Broadly speaking, what distinguishes the man who knows from the ignorant man is an ability to teach, and this is why we hold that art and not experience has the character of genuine knowledge episteme --namely, that artists can teach and others i. Those who can, do. Those who understand, teach. For a subject which would not bear raillery is suspicious; and a jest which would not bear a serious examination is certainly false wit.
Those who understand: Knowledge growth in teaching. Educational Researcher , 15 2 , 4 - Wikipedia has an article about: Aristotle. Wikisource has original works written by or about: Aristotle. Wikimedia Commons has media related to: Aristotle. Namespaces Page Discussion. Views Read Edit View history.
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Skip to main content. Numbers 3: Infinity. Praise for Numbers 3: Infinity "A little violent, a little supernatural, a little mysterious, a lot sentimental; fans of the trilogy won't be disappointed as this story edges toward magical thriller. In this sense the hyperreal line is the extension of the reals to the hyperreals. The development of analysis via infinitesimals creates a nonstandard analysis with a hyperreal line and a set of hyperreal numbers that include real numbers.
Sums and products of infinitesimals are infinitesimal. Sentences about the standardly-described reals are true if and only if they are true in this extension to the hyperreals. Nonstandard analysis allows proofs of all the classical theorems of standard analysis, but it very often provides shorter, more direct, and more elegant proofs than those that were originally proved by using standard analysis with epsilons and deltas. Objections by practicing mathematicians to infinitesimals subsided after this was appreciated. Mathematics is apparently about mathematical objects, so it is apparently about infinitely large objects, infinitely small objects, and infinitely many objects.
Mathematicians who are doing mathematics and are not being careful about ontology too easily remark that there are infinite dimensional spaces, the continuum, continuous functions, an infinity of functions, and this or that infinite structure. Do these infinities really exist? The philosophical literature is filled with arguments pro and con and with fine points about senses of existence. When axiomatizing geometry, Euclid said that between any two points one could choose to construct a line. Assertions require proofs. The constructivist believes that to justifiably assert the negation of a sentence S is to prove that the assumption of S leads to a contradiction.
So, legitimate mathematical objects must be shown to be constructible in principle by some mental activity and cannot be assumed to pre-exist any such construction process nor to exist simply because their non-existence would be contradictory.
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A constructivist, unlike a realist, is a kind of conceptualist, one who believes that an unknowable mathematical object is impossible. Most constructivists complain that, although potential infinites can be constructed, actual infinities cannot be. There are many different schools of constructivism. The first systematic one, and perhaps the most well known version and most radical version, is due to L.
Numbers are human creations. The number pi is intuitionistically legitimate because we have an algorithm for computing all its decimal digits, but the following number g is not legitimate: The following number g is illegitimate. It is the number whose nth digit is either 0 or 1, and it is 1 if and only if there are n consecutive 7s in the decimal expansion of pi. No person yet knows how to construct the decimal digits of g.
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Brouwer argued that the actually infinite set of natural numbers cannot be constructed using intuitions and so does not exist. The best we can do is to have a rule for adding more members to a set. So, his concept of an acceptable infinity is closer to that of potential infinity than actual infinity. Hermann Weyl emphasizes the merely potential character of these infinities:. Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers….
The sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation, but is not a closed realm of things existing in themselves. Weyl is quoted in Kleene , p. It is not legitimate for platonic realists, said Brouwer, to bring all the sets into existence at once by declaring they are whatever objects satisfy all the axioms of set theory.
Brouwer believed realists accept too many sets because they are too willing to accept sets merely by playing coherently with the finite symbols for them when sets instead should be tied to our experience. For Brouwer this experience is our experience of time. He believed we should arrive at our concept of the infinite by noticing that our experience of a duration can be divided into parts and then these parts can be further divided, and so.
This infinity is a potential infinity, not an actual infinity. For the intuitionist, there is no determinate, mind-independent mathematical reality which provides the facts to make mathematical sentences true or false. This metaphysical position is reflected in the principles of logic that are acceptable to an intuitionist.
And it is false only if we have proved that some x does not have property F. Otherwise, it is neither true nor false. The intuitionist does not accept the principle of excluded middle: For any sentence S, either S or the negation of S. Finitists, even those who are not constructivists, also argue that the actually infinite set of natural numbers does not exist. They say there is a finite rule for generating each numeral from the previous one, but the rule does not produce an actual infinity of either numerals or numbers.
