essays On Fictional! Since the first philosopher who developed an argument of essay this general form was Frege, it will be referred to as the Fregean argument . But the template is essays on lies me general and relationship abstracts away from most specific aspects of told Frege's own defense of the relationship, existence of mathematical objects, such as his view that arithmetic is reducible to logic. Fregean logicism is just one way in which this template can be developed; some other ways will be mentioned below. The Fregean argument is based on essays my teacher told me, two premises, the first of relationship which concerns the semantics of the language of mathematics: The singular terms of the essays on artists, language of relationship mathematics purport to refer to mathematical objects, and motivation dissertation its first-order quantifiers purport to range over such objects. The word ‘purport’ needs to relationship essay be explained. When a sentence S purports to burning barn faulkner refer or quantify in relationship, a certain way, this means that for S to be true, S must succeed in referring or quantifying in this way.
The second premise does not require much explanation: Most sentences accepted as mathematical theorems are true (regardless of their syntactic and semantic structure). Consider sentences which are accepted as mathematical theorems and which contain one or more mathematical singular terms. By Truth , most of these sentences are true. 9 Let S be one such sentence. By Classical Semantics , the truth of S requires that its singular terms succeed in referring to mathematical objects. Hence there must be mathematical objects, as asserted by Existence . 10. Classical Semantics claims that the language of mathematics functions semantically much like language in general functions (or at least has traditionally been assumed to function): the essays on future, semantic functions of singular terms and quantifiers are, respectively, to essay refer to objects and to range over objects.
This is a broadly empirical claim about the workings of a semi-formal language used by the community of burning barn faulkner professional mathematicians. (In the widely adopted terminology of Burgess Rosen 1997, pp. 67, Classical Semantics is a hermeneutic claim; that is, it is a descriptive claim about how a certain language is actually used, not a normative claim about how this language ought to be used.) Note also that Classical Semantics is compatible with most traditional views on semantics; in particular, it is relationship essay compatible with all the standard views on the meanings of sentences, namely that they are truth-values, propositions, or sets of essays technology possible worlds. Classical Semantics enjoys strong prima facie plausibility. For the language of mathematics strongly appears to have the same semantic structure as ordinary non-mathematical language. As Burgess (1999) observes, the following two sentences appear to have the same simple semantic structure of relationship essay a predicate being ascribed to essays characters a subject (p. 288): (4) Evelyn is prim. (5) Eleven is prime. This appearance is also borne out by the standard semantic analyses proposed by linguists and semanticists. Classical Semantics has nevertheless been challenged, for instance by nominalists such as Hellman (1989) and by Hofweber (2005). (See also Moltmann (2013) for essay some challenges concerned with arithmetical vocabulary in natural language.) This is not the place for an extended discussion of such challenges.
Let me just note that a lot of work is needed to substantiate this sort of challenge. The challenger will have to argue that the apparent semantic similarities between mathematical and non-mathematical language are deceptive. And these arguments will have to be of the on fictional characters, sort that linguists and semanticistswith no vested interest in relationship, the philosophy of mathematicscould come to on future recognize as significant. 11. Truth can be defended in a variety of different ways. Common to all defenses is that they first identify some standard by which the truth-values of relationship mathematical statements can be assessed and then argue that mathematical theorems meet this standard.
One option is to appeal to a standard that is more fundamental than that of mathematics itself. Logicism provides an example. Frege and other logicists first claim that any theorem of pure logic is true. Then they attempt to on lies my teacher show that the theorems of certain branches of mathematics can be proved from relationship, pure logic and definitions alone. Another option is to appeal to the standards of essays on artists empirical science. The Quine-Putnam indispensability argument provides an example.
First it is argued that any indispensable part of empirical science is likely to relationship be true and therefore something we are justified in believing. Then it is argued that large amounts of drawing conclusions essays mathematics are indispensable to empirical science. If both claims are correct, it follows that Truth is likely to relationship be true and that belief in Truth therefore is justified. A third option is to drawing essays appeal to the standards of mathematics itself. Why should one have to appeal to non-mathematical standards, such as those of logic or empirical science, in order to defend the truth of mathematical theorems? When we defend the truth of the claims of logic and physics, we do not need to appeal to standards outside of respectively logic and physics. Rather we assume that logic and physics provide their own sui generis standards of justification. Relationship! Why should mathematics be any different? This third strategy has received a lot of attention in recent years, often under the heading of ‘naturalism’ or ‘mathematical naturalism’. Essays On Fictional Characters! (See Burgess Rosen 1997, Maddy 1997, and, for critical discussion, see the relationship, entry on naturalism in the philosophy of mathematics.) Here is an example of essays technology how a naturalistic strategy can be developed.
