About site: Philosophy/Philosophy of Logic - Logic and Mathematics

About site: http://www.math.psu.edu/simpson/papers/philmath/

Title: Philosophy/Philosophy of Logic - Logic and Mathematics Essay on the nature of logic and its relationship to the philosophy of mathematics, written by Stephen G. Simpson.

White_Balloon Includes views about sexual assault, approaches to assault, survivor support and resources for residents of Queensland.

blogbook_org An online resource for readers and writers of law-oriented Blogs. [RSS]

Super_Tape_and_Discs Offer devotional music and Sanskrit chants in cassettes and compact discs.

Marks_Mind This is site of quotes and ideas with a Zen twist. Of stories, poetry, and inspiration; where I explore life and truth.

Morris,_Donna_Michelle Musings of this Toronto lesbian comedian.

Encyclopedia_Mythica__Principal_Gods_of_Norse_Mythology Provides a brief genealogy table.

Alexa statistic for http://www.math.psu.edu/simpson/papers/philmath/

Please visit: http://www.math.psu.edu/simpson/papers/philmath/

Related sites for http://www.math.psu.edu/simpson/papers/philmath/

Lawtech_Australia Calculators for personal injury, discount multipliers, time lapse life expectancy, superannuation, tax calculator, interest calculator.

UK_People_Search A UK based investigator offering skip tracing, adoption tracing, and background checks. International clients accepted.

Kira_Chase\'s_Fiction_&_Poetry Sizzling romance novels for women loving women.

Aamon From Wikipedia. Information about the demon Aamon.

Petition_to_Revive_Federal_Parole Online petition to revive Federal parole for US inmates.

Mcandrew,_Sean Research engineer. Personal information, news, writings, and links.

Speck,_Jennifer_L__-_Britapalooza_00 An American anglophile's view of Britain with stories, photos, links, and resources about the UK.

National_Conference_of_State_Social_Security_Administrators Information for federal, state, and local government agencies concerning Social Security coverage, tax policy and benefits for government employees.

Reese Family history of Christopher Boone Reese, descendent of Daniel Boon. Includes readable copy of the book 'The Adventures of Col. Daniel Boon' by biographer John Filson.

Sydney_Outdoor_Lighting_Improvement_Society_[SOLIS] Includes a range of general information about how anyone can improve their lighting as well as information on how the group is combating the problem in Sydney.

Terrorism-Drug_Connection Writer Mark S. Steinitz chronicles the relationship between terrorist groups, drug cartels and government agencies in Latin America's Andean region. [PDF]

Creighton_Students_For_Life Pro-life group features its mission, officers, and upcoming activities. Resources and student reports available.

The_Journey_of_Alvar_Nuñez_Cabeza_De_Vaca A detailed historical account.

Xavier_Gold__Me_Against_The_World A blog with posts about the (fictional) life of Xavier Gold and the bizarre things that happen to him, and also his views on current events and other subjects.

Against_Politics Libertarian theory from the Anarcho-Capitalist point of view.

University_Baptist_Church Austin, Texas (American Baptist). Their Open Circle ministry hosts an annual fall retreat open to all gay, lesbian, bisexual, transgendered and straight but not narrow people of all faiths.

The_State_of_Conservatism An Arizona college student's commentary and thoughts on conservative ideology and relevant news.

The_(tm) Outdoing Leonardo Finance by trademarking the word "the".

Children_with_Hairloss Organization offers assistance in the form of hair replacement to children under the age of 21 years.

Hall,_Debby__Heart_of_Gratitude Testimony, poetry, and links for kids. Background music.

This is websites2007.org cache of m/ as retrieved on 2008.07.24 websites2007.org's cache is the snapshot that we took of the page as we crawled the web. The page may have changed since that time.

Logic and Mathematics $next_inactive$ $up$ $previous$

Logic and Mathematics

Stephen G. Simpson Department of MathematicsApril 30, 1999 Pennsylvania State UniversityThis article is an overview of logic and the philosophy ofmathematics. It is intended for the general reader. It has appearedin the volume The Examined Life: Readings from Western Philosophy from Plato to Kant, edited by Stanley Rosen, publishedin 2000 by Random House.

LogicAristotelean logicThe predicate calculusFoundations of mathematicsThe geometry of EuclidFormal theories for mathematicsPhilosophy of mathematicsPlato and AristotleThe 20th centuryThe futureBibliography

Logic

Logic is the science of formal principles of reasoning orcorrect inference. Historically, logic originated with the ancientGreek philosopher Aristotle. Logic was further developed andsystematized by the Stoics and by the medieval scholasticphilosophers. In the late 19th and 20th centuries, logic sawexplosive growth, which has continued up to the present.One may ask whether logic is part of philosophy or independent of it.According to Bochenski [2, §10B], this issue isnowhere explicitly raised in the writings of Aristotle. However,Aristotle did go to great pains to formulate the basic concepts oflogic (terms, premises, syllogisms, etc.) in a neutral way,independent of any particular philosophical orientation. ThusAristotle seems to have viewed logic not as part of philosophy butrather as a tool or instrument1 to be used by philosophers and scientistsalike. This attitude about logic is in agreement with the modernview, according to which the predicate calculus (see 1.2below) is a general method or framework not only for philosophicalreasoning but also for reasoning about any subject matter whatsoever.Logic is the science of correct reasoning. What then is reasoning?According to Aristotle [13, Topics, 100a25], reasoning isany argument in which certain assumptions or premises are laiddown and then something other than these necessarily follows. Thuslogic is the science of necessary inference. However, when logic isapplied to specific subject matter, it is important to note that notall logical inference constitutes a scientifically validdemonstration. This is because a piece of formally correct reasoningis not scientifically valid unless it is based on a true and primarystarting point. Furthermore, any decisions about what is true andprimary do not pertain to logic but rather to the specific subjectmatter under consideration. In this way we limit the scope of logic,maintaining a sharp distinction between logic and the other sciences.All reasoning, both scientific and non-scientific, must take placewithin the logical framework, but it is only a framework, nothingmore. This is what is meant by saying that logic is a formalscience.For example, consider the following inference:Some real estate will increase in value.Anything that will increase in value is a good investment.Therefore, some real estate is a good investment.This inference is logically correct, because the conclusion ``somereal estate is a good investment'' necessarily follows once we acceptthe premises ``some real estate will increase in value'' and``anything that will increase in value is a good investment''. Yetthis same inference may not be a demonstration of its conclusion,because one or both of the premises may be faulty. Thus logic canhelp us to clarify our reasoning, but it can only go so far. The realissue in this particular inference is ultimately one of finance andeconomics, not logic.We shall now briefly indicate the basics of Aristotelean logic.

