Philosophy and the infinite
This page is a selection of philosophical writings about infinity with links to other resources on the net. I have tried to summarise and explain quotations and to translate Latin where possible, and to write around the subject, and in general to avoid it being simply a list of quotations. However, the page is still under construction., and it is mostly no more than a list of quotations.
Few, very few, for a hundred years past, have broken the repose of the immense works of the schoolmen. - Hallam
This page is contains only philosophical writing on infinity. (How that is distinguished from mathematical writing is a question I will not attempt to answer). I have included the writing of mathematicians such as Dekedind, Frege and Cantor where they have foundational or philosophical issues. I have included the writing of philosophers like Suarez where they touch upon mathematics.
My aim is (i) to provide links to primary sources now available on the net, that would once have been difficult to locate even with access to a university library; (ii) to link those sources directly to others, where possible. For example, where Cantor cites LII de natura loci as evidence for a medieval scholastic view, we can now examine the same passage, to see if he is right; (iii) to provide evidence for a greater continuity between medieval, early modern and modern philosophy than is commonly supposed.
A future ambition is to provide translations for the medieval sources, most of which are in Latin. Any offers of help would be appreciated, contact me here.
Aristotle (384-322 B.C.) is generally considered to be the greatest philosopher of antiquity. His views exerted considerable influence in the medieval period.
A question that informs every discussion of the infinite (including Cantor's), is whether the Infinite is actual or potential. Aristotle seems to have been the first to formulate this distinction. If every part of the continuum, or every number, or every moment in time actually exists, then (presumably) all such parts exist. But he believed (and most of his followers believed) that there were strong philosophical reasons against all of the parts existing as a whole. The parts were therefore thought to exist potentially: They do not exist until the extended object is broken into parts. (Of course, it is not clear how this resolves any problem. We are still free to quantify over parts that have not been actually divided; we can talk of any point at which the extension could be divided. And if we can take any such point, why not take all of them?)
In Physics Bk III.6 (207a7) he gives his definition of infinity as potential. "A quantity is infinite if it is such that we can always take a part [of it] outside what has already been taken". (Thus we cannot take all the parts). Thus (III.7, 207b12) "the infinite is potential, never actual: the number of parts that can be taken always surpasses any assigned number." (See also Aquinas, III Phys., Lect 11, cf. I de Caelo Lect.3). "Infinity is not a permanent actuality, but consists in a process of coming to be, like time and the number of time". A line consists of two halves, but only potentially, for the actualisation of the halves divides them from one another (Met. 1039a3 ff).
Aristotle also raised the problem of the nature of continuous quantity or the continuum ("continuum" being the Latin noun formed from the adjective "continuus", meaning joining, connected, uninterrupted or unbroken). Is the continuum composed of indivisible elements (points, puncta), or not? Most philosophers before Cantor (with the exception of some seventeenth century scholastics) followed Aristotle in holding that the continuum was not composed of indivisible elements.
Aristotle's argument that the continuum cannot be composed of indivisibles is as follows. He first (in Bk V.3 of the Physics)defines the following terms:
· Together means being in one place, apart means being in different places, in contact means having extremities together.
· Being between implies the existence of something that is before, something that is after, and something which after the first thing and before the second thing, i.e. "between" them.
· Something is the successor of another when it is after the beginning, and when there is nothing of the same kind between it and the object it is the successor of (226b34).
· A thing that is in succession and touches is contiguous.
· The continuous is a subdivision of the contiguous. "Things are called continuous when the touching limits of each become one and the same and are contained in each other. Continuity is impossible if these extremities are two". "Continuity belongs to things that naturally in virtue of their mutual contact form a unity. And in whatever way that which holds them together is one, so too will the whole be one, e.g. by a rivet or glue or contact or organic union".
Then (in Book VI.1 of the Physics) he argues that the extremities of two points cannot be one (since an indivisible cannot have extremities), nor together (since that which has no parts has no extremity). If the continuum is composed of points, they must be either continuous or in contact with one another. But if they are in contact, they will not have distinct parts, whereas the continuum does have distinct parts. Nor can a point be in succession to a point, for things are in succession if there is nothing of the same kind between them. But between two points there is always a line. Thus the continuum is divisible only into parts that are infinitely divisible (231 b15). See Metaphysics K.12 (1068b25), for a similar argument.
Most of Aristotle's works are available in digital form (though in out of copyright and therefore somewhat old translations). For example On Generation and Corruption, Prior Analytics Book I and II, Metaphysics, Physics. There are no sites with original Greek texts, in the way that there are many sites with Latin texts (or if there are, please let me know).
See the Internet Classics Archive for more ancient texts.
But even ten thousand years, or the greatest number you will, cannot even be compared with eternity. For there will always be r atio between finite things, but between the finite and the infinite there can never be any comparison. Wherefore, however long drawn out may be the life of your fame, it is not even small, but it is absolutely nothing when compared with eternity.
Thomas Aquinas (1225-1274) was the angelic doctor of the Catholic church.
He discusses the Infinity of God in Question vii of Book I of the Summa (I Q.7 in English here), and discusses the question whether an actually infinite magnitude can exist. He argues that it cannot, for the following reasons:
· Every material body has some determinate form, and hence a determinate quantity.