The ultrafinitist considers the classical finitist to be too liberal because finite numbers such as 2 and 2 can never be accessed by a human mind in a reasonable amount of time. Only the numerals or symbols for those numbers can be coherently manipulated. One challenge to ultrafinitists is that they should explain where the cutoff point is between numbers that can be accessed and numbers that cannot be. Ultrafinitsts have risen to this challenge.
The mathematician Harvey Friedman says:. I raised just this objection [about a cutoff] with the extreme ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific.
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He virtually immediately said yes. Then I asked about 2 2 , and he again said yes, but with a perceptible delay. Then 2 3 , and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2 times as long to answer yes to 2 than he would to answering 2 1. There is no way that I could get very far with this. Elwes , This battle among competing philosophies of mathematics will not be explored in depth in this article, but this section will offer a few more points about mathematical existence.
If a set of axioms is consistent, and so is its corresponding axiomatic theory, then the theory defines a class of models, and each axiom is true in any such model, but it does not follow that the axioms are really true.
The formal theory using these axioms is consistent and has a model, but it does not follow that either axiom is really true. Quine objected to Hilbert's criterion for existence as being too liberal. Mathematical theories which imply the existence of some actually infinite sets are indispensable to all these scientific theories, and their referring to these infinities cannot be paraphrased away. All this success is a good reason to believe in some actual infinite sets and to say the sentences of both the mathematical theories and the scientific theories are true or approximately true since their success would otherwise be a miracle.
But, he continues, of course it is no miracle. See Quine chapter 7. Quine points out that reference to mathematical entities is vital to science, and there is no way of separating out the evidence for the mathematics from the evidence for the science. This famous indispensability argument has been attacked in many ways.
Not even set theory itself can tell us how the existence of a set e. Cantor initially thought of a set as being a collection of objects that can be counted, but this notion eventually gave way to a set being a collection that has a clear membership condition. The acceptance was based on three reasons. Mathematics can be modeled in set theory; it can be given a basis in set theory.
Notice that one of the three reasons is not that set theory provides a foundation to mathematics in the sense of justifying the doing of mathematics or in the sense of showing its sentences are certain or necessary. Instead, set theory provides a basis for theories only in the sense that it helps to organize them, to reveal their interrelationships, and to provide a means to precisely define their concepts.
The first program for providing this basis began in the late 19 th century. Peano had given an axiomatization of the natural numbers. It can be expressed in set theory using standard devices for treating natural numbers and relations and functions and so forth as being sets. For example, zero is the empty set, and a relation is a set of ordered pairs.
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Then came the arithmetization of analysis which involved using set theory to construct from the natural numbers all the negative numbers and the fractions and real numbers and complex numbers. Along with this, the principles of these numbers became sentences of set theory.
In this way, the assumptions used in informal reasoning in arithmetic are explicitly stated in the formalism, and proofs in informal arithmetic can be rewritten as formal proofs so that no creativity is required for checking the correctness of the proofs. Once a mathematical theory is given a set theoretic basis in this manner, it follows that if we have any philosophical concerns about the higher level mathematical theory, those concerns will also be concerns about the lower level set theory in the basis.
One popular one is to define a finite set as a set onto which a one-to-one function maps the set of all natural numbers that are less than some natural number n. That finite set contains n elements. An infinite set is then defined as one that is not finite. Dedekind, himself, used another definition; he defined an infinite set as one that is not finite, but defined a finite set as any set in which there exists no one-to-one mapping of the set into a proper subset of itself.
The philosopher C. Peirce suggested essentially the same approach as Dedekind at approximately the same time, but he received little notice from the professional community. For more discussion of the details, see Wilder , p. Set theory implies quite a bit about infinity.
First, infinity in ZF has some very unsurprising features. If a set A is infinite and is the same size as set B, then B also is infinite. If A is infinite and is a subset of B, then B also is infinite. Using the axiom of choice, it follows that a set is infinite just in case for every natural number n, there is some subset whose size is n.