Call the relationship essay, attitude that mathematicians take to the theorems of mathematics ‘acceptance’. Then the motivation to complete, following claims seem plausible: (6) Mathematicians are justified in accepting the theorems of relationship mathematics. (7) Accepting a mathematical statement S involves taking S to be true. (8) When a mathematician accepts a mathematical statement S , the content of this attitude is in general the motivation, literal meaning of S . From these three claims it follows that mathematical experts are justified in taking the theorems of mathematics to relationship essay be literal truths. By extension the rest of us too are justified in believing Truth . Note that the on artists, experts with whom (6) is concerned need not themselves believe (7) and essay (8), let alone be justified in any such belief. What matters is that the three claims are true. The task of establishing the truth of (7) and (8) may fall to linguists, psychologists, sociologists, or philosophers, but certainly not to mathematicians themselves.
2.4 The notion of ontological commitment. Versions of the Fregean argument are sometimes stated in terms of the notion of ontological commitment. Assume we operate with the standard Quinean criterion of ontological commitment: A first-order sentence (or collection of such sentences) is ontologically committed to such objects as must be assumed to be in the range of the variables for the sentence (or collection of sentences) to be true. Then it follows from essays on artists, Classical Semantics that many sentences of mathematics are ontologically committed to mathematical objects. To see this, consider a typical mathematical theorem S , which involves some normal extensional occurrence of either singular terms or first-order quantifiers. By Classical Semantics , these expressions purport to refer to or range over mathematical objects. For S to be true, these expressions must succeed in doing what they purport to do.
Consequently, for S to be true, there must be mathematical objects in relationship, the range of the variables. By Quine's Criterion this means that S is ontologically committed to burning faulkner mathematical objects. Quine and many others take Quine's Criterion to be little more than a definition of the relationship essay, term ‘ontological commitment’ (Quine 1969 and Burgess 2004). But the criterion has nevertheless been challenged. Some philosophers deny that singular terms and first-order quantifiers automatically give rise to ontological commitments. Perhaps what is “required of the to complete, world” for the sentence to be true involves the existence of some but not all of the relationship, objects in to complete, the range of the quantifiers (Rayo 2008). Or perhaps we should sever the link between the essay, first-order existential quantifier and the notion of ontological commitment (Azzouni 2004 and Hofweber 2000).
One response to these challenges is to observe that the Fregean argument was developed above without any use of the term ‘ontological commitment’. Any challenge to the definition of ‘ontological commitment’ provided by motivation to complete Quine's Criterion therefore appears irrelevant to the version of the Fregean argument developed above. Essay! However, this response is unlikely to satisfy the challengers, who will respond that the conclusion of the conclusions essays, argument developed above is too weak to have its intended effect. Recall that the conclusion, Existence , is formalized in our philosophical meta-language L P as ‘∃ x Mx ’. So this formalization will fail to essay have its intended effect unless this meta-language sentence is of the sort that incurs ontological commitment. But that is precisely what the challengers dispute. This controversy cannot be pursued further here. For now, we simply observe that the challengers need to provide an account of why their non-standard notion of essays on lies me ontological commitment is better and theoretically more interesting than the standard Quinean notion. 3. From Existence to Mathematical Platonism.
Recall that mathematical platonism is the result of adding to Existence the essay, two further claims Abstractness and Independence . By the standards of philosophy, Abstractness has remained relatively uncontroversial. Among the few philosophers to have challenged it are Maddy (1990) (concerning impure sets) and Bigelow (1988) (concerning sets and drawing essays various kinds of relationship numbers). Essays! This relative lack of relationship controversy means that few explicit defenses of Abstractness have been developed. But it is not hard to see how such a defense might go. Here is motivation dissertation one idea. It is a plausible prima facie constraint on any philosophical interpretation of mathematical practice that it should avoid ascribing to relationship essay mathematics any features which would render actual mathematical practice misguided or inadequate. This constraint makes it hard to deny that the essays told, objects of pure mathematics are abstract.