· Every material body has some natural movement, and can change its place. But an infinite body would occupy every place, and every place would be its own place.
· Every mathematical body must be imagine as having some shape. But shape is defined by some term or boundary, and nothing infinite can have a boundary.
He also discusses (I Q7.a4, English here) whether an infinite multitude can exist. He says that it cannot, because multitude is measured by numbers. Since every number is multitude measured by units, no number is infinite; for. See also I.Q46a2 (English here).
Cantor disparaged this argument. However, it is not certain that Aquinas always held this view, or that he was absolutely certain about it. In a later essay De Aeternitate Mundi (1270) he says that no proof that an infinite multitude has yet been given. (Et tamen non est adhuc demonstratum quod Deus non possit facere infinita esse in actu), and in Phys. III lect. 8 the argument is said to be probable only. In the Questiones Quodlibetales (IX q1, XII a2) he distinguishes between the absolute and ordinated power of God, and suggests (respecting the former) that an actually infinite multitude may be possible.
(Note also that many of the arguments that Cantor attributes to Aquinas are not of his nad, but the works of other medieval writers. For example Opuscula LII de natura loci, XXXII de natura materiae et de dimensionibus interminatis and others).
On the continuum, he says that only a "mathematical body" can be divided infinitely. In the physical continuum we come to a minimum quantity of matter necessary to support a given form.
Corpus mathematicum est divisible in infinitum, in quo consideratur sola ratio quantitatis in qua nihil est repugnans divisioni infinitae. Sed corpus naturale, quod consideratur sub tota forma, non potest in infinitum dividi, quia, quando iam ad minimum deducitur, statim propter debilitatem virtutis convertitur in aliud. Unde est invenire minimam carnem, sicut dicitur in I. Physicorum, nec tamen corpus naturale componitur ex mathematicis. (De Sensu et Sensato, Tr.I, Lect. 15. Cf. Summa Theologica, Ia.Q48a4 (English here), Ib Q.85a2 (English here); II Sent. dist. XXX, Q2.a2.0; III Phys., Lect. 10a9; de Potentia IV, a1; IV Sent. Dist. XII Q1a2.
(If this seems surprising, note that the mathematician John Mayberry proposed a similar idea in a recent book).
Aquinas also discusses the definition of continuity in the following places: V Phys. Lect. 5; VI Phys. Lect. 1; and in XI Met., Lect. 13. He generally agrees with Aristotle that the whole consists of its parts only potentially. For example, see his commentaries on Met. 1039a3 (Lect. 13, 16, and Bk V Lect. 21). "Partes sunt in potentia in toto continuo". See also VII Phys. Lect. 9 "Pars autem prout est in toto, non est divisa in actu, sed in potentia tantum"; IV Sent. dist. X Q1a3 ("Partes alicuius homogenei continui ante divisionem non habent esse actu sed potentia tantum"); I de Caelo Lect.3 sect. 6; I Phys. Lect.9, sect. 8; Summa Theol., III.Q76 a3 ad1um (English here); Ia.Q85 a8 ad 2um; Quodlibet I q2a1.
Sicut aliquid est ens, sicut est unum, sicut est unum: unum autem est quod est in se indivisum at ab aliis divisum: pars autem prout est in toto, non est divisa in actu sed in potentia tantum, unde non est actu ens neque una, sed in potentia tantum.
John Duns Scotus (1270-1308), called the "subtle doctor" was a Franciscan. His fame as a theologian derives from his proof of the Immaculate Conception, which became Roman Catholic dogma in 1854. It is commonly supposed that the scholastic philosophers believed (following Aristotle) in the idea that the infinite was potential, not actual. Here is a passage by Scotus suggesting that they did not.
O Lord God, are not the things that can be known infinite in number and are they not all known actually by an intellect which knows all things? Therefore, that intellect is infinite which, at one and the same moment, has actual knowledge of all these things. Our God, yours is such an intellect (from the eighth conclusion just above). The nature that is identical with it then is also infinite.—I show the antecedent and consequence of this enthymeme. The antecedent: Things potentially infinite in number (i.e. things, which if taken one at a time are endless) become actually infinite if they exist simultaneously. Now what can be known is of such a nature so far as a created intellectual is concerned, as is sufficiently clear. Now all that the created intellect knows successively, your intellect knows actually at one and the same time. There, then, the actually infinite is known. I prove the major of this syllogism, although it seems evident enough. Consider these potentially infinite things as a whole. If they exist all at once, they are either actually infinite or actually finite. If finite, then if we take one after the other, eventually we shall actually know them all. But if we cannot actually know them all in this way, they will be actually infinite if known simultaneously. The consequence of this enthymeme I prove as follows. Whenever a greater number requires or implies greater perfection than does a smaller one, numerical infinity implies infinite perfection. For example, greater motive power is required to carry ten things than to carry five. Therefore, an infinite motive power is needed to carry an infinity of such things. Now in the point at issue, since the ability to know two things distinctly implies a greater perfection of intellect than the ability to know only one, what we propose to prove follows. This last I prove to be so because the intellect must apply itself and concentrate if it is to understand the intelligible distinctly. If then it can apply itself to more than one, it is not limited to any one of them and if it can apply itself to an infinity of such it is completely unlimited.
