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Volume XXII June, 1915 Number 6 


Phases in the Development of the Theory of Limits. 
By FLORIAN CAJORI, Colorado College. 


7. Kant and Other Pre-Cantorian Discussion. 

We now come to a commanding figure in philosophic thought — Emmanuel 
Kant. He took Zeno's dialectics more seriously than had been the custom before. 
Kant says that critics charged Zeno with a complete denial of both of two self- 
contradictory propositions. "But," says Kant, "I do not think that he can 
be rightly charged with this." 1 Zeno was not as much of a skeptic as has been 
pretended. Kant did not write on Zeno's arguments on motion, but he touched 
on other arguments of Zeno. Kant's first antinomy, or "the first conflict of the 
transcendental ideas," contains parts which remind one of the following annihila- 
tion of the notion of space, as given by Zeno : If there is space, it is in something, 
for every thing that is, is in something; but that which is in something, is also in 
space. Space, then, must also be in space, and so on infinitely : therefore there is 
no space. While Kant did not contribute directly to a clearer understanding of 
Zeno's arguments on motion, the effect of his writings was a more painstaking 
and searching examination of that subject. 

In 1794 there appeared in Halle a monograph on Zeno's arguments on motion 
by C. H. E. Lohse, which is permeated by the atmosphere of Kantian philosophy. 
It is the earliest publication on our topic which appeared in the form of a mono- 
graph. 2 Of its four parts, the first deals with Zeno's system in general, the second 
gives his arguments against motion, the third elucidates Aristotle's refutation of 
Zeno, the fourth deals with "the only way" of refuting Zeno. The last argument, 
the "stade," is not discussed at all. Aristotle's distinction between a potential 

1 Kant's Werke, Bd. Ill, "Kritik der reinen Vernunft," 2. Aufl. (1787), Berlin, 1904, p. 345. 

2 Car. Henr. Erdm. Lohse, Diss, (praeside Hoffbauer) de argumentis, quibus Zeno Eleates 
nullum essi motum demonstravit et de unica horum refutandorum ratione. Halle, 1794. All our 
information on Lohse's paper is taken from E. Wellmann, op. cit., pp. 12-14. 



and an actual division to infinity is pronounced arbitrary. Whatever can be 
divided to infinity, says Lohse, actually consists of an infinite number of parts 
which exist even before division. He decides on this point against Aristotle 
and in favor of Zeno, as Bayle had done, though he does not mention Bayle. 
Lohse claims that Zeno's fundamental error lay in a wrong conception of time 
and space. These are not qualities subject to our senses, but are forms which 
determine the manner in which our senses are affected; they are a priori ideas. 
Time and space can both be divided to infinity, but one cannot consider time 
as made up of indivisible points in the manner of Zeno, else what happens in a 
moment of time would happen in no time. Rest is not, as Zeno and his followers 
claim, the absence of motion; it is the least velocity of succession. A body can 
be perceived only as it moves. "Without doubt," says Lohse, "all mistakes of 
their system sprang from that error. Thence it came that reason and the senses 
seemed to contradict each other." 

Presiding at the time when Lohse presented his dissertation for an academic 
degree at Halle was Joh. Christoph Hoffbauer (1766-1827) who, many years 
later, prepared a cyclopaedia article, "Achilles (Der Trugschluss)." 1 After 
expressing his disapproval of Facciolati's argument (previously referred to) he 
states that Zeno's argument is true only on a condition which has not been 
stated explicitly: Zeno's contention that the faster runner will always only 
arrive at the places where the slower has been, and will be behind the slower 
runner, is true only on condition that the faster has not overtaken the slower. 
The only thing proved by Zeno is therefore that the faster runner cannot have 
overtaken the slower as long as the slower is still in advance ! 

A reply to Hoffbauer's argument was made by Christian Ludwig Gerling, 
professor of mathematics, astronomy, and physics at the University of Marburg, 
in a prorectorat address. 2 The claim that Zeno's argument is valid only for 
certain points, not for all, is no objection at all, unless it is first shown to be a 
mistake to assert as true for all points what is in fact true of an infinite number of 
points; a defender of Zeno may always demand that the points be shown, for 
which the proof does not hold. Gerling insists that Hoffbauer himself reasons 
in a circle when he accuses Zeno of reasoning in a circle, for whoever has still to 
prove the possibility of an overtaking is not yet permitted to speak of the time 
before or after which the overtaking takes place. 

