Shopping on line can be easy, simple and save you lots of money. It can also take a lot of your time, frustrate you, and result in unwanted purchases. Now the same can be said for regular high street shopping, but with the vast opportunity presented by the Internet it will pay you to spend a few minutes reading this and understanding how to better optimize your Concept shopping experience:

1. Compare - without doubt the biggest advantage that the Concept offers shoppers today is the ability to compare thousands of Concept at a time. This is a great thing, but not necessarily all the time! Too much can be daunting at times so take advantage of the great comparison sites and where possible let them do the hard work for you.

2. Research - if it has been said it will be on the internet. Ignorance is no longer a justifiable reason for buying the wrong thing. Take the time to research in detail everything that you could possible want to know about

3. Testimonials - don't know anybody that has bought a Concept? Wrong! If the Concept is good the internet will let you know. Use the Internet as a friend and get testimonials before you buy.

4. Questions - Got a question about Concept then search the Forums, FAQ's, Blogs etc. Don't be afraid to ask .....

5. Reputation - Never heard of the company selling Concept? Don't worry, no reason why you should know every company in the world, but you know someone that does! Use the internet to find out what people are saying about Concept and build up a picture of their reputation for sales, returns, customer service, delivery etc.

6. Returns - still worried that even after all of the above your Concept wont be what you want? Check out the returns policy. There is so much competition now that someone, somewhere is bound to offer the terms that you are comfortable with.

7. Feedback - happy with your Concept then let people know, after all you are depending on others people input in your buying decision, so why not give a little back.

8. Security - check for the yellow padlock on the Concept site before you buy, and the s after http:/ /i.e. https:// = a secure site

9. Contact - got a question about Concept, or want to leave a comment then check out the sites contact page. Reputable companies have them and respond.

10. Payment - ready to pay for your Concept, then use your credit card or PayPal! Be aware of companies that don't accept them, there may be genuine reasons but given the huge amount of choice you have when buying online there is no reason at all not to buy via credit card or PayPal.



As the term is used in mainstream cognitive science and philosophy of mind, a concept is an abstraction idea or a mental symbol, typically associated with a corresponding representation in and language or symbology.

A vast array of accounts attempt to explain the nature of concepts. According to classical accounts, a concept denotes all of the entities, phenomena, and/or relations in a given Categories (Aristotle) or class (philosophy) by using definitions. Concepts are abstract in that they omit the differences of the things in their extension (semantics), treating the members of the extension as if they were identical. Classical concepts are universal in that they apply equally to every thing in their extension. Concepts are also the basic elements of propositions, much the same way a word is the basic semantics element of a sentence (linguistics). Unlike perceptions, which are particular images of individual objects, concepts cannot be visualized. Because they are not themselves individual perceptions, concepts are discursive and result from reason.

Concepts are expected to be useful in dealing with reality. Generally speaking, concepts are taken to be (a) acquired dispositions to recognize perceived objects as being of this kind or of that ontological kind, and at the same time (b) to understand what this kind or that kind of object is like, and consequently (c) to perceive a number of perceived particulars as being the same in kind and to discriminate between them and other sensible particulars that are different in kind. In addition, concepts are acquired dispositions to understand what certain kinds of objects are like both (a) when the objects, though perceptible, are not actually perceived, and (b) also when they are not perceptible at all, as is the case with all the conceptual constructs we employ in physics, mathematics, and metaphysics. The impetus to have a theory of concepts that is ontologically useful has been so strong that it has pushed forward accounts that understand a concept to have a deep connection with reality.

On some accounts, there may be agents (perhaps some animals) which don't think about, but rather use relatively basic concepts (such as demonstrative and perceptual concepts for things in their perceptual field), even though it is generally assumed that they do not think in symbols.. On other accounts, mastery of symbolic thought (in particular, language) is a prerequisite for conceptual thought.Damasio, Antonio R. "Descartes' Error: Emotion, Reason, and the Human Brain," Avon 1994, p. 106 ff.

Concepts are bearers of Meaning (linguistic), as opposed to agent (grammar)s of meaning. A single concept can be expressed by any number of languages. The concept of DOG can be expressed as dog in English language, Hund in German language, as chien in French language, and perro in Spanish language. The fact that concepts are in some sense independent of language makes translation possible - words in various languages have identical meaning, because they express one and the same concept.

A terminology labels or designates concepts. Several partly or fully distinct concepts may share the same term. These different concepts are easily confused by mistakenly being used interchangeably, which is a fallacy. Also, the concepts of term and concept are often confused, although the two are not the same.