From A Treatise On God As First Principle John Duns Scotus 4.48.
William of Ockham (1280-1349) was an English Franciscan who made significant contributions to logic and philosophy. He is best known for his advocacy of nominalism. His main work is the Summa Totius Logicae, of which you can find nearly all the text (in Latin) here, and an English translation by Spade of selections from book I here. For a good overview of Ockham's philosophy by Spade, see here.
Like Leibniz, Ockham defended the idea that infinity is syncategorematic but actual.
Sed omne continuum est actualiter existens. Igitur quaelibet pars sua est vere existens in rerum natura. Sed partes continui sunt infinitae quia non tot quin plures, igitur partes infinitae sunt actualiter existentes. (Exp. phys. III, 10, §2, OP IV, 521, 13, §9, OP IV, 555, VI, 13, §6, OP V, 562.) ("OP" is Guillelmi de Ockham Opera Philosophica et Theologica (7 vols.), New York: St. Bonaventure, 1967-1986.)
But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent. (transl by Richard Arthur).
See also Murdoch, "William of Ockham and the Logic of Infinity and Continuity", pp. 165-206 in Infinity and Continuity in Ancient and Medieval Thought, ed. Norman Kretzmann, Ithaca and London, 1982: p. 189). Gregory of Rimini wrote:
Constat quod omne continuum habet plures partes, et non tot finitas numero quin plures, et omnes suas partes actualiter et simul habet, igitur omne continuum simul et actualiter habet partes infinitas. (In. l. sent., d. 42-44, q. 4, GL III, 441), (GL = Gregorii Ariminensis OESA lectura super primum et secundum sententiarum. Ed. A Damasus Trapp OSA and Venicio Marcolino, 7 vols., Berlin/New York, 1979-87) according to Beeley.
It is established that every continuum has further parts, and not so many parts finite in number that there are not further parts, and has all its parts actually and simultaneously, and therefore every continuum has simultaneously and actually infinitely many parts. (transl. Arthur)
See also chapter 44 of the Summa.
Item, si linea sit alia res a superficie et punctus a linea, igitur poterit Deus conservare lineam et destruere punctum. Quo facto, quaero: aut linea est finita aut infinita. Non infinita, manifestum est, igitur finita, et tamen sine puncto. Igitur frustra ponitur punctus terminans lineam. Similiter, posset Deus conservare lineam destruendo omnia puncta. Quo facto adhuc linea esset linea, et per consequens quantitas; et non quantitas discreta, igitur continua; igitur vere esset continua, quamvis non esset ibi aliqua alia res a partibus lineae copulans partes ad invicem. Frustra igitur ponuntur talia puncta distincta a linea. Et eadem ratione frustra ponuntur lineae distinctae a superficiebus, et eadem ratione frustra ponuntur superficies distinctae a corporibus.
Ockham is arguing against the "modern view" that a point is something distinct from a line. If it exists absolutely and in its own right, it would possible for God to destroy everything else and leave the point. But a point is simply the limit of a line. Conversely, it would not be possible for God to destroy all the points of the line, and leave the line.
This is all mixed up in an interesting way with complex theological arguments about the Holy Sacrament and the Eucharist. Is the "quantity" e.g. the length of a piece of wood, just all the bits of the wood taken collectively? Or is it a quality of a single substance, this "piece" of wood? The first case is like all the different bits of bread before the consecration [in thousands of different churches through Christendom, perhaps]. The second is like the after the consecration, when all the different pieces of bread become a part of the body of Christ. But we cannot suppose the second case is a mere external relation between Christ and the different bits of bread. Otherwise, "such a quantity intermediate between substance and quality would seem to be altogether superfluous". So, when all the bits of an extended object come together to make the whole, the bits are as it were annihilated, just as the separate bits of bread [in different churches, presumably] are, as bits of bread, annihilated, to become the whole body of Christ.
Therefore the theologians say "quod nulla quantitas est alia a substantia et qualitate" that quantity is not something separate from a substance. This whole question was later discussed in detail by Suarez.
Francisco Suarez (1548-1617) is considered to be the last great Scholastic philosopher. Descartes is known to have studied him (having studied at the Jesuit college of La Fleche) and also to have borrowed some of his terminology. Leibniz is also thought to have been influenced by him. Suarez is also a near contemporary of the mathematician Cavalieri, whose theory of indivisibles is supposed to have been a precursor of the modern view of the continuum.
The Disputations served as the official text of philosophy in almost all German universities during the 17th and a large part of the 18th century.
Here is a digitised version of the Disputationes Metaphysicae (originally published in 1597). Of particular interest is disputation 40 ("De Quantitate Continua") which contains a discussion of the arguments by Ockham (I.44 of the Summa, and elsewhere) mentioned above. Suarez argues against the nominalist position that continuous quantity is an accident in between substance and quality (quod quantitas continua est unum accidens medium inter substantiani et qualitatem), and defends the idea of terminating indivisibles (indivisibilia terminantia) and continuing indivisibles (indivisibilia continuantia).
In Section IV, he argues that both quantity and substance have extension in themselves. Substance has "entitative extension" (i.e. has integrating parts which taken together make up the whole). It has such parts before it has quantity. The function of quantity is to make the parts capable of filling a place, making the body impenetrable.
