http://www-history.mcs.st-andrews.ac.uk/Extras/Russell_Euclid.html

It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not always define, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid's earlier proofs fail before this test.

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http://www-history.mcs.st-andrews.ac.uk/Extras/Russell_Euclid.html

It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not always define, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid's earlier proofs fail before this test.

I shall examine Russel on these points.

it seems extremely harsh to criticise Euclid in the way that Russell does. As someone once said, Euclid's main fault in Russell's eyes is that he hadn't read the work of Russell.

The human ego of russel must always be borne in mind. What better way to establish his credibility than to discredit Euclid. However Russel was not attacking Euclid, but the mathematical establishment. Both He and Whitehead, inspired by German criticism and critical analyses set out on a crusade to reform British mathematics and bring it into line with the rest of europe. In so doing they belittled the english contribution to mathematics, considering it to be in comparison to germanic and european development, miniscule and stultified in a backwater.

While russel succeeded in shaking british mathematics to its foundations, by and large Whitehead and he were not able to demonstrate successfully the fruitfulness of their mathematical notions. In the main their contribution has led to unnecessary abstractness and emptiness of content disguised as high mathematical logic, or foundational mathematical concepts. Whitehead, at the last became silent on the successfulness of his approach in his latter days, as if defeated by the magnitude of the difficulties he had brought into being.

Axiomatics. Everything came to swivel around this "axis", but no one drives the points home! Euclid "Itemised" a set of demands of his students: They had to be able to draw very accurately, and they had to be able to recognise dual right angles as "one", and two angles "less than right", when constructed on a single "good" line with the same inside space "between" them they had to extend to meet away from the parts of the "good" line with the two "less than right" angles.

Meeting these demands meant you qualified to take the course , These were not axioms, but entrance requirements, however they give the lie to the axiomatic interpretation of Euclid's Stoikeioon.

These Aitema or demands were the basis of practical skills for using the notions in engineering, architecture, surveying,construction of military weapons etc. For example surveying and trigonometry are not possible without demand number 5. Why it became confused with a parallel postulate perhaps god only knows. Clearly Euclid defines "parallel" in the definitions. And do not be misled into thinking that "Non Euclidean" geometry is an unusual geometry. Realise that "Non Euclidean" geometry represents a reinventing of the wheel, as many "non flat" geometries existed before Lobachevski and Bolyai and Gauss.Western Mathematicians will hardly admit it, but there is no "non Euclidean" geometry.It is one of those human quirks that we often reinvent the wheel in ignorance or arrogance or both.

And {if} between two good lines a single good line is implanted in the innersides, and on these self/same parts of the innersides, angles, two that are less than right, are formed, (you must be able to)extend the two good lines as long as it takes to bring them together, extending away from those parts where the two angles less than right are

What follows these demands are the common inferences and judgements about magnitudes and comparison of magnitudes. these are not all the inferences because Euclid was not writing an axiomatic treatise for highbrow "russelians" but a pragmatic introduction to greek philosophical thinking.

The plan that is so admired in Euclid, particularly in book 1 was a devotional reflection of the work of the Goddess Isis.Thus it was not logical consistency that motivated Euclid but systemic and systematic connectedness.

The most obvious fact about the Stoikeioon is that it is not a book about Geometry. It is clear that it is a book about space, and it is a book about dynamic space. However it is also only an introductory book. The clue is in the Title: Stoikeioon.

As an introductory text it of course sets things out in an ordered way starting out simply and building up systematically to the more complex. Its subject matter is Pythagorean Theurgy as Plato understood it. Its modus operandi: synthesise that which has been exhaustively analysed philosophically. That which the Pythagoreans exhaustively analysed was form, and the principal form was the Sphere.

The sphere was in fact an idealisation of the Ouranos, the heavenly gods, and in fact it was what we may now call a fractal relationship with the ideal god, The Monad. Thus a Kosmological philosophy was what The Stoikeioon was serving as an introductory course to.

The many Henadic relationships that were common in the Pythagorean Theurgy al had their own symbola and sunthemata. The common sunthemata are the internal and external relationships between forms and the forms are in fact the sumbola. Even this simplified explanation belies the fractal complexity of the relationships in Pythagorean Theurgy and greel philosophical and religious thought. Clearly any form may be a sumbola. In addition a sumbola may be a sumbola of another sumbola, and so on. Thus the most amazing complexity is at all times possible. Further: the Stoikeioon is purposely set out on the basis of a graphical skill. But each sumbola also has a phonetic of musical association, an auditory "symbolism" if you follow. Thus a theme tune or a musical phrase, as well as a spoken word are all analogously included in this Pythagorean approach.

