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Topic: [HM] Irrational numbers
Replies: 7   Last Post: Apr 5, 2004 12:14 PM

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Julio Gonzalez Cabillon

Posts: 1,353
Registered: 12/3/04
Re: [HM] Irrational numbers
Posted: Apr 4, 2004 9:17 PM
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Dear All,

I thought that the following old exchanges might be of some interest
now, especially after Yakov Kupitz' latest proposal. I've taken the
liberty of editing some postings for the benefit of interested readers.
Although some exchanges could be deemed as too naive or shallow, the
whole bunch is worth the time, I think. Further reactions are, as
always, most welcome. Enjoy!

All the best, Julio GC


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Date: Wed, 31 Jul 1996 14:11:20 -0400
From: Kathy Davis <davis@fireant.ma.utexas.edu>

This post is a solicitation of advice/assistance/references, on
the early history of the use of the word logos. The target period
would be Pythagorean or pre-Pythagorean.

The issue arises from a reading of Taisbak & Szabo, conjectural
reconstructions of the path through which 'logos' came to mean
ratio. Szabo gives several references and translations in which
logos is used as "in the ranks" "belonging to the ranks" etc, etc.
Taisbak (Division and Logos) of course reconstructs Euclidean number
theory (Book VII), conjecturing that it ultimately derived from
tables, like multiplication tables. In Taisbak's reconstruction
it is even more tempting to translate the phrase "in the same logos"
as meaning, roughly, "occur together in a list".

So much for background; the question is whether there is any evidence
for jumping to such a conclusion (I should add that neither Szabo nor
Taisbak make such a leap).

Evidence would be early use of the word logos to refer to lists or
organized sets of numerical data. One would expect to find this in
account books (cf the secondary meaning of logos: account) or
inventory lists.

Does anyone know of such usages, or have a good reference to the
structure of early Greek accounting practices?

Thanks

Kathy Davis
Mathematics
University of Texas at Austin

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Date: Mon, 5 Aug 1996 12:59:07 -0400
From: John Conway <conway@math.Princeton.EDU>

The Greek verb "legein" and its Latin cognate "legere" both have
this idea of "listing" among their many meanings.

The Latin is better known to most English speakers, so I'll start
there - it means to gather , assemble, choose, and read, roughly
in that order of sense-development. We see it simply in the English words
"collect", "select", "elect", "lecture", and in many more complex forms
(eg., in "intellect", which comes from "choosing between").

The Greek verb has an overlapping but not identical set of meanings,
to gather, count or recount, speak, reason; and most of these carry
over to the noun "logos". We see its meanings of "words", "study",
"reason", "calculate" in an enormous number of English words, for
example "prologue", "physiology", "logic", "logistics".

Any good etymological dictionary will furnish many more examples -
my favorite is Partridge's "Origins", which has often been my bedtime
reading!

John Conway

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Date: Thu, 8 Aug 1996 00:51:03 -0400
From: Julio Gonzalez Cabillon <jgc@adinet.com.uy>

The primary sense of the Greek verb "legein" contains the idea of the
English verbs 'gather', '(re)collect' or 'select'. One gathers,
collects or selects 'words' in the same fashion one 'reads' (Lat. legere,
It. leggere, Germ. lesen, Sp. leer). In that fashion, one gathers the
"reason", the meaning of a speech or discourse. The "Logos" appears
like a 'harvest', as a result of a particular 'selection'.
That 'selection' should be 'intelligible'. Our power of mind to reason
("reri", "ratus sum")-- our 'intellect' -- distinguish us from other
objects.
It seems that ancient Greeks (Pythagoreans and possibly pre-Pythagoreans)
did not stop there.

On the one hand, Heracleitus, for instance, put forward the idea of
"Logos" as a cosmic reason that in ancient Greek philosophy provided
order and form to the world.

"Listening not to me but the Logos it is wise to agree that all things
are one."

On the other hand, as it seems, Pythagorean school (e.g. Philolaus) put
forward the idea that "all things which can be known have number; since
it is not possible that anything can either be conceived or known
without number."

Aristotle saw in the Pythagorean doctrine that "not only all things do
possess numbers, but also all things are numbers".

The relation between "Logos" and "Numbers" seems clear. "Number" was a
cosmic reason to the Pythagorean period.

We know that the stoics borrowed the word Logos from Heracleitus, but
as Cornford points out "no ancient stoic could have told you how much
meaning it had lost or gained in the two centuries between Heracleitus
and Zeno. He would certainly assume that it meant to H just what it
meant to him. We may be sure that this is not true; but we, who think
in English, have to contend with the further difficulty, that no modern
word covers more than a fraction of the meaning that 'logos' had for
either Heracleitus or the Stoic."

The translation of *abstract* words into, and out of, the Greek and
Latin languages has always shown insurmountable difficulties, since such
words have the particular bad habit of shifting their significances.
This peculiarity does not seem to occur often with names of *concrete*
objects.

We all know that 'geometry' and 'geodesy' meant different things. The
same apply to 'arithmetic' and 'logistic', although both subjects
involve numbers.

According to Plato, 'arithmetic' is the science of numbers--very close
to our Number Theory. So "arithmus" means 'number' from an abstract
viewpoint.
The art of calculation (quite close to our basic arithmetic) was Plato's
'logistic'. Logistic included the so-called Greek and Egyptian methods
in multiplication and division.

I never saw the Greek verb "legein", or its Latin cognate, "legere",
meaning "listing".

A few references (apropos of Kathy's query):

1) Christian Marinus Taisbak: "Division and Logos: A Theory of Equivalent
Couples and Sets of Integers", Acta Historica Scientiarum Naturalium et
Medicinalium, 25. 129 pages. 1971. Odense University Press,
ISBN 87-7492-045-6.

2) Szabo Arpad: "Existentia, studia philosophorum" Budapest, Societas
philosophia classica, 858 pages, Wirtschaftswissenschaftliche Beitraege,
1994.

3) Szabo Arpad: "Anfaenge der griechischen Mathematik", Oldenbourg,
Munich, 1969. ("The beginnings of Greek mathematics" in English / Szabo
Arpad, translated from the German by A. M. Ungar., Hingham, Mass.:
D. Reidel Pub., 358 pages, 1978. Series: Synthese historical library;
v. 17 ISBN:9027708193. Includes bibliographical references and indexes).