Against Waldin's argument, advanced at this same university (Marburg) 
forty-three years previous, to the effect that Zeno assumes the existence of motion, 
the very thing that is in dispute, Gerling argues that Zeno's argument is an 
indirect one, a reductio ad absurdum, the form of which is quite valid. 

1 Allg. Encycl. d. Wissensch. u. Kilnste, von J. S. Ersch u. J. G. Gruber, Leipzig, 1818. 

2 De Zenonis Eleatici paralogismis motum spectantibus, Disseriatio auctore Chr. Lud. Gerling. 
Marburg, 1825. We know this dissertation only from the description of it given by E. Wellmann, 
op. <At., pp. 14, 15, and by Dr. Johann Heinrich Loewe, "Ueber die Zenonischen EinwUrfe gegen 
die Bewegung," in Bbhm. GeseUsch. d. Wissensch., VI Folge, 1 Bd., 1867, pp. 30, 34. In Poggen- 
dorff's Handworterbuch, the date of Gerling's dissertation is given as 1830. We have seen refer- 
ences to an edition in German of the year 1846. From this we infer that several editions of it 
have appeared, and that it enjoyed a considerable circulation. 


Lohse's metaphysical apparatus Gerling declares needless and useless. In the 
constructive part of his dissertation, Gerling dwells on the distinction between 
continuous and discrete quantity, admits the infinite divisibility of space and 
time, and constructs the infinite geometric progressions whose sums give respec- 
tively the distance and the time of running, before Achilles overtakes the tortoise. 
Gerling here repeats what Gregory St. Vincent had done long before, only Gerling 
uses letters, while Gregory assumed a special numerical case. Gregory is no- 
where mentioned by Gerling. The sums of the two geometric progressions are 
values which in no way conflict with the estimate obtained from sensuous per- 
ception; Zeno's paradox, as interpreted by aid of the mathematical formulas, 
conflicts in no way with experience. Hence the puzzle is solved. Though a 
mathematician, Gerling does not feel the need of explaining the possibility of a 
variable reaching its limit. 

As to the "Arrow" a sharp distinction between the continuous and the dis- 
crete is sufficient. In continuous quantity the number of possible subdivisions 
is arbitrary, and each subdivision is itself continuous. Hence Zeno's alleged 
denial of the infinite divisibility does not follow. Gerling treats the "Stade" 
with more than customary respect, and admits that, if one assumes with Zeno 
that space and time be not infinitely divisible, then it follows, as Zeno says, that 
half the time is equal to the whole time. 

An entirely different type of discussion, more along the lines of Kant, pro- 
found and obscure, is given by Georg Wilhelm Friedrich Hegel. He holds the 
view that "Zeno's dialectic of matter has not been refuted to the present day; 
even now we have not got beyond it, and the matter is left in uncertainty." 1 
He protects Aristotle against Bayle who objected to Aristotle's distinction between 
a potential and an actual subdivision of a line to infinity. Hegel keenly realizes 
the speculative importance of Zeno's paradoxes and points out that the dialec- 
tician of Elea had analyzed our concepts of time and space and had pointed out 
the contradictions involved therein; "Kant's antinomies do no more than Zeno 
did here." 2 Movement appears "in its distinction of pure self-identity and pure 
negativity, the point as distinguished from continuity." 3 This continuity is an 
absolute hanging together, an annihilation of all differences, of being by itself; 
the point on the other hand is pure existence by itself, the absolute distinctness 
from others, the suspension of all self-identity and all hanging together. In time 
and space the opposites are united in one, hence the contradiction as exhibited 
in motion. Hegel's position is a long way, still, from Georg Cantor's continuum, 
with its skilful union of continuity and discreteness. In the "dichotomy" the 
assumption of half a space is incorrect, says Hegel, "there is no half of space, 
for space is continuous; a piece of wood may be broken into two halves, but not 
space, and space only exists in movement." 4 Motion is connectivity, disintegra- 
tion into an indefinite number of aggregates is its opposite. 

