Tuesday, October 18, 2011

Proclus on Euclid

Proclus wrote some commentaries on Euclid's Elements and found a pattern to the presentation of material:

Every Problem and every Theorem that is furnished with all its parts should contain the following elements: an enunciation, an exposition, a specification, a construction, a proof, and a conclusion. Of these enunciation states what is given and what is being sought from it, a perfect enunciation consists of both these parts. The exposition takes separately what is given and prepares it in advance for use in the investigation. The specification takes separately the thing that is sought and makes clear precisely what it is. The construction adds what is lacking in the given for finding what is sought. The proof draws the proposed inference by reasoning scientifically from the propositions that have been admitted. The conclusion reverts to the enunciation, confirming what has been proved.

Lincoln used this approach in his rhetoric, as Hirsch & Van Haften notes in their book Abraham Lincoln and the Structure of Reason. In an interview, they explained each step.

For the enunciation, think in terms of: Why are we here. It contains short, indisputable facts. They are part of the given. It also includes a sought. This is a high level statement of the general issue being discussed.

For the exposition, think in terms of: What do we need to know relating to what is given. These are additional facts, generally fairly simple, and indisputable. These facts take what was in the enunciation’s given, and prepare for use in the investigation (in the construction).

For the specification, think: What are we trying to prove. The specification is a more direct restatement of the enunciation’s sought. While the sought is frequently neutrally stated, the specification is a direct statement of the proposition to be proved.

For the construction, think: How do the facts lead to what is sought. The construction adds what is lacking in the given for finding what is sought.

For the proof, think in terms of: How does the admitted truth confirm the proposed inference. The proof draws the proposed inference by reasoning scientifically from the propositions that have been admitted.

For the conclusion, think: What has been proved. The conclusion reverts back to the enunciation confirming what has been proved. The conclusion should be straightforward, forceful, and generally short.

It might be useful if anyone ever goes into mathematics...or Law...

Addendum: Modern Mathematics

It seems that this format can be used in modern mathematics. Andrei Rodin's "Doing and Showing" (arXiv:1109.4298 [math.HO]) notes on page 25 how a modern theorem/proof can be formulated in the Euclidean tradition:

Theorem 3:
Any closed subset of a compact space is compact

Proof:
Let F be a closed subset of compact space T and {Fα} be an arbitrary centered system of closed subsets of subspace FT. Then every Fα is also closed in T, and hence {Fα} is a centered system of closed sets in T. Therefore ∩Fα = ∅. By Theorem 1 it follows that F is compact.

Although the above theorem is presented in the usual for today's mathematics form "proposition-proof", its Euclidean structure can be made explicit without re-interpretations and paraphrasing:

[enunciation:]
Any closed subset of a compact space is compact

[exposition:]
Let F be a closed subset of compact space T

[specification: absent].

[construction:]
[Let] {Fα} [be] an arbitrary centered system of closed subsets of subspace FT.

[proof :] [E]very Fα is also closed in T, and hence {Fα} is a centered system of closed sets in T. Therefore ∩Fα=∅. By Theorem 1 it follows that F is compact.

[conclusion: absent ].

The absent specification can be formulated as follows:

"I say that F is a compact space"

while the absent conclusion is supposed to be a literal repetition of the enunciation of this theorem.

4 comments:

  1. > It might be useful if anyone ever
    > goes into mathematics...or Law...

    Actually it is useful for anyone who needs to demonstrate or prove *anything*. Selling, persuading, convincing, speaking, writing -- you name it. This technique was nearly lost in the dustbin of history. Turns out, properly used, it is ironclad. In addition to Lincoln, Thomas Jefferson used it at least twice. Isaac Newton used it in The Principia. Anyone can use this technique with practice. It is elegant. Simple but textured.

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  2. Well, when applied to public speaking of any form, it does seem quite related to Cicero's 6 components of a speech (exordium, narratio, partitio, confirmatio, refutatio, peroratio).

    In fact, to play devil's advocate, Lincoln's Gettysburg address seems to be modeled more after Pericles' funeral oration than a geometric proof.

    Again, continuing as devil's advocate, in modern America I don't believe that a logical argument would hold water when an emotional argument is given (e.g., the whole "death panels" spin in the health care debate). Thoughts?

    But, being a mathematician, I'm more interested in examining *modern* mathematics in this light.

    I think tacitly mathematicians do it anyways...but consciously knowing about this pattern enables me to write mathematics better (at least stating theorems and doing proofs)...

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  4. Cicero and Euclid are different. Said another way, Cicero's six do not equal Euclid's.

    Take a look at:

    http://www.thestructureofreason.com/the-gettysburg-address/the-gettysburg-address-demarcated

    Notice the connection between the Sought, the Specification, and the Conclusion.

    Then notice the connection between the Given and the Exposition.

    Then observe how the Given and the Exposition lead to the Construction, setting up the Proof.

    Just like Euclid.

    -David Hirsch

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