Aristotelean logic

Aristotle's collection of logical treatises is known as the Organon.Of these treatises, the Prior Analytics contains the most systematicdiscussion of formal logic. In addition to the Organon, theMetaphysics2also contains relevant material. See Aristotle [13] andRoss [19].Subjects and predicatesAristotelean logic begins with the familiar grammatical distinctionbetween subject and predicate. A subject is typically anindividual entity, for instance a man3 or a house or acity. It may also be a class of entities, for instance all men. Apredicate is a property or attribute or mode of existence whicha given subject may or may not possess. For example, an individualman (the subject) may or may not be skillful (the predicate), and allmen (the subject) may or may not be brothers (the predicate).The fundamental principles of predication are:Identity. Everything is what it is and acts accordingly. In symbols: $A$ is $A$ . For example, an acorn will grow into an oak tree and nothing else.Non-contradiction. It is impossible for a thing both to be and not to be. A given predicate cannot both belong and not belong to a given subject in a given respect at a given time. Contradictions do not exist. Symbolically: $A$ and non- $A$ cannot both be the case. For example, an honest man cannot also be a thief.Either-or. Everything must either be or not be. A given predicate either belongs or does not belong to a given subject in a given respect at a given time. Symbolically: Either $A$ or non- $A$ . For example, a society must be either free or not free.These principles have exercised a powerful influence on subsequentthinkers. For example, the 20th century intellectual Ayn Rand titledthe three main divisions of her best-selling philosophical novelAtlas Shrugged4[18] after the three principles above, in tribute toAristotle.SyllogismsAccording to Aristotelean logic, the basic unit of reasoning is thesyllogism. For example, the real estate inference which waspresented above is a syllogism. It is of the formSome $A$ is $B$ .All $B$ is $C$ .Therefore, some $A$ is $C$ .Here $A$ denotes real estate, $B$ denotes increase in value, and $C$ denotes a good investment. Just as in the case of this example, everysyllogism consists of two premises and one conclusion. Each of thepremises and the conclusion is of one of four types:universal affirmative:All $A$ is $B$ . universal negative:No $A$ is $B$ . particular affirmative:Some $A$ is $B$ . particular negative:Some $A$ is not $B$ . The letters $A$ , $B$ , $C$ are known as terms. Every syllogismcontains three terms. The two premises always share a common termwhich does not appear in the conclusion. This is known as themiddle term. In our real estate example, the middle term is $B$ , i.e., that which increases in value.In order to classify the various types of syllogisms, one must takeaccount of certain symmetries. In particular, ``no $A$ is $B$ '' and``no $B$ is $A$ '' are equivalent, as are ``some $A$ is $B$ '' and``some $B$ is $A$ ''. Furthermore, the order of the two premises in asyllogism does not matter. Allowing for these symmetries, we canenumerate a total of $126$ possible syllogistic forms. Of these $126$ , only $11$ represent correct inferences. For example, the formall $A$ is $B$ , all $B$ is $C$ , therefore all $A$ is $C$ represents a correct inference, whileall $A$ is $B$ , all $C$ is $B$ , therefore some $A$ is $C$ does not. The classification of syllogisms leads to a rather complex theory.Medieval thinkers perfected it and developed ingenious mnemonics toaid in distinguishing the correct forms from the incorrect ones. Thisculminated in the famous pons asinorum (``bridge of asses''),an intricate diagram which illustrates all of the syllogistic forms bymeans of a contrast between the good and the pleasurable. SeeBochenski [2, §24H, §32F].

