# 1 (number)

natural number

**1** (one, also called **unit**, **unity**, and **multiplicative identity**) is a number, a numeral, and glyph. It represents a single entity, the unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1.

## Quotes edit

- Things consisting of fewer Principles are more accurate, than those understood by Addition [of more principles], as Arithmetic is more accurate than Geometry. ...That Science is more accurate which consists of fewer Principles, than what is only to be understood by Addition, as Arithmetic is more accurate than Geometry. I say by Addition, as Unity is a Being understood without Position, but a Point is to be understood only by Position.
- Aristotle,
*Posterior Analytics*(ca. 350 BC) as quoted by Isaac Barrow,*The Usefulness of Mathematical Learning Explained and Demonstrated*(1734)

- Aristotle,

- Every quantity is recognized as quantity through the one, and that by which quantities are primarily known is the one itself; therefore the one is the source of number as number.
- Aristotle,
*Metaphysics*M 9, 1085 b 22, as quoted by Jacob Klein,*Greek Mathematical Thought and the Origin of Algebra*(1968)

- Aristotle,

- More knowable than the number is the unit; for it is prior and the source of every number.
- Aristotle,
*Topics*Z4, 141 b 5 ff, as quoted by Jacob Klein,*Greek Mathematical Thought and the Origin of Algebra*(1968)

- Aristotle,

- We find also the Famous
*Simon Stevin*, Mathematician to the*Prince of Orange*, having defined Number to be,*That by which is explained the quantity of every Thing*, he becomes so highly inflamed against those that will not have the*Unit*to be a*Number*, as to exclaim against Rhetoric, as if he were upon some solid Argument. True it is that he intermixes in his Discourses a question of some Importance, that is, whether a*Unit*be to*Number*, as a*Point*is to a*Line*. But here he should have made a distinction, to avoid the confusing together of two different things. To which end these two*questions*were to have been treated apart;*whether a Unit be Number*, and*whether a Unit be to Number, as a Point is to a Line*; and then to the first he should have said, that it was only a Dispute about a Word, and that an*Unit*was, or was not a Number, according to the Definition, which a Man would give to Number. That according to*Euclid's*Definition of*Number*;*Number is a Multitude of Units assembled together:*it was visible, that a*Unit*was no Number. But in regard this Definition of*Euclid*was arbitrary, and that it was lawful to give another Definition of*Number*,*Number*might be defined as*Stevin*defines it, according to which Definition a*Unit*is a*Number*; so that by what has been said, the*first question*is resolved, and there is nothing farther to be alleged against those that denied the Unit to be a Number, without a manifest begging of the question, as we may see by examining the pretended Demonstrations of*Stevin*. The first is,*The Part is of the same Nature with the whole,*

The Unit is a Part of a Multitude of Units,

Therefore the Unit is of the same Nature with a MuItitude of Units, and consequently of Number.

This Argument is of no validity. For though the part were always of the same nature with the whole, it does not follow that it ought to have always the same name with the whole; nay it often... has not the same Name. A Soldier is part of an Army, and yet is no Army... a Half-Circle is no Circle... if we would we could not... give to Unit more than its name of Unit or part of Number.

The Second Argument which*Stevin*produces is of no more force.*If then the Unit were not a Number, Subtracting one out of three, the Number given would remain, which is absurd.*

But... to make it another Number than what was given, there needs no more than to subtract a Number from it, or a part of a Number, which is the Unit. Besides, if this Argument were good, we might prove in the same manner, that by taking a half Circle from a Circle given, the Circle given would remain, because no Circle is taken away. ...

But the second Question,*Whether an Unit be to Number, as a Point is to a Line*, is a dispute concerning the thing? For it is absolutely false, that an Unit is to number as a point is to a Line. Since an Unit added to number makes it bigger, but a Line is not made bigger by the addition of a point. The Unit is a part of Number, but a Point is no part of a Line. An Unit being subtracted from a Number, the Number given does not remain; but a point being taken from a Line, the Line given remains.

Thus doth*Stevin*frequently wrangle about the Definition of words, as when he perplexes himself to prove that Number is not a quantity discreet, that Proportion of Number is always Arithmetical, and not Geometrical, that the Root of what Number soever, is a Number, which shews us that he did not properly understand the definition of words, and that he mistook the definition of words, which were disputable, for the definition of things that were beyond all Controversy.- Antoine Arnauld, Pierre Nicole,
*La logique ou l'art de penser contenant outre les règles communes, plusieurs observations nouvelles, propres à former le jugement*(1683)*The Port Royal Art of Thinking: In Four Parts*(1818) Translation, pp. 239-240.