4) Szabo Arpad: "Die Entfaltung der griechischen Mathematik", Mannheim:
B.I. Wissenschaftsverlag, 471 pages, c1994. Series: Lehrbeucher und
Monographien zur Didaktik der Mathematik; Band 26, ISBN: 3411169214.
Includes bibliographical references (pages 459-465) and indexes.

5) Szabo Arpad: "Die Fruehgriechische Proportionenlehre im Spiegel
ihrer Terminologie", AHES, vol. 2, pp. 197-270.

Julio Gonzalez Cabillon

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Date: Wed, 14 Aug 1996 11:34:02 -0400
From: Kathy Davis <davis@fireant.ma.utexas.edu>

The original question was whether anyone knew of evidence to support
Taisbak's conjecture that the word 'logos' came to mean 'ratio' because of
the structure of Greek tables of products.

Several people have replied, suggested references, etc. What I've been
able to track down is very largely negative. Some aspects (primarily
Sumerian) seem to have interest, hence this note.

Several suggested papyrological research; two articles of David Fowler in
Zeitschrift fur Papyrologie und Epigraphic give references where
mathematical tables may be found. (I can't resist remarking how nice a
resource this is, and how very well done). The most recent article is
105(1955) 225-228, with references to the earlier article.

The early multiplication tables I've seen are not arranged in a
'matrix' form; instead what we would write as separate columns are run
together and continued into other columns, though the boundary between
series of products is demarcated by a horizontal line. (significance
problematic; cf discussion of Sumerian accounting techniques and the word
NIGIN, below)

Accounting as a source is, I think, a complete wash-out. Those accounts
I've seen are not structured in any way that suggests a visual patterning.
This is consistent with remarks in the literature; cf G.E.M. de Ste.
Croix's article, Greek and Roman Accounting, in the book 'Studies in the
History of Accounting', edited by A.C. Litleton and B. S. Yamey. See also
very brief discussion in Rosalind Thomas, 'Literacy and Orality in Ancient
Greece'.

One suggestion was to do a bit more etymological research. I used mostly
Liddell and Scott's Greek Lexicon, as well as Julius Pokorny's
IndoGermanisches Etymologisches Worterbuch. See also Herbert Boeder: Der
Fruhgriechische Wortgebrauch von Logos und Aletheia, ....

One way to approach the etymological problem is to ask: why there are two
words -- logos and arithmos -- with very similar meaning in early (Odyssey)
Greek. (noting here that by the time of Plato, the meanings were
distinct). Might one of the meanings be a foreign loanword, or loan
concept?

Logos derives from lego, to pick up, select, choose, deriving in turn from
IndoEuropean legh, 'zusammenlesen, sammeln' probably derived from lie down.
(Pokorny p658). Arithmos, in contrast, comes from the Greek ararisko, to
join or fit together, with IE root ar, to fasten (Pokorny p61).This word can
probably be taken back a bit further, but needs a better linguist than I.

There is, interestingly enough, a possible Babylonian cognate for logos:
lequ^ (von Soden or Civil et al), 'to pick up an object, to take an object
with one's hands', and related meanings. As far as I can tell, however,
there are no mathematical usages, and the standard references (E. Masson,
Les plus anciens emprunts semitiques en greq, also P. Chantraine, and te
reviews of O. Szemerenyi) give no indication that these words are
connected. Nice Try, tho.

So far, dead end. There are, however, some suggestive Sumerian and
Babylonian words.

First, a nice overview-summary in Jens Hoyrup, "Algebra and Naive Geometry.
An Investigation of Some Basic Aspects of Old Babylonian Mathematical Thought
I', Altorientalische Forschungen 17(1990) 27-69. Hoyrup gives two basic
Babylonian terms for sum; one of which is kamarum, 'to accumulate', and, in
Civil et al, kama-ru, to heap up, to spread (dates for sorting) to add, to
accumulate; kummuru, to heap up, pile up; kumurru^, sum, total, sorting of
the date harvest, building for storing dates (I couldn't resist).

Besides these meanings, Hoyrup suggests that we are to understand the word
in the sense that the number representing the sum of a collection of
objects is to be taken as representing the collection itself. Thus it
ought to be no surprise to see the word logos referring to a collection and
a sum.

Actually, even an amateur can trace the history back, thanks to the text of
Hans J. Nissen, Peter Damerow and Robert K. Englund, 'Archaic Bookkeeping,
Writing and Techniques of Economic Administration in the Ancient Near East'.

Very early tablets -- Uruk IV, 3100 BC -- might have a list of
entries one one side; the reverse side would then have the total. In later
tablets (Uruk III, 3000 BC) the total appears in a little boxed-off area of
its own. Finally, old Sumerian (2500-2250) starts to show a cuneiform
symbol, which Nissen et al translate as 'together' appearing with the
total. The Sumerian is S^U-NIGIN: the cuneiform for hand, followed by a
cute little square drawn of four wedges. A little reference work
(Friedrich Delitzsch, Sumerisches Glossar) gives us u.a.,

NIGI: a) rings umschleissen, d) Vereinigung, gesamtheit, equivalent to
napharu, puhhurum. These appear in von Soden as, bzW: "Gesamtheit, Summe"
and "versammelt". One is also invited to vgl. u. II, that is,
NIGIN = kummu, which presumably brings us back to Babylonian kamarum.

I think one has to be a little cautious here; the collecting or closing off
referred to here could conceivably have been done on a counting board, and
only copied onto a clay tablet. Still, it is a promising start.

Even more interesting is the Sumerian/Babylonian term for the accounting
tablet itself. Nissen et. al. translate the word as 'account'; the
Sumerian is S^ID, possibly derived from SANGA, logogram for an accounting
tablet (a cute little picture of a tablet ruled off into rows and columns).
The derivation isn't so important as the Babylonian meanings; again, from
Delitzsch: zahlen, zuzahlen, and the Babylonian manu^. Back to Civil et
al,
manu^tu: currency, standard
mana^tu: accounting manu^: mina one
sixteenth of a talent (currency); one third a shekel; unit of time (cf
minute)
manu^: to count; to count and list individual objects; to recite,
to recount events, and other meanings.
mani-tu: normal size of an object,
normal strength or measure, (both seem to refer to divination) account,
number, length, accounting; sum of all parts of the body: limbs; body;
shape, size; proportions
mi-nu: number, amount; accounting
minu^tu: amount; contingent of soldiers; standard of coinage; string of beads;
counting; recitation.


