Jon, List, All the commentary, quotations, and citations below by both of us are irrelevant to mathematical practice from ancient times to the present. Following is a definitive statement of mathematical practice from Euclid to the present:
In mathematics, the distinction between axioms, postulates, and theorems is stated by metalanguage. That is a convenience for classifying, citing, and finding them. But during any proof or reasoning of any kind, all relevant statements are placed in a single common area (paper, blackboard, whiteboard, computer storage, or phemic sheet). The metalanguage that distinguishes axioms, postulates, and theorems is useful information for finding and citing the sources in a published proof. But during a proof, the axioms, postulates, and previously proved theorems are copied into the current workspace (or phemic sheet). They are used in exactly the same way, independent of their source or what they may be called. This is not a debatable opinion. It is a fact that can be verified by looking at any proof in any well-edited textbook of any branch of mathematics from the ancient past to the latest and greatest. John PS: I majored in mathematics and related topics (logic and physics) at MIT and Harvard. I later taught, lectured, and published books and articles on mathematics, logic, computer science, and computational linguistics. The above facts are indisputable. Peirce knew them very well. He would not make a careless mistake about them. ---------------------------------------- From: "Jon Alan Schmidt" <[email protected]> Sent: 3/23/24 10:51 PM John, List: JFS: I am happy to say that I completely agree with Jon's note below. However, the following passage from another note is misleading about Peirce, Euclid, and mathematical practice from ancient times to the present. The quoted passage is from my same note below. JFS: In mathematical texts, it's common to say "Given A1, A2, A3..., it follows THAT T1, T2, T3... where the A's are axioms, and the T's are theorems that follow from the axioms. Note the word 'that'. It is a sign of METALANGUAGE, between two clauses of a sentence. It is not a sign of implication. The word 'follows' or more precisely 'my be proved from' indicate the steps of a proof. "Given A, it follows that T" is logically equivalent to "if A, then T," which is logically equivalent to "A implies T." A is the premiss or antecedent (e.g., postulates), and T is the conclusion or consequent (e.g., theorems). There is no need for metalanguage to express this in EGs since it is represented by a scroll or nested cuts or a ring-shaped shaded area--including a sheet with a red line drawn just inside its edges (or a shaded margin). JFS: As for the notations in R514 and L376, Peirce made another distinction: postulates are propositions on which the utterer and the interpreter agree. The choice of postulates is the result of an AGREEMENT between the utterer and the interpreter. The results inside the red line are the result of an INVESTIGATION that may be far more complex than an mathematical proof. In R 514, Peirce only states that "in the margin outside the red line, whatever is scribed is merely asserted to be possible. Thus, if the subject were geometry, I could write in that margin the postulates, and any pertinent problems stated in the form of postulates." He does not say anything about "an agreement between the utterer and the interpreter," nor about the "results inside the red line" before the text breaks off in mid-sentence. In R L376, Peirce does not say anything about "the red line" nor "an investigation," complex or otherwise. The "agreement between the utterer and the interpreter" is on the subject of the graphs scribed on each piece of paper, which represents a portion of the overall universe of discourse that "is before the common attention" of both parties at one time or another. He only discusses postulates in the paragraph right before the section on "The Phemic Sheet," and only for the purpose of explaining why "any valid deductive conclusion" is not "instantaneously evident upon an examination of the premisses." Here is what he says. CSP: My second reason is found in the peculiar character of mathematical postulates. These pronounce that certain things are possible. But these possibles are not, of course, single things, for a single thing must be more or less than possible: they embrace whole infinite series of infinite series of objects in each postulate; and it is upon the statement of the possibility of one single one of those objects or single one for each set of certain others, that some essential part of the conclusion is founded. How many demonstrations, for example, and very simple ones too, as mathematics goes, depend, each of them, upon the possibility of a single straight line; while this possibility is only asserted in the postulate that there is, or may be, a straight line through any two points of space. In that statement the possibility of every single straight line in space is asserted, including the single one whose existence is pertinent and concerning which a similar postulate directly or mediately asserts something which is an essential ingredient of the conclusion. Consistent with R 514, postulates "pronounce that certain things are possible." Moreover, the only kind of investigation that Peirce discusses here is a mathematical demonstration. JFS: The complexity of the investigation is the reason why Delta graphs are a completely new branch of EGs. Again, Peirce's only stated reason for needing "to add a Delta part" to EGs is "in order to deal with modals"--not for metalanguage, and not for complex investigations. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Sat, Mar 23, 2024 at 4:46 PM John F Sowa <[email protected]> wrote: Jerry, Jon, List, JLRC: If the critical concept that is under scrutiny here the issue of “graphs of graphs” , how is this related to the arithmetical notion of division? I agree with Jon's explanation below that Peirce did not use the word "division" to mean the numerical operation of dividing two numbers. He was talking about dividing different parts of a text. As for the phrase "graph of graphs", that excerpt occurred in the introductory paragraphs of Lecture V of Peirce's Lowell lectures of 1903. Immediately before that, he used the synonym "graphs about graphs''. Since the word 'metalanguage' had not yet been introduced in English, the phrase "graphs about graphs" is his best and clearest term. It he had used his Greek, he might have coined the word 'metagraph'. I am happy to say that I completely agree with Jon's note below. However, the following passage from another note is misleading about Peirce, Euclid, and mathematical practice from ancient times to the present. JAS: The "red pencil" notation (1909) is entirely different from this--a red line is drawn just inside the physical edge of the sheet, and postulates are written in the resulting margin. These are not propositions about the propositions written inside the red line (metalanguage), they are premisses (antecedent) from which the propositions written inside the red line follow necessarily as deductive conclusions (consequent). For example, if the EGs for Euclid's five postulates are scribed in the margin, then they can be iterated to the interior, where the EGs for all the theorems of Euclidean geometry can be derived from them in accordance with the usual permissions. It's true that postulates are iterated (or copied) during the process of proving a theorem. But it's also possible to iterate a statement from a that-clause of metalanguage to a collection of statements that are being discussed in other ways. In mathematical texts, it's common to say "Given A1, A2, A3..., it follows THAT T1, T2, T3... where the A's are axioms, and the T's are theorems that follow from the axioms. Note the word 'that'. It is a sign of METALANGUAGE, between two clauses of a sentence. It is not a sign of implication. The word 'follows' or more precisely 'my be proved from' indicate the steps of a proof. As for the notations in R514 and L376, Peirce made another distinction: postulates are propositions on which the utterer and the interpreter agree. The choice of postulates is the result of an AGREEMENT between the utterer and the interpreter. The results inside the red line are the result of an INVESTIGATION that may be far more complex than an mathematical proof. The complexity of the investigation is the reason why Delta graphs are a completely new branch of EGs. In summary, metalanguage is the "secrete sauce" that makes Gamma graphs a third branch. But investigation makes Delta graphs the fourth branch. That difference is very important. John
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