According to the fictionalist, the number 6 is the same kind of thing as Dr. Watson or Miss Marple. According to Field, mathematical language should be understood at face value. Of course, Field does not exhort mathematicians to settle their open questions via this vacuity. Unlike Quine, Field has no proposals for changing the methodology of mathematics. His view concerns how the results of mathematics should be interpreted, and the role of these results in the scientific enterprise.
For Field, the goal of mathematics is not to assert the true. The only mathematical knowledge that matters is knowledge of logical consequences see Field []. As we have seen, more traditional philosophers—and most mathematicians—regard indispensability as irrelevant to mathematical knowledge. In contrast, for thinkers like Field, once one has undermined the indispensability argument, there is no longer any serious reason to believe in the existence of mathematical objects.
As Quine and Putnam pointed out, most of the theories developed in scientific practice are not nominalistic, and so begins the indispensability argument. The first aspect of Field's program is to develop nominalistic versions of p. Of course, Field does not do this for every prominent scientific theory. To do so, he would have to understand every prominent scientific theory, a task that no human can accomplish anymore.
Field gives one example—Newtonian gravitational theory—in some detail, to illustrate a technique that can supposedly be extended to other scientific work. The second aspect of Field's program is to show that the nominalistic theories are sufficient for attaining the scientific goal of determining truths about the physical universe i.
Thus, if the mathematical theory is conservative over the nominalist one, then any physical consequence we get via the mathematics we could get from the nominalistic physics alone. This would show that mathematics is dispensable in principle, even if it is practically necessary. The sizable philosophical literature generated by Field [] includes arguments that Field's technique does not generalize to more contemporary theories like quantum mechanics Malament [] ; arguments that Field's distinction between abstract and concrete does not stand up, or that it does not play the role needed to sustain Field's fictionalism Resnik [] ; and arguments that Field's nominalistic theories are not conservative in the philosophically relevant sense Shapiro [].
The collection by Field [] contains replies to some of these objections. The philosopher understands mathematical assertions to be about what is possible, or about what would be the case if objects of a certain sort existed.
The formal language developed in Chihara [] includes variables that range over open sentences i. With keen attention to detail, p.
Mathematics comes out objective, even if it has no ontology. Chihara's program shows initial promise on the epistemic front. Perhaps it is easier to account for how the mathematician comes to know about what is possible, or about what sentences can be constructed, than it is to account for how the mathematician knows about a Platonic realm of objects.
See chapters 15 and 16 in this volume. Unlike fictionalists, traditional intuitionists , such as L. Brouwer e. Natural numbers and real numbers are mental constructions or are the result of mental constructions.
In mathematics, to exist is to be constructed. Some of their writing seems to imply that each person constructs his own mathematical realm. Communication between mathematicians consists in exchanging notes about their individual constructive activities. This would make mathematics subjective. It is more common, however, for these intuitionists, especially Brouwer, to hold that mathematics concerns the forms of mental construction as such see Posy []. This follows a Kantian theme, reviving the thesis that mathematics is synthetic a priori.
This perspective has consequences concerning the proper practice of mathematics. For the intuitionist, every mathematical assertion must correspond to a construction. For example, let P be a property of natural numbers. The latter formula cannot be inferred from the former because, clearly, it is possible to show that a property cannot hold universally without constructing a number for which it fails.
From the realist's point of view, the two formulas are equivalent. Some contemporary intuitionists, such as Michael Dummett [ , ] and Neil Tennant [ , ] , take a different route to roughly the same revisionist conclusion. Their proposed logic is similar to that of Brouwer and Heyting, but their supporting arguments and philosophy are different.
Dummett begins with reflections on language acquisition and use, and the role of language in communication. One who understands a sentence must grasp its meaning, and one who learns a sentence thereby learns its meaning. Someone who understands the meaning of any sentence of a language must be able to manifest that understanding in behavior.
Since language is an instrument of communication, an individual cannot communicate what he cannot be observed to communicate. Most semantic theories are compositional in the sense that the semantic content of a compound statement is analyzed in terms of the semantic content of its parts.
