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I thought to avoid working with irrational numbers ancient Greeks introduced the concept of magnitudes. This allows mathematicians to do a rigorous mathematics based on geometry for 2000 years. However, what I do not understand is that how could they rigorously define things like addition and multiplication for magnitudes without relying on numbers. By rigorously I mean to only use the concept of magnitudes and nothing else.

For example for magnitude x, why was it acceptable to define x+x=2x? shouldn't 2 in this case be defined in terms of magnitudes?

abk
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  • What do you mean by "define $x+x=2x$"? Certainly you will find no statement written in this notation in Euclid. And what is this equation meant to be a definition of, exactly? – Eric Wofsey Apr 03 '17 at 04:24
  • @EricWofsey, Thanks! Yes you are absolutely right. This is meant to be the definition of adding a magnitude by itself. I am trying to make sense of the definitions given here (http://aleph0.clarku.edu/~djoyce/java/elements/bookV/defV5.html) but I do not understand how the concept of numbers is freely used in these definitions. – abk Apr 03 '17 at 04:40
  • What on earth makes you think that the ancient Greeks avoided working with irrational numbers? Have you considered actually reading sources like Euclid's Elements rather than asking ill-informed questions on MSE about Greek mathematics? If you did, you would find that Euclid devotes all of Book X to incommensurate magnitudes: i.e., magnitudes whose ratio is irrational. – Rob Arthan Apr 03 '17 at 20:07
  • @RobArthan, there's no need to be rude. The whole point of this site is for people to ask questions. – mweiss Apr 03 '17 at 21:27
  • @mweiss: to whom have I been rude? The question begins with an incorrect statement and my comment to abk is correcting that statement. My comment to abk is a little sarcastic, but it is not rude. – Rob Arthan Apr 03 '17 at 21:36
  • @RobArthan Thanks for the comment. I mention that this is what I thought about mathematics of ancient Greeks. I haven't read the entire Euclid's elements. I am just trying to make sense of some definitions given in there. As you pointed out Euclid treats irrational numbers as incommensurate magnitudes. So irrational numbers weren't treated as numbers but as magnitudes (avoid treating them as numbers). I was just wondering how other algebraic operations were defined using magnitudes without relying on the number concepts. – abk Apr 03 '17 at 23:05
  • @abk: as regards Euclid's Elements, the "reading plan" in my answer should only involve a few pages for you to read. – Rob Arthan Apr 03 '17 at 23:24

2 Answers2

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Magnitude is not really a defined term in the first few books of Euclid, despite the fact that the concept is used informally beginning with the very first proposition of Book I; it is used to capture the informal notion of (variously) length, area, or volume. If $AB$ is a line segment, for example, then an expression like "twice $AB$" simply means "a line segment formed by adjoining two segments, each separately congruent to $AB$". If $ABCD$ is a rectangle then "twice $ABCD$" means "a rectangle formed by adjoining two rectangles, each separately congruent to $ABCD$". When Euclid says something like "rectangle $ABCD$ is equal to twice triangle $PQR$" he means that two identical copies of triangle $PQR$ have area equal to rectangle $ABCD$. Proving such claims does not require that magnitudes be interpreted as numerical quantities; it does, however, require some axioms about magnitudes (things like a whole being equal to the sum of its parts, for instance).

mweiss
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  • thanks that makes sense but shouldn't expressions like "twice AB" be well defined. These statements do not seem to be coming from Euclid's axioms of geometry. – abk Apr 03 '17 at 05:13
  • What do you mean by "well defined"? Euclid's Elements are not rigorous by modern standards. – mweiss Apr 03 '17 at 05:19
  • Right got it! Thanks. I thought he needed to have everything built up on his axioms, but probably they didn't have or need that level of rigor back then. – abk Apr 03 '17 at 05:25
  • He thought he was being rigorous. But whippersnappers since have been poking holes and pointing the rigor failures ever since. – fleablood Apr 03 '17 at 07:00
  • @mweiss: your answer is on the border-line between plain wrong and just badly mis-informed. Have you read the first few paragraphs of Book VI of Euclid's Elements? See my answer below for more information about why you really ought to read some more Euclid. – Rob Arthan Apr 03 '17 at 21:00
  • @abk: your conclusion from mweiss's answer is wrong. Rigour was extremely important to the Greek mathematicians. However, they were trying to formalise difficult concepts like number and magnitude. The approach they took shows deep and subtle insights into these concepts. – Rob Arthan Apr 03 '17 at 21:13
  • @RobArthan I am well aware of what is in the later books of Euclid, thank you very much. You have misunderstood my response, and (I think) the OP's question. Please see my comments on your answer. – mweiss Apr 03 '17 at 21:23
  • But are you are aware of what is in the lost works of Eudoxus, Theaetus or Archimedes or other ancient Greek mathematicians? The question is about ancient Greek mathematics and not just Euclid (even if the OP thinks that ancient Greek mathematics is just Euclid). To be read fairly and constructively, Euclid's Elements should be taken as evidence for the mathematical insights of the ancient Greeks and not as a source for trivial jibes. – Rob Arthan Apr 03 '17 at 22:08
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I am treating this as a reference request. The first reference you want to consult is Euclid's Elements. If you can get Heath's translation, then it contains some excellent commentary, but there are good versions online too, e.g., a nice translation by R. Fitzpatrick.

