## (2,4): The Book of Numbers ### Or: Peano Arithmetic - Show Peano using backwards c for implication ("inverse contains"), upside down semicolons, and upside down square root for power. - By volume 5, Peano is now writing in Latin instead of French. ![[history-of-logic-notation-20.png]] ![[history-of-logic-notation-21.png]] ![[history-of-logic-notation-22.png]] ![[history-of-logic-notation-23.png]] ![[history-of-logic-notation-24.png]] ![[history-of-logic-notation-25.png]] ## Frege - Frege's symbolism is called "repulsive" and it's said that he basically influenced nobody except Bertrand Russell. - Wonderful quote from Frege about how to convince oneself that no intuition has accidentally ended up in an argument. The original desire of mechanistic reasoning. - Frege's notation (or the history of logic notation section in general) is a perfect way to introduce the idea of the "Anti-Gist." There would be little value in trudging through every one of these books, but skimming a "best of" or "greatest hits" that _shows the technical details_ while giving less technical commentary is a much better way to summarize an entire era, subject, or author's writing style than simple describing the same things in the form of the standard "gist" in natural language. ![[history-of-logic-notation-17.png]] ![[history-of-logic-notation-18.png]] ![[history-of-logic-notation-19.png]] ![[history-of-logic-notation-26.png]] ## Russell and Whitehead - The quote from Russell and Whitehead here is fantastic. - Include the line about "Paternal Aunt is the Relative Product of Sister and Father." - Two lines below this they mention that "Peano and Frege showed that the class whose only member is x is not identical with x." The fact that they feel the need to CITE someone on this point, let alone someone so recent, really shows how _informal_ mathematics had been for the several thousand years of its history until then. - At the top of 311 they mention that "types" are introduced to avoid contradictions. - On 312, below the boobs, the thousand page book on the History of Mathematical Notations says of Russel and Whitehead's Principia Mathematica that: "We don't have enough space to list all the notations in this book." (LOL) - On page 314, in the final page of the Logic section in the book, the author mentions that no topic in mathematics comes close to logic in terms of its ability to provide a universal language, no group has a harder notation problem to solve, and no group has thought about that problem more deeply. This fact continues today into the culture of programming, where the need to maintain large codebases which actually execute (unlike mathematics in a textbook) has led to intuitions about what constitutes a good notation that go so far beyond mathematics that it's scarcely possible to compare the two. Perhaps the most illustrative example is the fact that it was Unix, not mathematics, that finally had the sense to write function composition in the right order: f | g. ![[history-of-logic-notation-27.png]] ![[history-of-logic-notation-28.png]] ![[history-of-logic-notation-29.png]] ![[history-of-logic-notation-30.png]] ![[history-of-logic-notation-31.png]] ![[history-of-logic-notation-32.png]] ![[history-of-logic-notation-33.png]] ![[history-of-logic-notation-34.png]] ![[history-of-logic-notation-35.png]] ![[history-of-logic-notation-36.png]] ![[history-of-logic-notation-37.png]] ![[history-of-logic-notation-38.png]] ![[principia-mathematica-01.png]] ![[principia-mathematica-02.png]] ![[principia-mathematica-03.png]] ![[principia-mathematica-04.png]] ![[principia-mathematica-05.png]] ![[principia-mathematica-06.png]] ![[principia-mathematica-07.png]] ![[principia-mathematica-08.png]] ![[principia-mathematica-09.png]] ![[principia-mathematica-10.png]] ![[principia-mathematica-11.png]] ![[principia-mathematica-12.png]] ![[principia-mathematica-13.png]] ## The Lost Generation 0: WWI vaporized a generation. All across Europe, but especially in Germany. After that, the center of logic moves from Germany to the US. | Name | Birth Year | Age in 1931 | Birth Country / Empire | Residence During WW I | Notes / Major Contributions | | -------------------------- | ---------- | ------------- | ---------------------------------------------- | -------------------------------- | -------------------------------------------- | | **Georg Cantor** | 1845 | ✝ (Dead 1918) | Russian Empire (St Petersburg – German family) | Germany | Set theory, transfinite numbers | | **Gottlob Frege** | 1848 | ✝ (Dead 1925) | Germany | Germany | Predicate logic, quantifiers | | **Giuseppe Peano** | 1858 | 73 | Italy | Italy | Peano axioms, symbolic logic | | **Alfred North Whitehead** | 1861 | 70 | England | England | *Principia Mathematica* (with Russell) | | **David Hilbert** | 1862 | 69 | Germany | Germany | Formalism, Hilbert program | | **Ernst Zermelo** | 1871 | 60 | Germany | Germany | Zermelo set theory, axiom of choice | | **Bertrand Russell** | 1872 | 59 | England | England | Type theory, analytic philosophy | | **Leopold Löwenheim** | 1878 | 53 | Germany | Germany (teacher during WW I) | First model-theoretic results | | **L. E. J. Brouwer** | 1881 | 50 | Netherlands | Netherlands (neutral) | Intuitionism | | **Thoralf Skolem** | 1887 | 44 | Norway | Norway (neutral) | Löwenheim–Skolem theorem | | **Paul Bernays** | 1888 | 43 | Germany (left in 1912 before WWI) | Zurich, Switzerland (neutral) | Hilbert collaborator | | **Howard Aiken** | 1900 | 31 | USA | USA | Harvard Mark I electromechanical computer | | **Haskell Curry** | 1900 | 31 | USA | USA | Combinatory logic | | **Alfred Tarski** | 1901 | 30 | Poland (Russian Empire) | Poland | Model theory, semantics of truth | | **John von Neumann** | 1903 | 28 | Hungary (Austria-Hungary) | Hungary | Set theory, later computing architecture | | **Alonzo Church** | 1903 | 28 | USA | USA | Lambda calculus, logic | | **Kurt Gödel** | 1906 | 25 | Austria-Hungary (Brünn, now Czechia) | Austria-Hungary / Czechoslovakia | Completeness, incompleteness | | **Barkley Rosser** | 1907 | 24 | USA | USA | Kleene Rosser paradox, Church Rosser theorem | | **Stephen Cole Kleene** | 1909 | 22 | USA | USA | Recursive functions | | **Konrad Zuse** | 1910 | 21 | Germany | Germany | Z3 computer (1930s–40s) | | **Alan Turing** | 1912 | 19 | England | England | Turing machines, computability | | **Claude Shannon** | 1916 | 15 | USA | USA | Information theory, Boolean circuits | Exceptions that prove the rule: | Name | Birth Year | Age in 1931 | Birth Country / Empire | Residence During WW I | Notes / Major Contributions | | --------------------- | ---------- | ----------- | ----------------------- | --------------------------------------- | ----------------------------- | | **Emil Post** | 1897 | 34 | Poland (Russian Empire) | USA (New York) | Recursion, Post problem | | **Arend Heyting** | 1898 | 33 | Netherlands (neutral) | Netherlands (neutral) | Formalized intuitionism | Emil Post was born Feb 11th, 1897, so the first and second rounds of US conscription for WWI just missed his age group by 6 months. Further, he was missing his left arm from a car crash at age 12 in 1909. Heyting was the right age too, but Netherlands was neutral and did not enter WWI. First round of US conscription grabbed anyone born between 1887 and 1896. Second got everyone born after June 5, 1897. > The first, on June 5, 1917, was for all men between the ages of 21 and 30. · The second, on June 5, 1918, registered those who attained age 21 after June 5, 1917. > Cantor retired in 1913, and lived in poverty and suffered malnourishment during World War I. > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge > In 1954 Dummett studied the transcriptions of Frege's Nachlass that had survived the Second World War, including fragments of a 1924 diary. Dummett, an anti-racism activist as well as a Frege scholar, later recounted how he had been deeply shocked to discover from this that the man he had "revered" as "an absolutely rational man" was, at the end of his life, a 'virulent anti-Semite' of "extreme right-wing opinions". > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge Note that Frege had lived in Germany during the first world war, and these notes were taken during the inter-war period. ## Judgement Stroke 0: Now, my dearest one, after all of the above, do you--- 1: Yes. For the love of god yes. 0: What question did you think I was about to ask? 1: Creation story. I see the point. Holy hell that was a lot. 0: Good, because--- 1: You don't need to convince me! I got it. I mean the horse logic alone probably would have convinced me. 0: In summary? 1: I can summarize. 0: Go ahead. 1: In summary, a creation story isn't the worst thing in the world. Actual history is worse. History is literally everything that's ever happened. If I learned anything from this file, it's that bibles probably wouldn't be so popular if they started with everything that ever happened. I mean Christ! Who wants to hear that? 0: Oh c'mon, you had fun in this file. 1: I know, I know. But still. Please no more history without a narrative. It hurts. 0: No more horse logic? 1: No more horse logic. 0: Any suggestions on where we should go next? 1: Let's get back to the creation story. 0: Say the magic words. 1: Please? 0: Not magical enough. _(Narrator: 1 flips back through the current file for a moment.)_ 1: Here. ![[magic-words-frege.png]] 0: Still not magic enough. 1: I hate you. _(Narrator: 1 leaves, heading in the direction of the the previous file, before returning with something in hand.)_ 0: What do you have to say for yourself? 1: The most magical words I know. 0: I'm listening. 1: ![[magic-words-horse-logic.png]] _(Narrator: 0 and 1 are immediately swept away to the next file, as if by magic.)_ goto: [[lost+found/2/5]]