The acquisition of concepts is studied in machine learning as supervised classification and unsupervised classification, and in psychology and cognitive science as concept learning and category formation. In the philosophy of Kant, any purely empiricism theory dealing with the acquisition of concepts is referred to as a noogony.

Origin and acquisition of concepts A posteriori abstractions John Locke's description of a general idea corresponds to a description of a concept. According to Locke, a general idea is created by abstracting, drawing away, or removing the common characteristic or characteristics from several particular ideas. This common characteristic is that which is similar to all of the different individuals. For example, the abstract general idea or concept that is designated by the word "red" is that characteristic which is common to apples, cherries, and blood. The abstract general idea or concept that is signified by the word "dog" is the collection of those characteristics which are common to Airedales, Collies, and Chihuahuas.

In the same tradition as Locke, John Stuart Mill stated that general conceptions are formed through abstraction. A general conception is the common element among the many images of members of a class. "...hen we form a set of phenomena into a class, that is, when we compare them with one another to ascertain in what they agree, some general conception is implied in this mental operation" (A System of Logic, Book IV, Ch. II). Mill did not believe that concepts exist in the mind before the act of abstraction. "It is not a law of our intellect, that, in comparing things with each other and taking note of their agreement, we merely recognize as realised in the outward world something that we already had in our minds. The conception originally found its way to us as the result of such a comparison. It was obtained (in metaphysical phrase) by abstraction from individual things" (Ibid.).

For Schopenhauer, empirical concepts "...are mere abstractions from what is known through intuitive perception, and they have arisen from our arbitrarily thinking away or dropping of some qualities and our retention of others." (Parerga and Paralipomena, Vol. I, "Sketch of a History of the Ideal and the Reality"). In his On the Will in Nature, "Physiology and Pathology," Schopenhauer said that a concept is "drawn off from previous images ... by putting off their differences. This concept is then no longer intuitively perceptible, but is denoted and fixed merely by words." Nietzsche, who was heavily influenced by Schopenhauer, wrote: "Every concept originates through our equating what is unequal. No leaf ever wholly equals another, and the concept 'leaf' is formed through an arbitrary abstraction from these individual differences, through forgetting the distinctions… .""On Truth and Lie in an Extra–Moral Sense," The Portable Nietzsche, p. 46

By contrast to the above philosophers, Immanuel Kant held that the account of the concept as an abstraction of experience is only partly correct. He called those concepts that result of abstraction "a posteriori concepts" (meaning concepts that arise out of experience). An empirical or an a posteriori concept is a general representation (Vorstellung) or non-specific thought of that which is common to several specific perceived objects. (Logic, I, 1., §1, Note 1)

A concept is a common feature or characteristic. Kant investigated the way that empirical a posteriori concepts are created.

Kant's description of the making of a concept has been paraphrased as "… to conceive is essentially to think in abstraction what is common to a plurality of possible instances… ." (H.J. Paton, Kant's Metaphysics of Experience, I, 250). In his discussion of Kant, Christopher Janaway wr ote: "… generic concepts are formed by abstraction from more than one species."Christopher Janaway, Self and World in Schopenhauer's Philosophy, Ch. 3, p. 112, Oxford, 2003, ISBN 0-19-825003-7

A priori concepts Kant declared that human minds possess pure or A priori and a posteriori (philosophy) concepts. Instead of being abstracted from individual perceptions, like empirical concepts, they originate in the mind itself. He called these concepts Category (Kant), in the sense of the word that means predicate (grammar), attribute, characteristic, or quality. But these pure categories are predicates of things in general, not of a particular thing. According to Kant, there are 12 categories that constitute the understanding of phenomenal objects. Each category is that one predicate which is common to multiple empirical concepts. In order to explain how an a priori concept can relate to individual phenomena, in a manner analogous to an a posteriori concept, Kant employed the technical concept of the Schema (Kant).

Conceptual structure It seems intuitively obvious that concepts must have some kind of structure. Up until recently, the dominant view of conceptual structure was a containment model, associated with the classical view of concepts. According to this model, a concept is endowed with certain necessary and sufficient conditions in their description which unequivocally determine an extension. The containment model allows for no degrees; a thing is either in, or out, of the concept's extension. By contrast, the inferential model understands conceptual structure to be determined in a graded manner, according to the tendency of the concept to be used in certain kinds of inferences. As a result, concepts do not have a kind of structure that is in terms of necessary and sufficient conditions; all conditions are contingent. (Margolis:5)

However, some theorists claim that primitive concepts lack any structure at all. For instance, Jerry Fodor presents his Asymmetric Dependence Theory as a way of showing how a primitive concept's content is determined by a reliable relationship between the information in mental contents and the world. These sorts of claims are referred to as "atomistic", because the primitive concept is treated as if it were a genuine atom.