In Section V (Utrum in quantitate continua sint puncta, lineae et superficies quae sint verae res, inter se et a corpore quanto realiter distinctae - whether in continuous quantity there exist points. lines and surfaces which are truly things, distinct from one another and from bodily quantity) he responds to the arguments against terminating indivisibles (paras 40-41) and against continuing indivisibles (paras 42-49).
The arguments were about whether the continuum was a mixture of extended and indivisible elements (indivisibilia). Indivisibles were thought to be of two kinds: the terminating indivisibles (indivisibilia terminantia), such as the points at each end of a line, and the continuing indivisibles (indivisibilia continuantia), which join together the parts which can be separated at them. Terminating indivisibles are simply the points that terminate the end of a line, required to stop bodies interpenetrating when they touch. Continuing indivisibles join together the parts of the line. Nominalists denied the existence of the latter, seventeenth century scholastics like Suarez and John of St Thomas upheld them.
According to Mancosu (Philosophy of Mathematics & Mathematical Practice in the Seventeenth Century, p. 218) there was a general hostility amongh the Jesuits to Cavalieri's idea of indivisibles. He says that Festa provides archival evidence to show that the teaching of indivisibilist techniques in geometry was forbidden in Jesuit schools. The hostility was motivated by theological concerns about the Eucharist.
However, this does not accord with what Suarez says in sections IV and V of Disputation XL, where he demonstrates that the idea of indivisibilia is consistent with his theology of the Eucharist, and is indeed required by it.
John Locke (1632-1704) is best known to philosophers for his Essay Concerning Human Understanding. There are many versions of the Essay available in digital form, for example here. He discusses the idea of infinity in chapter XVII. He says that if we take an idea of a thing of any size whatever, we find we can enlarge it and mentally add to its size as much as we please, and have no more reason to stop than we were at the beginning. This gives us our idea of infinite space. He adds that there is something contradictory about our idea of infinity.
I think it is not an insignificant subtilty, if I say, that we are carefully to distinguish between the idea of the infinity of space, and the idea of a space infinite. The first is nothing but a supposed endless progression of the mind, over what repeated ideas of space it pleases; but to have actually in the mind the idea of a space infinite, is to suppose the mind already passed over, and actually to have a view of all those repeated ideas of space which an endless repetition can never totally represent to it; which carries in it a plain contradiction. (My emphasis).
Gottfried Leibniz (1646-1716) was a mathematician and philosopher, whose best known idea was that the universe is composed of an infinity of individisible, immaterial monads, the highest of which is God.. He had a great influence, via Russell, on modern logic.
In an argument that would be used later by Malezieu, Leibniz argues that the existence of an aggregate presupposes the existence of things that have a true unity "for [the aggregate only takes its reality from the reality of those of which it is composed, so that it will not have any at all, if each entity of which it is composed is itself an entity by aggregation. [Morris p. 78] . "The plural presupposes the singular, and where there is not a thing still less can there be several things". See also the Monadology, where he says that "… there must be simple substances, because there are compounds: for the compound is nothing but a collection or aggregatum of simples." Leibniz, Monadology, in Philosophical Writings G. H. R. Parkinson (London: J. M. Dent, 1973 p.179.
Malezieu applied this argument to the continuum. However, Philip Neujahr argues that Leibniz would not have used the argument in this way, "because it seemed obvious to him that any area of space or duration of time , however small, could be divided. This contributed to Leibniz' view that space and time are phenomena bene fundata."
Richard Arthur has argued that Leibniz believed in an actual infinity, but not in the Cantorean sense. Leibniz: "accurately speaking, instead of an infinite number, we ought to say that there are more than any number can express, and instead of an infinite straight line that it is a straight line continued beyond any magnitude that can be assigned, so that a larger and larger straight line is always available." (Leibniz to Des Bosses, 11 March 1706, G.ii.304-05). Arthur defends an account of the actual infinite that is a rival to the Cantorean account, but which eschews infinite sets.
According to Arthur, Leibniz advocates an infinite that is syncategorematic but actual. Thus there is, according to him, an actual infinity of parts into which any piece of matter is actually (not merely potentially) divided, but there is no totality or collection of all these parts. That is, every part is divided into further parts (“there is no part so small that it is not further divided.”) Whatever finite number is proposed, there are more parts than this but it may not be expressed collectively.
Under the spell of Cantor, many modern commentators cannot envision that the first way of expressing the matter does not automatically entail the second, even though the logical distinction between syncategorematic and categorematic infinities was well known in the middle ages. [Arthur, here]
Little is known of Malezieu except that he was a French geometer and philosopher (and a courtesan of Louis XIV). The assumption that I have elsewhere called Malezieu's principle is that existence belongs only to individual things, so that a set of things exists only insofar as its elements do. I have no direct source for the principle except where it is mentioned in the second part of book I of the Treatise where Hume argues against the infinite divisibility of space and time. Hume writes:
I may subjoin another argument proposed by a noted author [Mons. MALEZIEU], which seems to me very strong and beautiful. It is evident, that existence in itself belongs only to unity, and is never applicable to number, but on account of the unites, of which the number is composed. Twenty men may be said to exist; but it is only because one, two, three, four, &c. are existent, and if you deny the existence of the latter, that of the former falls of course. It is therefore utterly absurd to suppose any number [of things] to exist, and yet deny the existence of unites; and as extension is always a number, according to the common sentiment of metaphysicians, and never resolves itself into any unite or indivisible quantity, it follows, that extension can never at all exist.