- Antoine Arnauld, Pierre Nicole,

- What Science can be more accurate than Geometry? What Science can afford Principles more evident, more certain, yea I will add, more simple than Geometrical Axioms, or exercises a more strictly accurate Logic in drawing its Conclusions? But
(they had more rightly said Numbers)*Aristotle*and*Proclus*affirm that*Unity***the Principle of Arithmetic, is more simple than a***Point*which is the Principle of Geometry, or rather of Magnitude. Because a Point implies Position, but Unity does not.*A Point*, says*Aristotle, and Unity are not to be divided, as Quantity: Unity requires no Position, a Point does*. But this Comparison of a*Point*in Geometry with*Unity*in Arithmetic is of all the most unsufferable, and derives the worst Consequences upon Mathematical Learning. For*Unity*answers really to some Part of every Magnitude, but not to a*Point*: Thus if a Line be divided into six equal Parts, as the whole Line answers to the Number six, so every sixth Part answers to*Unity*, but not to a*Point*which is no Part of this Right Line. A*Point*is rightly termed Indivisible, not*Unity*. (For how ex. gr. can equal*Unity*, if*Unity*be indivisible, and incomposed, and represent a Point) but rather only*Unity*is properly divisible, and Numbers arise from the Division of*Unity*. A Geometrical Point is much better compared to a*Cypher*or*Arithmetical Nothing*, which is really the Bound of every Number, coming between it and the Numbers next following, but not as a Part. A*Cypher*being added to or taken from a Number does neither encrease nor diminish it; from it is taken the Beginning of Computation, while itself is not computed; and it bears a manifest Relation to the principal Properties of a Geometrical Point. Nor is that altogether unexceptionable, which is said of*Position*; for a*Point*taken universally is not less indeterminate, and void of*Position*, than*Unity*taken the same Way: But*Unity*taken particularly implies a definite*Position*, and all other particular Circumstances, as well as a particular*Point*. Lastly, the Accuracy of Arithmetic and Geometry is so far from being different that it is altogether the same, drawn from the same Principles, and employed about the same Things. I might here annex many Observations and Consequences drawn from hence; but

Not to be too tedious and prolix, I judge it will appear plain enough to every one who duly weighs what I have suggested, that, in reality,*Number*(at least that treated of by Mathematicians) differs nothing from*continued Magnitude*it self, nor seems to have any other Properties (Composition, Division, Proportion, and the like) than either from, or in respect to it, as it represents, or supplies its Place; nor consequently that it is any Species of Quantity distinct from Magnitude, or the Object of any Science but Geometry (which is conversant about Magnitude in general): In sum, that Number includes in it every Consideration pertaining to Geometry. Therefore the*Element Writer*(whatsoever*Ramus*can object, who taunts him with that Name) did not unadvisedly, in inserting Arithmetical Speculations among the Elements of Geometry, nay rather he did great Service to the Mathematics, and merited highly in not permitting these Sciences to be separated from one another, as if they were separate in Nature, but assigning to Arithmetic a suitable Place in Geometry.- Isaac Barrow, Cambridge mathematical lecture (ca. 1665) published as
*Lectiones Mathematicae*(1683) and translated in*The Usefulness of Mathematical Learning Explained and Demonstrated*(1734) Tr. John Kirkby, pp. 47-49.

- Isaac Barrow, Cambridge mathematical lecture (ca. 1665) published as

- [A]s the great extreme of dimension is sublime, so the last extreme of littleness is in the same measure sublime... when we attend to the infinite divisibility of matter, when we pursue animal life into these excessively small, and yet organized beings... when we push our discoveries yet downward... in tracing which the imagination is lost as well as the sense; we become amazed and confounded at the wonders of minuteness; nor can we distinguish in its effects this extreme of littleness from the vast itself. For division must be infinite as well as addition; because the idea of a perfect unity can no more be arrived at, than that of a complete whole, to which nothing can be added.
- Edmund Burke,
*A Philosophical Enquiry into the Origin of Our Ideas of the Sublime and Beautiful*(1757)*The Works of the Right Honorable Edmund Burke*(1889) Vol. 1, pp. 100-101.

- Edmund Burke,

- I will now say something which may perhaps astonish you; it refers to the possibility of dividing a line into its infinitely small elements by following the same order which one employs in dividing the same line into forty, sixty, or a hundred parts, that is, by dividing it into two, four, etc. He who thinks that, by following this method, he can reach an infinite number of points is greatly mistaken; for if this process were followed to eternity there would still remain finite parts which were undivided. ...