For if these objects had spatiotemporal locations, then actual mathematical practice would be misguided and inadequate, since pure mathematicians ought then to essay take an interest in the locations of their objects, just as physicists take an interest in essays on artists, the locations of theirs. Relationship Essay! The fact that pure mathematicians take no interest in this question suggests that their objects are abstract. Independence says that mathematical objects, if there are any, are independent of intelligent agents and their language, thought, and practices. This claim has received relatively little explicit attention in recent decades. (Among the exceptions are philosophers of intuitionist and constructivist leanings, such as Dummett. See also Cole 2009.) The claim appears to have been tacitly accepted by most analytic philosophers, not always because they are moved by any arguments in its favor, but often only because they don't understand what it would be for to complete dissertation the claim to fail. Ordinary physical objects provide a good model for what it is for an object to be independent of us and our activities. But it remains unclear what it would be for an object not to essay be thus independent. However, a failure to see any clear alternatives to a view is not a defense of the on artists, view. Can one do better?
One strategy is to relationship essay search for a route from working realism to Independence . Assume that the methodology of classical mathematics is justified. Could it be that the best explanation of this fact is that Independence is barn faulkner true? One such argument is suggested by Gödel, who claims that the legitimacy of impredicative definitions is best explained by the truth of some form of platonism, including something like our claim Independence (Gödel 1944, pp. 455457; see also Bernays 1935 for a related but weaker claim). However, although it is widely agreed that Independence would support the legitimacy of relationship essay impredicative definitions, it remains an open question whether the converse implication is defensible. Another option is to proceed from the methodology of contemporary set theory to Independence (Gödel 1964). Much of the search for new axioms in set theory is today based on technology, so-called “extrinsic justifications”, where candidate axioms are assessed not just for essay their intrinsic plausibility but also for their capacity to essays on artists explain and systematize more basic mathematical facts.
Perhaps this methodology can somehow be used to motivate Independence . However, it remains an open question whether this suggestion can be developed into a convincing argument. (See Maddy 1988 for a discussion of relationship extrinsic justifications in characters, set theory. See also Parsons 1995 for a discussion of Gödel's platonism.) 4. Objections to relationship essay Mathematical Platonism. A variety of barn objections to mathematical platonism have been developed. Relationship! Here are the most important ones.
The most influential objection is probably the one inspired by Benacerraf (1973). What follows is an improved version of Benacerraf's objection due to motivation Field (1989). 12 This version relies on the following three premises. If these three premises are correct, it will follow that mathematical platonism undercuts our justification for believing in mathematics. But are the essay, premises correct? The first two premises are relatively uncontroversial. Most platonists are already committed to Premise 1. Conclusions! And Premise 2 seems fairly secure. If the relationship, reliability of some belief formation procedure could not even in principle be explained, then the conclusions essays, procedure would seem to work purely by chance, thus undercutting any justification we have for relationship the beliefs produced in this way. Premise 3 is more controversial.
Field defends this premise by observing that “the truth-values of our mathematical assertions depend on facts involving platonic entities that reside in a realm outside of space-time” (Field 1989, p. To Complete Dissertation! 68) and relationship essay thus are causally isolated from us even in drawing essays, principle. However, this defense assumes that any adequate explanation of the reliability in question must involve some causal correlation. This has been challenged by a variety of philosophers who have proposed more minimal explanations of the essay, reliability claim. Essays On Lies Told Me! (See Burgess Rosen 1997, pp. 4149 and Lewis 1991, pp. 111112. See Linnebo 2006 for a critique.) 13. Another famous article by Benacerraf develops a metaphysical objection to mathematical platonism (Benacerraf 1965, Kitcher 1978). Relationship! Although Benacerraf focuses on arithmetic, the objection naturally generalizes to most pure mathematical objects. Benacerraf opens by defending what is now known as a structuralist view of the natural numbers, according to which the natural numbers have no properties other than those they have in virtue of being positions in an ω-sequence. For instance, there is nothing more to essays on lies told being the number 3 than having certain intrastructurally defined relational properties, such as succeeding 2, being half of essay 6, and being prime.
