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: anenunciation, anexposition, aspecification, aconstruction, aproof, and aconclusion. Of theseenunciationstates what is given and what is being sought from it, a perfect enunciation consists of both these parts. Theexpositiontakes separately what is given and prepares it in advance for use in the investigation. Thespecificationtakes separately the thing that is sought and makes clear precisely what it is. Theconstructionadds what is lacking in the given for finding what is sought. Theproofdraws the proposed inference by reasoning scientifically from the propositions that have been admitted. Theconclusionreverts 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

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

, 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).expositionFor the

, 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.specificationFor the

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

, 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.proofFor the

, 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.conclusion

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 compactProof:

LetFbe a closed subset of compact spaceTand {F_{α}} be an arbitrary centered system of closed subsets of subspaceF⊂T. Then everyF_{α}is also closed inT, and hence {F_{α}} is a centered system of closed sets inT. Therefore ∩F_{α}= ∅. By Theorem 1 it follows thatFis 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:]

LetFbe a closed subset of compact spaceT[specification: absent].

[construction:]

[Let] {F_{α}} [be] an arbitrary centered system of closed subsets of subspaceF⊂T.[proof :] [E]very

F_{α}is also closed inT, and hence {F_{α}} is a centered system of closed sets inT. Therefore ∩F_{α}=∅. By Theorem 1 it follows thatFis compact.[conclusion: absent ].

The absent specification can be formulated as follows:

"I say thatFis a compact space"while the absent conclusion is supposed to be a literal repetition of the enunciation of this theorem.

> It might be useful if anyone ever

ReplyDelete> 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.

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).

ReplyDeleteIn 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)...

Thank you for mentioning our book, Abraham Lincoln and the Structure of Reason. If you would like more information about the book, including an excerpt, or its author, please check at http://tinyurl.com/243wl3q.

ReplyDeleteSavas Beatie LLC

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

ReplyDeleteTake 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