Indeed by such a method one is very far from reaching the goal of indivisibility; on the contrary he recedes from it and while he thinks that, by continuing this division and by multiplying the multitude of parts, he will approach infinity, he is... getting farther and farther away from it. My reason is this. In the preceding discussion we concluded that, in an infinite number, it is necessary that the squares and cubes should be as numerous as the totality of the natural numbers [*tutti i numeri*], because both of these are as numerous as their roots which constitute the totality of the natural numbers. Next we saw that the larger the numbers taken the more sparsely distributed were the squares, and still more sparsely the cubes; therefore it is clear that the larger the numbers to which we pass the farther we recede from the infinite number; hence it follows that since this process carries us farther and farther from the end sought, if on turning back we shall find that any number can be said to be infinite, it must be unity. Here indeed are satisfied all those conditions which are requisite for an infinite number; I mean that unity contains in itself as many squares as there are cubes and natural numbers [*tutti i numeri*]. ...

There is no difficulty in the matter because unity is at once a square, a cube, a square of a square, and all the other powers [*dignitā*]; nor is there any essential peculiarity in squares or cubes which does not belong to unity; as, for example, the property of two square numbers that they have between them a mean proportional; take any square number you please as the first term and unity for the other, then you will always find a number which is a mean proportional. Consider the two square numbers, 9 and 4; then 3 is the mean proportional between 9 and 1 [ ]; while 2 is a mean proportional between 4 and 1 [ ]; between 9 and 4 we have 6 as a mean proportional [ ]. A property of cubes is that they must have between them two mean proportional numbers; take 8 and 27; between them lie 12 and 18 [ ]; while between 1 and 8 we have 2 and 4 intervening [ ]; and between 1 and 27 there lie 3 and 9 [ ]. Therefore we conclude that unity is the only infinite number. These are some of the marvels which our imagination cannot grasp and which should warn us against the serious error of those who attempt to discuss the infinite by assigning to it the same properties which we employ for the finite, the natures of the two having nothing in common.- Galileo Galilei as Salviati,
*Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze*or*Discourses and Mathematical Demonstrations Relating to Two New Sciences*(1638) and as*Dialogues Concerning Two New Sciences*(1914) Tr. Henry Crew, Alfonso de Salvio.

- Galileo Galilei as Salviati,

- When do centuries end?—at the termination of years marked '99 (as common sensibility suggests), or at the termination of years marked '00 (as the narrow logic of a peculiar system dictates)?... the source of all our infernal trouble about the ends of centuries may be laid at the doorstep of a sixth-century monk named Dionysius Exiguus, or (literally) Dennis the Short. ...Dennis neglected to begin time with year zero, thus discombobulating all our usual notions of counting. During the year that Jesus was one year old, the time system that supposedly started with his birth was two years old. (Babies are zero years old until their first birthday; modern time was already one year old at its inception.) The absence of a year zero also means that we cannot calculate algebraically (without making a correction) through the B.C.-A.D. transition. ...The problem of centuries starts from Dennis's unfortunate decision to start with year one, rather than year zero... logic and sensibility do not coincide, and since both have legitimate claims upon our decision, the great and recurring debate about century boundaries simply cannot be resolved. ...One might argue that humans, as creatures of reason, should be willing to subjugate sensibility for logic; but we are, just as much, creatures of feeling. And so the debate has progressed at every go-round.
- Stephen Jay Gould,
*Dinosaur in a Haystack*(1995) "Dousing Diminutive Dennis' Debate (or DDDD = 2000)"

- Stephen Jay Gould,

- We may... go to our... statement from Aristotle's treatise on the Pythagoreans, that according to them the universe draws in from the Unlimited time and breath and the void. The cosmic nucleus starts from the unit-seed, which generates mathematically the number-series and physically the distinct forms of matter. ...it feeds on the Unlimited outside and imposes form or limit on it. Physically speaking this Unlimited is [potential or] unformed matter... mathematically it is extension not yet delimited by number or figure. ...As
*apeiron*in the full sense, it was... duration without beginning, end, or internal division—not time, in Plutarch's words, but only the shapeless and unformed raw material of time... As soon... as it had been drawn or breathed in by the unit, or limiting principle, number is imposed on it and at once it is time in the proper sense. ...the Limit, that is the growing cosmos, breathed in... imposed form on sheer extension, and by developing the heavenly bodies to swing in regular, repetitive circular motion... it took in the raw material of time and turned it into time itself.- W. K. C. Guthrie,
*A History of Greek Philosophy*Vol. 1, "The Earlier Presocratics and the Pythagoreans" (1962)

- W. K. C. Guthrie,

- Aristotle observes that the One is reasonably regarded as not being itself a number, because a measure is not the things measured, but the measure or the One is the beginning (or principle) of number. This doctrine may be of Pythagorean origin; Nicomachus has it; Euclid implies it when he says that a unit is that by virtue of which each of existing things is called one, while a number is 'the multitude made up of units'; and the statement was generally accepted. According to Iamblichus, Thymaridas (an ancient Pythagorean, probably not later than Plato's time) defined a unit as 'limiting quantity'... or, as we might say, 'limit of fewness', while some Pythagoreans called it 'the confine between number and parts', i.e. that which separates multiples and submultiples. Chrysippus (third century B.C.) called it 'multitude one',... a definition objected to by Iamblichus as a contradiction in terms, but important as an attempt to bring 1 into the conception of number.
- Sir Thomas Little Heath,
*A History of Greek Mathematics*(1921) Vol. 1 p. 69.