Tarskian semantics, for example, is compositional, because the satisfaction of a complex formula is defined in terms of the satisfaction of its subformulas. Dummett's proposal is that the lessons of the manifestation requirement be incorporated into a compositional semantics.
Instead of providing satisfaction conditions of each formula, Dummett proposes that the proper semantics supplies proof or computation conditions. Heyting and Dummett argue that on a semantics like this, the law of excluded middle is not universally upheld. A large body of research in mathematical logic shows how intuitionistic mathematics differs from its classical counterpart.
Many mathematicians hold that the intuitionistic restrictions would cripple their discipline see, e. For some philosophers of mathematics, the revision of mathematics is too high a price to pay. If a philosophy entails that there is something wrong with what the mathematicians do, then the philosophy is rejected out of hand. According to them, intuitionism can be safely ignored. A less dogmatic approach would be to take Dummett's arguments as a challenge to answer the criticisms he brings.
Dummett argues that classical logic, and mathematics as practiced, do not enjoy a certain kind of justification, a justification one might think a logic and mathematics ought to have. We leave the debate at this juncture. See chapters 9 and 10 in this volume.
According to another popular philosophy of mathematics, the subject matter of arithmetic, for example, is the pattern common to any infinite system of objects that has a distinguished initial object, and a successor relation or operation that satisfies the induction principle. The arabic numerals exemplify this natural number structure , as do sequences of characters on a finite alphabet in lexical order, an infinite sequence of distinct moments of time, and so on.
A natural number, such as 6, is a place in the natural number structure, the seventh place if the structure starts with zero. According to the structuralist, the application of mathematics to science occurs, in part, by discovering or postulating that certain structures are exemplified in the material world. Mathematics is to material reality as pattern is to patterned. There are several ontological views concerning structures, corresponding roughly to traditional views concerning universals.
One is that the natural number structure, for example, exists independent of whether it has instances in the p. Let us call this ante rem structuralism, after the analogous view concerning universals see Shapiro [] and Resnik [] ; see also Parsons []. Another view is that there is no more to the natural number structure than the systems of objects that exemplify this structure. Destroy the systems, and the structure goes with them.
Views like this are sometimes dubbed eliminative structuralism see Benacerraf []. According to ante rem structuralism, statements of mathematics are understood at face value. For the eliminative structuralist, by contrast, these apparent singular terms are actually bound variables. Taken at face value, eliminative structuralism requires a large ontology to keep mathematics from being vacuous.
For example, if there are only finitely many objects in the universe, then the natural number structure is not exemplified, and thus universally quantified statements of arithmetic are all vacuously true. Real and complex analysis and Euclidean geometry require a continuum of objects, and set theory requires a proper class or at least an inaccessible cardinal number of objects. He concluded from this that numbers are not objects.
This conclusion, however, depends on what it is to be an object—an interesting philosophical question in its own right. That is, ante rem structuralism accounts for the fact that mathematical structures are exemplified by other mathematical objects. Indeed, the natural number structure is exemplified by various systems of natural numbers , such as the even numbers and the prime numbers. From the ante rem perspective, this is straightforward: the natural numbers, as places in the natural number structure, exist.
Some of them are organized into systems, and some of these systems exemplify the natural number structure.
On the ante rem view, the main epistemological question becomes: How do we know about structures? On the eliminative versions, the question is: How do p.
Structuralists have developed several strategies for resolving the epistemic problems. The psychological mechanism of pattern recognition may be invoked for at least small, finite structures. By encountering instances of a given pattern, we obtain knowledge of the pattern itself. None of the structuralisms invoked so far provide for a reduction of the ontological burden of mathematics. The ontology of ante rem structuralism is as large and extensive as that of traditional realism in ontology.
Indeed, ante rem structuralism is a realism in ontology. Only the nature of the ontology is in question. Eliminative structuralism also requires a large ontology to keep the various branches of mathematics from lapsing into vacuity.
Surely there are not enough physical objects to keep structuralism from being vacuous when it comes to functional analysis or set theory. Instead of asserting that arithmetic is about all systems of a certain type, the modal structuralist says that arithmetic is about all possible systems of that type. The modal structuralist faces an attenuated threat of vacuity similar to that of the eliminative structuralist.