The power set axiom which says every set has a power set, namely a set of all its subsets then generates many more infinite sets of larger cardinality, a surprising result that Cantor first discovered in In ZF, there is no set with maximum cardinality, nor a set of all sets, nor an infinitely descending sequence of sets x 0 , x 1 , x 2 , There is however, an infinitely ascending sequence of sets x 0 , x 1 , x 2 , In ZF, a set exists if it is implied by the axioms; there is no requirement that there be some property P such that the set is the extension of P.
One especially important feature of ZF is that for any condition or property, there is only one set of objects having that property, but it cannot be assumed that for any property, there is a set of all those objects that have that property. For example, it cannot be assumed that, for the property of being a set, there is a set of all objects having that property. In ZF, all sets are pure. A set is pure if it is empty or its members are sets, and its members' members are sets, and so forth. In informal set theory, a set can contain cows and electrons and other non-sets.
NBG is designed to have proper classes, classes that are not sets, even though they can have members which are sets. The intuitive idea is that a proper class is a collection that is too big to be a set. There can be a proper class of all sets, but neither a set of all sets nor a class of all classes. Are philosophers justified in saying there is more to know about sets than is contained within ZF set theory? If V is the collection or class of all sets, do mathematicians have any access to V independently of the axioms?
This is an open question that arose concerning the axiom of choice and the continuum hypothesis. Consider whether to believe in the axiom of choice. However, the axiom does not say how to do the choosing. For some sets there might not be a precise rule of choice. If the collection is infinite and its sets are not well-ordered in any way that has been specified, then there is in general no way to define the choice set.
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The axiom is implicitly used throughout the field of mathematics, and several important theorems cannot be proved without it. Mathematical Platonists tend to like the axiom, but those who want explicit definitions or constructions for sets do not like it. The dispute can get quite intense with advocates of the axiom of choice saying that their opponents are throwing out invaluable mathematics, while these opponents consider themselves to be removing tainted mathematics.
A set is always smaller than its power set. How much bigger is the power set? The generalized continuum hypothesis is more general; it says that, given an infinite set of any cardinality, the cardinality of its power set is the next larger cardinal and not some even higher cardinal. Cantor believed the continuum hypothesis, but he was frustrated that he could not prove it. The philosophical issue is whether we should alter the axioms to enable the hypotheses to be proved.
In this sense, both the continuum hypothesis and the axiom of choice are independent of ZF. So, how do we decide whether to believe the axiom of choice and continuum hypothesis, and how do we decide whether to add them to the principles of ZF or any other set theory?
Most mathematicians do believe the axiom of choice is true, but there is more uncertainty about the continuum hypothesis. The independence does not rule out our someday finding a convincing argument that the hypothesis is true or a convincing argument that it is false, but the argument will need more premises than just the principles of ZF.
At this point the philosophers of mathematics divide into two camps. The realists, who think there is a unique universe of sets to be discovered, believe that if ZF does not fix the truth values of the continuum hypothesis and the axiom of choice, then this is a defect within ZF and we need to explore our intuitions about infinity in order to uncover a missing axiom or two for ZF that will settle the truth values. These persons prefer to think that there is a single system of mathematics to which set theory is providing a foundation, but they would prefer not simply to add the continuum hypothesis itself as an axiom because the hope is to make the axioms "readily believable," yet it is not clear enough that the axiom itself is readily believable.
The second camp of philosophers of mathematics disagree and say the concept of infinite set is so vague that we simply do not have any intuitions that will or should settle the truth values. According to this second camp, there are set theories with and without axioms that fix the truth values of the axiom of choice and the continuum hypothesis, and set theory should no more be a unique theory of sets than Euclidean geometry should be the unique theory of geometry. So far there is no agreement among researchers about the acceptability of any of the new axioms. See Wolf , pp. The infinite appears in many interesting ways in formal deductive logic, and this section presents an introduction to a few of those ways.
Among all the various kinds of formal deductive logics, first-order logic the usual predicate logic stands out as especially important, in part because of the accuracy and detail with which it can mirror mathematical deductions. First-order logic also stands out because it is the strongest logic that has a proof for every one of its infinitely numerous logically true sentences, and that is compact in the sense that if an infinite set of its sentences is inconsistent, then so is some finite subset. But first-order logic has expressive limitations:.