Aren't you a bit misleading here when you shoot for a
"translation," where the very word "measure" is itself fraught with
ambiguity?

A)/logos for Pythagoras seems to have been situated in the context of
"irresolvable into counting numbers." "Seems," to mean means "Caution:
Niagara Falls around the next turn!" There are many other contexts for
the usage in Plato, sometimes with the word, sometimes only
implicitly, e.g., my favorite joke at the beginning of the Statesman.
Then we also have Timaeus' irresolvable triangles, and we have the
Athenian Stranger in the Nomoi with his supposition that his partners
believe that "any line can be measured with any other line, each
surface can be measured with any other surface, and any solid....,"
etc., and then proclaims it ain't so, 'cause there are
incommensurables. -- That is a long way from "uncountable," and it is
also a peculiar notion of "measure," because the "measure" will not be
expressible in units of any kind: here measure would seem to have to
do with symmetry or geometrical similarity, but extended to the
"line," and then inherited across the two higher dimensions, that is
problematic also: it would seem that geometrical similarity would
apply to any line, but the Stranger evidently doesn't think so, or he
is not speaking of what we would think of by geometrical similarity.
The Stranger is dealing with a "measure" which encompasses all three
dimensions, in which incommensurability has some common feature in
each of them, so that's *weird*! Theatetus (I believe this is in
Euclid somewhere) introduced the notion of "expressible (or
"rational") in the square" for first-order incommensurables like the
hypotenuse, and reserved the notion of a)/logos for higher-order
incommensurables. In Euclid, things get hairy in bks X, XI, but there
are contexts in which incommensurable means "irresolvable by Eudoxian
exhaustion."

So, my vote goes for "no translation possible, and it wouldn't help if
there were one by someone's version of an historical dictionary." Very
careful attention has to be paid to the span of thinkers in which this
first became an explosive problem, and, to the extent that there are
"primary source" testimonies (i.e., so that one need not exclusively
rely on Aristotle's gossip), one would have to examine the context of
the employment in each thinker. -- At that time, at least, this was
not a problem of generating a consensus of definition, so there can be
no such consensus which we might seek to translate.

Regards
Emile Nowis

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From: Luigi Borzacchini <gibi@pascal.dm.uniba.it>
Subject: [HM] Music and Incommensurability
Date: Mon, 12 Jul 1999 11:39:54 +0200


Dear HM-list members,

I would like to propose for this hot summer a question concerning the
earlier theory of proportions and commensurability, more precisely the
respective roles of music and geometry in the discovery of
incommensurability.
The problem of the role played by music in the discovery was already
addressed at the end of the last century by P. Tannery. However, usually the
genesis of the theory of the incommensurable magnitudes is placed in
geometric questions (diagonal and side of the square, pentagon, golden
section, etc.).
In the following I try to give the main arguments about the topic, and even
to raise some foundational issues from the historical analysis. .

1. SZABO'S ANALYSIS

The question was raised by Arpad Szabo in the II part of his "The beginnings
of Greek Mathematics" (D.Reidel, '74), where he sets out to show how the
pre-Eudoxian theory of proportions initially took place in the Pythagorean
theory of music.
He supported this thesis with a deep analysis of the Greek technical terms
of the theory (diasteeme, oroi, analogon, logos, etc.) and their recognition
in the supposedly Pythagorean experimental practice of a string stretched
across a ruler, the socalled "canon", divided in 12 parts. Here the
Pythagoreans found three main consonances: the 'octave' (diplasion, 12:6),
the 'fourth' (epitriton, 12:9 or 8:6) and the 'fifth' (emiolion, 12:8 or
9:6).

This musical theory was explicitly connected by Architas with the means:
"Now there are three means in music...": arithmetic a-b=b-c, geometric
a:b=b:c, harmonic or subcontrary a-b:a = b-c:c. (DK 47 B2)
However the first and third can be easily recognized in the fourth (such as
C-F) and the fifth (such as C-G) consonances, whereas the geometric mean
does not correspond to any consonance on the "canon", for it would yield
12:(6*radical(2)) or (6*radical(2)):6. With the word of Szabo (174): "an
octave cannot be divided into two equal subintervals by a number". Remark
than here 'equal' (in consonance) means 'proportional' (on the canon's
intervals), according to the logarithmic relationship between lengths and
tones on a string in our "equal temperament".
I want to remark however that the geometric mean is the natural
relationship between the different octaves and hence seems even a sort of
natural consonance, and that Aristotle (Rose, fragm. 43) recognized instead
that only the arithmetical and the harmonic means were related to musical
harmony.

Szabo did not ascribe to the Pythagorean theory of music "more than a start"
(171): the full development of the theory of proportions had to be found in
the "geometrical arithmetic of the Pythagoreans", where a crucial role was
played by the definition of "similar (omoioi) plane numbers" as "those which
have their sides proportional (analogon)" (Euclides' Elements , def VII,21).
Szabo moreover conjectures that "the concepts of the musical theory of
proportions were applied first of all in arithmetic... Furthermore the
application of this theory to geometrical arithmetic contributed towards an
understanding of the problem of geometric similarity, and this problem in
turn soon led to the problem of linear incommensurability" (173-4).
This displacement of the problem from music to geometry, to be ascribed in
my opinion probably to Archytas himself, was even the rationale of the name
of the 'geometrical' mean, because it could produce no musical consonances,
whereas had precise and easy geometrical instances.

I think that probably it is wrong to look for the 'first' proof of the
incommensurability, even because the numerical 'fact' of some
incommensurable results (for example side and diagonal of the square) was
already known to the Babylonians, whereas a rigorous proof required instead
the introduction of the "absurd" proof in an earlier visual and constructive
mathematics, so that the theorem had to be proved together with the
establishment of its method of proof.
However it is of great relevance the hypothesis of a 'musical' background
for the discovery. And actually a 'negative' proposition, asserting the
incommensurability in musical and arithmetic terms, can be found in the
prop.3 of the "Sectio Canonis", ascribed to Euclid (Appendix).
The same proposition can be found in Boethius' "De Institutione Musica"
iii,11 (DK 47 A19), ascribed to Archytas: "No mean proportional number can
ever be found between two numbers in a ratio superparticularis". 'Ratio
superparticularis' is the 'epimorion diasteema (logos)', i.e. the ratio
n+1:n or mn+m:nm, for n,m integers (Appendix).
In such proof some theorems in the arithmetic books of Euclid (VII, VIII and
IX usually partially ascribed to the Pythagoreans) are implicit, such as
VIII, 20: "If one mean proportional number fall between two numbers, the
numbers will be similar plane numbers".
A possible reconstruction of the missing steps could be found if we
cut-and-paste of the proofs of the propositions VIII.9, VIII.11 and VIII.20,
to prove something like "If one mean proportional number fall between two
numbers, the least numbers of those which have the same ratio of these two
numbers must be square numbers", and furthermore it is easy to show that two
consecutive numbers can not be both square, and this completes the proof.
A similar and detailed reconstruction of the proof is in Knorr's "The
evolution of the Euclidean Elements" (VII.1).

