stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2006-11-11 05:11 pm
stoutfellow) wrote2006-11-11 05:11 pm
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Ramble, Part 4: What Euclid Didn't Do
In the last Ramble post, I pointed out three major effects of Euclid's work on the history of mathematics. On each point, though, I'd like to offer some caveats, under the cut.
The first point had to do with the axiom-theorem-proof model of mathematics. It is true that Euclid established the standard in this respect, but it also has to be admitted that he fell short of that standard, both in conception and in execution. On the one hand, though he did realize the necessity of starting with unproven axioms, he did not break through to the other half - the need to start with undefined terms. He can't really be faulted for this, though; that realization didn't come until the mid-nineteenth century, and had enormous consequences, as we'll see when we get there. On the other hand, there were flaws in how he carried out the program. Specifically, some of his proofs are logically flawed - they do not follow, in full, from his axioms. For example, many of them rely on the points of intersection between a circle and a line, or between two circles. Unfortunately, though his axioms do establish conditions under which lines intersect each other, they do not do the same with regard to circles. This flaw, and others like it, were not corrected until the late nineteenth century; there were many other things which had to happen before that became possible.
The second point involved the subordination of arithmetic to geometry. Though this was true for the most part, there were mathematicians, most importantly Diophantus of Alexandria, who studied numbers in their own right, without reference to geometry. His work was strongly influenced by the Pythagoreans, and some of his methods can be traced back to ancient Mesopotamia. I'll have more to say about him later, but for the moment I'll just point out that his work did not follow the axiomatic pattern laid out by Euclid. Indeed, the axiomatic approach was not applied to algebra until, again, the nineteenth century.
The third point concerned the dominant position of straightedge-and-compass constructions - i.e., reliance on the line and the circle as fundamental curves. The classical mathematicians were certainly aware of other curves; Apollonius of Perga wrote a treatise on conic sections that's still worth study, and Archimedes wrote a treatise on spirals. Conic sections were used to solve some problems that couldn't be solved by straightedge and compass alone. (In algebraic terms, straightedge and compass can be used to add, subtract, multiply, divide, and extract square roots, but cube roots are out of reach; conic sections allow the extraction of cube roots. More on that later.) Still, the difficulty of incorporating them into Euclid's axiom system seems to have limited their use.
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The first point had to do with the axiom-theorem-proof model of mathematics. It is true that Euclid established the standard in this respect, but it also has to be admitted that he fell short of that standard, both in conception and in execution. On the one hand, though he did realize the necessity of starting with unproven axioms, he did not break through to the other half - the need to start with undefined terms. He can't really be faulted for this, though; that realization didn't come until the mid-nineteenth century, and had enormous consequences, as we'll see when we get there. On the other hand, there were flaws in how he carried out the program. Specifically, some of his proofs are logically flawed - they do not follow, in full, from his axioms. For example, many of them rely on the points of intersection between a circle and a line, or between two circles. Unfortunately, though his axioms do establish conditions under which lines intersect each other, they do not do the same with regard to circles. This flaw, and others like it, were not corrected until the late nineteenth century; there were many other things which had to happen before that became possible.
The second point involved the subordination of arithmetic to geometry. Though this was true for the most part, there were mathematicians, most importantly Diophantus of Alexandria, who studied numbers in their own right, without reference to geometry. His work was strongly influenced by the Pythagoreans, and some of his methods can be traced back to ancient Mesopotamia. I'll have more to say about him later, but for the moment I'll just point out that his work did not follow the axiomatic pattern laid out by Euclid. Indeed, the axiomatic approach was not applied to algebra until, again, the nineteenth century.
The third point concerned the dominant position of straightedge-and-compass constructions - i.e., reliance on the line and the circle as fundamental curves. The classical mathematicians were certainly aware of other curves; Apollonius of Perga wrote a treatise on conic sections that's still worth study, and Archimedes wrote a treatise on spirals. Conic sections were used to solve some problems that couldn't be solved by straightedge and compass alone. (In algebraic terms, straightedge and compass can be used to add, subtract, multiply, divide, and extract square roots, but cube roots are out of reach; conic sections allow the extraction of cube roots. More on that later.) Still, the difficulty of incorporating them into Euclid's axiom system seems to have limited their use.
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Ramble Contents
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Thanks again for doing these!
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