The predicate calculus

In 1879 the German philosopher Gottlob Frege published a remarkabletreatise, the Begriffsschrift (``concept script'') [22]. Thisbrilliant monograph is the origin of modern logical theory. However,Frege's account was defective in several respects, and notationallyawkward to boot. Instead of Frege's system, we shall present astreamlined system known as first-order logic or the predicate calculus.The predicate calculus dates from the 1910's and 1920's. It is basicfor all subsequent logical research. It is a very general system oflogic which accurately expresses a huge variety of assertions andmodes of reasoning. We shall see that it is much more flexible thanthe Aristotelean syllogistic.Predicates and individualsIn the predicate calculus, the subject/predicate distinction is drawnsomewhat differently from the way it is drawn in Aristotelean logic.The main point here is that, in the predicate calculus, a subject isalways an individual entity, never a class of entities. For example,an individual man can be treated as a subject, but the class of allmen must be treated as a predicate. Since a subject in the predicatecalculus is always an individual entity, it is usual to speak ofindividuals rather than subjects. We shall follow thiscustomary practice.The predicate calculus makes heavy use of symbolic notation.Lower-case letters $a$ , $b$ , $c$ , ..., $x$ , $y$ , $z$ , ... areused to denote individuals. Upper-case letters $M$ , $N$ , $P$ , $Q$ , $R$ , ... are used to denote predicates. Simple assertions may beformed by juxtaposing a predicate with an individual.For example, if $M$ is the predicate ``to be a man'' and $a$ is theindividual ``Socrates'', then $Ma$ denotes the assertion ``Socrates isa man''. The symbol $a$ is called an argument of $M$ . Thepredicate $M$ may be applied to any individual, and that individual isthen an argument of $M$ . If $b$ is the individual ``New York'', then $Mb$ asserts, falsely, that New York is a man. In general, if $x$ isany individual whatsoever, then $Mx$ is the assertion that $x$ is aman. This assertion may or may not be true, depending on what $x$ is.The expression $Mx$ is called an atomic formula of thepredicate calculus.Some predicates require more than one argument. For example, if $B$ is the predicate ``bigger than'', then $Bxy$ denotes the assertion`` $x$ is bigger than $y$ ''. Thus $B$ requires two arguments, and $Bxy$ is an atomic formula. If we try to use $B$ with only oneargument, we obtain something like $Bx$ , i.e., `` $x$ is bigger than''.This is not an atomic formula or any other kind of assertion. It isonly a meaningless combination of symbols. In analogy with Englishgrammar, we could say that $Bxy$ is like a grammatically correctsentence, while $Bx$ is merely a sentence fragment. Such fragmentsplay no role in the predicate calculus.Let us now go into more detail about the role of individuals in thepredicate calculus. We have already said that lower-case lettersdenote individuals. We now divide the lower-case letters into twogroups: $a$ , $b$ , $c$ , ... near the beginning of the alphabet, and $x$ , $y$ , $z$ , ... near the end of the alphabet. We insist on animportant grammatical or logical distinction between these two groups.Letters of the first group are known as individual constants orsimply constants. As in the above examples, we think of themas denoting specific individuals, such as Socrates or New York.Letters of the second group are known as individual variablesor simply variables. For example, $x$ is a variable. We thinkof $x$ as denoting not a specific individual but rather an arbitraryor unspecified individual.5Formulas and logical operatorsWe have already mentioned two kinds of symbols: lower-case letters forindividuals (constants and variables), and upper-case letters forpredicates. In addition to these, the predicate calculus employsseven special symbols known as logical operators6: $\& \qquad\lor\qquad\lnot\qquad\supset \qquad\equiv \qquad\forall\qquad\exists$ The names and meanings of the logical operators are given bysymbolnameusagemeaning $\&$ conjunction $\dots \& \dots$ ``both ... and ...'' $\lor$ disjunction $\dots\lor\dots$ ``either ... or ... (or both)'' $\lnot$ negation $\lnot\dots$ ``it is not the case that ...'' $\supset$ implication7 $\dots\supset \dots$ ``if ... then ...'' $\equiv$ bi-implication8 $\dots\equiv \dots$ ``... if and only if ...'' $\forall$ universal quantifier $\forall x \dots$ ``for all $x$ , ...'' $\exists$ existential quantifier $\exists x \dots$ ``there exists $x$ such that ...'' Here $x$ is anyvariable.A formula is a meaningful expression built up from atomicformulas by repeated application of the logical operators. In theabove table, an ellipsis mark ... stands for a formula within alarger formula.For example, suppose we have a predicate $M$ meaning ``is a man'',another predicate $T$ meaning ``is a truck'', and another predicate $D$ meaning ``drives''. Here $M$ and $T$ are predicates which requireonly one argument apiece. The predicate $D$ requires two arguments:the driver, and the vehicle being driven. Thus $Mx$ , $Ty$ , and $Dxy$ are atomic formulas meaning `` $x$ is a man'', `` $y$ is a truck'', and`` $x$ drives $y$ '', respectively. A typical formula built from theseatomic formulas is $\forall x (Mx\supset \exists y (Ty \& Dxy))$ which we can translate as ``for all $x$ , if $x$ is a man then thereexists $y$ such that $y$ is a truck and $x$ drives $y$ ''. In otherwords,Every man drives at least one truck.Similarly, the formula $\forall y (Ty\supset \exists x (Mx \& Dxy))$ translates toEvery truck is driven by at least one man.In writing formulas, we often use parentheses as punctuation marks toindicate grouping and thereby remove ambiguity. If parentheses werenot used, one could construe the formula $\lnot Ty \& Dxy$ in twologically inequivalent ways: as $(\lnot Ty) \& Dxy$ (`` $y$ is not atruck, and $x$ drives $y$ ''), or as $\lnot (Ty \& Dxy)$ (`` $y$ is nota truck that $x$ drives''). The parentheses allow us to choose themeaning that we intend.The predicate calculus is very rich in expressive power. For example,the four Aristotelean premise types discussed in 1.1.2 caneasily be rendered as formulas of the predicate calculus. Letting $A$ and $B$ be predicates which require one argument apiece, we haveuniversal affirmativeall $A$ is $B$ $\forall x (Ax\supset Bx)$ universal negativeno $A$ is $B$ $\forall x (Ax\supset \lnot Bx)$ particular affirmativesome $A$ is $B$ $\exists x (Ax \& Bx)$ particular negativesome $A$ is not $B$ $\exists x (Ax \& \lnot Bx)$ In the second line of this table, the universal negative ``no $A$ is $B$ '' could have been rendered equivalently as $\lnot\existsx (Ax \& Bx)$ , or as $\forall x (Bx\supset \lnot Ax)$ .The above table may tend to gloss over a subtle but philosophicallysignificant difference between Aristotelean logic and the predicatecalculus. Namely, where Aristotelean logic views $A$ as a subject and $B$ as a predicate, the predicate calculus views both $A$ and $B$ aspredicates. This is typical of the different perspectives involved.Aristotelean logic emphasizes the universal essences of subjects orentities, while the predicate calculus elevates predicates to aposition of supreme importance.Logical validity and logical consequenceA formula of the predicate calculus is said to be logically valid if it is necessarily always true, regardless of the specificpredicates and individuals involved. For example, the threefundamental