1 G. W. F. Hegel, History of Philosophy, transl. by E. S. Haldane, Vol. I, London, 1892, p. 265. 
See also Hegel's Samtt. Werke, Bd. 13, 1833, pp. 314-327. 

2 Hegel [ed. Haldane], Vol. I, p. 277. 

3 Hegel, op. tit., Vol. I, p. 268. 

4 Hegel, op. tit., Vol. I, p. 271. 


Somewhat more specific and comprehensible are the ideas set forth by his 

philosophical opponent, Johann Friedrich Herbart. Zeno's paradoxes are taken 

up by him in two works, his popular Einleitung in die Philosophic (1813) and his 

more technical and scientific Allgemeine Metaphysik (1828-9). Only in the 

latter work is the solution of the contradictions attempted. From it we quote : x 

"The argument inevitably confuses those, who admit the infinite divisibility of the path 
and then console themselves with a corresponding infinite divisibility of the time, to such a degree 
that though at first willing to consider the process of dividing, which must continue to infinity, 
they soon in one leap consider the infinite number of time intervals as passed over, since they see 
that they must combine the infinite number of parts of the time as well as of the path to the place 
of overtaking, which they cannot do. The leap and the doubly infinite division are both faulty 
and amount to naught." 

Thus, this infinite subdivision of the time and space is rejected by Herbart, 
because the mind is not able to imagine all the steps in the process. Imaginability 
is made the criterion of truth or error. This criterion throws out infinity at once; 
it throws out non-euclidean geometry and other parts of mathematics. We 
cannot really imagine things which we have never seen. Our senses are inaccurate, 
our intuitions are crude; hence it would seem to us impossible to build up sound 
mathematical theory, if everything unimaginable were to be cast aside. Herbart 
tries to explain motion by the concept of velocity, which seems itself to involve 
a contradiction, that Herbart endeavors to resolve by his theory of a "rigid line," 
a sort of continuum, which might have given rise to great possibilities upon more 
careful development. As it is, it offers greater obstacles by far than does the 
original "Dichotomy" or "Achilles" which it is intended to explain. 

A still different attitude toward Zeno's paradoxes is taken by Friedrich Adolf 
Trendelenburg of the University of Berlin, in his Logische TJntersuchungen, 
1840, where he constructs his philosophic system upon the concept of motion. 
Constructive motion is common to the external world of being and to the internal 
world of thought, so that thought, as the counterpart of external motion, pro- 
duces from itself space, time, and the categories. Motion is undefinable. In 
accordance with this view it is only through motion that Zeno's arguments 
against motion have come to be. For they depend upon the division of time and 
space, and the synthesis of those divisions. But division and synthesis are 
nothing but special forms of motion. What the proofs combat, they themselves 
use as the means of combat, and thereby testify to the controlling nature of mo- 
tion. Trendelenburg and Kant evidently begin at opposite ends; Kant takes 
time and space as a priori ideas, and motion as secondary and dependent upon 
them; Trendelenburg makes motion the a priori idea, and pretends to derive 
time and space from it. 

Friederich Ueberweg of the University of Konigsberg 2 refers to our subject 
in different parts of his Logik. He says in one place that the " Achilles " proves 
too little; it proves merely that the tortoise cannot be overtaken within a definite 
series, and then claims that the tortoise cannot be overtaken anywhere and at 

1 J. T. Herbart, Sdmtl. Werke, herausgeg. von Karl Kehrbach, Langensalza, Vol. VIII, 1893, 
p. 177. 

2 F. Ueberweg, System der Logik, 2. AufL, p. 387 ff. 


any time. True as this criticism may be, it does not illuminate the matter 
sufficiently to satisfy the reader. 

Of the same type, but fuller in statement, is a criticism by Carl Prantl, 1 
professor at the University of Munich. He claims that Zeno discarded the con- 
cept of continuity by considering only some particular points on a line and only 
some particular moments in time. By drawing his inferences from these dis- 
integrated fragments of time and space, Zeno was able to advance contradictions 
in a picturesque manner. This conversion of the general and continuous into 
the particular and momentary will be encountered often, says Prantl, in those 
who care more for rhetorical form than for true philosophy. 