The Lowenheim-Skolem theorems entail that no infinite structure can be characterized up to isomorphism in a first-order language. Let's be clearer about just what first-order logic is. To answer this and other questions, it is helpful to introduce some technical terminology. Here is a chart of what is ahead:. A first-order theory is a set of sentences expressed in a first-order language which will be defined below.
A first-order formal system is a first-order theory plus its deductive structure method of building proofs. Intuitively and informally, any formal system is a system of symbols that are manipulated by the logician in game-like fashion for the purpose of more deeply understanding the properties of the structure that is represented by the formal system. The symbols denote elements or features of the structure the formal system is being used to represent.
It can mean a first-order language with its deductive structure, or a first-order language with its semantics, or the academic discipline that studies first-order languages and theories. Classical first-order logic is classical predicate logic with its core of classical propositional logic. This logic is distinguished by its satisfying certain classically-accepted assumptions: that it has only two truth values some non-classical logics have an infinite number of truth values , every sentence that is, proposition gets exactly one of the two truth values; no sentence can contain an infinite number of symbols; a valid deduction cannot be made from true sentences to a false one; deductions cannot be infinitely long; the domain of an interpretation cannot be empty but can have any infinite cardinality; an individual constant name must name something in the domain; and so forth.
It has a denumerable list of variables. A first-order language has a countably finite or countably infinite number of predicate symbols and function symbols, but not a zero number of both. First-order languages differ from each other only in their predicate symbols or function symbols or constants symbols or in having or not having the equality symbol. See Wolf , p. There are denumerably many terms, formulas, and sentences. Also, because there are uncountably many real numbers, a theory of real numbers in a first-order language does not have enough names for all the real numbers.
To carry out proofs or deductions in a first-order language, the language needs to be given a deductive structure. There are several different ways to do this via axioms, natural deduction, sequent calculus , but the ways are all independent of which first-order language is being used, and they all require specifying rules such as modus ponens for how to deduce wffs from finitely many previous wffs in the deduction. To give some semantics or meaning to its symbols, the first-order language needs a definition of valuation and of truth in a valuation and of validity of an argument.
In a propositional logic, the valuation assigns to each sentence letter its own single truth value; in predicate logic each term is given its own denotation its extension , and each predicate is given a set of objects its extension in the domain that satisfy the predicate. The valuation rules then determine the truth values of all the wffs.
The domain may be of any finite or transfinite size, but the variables can range only over objects in this domain, not over sets of those objects. Tarski, who was influential in giving an appropriate, rigorous definition to first-ordr language, was always bothered by the tension between his nominalist view of language as the product of human activity, which is finite, and his view that intellectual progress in logic and mathematics requires treating a formal language as having infinite features such as an infinity of sentences.
This article does not explore how this tension can be eased, or whether it should be. P is called a predicate variable or property variable. Every valid deduction in first-order logic is also valid in second-order logic. A language is third-order if it has quantifiers on variables that range over properties of properties of objects or over sets of sets of objects , and so forth.
A language is called higher-order if its order is second-order or higher. The definition of first-order theory given earlier in this section was that it is any set of wffs in a first-order language. A more ordinary definition adds that it is closed under deduction. This additional requirement implies that every deductive consequence of some sentences of the theory also is in the theory.
Since the consequences are countably infinite, all ordinary first-order theories are countably infinite. If the language is not explicitly mentioned for a first-order theory, then it is generally assumed that the language is the smallest first-order language that contains all the sentences of the theory. Valuations of the language in which all the sentences of the theory are true are said to be models of the theory.
If the theory is axiomatized, then in addition to the logical axioms there are proper axioms also called non-logical axioms ; these axioms are specific to the theory and so usually do not hold in other first-order theories. See Wolf, , pp. None of these truths so far are known to lie in mainstream mathematics. But they might. And there is another reason to worry about the limitations of PA. Because the set of sentences of PA is only countable, whereas there are uncountably many sets of numbers in informal arithmetic, it might be that PA is inadequate for expressing and proving some important theorems about sets of numbers.