For n=1 in the ratio superparticularis the above results prove the
inexistence of an integer x such that 2m:x=x:m and then such that
2:x/m=x/m:1.
This 'negative' result in Archytas' form derived probably by some 'evident'
impossibility, which was credibly progressively substituted by a rigorous
'proof by absurd' according to the Euclidean style, such as the spurious
X.117 of the Elements (Appendix).
Szabo's conclusion is that "from a historical point of view the Greeks
originally thought of the problem of irrationality as belonging to the
theory of proportions. I hope that part II has shown how the theory of
proportions, whose initial development took place in the Pythagorean theory
of music, may in fact have led by way of arithmetic to the problem of
geometrical similarity and thence to problem of linear commensurability...We
have not yet shown how they reached the stage of being able to prove the
existence of linear incommensurability in a rigorous manner" (181)

2. THE MUSICAL-LOGISTIC APPROACH TO INCOMMENSURABILITY

The final words of this remark pave the road to the III part of Szabo's
book, aimed to show the crucial role of the Eleatism for the embedding of
mathematics within a deductive framework. Even this last thesis is very well
argued, and it is not my goal to discuss it now.
I would like instead to stress the 'musical-arithmetical' origin of the
problem of incommensurability even in its theoretical establishment. More
precisely, I believe that at least in Archytas the core of mathematics was
in "logistic", which was not simply a "practical art of computation", but
the "science of the relationships between numbers", the "theory of the
logoi", including even a pre-Eudoxian theory of proportions about numbers,
theoretically solid enough to deal with the theoretical aspects of
incommensurability and give us at least the negative result. To this aim I
remark that music and logistic were considered the main Archytas' fields of
interest, and even fragment DK 47 b4 ("logistic seems superior to the other
sciences ... even to geometry") supports the same hypothesis.
With the words of Tannery about the Pythagoreans: "It is characteristic of
this tradition that it apprehends the numbers themselves directly in the
visible world, but their ratios in the audible world".

In order to underline the role of Archytas' logistic it is sufficient to
remind Klein's analysis ("Greek Mathematical Though and the origin of
Algebra", MIT Press '68, first german edition '34-'36) about the
relationship between arithmetic and logistic, which overcomes the common
idea of a distinction/opposition between (a purely practical) logistic and
(a purely theoretical) arithmetic.

The Archytas' arithmetical proof applied for n=1 can be consiederd as the
first step of a long evolution ending with the proof (of the
incommensurability of diagonal and side of a square) that can be found in
the proposition X.117 (Appendix) and in Aristotle (Anal. Priora I.23, 41a
26-7).
To this aim, I remark, following Knorr, a notational aspect common to
Boethius' fragment ascribed to Archytas, the Euclidean prop.3 of the Sectio
Canonis and the post-Euclidean proof of the incommensurability of side and
diagonal of the square (XII.117 of the Elements and so far the first
rigorous proof we know): the two incommensurable numbers/segments are not
homogeneously denoted, the larger is denoted by two letters, the smaller by
just one. The rationale in Boethius is clear: the greater number (or
numbered part of the Canon, according to Szabo) is the sum of the smaller
and one of its parts. In the Sectio Canonis and in Elements XII.117 the pair
of letters are the extreme of a number/interval and the single letter is a
number tout court: the first number/interval is divided during the proof in
two intervals by one point (and this use is not irritual in Euclides and can
be found for example in the proposition IX,35). The difference is that in
the former we deal with numbers represented as segments, in the latter the
starting point is explicitly geometric. However, the permanence of the
syntax with changing meanings is intriguing and seems to mark a historical
evolution which changed only the 'embedding' of the proof from
musical-arithmetical to geometrical form, preserving its basic arithmetic
structure, and whose origin can be found in the Boethius/Archytas fragment.

In addition, I underline that in the ancient Pythagorean philosophy, music
was something absolutely central, much more than geometrical metrical
constructions, which could appear in special problems as the duplication of
the cube or in the construction of solids, e.g. for astronomical modelling.
On the other side the "numerical atomism" never was a theory about some kind
of 'least magnitude', and the "indivisible lines" are ascribed by Aristotle
to Democritus or Plato or Xenocrates, whereas the idea of a 'least interval'
was common in the ancient music theories. Square, triangular and polygonal
numbers were studied from Philolaus to Diophantus without any trace of
'incommensurability' troubles, and Aristotle reminds us that Euritos drew
pictures of animals and plants as others did for triangles and squares
(Metaphysica 1092b): all of them were full of incommensurable magnitudes,
diagonal of squares and heights of equilateral triangles, but nobody
worried.

Music was instead the core of the Pythagorean philosophy until Plato. We can
find even in the Republica (424c) the echo of Damon's learning: "never
musical modes can be changed without changing the most important laws of the
polis". Music was the ground of the Platonic theory of education, and
appeared both in elementary (with gymnastic) and superior (part of the
Quadrivium) education. In Philolaus the same structure of kosmos' harmony
reflected musical consonances (DK 44,B 6).
Moreover, cutting the musical intervals by the geometric mean of the
superparticular ratios meant to find the way to connect those seven modes of
greek music (dorian, phrigian, lydian, etc) which were considered by the
Pythagoreans (and even by Plato) basic for the harmonic behaviour of the
citizen and the city as well. Plutarchus (de an. procr. in Tim. c17, 1020E)
reminds that the crucial problem was the division of the tone (9/8, i.e.the
interval between the fourth and the fifth) in teo 'equal', i.e.
'proportional', parts, and that the Pythagoreans discovered it to be
impossible because 9/8 is epimoric. Equivalently the octave could be divided
in 6 tones (according to Aristoxenus) or 5 tones and 2 not joinable
semitones (according to the Pythagoreans).
We can even remember the words of Plato in Republica 531 a-c against the
musicians who try to find the "least interval" by ear, "placing the ear
before the mind", and, at the opposite of the Pythagoreans, Aristoxenus'
defence of the musical practice to reveal the real consonances, and his
research of something like our "equal temperament" (without the relative
mathematics) to allow the connection in one framework of the different
modes: "there is no least interval" (Elem. Harm. II,46).
In Aristoxenus music had almost completely lost its Pythagorean awe and was
instead placed in the realm of the "aisthesis". Aristoxenus was proud of his
Pythagorean education, but his idea of science and music was thoroughly
Aristotelean: in Aristotle music had lost any ethic or cosmological flavour,
and was just a psychological tool for politics, for the pleasure it gave
(Politica, VIII).

