Thesis on mathematical induction in ancient Greek mathematics is twofold. a) to record and evaluate that of scientists, especially Freudenthal (1953), Mueller (1981), Unguru (1991), Fowler (1994), Acerbi (2000), on the question of the presence and exact form of the deductive method. complete mathematical induction in Greek mathematics, and. We also study the basic medieval sources related to mathematical induction, including the Arabs Abu-Kamil, Al-Karaji, Al-Samaw'al and Ibn al-Haytham, the Franco-Jewish Levi Ben Gerson and Francisco Maurolico, of Greek origin. 2“unlike the principle of induction, the principle of least numbers or the denial of infinitely decreasing chains does not seem to depend on a genuine structural conception of the positive integers.
Because these principles can only be understood in terms of the Euclidean conception of numbers as finite relations of units". 6 «In my research on the history of mathematical induction, I came across an example of the "quasi-general" form of induction used by Abu Kamil Shuja' Ibn Aslam.
Πρόταση IX 8 και IX 9
Τα επιχειρήματα του Unguru κατά της παρουσίας μαθηματικής επαγωγής στην Πρόταση IX 8 και
And this, to my way of thinking, is not an example of a proof by mathematical induction, a method that does not need to be reinforced and supplemented by one. Mathematical induction, whose substance consists in the conclusion from n to n+1, is not an erroneous method of proof.
Κριτική στο επιχείρημα του Unguru για την απόδειξη της Πρότασης IX 9
Η έννοια του αριθμού στα Αρχαία Ελληνικά μαθηματικά κατά τον Jacob Klein
The "unity" as such is no arithmetic, a fact which seems strange only if we assume the idea of the "natural number series". The smallest number of things or units is: two things or units. It is only because this is the case that it has the character of a "beginning" or "source" that makes something of the nature of "counting" originally possible.» (page 48). Only a careful consideration of the fact that it is really necessary to suppose that there.
Thus an unlimited field of "pure" entities presents itself to the view of the "scientific." The problem of the "general" applicability of a method is therefore for the ancients the problem of the "generality" (of the mathematical objects themselves, and this problem they can only solve on the basis of an ontology of mathematical objects.
Το βασικό θεωρητικό επιχείρημα του Unguru κατά της δυνατότητος ύπαρξης μαθηματικής
Thus, we can say that the integers themselves are not designed in a structural way suitable for the use of induction. In this way, Mueller denies that Euclid could consciously use the KI principle, but he does not claim that it would be impossible for him to present evidence in the KI scheme. Proofs in the CI scheme that work on the number of terms involved in the statement rather than on the integers as such—even proofs that the steps A through D described above are verbalized and formalized in a way consistent with what we have the idea that the ontology underlying "ancient Greek mathematics" - seems to be within the realm of possibility."
9 “Proofs in the scheme of CI [=Complete Induction], working on the number of terms involved in a statement” (σελ. 64). 10 'On the other hand, there is other evidence in the arithmetic books that does suggest a conception of the idea of induction.
- Ορισμός της άπειρης αναδρομικής ακολουθίας των ανθυφαιρετικών υπολοίπων
- Ορισμός της άπειρης αναδρομικής ακολουθίας των πλευρικών-διαμετρικών αριθμών (Θέων,
- Κριτική του θεωρητικού επιχειρήματος Unguru στο Κεφάλαιο 3
- Παράρτημα
Mathematical induction is implicit in some of Euclid's proofs, for example in the proof that there are infinitely many prime numbers.». It is easy to see how some slight change in the mode of presentation or in the point of view would produce the modern mathematical induction. The repetitive modes of derivation of the Hindus and Greeks are more nearly the modern process of.
Needless to say, the concept of 'all natural numbers' is foreign to Greek mathematics." (page 278). 13 Unguru (1991, p.285) 'side and diameter numbers are a series of numbers obtained by a certain recursive relation'; (p.287) 'In fact, it is not at all difficult for us to see that the recursive relation between sn and dn is wonderfully provable by induction.'