Note that the word "number" here does not mean a numeral or mathematical entity, but a number of things, as when we say "a number of people are at the door". (This is a much older conception of a set, mentioned by Aristotle, for example, in Met. 1088a14).
The principle is that the existence of a number-of-things - a plurality – is no more than the existence of the individuals that constitute it. The existence of Alice and Bob is simply the existence of Alice and the existence of Bob, the existence of the twenty men amounts to the existenc of the first, the second, &c. We cannot suppose that the elements of a set exist, but not the set itself (and conversely). Set theory would be impossible on this assumption, for it requires that there is a domain D containing every set, but which cannot contain the totality of all sets - the set hierarchy. (For any such totality would itself have to be a set, thus lying somewhere within the hierarchy and thus failing to contain every set).
As Neujahr has noted, the Hume/Malezieu argument is similar to an argument by Leibniz in the Monadology (and also to arguments he gives in his correspondence with Arnauld).
George Berkeley (1685-1753) was an Irish Anglican bishop who also wrote philosophy. He is best known for the (according to him) commonsensical doctrine that the material world does not exist, only our perceptions do, given by God. Since every finite extension is just an perception in our mind, and since we cannot distinguish infinitely many parts in the perception, it follows (according to Berkeley) that extension itself is not infinitely divisible. A Treatise Concerning the Principles of Human Knowledge (1710) is here, from which the following passage is taken.
Every particular finite extension which may possibly be the object of our thought is an idea existing only in the mind, and consequently each part thereof must be perceived. If, therefore, I cannot perceive innumerable parts in any finite extension that I consider, it is certain they are not contained in it; but, it is evident that I cannot distinguish innumerable parts in any particular line, surface, or solid, which I either perceive by sense, or figure to myself in my mind: wherefore I conclude they are not contained in it. Nothing can be plainer to me than that the extensions I have in view are no other than my own ideas; and it is no less plain that I cannot resolve any one of my ideas into an infinite number of other ideas, that is, that they are not infinitely divisible. If by finite extension be meant something distinct from a finite idea, I declare I do not know what that is, and so cannot affirm or deny anything of it. But if the terms "extension," "parts," &c., are taken in any sense conceivable, that is, for ideas, then to say a finite quantity or extension consists of parts infinite in number is so manifest a contradiction, that every one at first sight acknowledges it to be so; and it is impossible it should ever gain the assent of any reasonable creature who is not brought to it by gentle and slow degrees, as a converted Gentile to the belief of transubstantiation. Ancient and rooted prejudices do often pass into principles; and those propositions which once obtain the force and credit of a principle, are not only themselves, but likewise whatever is deducible from them, thought privileged from all examination. And there is no absurdity so gross, which, by this means, the mind of man may not be prepared to swallow (section 124).
There are other arguments against infinite extension in sections 124-133.
Apart from the argument that Hume quotes from Malezieu, Hume gives other arguments against the infinite infinite divisibility of space and time in the second part of book I. However, these are a consequence of his representationalism, similar to those advanced by Berkeley.
Bernard Bolzano (1781-1848) was a Czech philosopher, mathematician, and theologian whose ideas about sets and infinity both influenced and anticipated Cantor. He says (Paradoxes of the Infinite 1851, § 3) that a set is "a whole consisting of certain parts" or "a system (Inbegriff) of certain things". The idea of a set is given by one of the "simplest conceptions of our understanding", namely by the meaning of the word and when we say "The sun, the earth, and the moon act upon one another," "The rose, and the conception of a rose, are two very different things".
But we must add that every arbitrary object A can be combined in a system with any others B, C, D, …, or (still speaking more correctly) already forms a system [an aggregate] by itself [without our intervention – an sich selbst schon] i.e. the combination of A with B, C, D … already forms a system [an aggregate], of which some more or less important truth can be enunciated, provided only that each of the presentations A, B, C, D, … in fact represents a different object, or in so far as none of the propositions 'A is the same as B,' 'A is the same as C,' 'A is the same as D,' etc., is true. For if e.g. A is the same as B, then it is certainly unreasonable to speak of a system [an aggregate] of the things A and B.
Firstly: "Nowhere can an infinite set exist", they say, "for the simple reason that an infinite set can never be united to form a whole, never be collected together in thought." I must stigmatise this assertion as a mistake, and as a mistake engendered by the false opinion that a whoe consisting of certain objects a, b, c, d, … cannot be constructed in thought unless one first forms separate mental representations [Vorstellungen] of its separate component objects. This is by no means true. I can thnk of the set, of the aggregate, or if you prefer it, the totality, of the inhabitants of Prague or of Peking without forming a separate representation of each separate inhabitant. § 21 (Ewald p. 258)
Richard Dedekind (1831-1916) was a German mathematician, a friend and contemporary of Cantor. He is famous for his definition of continuity:
If all points of the straight line fall into two classes such that every point of the first class (Klasse) lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.