Instead of asserting that there are systems satisfying the natural number structure, for example, the modalist needs to affirm that such systems are possible.
The key issue here is to articulate the underlying modality. See chapters 17 and 18 in this volume. The above survey broached a number of issues concerning logic and the philosophy of logic.
The debate over intuitionism invokes the general validity, within mathematics, of the law of excluded middle and other inferences based on it p. There is traffic in the other direction as well, from logic to the philosophy of mathematics. Perhaps the primary issue in the philosophy of logic concerns the nature, or natures, of logical consequence. A similar, perhaps identical, idea underlies Frege's development of logic in defense of logicism, and occurs also in neologicism.
Of course, the resolution of these issues depends on prior questions concerning the nature of logic and the goals of logical study chapters 21 and 22 in this volume.
If both notions of consequence are legitimate, we can ask about relations between them. Perish the thought. However, the converse seems less crucial. It may well be that there is a semantically valid argument whose conclusion cannot be deduced from its premises. What does this say about the underlying nature of mathematics?
See chapters 25 and 26 in this volume. One of these is the thesis that a logical truth follows from any set of premises whatsoever, and another is ex falso quodlibet , the thesis that any conclusion follows from a contradiction. The extent to which such inferences occur in mathematics is itself a subject of debate chapters 23 and 24 in this volume.
Acknowledgment Some of the contents of this chapter were culled from Shapiro [b] and [b]. Aspray, W. A wide range of articles, most of which draw philosophical morals from historical studies. Find this resource:. Azzouni, J. Fresh philosophical view. Balaguer, M. Account of realism in ontology and its rivals. Benacerraf, P. One of the most widely cited works in the field; argues that numbers are not objects, and introduces an eliminative structuralism.
Putnam editors [], Philosophy of mathematics , second edition, Cambridge, Cambridge University Press. Bernays, P. Boolos, G. Criticisms of the claims of neologicism concerning the status of abstraction principles.
Brouwer, L. Burgess, J. Early critique of nominalism. Extensive articulation and criticism of nominalism. Chihara, C. Defense of a modal view of mathematics, and sharp criticisms of several competing views. Coffa, A. Colyvan, M. Elaboration and defense of the indispensability argument for ontological realism. Curry, H. Dummett, M. Influential defense of intuitionism.
Detailed introduction to and defense of intuitionistic mathematics. A collection of Dummett's central articles in metaphysics and the philosophy of language. Field, H. A widely cited defense of fictionalism by attempting to refute the indispensability argument. Reprints of Field's articles on fictionalism. Frege, G.
Austin, second edition, New York, Harper, Classic articulation and defense of logicism. Pohle; reprinted Hildesheim, Olms, More technical development of Frege's logicism. Detailed development of neologicism, to support Wright []. Hart, W. Collection of articles first published elsewhere. Hellman, G. Articulation and defense of modal structuralism. Develops deductive system and semantics for intuitionistic mathematics. Heyting, A. Readable account of intuitionism. Hodes, H. Another roughly Fregean logicism.
Kitcher, P. Articulation of a constructivist epistemology and a detailed attack on the thesis that mathematical knowledge is a priori. Lakatos, I. Contains many important papers by major philosophers and logicians. Worrall and E. Zahar, Cambridge, Cambridge University Press. A much cited study that is an attack on the rationalist epistemology for mathematics.
MacLane, S. The Golden Ratio is seen in the proportions in the sections of a finger. It is also worthwhile to mention that we have 8 fingers in total, 5 digits on each hand, 3 bones in each finger, 2 bones in 1 thumb, and 1 thumb on each hand. The ratio between the forearm and the hand is the Golden Ratio! The cochlea of the inner ear forms a Golden Spiral. Thus, science is highly enriched with patterns from the scientific observations of an experiment to the movement of an object, everywhere pattern is involved.
We also know that finding pattern and getting out some solutions out of them is the aspect of Mathematics. Thus, we say that the Mathematics is science of patterns without which science is not just incomplete also inefficient.
Golden Ratio Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Like every artist like a painter leaves behind his signature marking in their artwork similarly, God is said to be leaving behind the Golden-Ratio in all his creations, as it is found most widely in the nature.. The Golden Ratio appears extensively in the human face, as demonstrated in a university study on attractiveness.