Much confidence in his ability to clear up the mystery of Zeno's paradoxes is 
displayed by Eugen Carl Duhring in his Kritische Geschickte der Philosophie, 
first edition, 1869. Three concepts are necessary here: rest, motion and position. 
Usually only the first two are considered. At each moment (point) of time a 
moving body has a definite position but no motion. This fact makes it difficult 
to explain motion. He says further: 2 "The compelling force and real conclusive- 
ness of the Eleatic contentions is to be found chiefly and almost exclusively in 
the logical necessity which does not permit the infinite to be thought of as com- 
pleted, as enumerated so to speak, and concluded. ... It is the concept of 
infinity which proves itself everywhere and also where it is not readily recognized, 
as the true cause of the contradictions." Duhring discusses infinity in several 
places of his works. He believes in the infinity usually set forth in the study of 
the calculus,— a variable which increases without limit, but at any moment has 
really a finite value. He makes war against the concept of an actual infinity — 
"jene wiiste, sich widersprechende Unendlichkeit." "The infinite divisibility 
indicates . . . only this, that I can conceive the division of a quantity as far as 
I choose, without limit. If, on the other hand, I consider the division to infinity 
as really existing outside of my presentation of it, then there soon result the most 
manifold contradictions. ... As regards motion, it must be recognized that it 
belongs to the empirical concepts, i. e., in our thinking there remains here always 
an unrecognizable residue, for we must give up the attempt to penetrate to the 
reasons of the phenomena." Georg Cantor criticizes Duhring in these words: 

"The proofs of Duhring against the properly-infinite could be given in much fewer words 
and appear to me to amount to this, either that a definite finite number, however large it may be 
thought to be, can never be an infinite number, as follows immediately from the concept of it, or 
else that the variable, an unlimitedly large finite number, cannot be thought of with the quality 
of definiteness and therefore not with the quality of existence, as follows again from the nature 
of the variability. That not the least is hereby established against the conceivability of trans- 
finite number, I feel certain; and yet, those proofs are taken as proofs against the reality of trans- 
finite numbers. To me this mode of argumentation appears the same as if, from the existence of 
innumerable shades of green, we were to conclude that there can be no red." 3 

Diihring's explanation of infinity and of Zeno is accepted by Eduard Well- 
mann, in his historical monograph 4 of 1870. Another research, partly historical 

1 Carl Prantl, Qeschichte der Logih im Abendlande, 1. Bd., Leipzig, 1855, pp. 10, 11. 

2 Kritische Oesch. d. Philosophie, Dr. E, Duhring, Leipzig, 1894, p. 49. 

3 Georg Cantor, Grundlagen einer allg. Mannichfaltigkeitslehre, Leipzig, 1883, p. 44. 

4 E. Wellmann, op. tit., p. 23. 


and partly expository, was published in 1867, by Johann Heinrich Loewe, a 
pupil of the philosopher, Anton Gtinther of Vienna. It is referred to by Knauer 1 
as the most acute and satisfactory explanation that has yet been offered. "The 
solution of the riddle," says Loewe, 2 "appears to us to lie in the knowledge that 
contradictions must arise inevitably, as soon as space, time, and motion are con- 
sidered at the same time from the stand-point of sensuous presentation and of 
non-sensuous conceptual reasoning." One point of view appeals to the imagina- 
tion; the other to abstract thought. Sensuous perception can follow the process 
of infinite division only a little way, everything beyond is a matter of pure 
reason. Gerling's presentation of "Achilles" is an appeal to reason. As long 
as one considers the infinite multiplicity of small distances and of time-intervals, 
one approaches the riddles from the standpoint of abstract thought; when one 
appeals to the imagination, then the finite time and the finite length of the race 
stand out. Loewe seems still to hold to the old view that thought can recognize 
no end to a motion which extends over an infinite process. Hence the contradic- 
tion must stand, the antinomy is evident. 

Thus we see that German philosophy down to the last quarter of the nine- 
teenth century continually accentuates the existence of contradictions in the 
problem of motion. 

Some English thinkers of the nineteenth century, who were interested in 
Zeno's arguments, came under the influence of Kantian philosophy. The 
Kantian attitude toward Zeno is described in the article "Zeno" in the eighth 
edition of the Encyclopaedia Britannica (1860) thus: 

"He brought a most powerful mind to his task, and, curious to say, subsequent thinkers have 
very generally agreed in misunderstanding both his reasoning and his method, and it is only of 
late years that Kant, in his Antinomies of the Pure Reason (see Kritik der Reinen Vernunft) seized 
upon the much maligned doctrines of the Eleatic, and held them up to the admiration of all true 
thinkers as rare examples of acute and just thought. Bayle, in a clever paper on Zeno, in his 
Dictionnaire, makes him, according to custom, a sceptic. Brucker finds that Zeno surpasses 
his intelligence, and he is content to make him a pantheist. Others again, have charged him with 
nihilism. Zeno, fortunately, can afford to sit quite easy to all those affronts offered to his reason 
. . . they [arguments against motion] all take their rise, as Kant and Hamilton (Lectures on 
Metaphysics) have shown, from the inability of the mind to conceive either the ultimate indivisi- 
bility, or the endless divisibility, of space and time, as extensive and as protensive quantities. 
The possibility of motion, however certain as an observed fact, is thus shown to be inconceivable. 
To have discovered this peculiarity of our mental constitution, and to have stated it with eminent 
clearness, belongs to Zeno the Eleatic, and to him alone." 

Sir William Hamilton puts this matter thus : 3 

"Time is a protensive quantity, and, consequently, any part of it, however small, cannot, 
without a contradiction, be imagined as not divisible into parts, and these parts into others ad 
infinitum. But the opposite alternative is equally impossible; we cannot think this infinite 
division. One is necessarily true; but neither can be conceived possible. It is on this inability 
of the mind to conceive either the ultimate indivisibility, or the endless divisibility of space and 
time, that the arguments of the Eleatic Zeno against the possibility of motion are founded, — 
arguments which at least show, that motion, however certain as a fact, cannot be conceived pos- 

1 Vincenz Knauer, Die Hauptprobleme der Philosophic, Wien u. Leipzig, 1892, p. 54. 

2 J. H. Loewe, op. cit., p. 32. 

3 Lectures on Metaphysics and Logic, by Sir William Hamilton, Vol. I, Boston, 1863, Lecture 
38, p. 530. 


sible, as it involves a contradiction. . . . Now the law of mind, that the conceivable is in every 
relation bounded by the inconceivable, I call the Law of the Conditioned." 

John Stuart Mill, in his Logic, 1 refers to Thomas Brown who considered the 
"Achilles" insoluble, and then offers a solution to the invention of which he lays 
no claim. It presents no new points of view. Herbert Spencer discusses ques- 
tions of time and space in his First Principles 2 and concludes in general that 
"ultimate scientific ideas, then, are all representative of realities that cannot be 
comprehended." In particular, "halve and again halve the rate of movement 
for ever, yet movement still exists; and the smallest movement is separated by 
an impassable gap from no movement." 

It is readily seen that the nineteenth century philosophers had penetrated 
deeper than most of their predecessors and had encountered difficulties previously 
neglected by Hobbes and others who seemed to think that they had solved the 
"Achilles" paradox by the mere statement that time, as well as space, was infi- 
nitely divisible. What came to be thoroughly realized since the time of Kant 
was the impossibility of imagining the "Achilles" from the standpoint of infinite 
divisibility of a distance, that all appeals to intuition were futile. When Spencer- 
says that infinite divisibility cannot be " comprehended," and Thomas Brown and 
Sir William Hamilton say that motion is "insoluble" and "inconceivable," I 
take it that they mean simply that these processes are unimaginable, that they 
are beyond the reach of our sensual intuitions. I do not interpret them to mean 
that these processes are beyond the reach of logic, beyond the reach of the 
reasoning faculty so as to be, and forever remain, wholly mysterious. Mathe- 
matics includes among its results numerous teachings which one cannot "imag- 
ine." Probably no one claims to be truly able to visualize to himself the non- 
euclidean geometries; analysts do not claim to be able to imagine or see a con- 
tinuous curve which has no tangent line at any of its points. Yet no modern 
mathematician rejects non-euclidean geometries and non-differentiable continuous 