Conceptual content Content as pragmatic role A concept may be abstracted from several perceptions, but that is only its origin. In regard to its meaning or its truth, William James proposed his Pragmatic Rule. This rule states that the meaning of a concept may always be found in some particular difference in the course of human experience which its being true will make (Some Problems of Philosophy, "Percept and Concept -- The Import of Concepts"). In order to understand the meaning of the concept and to discuss its importance, a concept may be tested by asking, "What sensible difference to anybody will its truth make?" There is only one criterion of a concept's meaning and only one test of its truth. That criterion or test is its consequences for human behavior.

In this way, James bypassed the controversy between rationalists and empiricists regarding the origin of concepts. Instead of solving their dispute, he ignored it. The rationalists had asserted that concepts are a revelation of Reason. Concepts are a glimpse of a different world, one which contains timeless truths in areas such as logic, mathematics, ethics, and aesthetics. By pure thought, humans can discover the relations that really exist among the parts of that divine world. On the other hand, the empiricists claimed that concepts were merely a distillation or abstraction from perceptions of the world of experience. Therefore, the significance of concepts depends solely on the perceptions that are its references. James's Pragmatic Rule does not connect the meaning of a concept with its origin. Instead, it relates the meaning to a concept's purpose, that is, its function, use, or result.

Embodied content In Cognitive linguistics, abstract concepts are transformations of concrete concepts derived from embodied experience. The mechanism of transformation is structural mapping, in which properties of two or more source domains are selectively mapped onto a blended space (Fauconnier & Turner, 1995; see conceptual blending). A common class of blends are metaphors. This theory contrasts with the rationalist view that concepts are perceptions (or recollections, in Plato's term) of an independently existing world of ideas, in that it denies the existence of any such realm. It also contrasts with the empiricist view that concepts are abstract generalizations of individual experiences, because the contingent and bodily experience is preserved in a concept, and not abstracted away. While the perspective is compatible with Concept#William James's truth (above), the notion of the transformation of embodied concepts through structural mapping makes a distinct contribution to the problem of concept formation.

Philosophical implications Concepts and metaphilosophy A long and well-established tradition in philosophy posits that philosophy itself is nothing more than conceptual analysis. This view has its proponents in contemporary literature as well as historical. According to Deleuze and Guattari's What Is Philosophy? (1991), philosophy is the activity of creating concepts. This creative activity differs from previous definitions of philosophy as simple reasoning, communication or contemplation of Universal (metaphysics). Concepts are specific to philosophy: science has got "percepts", and art "Affect (philosophy)". A concept is always signed: thus, Descartes' Cogito ergo sum or Kant's "transcendence (philosophy)#"Transcendence" in Modern Philosophy". It is a Mathematical singularity, not an universal, and connects itself with others concepts, on a "plane of immanence" traced by a particular philosophy. Concepts can jump from one plane of immanence to another, combining with other concepts and therefore engaging in a "becoming-Other."

Concepts in epistemology Concepts are vital to the development of scientific knowledge. For example, it would be difficult to imagine physics without concepts like: energy, force, or acceleration. Concepts help to integrate apparently unrelated observations and phenomena into viable hypothesis and theories, the basic ingredients of science. The concept map is a tool that is used to help researchers visualize the inter-relationships between various concepts.

Ontology of concepts Although the mainstream literature in cognitive science regards the concept as a kind of mental particular, it has been suggested by some theorists that concepts are real things. (Margolis:8) In most radical form, the realist about concepts attempts to show that the supposedly mental processes are not mental at all; rather, they are abstract entities, which are just as real as any mundane object.

Plato was the starkest proponent of the realist thesis of universal concepts. By his view, concepts (and ideas in general) are innate ideas that were instantiations of a transcendental world of pure forms that laid behind the veil of the physical world. In this way, universals were explained as transcendent objects. Needless to say this form of realism was tied deeply with Plato's ontological projects. This remark on Plato is not of merely historical interest. For example, the view that numbers are Platonic objects was revived by Kurt Godel as a result of certain puzzles that he took to arise from the phenomenological accounts.