3. NEGATIVE ARGUMENTS AND THE EPINOMIS' PASSAGE

The main reason for the oblivion of the 'musical' hypothesis about the
discovery of incommensurability is probably that in Plato and Aristotle there
seems to be no trace, whereas the reference, if any, is to geometrical
questions.
In particular, in Plato the proportions theory seems to belong to geometry.
However, I think we can proof that we can find in the Platonic Academy the
trace of this approach, with the tight connection between music, numeric
means and similarity, and without references to geometric figures, such as
square or pentagon. To this aim I give in Appendix a translation of a
passage in Plato's Epinomis (990d-991b), which is usually read with non
'technical' translations, so that 'omoios' is 'likening' or even sometimes
'commensurable', 'analogia' is 'analogy', and 'summetron' is 'proportioned'.
In my translation I employ instead the technical terms, respectively:
'similar', 'proportion', 'commensurable'. Epinomis is usually ascribed to
Philippus of Opus, a member of the Academy, approximately contemporary of
Aristotle and Eudoxus.

Knorr in "The evolution of the Euclidean Elements" gives another translation
of the passage about 'similarity' (pag.93 + note 109 pag. 107), whose
syntactic construction I follow (see appendix), but does not follow the
Euclidean terminology", justifying his choice by an analogous usage in
Thales and Nichomachus.
In addition he underevaluates the value of the passage because "most of the
mathematics Plato brings up in his dialogues does not correspond with the
style or the subject matter characteristic of Archytas' studies. Moreover,
Plato's contacts with Archytas were not such to have enabled a significant
influence on Plato's earlier conceptions" (89).
This is true. By and large Plato was well acquainted with the recent
technical results of geometric incommensurability, but his general approach
was more Philolaus-like than Archytas-like. In particular he did not seem to
appreciate the tendency to employ mathematics as representation and model of
physical reality, as done by Archytas (musical tone as speed of vibration)
and Eudoxus (cycles and epicycles in astronomy). He seems to explicitly
reject this approach in Republica 531c, where he criticizes those who in
astronomy and music analyze the numerical ratios without looking to the
'problems' and 'causes'.
However, I remark that Philippus of Opus was contemporary of Eudoxus who was
near to the Academy and credibly a scholar of Archytas, so that Archytas'
studies could have been well known in the Academy; on the other side he is
very far from Thales and Nichomachus: the acquaintance with Archytas in
Philippus could then be greater than in Plato.
Anyway the translation gets a very clear meaning if we assign to 'summetros'
and 'omoios' the Euclidean meaning (except in one case for 'omoios' where
for the context I incline toward a more general meaning) .
The passage is based on the reduction of the problem of commensurability to
the similarity and includes at the beginning the statement that (linear)
different numbers can not be similar and that similarity can be instead
recognized among planar numbers. The passage continues considering that in
stereometry we can make similar two numbers by two mean proportionals
(according to Archytas' algorithm, DK 47A 14). Then, after a (likely)
reference to the general role of dichotomy in the framework of dialectics,
the passage continues with a reference to the Pythagorean means on the
canon, and ends with the recognition that the musical harmony must be
restricted only to commensurable intervals.

If my translation is (quite) correct the above is the Platonic fragment most
influenced by Archytas and by the earlier theory of proportions we know, and
it is possible thereafter to conclude that in Plato's and Archytas' time the
strict geometric version of commensurability (as in Plato's Meno) played an
increasing role, probably justified by the geometric model of the concept of
similarity, whereas the earlier interpretation was still developed in the
musical and arithmetic theory of proportions, but was quickly vanishing
together with the main stream of Pythagorean philosophy.

In Boethius we can read the evolution of this musical road to the discovery
of incommensurability from Hippasus (DK 18,14) , where we find the beginning
of the analysis of the problem in terms of superparticular numbers, to the
development of the approach by a pre-eudoxian theory of proportions, in
Philolaus where we can find the statement of the problem in terms of
dicotomy of the tone, of the comma and of the diesis (DK 44B6), and finally
to Archytas when we read the discovery that such dicotomies are impossible
because all these intervals are superparticular (DK 47 A19). Other
fragments from Boethius (DK 44 A26) and Porphirius (DK 47 A17) reveal the
existence even of an earlier 'naive' purely numerologic approach, echoed
even in Plato's Timeus, where we can recognize the deep connections
Pythagoreans instituted between numbers, proportions, music, astronomy and
human knowledge.
In front of this rich Pythagorean musical tradition about the
incommensurability, we can find in the Pythagoreans' fragments no traces of
incommensurability deriving from geometrical metrical constructions on square
or pentagon, whereas in Plato and Aristotle the reference of
incommensurability becomes exclusively geometrical.
The "secret of the sect"? So well kept by Archytas on the geometrical side
and betrayed on the musical side, rigorous in Taras while at the same time
it was a favorite theme in Plato's Academy and Theodorus taught it in
Athens?
In my opinion we can instead explain this as revealing a sharp passage from
the musical to the geometrical framework on the line from Archytas to
Eudoxus, with the vanishing of the earlier approach. The translation was
easy because geometric similarity was well known, and the connection between
duplication of the square and the mean proportional between 1 and 2 was
known as well, if Hippocrates of Chios could reduce the duplication of the
cube to two mean proportionals and Archytas accomplished it geometrically.

