Achilles and the turtle

Graphic illustration of the paradox

The paradox of Achilles and the turtle is one of several well-known fallacies attributed to the Greek philosopher Zeno de Elea (5th century BC) and one of four paradoxes that Aristotle describes in his treatise Physics .

Paradox

The paradox is about a race between Achilles, who is known for his speed, and a slowly moving turtle . Both start at the same time, but the turtle gets a head start initially. Although Achilles is faster, he can never catch up with her.

Zeno's argument is based on the assumption that Achilles must first reach the point where the turtle started, by which time the turtle will have moved, if only a small distance, to another point; by the time Achilles has traveled the distance to that point, the turtle will have advanced to another point, and so on.

Solution from Aristotle

The Achilles Paradox illustrates the problem of the continuum . Aristotle's solution to this problem was to treat the segments of Achilles motion as only potential and not real, since he never realizes them by stopping. Anticipating modern measure theory , Aristotle argued that an infinity of subdivisions of a segment that is finite does not preclude the possibility of traversing that segment, since the subdivisions do not actually exist unless something is done to them, in this case stopping at them.

Math solution

In fact, a faster person will always catch up with a slower one if he has enough time to do so. The time required to retrieve is proportional to the lead and inversely proportional to the difference in the speeds of the two runners and, if the ratio of these two speeds remains the same, inversely proportional to each of them.

A geometric proof by means of the ray theorem , which was also possible for the Greeks . (It is best to choose a 45 ° angle at the origin for Achilles.)

turtle

Achilles

Zeno's fallacy is based on two mistakes:

It does not take into account that an infinite series can have a finite sum.
The path that Achilles covers from his starting point to the meeting with the turtle can be divided as often as desired - formally infinitely often - into projections of the turtle. From the fact that this act of division can be carried out as often as desired, however, it does not follow that the distance to be covered would be infinite or that an infinite amount of time would be required to cover it.

Zeno's Paradoxes

There are different views of what Zeno wanted to show with his "paradoxes". It is often assumed that they should support the Eleatic thesis (see Parmenides von Elea ), according to which there is no multiplicity in reality , but only a single immutable and indestructible whole , and that the everyday perception of diversity and movement is mere appearance. What is certain, however, is that this ancient consideration contributed to the formation of the concept of infinity and is still used today as a teaching example.

The paradox is not passed down directly, but can be found in Aristotle 's Physics and Simplikios ' commentary on it.

Related paradoxes ascribed to Zeno are the division paradox and the arrow paradox . Content not related to the Zeno paradox is one of Lewis Carroll in his brief dialogue What the Tortoise Said to Achilles ( What the tortoise said Achilles ) imagined argument with which he the difference between object and metalingual implication discussed and occasionally as Carroll -Paradox is called.

Remarks

↑ Let the time that elapses from the start of the race to the time that Achilles overtakes the turtle, the path that Achilles covers during time . the distance that the turtle covers during the time , the lead of the turtle at the start of the race, the speed of Achilles, the speed of the turtle. Then t can be calculated as follows:

{\ displaystyle t}

{\ displaystyle s}

{\ displaystyle t}

{\ displaystyle s'}

{\ displaystyle t}

{\ displaystyle s_ {0}}

{\ displaystyle v_ {a}}

{\ displaystyle v_ {s}}

{\ displaystyle v_ {a} \ cdot t = s = s_ {0} + s' = s_ {0} + v_ {s} \ cdot t}

, so ; with follow after Division: .

{\ displaystyle s_ {0} = v_ {a} \ cdot t-v_ {s} \ cdot t = (v_ {a} -v_ {s}) \ cdot t}

{\ displaystyle v_ {a} -v_ {s} \ neq 0}

{\ displaystyle t = {\ frac {s_ {0}} {v_ {a} -v_ {s}}}}

The latter shows the proportionality of the time claimed in the text to the lead of the turtle and the inverse proportionality of to the speed difference .