Gottlob Frege, founder of the analytic tradition in philosophy, famously argued for the analysis of language in terms of sense and reference. For him, the sense of an expression in language describes a certain state of affairs in the world, namely, the way that some object is presented. Since many commentators view the notion of sense as identical to the notion of concept, and Frege regards senses as the linguistic representations of states of affairs in the world, it seems to follow that we may understand concepts as the manner in which we grasp the world. Accordingly, concepts (as senses) have an ontological status. (Morgolis:7)

According to Carl Benjamin Boyer, in the introduction to his The History of the Calculus and its Conceptual Development, concepts in calculus do not refer to perceptions. As long as the concepts are useful and mutually compatible, they are accepted on their own. For example, the concepts of the derivative and the integral are not considered to refer to spatial or temporal perceptions of the external world of experience. Neither are they related in any way to mysterious Limit (mathematics) in which quantities are on the verge of nascence or evanescence, that is, coming into or going out of appearance or existence. The abstract concepts are now considered to be totally autonomous, even though they originated from the process of abstracting or taking away qualities from perceptions until only the common, essential attributes remained.

See also

References

External links



As the term is used in mainstream cognitive science and philosophy of mind, a concept is an abstraction idea or a mental symbol, typically associated with a corresponding representation in and language or symbology.

A vast array of accounts attempt to explain the nature of concepts. According to classical accounts, a concept denotes all of the entities, phenomena, and/or relations in a given Categories (Aristotle) or class (philosophy) by using definitions. Concepts are abstract in that they omit the differences of the things in their extension (semantics), treating the members of the extension as if they were identical. Classical concepts are universal in that they apply equally to every thing in their extension. Concepts are also the basic elements of propositions, much the same way a word is the basic semantics element of a sentence (linguistics). Unlike perceptions, which are particular images of individual objects, concepts cannot be visualized. Because they are not themselves individual perceptions, concepts are discursive and result from reason.

Concepts are expected to be useful in dealing with reality. Generally speaking, concepts are taken to be (a) acquired dispositions to recognize perceived objects as being of this kind or of that ontological kind, and at the same time (b) to understand what this kind or that kind of object is like, and consequently (c) to perceive a number of perceived particulars as being the same in kind and to discriminate between them and other sensible particulars that are different in kind. In addition, concepts are acquired dispositions to understand what certain kinds of objects are like both (a) when the objects, though perceptible, are not actually perceived, and (b) also when they are not perceptible at all, as is the case with all the conceptual constructs we employ in physics, mathematics, and metaphysics. The impetus to have a theory of concepts that is ontologically useful has been so strong that it has pushed forward accounts that understand a concept to have a deep connection with reality.

On some accounts, there may be agents (perhaps some animals) which don't think about, but rather use relatively basic concepts (such as demonstrative and perceptual concepts for things in their perceptual field), even though it is generally assumed that they do not think in symbols.. On other accounts, mastery of symbolic thought (in particular, language) is a prerequisite for conceptual thought.Damasio, Antonio R. "Descartes' Error: Emotion, Reason, and the Human Brain," Avon 1994, p. 106 ff.

Concepts are bearers of Meaning (linguistic), as opposed to agent (grammar)s of meaning. A single concept can be expressed by any number of languages. The concept of DOG can be expressed as dog in English language, Hund in German language, as chien in French language, and perro in Spanish language. The fact that concepts are in some sense independent of language makes translation possible - words in various languages have identical meaning, because they express one and the same concept.

A terminology labels or designates concepts. Several partly or fully distinct concepts may share the same term. These different concepts are easily confused by mistakenly being used interchangeably, which is a fallacy. Also, the concepts of term and concept are often confused, although the two are not the same.

The acquisition of concepts is studied in machine learning as supervised classification and unsupervised classification, and in psychology and cognitive science as concept learning and category formation. In the philosophy of Kant, any purely empiricism theory dealing with the acquisition of concepts is referred to as a noogony.

Origin and acquisition of concepts A posteriori abstractions John Locke's description of a general idea corresponds to a description of a concept. According to Locke, a general idea is created by abstracting, drawing away, or removing the common characteristic or characteristics from several particular ideas. This common characteristic is that which is similar to all of the different individuals. For example, the abstract general idea or concept that is designated by the word "red" is that characteristic which is common to apples, cherries, and blood. The abstract general idea or concept that is signified by the word "dog" is the collection of those characteristics which are common to Airedales, Collies, and Chihuahuas.

In the same tradition as Locke, John Stuart Mill stated that general conceptions are formed through abstraction. A general conception is the common element among the many images of members of a class. "...hen we form a set of phenomena into a class, that is, when we compare them with one another to ascertain in what they agree, some general conception is implied in this mental operation" (A System of Logic, Book IV, Ch. II). Mill did not believe that concepts exist in the mind before the act of abstraction. "It is not a law of our intellect, that, in comparing things with each other and taking note of their agreement, we merely recognize as realised in the outward world something that we already had in our minds. The conception originally found its way to us as the result of such a comparison. It was obtained (in metaphysical phrase) by abstraction from individual things" (Ibid.).

For Schopenhauer, empirical concepts "...are mere abstractions from what is known through intuitive perception, and they have arisen from our arbitrarily thinking away or dropping of some qualities and our retention of others." (Parerga and Paralipomena, Vol. I, "Sketch of a History of the Ideal and the Reality"). In his On the Will in Nature, "Physiology and Pathology," Schopenhauer said that a concept is "drawn off from previous images ... by putting off their differences. This concept is then no longer intuitively perceptible, but is denoted and fixed merely by words." Nietzsche, who was heavily influenced by Schopenhauer, wrote: "Every concept originates through our equating what is unequal. No leaf ever wholly equals another, and the concept 'leaf' is formed through an arbitrary abstraction from these individual differences, through forgetting the distinctions… .""On Truth and Lie in an Extra–Moral Sense," The Portable Nietzsche, p. 46

By contrast to the above philosophers, Immanuel Kant held that the account of the concept as an abstraction of experience is only partly correct. He called those concepts that result of abstraction "a posteriori concepts" (meaning concepts that arise out of experience). An empirical or an a posteriori concept is a general representation (Vorstellung) or non-specific thought of that which is common to several specific perceived objects. (Logic, I, 1., §1, Note 1)

A concept is a common feature or characteristic. Kant investigated the way that empirical a posteriori concepts are created.

Kant's description of the making of a concept has been paraphrased as "… to conceive is essentially to think in abstraction what is common to a plurality of possible instances… ." (H.J. Paton, Kant's Metaphysics of Experience, I, 250). In his discussion of Kant, Christopher Janaway wr ote: "… generic concepts are formed by abstraction from more than one species."Christopher Janaway, Self and World in Schopenhauer's Philosophy, Ch. 3, p. 112, Oxford, 2003, ISBN 0-19-825003-7

A priori concepts Kant declared that human minds possess pure or A priori and a posteriori (philosophy) concepts. Instead of being abstracted from individual perceptions, like empirical concepts, they originate in the mind itself. He called these concepts Category (Kant), in the sense of the word that means predicate (grammar), attribute, characteristic, or quality. But these pure categories are predicates of things in general, not of a particular thing. According to Kant, there are 12 categories that constitute the understanding of phenomenal objects. Each category is that one predicate which is common to multiple empirical concepts. In order to explain how an a priori concept can relate to individual phenomena, in a manner analogous to an a posteriori concept, Kant employed the technical concept of the Schema (Kant).

Conceptual structure It seems intuitively obvious that concepts must have some kind of structure. Up until recently, the dominant view of conceptual structure was a containment model, associated with the classical view of concepts. According to this model, a concept is endowed with certain necessary and sufficient conditions in their description which unequivocally determine an extension. The containment model allows for no degrees; a thing is either in, or out, of the concept's extension. By contrast, the inferential model understands conceptual structure to be determined in a graded manner, according to the tendency of the concept to be used in certain kinds of inferences. As a result, concepts do not have a kind of structure that is in terms of necessary and sufficient conditions; all conditions are contingent. (Margolis:5)

However, some theorists claim that primitive concepts lack any structure at all. For instance, Jerry Fodor presents his Asymmetric Dependence Theory as a way of showing how a primitive concept's content is determined by a reliable relationship between the information in mental contents and the world. These sorts of claims are referred to as "atomistic", because the primitive concept is treated as if it were a genuine atom.

Conceptual content Content as pragmatic role A concept may be abstracted from several perceptions, but that is only its origin. In regard to its meaning or its truth, William James proposed his Pragmatic Rule. This rule states that the meaning of a concept may always be found in some particular difference in the course of human experience which its being true will make (Some Problems of Philosophy, "Percept and Concept -- The Import of Concepts"). In order to understand the meaning of the concept and to discuss its importance, a concept may be tested by asking, "What sensible difference to anybody will its truth make?" There is only one criterion of a concept's meaning and only one test of its truth. That criterion or test is its consequences for human behavior.