Why this sharp passage? Sure, musical proof was only negative, whereas the
geometrical approach allowed the effective construction of incommensurable
magnitudes.
This argument is employed by Knorr to claim "that the harmonic theory of
irrationality was a derivative from the geometric theory rather than the
converse" (216). Frankly, it seems a non-sequitur. The argument would rather
explain the quick prevailing of the geometric approach on the arithmetical
one. Even because a purely negative result (speaking about "something which
is not") had to fall under the blows of the "negative judgement paradox", so
important on the line from Parmenides to the Sophists and Plato, and
widespread at the beginning of the IV century in Athens.
In addition, Knorr argues that some logical flaw could there be in Archytas'
proof. Maybe. Even more likely, some steps could have been proved bi simple
'evidence', to be then translated in proofs by absurd within a more rigorous
style. Anyway I stress that such a flaw was not a limit inherent in the
musical-arithmetical approach, because a rigorous proof could be built with
the techniques of the VIII book of the Elements, and without reference to
geometric similarity.

4. FOUNDATIONAL ISSUES

If this interpretation is valid, even if we accept Szabo's cautions and
Knorr's arguments, we must face two historical problems of great
foundational relevance.
The first is that incommensurability was not the result of simply technical
philosophical or mathematical problems, but rose instead from the basic
anthropological and political questions of the (not only Pythagorean, but
tout court) ancient Greek culture, where music and the relationship between
musical modes and state played a role that today for us is difficult to
understand, in any case were the essential ingredients of both society and
culture.
The second is that there was between Archytas' and Plato's schools a sharp
rupture causing a shift from musical to geometrical incommensurability, so
that we can find nothing about the geometric approach in the extant
Pythagorean fragments, nothing about the musical in Plato or Aristotle.
The break was maybe produced by the 'negative judgement paradox', but the
issue reminds the problem of the nature of the opposition between arithmetic
and geometry in ancient Greek mathematics. We must underline even that it
was perfectly fitted in the discrete/continuous opposition in the Quadrivium
(I remark even that the V book of the Elements is frequently referred in
the VI book, but never in the arithmetical VII, VIII, IX), so that
incommensurability sounded more as a great confirmation than as a
Grundlagenkrisis.

From the comparative historical point of view we can observe that Nicomachus
(DK 44 A24) refers that the musical proportions were introduced by
Pythagoras but their origin was Babylonian. Needham's (Science and
Civilization in China, IV, 177) opinion is that even Chinese musical theory
concerning pitches had the same origin, even if in China they were organized
not as a "scale", but as a "court" and a "spiral of fifths".
But the most intriguing comparison can be achieved when we consider even the
mathematical aspects. Chinese mathematicians did not experience the
opposition discrete/continuous typical of the Greek mathematics and,
regarding the problem of incommensurability, they were "neither attracted
nor perplexed by irrationals, if indeed they appreciated their separate
existence" (Needham, S&CiC, III,90). On the other side they discovered the
mathematical computation of the 'equal temperament' in 1584 (Needham, S&CiC,
IV, 220), whereas in the West the discovery had to wait for Stevin or
Mersenne.
From Greece to China music and astronomy, "sister sciences" faced with a
similar problem: to fit a set of regularities, basic for the whole social
and intellectual life, from calendars to the measuring systems and the
musical practice, within numerical schemes. Incommensurability was the greek
and european answer which grounded european mathematics on the Quadrivium
and on the opposition between discrete and continuous.
I should say that the opposition between discrete and continuous for two
thousand years created big troubles to the development of western science
(that Chinese culture avoided) until the construction of the real numbers;
but, when such construction was accomplished in the XVII century, it
triggered an epochal change that the Chinese science never experienced.
And in music the incommensurability implied a musical theory that will be
'bad' tempered until the XVII century, whereas musical practice was
constrained to require 'irrational' ratios: even Bach needed the emergence
of the real numbers' theory! But this is another story.

Appendices.

Boethius' "De Institutione Musica" iii,11.

Archytas' proof that a superparticular ratio cannot be divided into equal
parts.
A superparticular ratio [proportio] cannot be divided into equal parts by
the interpolation of a mean proportional number... Let A B be a
superparticular ratio; I take the least terms C, DE in the same ratio.
Since C DE are the least in the same ratio and are superparticulars, the
number DE exceeds the number C by a part of itself. Let this be D. I say
that D will not be a number, but a unit. For if the number D is also part of
DE, the number D measures DE; whence il measures the number E, so that it
also measures C. Thus the number D measures both of the numbers C and DE,
which is impossible. For those numbers which are the least in the same ratio
as any other numbers are relatively prime, and they have the unit as sole
difference. Thus D is the unit. Thus DE exceeds C by a unit. Whence between
them no mean number lies which cuts their ratio equally. Thus, neither
between those which have the same ratio can there be found a mean number
which cuts the same ratio equally.

prop. 3, "Sectio Canonis"

No mean number, neither one nor several, may be interpolated in proportion
[analogon] in an epimoric interval [diasteema].
Let BG be an epimoric interval, and let the least terms in the same ratio be
DZ, T. Then they are measured only by the unit as common measure. Let HZ
equal to T be subtracted, (and since DZ of T is epimoric) the excess DH is a
common measure of DZ and T; it is thus the unit. Whence, there may not be
interpolated between DZ, T any mean number; for the mean would have to be
less than DZ but greater than T, (so that the unit would be divided, which
is impossible. Thus, none may be interpolated between DZ, T). Now however
many numbers may be interpolated in proportion between the least terms, the
same number may be interpolated in proportion between those having the same
ratio. But no number may be interpolated between DZ, T; whence none may be
interpolated between B,G.