Pause for a moment, and think: if writing a book you cannot hear a sound reference, you can only see a graphical figure which may vicariously elicit the intended sound. Thus a triangle is a well known form, and yet you cannot see one or even the one i am referring to. What you can see are the symbol "triangle", and within that the symbols"t,r,i,a,n,g,l,e". And within those symbols you may see the distinguishing forms of each of them, the colour,and he sequence of construction. In addition you may internally voice some sound associated to each distinction you have just made.

To assume that this level of sophisticated thought was not available to our forefathers is the reason why we do not understand even the simplest of their Sophias!

The Stoikeioon is therefore an introductory text into all of the fractal complexity of Pythagorean Theurgy.

This claim, however, vanishes on a close inspection. His definitions do not always define, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious.

In this point, therefore Russel shows himself to be ignorant or "provocative".

The first proposition assumes that the circles used in the construction intersect – an assumption not noticed by Euclid because of the dangerous habit of using a figure. We require as a lemma, before the construction can be known to succeed, the following:

If A and B be any two given points, there is at least one point C whose distances from A and B are both equal to AB.

Russel is of course speaking ignorantly or hoping to persuade the ignorant. The word does not mean stupid, it means "not knowing". What is it that is to be known of Euclid, either by Russel or by his audience? Simply that Euclid does not start his book with the propositions, but with his definitions. His definitions cover the notion of a seemeioon and a gramme and a Epiphaneia, and a kuklos. Thus it is possible to deduce the lemma from these definitions easily. Alternatively one may just draw a diagram.

Most people would see the lemma and not understand it, but would understand the drawing or figure. Most people would understand that what was being communicated was the same thing, and in fact even more than the same thing. Most would say that the diagram was easier to apprehend than the spoken or written explanation or lemma; and if elegance means the powerful communication of an idea in simple apprehensible form, that the figure was more elegant and more aesthetic. Most would say that Russel was being pedantic and a pedagogue.

WE have to understand where Russel is coming from, particularly when he says that written coded text retains its demonstrative force over a figure. Clearly there are many cases where this is simply not the case! Thus if one cannot read the assertion fails, and if one is blind and reading through braille the assertion fails. Russel is guilty of assuming that his profound abilities are the benchmark for all other intelligent communication, and believing the written word had a biblical authority as the repository of "truth". The slippery nature of words written or spoken he believes can be overcome by rigorous definition, but in fact his lifelong work with his friend and Mentor A.N. Whitehead demonstrated the fallacy of this notion. It turns out that a well drawn figure retains its communication content better than any word description.

The fourth proposition is a tissue of nonsense. Superposition is a logically worthless device; for if our triangles are spatial, not material, there is a logical contradiction in the notion of moving them, while if they are material, they cannot be perfectly rigid, and when superposed they are certain to be slightly deformed from the shape they had before. What is presupposed, if anything analogous to Euclid's proof is to be retained, is the following very complicated axiom:

Given a triangle ABC and a straight line DE, there are two triangles, one on either side of DE, having their vertices at D, and one side along DE, and equal in all respects to the triangle ABC.

(This axiom presupposes the definition of the two sides of a line, for which see below.) When the existence of a triangle thus equal in all respects to ABC is assured, we can prove that the triangle considered in the fourth proposition is this triangle.……………………………..This is an example of Russel's sophistry, and his use of presupposition to persuade.

When i started, i had a bee in my Bonnet about Russel, and i have apologised about the mistaken things i believed about his work. Similarly i was told Euclid was teaching geometry, which he clearly was not. It is important therefore to retain an open mind and an analytical one at that to be able to distinguish fact from fiction. That is not the same as saying there is a Truth and a Lie, because those words propagandise,and the proper apprehension of ones experience is obscured by their use. A lie is not the opposite or the contra of a truth and vice versa. If one determines that something is not true, it still remains to determine whether it is a lie. And in doing so we enter a course of defining referents to the terms which take them away from any practical use.

Many more general criticisms might be passed on Euclid's methods, and on his conception of Geometry; but the above definite fallacies seem sufficient to show that the value of his work as a masterpiece of logic has been very grossly exaggerated.

B RUSSELL

How this reads aptly if you exchange Euclid and Bertrand Russel.:jester: