Intuitive Notion Meaning


Welcome to the starting point for learning about derivatives. This page serves as a hub for all the pages on derivatives you'll find on this site. This is the place to start if you want to learn about this topic.

Intuitive Notion of Limits The notion of a limit of a process — what happens to the process as the parameter of the process approaches some value or tends toward infinity — will be introduced here. We have already seen that such a notion arises naturally in various problems. Find 20 ways to say INTUITIVE, along with antonyms, related words, and example sentences at, the world's most trusted free thesaurus. Adverb by means of direct perception, an instinctive inner sense, or gut feeling rather than rational thought:They’ve been married so long, they know intuitively how best to support each other.

The first question we'll try to answer is the most basic one: what is the derivative? I'll begin with an intuitive introduction to derivatives that will lead naturally to the mathematical definition using limits.

Maybe you aren't aware of it, but you already have an intuitive notion of the concept of 'derivative'. You probably use it almost everyday. The word 'derivative' doesn't serve as a very good description of it, I think.

Basically, there are two ways of thinking about the derivative of a function.

The first way is as an instantaneous rate of change. We use this concept all the time, for example, when talking about velocity. Velocity is the instantaneous rate of change of position.

There is also a geometric interpretation of the concept of derivative. The derivative can be thought of as the slope of a curve.

It is easy to define the slope of a straight line. That is usually learned in an algebra course. If you don't remember, or never have seen how that is done, don't worry, we'll review that.

I think it is important to review it even if you have already seen it before, because it is very important for the understanding of derivatives. That is because the derivative of a function is a generalization of that concept to any curve, not necessarily a straight line.

With calculus we realize that these two concepts are the same: the slope of a curve represents the instantaneous rate of change of a function.

In the rest of this page I give an overview of everything you need to learn about derivatives and I give links to pages that go more in depth in each topic.

Intuition Behind the Derivative

As I said above, there are two ways of thinking about the derivative: one is a physical approach (rate of change) and the other is geometric (slope).

Intuitive Notion Meaning

It is essential to understand both perspectives, because one is more useful than the other in different situations.

The essential idea behind the geometric approach is captured in the following picture. This picture shows how to calculate the slope of a straight line.

This image shows how to calculate the slope of a line

In this picture we see the graph of a straight line, in green. This line is the graph of a linear function. We take two points on the x axis, a and b, and the corresponding values of the function, f(a) and f(b).

Δx is the difference between b and a (the little triangle is spelled as 'delta', so Δx is spelled 'delta x'). And we call the difference between f(b) and f(a) Δy. The slope of the green line in the graph is defined as the quotient between these two deltas:

From the graph, we can see that:

All of this may seem incomprehensible to you if you haven't seen the concept of slope before. It that is so, I recommend you to go straight to the page The Definition of Derivative: The Intuition Behind It. There we carefull review the concept of slope of a straight line and we go on to define the derivative using the intuitive notions.

When we have a curve instead of a straight line, we take the same approach to try to calculate the slope. To actually calculate the slope at a single point we take a limit, and we arrive to the definition of the derivative.

For the physical approach, we consider a more concrete problem: a train travelling at varying speed. We imagine that we measure the position of the train at different times, and then plot position vs. time.

Recalling the definition of speed from physics, we realize that the calculation of speed is equivalent to the calculation of the slope of the curve given by the plot position vs. time. With this we realize that speed and slope are connected by a single concept: the derivative.

In the following page we study in depth these two intuitive notions that lead to the same definition of derivative: The Definition of Derivative: The Intuition Behind It.

Derivative Rules

After you understand both the geometric and physical intuition for the derivative of a function, the next step is learning how to calculate derivatives. After all, a definition is good only if you can use it. And your problem solving skills are what will be most heavily tested in any course or exam.

We'll begin by directly applying the definition of the derivative of a function. To calculate the derivative of a function we need to solve a limit. All your limit-skills will come into play here.

In the following page we'll learn the different tricks required to apply the limit definition of derivative to actually calculate the derivative of functions: Calculating the Drivative by Definition.

After finding the derivative of some functions using the definition, we realize that it is a lot of work to do it that way. We then start to deduce some general rules that will serve us neat shortcuts to calculate derivatives more quickly. These are the differentiation rules.

With these rules you can calculate the derivative of any function you know: trigonometric, exponential, logarithmic, etc.

In the following pages we learn the intuition for each rule and solve some problems. It is better to go through the following pages in the order they are listed here. This is because we usually use a previously learned rule to deduce another.

Solving Derivatives

Armed with the differentiation rules, you will be able to find the derivative of 'any function'. You will find that, after you learn the basic rules, solving derivative problems is very similar to solving algebra problems. You just follow the rules of the game.

The following pages cover some of the most important examples of derivatives. In these, you will learn some additional tricks that are good to know.

Finally, on the page solving derivatives, we review all the tricks and techniques for solving problems where you need to calculate the derivative of something.

Applications of the Derivative

After you learn to calculate derivatives, the next step is learning something even more fun and useful: how to apply derivatives to the solution of other problems. Derivatives have many applications in calculus itself and many other disciplines.

Derivatives have applications in physics, computer science, chemistry, biology, economics and archeology, just to name a few.

Right now I'm writing the pages on basic applications of the derivative. Some of them are ready, and you can access them below:

  • Optimization Problems
  • Finding Roots of Equations

If you have a doubt, a suggestion for a new topic, or just want to say hi, leave me a comment below. :-)

New! Comments

Do you have a doubt, or want some help with a problem? Leave a comment in the box below.
Word and Object
AuthorWillard Van Orman Quine
CountryUnited States
SubjectsEpistemology, language
PublisherMIT Press
Publication date
Media typePrint (Hardcover and Paperback)
Intuitive system definitionIntuitive Notion Meaning

Word and Object is a 1960 work by the philosopher Willard Van Orman Quine, in which the author expands upon the line of thought of his earlier writings in From a Logical Point of View (1953), and reformulates some of his earlier arguments, such as his attack in 'Two Dogmas of Empiricism' on the analytic–synthetic distinction.[1] The thought experiment of radical translation and the accompanying notion of indeterminacy of translation are original to Word and Object, which is Quine's most famous book.[2]


Quine emphasizes his naturalism, the doctrine that philosophy should be pursued as part of natural science.[3] He argues in favor of naturalizing epistemology, supports physicalism over phenomenalism and mind-body dualism, and extensionality over intensionality, develops a behavioristic conception of sentence-meaning, theorizes about language learning, speculates on the ontogenesis of reference, explains various forms of ambiguity and vagueness, recommends measures for regimenting language to eliminate ambiguity and vagueness as well as to make perspicuous the logic and ontic commitments of theories, argues against quantified modal logic and the essentialism it presupposes, argues for Platonic realism in mathematics, rejects instrumentalism in favor of scientific realism, develops a view of philosophical analysis as explication, argues against analyticity and for holism, against countenancing propositions, and tries to show that the meanings of theoretical sentences are indeterminate and that the reference of terms is inscrutable.[2]


Central to Quine's philosophy is his linguistic behaviorism. Quine has remarked that one may or may not choose to be a behaviorist in psychology, but one has no choice but to be a behaviorist in linguistics.[4]

This influence can be seen in Word and Object. In chapter 2 a linguist has to translate a native's unknown language into English. What is so specifically behavioristic there is that the linguist has nothing to go on but verbal behavior from the native and the visible environment the native interacts with. The same view is displayed in chapter 3 where Quine describes how a baby learns its first words. In this chapter Quine also mentions B.F. Skinner, a well known behaviorist, as one of his influences. The opposite view to Quine's and Skinner's in philosophy of language is represented by Noam Chomsky's linguistic nativism.[5]:73

Translation and meaning[edit]

In the second chapter of Word and Object, Quine investigates the concept of meaning. He shows to what extent his own, empirical, notion of meaning can give an account for our intuitive concept of meaning: 'what a sentence shares with its translation'.[5]:29 Quine also introduces his famous principle of indeterminacy of translation, with the help of the thought experiment of radical translation, i.e. translation of a hitherto unknown language (called Jungle by Quine) into English. The point of this thought experiment is to show that a behavioristic account of meaning does not allow for the determination of the right manual for translating one language into another, as there is no such single right translation manual.[6]

A linguist desiring to translate Jungle has to set up his translation manual based only on the events happening around him/her, the stimulations, combined with the verbal and non-verbal behaviour of Jungle natives.[7] The linguist can thus only use empirical information, therefore, radical translation will tell us which part of our language can be accounted for by stimulus conditions. In the experiment, Quine assumes that functional Jungle equivalents of 'Yes' and 'No' are relatively easy to be found. This allows the linguist to actively query the utterances of the natives, by repeating words (s)he has heard the native utter, and to subsequently record the native's reaction of assent or dissent.