principles of Aristotelean logic (see 1.1.1above) correspond to formulas as follows:Identity: $\forall x (Ax\equiv Ax)$ .Non-contradiction: $\lnot\exists x (Ax \& \lnot Ax)$ .Either-or: $\forall x (Ax\lor\lnot Ax)$ .These formulas are logically valid, because they are ``necessarily''or ``automatically'' or ``formally'' true, no matter what predicatemay be denoted by the symbol $A$ .The predicate calculus concept of logical validity subsumes theAristotelean syllogism. Each syllogism corresponds to a logicallyvalid implication $(\Phi_1 \& \Phi_2)\supset \Psi$ where $\Phi_1$ and $\Phi_2$ are formulas expressing the two premisesand $\Psi$ expresses the conclusion. For example, the syllogismsome $A$ is $B$ , all $B$ is $C$ , therefore some $A$ is $C$ has a predicate calculus rendition $((\exists x (Ax \& Bx)) \& (\forall x (Bx\supset Cx))) \supset (\exists x (Ax \& Cx))$ and this formula is logically valid.More generally, a formula $\Psi$ is said to be a logical consequence of a set of formulas $\Phi_1$ , ..., $\Phi_n$ just incase $(\Phi_1 \& \cdots \& \Phi_n)\supset \Psi$ is logically valid. Here $\Phi_1$ , ..., $\Phi_n$ are premises and $\Psi$ is a conclusion. This is similar to the Aristoteleansyllogism, but it is of wider applicability, because the premises andthe conclusion can be more complex. As an example, the 19th centurylogician Augustus DeMorgan noted9 that the inferenceall horses are animals,therefore, the head of a horse is the head of an animalis beyond the reach of Aristotelean logic. Yet this same inferencemay be paraphrased as ``if all horses are animals, then for all $x$ ,if $x$ is the head of some horse then $x$ is the head of someanimal'', and this corresponds to a logically valid formula $(\forall y (Hy\supset Ay)) \supset (\forall x ((\exists y (Rxy \& Hy))\supset (\exists y (Rxy \& Ay))))$ of the predicate calculus. Here $H$ , $A$ , $R$ denote ``is a horse'',``is an animal'', ``is the head of'', respectively. Thus DeMorgan'sconclusion is indeed a logical consequence of his premise.The completeness theoremFormulas of the predicate calculus can be exceedingly complicated.How then can we distinguish the formulas that are logically valid fromthe formulas that are not logically valid? It turns out that there isan algorithm10 for recognizing logicallyvalid formulas. We shall now sketch this algorithm.In order to recognize that a formula $\Phi$ is logically valid, itsuffices to construct what is known as a proof tree for $\Phi$ ,or equivalently a refutation tree for $\lnot\Phi$ . This is atree which carries $\lnot\Phi$ at the root. Each node of the treecarries a formula. The growth of the tree is guided by the meaning ofthe logical operators appearing in $\Phi$ . New nodes are added to thetree depending on what nodes have already appeared. For example, if anode carrying $\lnot (\Phi_1 \& \Phi_2)$ has appeared, we create twonew nodes carrying $\lnot\Phi_1$ and $\lnot\Phi_2$ respectively. Thethought behind these new nodes is that the only way for $\lnot (\Phi_1 \& \Phi_2)$ to be the case is if at least one of $\lnot\Phi_1$ or $\lnot\Phi_2$ is the case. Similarly, if a nodecarrying $\lnot\forall x \Psi$ has already appeared, we create a newnode carrying $\lnot\Psi'$ , where $\Psi'$ is the result ofsubstituting a new constant $a$ for the variable $x$ throughout theformula $\Psi$ . The idea here is that the only way for the universalstatement $\forall x \Psi$ to be false is if $\Psi$ is false for someparticular $x$ . Since $a$ is a new constant, $\Psi'$ is a formulawhich may be considered as the most general false instance of $\Psi$ .Corresponding to each of the seven logical operators, there areprescribed procedures for adding new nodes to the tree. We applythese procedures repeatedly until they cannot be applied any more. Ifexplicit contradictions11 are discoveredalong each and every branch of the tree, then we have a refutationtree for $\lnot\Phi$ . Thus $\lnot\Phi$ is seen to be logicallyimpossible. In other words, $\Phi$ is logically valid.The adequacy of proof trees for recognizing logically valid formulasis a major insight of 20th century logic. It is a variant of thefamous completeness theorem, first proved in 1930 by the greatlogician Kurt Gödel [5,22].On the other hand, the class of logically valid formulas is known tobe extremely complicated. Indeed, this class is undecidable:there is no algorithm12 which accepts as input an arbitrary formula $\Phi$ and outputs ``yes'' if $\Phi$ is logically valid and ``no'' if $\Phi$ is not logically valid. In this sense, the concept of logicalvalidity is too general and too intractable to be analyzed thoroughly.There will never be a predicate calculus analog of the pons asinorum.Formal theoriesThe predicate calculus is a very general and flexible framework forreasoning. By choosing appropriate predicates, one can reason aboutany subject whatsoever. These considerations lead to the notion of aformal theory.In order to specify a formal theory, one first chooses a smallcollection of predicates which are regarded as basic for a given fieldof study. These predicates are the primitives of the theory.They delimit the scope of the theory. Other predicates must bedefined in terms of the primitives. Using them, one writes downcertain formulas which are regarded as basic or self-evident withinthe given field of study. These formulas are the axioms of thetheory. It is crucial to make all of our underlying assumptionsexplicit as axioms. Once this has been done, a theorem is anyformula which is a logical consequence of the axioms. A formal theory is this structure of primitives, axioms, and theorems.As a frivolous example, we could envision a theory of cars, trucks,and drivers. We would begin with some primitives such as $C$ (``is acar''), $T$ (``is a truck''), $D$ (``drives''), $M$ (``is a man''),etc. We could then write down certain obvious or self-evident axiomssuch as $\forall x (Mx\supset \lnot Cx)$ (``no man is a car''), $\forallx ((\exists y Dxy)\supset Mx)$ (``every driver is a man''), etc.Then, within the constraints imposed by the axioms, we couldinvestigate the logical consequence relationships among variousnon-obvious assertions, such as $\lnot\exists x (Mx \& \exists y (Dxy \& Cy) \& \existsz (Dxz \& Tz))$ (``nobody drives both a car and a truck''). Additional predicates $V$ (``is a vehicle'') and $P$ (``is a driver'') can be defined in termsof the primitives. The defining axioms for $V$ and $P$ would be $\forall y (Vy\equiv (Cy\lor Ty))$ and $\forall x (Px\equiv \existsy Dxy)$ , respectively. In this fashion, we could attempt to codifyall available knowledge about vehicles and drivers.More seriously, one could try to write down formal theoriescorresponding to various scientific disciplines, such as mechanics orstatistics or law. In this way one could hope to analyze the logicalstructure of the respective disciplines.The process of codifying a scientific discipline by means ofprimitives and axioms in the predicate calculus is known asformalization. The key issue here is the choice of primitivesand axioms. They cannot be chosen arbitrarily. The scientist whochooses them must exercise a certain aesthetic touch. They must besmall in number; they must be basic and self-evident; and they mustaccount for the largest possible number of other concepts and facts.To date, this kind of formal theory-building has been convincinglycarried out in only a few cases. A survey is in Tarski [21].The most notable successes have been in mathematics.