No matter how hard we study arithmetic and essays on fictional set theory, we will never know whether 3 is relationship essay identical with the fourth von Neumann ordinal, or with the corresponding Zermelo ordinal, or perhaps, as Frege suggested, with the class of all three-membered classes (in some system that allows such classes to exist). Benacerraf now draws the following conclusion: Therefore, numbers are not objects at all, because in giving the properties of numbers you merely characterize an abstract structureand the distinction lies in the fact that the barn, “elements” of the structure have no properties other than those relating them to other “elements” of the relationship essay, same structure. (Benacerraf 1965, p. 291) In other words, Benacerraf claims that there can be no objects which have nothing but structural properties. All objects must have some non-structural properties as well. (See Benacerraf 1996 for essays on lies some later reflections on relationship essay, this argument.) Both of the steps of Benacerraf's argument are controversial. The first stepthat natural numbers have only structural propertieshas recently been defended by essays a variety of relationship essay mathematical structuralists (Parsons 1990, Resnik 1997, and Shapiro 1997). But this step is on future denied by logicists and neo-logicists, who claim that the natural numbers are intrinsically tied to the cardinalities of the collections that they number. And the second stepthat there can be no objects with only structural propertiesis explicitly rejected by all of the structuralists who defend the first step. Essay! (For some voices sympathetic to the second step, see Hellman 2001 and MacBride 2005. See also Linnebo 2008 for on future technology discussion.)
In addition to Benacerraf's, a variety of metaphysical objections to mathematical platonism have been developed. One of the more famous examples is an argument of Nelson Goodman's against set theory. Goodman (1956) defends the Principle of Nominalism , which states that whenever two entities have the essay, same basic constituents, they are identical. This principle can be regarded as a strengthening of the essays on lies my teacher me, familiar set theoretic axiom of relationship essay extensionality. The axiom of extensionality states that if two sets x and y have the on future, same elementsthat is, if ∀ u ( u ∈ x ↔ u ∈ y )then they are identical. Relationship Essay! The Principle of Nominalism is obtained by replacing the membership relation with its transitive closure. 14 The principle thus states that if x and y are borne ∈* by the same individualsthat is, if ∀ u ( u ∈* x ↔ u ∈* y )then x and conclusions essays y are identical. By endorsing this principle, Goodman disallows the relationship, formation of sets and classes, allowing only the formation of mereological sums and the application to the standard mereological operations (as described by his “calculus of individuals”).
However, Goodman's defense of the Principle of Nominalism is now widely held to be unconvincing, as witnessed by the widespread acceptance by philosophers and mathematicians of set theory as a legitimate and valuable branch of mathematics. Anti-nominalism says there exist abstract mathematical objects, whereas platonism adds Independence , which says that mathematical objects are independent of intelligent agents and their language, thought, and motivation practices. Alternatives are available; this final section surveys some lightweight forms of platonism which belong somewhere in the territory between anti-nominalism and full-fledged platonism. One lightweight form of platonism (its name notwithstanding) is the “full-blooded platonism” of Balaguer 1998. This form of platonism is characterized by a plenitude principle to the effect that any mathematical objects that could exist actually do exist. For instance, since the Continuum Hypothesis is independent of the standard axiomatization of set theory, there is a universe of sets in which the hypothesis is true and essay another in which it is false. And neither universe is metaphysically privileged. Essays Characters! By contrast, traditional platonism asserts that there is essay a unique universe of essays sets in which the Continuum Hypothesis is either determinately true or determinately false. 15. One alleged benefit of this plenitudinous view is in the epistemology of mathematics. If every consistent mathematical theory is true of some universe of mathematical objects, then mathematical knowledge will, in some sense, be easy to relationship essay obtain: provided that our mathematical theories are consistent, they are guaranteed to be true of some universe of mathematical objects.
However, “full-blooded platonism” has received much criticism. Colyvan and Zalta 1999 criticize it for on future undermining the possibility of relationship essay reference to mathematical objects, and Restall 2003, for lacking a precise and coherent formulation of the plenitude principle on which the view is based. Martin (2001) proposes that different universes of sets be amalgamated to yield a single maximal universe, which will be privileged by fitting our conception of set better than any other universe of sets. A different version of plenitudinous platonism is developed in Linsky Zalta 1995 and a series of essays told further articles. (See, for relationship essay instance, Linsky Zalta 2006 and other articles cited therein.) Traditional platonism goes wrong by “conceiv[ing] of abstract objects on faulkner, the model of physical objects” (Linsky Zalta 1995, p. Relationship Essay! 533), including in particular the on artists, idea that such objects are “sparse” rather than plenitudinous. Linsky Zalta develop an essay alternative approach on the basis of the second author's “object theory”. The main feature of object theory is a very general comprehension principle which asserts the essays, existence of a plenitude of essay abstract objects: for any collection of properties, there is an abstract object which “encodes” precisely these properties. In object theory, moreover, two abstract objects are identical just in case they encode precisely the same properties. Object theory's comprehension principle and identity criterion are said to “provide the link between our cognitive faculty of understanding and abstract objects” ( ibid ., p. Essays On Fictional! 547). (See Ebert Rossberg 2007 for critical discussion.) Assume that anti-nominalism is true. For convenience, assume also Classical Semantics . Essay! These assumptions ensure that the singular terms and quantifiers of mathematical language refer to and range over abstract objects. Given these assumptions, should one also be a mathematical platonist?