In determining the translation of the Jungle sentence 'Gavagai' (whose English equivalent would be 'Look, a rabbit'), the linguist first has to determine which stimulation prompt the native to assent, and which prompt him to dissent to the linguist uttering 'Gavagai'. For example, if the linguist sees a rabbit, and the native says 'Gavagai', the linguist may think that 'Gavagai' means 'Rabbit'. (S)he will then try the sentence 'Gavagai' in different situations caused by the stimulation of a rabbit, to see whether the native assents or dissents to the utterance. The native's reaction is elicited by the linguist's question and the prompting stimulation together. It is the stimulation that prompts the assent or dissent, not the object in the world, because an object in the world can be replaced by a replica, but then the stimulation stays the same. 'The class of all the stimulations [..] that would prompt his assent'[5]:29 is the affirmative stimulus meaning of a certain sentence for a given speaker. Negative stimulus meaning is defined likewise, with assent and dissent interchanged. Quine calls these affirmative and negative stimulus meaning combined the stimulus meaning of the sentence. However, since we want to account for the fact that a speaker can change the meaning of a concept, we add the modulus to the definition of stimulus meaning: the time frame in which the stimulations take place. Once the stimulus meaning has been found, the linguist can then compare it to the stimulus meanings of sentences in English. The English sentence with (near-) identical stimulus meaning to 'Gavagai' functions as a translation of 'Gavagai'.

After Quine has set out the concept of stimulus meaning, he continues by comparing it with our intuitive notion of meaning.[8]:100 For this, he distinguished two kinds of sentences: occasion sentences and standing sentences. Occasion sentences are the sentences that are only affirmed or dissented after an appropriate stimulation,[5]:32–33 e.g. 'Look, a rabbit walks by!' On the other hand, there are standing sentences, which do not rely on stimulation for assent or dissent; they can be prompted by stimulation, but they don't have to be, e.g. 'Rabbits are mammals'. Thus, the stimulus meaning is less useful to approximate the intuitive meaning of standing sentences. However, the difference between occasion and standing sentences is only a gradual difference. This difference depends on the modulus because 'an occasion sentence modulo n seconds can be a standing sentence modulo n – 1'.[5]:32

Since stimulus meaning cannot really account for the intuitive concept of meaning for standing sentences, the question remain whether it can account for the intuitive concept of meaning for observation sentences. Quine approaches this question by investigating whether, for occasion sentences, the intuitive notion of synonymy (sameness of meaning) is equivalent to the notion of stimulus synonymy (sameness of stimulus meaning).[8]:100 For this question, he uses the notion of observationality. A special subclass of occasion sentences are the observations sentences. Their stimulus meaning is least influenced by collateral information, extra information that is hidden for the linguist, and does not vary over the population. Therefore, observation sentences belong to the sentences that are directly translatable by the linguist.[9] However, it is exactly this point of collateral information that poses problems for equating the intuitive notion of synonymy with the notion of stimulus synonymy. For even sentences that are supposedly highly observational, like 'Gavagai!', can be affected by collateral information. Quine uses the example of a rabbit-fly: assume that there is a fly that is unknown to the linguist, that only occurs in the presence of rabbits. Seeing such a rabbit-fly in the grass would thus make the native assent to the sentence 'Gavagai', because the native can be sure that there is a rabbit nearby. However, the rabbit-fly is not part of the stimulus meaning of 'Rabbit' for the linguist. Thus, even for the most observational occasion sentences, it is not possible to equate the intuitive notion of synonymy with stimulus synonymy. From this, Quine concludes that we cannot make sense of our intuitive notions of meaning. As Becker formulates it:

From Quine's perspective, the conclusion to be drawn from our failure to reconstruct intuitive semantics is not that the attempt was misconceived but that our ordinary notions about meaning cannot be made intelligible. More particularly, intuitive semantics is committed to a distinction—between semantic information, information about meanings, and factual (or collateral) information, information not about meanings—which we cannot make sense of even in the case of sentences like 'Rabbit', let alone for sentences in general.[8]:109

Indeterminacy of translation[edit]

Having taken the first steps in translating sentences, the linguist still has no idea if the term 'gavagai' is actually synonymous to the term 'rabbit', as it is just as plausible to translate it as 'one second rabbit stage', 'undetached rabbit part', 'the spatial whole of all rabbits', or 'rabbithood'. Thus, the identical stimulus meaning of two sentences 'Gavagai' and 'Rabbit' does not mean that the terms 'gavagai' and 'rabbit' are synonymous (have the same meaning). In fact, we cannot even be sure that they are coextensive terms,[8]:159 because 'terms and reference are local to our conceptual scheme',[5]:48 and cannot be accounted for by stimulus meaning. It appears therefore impossible to determine a unique correct translation of the term 'gavagai', since the linguist can take any of the mentioned possibilities and have it correspond to the stimulus meaning through adaptation of logical connectives. This implies there is no matter of fact to which the word refers. Quine calls this the inscrutability of reference.[10]

This inscrutability leads to difficulties in translating sentences, especially with sentences that have no direct connection to stimuli. For example, the tautological Jungle sentence 'Gavagai xyz gavagai' could be translated (in accordance with stimulus meaning) as 'This rabbit is the same as this rabbit'. However, when 'gavagai' is taken as 'undetached rabbit part' and 'xyz' as 'is part of the same animal as', the English translation could also run 'This undetached rabbit part is part of the same animal as this undetached rabbit part'. The Jungle sentence and its two English translations all have the same stimulus meaning and truth conditions, even though the two translations are clearly different. Quine concludes that the linguist can set up his translation manual in different ways, that all fit the native's speech behaviour yet are mutually incompatible.[5]:24 This is called holophrastic indeterminacy. There is no one correct translation of Jungle: translation is indeterminate.[10]

Analytical hypotheses[edit]

Quine sums up the first steps of the radical translation:

(1) Observation sentences can be translated. There is uncertainty, but the situation is the normal inductive one. (2) Truth functions can be translated. (3) Stimulus-analytic sentences can be recognized. So can the sentences of the opposite type, the 'stimulus-contradictory' sentences, which command irreversible dissent. (4) Questions of intrasubjective stimulus synonymy of native occasion sentences even of non-observational kind can be settled if raised, but the sentences cannot be translated.

To go beyond these boundaries of translation by stimulus meaning, the linguist uses analytical hypotheses, in which he uses the steps (1)–(4) to equate parts of the native sentences to English words or phrases. Using the analytical hypotheses the linguist can form new sentences and create a translation manual.

Counter Intuitive Meaning Synonyms


In Chapter 2 of Word and Object, Quine shows that the total apparatus of grammatical and semantic devices in a language is not objectively translatable into foreign languages. Therefore, in Chapter 3, he proposes to investigate a language's devices relative to each other. For this, he first describes a child's process of acquiring reference, by showing the order in which children learn grammatical devices. In Chapter 4 he then turns away from language acquisition, to investigate the vagaries of reference in a particular language (English). In Chapter 5, Quine proposes a system for regimentation, which should help us understand how reference in language works and should clarify our conceptual scheme. He calls this system the canonical notation; it is a system with which we can investigate the grammatical and semantic devices of English by paraphrase.

Acquiring reference[edit]

In order to learn a language, a child has to learn how the language expresses reference grammatically. Quine presents a behavioral theory in which the child acquires language through a process of conditioning and ostension.[11] This process consists of four phases.[5]:98–100 In the first phase the child starts babbling. This behavior gets rewarded or not, dependent on the situation in which it occurs. Terms are learned by a process of reinforcement and extinction. In this phase, the child is not aware yet of objects, it just reacts to stimulations. This is a form of operant conditioning. In the second phase, the child acquires general terms, and demonstrative singular terms (this, that) and singular description, sentences that name (or purport to name) only one object. In this phase the child also learns terms that do not have reference, like 'unicorn'. Furthermore, the child learns divided reference of general terms (that general terms refer to more than one object), and with that it has access to a conceptual scheme that includes 'enduring and recurring objects'.[5]:86 With this, the child has acquired the important distinction between singular and general terms. This distinction entails that a singular term 'purports to refer to one object' while a general term does not purports to refer to an object.[5]:87

As Quine points out: 'The basis combination in which general and singular terms find their contrasting roles is that of predication.'[5]:87 Predication combines general terms with singular terms, in a sentence that is true or false just as the general term ('F') is true or false of the object to which the singular term ('a') refers. Predication is thus logically represented as 'Fa'. In the third phase, the child learns composite general terms, which are formed by joining two general terms. In the fourth phase, the child learns how to talk about new objects. The child can now apply relative terms to singular or general terms. A relative term is a term that is true of two (or more) objects in relation to each other, like 'bigger than'. The child can now make use of terms that refer to objects that cannot be seen, for example 'smaller than that speck' to refer to a neutrino.[5]:100 This phase thus gives a new dimension to the child's conceptual scheme.

Vagaries of reference and referential transparency[edit]

In Chapter 4 of Word and Object, Quine looks at the indeterminacies of reference that are inherent to the (English) language system. A term is vague if the boundaries of its reference are not clear. For a singular term this means that the boundaries of the object it refers to are not clear, e.g. with the 'mountain': for two neighboring mountains it is not clear where the first mountain stops and the second one begins. General terms can be vague in this same way, but also in yet another way, namely that there are some objects of which it is not clear whether or not they should be included in the reference of the term. For example, the term 'blue' is vague insofar as it is not clear whether or not some objects are blue or green. A second vagary of reference is ambiguity. Ambiguity differs from vagueness in that for a vague term the (boundaries of) its reference are unsettled, whereas ambiguous terms do refer to clearly to objects, however they are clearly true and clearly false of the same objects. For example, the term 'light' is clearly true of a dark feather, but at the same time clearly false of it.

Quine also introduces the term 'referential transparency'. Quine wants to make explicit the ambiguities in language, and to show different interpretations of sentences, therefore, he has to know where the terms in a sentence refer to. A term is used in purely referential position if its only purpose is to specify its object so that the rest of the sentence can say something about it. If a term is used in purely referential position, it is subject to the substituitivity of identity: the term can be substituted by a coextensive term (a term true of the same objects) without changing the truth-value of the sentence. In the sentence, 'Amsterdam rhymes with Peter Pan' you cannot substitute 'Amsterdam' with 'the capital of the Netherlands'. A construction, a way in which a singular term or a sentence is included in another singular term or sentence, has referential transparency: it is either referentially transparent or referentially opaque. A construction is referentially transparent if it is the case that if an occurrence of a term is purely referential in a sentence then it is purely referential also in the containing sentence. However, Quine's goal is to make clear which positions in a sentence are referentially transparent, not to make them all transparent.

Canonical notation[edit]

In Chapter 5 of Word and Object Quine proposes a system of regimentation: the paraphrasing of sentences into a 'canonical notation', that we can use to understand how reference works in a language. Since we use language for science, the reductions that we make in the complexity of the structure of sentences will also simplify the conceptual schema of science. In the canonical notation, a sentence S is paraphrased as S'. S' is a paraphrase of S that should clarify its reference, which means that it often resolves ambiguities, and is therefore by definition not synonymous with S. However, S' should express the intended meaning of the speaker. Therefore, it should always be the original speaker who does the paraphrasing. The canonical notation consists of: atomic sentences (sentences that do not have sentences as a part) that have a general term in the predicate position, with one or more variables: 'Fa' or 'Fab,' etc. Non-atomic sentences are built from atomic sentences by using truth functions, quantifiers, and some other devices, like the four variable-binding operators. Quine drops tense, and instead uses the present as temporally neutral. We can express time with the use of 'a at t', where x is a spatiotemporal object. In his canonical notation, Quine has eliminated all singular terms other than variables. This greatly simplifies his logical theory, in the sense that there is economy in the roots of the theory: there is a very limited number of elements. In some situations, however, short paraphrases are very useful, for example in mathematic deductions. For these cases, Quine introduces definitions: we can define singular terms relative to the canonical notation. In that way, we can still use singular terms, without having to include them in our theory.