- Sir Thomas Little Heath,

- The first definition of number is attributed to Thales, who defined it as a collection of units... following the Egyptian view. The Pythagoreans 'made number out of one' [Aristotle,
*Metaph*. A. 5, 986 a 20]; some of them called it 'a progression of multitude beginning from a unit and a regression ending in it' [Theon of Smyrna, p. 18. 3-5]. Stobaeus credits Modoratus, a Neo-Pythagorean of the time of Nero, with this definition.) Eudoxus defined number as a 'determinate multitude'... Nicomachus has yet another definition, 'a flow of quantity made up of units'... Aristotle gives a number of definitions equivalent to one or other of those just mentioned, 'limited multitude', 'multitude (or combination) of units', 'multitude of indivisibles', 'several ones'... 'multitude measurable by one', 'multitude measured', and 'multitude of measures' (the measure being the unit).- Sir Thomas Little Heath,
*A History of Greek Mathematics*(1921) Vol. 1 pp. 69-70.

- Sir Thomas Little Heath,

- Let us take as the basis of our consideration first of all a thought-thing 1 (one).
- David Hilbert, "The Foundations of Logic and Arithmetic," (1904)
*Monist*XV, Tr. G. B. Halsted, p. 341, as quoted by Henri Poincaré,*The Foundations of Science*(1913).

- David Hilbert, "The Foundations of Logic and Arithmetic," (1904)

- The observable efforts of Greek philosophy were generally directed toward the resolution of multiplicity into unity. Empedocles is reported to have said, "the universe is alternately in motion and at rest—in motion when love is making one out of many, or strife is making many out of one, and at rest in the intermediate periods of time." Even here, where two states are posited, the unifying impulse is obviously felt to be the more desirable. Hence it is very natural that the Pythagoreans should have considered the monad as the first principle from which the other numbers flow. Itself not a number, it is an essence rather than a being and is sometimes, like the duad, designated as a potential number, since the point, though not a plane figure, can originate plane figures. As first originator, the monad is good and God. It is both odd and even, male and female... It is the basis and creator of number... In short, it is always taken to represent all that is good and desirable and
*essential*, indivisible and uncreated.- Vincent Foster Hopper,
*Medieval Number Symbolism*(1938)

- Vincent Foster Hopper,

- Nicomachus gives three definitions of number. ...The third stream of quantity composed of units Philoponus explains as another attempt to distinguish the particular kind of quantum treated in Arithmetic. The Unit was conceived either mystically as an Idea whose "essence" passes in some way into concrete individuals and even into the Ideas to organize them, or spatially and temporally as the boundary of individuals. The former conception gave rise to fantastic speculations on the cosmic meaning of number, examples of which Nicomachus has... given us; the latter gave rise to the figurate numbers...
- George Johnson,
*The Arithmetical Philosophy of Nicomachus of Gersa*(1916)

- George Johnson,

- Those numbers... independent of the particular things which happen to undergo counting—
*of what*are these... ? To pose this question means to raise the problem of "scientific" arithmetic or logistic. ...we are no longer interested in the requirements of daily life ...now our concern is rather with understanding the very possibility of this activity, with understanding... that*knowing*is involved and that there must... be a corresponding*being*which possesses that*permanence*of condition which first makes it capable of being "known." But the soul's turning away from the things of daily life, the changing of the direction... the "conversion" and "turning about"... leads to a further question... What is required is an object which has a purely noetic character and which exhibits at the same time... the*countable*... This requirement is exactly fulfilled by the "pure" units, which are "nonsensual," accessible only to the understanding, indistinguishable from one another, and resistant to all participation. The "scientific" arithmetician and logistician deals with*numbers of pure monads*. And... Plato stresses emphatically that there is "no mean difference" between these and the ordinary numbers. ...Only a careful consideration of the fact... forces us into the further supposition that there must indeed be a special "nonsensual" material to which these numbers refer. The immense propaedeutic importance... within Platonic doctrine is immediately clear, for is not a continual effort made in this doctrine to exhibit as the true object of knowing that which is*not*accessible to the senses? Here we have indeed a "learning matter"... "capable of hauling [us] toward being". It forces the soul to study, by thought alone, the truth as it shows itself by itself. ...ability to count and to calculate*presupposes*the existence of "nonsensual" units.