It seems that all the important theorems of arithmetic and the rest of mathematics can be expressed and proved in another first-order theory, Zermelo-Fraenkel set theory with the axiom of choice ZFC. Unlike first-order Peano Arithmetic, ZFC needs only a very simple first-order language that surprisingly has no undefined predicate symbol, equality symbol, relation symbol, or function symbol, other than a single two-place binary relation symbol intended to represent set membership.
The domain is intended to be composed only of sets but since mathematical objects can be defined to be sets, the domain contains these mathematical objects. In the process of axiomatizing a theory, any sentence of the theory can be called an axiom. When axiomatizing a theory, there is no problem with having an infinite number of axioms so long as the set of axioms is decidable, that is, so long as there is a finitely long computation or mechanical procedure for deciding, for any sentence, whether it is an axiom.
Logicians are curious as to which formal theories can be finitely axiomatized in a given formal system and which can only be infinitely axiomatized. Group theory is finitely axiomatizable in classical first-order logic, but Peano Arithmetic and ZFC are not. Peano Arithmetic is not finitely axiomatizable because it requires an axiom scheme for induction. The first-order theory of Euclidean geometry is not finitely axiomatizable, and the second-order logic used in Field to reconstruct mathematical physics without quantifying over numbers also is not finitely axiomatizable.
See Mendelson for more discussion of finite axiomatizability. In the languages of classical first-order logic, every formula is required to be only finitely long, but an infinitary logic might relax this. The original, intuitive idea behind requiring finitely long sentences in classical logic was that logic should reflect the finitude of the human mind.
But with increasing opposition to psychologism in logic, that is, to making logic somehow dependent on human psychology, researchers began to ignore the finitude restrictions. In , Alfred Tarski and Dana Scott explored permitting the operations of conjunction and disjunction to link infinitely many formulas into an infinitely long formula. Tarski also suggested allowing formulas to have a sequence of quantifiers of any transfinite length. William Hanf proved in that, unlike classical logics, these infinitary logics fail to be compact.
See Barwise for more discussion of these developments. Classical formal logic requires proofs to contain a finite number of steps. In the mid th century with the disappearance of psychologism in logic, researchers began to investigate logics with infinitely long proofs as an aid to simplifying consistency proofs. See Barwise One reason for permitting an infinite number of truth values is to represent the idea that truth is a matter of degree. One of the simplest infinite-valued semantics uses a continuum of truth values. Its valuations assign to each basic sentence a formal sentence that contains no connectives or quantifiers a truth value that is a specific number in the closed interval of real numbers from 0 to 1.
It assigns to the disjunction P v Q the maximum of the truth values of P and of Q, and so forth. One advantage to using an infinite-valued semantics is that by permitting modus ponens to produce a conclusion that is slightly less true than either premise, we can create a solution to the paradox of the heap, the sorites paradox.
One disadvantage is that there is no well-motivated choice for the specific real number that is the truth value of a vague statement. Is the truth value 0. This latter problem of assigning truth values to specific sentences without being arbitrary has led to the development of fuzzy logics in place of the simpler infinite-valued semantics we have been considering. Lofti Zadeh suggested that instead of vague sentences having any of a continuum of precise truth values we should make the continuum of truth values themselves imprecise.
His suggestion was to assign a sentence a truth value that is a fuzzy set of numerical values, a set for which membership is a matter of degree. For more details, see Nolt , pp. A countable language is a language with countably many symbols. If a first-order theory in a countable language has an infinite model, then it has a countably infinite model. This is a surprising result about infinity.
Would you want your theory of real numbers to have a countable model? Strictly speaking it is a puzzle and not a paradox because the property of being countably infinite is a property it has when viewed from outside the object language not within it. The theorem does not imply first-order theories of real numbers must have no more real numbers than there are natural numbers. This is a limitation on first-order theories; they do not permit having a categorical theory of an infinite structure.
A formal theory is said to be categorical if any two models satisfying the theory are isomorphic. So, if you create a first-order theory intended to describe a single infinite structure of a certain size, the theory will end up having, for any infinite size, a model of that size. See Enderton , pp. Because of this limitation, many logicians have turned to second-order logics. There are second-order categorical theories for the natural numbers and for the real numbers. Unfortunately, there is no sound and complete deductive structure for any second-order logic having a decidable set of axioms; this is a major negative feature of second-order logics.