{\ displaystyle t}

{\ displaystyle s_ {0}}

{\ displaystyle t}

{\ displaystyle v_ {a} -v_ {s}}

↑ (With ) is further the ratio of the speeds, so that (with ) also . Because is , and the expression for can be further transform: ; for a constant ratio of the two speeds, the last two fractions show the inverse proportionality of time to or, as claimed in the text . The inverse proportionality of to means that Achilles is more likely to hit the turtle if it runs faster . At first this might surprise you; It is assumed here, however, that in this case Achilles also runs faster by the same factor as the turtle (since it is assumed to be constant). ${\ displaystyle v_ {a} \ neq 0}$ ${\ displaystyle q = {\ frac {v_ {s}} {v_ {a}}}}$ ${\ displaystyle v_ {s} = q \ cdot v_ {a}}$ ${\ displaystyle v_ {s} \ neq 0}$ ${\ displaystyle {\ frac {1} {v_ {a}}} = {\ frac {q} {v_ {s}}}}$ ${\ displaystyle v_ {a} -v_ {s} \ neq 0}$ ${\ displaystyle q \ neq 1}$ ${\ displaystyle t}$ ${\ displaystyle t = {\ frac {s_ {0}} {v_ {a} -v_ {s}}} = {\ frac {s_ {0}} {v_ {a} -q \ cdot v_ {a}} } = {\ frac {s_ {0}} {v_ {a} \ cdot (1-q)}} = {\ frac {q \ cdot s_ {0}} {v_ {s} \ cdot (1-q) }}}$ ${\ displaystyle q}$ ${\ displaystyle t}$ ${\ displaystyle v_ {a}}$ ${\ displaystyle v_ {s}}$ ${\ displaystyle t}$ ${\ displaystyle v_ {s}}$ ${\ displaystyle q}$

↑ It is - today - possible to calculate the time after which Achilles overtakes the turtle using Zeno's approach . - As above, be the turtle's lead at the start of the race, the time it takes Achilles to cover . Furthermore, the turtle is times slower than Achilles. Then Achilles catches up with the turtle one more time after time, a third time after time, and so on. With is the sum of all times observed by Zenon that Achilles covers: ${\ displaystyle t}$ ${\ displaystyle s_ {0}}$ ${\ displaystyle t_ {0}}$ ${\ displaystyle s_ {0}}$ ${\ displaystyle q}$ ${\ displaystyle t_ {0} \ cdot q}$ ${\ displaystyle (t_ {0} \ cdot q) \ cdot q = t_ {0} \ cdot q ^ {2}}$ ${\ displaystyle q ^ {0} = 1}$
${\ displaystyle t = t_ {0} \ cdot \ sum _ {n = 0} ^ {\ infty} q ^ {n} = t_ {0} \ cdot \ lim _ {n \ to \ infty} \ sum _ { k = 0} ^ {n} q ^ {k} = t_ {0} \ cdot \ lim _ {n \ to \ infty} {\ frac {1-q ^ {n + 1}} {1-q}} = {\ frac {t_ {0}} {1-q}}}$ .
It is possible, but not absolutely necessary, to be understood as the quotient of two speeds , as above . Then continue with : ${\ displaystyle q}$ ${\ displaystyle {\ frac {v_ {s}} {v_ {a}}}}$ ${\ displaystyle t_ {0} = {\ frac {s_ {0}} {v_ {a}}}}$
${\ displaystyle t = {\ frac {t_ {0}} {1-q}} = {\ frac {s_ {0}} {v_ {a} -q \ cdot v_ {a}}} = {\ frac { s_ {0}} {v_ {a} -v_ {s}}};}$
the convergent geometric series thus gives the same result for as the calculation in note 1 without decomposing according to Zenon's approach. Because of this, the series fulfills a convergence criterion , so that limit value calculation assigns it to exactly one (exact, called "limit value") number that it reaches at infinity . Zeno was evidently not aware of such mathematics . ${\ displaystyle t_ {0} \ cdot \ sum _ {n = 0} ^ {\ infty} q ^ {n}}$ ${\ displaystyle t}$ ${\ displaystyle t}$ ${\ displaystyle 0 <q <1}$

↑ Sainsbury's paradoxes show the vagueness of the problem on the basis of the division into two: length two is halved, two lengths one, then one length one in two halves, half of that in two quarters and so on. It is obvious that the two is not exceeded, nor does time stretch. Rather, the remainder is always clear: identical to the last division link (a quarter above). (It therefore does not seem to be Zeno's aim to show that the race is eternal and indefinitely long. As an argument, similar to the arrow paradox, the impossibility remains (in accordance with the missing mathematical statements about infinity or possibly zero ) to reach the goal.)