These unimaginable mathematical creations are admitted into the science 
as a matter of necessity. Felix Klein states the issue as follows : " As the subjects- 
of abstract geometry cannot be sharply comprehended through space intuition,, 
one cannot rest a rigorous proof in abstract geometry upon mere intuition, but 
must go back to a logical deduction from axioms assumed to be exact." 3 

It so happens that England's two famous opium eaters, Thomas De Quincey 
and Samuel Taylor Coleridge, were interested in the "Achilles." Coleridge's 
critical powers were set forth by De Quincey in the following terms: 4 

"I had remarked to him that the sophism, as it is usually called, but the difficulty, as it 
should be called, of Achilles and the Tortoise, which had puzzled all the sages of Greece, was, in 
fact, merely another form of the perplexity which besets decimal fractions; that, for example, if' 

1 A System of Logic, Vol. II, London, 1851, p. 381. 

2 H. Spencer, First Principles of a New System of Philosophy, New York, 1882, pp. 47-67.- 

3 F. Klein, Anwendung der Differential- und Integralrechnung auf Geometric Leipzig, 1907 r 
p. 19. 

4 Tait's Magazine, Sept. 1834, p. 514. 


you throw f into a decimal form, it will never terminate, but be .666666, etc., ad infinitum. 'Yes,' 
Coleridge replied, 'the apparent absurdity in the Grecian problem arises thus, — because it assumes 
the infinite divisibility of space, but drops out of view the corresponding infinity of time.' There 
was a flash of lightning, which illuminated a darkness that had existed for twenty-three centuries." 

As a matter of fact, Aristotle had seen that far. But Coleridge proceeded 
somewhat farther in an essay on Greek sophists in The Friend, 1 where he says: 

"The few remains of Zeno the Eleatic, his paradoxes against the reality of motion, are mere 
identical propositions spun out into a sort of whimsical conundrums, as in the celebrated paradox 
entitled Achilles and the Tortoise, the whole plausibility of which rests on the trick of assuming a 
minimum of time while no minimum is allowed to space, joined with that of exacting from intelligi- 
bilia, vobfieva, the conditions peculiar to objects of the senses <pcu.vbjitva or alvdavbueva." 

What belongs to Coleridge himself in this passage is the contention that the 
sophism consists in applying to an idea conditions only properly applicable to 
sensuous phaenomena. Coleridge's argument was elaborated many years later 
in dialogue form, by Shadworth H. Hodgson. We give the critical part of the 
discussion: 2 

"... being infinitely divisible is not the same thing as being infinitely divided. Actually to 
divide to infinity that hundredth part of a minute, in which (phenomenally as you say) Achilles 
overtakes the tortoise, is an infinitely long operation. . . . And this division you call upon 
Achilles to perform, before the tortoise can be overtaken, and to perform phenomenally. . . . You 
require that Achilles shall exhibit to the senses the infinite divisibility of time and space, which 
appertains to them truly indeed, but only as objects of imagination and thought. . . . The world 
of thought and reality is not a world apart, but is identical with the phenomenal world, only dif- 
ferently treated. . . . Neither is there any contradiction between them. Phenomenal motion is 
as infinitely divisible in thought as time and space are." 

This explanation does not explain. Even as "objects of the imagination" 
the infinite divisibility of time and space is a source of perplexity. Our imagina- 
tion is unable to follow Achilles to the end, through the infinities of time and 
space intervals. Moreover, "thought and reality" are indeed worlds "apart" 
whenever the time intervals, corresponding to the space-intervals passed over by 
Achilles, are so taken that they form together an infinite series that is divergent, 
so that, in thought, Achilles never overtakes the tortoise; in Zeno's traditoinal 
argument, "thought and reality" were "apart." 

1 Complete Works of S. T. Coleridge, Vol. II, New York, 1856, p. 399. 

2 Mind, London, Vol. V, 1880, pp. 386-388. 

[The remaining parts of this series are: D. Viewed in the Light of an Idealistic Contin- 
uum (G. Cantor); E. Post-Cantobian Dissension.]