"Elements", X, 117

Let it be proposed to us to prove that in square figures the diameter is
incommensurable in length with the side.
Let ABCD be a square, of which AC is the diameter. I say that AC is
incommensurable in length with AB. <the figure includes a square ABCD with
the diagonal AC, and two segments, the first with extremes EZ and with the
letter T in the middle, the second with the letter H in the middle>.
For if possible, let it be commensurable. I say that it will follow that the
same number is odd and even. Now it is manifest that the square on AC is the
double of that on AB. Since AC is commensurable with AB, then AC will have
the ratio [logon] to AB of one number to another. Let these number be EZ and
H, and let them be the least numbers in this ratio. Then EZ is not a unit.
For if EZ is a unit and has the ratio to H which AC has to AB, and AC is
greater than AB, then EZ is greater than H, which is impossible. Thus EZ is
not a unit; hence it is a number. And since AC is to AB as EZ is to H, so
also the square on AC is to that on AB as the square of EZ is to that of H.
The square on AC is double that on AB, so the square of EZ is double that of
H. The square of EZ is thus an even number; thus EZ itself is even. For if
it were odd, the square on it would also be odd; since, if an odd number of
odd terms is summed, the whole is odd. So EZ is even. Let it be divided
[tetmeesthoo] in half by T. Since EZ and H are the lest numbers of those
having this ratio, they are relatively prime. And EZ is prime; so H is odd.
For if it were even, the dyad would measure EZ and H. For an even number has
a half part. Yet they are relatively prime; so this is impossible. Thus H is
not even; it is odd. Since EZ is double ET, the square of EZ is four times
the square of ET. But the square of EZ is the double of that of H; so the
square of H is double that of ET. So the square of H is even, and H is even
for the reasons already given. But it is also odd, which is impossible.
Hence, AC is incommensurable in length with AB. This was to be proved.


Epinomis' fragment (Loeb translation, some passages have two
translations: <Loeb translation>and [my translation]).

When he has learnt these things, there comes next after these what they call
by the very ridiculous name of geometry,
<when it proves to be a manifest likening of numbers not like one another by
nature in respect of the province of planes;>
[when it proves of numbers not being similar to one another by nature,
similitude can be manifest in respect of the province of planes;]
and this will be clearly seen by him who is able to understand it to be a
marvel not of human, but of divine origin.
<And then, after that, the numbers thrice increased and like to the solid
nature, and those again which have been made unlike, he likens by another
art,>
[And then , after that, the numbers thrice increased similar to the solid
nature, those which are not similar made similar by another art]
namely, that which its adepts called stereometry; and a divine and
marvellous thing it is to those who envisage it and reflect, how the whole
of nature is impressed with species and class according to each <analogy>
[proportion], as power and its opposite continually turn upon the double.
Thus the first <analogy> [proportion] is of the double in point of number,
passing from one to two in order of counting, and that which is according to
power is double; that which passes to the solid and tangible is likewise
again double, having proceeded from one to eight; but that of the double has
a mean, as much more than the less as it is less than the greater, while its
other mean exceeds and is exceeded by the same portion of the extremes
themselves. Between six and twelve comes the whole-and-a-half(9=6+3)and
whole-and-a-third(8=6+2): turning between these very two, to one side or the
other, this power(9)assigned to men an accordant and
<proportioned>[commensurable] use for the purpose of rhythm and harmony in
their pastimes, and has been assigned to the blessed dance of the Muses.

Luigi Borzacchini


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Date: Wed, 14 Jul 1999 08:54:04 +0100
From: David Fowler <david.fowler@maths.warwick.ac.uk>
Subject: Re: [HM] Music and Incommensurability


Dear HM members, and especially Luigi Borzacchini,

I would like to react to Luigi's very illuminating and well-organised
posting on Greek music theory and its general influence in Greek
mathematics (12/7/99), and suggest that, by playing down considerably one
important topic that runs through his argument, his description might then
be even more based on our direct sources of evidence, and may even show how
his material can then fit together in ways that could give different kinds
of insights. That topic is the role of incommensurability in the story -
or, rather, our feeling today about what that role should be. I am not in
any way denying that the early Greeks knew about incommensurability; the
issue is how big a driving-force its discovery played in their development
of mathematics. I offer, to begin with. just one little illustration of my
point of view: the Greeks tell us about this kind of issue, the creation
and understanding of mathematics, and of how important *problems* were in
it -- but nobody ever mentions incommensurability as posing such a problem.

I posted a couple of pieces about the evidence about incommensurability and
its role in early Greek mathematics some time ago (reference?), and more
can be found in my book The Math of Plato's Academy, 1st ed 1987, 2nd ed
just out, Oxford UP. To be more accurate, there is not much about
incommensurability in the 1st ed of the book before its very last section
because I try to discuss the interpretation of our surviving *early
evidence from what seem to be reliable sources*, but there is a bit more
discussion of this absence in the new material added to the 2nd ed, from
its first section that I propose should be treated as a new Introduction.

What I shall do is take a few extracts from Luigi's mail which contain
references to incommensurability, and comment on them. I identify where
they come from by numbering the paragraphs and quoting the first few words,
thus
1 >The question was raised by Arpad Szabo ...



3 >Szabo did not ascribe to the Pythagorean theory of music...

>Szabo moreover conjectures that "the concepts of the musical theory of
>proportions were applied first of all in arithmetic... Furthermore the
>application of this theory to geometrical arithmetic contributed towards an
>understanding of the problem of geometric similarity, and this problem in
>turn soon led to the problem of linear incommensurability" (173-4).


See the argument: we start from a '[Pythagorean] musical theory of
proportions ... in arithmetic' (we do have early Pythagorean examples of
consonant ratios, but not, I think, any coherent theory of proportion),
translate this into a theory of geometrical similarity (what is that? what
evidence do we have about it?), and then on to incommensurability. All is
conjecture, as Szabo says - plausible for some, less so for others, but
based on little or no tangible surviving evidence.


5 >For n=1 in the ratio superparticularis...

>Szabo's conclusion is that "from a historical point of view the Greeks
>originally thought of the problem of irrationality as belonging to the
>theory of proportions. I hope that part II has shown how the theory of
>proportions, whose initial development took place in the Pythagorean theory
>of music, may in fact have led by way of arithmetic to the problem of
>geometrical similarity and thence to problem of linear commensurability...We
>have not yet shown how they reached the stage of being able to prove the
>existence of linear incommensurability in a rigorous manner" (181)


We have no real evidence that "the Greeks originally thought of the problem
of irrationality as belonging to the theory of proportions" - none of our
early accounts of proportionality say anything about this.
A sophisticated illustration: Proclus, in his commentary on Elements I,
mentions incommensurability several times, and refers to Eudemus (a
contemporary of Eudoxus and author of the first history of mathematics, now
lost) several times, but he never mentions Eudemus saying anything about
incommensurability. And another similar example: our edition of the
surviving fragments of Eudoxus by Laserre does not seem to contain any of
the words connected with (in)commensurability: (a)summetros, (ar)rhetos,
alogos.. I offer these only to suggest that we should reflect on an
interpretation in terms of a proposal of a close relation between Greek
proportion theory and incommensurability, but with no view here to building
any theory on it.