Particularly important to dispel your misunderstandings are:

Book V, which expounds Eudoxus's theory of proportion. This begins with what is effectively a relative definition of the concept of magnitude: a magnitude is something like a length or an area that you can add, subtract and compare geometrically. Particularly important is definition V.5, which tells you how to compare two ratios between arbitrary magnitudes. Dedekind's approach to defining the real numbers is a direct descendant of this definition.

Book VII, which is about elementary number theory. It begins with a definition of number, i.e., positive integer. I don't believe that numbers where intended to be thought of as magnitudes, but the theory of proportions also applies to numbers and so numbers and magnitudes can be compared. Euclid has no notion of algebraic expression, but he can say of magnitudes $x$ and $y$ that "$x$ is to $y$ as $1$ is to $2$", meaning $y = 2x$.

Book X, which is about incommensurable magnitudes, i.e., magnitudes $x$ and $y$ such that $x/y$ is irrational. It begins with a definition of Euclid's algorithm for arbitrary magnitudes. Theorem X.2 analyses the termination condition for this algorithm and concludes that it terminates on inputs $x$ and $y$ iff $x/y$ is rational.

You should then bear in mind that Euclid's Elements is not an enyclopaedic compendium of all ancient Greek mathematics. It was probably intended as something more like a graduate textbook. There are lots of good sources on the history of mathematics that you should look into if you are interested in the subject. The MacTutor page on Euclid could be a good place to start in this case.

Rob Arthan
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    While this is all correct, it is also correct that Euclid begins using the informal notions of "whole number multiple of..." (and the related notion of "unit fraction of..." in Book I. For example in I.9 we have "to cut a rectilinear angle in half", and in I.10 we have "to cut a straight line [segment] in half." At this point in the text the notion of "half" as a numerical quantity is not formally defined; from context, it clearly means "two parts that are equal", and yet "two" is not defined either. (contd) – mweiss Apr 03 '17 at 21:19
  • Likewise Prop I.13 refers to "angles that are equal to two right angles". Prop I.47 (the Pythagorean Theorem) speaks of a square being equal to the sum of two other squares. All of this precedes the formal definitions of proportion and number found in the later books. (contd) – mweiss Apr 03 '17 at 21:21
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    This kind of arithmetic of magnitudes seemed to me to be precisely what the OP was asking about, and what I was referring to in my response. – mweiss Apr 03 '17 at 21:22
  • @mweiss: your answer begins "Magnitude is not really a defined term in Euclid". That statement is hard to justify given the depth of the analysis of the concept of magnitude in the later books. It is grossly unfair to the Greek mathematicians to draw snap conclusions based on a reading of the first few books of Euclid with 2,000 and more years of hindsight. – Rob Arthan Apr 03 '17 at 21:32
  • How would you feel about "Magnitude is not really a defined term in the first few books of Euclid, despite the fact that the concept is used informally beginning with the very first proposition of Book I"? – mweiss Apr 03 '17 at 21:34
  • That would be better. However, the question is about Greek mathematics and not Euclid's Elements. Eudoxus preceded Euclid by a generation or two. Given the haphazard way that a minute fraction of the literature of the ancient world survived until today, making harsh judgments based on the ordering of the scrolls in the Elements is pretty unfair. And your judgment is harsh. – Rob Arthan Apr 03 '17 at 21:42
  • Here's another good one: II.8 asserts (Fitzpatrick's translation): "If a straight-line is cut at random, then four times the rectangle contained by the whole (straight-line), and one of the pieces (of the straight-line), plus the square on the remaining piece, is equal to the square described on the whole and the former piece, as on one (complete straight- line)." (emphasis added) "Four times", "plus" and "is equal" are all undefined terms. – mweiss Apr 03 '17 at 21:42
  • @mweiss: that is all irrelevant to a question that is about ancient Greek mathematics and not its presentation in Euclid's Elements (interpreted in your harsh way as if it were a 21st century homework exercise by a student that you were keen to fail), As we don't know what Eudoxus wrote, but we do have Euclid's Book V to tells us how his very beautiful theory of proportion worked, I think we can conclude that the ancient Greeks had a very sound idea of the notion of number and magnitude despite your quibbles about Euclid's presentation. – Rob Arthan Apr 03 '17 at 21:54
  • Where in the world are you getting the mistaken notion that my view of Euclid is harsh, or critical, or in any way pejorative? All I said was that it's not rigorous by modern standards, and I think that's obvious on its face. I never said it should be. – mweiss Apr 03 '17 at 22:50
  • "Here's another good one ..." doesn't read like an introduction to an unbiased criticism to me. It reads like "here you go, those idiot ancient Greeks got it wrong again". – Rob Arthan Apr 03 '17 at 23:03
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    @RobArthan Thanks for the references. – abk Apr 03 '17 at 23:09
  • No, it's more like: "Here's another good example that you are wrong in assuming that the formal view introduced in the later books of Euclid is used consistently even in elementary contexts when it is unnecessary." – mweiss Apr 03 '17 at 23:44
  • @mweiss: duh? I think you have lost the plot. I have made no assumptions about logically consistency between the books of Euclid's Elements and I am fully aware that the Elements contain many inconsistencies and non sequiturs. However, the Elements contain a huge amount of important mathematical insight which you want to dismiss because it's not been "properly written up" according to our modern standards. – Rob Arthan Apr 04 '17 at 00:03