To illustrate one more surprise regarding infinity in formal logic, notice that the quantifiers are defined in terms of their domain, the domain of discourse. Unfortunately, in ZF there is no set of all sets to serve as this domain. According to Tarski, since no single language has a global truth predicate, the best approach to expressing truth formally within the language is to expand the language into an infinite hierarchy of languages, with each higher language the metalanguage containing a truth predicate that can apply to all and only the true sentences of languages lower in the hierarchy.
This process is iterated into the transfinite to obtain Tarski's hierarchy of metalanguages. Some philosophers have suggested that this infinite hierarchy is implicit within natural languages such as English, but other philosophers, including Tarski himself, believe an informal language does not contain within it a formal language. To handle the concept of truth formally, Saul Kripke rejects the infinite hierarchy of metalanguages in favor of an infinite hierarchy of interpretations that is, valuations of a single language, such as a first-order predicate calculus, with enough apparatus to discuss its own syntax.
At the first step in the hierarchy, all predicates but the single one-place predicate T x are interpreted. T x is completely uninterpreted at this level. As we go up the hierarchy, the interpretation of the other basic predicates are unchanged, but T is satisfied by the names of sentences that were true at lower levels. At each step in the hierarchy, more sentences get truth values, but any sentence that has a truth value at one level has that same truth value at all higher levels. T almost becomes a global truth predicate when the inductive interpretation-building reaches the first so-called fixed point level.
At this countably infinite level, although T is a truth predicate for all those sentences having one of the two classical truth values, the predicate is not quite satisfied by the names of every true sentence because it is not satisfied by the names of some of the true sentences containing T.
At this fixed point level, the Liar sentence of the Liar Paradox is still neither true nor false. See Kripke, Yablo produced a semantic paradox somewhat like the Liar Paradox. Notice that no sentence overtly refers to itself. There are many aspects of the infinite that this article does not cover. For more discussion of these latter three programs, see Maddy Bradley Dowden Email: dowden csus. The Infinite Working with the infinite is tricky business.
Actual, Potential, and Transcendental Infinity The Ancient Greeks generally conceived of the infinite as formless, characterless, indefinite, indeterminate, chaotic, and unintelligible. Here is an example of how Gregory of Rimini argued in the fourteenth century for the coherence of the concept of actual infinity: If God can endlessly add a cubic foot to a stone—which He can—then He can create an infinitely big stone. Moore , 53 Leibniz envisioned the world as being an actual infinity of mind-like monads, and in Leibniz he freely used the concept of being infinitesimally small in his development of the calculus in mathematics.
Cantor The new idea is that the potentially infinite set presupposes an actually infinite one. Here is why some mathematicians believe the set-theoretic basis is so important: Just as chemistry was unified and simplified when it was realized that every chemical compound is made of atoms, mathematics was dramatically unified when it was realized that every object of mathematics can be taken to be the same kind of thing [namely, a set]. Infinity and the Mind Can humans grasp the concept of the infinite?
In regard to mentally grasping an infinite set or any other set, Cantor said: A set is a Many which allows itself to be thought of as a One. Infinity in Metaphysics There is a concept which corrupts and upsets all others. Infinity in Physical Science From a metaphysical perspective, the theories of mathematical physics seem to be ontologically committed to objects and their properties.
Infinitely Small and Infinitely Divisible Consider the size of electrons and quarks, the two main components of atoms. Penrose , b. Singularities There is a good reason why scientists fear the infinite more than mathematicians do. Idealization and Approximation Scientific theories use idealization and approximation; they are "lies that help us to see the truth," to use a phrase from the painter Pablo Picasso who was speaking about art, not science.
Infinity in Mathematics The previous sections of this article have introduced the concepts of actual infinity and potential infinity and explored the development of calculus and set theory, but this section probes deeper into the role of infinity in mathematics. In recommending how to use the concept of infinity coherently, Bertrand Russell said pejoratively: The whole difficulty of the subject lies in the necessity of thinking in an unfamiliar way, and in realising that many properties which we have thought inherent in number are in fact peculiar to finite numbers.
Infinitesimals and Hyperreals There has been considerable controversy throughout history about how to understand infinitesimal objects and infinitesimal changes in the properties of objects. This derivative was defined by Leibniz to be where h is an infinitesimal.