↑ Using Zeno's approach, the path that Achilles covers in the period (from his starting point to the time the turtle is caught up) can also be calculated . - In the invoice in note 3, only or is to be replaced by or :

{\ displaystyle s}

{\ displaystyle t}

{\ displaystyle t}

{\ displaystyle t_ {0}}

{\ displaystyle s}

{\ displaystyle s_ {0}}

{\ displaystyle s = s_ {0} \ cdot \ sum _ {n = 0} ^ {\ infty} q ^ {n} = {\ frac {s_ {0}} {1-q}}}

.

If velocities are introduced, it is possible to use the above representation of :

{\ displaystyle t}

{\ displaystyle s = v_ {a} \ cdot t = {\ frac {v_ {a} \ cdot s_ {0}} {v_ {a} -v_ {s}}}}

;

Excluding in the denominator and reducing it gives the same result as the convergent series.

{\ displaystyle v_ {a}}

literature

Max Black: Achilles and the Tortoise , in: Analysis 11 (1950), pp. 91-101.
Simon Blackburn: Practical Tortoise Raising , in: Mind 104 (1995), pp. 696-711.
S. Brown: What the Tortoise taught us , in: Mind 63 (1954), pp. 170-179.
Florian Cajori: The Purpose of Zeno's Arguments on Motion , in: Isis 3/1 (1920), pp. 7-20.
L. Carroll (CL Dogson): What the Tortoise said to Achilles , in: Mind 104 (1995), pp. 278-280.
M. Clark: Paradoxes, from A to Z , Routledge, London 2000.
Pascal Engel: Dummett, Achilles and the tortoise , in: L. Hahn / R. Auxier (eds.): The philosophy of Michael Dummett (Library of Living philosophers) , La Salle, Ill .: Open Court 2005.
Adolf Grünbaum: Modern Science and Zeno's Paradoxes , Middletown: Wesleyan University Press 1967.
Andrew Harrison: Zeno's Paper Chase , in: Mind 76/304 (1967), pp. 568-575.
JM Hinton / CB Martin: Achilles and the Tortoise , in: Analysis 14/3 (1954), pp. 56-68.
CV Jones: Zeno's paradoxes and the first foundations of mathematics (Spanish), in: Mathesis 3/1 1987.
S. Makin: Art. Zeno of Elea , in: Routledge Encyclopedia of Philosophy 9, London 1998, pp. 843-853.
R. Morris: Achilles in the Quantum Universe , Redwood Books, Trowbridge, Wiltshire 1997.
Jorge Luis Borges : Two Essays in Kabbala and Tango , S. Fischer Verlag, 1991.
Aloys Müller : The problem of the race between Achilles and the turtle , in: Archive for Philosophy 2 (1948), pp. 106–111.
Stanislaus Quan: The Solution of Zeno's First Paradox , in: Mind 77/306 (1968), pp. 206-221.
WD Ross: Aristotle's Physics , Oxford: Clarendon 1936, xi-xii Bibliography of older literature on the paradoxes of movement, pp. 70–85, etc. Commentary on the sections in Aristotle.
Bertrand Russell : Our Knowledge of the External World , Open Court, London / Chicago 1914, chap. 5 and 6.
Richard Mark Sainsbury: Paradoxien , Reclam, Stuttgart 2001 (= Reclams Universal Library 18135), ISBN 3-15-018135-6 .
Wesley C. Salmon (Ed.): Zeno's paradoxes , Hacket, Indianapolis 1970, reprint 2001, ISBN 0-87220-560-6 .
Wesley C. Salmon: Space, Time and Motion , Enrico, California and Belmont, California, Dickenson Publishing Co., Inc. 1975, chap. 2
T. Smiley: A Tale of Two Tortoises , in: Mind 104 (1995), pp. 725-736.
Roy Sorensen: A Brief History of the Paradox , Oxford University Press 2003.
LE Thomas: Achilles and the Tortoise , in: Analysis 12/4 (1952), pp. 92-94.
JF Thomson: What Achilles should have said to the Tortoise , in: Ratio 3 (1960), pp. 95-105.