Semantic ascent[edit]

In the last paragraph of Word and Object,[5]:56 Quine asks the question why, in a book titled Word and Object, we have talked more about words than about objects. He comes to the conclusion that this has to do with the distinction Rudolf Carnap makes between a material mode of speech and a formal one.[12] In the material mode we talk about objects themselves and usually this is unproblematic. However, when two people with completely different ideas of whether or not there are such entities as miles, are discussing miles as the objects themselves this discussion will be fruitless. It is in these instances that we see what Quine calls semantic ascent,[5]:249–254 the shift from the material mode of language to the formal one. In the formal mode of language we are at a different level. Rather than talking about miles as objects we are talking about what this word 'mile' even means, what it refers to and if it even refers at all. In the formal mode, people with different conceptual schemes might be able to have a reasonable discussion because they are talking about something their conceptual schemes have in common: language.

Intuitive Notion Meaning Synonyms

Quine differs from Carnap in applicability of semantic ascent.[5]:250 Carnap believes that talking in a formal mode is something that can only be done to some effect in philosophy. Quine, however, believes that semantic ascent is used in science as well. He argues that Einstein's theory of relativity wasn't just accepted by the scientific community because of what it had to say about 'time, light, headlong bodies and the perturbations of Mercury'[5]:251 in the material mode, but also because of its simplicity compared to other theories in the formal mode. The formal mode allows for a more distant approach to certain problems; however, we are not able to reach a vantage point outside of our conceptual scheme, to Quine 'there is no such cosmic exile'.[5]:254

See also[edit]

Intuitive Meaning In English

  • Neurath's boat – a philosophical analogy popularized by Word and Object

Intuitive Notion Meaning Synonym


  1. ^Quine, Willard Van Orman (1985). The Time of My Life: An Autobiography. Cambridge, Massachusetts: MIT Press. p. 392. ISBN978-0262670043.
  2. ^ abGibson, Roger F. (1999). Audi, Robert (ed.). The Cambridge Dictionary of Philosophy. Cambridge: Cambridge University Press. pp. 767–768. ISBN0-521-63722-8.
  3. ^Hookway, C. J. (2005). Honderich, Ted (ed.). The Oxford Companion to Philosophy. Oxford: Oxford University Press. p. 779. ISBN0-19-926479-1.
  4. ^The Cambridge Companion to Quine, Roger F. Gibson, Cambridge University Press, 2004, p. 199
  5. ^ abcdefghijklmnopqQuine, Willard Van Orman (2013) [1960]. Word and Object (New ed.). Cambridge, MA: MIT Press. doi:10.7551/mitpress/9636.001.0001. ISBN9780262518314. OCLC808006883. New edition with a foreword by Patricia Churchland.
  6. ^Harman, G. (2013). Harman, G.; Lepore, E. (eds.). A Companion to W.V.O. Quine. Hoboken, NJ: Wiley. pp. 236–237. ISBN9781118607992.
  7. ^Hookway, C. J. (1995). Honderich, Ted (ed.). The Oxford Companion to Philosophy. Oxford: Oxford University Press. p. 740. ISBN0-19-866132-0.
  8. ^ abcdBecker, E. (2012). The Themes of Quine's Philosophy: Meaning, Reference, and Knowledge. Cambridge University Press.
  9. ^Kirk, Robert. (2004). 'Indeterminacy of Translation'. In: Roger F. Gibson, Jr (ed.) The Cambridge Companion to Quine. pp. 151-180. Cambridge Companions to Philosophy. Cambridge: Cambridge University Press. p. 162.
  10. ^ abMarsoobian, A. T., Ryder, J. (2003). Marsoobian, A. T.; Ryder, J. (eds.). The Blackwell Guide to American Philosophy. Hoboken, NJ: Wiley-Blackwell. p. 251. ISBN978-0-631-21623-0.CS1 maint: multiple names: authors list (link)
  11. ^Murphey, M. The Development of Quine's Philosophy. Springer, 2011. Web. Boston Studies in the Philosophy of Science. p. 163
  12. ^Carnap, Rudolf, Logical Syntax of Language [1960]. The International Library of Philosophy: Philosophy of Mind and Language, Routledge, Reprint edition, 2010, pp. 63-64.

Intuitive Concept Meaning

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