That is, do the objects that mathematical sentences refer to and quantify over satisfy Independence or some similar condition? It will be useful to restate our assumptions in more neutral terms. We can do this by invoking the notion of a semantic value , which plays an important role in semantics and the philosophy of essays language. In these fields it is widely assumed that each expression makes some definite contribution to essay the truth-value of sentences in which the motivation, expression occurs. This contribution is known as the semantic value of the expression. It is widely assumed that (at least in extensional contexts) the semantic value of a singular term is just its referent.
Our assumptions can now be stated neutrally as the essay, claim that mathematical singular terms have abstract semantic values and that its quantifiers range over the kinds of item that serve as semantic values. Let's focus on essays on artists, the claim about singular terms. What is the philosophical significance of essay this claim? In particular, does it support some version of Independence ? The answer will depend on what is required for a mathematical singular term to have a semantic value. Some philosophers argue that not very much is required (Frege 1953, Dummett 1981, Dummett 1991a, Hale Wright 2000, and Wright 1983). It suffices for the term t to make some definite contribution to the truth-values of sentences in which it occurs. Burning Faulkner! The whole purpose of the notion of a semantic value was to represent such contributions.
It therefore suffices for a singular term to possess a semantic value that it makes some such suitable contribution. This may even open the way for a kind of reductionism (Dummett 1991a, Linnebo 2012). Although it is perfectly true that the mathematical singular term t has an relationship essay abstract object as its semantic value, this truth may obtain in virtue of more basic facts which do not mention or involve the relevant abstract object. Compare for instance the relation of ownership that obtains between a person and her bank account. On Future! Although it is perfectly true that the person owns the bank account, this truth may obtain in virtue of more basic sociological or psychological facts which do not mention or involve the bank account. If some lightweight account of semantic values is defensible, we can accept the assumptions of anti-nominalism and Classical Semantics without committing ourselves to any traditional or robust form of platonism. 5.3 What is mathematical platonism anyway? Do the lightweight forms of platonism deserve to relationship be called ‘platonist’?
Since the views clearly qualify as anti-nominalist, the question is whether they are sufficiently true to the claim Independence . A natural gloss on motivation to complete, Independence is the relationship, counterfactual conditional that, had there not been any intelligent agents, or had their language, thought, or practices been suitably different, there would still have been mathematical objects. If this is essays on fictional characters all that Independence amounts to, then the lightweight forms of platonism are likely to satisfy the relationship, claim and thus qualify as genuinely platonist. But it is doubtful that this gloss is acceptable. For Independence is meant to substantiate an analogy between mathematical objects and ordinary physical objects. Just as electrons and planets exist independently of us, so do numbers and essays sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects' perfectly objective properties, so are statements about numbers and sets. Since the lightweight forms of platonism explicitly distance themselves from this analogy, Independence should presumably be glossed in relationship essay, a way that makes it incompatible with the lightweight forms of platonism. The problem is the elusive nature of the analogy on which Independence is based.
Until we get a better grip on this analogy, it will remain unclear exactly how platonism is supposed to go beyond anti-nominalism. Azzouni, Jody, 2004, Deflating Existential Consequence: A Case for Nominalism , Oxford: Oxford University Press. Balaguer, Mark, 1998, Platonism and Anti-Platonism in Mathematics , Oxford: Oxford University Press. Balaguer, Mark, 2001, “A theory of mathematical correctness and mathematical truth”, Pacific Philosophical Quarterly , 82: 87114. Benacerraf, Paul, 1965, “What numbers could not be”, Philosophical Review , 74: 4773. Benacerraf, Paul, 1973, “Mathematical Truth”, Journal of Philosophy , 70(19): 661679. Benacerraf, Paul, 1996, “What mathematical truth could not be, i”, in essays, Benacerraf and essay His Critics , A. On Fictional! Morton and S. Stich, eds., Oxford: Blackwell.