Thus an*unlimited*field of "pure" units presents itself to the view of the "scientific" arithemetician and logistician.- Jacob Klein,
*Greek Mathematical Thought and the Origin of Algebra*(1968) pp. 49-51, citing Plato's*Republic*518 D, 524 E; 523 A; 526 A-B, and Aristotle's*Posterior Analytics*, A 10, 76 b 4 f. Compare René Descartes'*Meditations on First Philosophy*.

- Jacob Klein,

- And yet this dianoetic quarry, as it is brought in especially by the mathematicians, must first be handed over to the
*dialecticians*for proper use (*Euthydemus*290 C;*Republic*531 C-534 E). Only dialectic can open up the realm of true being, can give ground for the powers of the*dianoia*and can reveal Being and the One and the Good as they are—beyond all time and all opposition—in themselves and in truth.- Jacob Klein,
*Greek Mathematical Thought and the Origin of Algebra*(1968) p. 79, noting Cf. Theaetetus 177 C ff. and 186 A-B.

- Jacob Klein,

- Plus one, minus one, plus one, minus one, etc.

By adding the first two terms, the next two, and so forth, the result is converted into another for which each term is zero. Grandi, the Italian Jesuit, had concluded the possibility of creation from this series; because the result is always equal to ½, he saw the unborn fraction of infinitely many zeros, or nothingness. It was thus that Leibnitz saw in his binary arithmetic the image of creation. He imagined that Unity represented God, and Zero the void; and that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in this system of numeration. This conception was so pleasing to Leibnitz that he communicated it to the Jesuit, [Claudio Filippo] Grimaldi, president of the Chinese tribunal for mathematics, in the hope that this emblem of creation would convert the Emperor of China, since he was very fond of the sciences, to Christianity. I mention this merely to show how childhood prejudices may lead astray even the greatest men.- Pierre-Simon Laplace,
*Essai Philosophique sur les Probabilitésas*(1814) p. 82, partially quoted in Tobias Dantzig,*Number: The Language of Science*(1930) p. 15 and in Richard Courant, Herbert Robbins*What is Mathematics?*(1941) revised by Ian Stewart (1996)

- Pierre-Simon Laplace,

- To the Pythagoreans, numbers were both living entities and universal principles, permeating everything from the heavens to human ethics. In other words, numbers had two distinct, complementary aspects. On the one hand, they had a tangible physical existence; on the other, they were abstract prescriptions on which everything was founded. For instance, the
*monad*(the number 1) was understood both as a generator of all other numbers, an entity as real as water, air, and fire that participated in the structure of the physical world, and as an idea—the metaphysical unity at the source of all creation.- Mario Livio,
*Is God a Mathematician?*(2009)

- Mario Livio,

- A theory of the world was gradually developed on the fundamental conception of the unity of all things. Beneath the changes which seem to occur... there must be... some unchanging and unchangeable essence. A hidden thread of unity must run through all phenomena. To find that unchangeable essence, to discover that secret binding unity, became the quest of alchemy, the art which rested on the conception of
*the one and the all*.- M. M. Pattison Muir,
*A History of Chemical Theories and Laws*(1907) p. 4.

- M. M. Pattison Muir,

- Number is limited multitude or a combination of units or a flow of quantity made up of units; and the first division of number is even and odd.
- Nicomachus,
*Introduction to Arithmetic*(c. 100 AD) as translated by Martin Luther D'Ooge,*Nicomachus of Gerasa: Introduction to Arithmetic*(1926) Book I, Chapter VII.

- Nicomachus,

- If simple unity could be adequately perceived by the sight or by any other sense, then, there would be nothing to attract the mind towards reality any more than in the case of the finger... But when it is combined with the perception of its opposite, and seems to involve the conception of plurality as much as unity, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks "What is absolute unity?" This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of reality.
- Plato,
*Republic*(ca. 380 BC) Tr. Benjamin Jowett (1908)

- Plato,

- I come now to the capital work of Hilbert which he communicated to the Congress of Mathematicians at Heidelberg... of which...an English translation due to Halsted appeared in
*The Monist*. ...the author's aim is analogous to that of Russell, but on many points he diverges from his predecessor.

"But," he says, "on attentive consideration we become aware that in the usual exposition of the laws of logic certain fundamental concepts of arithmetic are already employed; for example, the concept of the aggregate, in part also the concept of number.

"We fall thus into a vicious circle and therefore to avoid paradoxes a partly simultaneous development of the laws of logic and arithmetic is requisite."

...what Hilbert says of the principles of logic*in the usual exposition*applies likewise to the logic of Russell. So for Russell logic is prior to arithmetic; for Hilbert they are 'simultaneous.' We shall find... other differences still greater... I prefer to follow step by step the development of Hilbert's thought...

"Let us take as the basis of our consideration first of all a thought-thing 1 (one)." Notice that in so doing we in no wise imply the notion of number, because it is understood that 1 is here only a symbol and that we do not at all seek to know its meaning. "The taking of this thing together with itself respectively two, three or more times ..." Ah! this time it is no longer the same; if we introduce the words 'two,' 'three,' and above all 'more,' 'several,' we introduce the notion of number; and then the definition of finite whole number which we shall presently find, will come too late. Our author was too circumspect not to perceive this begging of the question. So at the end of his work he tries to proceed to a truly patching-up process.

Hilbert then introduces two simple objects 1 and =, and and considers all the combinations of these two objects, all the combinations of their combinations, etc. It goes without saying that we must forget the ordinary meaning of these two signs and not attribute any to them.

Afterwards he separates these combinations into two classes, the class of the existent and the class of the non-existent... entirely arbitrary. Every affirmative statement tells us that a certain combination belongs to the class of the existent; every negative statement tells us that a certain combination belongs to the class of the non-existent.

Note now a difference of the highest importance. For Russell any object whatsoever, which he designates by*x*, is an object absolutely undetermined and about which he supposes nothing; for Hilbert it is one of the combinations formed with the symbols 1 and =; he could not conceive of the introduction of anything other than combinations of objects already defined. ...

Russell is faithful to his point of view, which is that of comprehension. He starts from the general idea of being, and enriches it more and more while restricting it, by adding new qualities. Hilbert on the contrary recognizes as possible beings only combinations of objects already known; so that (looking at only one side of his thought) we might say he takes the viewpoint of extension.- Henri Poincaré,
*The Foundations of Science*(1902-1908) Tr. George Bruce Halstead (1913) pp. 464-465.

- Henri Poincaré,

- But as the Pythagoreans define a point to be unity having position, let us consider what they mean. That numbers, indeed, are more immaterial and more pure than magnitudes, and that the principle of numbers is more simple than the principle of magnitudes, is manifest to every one: but when they say that a point is unity endued with position, they appear to me to evince that unity and number subsist in opinion: I mean monadic number. On which account, every number, as the pentad and the heptad, is one in every soul, and not many; and they are destitute of figure and adventitious form. But a point openly presents itself in the phantasy, subsists, as it were, in place, and is material according to intelligible matter. Unity, therefore, has no position, so far as it is immaterial, and free from all interval and place: but a point has position, so far as it appears seated in the bosom of the phantasy, and has a material subsistence. But unity is still more simple than a point, on account of the community of principles. Since a point exceeds unity according to position; but appositions in incorporeals produce diminutions of those natures, by which the appositions are received.

- But for the present we desire to contemplate, if possible, the nature of the pure and true one, which is not one from another, but from itself alone. It is therefore here requisite, to transfer ourselves on all sides to one itself, without adding any thing to its nature, and to acquiesce entirely in its contemplation; being careful lest we should wander from him in the least, and fall from one into two. But if we are less cautious we shall contemplate two, nor in the two possess the one itself; for they are both posterior to unity. And one will not suffer itself to be numerated with another, nor indeed to be numbered at all: for it is a measure free from all mensuration. Nor is it equal to any others, so as to agree with them in any particular, or it would inherit something in common with its connumerated natures; and thus this common something, would be superior to one though this is utterly impossible. Hence neither essential number, nor number posterior to this, which properly pertains to quantity can be predicated of one: not essential number whose essence always consists in intellection; nor that which regards quantity, since it embraces unity, together with other things different from one. For the nature pertaining to number which is inherent in quantity, imitating the nature essential to prior numbers, and looking back upon true unity, procures its own essence neither dispersing nor dividing unity, but while it becomes the duad, the one remains prior to the duad, and is different from both the unities comprehended by the duad, and from each apart. For why should the duad be unity itself? Or one unity of the duad rather than another, be one itself? If then neither both together, nor each apart is unity itself, certainly unity which is the origin of all number, is different from all these; and while it truly abides, seems after a manner not to abide. But how are those unities different from the one? And how is the duad in a certain respect one? And again, is it the same one, which is preserved in the comprehension of each unity? Perhaps it must be said that both unities, participate of the first unity, but are different from that which they participate: and that the duad so far as it is a certain one participates of one itself, yet not every where after the same manner: for an army, and a house are not similarly one; since these when compared with continued quantity, are not one, either with respect to essence, or quantity. Are then the unities in the pentad, differently related to one, from those in the decad? But is the one contained in the pentad, the same with the one in the decad? Perhaps also if the whole of a small ship, is compared with the whole of a large one, a city to a city, and an army to an army, there will be in these the same one. But if not in the first instance, neither in these. However, if any farther doubts remain, we must leave them to a subsequent discussion.
- Proclus (ca. 450 A.D.)
*The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements*(1791) Tr.Thomas Taylor, pp. 242-243.

- Proclus (ca. 450 A.D.)

- But let us return to unity itself, asserting that it always remains the same, though all things flow from it as their inexhaustible fountain. In numbers, indeed, while unity abides in the simplicitly of its essence, number producing another is generated according to this abiding one. But the one which is above beings, much more abides in ineffable station. But while it abides, another does not produce beings, according to the nature of one: for it is sufficient of itself to the generation of beings. But as in numbers the form of the first monad is preserved in all numbers, in the first and second degree while each of the following numbers do not equally participate of unity; so in the order of things, every nature subordinate to the first, contains something of the first, as it were his vestige or form in its essence. And in numbers, indeed, the participation of unity produces their quantity. But here the vestige of one gives essence to all the series of divine numbers, so that being itself, is as were the footstep of ineffable unity.
- Proclus (ca. 450 A.D.)
*The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements*(1791) Tr.Thomas Taylor, p. 243.

- Proclus (ca. 450 A.D.)

- Arithmetic is indeed more accurate than Geometry, for its Principles respect Simplicity. Unity implies no Position, a Point does, and a Point requiring Position is the Principle of Geometry, Unity of Arithmetic.
- Proclus (ca. 450 A.D.) as quoted by Isaac Barrow,
*The Usefulness of Mathematical Learning Explained and Demonstrated*(1734)

- Proclus (ca. 450 A.D.) as quoted by Isaac Barrow,

- Since each number is with respect to its own
*kind*one and without parts but with respect to its own*material*, as it were, divisible into parts, though not with respect to all of the material either; but rather what is ultimate [in it, i.e., the unit] is without parts even in the material, and in this ultimate thing [counting or calculation and, above all, partitioning] comes to a stop.- Proclus,
*Timaeum*(ca. 450 A.D.) Ernst Diehl, Vol. II, p. 138 as quoted by Jacob Klein,*Greek Mathematical Thought and the Origin of Algebra*(1968) Tr. Eva Brann.

- Proclus,

- God, creating the universe, neither made it perfectly like Himself, nor perfectly unlike, for He, being
*One*, has made the world as*not one*, from the diverse multiplicity of its innumerable parts, ordaining, nevertheless, that they should collect into a certain unity by their exact contiguity. The upper world has no connexion with this subject; the lower, and elementary world, owes this contiguity to the weight divinely impressed on its parts, aided by the subtle fluidity of some of its simple bodies. It is by this quality, with which the matter of the four elements is more or less invested, that they are separated from one another, and each transported to its proper place, as the generation of compounds, and the beauty of the universe requires.- John Rey,
*Essays*(1630) as quoted in Art IX.*A Translation of Rey's Essays on the Calcination of Metals, &c.*(1821)*The Quarterly Journal of Science, Literature, and the Arts*, Vol. XI (1821) pp. 80-83. A translation as communicated by John Goerge Children, Esq., F.R.S., &c. Ref: Rey J.,*Essays de Jean Rey, sur la recherche de la cause pour laquelle l'estain et le plomb augmentent de poids quand on les calcine*. Nouvelle édition revue sur l'exemplaire original et augmentée sur les manuscrits de la Bibliothèque du Roi et des Minimes, avec des notes par M. Gobet. Paris, Ruault, 1777.

- John Rey,

- In geometry we begin with the point, which is indimensional. This is the beginning of the first dimensional form, the line, and by movement the point generates the line. Now Nicomachus had a similar idea of the nature of multitude and number; they form a series, as it were a moving stream, which proceeds out of unity, the monad. Just as the point is not part of the line (for it is indimensional, and the line is defined as that which has one dimension), but is potentially a line, so the monad is not a part of multitude nor of number, though it is the beginning of both, and potentially both. The monad is unity, absence of multitude, potentiality; out of it the dyad first separates itself and 'goes forward' and then in succession follow the other numbers.
- Frank Egleston Robbins,
*Nicomachus of Gerasa: Introduction to Arithmetic*(1926) Tr. Martin Luther D'Ooge, with studies in Greek arithmetic by Frank Egleston Robbins and Louis Charles Karpinski, Ch. 8 Nicomachus' Philosophy of Number, p. 116.

- Frank Egleston Robbins,

- All of the sons of Adam are part of one single body,

They are of the same essence.

When time afflicts us with pain

In one part of that body

All the other parts feel it too.

If you fail to feel the pain of others

You do not deserve the name of man.

- We were so foolish, my friend, before you [Hippias] said what you did, that we had an opinion about me and you that each of us is one, but that we would not both be one (which is what each of us would be) because we are not one but two. But now, we have been instructed by you that if two is what we both are, two is what each of us must be as well... Then its not entirely necessary, as you said it was a moment ago, that whatever is true of both is also true of each, and that whatever is true of each is also true of both.
- Socrates, in Plato's
*Hippias Major*(c. 390 BC) 301 d 5-302 b 3, as quoted by Massimiliano Carrara, Alexandra Arapinis, Friederike Moltmann,*Unity and Plurality: Logic, Philosophy, and Linguistics*(2016)

- Socrates, in Plato's

- I cannot satisfy myself that, when one is added to one, the one to which the addition is made becomes two, or that the two units added together make two by reason of the addition. I cannot understand how, when separated from the other, each of them was one and not two, and now, when they are brought together the mere juxtaposition or meeting of them should be the cause of their becoming two...
- Socrates (c. 399 BC) as quoted by Plato,
*Phaedo*, in*The Dialogues of Plato*(1892) Tr. Benjamin Jowett, Vol. 2, p. 242.

- Socrates (c. 399 BC) as quoted by Plato,

- Number is of two kinds, the Intellectuall (or immateriall) and the Scientiall. The Intellectual is that
*eternall substance*of number, which*Pythagoras*and his Discourse concerning the Gods asserted to be*the principle most providentiall of all Heaven and Earth, and the nature which is betwixt them*. Moreover, it is*the root of divine Beings, and of Gods, and of Dœmons*. This is that which is termed*the principle, fountain, and root of all things*, and defined it to be*that which before all things exists in the divine mind; from which and out of which all things are digested into order, and remain numbred by an indissoluble series*.

For all things which are ordered in the world by nature according to an artificiall course in part and in whole appear to be distinguished and adorn'd by Providence and All-creating Mind, according to Number; the exemplar being established by applying (as the reason of the principle before the impression of things) the number præexistent in the Intellect of God, maker of the world. This only in intellectual, & wholly immaterial, really a substance according to which as being the most exact artificiall reason, all things are perfected, Time, Heaven, Motion, the Stars, and their various revolutions.- Thomas Stanley,
*The History of Philosophy*(1660) Vol. 3, Chap. I. Number, Its kinds; the first kind, intellectual in the divine mind.

- Thomas Stanley,

- Sciential Number is that which
*Pythagoras*defines as*the extension and production into act of the seminall reasons which are in the Monad, or a heap of Monads, or a progression of multitude beginning from Monad, and a regression ending in Monad*. ...

They make a difference between the Monad and One, conceiving the Monad to be that which exists in intellectualls; One, in numbers [or as Moderatus expresseth it, Monad among numbers, One amongst things numbred, one being divisible into infinite; thus Numbers and things numbred differ, as incorporealls and bodies] in like manner Two is amongst numbers. The Duad is indeterminate; Monad is taken according to equality and measure, Duad according to excess and defect: mean and measure cannot admit more and lesse, but excesse and defect (seeing that they proceed to infinite) admit it; therefore they call the Duad indeterminate, holding Number to be infinite, not that number which is separate and incorporeall, but that which is not separate from sensible things.- Thomas Stanley,
*The History of Philosophy*(1660) Vol. 3, Chap. II. The other kind of number, Scientiall; its principles.

- Thomas Stanley,

- Quite aside from the fact that mathematics is the necessary instrument of natural science, purely mathematical inquiry in itself... by its special character, its certainty and stringency, lifts the human mind into closer proximity with the divine than is attainable through any other medium.
*Mathematics is the science of the infinite*, its goal the symbolic comprehension of the infinite with human, that is finite, means. ...The connection between mathematics of the infinite and the perception of God was pursued most fervently by Nicholas of Cusa, the thinker who... intoned the new melody of thought which with Leonardo, Bruno, Kepler, and Descartes gradually swells into a triumphant symphony. He recognizes that the scholastic form of thinking, Aristotelian logic, which is based on... the excluded third, cannot... think the absolute, the infinite... every kind of "rational" theology is rejected, and "mystic" theology takes its place. ...The true love of God is*amor de intellectualis*[intellectual love]. ...Cusanus does not refer to the mystic form of contemplation, but rather to mathematics and its symbolic method. ...This urge finds its simplest expression in the sequence of numbers, which can be driven beyond any place by repeated addition of one.- Hermann Weyl, "The Open World: Three Lectures on the Metaphysical Implications of Science," (1932) as quoted in
*Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics*(2009) ed. Peter Pesic.

- Hermann Weyl, "The Open World: Three Lectures on the Metaphysical Implications of Science," (1932) as quoted in