6 >The final words of this remark...

>... including even a pre-Eudoxian theory of proportions about numbers,
>theoretically solid enough to deal with the theoretical aspects of
>incommensurability and give us at least the negative result. To this aim I
>remark that music and logistic were considered the main Archytas' fields of
>interest, and even fragment DK 47 b4 ("logistic seems superior to the other
>sciences ... even to geometry") supports the same hypothesis.
>With the words of Tannery about the Pythagoreans: "It is characteristic of
>this tradition that it apprehends the numbers themselves directly in the
>visible world, but their ratios in the audible world".


We have no surviving "pre-Eudoxian theory of proportions about numbers",
let alone one which is "theoretically solid enough to deal with the
theoretical aspects of incommensurability". The only surviving
pre-Euclidean mathematical remark about proportion/ratio is at Aristotle,
Topics 158-9.
(And, for a clever but far-fetched comment on Archytas & Plato,
proportionality, and politics, see Chapter 6, note 11 in my book.)



8 >The Archytas' arithmetical proof...

>To this aim, I remark, following Knorr, a notational aspect common to
>Boethius' fragment ascribed to Archytas, the Euclidean prop.3 of the Sectio
>Canonis and the post-Euclidean proof of the incommensurability of side and
>diagonal of the square (XII [read X].117 of the Elements and so far the *first
>rigorous proof we know*): the two incommensurable numbers/segments are not
>homogeneously denoted, the larger is denoted by two letters, the smaller...


Note that the texts under discussion here, the Sectio Canonis and Boethius'
De Institutione Musica, make no mention of (in)commensurability. Of course,
what they are doing is somehow connected with the topic, but their theory
is concerned with something else.
Knorr argues, convincingly to me, that this proof in X 117 (appended, out
of place, at the end of this very long Book X) was a later addition for the
benefit of Aristotlean commentators. The first such kind of proof is found
in Alexander of Aphrodisias, early 3rd century AD, writing on Prior
Analytics. Knorr also proposes that this proof had been transmitted down
from earlier times, but there is no further evidence for this.



9 >In addition, I underline that in the ancient Pythagorean philosophy, music
>was something absolutely central, much more than geometrical metrical
>constructions...


Just a remark that the Greek "geometrical metrical constructions" don't use
numbers in anything like the way that our geometry is now based on real
numbers. There is limited use of the arithmoi, 1, 2, 3,..., but their
geometry is mainly geometry.

> Aristotle reminds us that Euritos drew
>pictures of animals and plants as others did for triangles and squares
>(Metaphysica 1092b): all of them were full of incommensurable magnitudes,
>diagonal of squares and heigths of equilateral triangles, but nobody worried.


My point, indeed! Where do you find the *early* Greeks worrying about
incommensurability? It is the *late* commentators and others thereafter,
right up to today, who have problems.


13 >In Boethius we can read the evolution of this musical road to the discovery
>of incommensurability from Hippasus (DK 18,14) , where we find the beginning
>of the analysis of the problem in terms of superparticular numbers, to the
>development of the approach by a pre-eudoxian theory of proportions, in
>Philolaus...


Does Boethius talk about incommensurability and its development like this?
(I can't remember, and don't have a copy to hand.)
Pilolaus seems quite mathematical, but does he really present a pre-Eudoxan
theory of proportions? (Ditto.)



15 >If this interpretation is valid, even if we accept Szabo's cautions...

>The first is that incommensurability ...
>The second is that ...


It will need a lot of time for me to work out what to think of this
because, as it says, it depends on that opening statement: "If this
interpretation is valid..." What might happen to that interpretation if
that incommensurability issue is played down? A question to help pass the
long hot summer?

David Fowler

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From: Luigi Borzacchini <gibi@pascal.dm.uniba.it>
Date: Wed, 20 Sep 2000 09:07:54 +0200
Subject: [HM] Logos and ratio

I would like to add a point which could be interesting for HM: the
difference in the semantic fields of the Greek "logos" and the Latin
"ratio".
Even though they are supposed equivalent (ratio was the term for logos
in Latin), they stemmed from two different indoeuropean roots.
Logos, legein is from *leg (to collect, to pick up, to say), and had
no mathematical meaning in Homer, Hesiod and Greek Lyric.
At the opposite ratio, reor is from *ree (to count, to believe) was
first and foremost an arithmetical term and had no linguistic employment,
but got easily the meanings of reason, rationale, etc. (as in German
Rechnung, Recht). For the accountants, we must remember that in Latin
the accountant was the 'rationarius'.

This remark could be interesting to understand the evolution of the
concept of ratio from the Greek logos, which was not a number but a
relationship between magnitudes (or numbers), more a geometrical than
an arithmetical term, to the idea of rational number.
Probably this evolution was fostered in the Middle Ages by the
arithmetical nature of the Latin word ratio, so that it was naturally
in the hands of the Italian "ragionieri" and got easily a practical and
arithmetical meaning.

Luigi Borzacchini

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From: David Reed <dreed@math.duke.edu>
Date: Mon, 25 Sep 2000 20:33:30 -0400 (EDT)
Subject: [HM] Logos and ratio

Luigi Borzacchini makes some interesting remarks here regarding logos
and ratio. While ratio and logos have very different origins the
medieval philosophers put ratio together with oratio to mimic some of
the semantic breadth of logos. It is also interesting that while
proportion is a direct latinization of analogia, the Greek term has
little mathematical usage today (at least in English).

Corroborating Borzacchini's assertion that in Greek mathematics logos
is not a number,one sees in Euclid that ratios are not described as
"equal" but rather as "the same". Alogos or arhetos (unsayable) are
the words used for ratios which are not the same as ratios between
numbers.

On a related topic, there is a close parallel between the terminology
Aristotle uses in his Prior Analytics for what he calls "figures"
(schemata) in which syllogisms can be found (there are three) and the
terms used to describe proportions in Euclid which I have not seen
remarked anywhere. Aristotle's syllogisms have three terms (not major
and minor premises!!) two extremes and a middle...Euclid's proportions
have three or four terms (same word) and finding the "middle" term is
key to both. This points to a similarity in structure between
"syllogistic" reasoning and "analogical" reasoning, which are often
thought to be quite different from one another. Any references to
discussions of this would be gratefully received!

David Reed
Chapel Hill






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