In this way, James bypassed the controversy between rationalists and empiricists regarding the origin of concepts. Instead of solving their dispute, he ignored it. The rationalists had asserted that concepts are a revelation of Reason. Concepts are a glimpse of a different world, one which contains timeless truths in areas such as logic, mathematics, ethics, and aesthetics. By pure thought, humans can discover the relations that really exist among the parts of that divine world. On the other hand, the empiricists claimed that concepts were merely a distillation or abstraction from perceptions of the world of experience. Therefore, the significance of concepts depends solely on the perceptions that are its references. James's Pragmatic Rule does not connect the meaning of a concept with its origin. Instead, it relates the meaning to a concept's purpose, that is, its function, use, or result.

Embodied content In Cognitive linguistics, abstract concepts are transformations of concrete concepts derived from embodied experience. The mechanism of transformation is structural mapping, in which properties of two or more source domains are selectively mapped onto a blended space (Fauconnier & Turner, 1995; see conceptual blending). A common class of blends are metaphors. This theory contrasts with the rationalist view that concepts are perceptions (or recollections, in Plato's term) of an independently existing world of ideas, in that it denies the existence of any such realm. It also contrasts with the empiricist view that concepts are abstract generalizations of individual experiences, because the contingent and bodily experience is preserved in a concept, and not abstracted away. While the perspective is compatible with Concept#William James's truth (above), the notion of the transformation of embodied concepts through structural mapping makes a distinct contribution to the problem of concept formation.

Philosophical implications Concepts and metaphilosophy A long and well-established tradition in philosophy posits that philosophy itself is nothing more than conceptual analysis. This view has its proponents in contemporary literature as well as historical. According to Deleuze and Guattari's What Is Philosophy? (1991), philosophy is the activity of creating concepts. This creative activity differs from previous definitions of philosophy as simple reasoning, communication or contemplation of Universal (metaphysics). Concepts are specific to philosophy: science has got "percepts", and art "Affect (philosophy)". A concept is always signed: thus, Descartes' Cogito ergo sum or Kant's "transcendence (philosophy)#"Transcendence" in Modern Philosophy". It is a Mathematical singularity, not an universal, and connects itself with others concepts, on a "plane of immanence" traced by a particular philosophy. Concepts can jump from one plane of immanence to another, combining with other concepts and therefore engaging in a "becoming-Other."

Concepts in epistemology Concepts are vital to the development of scientific knowledge. For example, it would be difficult to imagine physics without concepts like: energy, force, or acceleration. Concepts help to integrate apparently unrelated observations and phenomena into viable hypothesis and theories, the basic ingredients of science. The concept map is a tool that is used to help researchers visualize the inter-relationships between various concepts.

Ontology of concepts Although the mainstream literature in cognitive science regards the concept as a kind of mental particular, it has been suggested by some theorists that concepts are real things. (Margolis:8) In most radical form, the realist about concepts attempts to show that the supposedly mental processes are not mental at all; rather, they are abstract entities, which are just as real as any mundane object.

Plato was the starkest proponent of the realist thesis of universal concepts. By his view, concepts (and ideas in general) are innate ideas that were instantiations of a transcendental world of pure forms that laid behind the veil of the physical world. In this way, universals were explained as transcendent objects. Needless to say this form of realism was tied deeply with Plato's ontological projects. This remark on Plato is not of merely historical interest. For example, the view that numbers are Platonic objects was revived by Kurt Godel as a result of certain puzzles that he took to arise from the phenomenological accounts.

Gottlob Frege, founder of the analytic tradition in philosophy, famously argued for the analysis of language in terms of sense and reference. For him, the sense of an expression in language describes a certain state of affairs in the world, namely, the way that some object is presented. Since many commentators view the notion of sense as identical to the notion of concept, and Frege regards senses as the linguistic representations of states of affairs in the world, it seems to follow that we may understand concepts as the manner in which we grasp the world. Accordingly, concepts (as senses) have an ontological status. (Morgolis:7)

According to Carl Benjamin Boyer, in the introduction to his The History of the Calculus and its Conceptual Development, concepts in calculus do not refer to perceptions. As long as the concepts are useful and mutually compatible, they are accepted on their own. For example, the concepts of the derivative and the integral are not considered to refer to spatial or temporal perceptions of the external world of experience. Neither are they related in any way to mysterious Limit (mathematics) in which quantities are on the verge of nascence or evanescence, that is, coming into or going out of appearance or existence. The abstract concepts are now considered to be totally autonomous, even though they originated from the process of abstracting or taking away qualities from perceptions until only the common, essential attributes remained.

See also

References

External links



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