The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we think continuity into the line. If space has a real existence at all it is not necesssary for it to be continuous; many of its properties would remain the same even if it were discontinuous. And if we knew for certain that space were discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps in thought and thus making it continuous; this filling up would consist in a creation of of new point-individuals and would have to be carried out in accordance with the above principle. (Stetigkeit und irrationale Zahlen. Vieweg, Braunschweig 1872. Transl. by W.W. Beman as "Continuity and irrational numbers", reprinted by W. Ewald, From Kant to Hilbert A Source Book In The Foundations Of Mathematics, Vols I&II, Oxford 1996.Vol II pp. 765-79).
Note, however that this definition of continuity is not wholly inconsistent with the one given by Aristotle. Dedekind is thought to have given the first precise definition of the real number system.
Here he discusses the problem of defining our concept of finitude.
An element n belongs to the sequence N, if and only if, starting with the element 1, and counting on and on steadfastly, that is, going through a finite number of iterations … I actually reach the element n at some time; by this procedure, however, I shall never reach an element t outside the sequence N. But this way of characterising the distinction between the elements t that are to be ejected from S and those elements n that alone are to remain is surely quite useless for our purpose; it would, after all, contain the most pernicious and obvious kind of vicious circle. The mere words "finally get there at some time", of course, will not do either; they would be of no more use that, say, the words "karam sipo tatura", which I invent at this instant without giving them any clearly defined meaning. (From a letter dated 27 February 1890 to Dr. Hans Keferstein, transl. Wang & Bauer-Mengelberg).
Georg Cantor (1845-1918). Like Peirce , Cantor also studied scholastic writing on infinity, but thought it confused, because it lacked an exact and complete definition of the concept of the continuum, and because the whole idea of continuity had not been thought out with clarity and completeness..
(Grundlagen §4, transl. Ewald) As is well known, throughout the Middle Ages the proposition "infinitum actu non datur", taken from Aristotle, was held to be incontrovertible by all the scholastics. But if one considers the reasons which Aristotle brought forward against the real existence of the infinite (see e.g. his Metaphysics, XI.10), it will be found that the main issue can be traced back to a presupposition which involves a petitio principii, namely, the presupposition that there are only finite numbers. [Cantor adds a footnote in which he refers to some sources, e.g. Zeller's Die Philosophie der Griechen for Aristotle and Plato, and also Der Cardinal Nicolaus von Cusa als vorgaenger Leibnizens (1852) by Zimmerman as a source on Nicholas of Cusa, and Giordano Brunos Weltanschauung und Verhaengnis (1882) by Brunhofer, on Giordano Bruno, the successor of Cusa.
He also refers disaparagingly to the "medieval-scholastic" point of view, in which the continuum is thought to be an unanalysable concept. "Timid natures thereby get the impression that with the "continuum" it is not a matter of a mathematically logical concept but rather of religious dogma" (Grundlagen §10).
He says that Aquinas held that the continuum consisted of neither finitely nor infintely many parts "[which] seems to me to contain less an explanation of the facts than a tactit confession that one has not got to the bottom of the matter and prefers to get genteelly out of its way". (He cites Aquinas Opuscula XLII, de natura generis cap. 19 & 20, LII de natura loci, XXXII de natura materiae et de dimensionibus interminatis, and a secondary source. C. Jourdain, La Philosophie de Satin Thomas d'Aquin, pp. 303-29, K.Werner, Der Heilige Thomas von Aquino (Regensburg 1859), vol. 2, pp. 177-201.
Note all these passages are now known (or suspected) not to have been written by Aquinas. Some were written by Thomas de Sutton: see here.
About Gottlob Frege (1848-1925), it is customary to say that he invented quantification theory, was the founding father of mathematical logic, and thus the greatest logician since Aristotle &c &c. In fact, a cursory look at the history of logic since 1870 suggests that Frege had a marginal influence on its development. See, e.g. Grattan-Guinness, I. The Search foir Mathematical Roots, 1870-1940, Princeton 2000 for a (reasonably) accurate survey of this period. The development of logic itself was more influenced by Frege's arch-rival Ernst Schroeder, whose notation (itself influenced by Peirce) was adopted by Zermelo for his formalisation of Cantor's theory. As Putnam unkindly writes " While, to my knowledge, no one except Frege ever published a single paper in Frege's notation, many famous logicians adopted Peirce-Schroeder notation, and famous results and systems were published in it."
Nonetheless, Frege's work had a huge impact on modern philosophy. It is also philsosophically better, by some way, than those of his contemporaries. Russell says that "Frege's work abounds in subtle distinctions, and avoids all the usual fallacies which best writers on Logic" (PoM § 475). Frege had a number of ideas.
· The syntax of natural language is logically useless, misleading and incoherent. Thus an adequate account of inference as expressed in ordinary language requires translation into a new idiom, expressly constructed for use by logicians. This led Frege to devise an unreadable (and unprintable) calculus for expressing logical ideas.
· A proposition like "Socrates is wise" consists of a name for an object ("Socrates"), and a name for a concept ("is wise"). The concept is like a function f, the object is like itse argument, and its value is a so-called "truth value". Thus is-wise(Socrates)= The True.
· Proper names and singular definite descriptions are names for objects. Thus the expression "the concept 'horse'" is the name of an object.
· Concepts have extensions. E.g. the concept "is a bald person" has as its extension all bald people.
Ueber Begriff und Gegenstand, maintained by Alain Blachair.
Bertrand Russell (1872-1970) was an English philosopher and mathematician who made important contributions to the early development of set theory (i.e. the decades of the twentieth century preceding the First World War). The following passages are taken from the The Principles of Mathematics (1903).
In an important passage (§ 183 of Principles) he says that philosophers who have argued against the reality of infinite numbers have simply been mistaken on a technical issue: they do not understand the difference between finite and infinite numbers. Finite numbers obey the law of mathematical induction; infinite numbers do not.
The principle may otherwise be stated thus: If every proposition which holds concerning 0, and also holds concerning the immediate successor of every number of which it holds, holds concerning the number n, then n is finite; if not, not. This is the precise sense of what may be popularly expressed by saying that every finite number can be reached from 0 by successive steps, or by successive additions of 1. This is the principle which the philosopher must be held to lay down as obviously applicable to all numbers, though he will have to admit that the more precisely his principle is stated, the less obvious it becomes.
It is in this principle alone, and in its consequences, that finite and infinite numbers differ. Thus, more than 2,000 years of philosophical worry about the infinite were the result of ignorance of a simple principle that could be taught to schoolchildren in a text book. Certainly, after the Frege-Russell period, few philosophers (apart from Wittgenstein) discussed problems of the infinite.
§ 21. The insistence on the distinction between e and the relation of whole and part between classes is due to Peano, and is of very great importance to the whole technical development and the whole of the applications to mathematics. In the scholastic doctrine of the syllogism, and in all previous symbolic logic, the two relations are confounded, except in the work of Frege. The distinction is the same as that between the relation of individual to species and that of species to genus, between the relation of Socrates to the class of Greeks and the relation of Greeks to men.
§ 69. The best formal treatment of classes in existence is that of Peano. But in this treatment a number of distinctions of great philosophical importance are overlooked. Peano, not I think quite consciously, identifies the class with the class-concept; thus the relation of an individual to its class is, for him, expressed by is a. For him, "2 is a number" is a proposition in which a term is said to belong to the class "number". Nevertheless he identifies the equality of classes, which consists in their having the same terms, with identity a proceeding which is quite illegitimate when the class is regarded as the class-concept. In order to perceive that man and featherless biped are not identical, it is quite unnecessary to take a hen and deprive the poor bird of its feathers.
§ 478. (Classes) … the extension of a Begriff, Frege says, is the range (Werthvelauf) of a function whose value for every argument is a truth-value (FuB. p.16). Ranges are things, whereas functions are not (p.19). There would be no null-class, if classes were taken in extension; for the null-class is only possible if a class is not a collection of terms (KB. pp 436-7). If x be a term, we cannot identify x, as the extensional view requires, with the class whose only member is x; for suppose x to be a class having more than one member, and let y, z be two different members of x; then if x is identical with the class whose only member is x, y and z will both be members of this class, and will therefore be identical with x and with each other, contrary to the hypothesis (p. 444). The extension of a Begriff has its being in the Begriff itself, not in the individuals falling under the Begriff (p. 451). When I say something about all men, I say nothing about some wretch in the centre of Africa, who is in no way indicated* (Bedeutet?) and does not belong to the indication (Bedeutung?) of man (p. 454). Begriffe are prior to their extension, and it is a mistake to attempt, as Schroeder does, to base extension on individuals; this leads to the calculus of regions (Gebiete) not to Logic (p. 455).
§ 100. A class-concept may or may not be a term of its own extension. "Class-concept which is not a term of its own extension" appears to be a class-concept. But if it is a term of its own extension, it is a class-concept which is not a term of its own extension, and vice versa. Thus we must conclude, against appearances, that "class-concept which is not a term of its own extension" is not a class-concept.
Wittgenstein is the most important and interesting philosopher of mathematics in the twentieth century, for the following reasons
· He is practically the only philosopher of mathematics in the twentieth century other than Russell. After the formalisation of set theory by Russell and Zermelo, philosophers regarded the mathematics as a closed book: a matter for technicians and not philosophers.
· He was absolutely opposed to set theory and mathematical logic, which he believed had "completely deformed" the thinking of mathematicians and of philosophers by its "superficial interpretation" of the forms of our everyday language.
· Mathematics is central to his philosophy. Remember that Wittgenstein's first exposure to philosophy (in 1911) was through Russell, just years after the discovery of an important contradiction in early mathematical logic. The contradiction made a tremendous impact on Wittgenstein, and his first (and only) work published in his life was an attempt to resolve it. (cf Tractatus 3.3333).
· Wittgenstein thought of himself as a philosopher of mathematics, and thought (at one point in his life – cf. Monk p.466) that his chief contribution to philosophy lay in that area.
· Concern about mathematics underlies his most important work, the Philosophical Investigations. The preface to the book lists the foundations of matheamtics as one of its subjects. The final paragraph is about the confusion barrenness of certain parts of mathematics. Just as psychology mixes together conceptual confusion with the methods of experimental science, so set theory joins conceptual confusion with methods of proof.
Philosophical clarity will have the same effect on the growth of mathematics as sunlight has on the growth of potato shoots. (In a cellar they grow yards long). (Philosophical Grammar p. 381).
Wittgenstein is sometimes dismissed as a disciple of the intuitionist Brouwer. While it is true that his return to philosophical work an the late 1920's was prompted by a lecture of Brouwer's, Wittgenstein's work is altogether more subtle than that. Would Wittgenstein have identifed what is true with what can be proved? In any case, he thought that intuitionism was "all bosh – entirely". (Lectures on the Foundations of Mathematics: Cambridge 1939 p.207).
Wittgenstein loathed the diagonal argument, and thought it was "hocus-pocus". In his 1938 Cambridge lectures on aesthetics he said it was only the charm of such proofs that gives them their interest. It fascinates us to be told there exist numbers that are (as it were) greater than infinity.
He believed he could put the proof in a way in which "it will lose its charm for a great number of people and certainly will lose its charm for me" [Lectures and Conversations p.28]. In reply to Hilbert's famous assertion that Cantor had created a paradise from which mathematicians shall never be expelled, Wittgenstein said "If one person can see it as a paradise of mathematicians, why should not another see it as a joke?) [Remarks on the Foundations of Mathematics, V. 7]. Nor would he dream of expelling anyone. He would show that it was not a paradise "so that you'll leave of your own accord". [Lectures on the Foundations of Mathematics: Cambridge 1939 p.103].
He said [Remarks on the Foundations of Mathematics II.22] that the proof ought to show that the concept of a real number has less analogy with the concept of cardinal number than it would seem at first sight. "But just the opposite happens: one pretends to compare the "set" of real numbers in magnitude with that of cardinal numbers. The difference in kind between the two conceptions is represented, by a skew form of expression, as difference of extension. I believe, and I hope, that a future generation will laugh at this hocus pocus".
In Philosophical Grammar, he says that the idea of correlating a set with one of its subsets (for example by the formula m=2n) uses "a misleading analogy to clothe a trivial sense in a paradoxical form".
And instead of being ashamed of this paradoxical form as something ridiculous, people plume themselves on a victory over all prejudices of the understanding." [p. 465].
It is nonsense to imagine writing the cardinal numbers side by side with the real numbers, so that one ends in mere endlessness, whereas the other goes beyond [p. 265].
Indeed [p. 465], we can't even imagine it. How can we take a subset of the row of trees and investigate whether it can be co-ordinated one-to-one to the whole class? It has no meaning. With a finite set, the attempt must eo ipso end in failure. With an infinite set, it will not end at all.
What set theory has to lose is rather the atmosphere of clouds of thought surrounding the bare calculus, the suggestion of an underlying imaginary symbolism, a symbolism which isn’t employed in its calculus, the apparent description of which is really nonsense. (In mathematics anything can be imagined, except for a part of our calculus.) [p. 469]
Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one. (PG § 138).
On the dots of ellipsis
W. believed that the expression "and so on" is nothing but the expression "and so on". It has a meaning via the rules that hold of it, and cannot say more than it shows. It is essential if we are to indicate endlessness, but only through the rules that govern its use. We can distinguish the series "1, 1+1, 1+1+1" from the series "1, 1+1, 1+1+1 and so on". "And this last sign and its use is no less essential for the calculus than any other." (Philosophical Grammar p. 283).
The expression "and so on" is nothing but the expression "and so on" (nothing, that is, but a sign in an calculus which can’t do more than have meaning via the rules that hold of it; which can't say more than it shows). Philosophical Grammar p. 282
For the sign "and so on", or some sign corresponding to it, is essential if we are to indicate endlessness - through the rules, of course, that govern such a sign. That is to say, we can distinguish the limited series "1, 1+1, 1+1+1" from the series "1, 1+1, 1+1+1 and so on". and this last sign and its use is no less essential for the calculus than any other. (ibid p. 283)
... the sign "1, 1+1, 1+1+1 ..." is to be taken as perfectly exact; governed by definite rules which are different from those for "1, 1+1, 1+1+1", and not a substitute for a series "which cannot be written down"." (p. 284)
A finite series is explained to me by examples of the type 1, 2, 3, 4, and [an?] infinite one by signs of the type "1, 2, 3, 4, and so on" or "1, 2, 3, 4 …" (p. 287)
On the sign for a set
§ 19 p. 332. A cardinal number is an internal property of a list.
The sign for the extension of a concept is a list. We might say, as an approximation, that a number is an external property of a concept, and an internal property of its extension (the list of objects that fall under it). A number is a schema for the extension of a concept.
I use such a list when I say "a, b, c, d fall under the concept F(x)": "a, b, c, d," is the list. Of course this proposition says the same as Fa.Fb.Fc.Fd; but the use of the list in writing the proposition shows its relationship to "(Ex,y,z,u).Fx.Fy.Fz.Fu" which we can abbreviate to "(E||||x.F(x)."
What arithmetic is concerned with is the schema ||||. – But does arithmetic talk about the lines that I draw with pencil on paper? -Arithmetic doesn't talk about the lines, it operates with them. (cf. PR § 119 )
The symbol for a class [i.e. a set] is a list.
A cardinal number is an internal property of a list.
Although James Joyce (1882 –1941) is better known as a writer of stories and as a pioneer of modernism in literature, his description of the infinity of hell (from the book A Portrait of the Artist as a Young Man) deservers a place here. Follow the link.
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