Foundations of mathematics

Mathematics is the science of quantity. Traditionally therewere two branches of mathematics, arithmetic and geometry, dealingwith two kinds of quantities: numbers and shapes. Modern mathematicsis richer and deals with a wider variety of objects, but arithmeticand geometry are still of central importance.Foundations of mathematics is the study of the most basicconcepts and logical structure of mathematics, with an eye to theunity of human knowledge. Among the most basic mathematical conceptsare: number, shape, set, function, algorithm, mathematical axiom,mathematical definition, mathematical proof.The reader may reasonably ask why mathematics appears at all in thisvolume. Isn't mathematics too narrow a subject? Isn't the philosophyof mathematics of rather specialized interest, all the more so incomparison to the broad humanistic issues of philosophy proper, issuessuch as the good, the true, and the beautiful?There are three reasons for discussing mathematics in a volume ongeneral philosophy:Mathematics has always played a special role in scientific thought. The abstract nature of mathematical objects presents philosophical challenges that are unusual and unique.Foundations of mathematics is a subject that has always exhibited an unusually high level of technical sophistication. For this reason, many thinkers have conjectured that foundations of mathematics can serve as a model or pattern for foundations of other sciences.The philosophy of mathematics has served as a highly articulated test-bed where mathematicians and philosophers alike can explore how various general philosophical doctrines play out13 in a specific scientific context.The purpose of this section is to indicate the role of logic in thefoundations of mathematics. We begin with a few remarks on thegeometry of Euclid. We then describe some modern formal theories formathematics.

The geometry of Euclid

Above the gateway to Plato's academy appeared a famous inscription:Let no one who is ignorant of geometry enter here.In this way Plato indicated his high opinion of geometry. Accordingto Heath [9, page 284], Plato regarded geometry as``the first essential in the training of philosophers'', because ofits abstract character. See also Plato [17, Republic, 527B].In the Posterior Analytics [13], Aristotle laid down thebasics of the scientific method.14 The essence of themethod is to organize a field of knowledge logically by means ofprimitive concepts, axioms, postulates, definitions, and theorems.The majority of Aristotle's examples of this method are drawn fromarithmetic and geometry[1,7,9].The methodological ideas of Aristotle decisively influenced thestructure and organization of Euclid's monumental treatise ongeometry, the Elements [8]. Euclid begins with 21definitions, five postulates, and five common notions. After that,the rest of the Elements are an elaborate deductive structureconsisting of hundreds of propositions. Each proposition is justifiedby its own demonstration. The demonstrations are in the form ofchains of syllogisms. In each syllogism, the premises are identifiedas coming from among the definitions, postulates, common notions, andpreviously demonstrated propositions. For example, in Book I of theElements, the demonstration of Proposition 16 (``in any triangle, ifone of the sides be produced, the exterior angle is greater thaneither of the interior and opposite angles'') is a chain of syllogismswith Postulate 2, Common Notion 5, and Propositions 3, 4 and 15 (``iftwo straight lines cut one another, they make the vertical anglesequal to one another'') occurring as premises. It is true that thesyllogisms of Euclid do not always conform strictly to Aristoteleantemplates. However, the standards of rigor are very high, andAristotle's influence is readily apparent.The logic of Aristotle and the geometry of Euclid are universallyrecognized as towering scientific achievements of ancient Greece.