Benacerraf, Paul and Putnam, Hilary (eds.), 1983, Philosophy of relationship Mathematics: Selected Readings , Cambridge: Cambridge University Press. Second edition. Drawing! Bernays, Paul, 1935, “On Platonism in essay, Mathematics”, Reprinted in Benacerraf and Putnam (1983). Bigelow, John, 1988, The Reality of Numbers: A Physicalist's Philosophy of Mathematics , Oxford: Clarendon. Burgess, John P., 1999, “Review of conclusions essays Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology”, Notre Dame Journal of relationship essay Formal Logic , 40(2): 28391. Burgess, John P., 2004, “Review of Jody Azzouni, Deflating Existential Consequence: A Case for Nominalism”, Bulletin of drawing essays Symbolic Logic , 10(4): 573577. Burgess, John P. and Rosen, Gideon, 1997, A Subject with No Object , Oxford: Oxford University Press. Cole, Julian C., 2009, “Creativity, freedom, and authority: A new perspective on the metaphysics of mathematics”, Australasian Journal of Philosophy , 87: 589608. Relationship! Colyvan, Mark and Zalta, Edward N., 1999, “Mathematics: Truth and Fiction?”, Philosophia Mathematica , 7(3): 336349. Dummett, Michael, 1978a, “The philosophical basis of intuitionistic logic”, in Truth and on artists Other Enigmas , Cambridge, MA: Harvard University Press, 215247.
Reprinted in Benacerraf and relationship Putnam (1983). Dummett, Michael, 1978b, Truth and Other Enigmas , Cambridge, MA: Harvard University Press. Essays My Teacher! Dummett, Michael, 1981, Frege: Philosophy of Language , Cambridge, MA: Harvard University Press, second ed. Dummett, Michael, 1991a, Frege: Philosophy of Mathematics , Cambridge, MA: Harvard University Press. Relationship! Dummett, Michael, 1991b, The Logical Basis of motivation to complete Metaphysics , Cambridge, MA: Harvard University Press. Ebert, Philip and Rossberg, Marcus, 2007, “What is the purpose of neo-logicism?”, Travaux de Logique , 18: 3361. Field, Hartry, 1989, Realism, Mathematics, and Modality , Oxford: Blackwell.
Frege, Gottlob, 1953, Foundations of Arithmetic , Oxford: Blackwell. Transl. by essay J.L. Essays On Fictional Characters! Austin. Gaifman, Haim, 1975, “Ontology and Conceptual Frameworks, Part I”, Erkenntnis , 9: 329353. Gödel, Kurt, 1944, “Russell's Mathematical Logic”, In Benacerraf and Putnam (1983).
Gödel, Kurt, 1964, “What is Cantor's Continuum Hypothesis?”, In Benacerraf and Putnam (1983). Gödel, Kurt, 1995, “Some basic theorems on the foundations of essay mathematics and their implications”, in Collected Words , S. Feferman et al, ed., Oxford: Oxford University Press, vol. Motivation Dissertation! III, 304323. Goodman, Nelson, 1956, “A World of Individuals”, reprinted. in P. Relationship Essay! Benacerraf and H. Putnam, eds., Philosophy of Mathematics: Selected Readings, 1st ed., Prentice-Hall. Hale, Bob, 1987, Abstract Objects , Oxford: Blackwell. Hale, Bob and Wright, Crispin, 2000, “Implicit Definition and the A Priori”, in essays, New Essays on the A Priori , Paul Boghossian and Christopher Peacocke, eds., Oxford: Oxford University Press. Reprinted in relationship, Hale and Wright (2001).
Hale, Bob and Wright, Crispin, 2001, Reason's Proper Study , Oxford: Clarendon. Hellman, Geoffrey, 1989, Mathematics without Numbers , Oxford: Clarendon. Hellman, Geoffrey, 2001, “Three Varieties of Mathematical Structuralism”, Philosophia Mathematica , 9(3): 184211. On Artists! Hilbert, David, 1996, “Mathematical problems”, in From Kant to relationship Hilbert , William Ewald, ed., Oxford: Oxford University Press, vol. 2, 10961105. Hofweber, Thomas, 2000, “Quantification and non-existent objects”, in Empty Names, Fiction and the Puzzle of Non-Existence , Anthony Everett and Thomas Hofweber, eds., Stanford, CA: CSLI Publications, 24973. Hofweber, Thomas, 2005, “Number determiners, numbers, and arithmetic”, Philosophical Review , 114(2): 179225. Isaacson, Daniel, 1994, “Mathematical intuition and objectivity”, in Mathematics and Mind , Alexander George, ed., Oxford: Oxford University Press, chap. 5. Kitcher, Philip, 1978, “The plight of the platonist”, Noûs , 12: 119136. Kreisel, Georg, 1958, “Review of Wittgenstein's remarks on the foundations of mathematics”, British Journal for to complete dissertation the Philosophy of Science , 9: 135158. Relationship Essay! Lewis, David, 1991, Parts of my teacher Classes , Oxford: Blackwell.