Web links

Christoph Bock: Elements of Analysis (PDF; 1.2 MB) p. 59f
Nick Huggett: Zeno's Paradoxes. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .
Entry in Wolfram's Math World (English, with further literature)
Peter Lynds : Zeno's Paradoxes: A Timely Solution (English with further literature; PDF file; 166 kB)
Math Planet Article
Ulrich Eckhardt: [1] (PDF; 320 kB)
Zeno and the slow motion. in: Wolfram Heinrich: Zenon, Achilles and the tortoise. The forgotten thinker Costabile Matarazzo. [2] @ theodor-rieh.de and [3] @ freitag.de

Individual evidence

Wikiquote: Arnfrid Astel - Quotes

^ According to Peter Janich: "Achilles and the Tortoise", in: Jürgen Mittelstraß (Ed.): Enzyklopädie Philosophie und Wissenschaftstheorie , Volume 1, Metzler Stuttgart 1995, reprint 2004, page 41, ISBN 3-476-02012-6
↑ VI, 9,239b14-240a18 in the formulation that “even the slowest animal in the course cannot be overtaken by the fastest, since the pursuer always has to get to where the fleeing animal has run away from, so that the slower one always has a head start keep ".
↑ Original Greek text in Aristotle: Physics. (see section 4 of the screen). Archived from the original on May 16, 2008 ; Retrieved October 16, 2013 .
↑ Simplicius: On Aristotle's Physics 1014,10, see: Readings in Ancient Greek Philosophy From Thales to Aristotle, ed. SM Chohen / P. Curd / CDC Reeve, Indianapolis / Cambridge: Hackett 1995, 58f
↑ Mind 1 (1895), pp. 278-280.
↑ see, for example, Pascal Engel: Dummett, Achilles and the tortoise , In: L. Hahn / R. Auxier (eds.): The philosophy of Michael Dummett (Library of Living philosophers) , La Salle, Ill .: Open Court 2005 .

[anm1-1] Let the time that elapses from the start of the race to the time that Achilles overtakes the turtle, the path that Achilles covers during time . the distance that the turtle covers during the time , the lead of the turtle at the start of the race, the speed of Achilles, the speed of the turtle. Then t can be calculated as follows: ${\ displaystyle t}$ ${\ displaystyle s}$ ${\ displaystyle t}$ ${\ displaystyle s'}$ ${\ displaystyle t}$ ${\ displaystyle s_ {0}}$ ${\ displaystyle v_ {a}}$ ${\ displaystyle v_ {s}}$
${\ displaystyle v_ {a} \ cdot t = s = s_ {0} + s' = s_ {0} + v_ {s} \ cdot t}$ , so ; with follow after Division: . ${\ displaystyle s_ {0} = v_ {a} \ cdot t-v_ {s} \ cdot t = (v_ {a} -v_ {s}) \ cdot t}$ ${\ displaystyle v_ {a} -v_ {s} \ neq 0}$ ${\ displaystyle t = {\ frac {s_ {0}} {v_ {a} -v_ {s}}}}$
The latter shows the proportionality of the time claimed in the text to the lead of the turtle and the inverse proportionality of to the speed difference . ${\ displaystyle t}$ ${\ displaystyle s_ {0}}$ ${\ displaystyle t}$ ${\ displaystyle v_ {a} -v_ {s}}$

[3] According to Peter Janich: "Achilles and the Tortoise", in: Jürgen Mittelstraß (Ed.): Enzyklopädie Philosophie und Wissenschaftstheorie , Volume 1, Metzler Stuttgart 1995, reprint 2004, page 41, ISBN 3-476-02012-6

[7] VI, 9,239b14-240a18 in the formulation that “even the slowest animal in the course cannot be overtaken by the fastest, since the pursuer always has to get to where the fleeing animal has run away from, so that the slower one always has a head start keep ".

[8] Original Greek text in Aristotle: Physics. (see section 4 of the screen). Archived from the original on May 16, 2008 ; Retrieved October 16, 2013 .

[9] Simplicius: On Aristotle's Physics 1014,10, see: Readings in Ancient Greek Philosophy From Thales to Aristotle, ed. SM Chohen / P. Curd / CDC Reeve, Indianapolis / Cambridge: Hackett 1995, 58f

[10] Mind 1 (1895), pp. 278-280.

[11] see, for example, Pascal Engel: Dummett, Achilles and the tortoise , In: L. Hahn / R. Auxier (eds.): The philosophy of Michael Dummett (Library of Living philosophers) , La Salle, Ill .: Open Court 2005 .