Formal theories for mathematics

A formal theory for geometryWith the advent of calculus in the 17th and 18th centuries,mathematics developed very rapidly and with little attention tological foundations. Euclid's geometry was still regarded as a modelof logical rigor, a shining example of what a well-organizedscientific discipline ideally ought to look like. But the prolificEnlightenment mathematicians such as Leonhard Euler showed almost nointerest in trying to place calculus on a similarly firm foundation.Only in the last half of the 19th century did scientists begin to dealwith this foundational problem in earnest. The resulting crisis hadfar-reaching consequences. Even Euclid's geometry itself came undercritical scrutiny. Geometers such as Moritz Pasch discovered whatthey regarded as gaps or inaccuracies in the Elements. Greatmathematicians such as David Hilbert entered the fray.An outcome of all this foundational activity was a thorough reworkingof geometry, this time as a collection of formal theories within thepredicate calculus. Decisive insights were obtained by Alfred Tarski.We shall sketch Tarski's formal theory for Euclidean15 planegeometry.16As his primitive predicates, Tarski takes $P$ (``point''), $B$ (``between''), $D$ (``distance''), $I$ (``identity''). The atomicformulas $Px$ , $Bxyz$ , $Dxyuv$ , and $Ixy$ mean `` $x$ is a point'',`` $y$ lies between $x$ and $z$ '', ``the distance from $x$ to $y$ isequal to the distance from $u$ to $v$ '', and `` $x$ is identical to $y$ '', respectively. Geometrical objects other than points, such asline segments, angles, triangles, circles, etc., are handled by meansof the primitives. For example, the circle with center $x$ and radius $uv$ consists of all points $y$ such that $Dxyuv$ holds.In geometry, two points $x$ and $y$ are considered identical if thedistance between them is zero. Tarski expresses this by means of anaxiom $\forall x \forall y \forall z (Dxyzz\supset Ixy)$ .Another axiom $\forall w \forall x \forall y \forall z ((Bwxy \& Bwyz)\supsetBxyz)$ expresses the fact that, given any four points, if the second isbetween the first and the third, and if the third is between the firstand the fourth, then the third is between the second and the fourth.A noteworthy axiom is $\forall x \forall y \forall z \forall u \forall v ((Dxuxv \& Dyuyv \& Dzuzv \& \lnot Iuv)\supset (Bxyz\lor Bxzy\lor Byxz))$ which says: any three points $x,y,z$ equidistant from two distinctpoints $u,v$ must be collinear. This axiom is typical oftwo-dimensional (i.e., plane) geometry and does not apply togeometries of dimension greater than two.Altogether Tarski presents twelve axioms, plus an additionalcollection of axioms expressing the idea that a line is continuous.The full statement of Tarski's axioms for Euclidean plane geometry isgiven at [10, pages 19-20]. Let $T_g$ be the formal theorybased on Tarski's axioms.Remarkably, Tarski has demonstrated that $T_g$ is complete.This means that, for any purely geometrical17 statement $\Psi$ , either $\Psi$ or $\lnot\Psi$ is a theorem of $T_g$ . Thus wesee that the axioms of $T_g$ suffice to answer all yes/no questions ofEuclidean plane geometry. Combining this with the completenesstheorem of Gödel, we find that $T_g$ is decidable: there isan algorithm18 which accepts as input an arbitrarystatement of plane Euclidean geometry, and outputs ``true'' if thestatement is true, and ``false'' if it is false. This is a triumph ofmodern foundational research.A formal theory for arithmeticBy arithmetic we mean elementary school arithmetic, i.e., thestudy of the positive whole numbers $1$ , $2$ , $3$ , ... along withthe familiar operations of addition ( $+$ ) and multiplication( $\times$ ). This part of mathematics is obviously fundamental, yet itturns out to be surprisingly complicated. Below we write down some ofthe axioms which go into a formal theory of arithmetic.19Our primitive predicates for arithmetic are $N$ (``number''), $A$ (``addition''), $M$ (``multiplication''), $I$ (``identity''). Theatomic formulas $Nx$ , $Axyz$ , $Mxyz$ , $Ixy$ mean `` $x$ is a number'',`` $x+y=z$ '', `` $x\times y=z$ '', `` $x=y$ '', respectively. Our axiomswill use the predicates $N$ , $A$ , $M$ , $I$ to assert that for anygiven numbers $x$ and $y$ , the numbers $x+y$ and $x\times y$ alwaysexist and are unique. We shall also have axioms expressing some wellknown arithmetical laws:substitution laws:if $x=y$ and $x$ is a number then $y$ is a number, etc.commutative laws: $x+y=y+x$ and $x\times y=y\times x$ .associative laws: $(x+y)+z=x+(y+z)$ and $(x\times y)\timesz=x\times (y\times z)$ .distributive law: $x\times(y+z)=(x\times y)+(x\times z)$ .comparison law: $x\ne y$ if and only if, for some $z$ , $x+z=y$ or $x=y+z$ .unit law: $x\times1=x$ . Our formal axioms for arithmetic are as follows.substitution laws: $\forall x Ixx$ $\forall x \forall y (Ixy\equiv Iyx)$ $\forall x \forall y \forall z ((Ixy \& Iyz)\supset Ixz)$ $\forall x \forall y (Ixy\supset (Nx\equiv Ny))$ existence and uniqueness of $x+y$ : $\forall x \forall y \forall z \forall u \forall v \forall w ((Ixu \& Iyv \& Izw)\supset (Axyz\equiv Auvw))$ $\forall x \forall y \forall z (Axyz\supset (Nx \& Ny \& Nz))$ $\forall x \forall y ((Nx \& Ny)\supset \exists w \forall z (Iwz\equiv Axyz))$ existence and uniqueness of $x\times y$ : $\forall x \forall y \forall z \forall u \forall v \forall w ((Ixu \& Iyv \& Izw)\supset (Mxyz\equiv Muvw))$ $\forall x \forall y \forall z (Mxyz\supset (Nx \& Ny \& Nz))$ $\forall x \forall y ((Nx \& Ny)\supset \exists w \forall z (Iwz\equiv Mxyz))$ commutative laws: $\forall x \forall y \exists z (Axyz \& Ayxz)$ $\forall x \forall y \exists z (Mxyz \& Myxz)$ associative laws: $\forall x \forall y \forall z \exists u \exists v \exists w (Axyu \& Auzw \& Ayzv \& Axvw)$ $\forall x \forall y \forall z \exists u \exists v \exists w (Mxyu \& Muzw \& Myzv \& Mxvw)$ distributive law: $\forall x \forall y \forall z \exists t \exists u \exists v \exists w (Ayzt \& Mxtw \& Mxyu \& Mxzv \& Auvw)$ comparison law: $\forall x \forall y ((Nx \& Ny)\supset (Ixy\equiv \lnot\exists z (Axzy\lor Ayzx)))$ unit law: $\exists z (Nz \& (\lnot\exists x \exists y Axyz) \& \forall w (Nw\supset Mwzw))$ Let $T_a$ be the formal theory specified by the above primitives andaxioms.It is known that $T_a$ suffices to derive many familiar arithmeticalfacts. For example, $2+2=4$ may be expressed, awkwardly20 to be sure, as $(1+1)+(1+1)=((1+1)+1)+1$ or $\exists x \exists y \exists z \exists w (Mxxx \& Axxy \& Axyz \& Axzw \& Ayyw)$ 21and this formula is indeed a theorem of $T_a$ , i.e., a logicalconsequence of the axioms of $T_a$ . Another theorem of $T_a$ is $\forall x \forall y \forall z \forall w (((Axzw \& Ayzw)\lor(Mxzw \& Myzw))\supset Ixy)$ expressing a familiar cancellation law: if either $x+z=y+z$ or $x\times z=y\times z$ , then $x=y$ .On the other hand, the axioms of $T_a$ are by no means exhaustive.They can be supplemented with other axioms expressing the so-calledmathematical induction or least number principle: ifthere exists a number having some well-defined property, then amongall numbers having the property there is a smallest one. Theresulting formal theory is remarkably powerful, in the sense that itstheorems include virtually all known arithmetical facts. But it isnot so powerful as one might wish. Indeed, any formal theory whichincludes $T_a$ is necessarily either inconsistent22 or incomplete. Thus there is no hope of writing downenough axioms or developing an algorithm to decide all arithmeticalfacts. This is a variant of the famous 1931 incompleteness theorem of Gödel [5,22]. There are severalmethods of coping with the incompleteness phenomenon, and thisconstitutes a currently active area of research in foundations ofmathematics.The contrast between the completeness of formal geometry and theincompleteness of formal arithmetic is striking. Both sides of thisdichotomy are of evident philosophical interest.A formal theory of setsOne of the aims of modern logical research is to devise a singleformal theory which will unify all of mathematics. Such a theory willnecessarily be subject to the Gödel incompleteness phenomenon,because it will incorporate not only $T_g$ but also $T_a$ .One approach to a unified mathematics is to straightforwardly embedarithmetic into geometry, by identifying whole numbers with evenlyspaced points on a line. This idea was familiar to the ancientGreeks. Another approach is to explain geometry in terms ofarithmetic and algebra, by means of coordinate systems, like latitudeand longitude on a map. This idea goes back to the 17th centurymathematician and philosopher René Descartes and the 19th centurymathematician Karl Weierstrass. Both approaches give rise toessentially the same formal theory, known as second-order arithmetic.23 This theory includes both $T_a$ and $T_g$ and is adequate for the bulk of modern mathematics. Thus the decisionabout whether to make geometry more fundamental than arithmetic orvice versa seems to be mostly a matter of taste.A very different approach to a unified mathematics is via set theory. This is a peculiarly 20th century approach. It is basedon one very simple-looking concept: sets. Remarkably, this oneconcept leads directly to a vast structure which encompasses all ofmodern mathematics.A set is a collection of objects called the elements ofthe set. We sometimes use informal notations such as $x=\{y,z,\dots\}$ to indicate that $x$ is a set consisting of elements $y$ , $z$ , .... The number of elements in a set can be arbitrarilylarge or even infinite. A basic principle of set theory is that a setis determined by its elements. Thus two sets are identical if andonly if they have the same elements. This principle is known asextensionality. For example, the set $\{a,b,c\}$ is consideredto be the same set as $\{c,a,b\}$ because the elements are the same,even though written in a different order.Much of the complexity of set theory arises from the fact that setsmay be elements of other sets. For instance, the set $\{a,b\}$ is anelement of the set $\{\{a,b\},c\}$ and this is distinct from the set $\{a,b,c\}$ .For a formal theory of sets, we use three primitives: $S$ (``set''), $I$ (``identity''), $E$ (``element''). The atomic formulas $Sx$ , $Ixy$ , $Exy$ mean `` $x$ is a set'', `` $x$ is identical to $y$ '', `` $x$ is an element of $y$ '', respectively. One of the ground rules of settheory is that only sets may have elements. This is expressed as anaxiom $\forall w \forall x (Ewx\supset Sx)$ . In addition there is anaxiom of extensionality $\forall x \forall y ((Sx \& Sy)\supset (Ixy\equiv \forallw (Ewx\equiv Ewy)))$ and an axiom $\exists x (Sx \& \lnot\exists w Ewx)$ expressing theexistence of the empty set, i.e. a set $\{ \}$ having noelements. A list of all the axioms of set theory may be found intextbooks [11,15]. Let $T_s$ be the formal theory ofsets based on these axioms.The set theory approach to arithmetic is in terms of the non-negativewhole numbers $0$ , $1$ , $2$ , $3$ , .... These numbers are identifiedwith specific sets. Namely, we identify $0$ with the empty set $\{ \}$ , $1$ with $\{\{ \}\}$ , $2$ with $\{\{ \},\{\{ \}\}\}$ , $3$ with $\{\{ \},\{\{ \}\},\{\{ \},\{\{ \}\}\}\}$ , etc. In general,we identify the number $n$ with the set of smaller numbers $\{0,1,\dots,n-1\}$ . Among the axioms of $T_s$ is an axiom of infinity asserting the existence of the infinite set $\omega=\{0,1,2,3,\dots\}$ . One can use the set $\omega$ to show that $T_s$ includes a theory equivalent to $T_a$ . After that, one canfollow the ideas of Descartes and Weierstrass to see that $T_s$ alsoincludes a theory equivalent to $T_g$ . It turns out that the rest ofmodern mathematics can also be emulated within $T_s$ . This includesan elaborate theory of infinite sets which are much larger than $\omega$ .The set-theoretical approach to arithmetic and geometry is admittedlysomewhat artificial. However, the idea of basing all of mathematicson one simple concept, sets, has exerted a powerfulattraction.24 The implications of thisidea are not yet fully understood and are a topic of current research.