- Ανακάλυψη πλευρικών διαμετρικών αριθμών και της Βασικής ιδιότητάς τους
- Η ιδέα των Πυθαγορείων για την απόδειξη της βασικής ιδιότητος με την μετατροπή της ρητής
- Η παρεμβολή του γεωμετρικού γλαφυρού θεωρήματος
- Το γλαφυρόν θεώρημα: η γεωμετρική μορφή του αναδρομικού ορισμού
- Η επινόηση της γεωμετρικής Πρότασης ΙΙ.10, αντίστοιχης της αριθμητικής ιδέας των
- Η απόδειξη του γλαφυρού θεωρήματος
- Η απόδειξη με επαγωγή της βασικής ιδιότητος των ρητών διαμέτρων
- Η αρχική ρητή διάμετρος δ 1 είναι -1 ελλείπουσα, 2,28,11-15
- Η δεύτερη ρητή διάμετρος δ2 είναι +1 πλεονάζουσα
These words clearly demonstrate Proclus's perception that the relationship between side and queen numbers he described was always valid, and they also underline the fact that there is not only no inductive proof here, but also no proof tout court.”
Πρόταση IX 20
Κριτική μελετητών για την απόδειξη της IX 20
Άπειρη αναδρομή και επαγωγή στην Πρόταση IX 20
Παρμενίδης 142b1-143a3
Παρμενίδης 149a4-c3
Συμπλίκιος Εις Φυσικά αποσπάσματα Β1, Β2, Β3 του Ζήνωνα
- Αποσπάσματα Β2, Β1-, B3- , B3+
- Η ερμηνεία του Β3+ ένεκα του παραρτήματος A
- A. Παράρτημα
- Το απόσπασμα Β1+ και η ανθυφαιρετική ερμηνεία αυτού, ανάλογη με το χωρίο 142d4-143a3
- A. Παράρτημα Η έννοια του «ἀπέχειν», «προέχειν» στο B1+, ως ελλιπές, πλεονάζον,
Ορισμοί του έργου Περί Ελίκων
Προτάσεις 10 και 11του έργου Περί Ελίκων
- Πρόταση 10
- Πρόταση 11
Προτάσεις 24 και 25του έργου Περί Ελίκων
Η Επαγωγική Πρόταση 27 του έργου Περί Ελίκων
Η Πρόταση 14
Οι Προτάσεις 15 και 16: λήμματα για την απόδειξη της Πρότασης 17
Πρόταση 17: επαγωγικό λήμμα για την απόδειξη της Πρότασης 18
Οι επαγωγικές Προτάσεις 18, 20 και παραλλαγές αυτών στο έργο Συναγωγή
The product of the three quantities, the number of terms plus one, increased by the number of terms, and the number of terms, divided by 6, [(n+1,2n+1.n)/6]. It seems that there were two schools in Arabia at that time which were opposed to each other in the sense that one favored Greek and the other Indian methods22. Alkarkhi23 was one of those who followed almost exclusively Greek models, and he has a proof of the theorem now in question by means of a figure with gnomons drawn in it, which provides an excellent example of the geometric algebra that is so characteristically Greek.
Abu Kamil
Αl-Karajī & Αl-Samaw’al
- Αναδρομικός ορισμός δυνάμεων στο Al-Fakhri
- Η μη επαγωγική απόδειξη της Αρχιμήδειας Πρότασης (12+22+…+n2): (1+2+…+n)= (2n+1)/3
- Η επαγωγική απόδειξη της Πρότασης (13+23+…+n3)= (1+2+…+n)2 κατά al-Karaji και al-
- Η επαγωγική απόδειξη της Αρχιμήδειας Πρότασης (12+22+…+n2): (1+2+…+n)= (2n+1)/3
- To διωνυμικό ανάπτυγμα
It is a good idea to see al-Samaw'al and al-Bahir. if we want to add the cubes of the number following each other from 1 [according to natural order] we multiply their sum by itself. The cube of a number is equal to the sum of its square and the double product of this number by the sum of the numbers following from one [according to natural order] to the predecessor of that same number." Για n=5 δεν αποδεικνύει το θεώρημα, αλλά γράφει: «he who has understood what we have just said can prove it for any divided number. in two parts its quadrato cube is equal to the sum of the quadrato cube of each of its parts, five times the product of each of its parts by the square of the other and ten. times the product of the square of each of them and the cube of the other.