Stephen Marquardt has studied human beauty for years in his practice of oral and maxillofacial surgery. Marquardt performed cross-cultural surveys on beauty and found that all groups had the same perceptions of facial beauty. He also analyzed the human face from ancient times to the modern day. This mask uses the pentagon and decagon as its foundation, which embody phi in all their dimensions. For more information and other examples, see his site at Marquardt Beauty Analysis.
Angelina Jolie, One of the globally renowned actress and one of the beautiful actress. A geometric figure such as a square, triangle, or rectangle. It is the first differentiating characteristics of an object. There are many different shapes in the world, like geometrical shapes like square, circle or the shape of a human or shape of a tree and many such billions of different shapes.
Each shape has its own significance and thus each and every shape is important. Each and every field requires the knowledge of geometry and thus shapes may it be physics, biology, chemistry, fashion technology, etc. For instance, polygons are classified according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into smaller categories; triangles can be equilateral, isosceles, obtuse, acute, scalene etc.
Other common shapes are points, lines, planes, and conic sections such as ellipses, circles, and parabolas. Among the most common 3-dimensional shapes are polyhedra, which are shapes with flat faces; ellipsoids, which are egg-shaped or sphere-shaped objects; cylinders; and cones. If an object falls into one of these categories exactly or even approximately we can use it to describe the shape of the object. Thus, we say that the shape of a manhole cover is a disk, because it is approximately the same geometric object as an actual geometric disk.
Isotopy: Two objects are isotopic if one can be transformed into the other by a sequence of deformations that do not tear the object or put holes in it. Circle is one of the most important shape of all shapes. Symmetry Symmetry is one of the most important key to all scientific researches. Infact, many of the vast theories are proved using symmetry.
For Example:- Buildings which are symmetric are stronger than non-symmetric one sand protected to earthquakes. Symmetry also has a vital role in the General Relativity Principles by Einstein. Numbers are the reasons which have made mathematics significant and each and every branch of mathematics meaningful, so, in practical terms, numbers are the basis of the mathematics. Also, they are the reason which makes mathematics interesting and interactive. Numbers are the reason for mathematics being part of each and every other field like physics, chemistry, biology, fashion technology, etc.
Main classification of Numbers:- Natural 0, 1, 2, 3, 4, It help us to get counting, location, speed, position, distance, etc.
The property of being prime or not is called primality. A simple but slow method of verifying the primality of a given number n is known as trial division.
It consists of testing whether n is a multiple of any integer between 2 and Root n. Algorithms much more efficient than trial division have been devised to test the primality of large numbers.
These include the Miller—Rabin primality test, which is fast but has a small probability of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of January , the largest known prime number has 22,, decimal digits. There is no known simple formula that separates prime numbers from composite numbers.
However, the distribution of primes, that is to say, the statistical behavior of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.
Many questions regarding prime numbers remain open, such as Goldbach's conjecture that every even integer greater than 2 can be expressed as the sum of two primes , and the twin prime conjecture that there are infinitely many pairs of primes whose difference is 2.
Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors.
Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as elements and prime ideals. Also, Mathematics is the science of quantity, measurement and spatial relations. However, each definition throws insight in to one or more aspects of Mathematics. Nature of Mathematics:- The nature of Mathematics can be made explicit by analyzing the chief characteristics of Mathematics.
According to A. The children must have opportunities for making their own discoveries of mathematical ideas, but they must also have the practice necessary to achieve accuracy in their calculations.
Today it is discovery techniques, which are making spectacular progress. They are being applied in two fields: in pure number relationships and in everyday problems like money, weights and measures. From this view points, Mathematics is mainly a matter of 8. Hence, the learner can check whether or not he has drawn the correct conclusions, permit the learner to begin with simple and very easy conclusions, gradually move over to more difficult and complex ones.
However, many conceive Mathematics as a very useful means to other ends, a powerful and incisive tool of wide applicability. This must come before rules are stated or formal operations are introduced.
The teacher has to foster intuition in our young children, by following the right strategies of teaching. Intuition when applied to Mathematics involves the concretization of an idea not get stated in the form of some sort of operations or examples. Intuition is to anticipate what will happen next and what to do about it. It implies the act of grasping the meaning or significance or structure of a problem without explicit reliance on the analytic mode of thought.
It is a form of mathematical activity which depends on the confidence in the applicability of the process rather than upon the importance of right answers all the time. It is up to the teacher to allow the child to use his natural and intuition way of thinking, by encouraging him to do so and honoring him when he does.
It is perhaps the only subject which can claim certainty of results. In Mathematics, the results are either right or wrong, accepted or rejected.
There is no midway possible between rights and wrong. Mathematic can decide whether or not its conclusions are right. Even when there is a new emphasis on approximation, mathematical results can have any degree of accuracy required. Mathematics learning always proceeds from simple It is a subject in which the dependence on earlier knowledge is particularly great. Thus gradation and sequence can be observed among topics in any selected branch of Mathematics.
The students can always verify the validity of mathematical rules and relationships by applying them to novel situations. Concept and principle become more functional and meaningful only when they are related to actual practical applications. Such a practice will make the learning of Mathematics more meaningful and significant. When the pupil evolves his own definitions, concept and theorems, he is making generalizations.
The generalizations and classification of Mathematics are very simple and obvious in comparison with those of other domains of thought and activity. However, the Mathematics teacher should take care to see that the final generalization into a rule should always be deferred until it is almost spontaneously suggested by the pupils themselves.
No such objects can be found in the physical world. Hence, mathematical concepts cannot be learned through experiences with concrete objects. Some concepts can be learned only through their definitions and they may not have concrete counter parts to be abstracted from. Most of the mathematical concepts are such concepts without concretization and hence they are abstract. The concept of prime numbers, the concept of probability, the concept of a function, the concept of limits, concept of continuous functions, to list few are all abstract in the sense that they can be learned Even when concretization is possible they are only representation of the concepts and not physical object themselves.
Therefore, a mathematical structure should be some sort of arrangement, formation or result of putting together of parts. For example, we take as the fundamental building units of a structure the members a ,b ,c,…….. A mathematical structure is a mathematical system with one or more explicitly recognized We may create a structure from mathematical systems by making specific recognition of one or more of the commutative, associative or distribution properties that the system may have.
With one or more basic structure at hand, one may construct other structures. Since plane analytic geometry is the study of subset or the Cartesian set Re X Re, where Re is the set of real numbers, plane analytic geometry may be considered as a superstructure based up on the structure know as real number system. Thus mathematics has got definite logical structure. These structures ensure the beauty and order of mathematics.
Number systems, group, field, ring vector, space … etc are all examples of mathematical structures. It governs the pattern of deductive proof through which mathematics is developed. Of course, logic was used in mathematics centuries ago. One face is a systematic deductive science. The first face has resulted in presenting mathematic as an axiomatic body of definitions, undefined terms, axioms and theorems.
In mathematics, granted the premises, conclusion follows inevitalely. Thus the process of deduction involves two steps: 1. Replacing the real premises by hypothetical ones. Making a mathematical inference from the hypothetical working premises.
Therefore to think mathematically is to free oneself by abstraction from any peculiarity of subject matter and to make inferences and deductions justified by fundamental premises. It involves logical reasoning. By reasoning, we prove that if something is true then something else must be true. However, the validity of the conclusions rests upon the validity and consistency of the assumptions and definitions upon which the conclusions are based.
The teacher should make the students realize this point of view. The generalizations follow as outcomes of the observations of mathematical phenomena and relationships. It is based on the principle that if a relationship holds good for some particular cases, it holds good for any similar case and hence the relationship can be generalized. For example, the student generalizes that the sum of the angles in a triangle is after having observation this property in a number of triangles.
Thus a generalization, a rule or a formula is arrived at, through the careful observation of particular facts, instances and examples. Many of the The teacher of Mathematics has to provide an adequate number of examples requiring the student to observe so that the relationships are explicit leading to the generalization.
Mathematical language and symbolism:- Mathematics has its own language, its own tools and mode of operations. The language for the communication of mathematical ideas is largely in terms of symbols and words which everybody cannot understand.
There is no proper terminology for talking about Mathematics. For example, the distinction between a number and a numeral could head the list.
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