Philosophy of mathematics

In this section we indicate some issues and trends in the philosophyof mathematics.

Plato and Aristotle

The objects that are studied in mathematics tend to be somewhatabstract and remote from everyday perceptual experience. Therefore,the existence and nature of mathematical objects present specialphilosophical challenges. For example, is a geometrical squaredifferent from a square floor tile? If so, then where is thegeometrical square? Is it on the floor, in our minds, or somewhereelse? And what about sets? Is a set of $52$ cards something otherthan the cards themselves?The ancient Greek philosophers took such questions very seriously.Indeed, many of their general philosophical discussions were carriedon with extensive reference to geometry and arithmetic. Plato seemedto insist that mathematical objects, like the Platonic forms oressences, must be perfectly abstract and have a separate, non-materialkind of existence. Aristotle[1,7,13,19] dissectedand refuted this view in books $M$ and $N$ of the Metaphysics.According to Aristotle, the geometrical square is a significant aspectof the square floor tile, but it can only be understood by discardingother irrelevant aspects such as the exact measurements, the tilingmaterial, etc. Clearly these questions provide much food forphilosophical analysis and debate.

The 20th century

In the 20th century, the advent of the predicate calculus and thedigital computer profoundly affected our view of mathematics. Thediscovery that all of mathematics can be codified in formal theoriescreated a huge stir. One expression of this excitement was the riseof an extreme philosophical doctrine known asformalism.25According to formalism, mathematics is only a formal game, concernedsolely with algorithmic manipulation of symbols. Under this view, thesymbols of the predicate calculus do not denote predicates or anythingelse. They are merely marks on paper, or bits and bytes in the memoryof a computer. Therefore, mathematics cannot claim to be any sort ofknowledge of mathematical objects. Indeed, mathematical objects donot exist at all, and the profound questions debated by Plato andAristotle become moot. Mathematics is nothing but a kind of blindcalculation.The formalist doctrine fits well with certain modern trends incomputer science, e.g., artificial intelligence. However, formalismhas proved inadequate as an integrated philosophy of mathematics,because it fails to account for human mathematical understanding, notto mention the spectacular applications of mathematics in fields suchas physics and engineering.By way of reaction against formalism, several alternative doctrineshave been advocated. One of these is constructivism, the ideathat mathematical knowledge can be obtained by means of a series ofpurely mental constructions. Under this view, mathematical objectsexist solely in the mind of the mathematician, so mathematicalknowledge is absolutely certain. However, the status of mathematicsvis a vis the external world becomes doubtful. An extreme version ofconstructivism is so solipsistic that it does not even allow for thepossibility of mathematical communication from one mind to another.An additional disturbing feature of constructivism is that it entailsrejection of the basic laws of logic. To see how this comes about,consider some specific mathematical problem or question26 of a yes/no nature, for which theanswer is currently unknown. (Mathematics abounds with suchquestions, and the Gödel incompleteness phenomenon suggests thatsuch questions will always exist.) Express the ``yes'' answer as aformula $\Psi$ and the ``no'' answer as the negated formula $\lnot\Psi$ . Since the answer is unknown, neither $\Psi$ nor $\lnot\Psi$ is in the mind of the mathematician. Therefore, accordingto the constructivists, the disjunction $\Psi\lor\lnot\Psi$ is not alegitimate mathematical assumption. Thus Aristotle's either-orprinciple (see 1.1.1 and 1.2.3 above) must beabandoned.27Constructivism has the merit of allowing human beings to possessmathematical knowledge. However, the constructivist rejection of theexternal world and of Aristotelean logic are highly unpalatable tomost mathematicians and mathematically oriented scientists. For thisreason, constructivism remains a fringe movement on the 20th centurymathematical landscape.Another 20th century philosophical doctrine has arisen fromset-theoretical foundations. The reliance on infinite sets suggestsmany perplexing questions. What do such sets correspond to inreality? Where are they, and how can the human mind grasp them? Inorder to boldly answer these questions, and as a reaction againstformalism, many researchers in axiomatic set theory have subscribed towhat is known as set-theoretical Platonism. According to thisvariant of the Platonic doctrine, infinite sets exist in anon-material, purely mathematical realm. By extending our intuitiveunderstanding of this realm, we will be able to cope with chaosissuing from the Gödel incompleteness phenomenon. The mostprominent and frequently cited authority for this kind of Platonism isGödel himself [5].There is a good fit between set-theoretical Platonism and certainaspects of 20th century mathematical practice. However, as aphilosophical doctrine, set-theoretical Platonism leaves much to bedesired. Many of Aristotle's objections to the Platonic forms arestill cogent. There are serious questions about how a theory ofinfinite sets can be applicable to a finite world.We have mentioned three competing 20th century doctrines: formalism,constructivism, set-theoretical Platonism. None of these doctrinesare philosophically satisfactory, and they do not provide muchguidance for mathematically oriented scientists and other users ofmathematics. As a result, late 20th century mathematicians havedeveloped a split view, a kind of Kantian schizophrenia, which isusually described as ``Platonism on weekdays, formalism on weekends''.In other words, they accept the existence of infinite sets as aworking hypothesis in their mathematical research, but when it comesto philosophical speculation, they retreat to a formalist stance.Thus they have given up hope of an integrated view which accounts forboth mathematical knowledge and the applicability of mathematics tophysical reality. In this respect, the philosophy of mathematics isin a sorry state.

The future

From the Renaissance through the 20th century, Aristotle's ideas aboutthe nature of mathematical objects have been neglected and ignored.Now the time seems ripe for a renovation of the philosophy ofmathematics, based on Aristotelean and neo-Aristotelean [16]ideas and bolstered by the techniques of modern logic, including thepredicate calculus.The great mathematician David Hilbert anticipated such a renovation inhis 1925 essay, On the Infinite [22]. Hilbert was aware that,according to modern physics, the physical universe is finite. Yetinfinite sets were playing an increasingly large role in themathematics of the day. Hilbert therefore recognized that the mostvulnerable chink in the armor of mathematics was the infinite. Inorder to defend what he called ``the honor of human understanding'',Hilbert proposed to develop a new foundation of mathematics, in whichformal theories of infinite sets, such as $T_s$ , would be rigorouslyjustified by reference to the finite. This is Hilbert's program offinitistic reductionism.28Although Hilbert did not cite Aristotle, we can imagine that Hilbertwould have profited from an examination of Aristotle's distinctionbetween actual and potential infinity. An actual infinity issomething like an infinite set regarded as a completed totality. Apotential infinity is more like a finite but indefinitely long,unending series of events. According to Aristotle, actual infinitiescannot exist, but potential infinities exist in nature and aremanifested to us in various ways, for instance the indefinite cycle ofthe seasons, or the indefinite divisibility of a piece of gold.In any case, it turned out that Hilbert had stated his program in toosweeping a fashion. The wholesale finitistic reduction which Hilbertdesired cannot be carried out. This follows from Gödel'sincompleteness theorem [5,22]. The remarkableresults obtained by Gödel in 1931 caused the philosophical ideasof Hilbert's 1925 essay to fall into disrepute. Hilbert's grandfoundational program appeared to be dead, broken beyond hope ofrepair.The last 20 years have seen a revival of Hilbert's program. Recentfoundational research [20] has revealed that, although $T_s$ is not finitistically reducible, there are other formal theories whichare finitistically reducible, in the precise sense envisionedby Hilbert. Moreover, these other formal theories turn out to beadequate for a very large portion of mathematics. They do notencompass actual infinities such as $\omega$ , but they do include themain results of arithmetic and geometry and allied disciplines.This new research has not yet had an impact on the philosophy ofmathematics or on mathematical practice. Philosophers andmathematicians are free to choose which directions to pursue and whichtechniques to emphasize. Only time will reveal the future evolutionof the philosophy of mathematics.

Bibliography

1Hippocrates G. Apostle.Aristotle's Philosophy of Mathematics.University of Chicago Press, 1952.X + 228 pages.2I. M. Bochenski.A History of Formal Logic.Chelsea Publishing Company, New York, 2nd edition, 1970.Translated and edited by Ivo Thomas, XXII + 567 pages.3Haskell B. Curry.Outlines of a Formalist Philosophy of Mathematics.Studies in Logic and the Foundations of Mathematics. North-Holland, 1951.VII + 75 pages.4Melvin Fitting.First-Order Logic and Automated Theorem Proving.Graduate Texts in Computer Science. Springer-Verlag New York Inc., 2nd edition, 1996.XVI + 326 pages.5Kurt Gödel.Collected Works.Oxford University Press, 1986-1995.Volume I, XVIII + 474 pages, 1986, Volume II, XVI + 407 pages, 1990, Volume III, XX + 532 pages, 1995.6Petr Hájek and Pavel Pudlák.Metamathematics of First-Order Arithmetic.Perspectives in Mathematical Logic. Springer-Verlag, 1993.XIV + 460 pages.7Thomas Heath.Mathematics in Aristotle.Clarendon Press, Oxford, 1949.XIV + 291 pages.8Thomas Heath.The Thirteen Books of Euclid's Elements.Dover Publications, New York, 2nd revised edition, 1956.Three volumes.9Thomas Heath.A History of Greek Mathematics.Dover Publications, New York, 1981.Volume I, XV + 446 pages, Volume II, XI + 586 pages.10Leon Henkin, Patrick Suppes, and Alfred Tarski, editors.The Axiomatic Method, With Special Reference to Geometry and Physics.Studies in Logic and the Foundations of Mathematics. North-Holland, 1959.XI + 488 pages.11Thomas Jech.Set Theory.Academic Press, 1978.XI + 621 pages.12Morris Kline.Why Johnny Can't Add.St. Martin's Press, 1973.173 pages.13Richard McKeon, editor.The Basic Works of Aristotle.Random House, 1941.XXXIX + 1487 pages.14Elliott Mendelson.Boolean Algebra and Switching Circuits.Schaum's Outline Series. 1970.213 pages.15Elliott Mendelson.Introduction to Mathematical Logic.Wadsworth, 3rd edition, 1987.IX + 341 pages.16Leonard Peikoff.Objectivism: The Philosophy of Ayn Rand.Dutton, New York, 1991.XV + 493 pages.17Plato.Dialogues.Oxford, Clarendon Press, 4th edition, 1964.Translated by Benjamin Jowett.18Ayn Rand.Atlas Shrugged.Random House, New York, 1957.V + 1168 pages.19David Ross.Aristotle.Barnes and Noble, 5th revised edition, 1964.XIV + 300 pages.20Stephen G. Simpson.Subsystems of Second Order Arithmetic.Perspectives in Mathematical Logic. Springer-Verlag, 1999.XIV + 445 pages.21Alfred Tarski.Introduction to Logic and to the Methodology of Deductive Sciences.Oxford University Press, 4th edition, 1994.XXII + 229 pages.22J. van Heijenoort, editor.From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931.Harvard University Press, 1971.Second Printing. XII + 660 pages.

About this document ...

Logic and MathematicsThis document was generated using theLaTeX2HTML translator Version 2002-2-1 (1.71)Copyright © 1993, 1994, 1995, 1996,Nikos Drakos, Computer Based Learning Unit, University of Leeds.Copyright © 1997, 1998, 1999,Ross Moore, Mathematics Department, Macquarie University, Sydney.The command line arguments were: latex2html -split 0 philmathThe translation was initiated by Stephen G Simpson on 2007-09-28Footnotes... instrument1The Greek word for instrument is organon. The collection of Aristotle's logical writings is known as the Organon....Metaphysics2The Metaphysics is Aristotle's treatise on the science of existence, i.e., being as such. It includes a detailed analysis of the various ways in which a thing can be said to be.... man3We use man in the traditional sense, equivalent to ``human being''. There is no intention to exclude persons of the female gender.... Shrugged4A survey conducted for the Book-of-the-Month Club and the Library of Congress in 1991 found that Atlas Shrugged is the most influential book in the United States of America, second only to the Bible. See http://www.lcweb.loc.gov/loc/cfbook/bklists.html.... individual.5The idea of using letters such as $x$ and $y$ as variables is of great value. Historically, the creators of the predicate calculus borrowed this idea from the mathematical discipline known as algebra. Recall that algebra is a kind of generalized arithmetic. In algebra there are constants, i.e., specific quantities such as $2$ , the square root of $10$ , etc., but there are also variables such as $x$ , $y$ , etc. The key idea of algebra is that a variable $x$ represents an unspecified or unknown quantity. It always stands for some quantity, but it may stand for any quantity. The use of variables makes algebra much more powerful than arithmetic. Variables help us to express and solve equations such as $2x+3y=11$ involving one or more unknown quantities. Variables can also be used to express arithmetical laws such as $x+y=y+x$ .... operators6The first five logical operators ( $\&$ , $\lor$ , $\lnot$ , $\supset$ , $\equiv$ ) are equivalent to so-called ``Boolean logic gates'' of electrical engineering. Formulas built from them may be viewed as representations of the binary switching circuits that control the operation of modern digital computers. See Mendelson [14,15].... implication7This is the so-called ``material implication'': $\Phi_1\supset \Phi_2$ is equivalent to $\lnot (\Phi_1 \& \lnot \Phi_2)$ .... bi-implication8This is called bi-implication because $\Phi_1\equiv \Phi_2$ is equivalent to $(\Phi_1\supset \Phi_2) \& (\Phi_2\supset \Phi_1)$ .... noted9See however Bochenski [2, §16E].... algorithm10The details of this algorithm are explained in modern logic textbooks. Variants of it have been programmed to run on digital computers. They form the basis of a system of computer logic. See Fitting [4].... contradictions11An explicit contradiction is a pair of formulas of the form $\Psi$ , $\lnot \Psi$ .... algorithm12The algorithms in question may be implemented as Turing machine programs. This undecidability result is known as Church's theorem. See Mendelson [15].... out13For example, philosophical intrinsicism may play out as mathematical Platonism. Philosophical subjectivism may play out as mathematical constructivism. Nominalism may play out as formalism.... method.14Our modern notion of a formal theory (see 1.2.5 above) is a variant of Aristotle's concept of scientific method.... Euclidean15Here ``Euclidean geometry'' refers to the familiar geometry in which the angles of a triangle sum to $180$ degrees, as distinct from the ``non-Euclidean'' (i.e., hyperbolic) geometry developed by Bolyai and Lobachevsky in the 19th century....geometry.16Tarski also showed how to handle non-Euclidean plane geometry, as well as Euclidean and non-Euclidean geometries of higher dimension, in a similar fashion.... geometrical17This means that all occurrences of variables $x$ within the formula $\Psi$ are within subformulas of the form $\forall x (Px\supset \dots)$ or $\exists x (Px \& \dots)$ . Thus we are restricting attention to the realm of geometry and excluding everything else.... algorithm18Such algorithms have been implemented as computer programs. They are useful in robotics and other artificial intelligence applications.... arithmetic.19Two recent studies of formal arithmetic are Hájek/Pudlák [6] and Simpson [20].... awkwardly20This kind of awkwardness can be alleviated by means of various devices. In particular, standard mathematical notation such as $x+2$ can be incorporated into the predicate calculus.... $\exists x \exists y \exists z \exists w (Mxxx \& Axxy \& Axyz \& Axzw \& Ayyw)$ 21In this formula, the variables $x$ , $y$ , $z$ , $w$ play the role of $1$ , $2$ , $3$ , $4$ respectively.... inconsistent22A formal theory is said to be inconsistent if its axioms logically imply an explicit contradiction. Such theories are of no scientific value.... arithmetic.23A recent study of second-order arithmetic is Simpson [20]....attraction.24The idea of set-theoretical foundations gave rise to the ``new math'' pedagogy of the 1960's. For a lively discussion, see Kline [12]....formalism.25See for example Curry [3].... question26For example, we could consider the following difficult question of Goldbach. Can every even number greater than $2$ be expressed as the sum of two prime numbers?...abandoned.27See the essays of Brouwer and Kolmogorov in [22].... reductionism.28Hilbert is often inaccurately described as a formalist. The details of Hilbert's program will not be presented here, but see [20,22]. Roughly speaking, a formal theory is said to be finitistically reducible if it can be embedded into some very restricted formal theory such as $T_a$ , which is physically meaningful and makes absolutely no reference to actual infinity. $next_inactive$ $up$ $previous$ Stephen G Simpson2007-09-28

Essay	on	the	nature
of	logic	and	its	relationship
to	the	philosophy	of	mathematics,
written	by	Stephen	G.	Simpson.

http://www.math.psu.edu/simpson/papers/philmath/

Logic and Mathematics 2008 July

dvd rental

dvd

Essay on the nature of logic and its relationship to the philosophy of mathematics, written by Stephen G. Simpson.

Rules

Recommended Sites: 1. Arts - Business - Computers - Games - Health - Home - Kids and Teens - News -

Logic and Mathematics

Contents

Logic

Aristotelean logic

The predicate calculus

Foundations of mathematics

The geometry of Euclid

Formal theories for mathematics

Philosophy of mathematics

Plato and Aristotle

The 20th century

The future

Bibliography

About this document ...

Essay

on

the

nature

of

logic

and

its

relationship

to

the

philosophy

of

mathematics,

written

by

Stephen

G.

Simpson.

http://www.math.psu.edu/simpson/papers/philmath/

Logic and Mathematics 2008 July

dvd rental

dvd

Rules