-KarajI said that to succeed we must place 'one' on a table and 'one' below the first 'one', move the [first] 'one' to another column, add the [first] 'one' to the one below it. This shows it for each number. composed of two numbers, if we multiply each of them by itself once - since the two extremes are 'one' and 'one' - and if we multiply each by each other twice - since the middle term is two - we get the square of this number. If we then transfer the 'one' in the second column to another column, then the 'one' adds [from the second column].
This teaches us that the cube of any number composed of two numbers is given by the sum of the cubes of each and three times the product of each by the square of the other. If we transfer the 'one' again from the third column to the second column, and if we add 'one' to the 'three' below it [from the third column], we get 'four', which is written under the 'one'; if we then add 'three' under it, we get 6, which is written under 'four'; if we then add another 'three' to that 'one' below it, we get 'four', which is written under 'six', and then put 'one' under 'four'; the result is another column with the numbers 'one', 'four', 6, 'four' and 'one'. This teaches us that the square of a number made up of two numbers is given by the square of each - since we have a 'one' at each end - then by the product of four times each number by the cube of the other - since the 'four' follows the 'one' at both ends - because the root multiplied by the cube is square-square, and finally six times the product of the square of each by the square of the other - because the product of the square by the square is square-square.
Then if we transfer the 'one' from the fourth column to the fifth column and add the 'one' to the 'four' below it, the 'four' to the 'six', the 'six' to the 'four' and the 'four' to the 'one', then we write the results under the carried 'one' in the manner indicated and finally we write the remaining 'one', we have a fifth column whose numbers are 'one', 5, 'ten', 'ten', 5 and 'one'. This teaches us that for any number divided into two parts, its square cube is equal to the square cube of each part - because both ends have 'one' and one - five times the product of each by the square of the square of the other - because 'five' succeeds at both ends on both sides, and six times the product of the square of each by the cube. second - because 'ten' follows every five. Each of these terms belongs to the series of a square cube, for the product of the square root and the product of the square cube gives a square cube; thus we can find the number of squares and cubes to the desired power.».
- Προτάσεις 26, 27,28
- Πρόταση 29
- Πρόταση 38. Η μη επαγωγική απόδειξη της Αρχιμήδειας Πρότασης (2n+1)Sn=3(n2+(n-
- Προτάσεις 41& 42. Η επαγωγική απόδειξη της Πρότασης 13+23+….n3=(1+2+…+n)2=Sn2 κατά
- Προτάσεις 63, 64 & 65. Η πρωτότυπη επαγωγική απόδειξη της Πρότασης Pn=n!, όπου Pn είναι το
Now the permutation of the third order is as much as the product of 5 by 6 by 7, since the excess of 7 over 2 is equal to 5. These factors number three and are in the order of whole numbers, the last being 7. Similarly, it is shown that the number of permutations of the fourth of the same order as the compound from 4.5.6.7 and similarly it was also shown for any number..».
Προτάσεις 4 και 6 του έργου Arithmeticorum libri duo
Προτάσεις 13 και 15 του έργου Arithmeticorum libri duo
Francisco Maurolicos erstes Buch der Arithmetik, geschrieben 1557 und gedruckt 1575: ein Schritt in Richtung einer Zahlentheorie. Lange, Gerson. (1909) Sefer Maassei Chosheb – Die Praxis des Rechners, Ein hebräisches Rechenwerk von Levi ben Gerschom aus dem Jahr 1321, Frankfurt am Main: Louis Golde.