Linnebo, Øystein, 2006, “Epistemological challenges to mathematical platonism”, Philosophical Studies , 129(3): 545574. Linnebo, Øystein, 2008, “Structuralism and the notion of dependence”, Philosophical Quarterly , 58: 5979. Linnebo, Øystein, 2012, “Reference by relationship abstraction”, Proceedings of the Aristotelian Society , 112: 4571. Linsky, Bernard and Zalta, Edward N., 1995, “Naturalized platonism versus platonized naturalism”, Journal of drawing conclusions Philosophy , 92(10): 525555. Linsky, Bernard and Zalta, Edward N., 2006, “What is neologicism?”, Bulletin of Symbolic Logic , 12(1): 6099. MacBride, Fraser, 2005, “Structuralism Reconsidered”, in Oxford Handbook of Philosophy of Mathematics and Logic , Stewart Shapiro, ed., Oxford: Clarendon, 563589.
Maddy, Penelope, 1988, “Believing the axioms, i and ii”, Journal of Symbolic Logic , 53: 481511 and 736764. Relationship! Maddy, Penelope, 1990, Realism in Mathematics , Oxford: Clarendon. Maddy, Penelope, 1997, Naturalism in Mathematics , Oxford: Clarendon. Essays On Fictional! Martin, Donald A., 2001, “Multiple universes of sets and indeterminate truth values”, Topoi , 20(1): 516. Relationship Essay! Moltmann, Friederike, 2013, “Reference to numbers in natural language”, Philosophical Studies , 162: 499536. Parsons, Charles, 1980, “Mathematical Intuition”, Proceedings of the Aristotelian Society , 80: 14568. Parsons, Charles, 1983, Mathematics in Philosophy , Ithaca, NY: Cornell University Press.
Parsons, Charles, 1990, “The Structuralist View of Mathematical Objects”, Synthese , 84: 303346. Parsons, Charles, 1995, “Platonism and mathematical intuition in Kurt Gödel's thought”, Bulletin of essays on fictional characters Symbolic Logic , 1(1): 4474. Quine, W.V., 1969, “Existence and quantification”, in Ontological Relativity and essay Other Essays , New York: Columbia University Press, 91113. Rayo, Agustín, 2008, “On specifying truth-conditions”, Philosophical Review , 117(3): 385443. Rees, D.A., 1967, “Platonism and the platonic tradition”, in on fictional characters, The Encyclopedia of Philosophy , Paul Edwards, ed., New York: Macmillan, vol. 5, 333341.
Resnik, Michael, 1980, Frege and the Philosophy of Mathematics , Ithaca, NY: Cornell University Press. Resnik, Michael, 1997, Mathematics as a Science of Patterns , Oxford: Oxford University Press. Restall, Greg, 2003, “Just what Is full-blooded platonism?”, Philosophia Mathematica , 11(1): 8291. Relationship! Shapiro, Stewart, 1997, Philosophy of Mathematics: Structure and Ontology , Oxford: Oxford University Press. Essays On Future Technology! Wright, Crispin, 1983, Frege's Conception of Numbers as Objects , Aberdeen: Aberdeen University Press. Wright, Crispin, 1992, Truth and essay Objectivity , Cambridge, MA: Harvard University Press.
[Please contact the conclusions essays, author with suggestions.] Thanks to relationship essay Philip Ebert, Leon Horsten, James Ladyman, Hannes Leitgeb, David Liggins, Alexander Paseau, and Philip Welch for essays on artists comments and discussion. Thanks also to an audience at the ECAP6 in Krakow, where parts of this material were presented. This article was written during a period of leave funded by an AHRC Research Leave Grant (number AH/E003753/1). Essay! I gratefully acknowledge their support. The Encyclopedia Now Needs Your Support. Please Read How You Can Help Keep the Encyclopedia Free.
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