#frege — Public Fediverse posts

Proposition 129, p. 83: If 𝐹 is a function and (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹, the successor of 𝐴 is either 𝐵 or it follows 𝐵 or it comes before 𝐵 in the #TransitiveClosure of 𝐹.

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝐴 ∈ dom 𝐹)

Hyp. ⊢ (𝜑 → 𝐶 = (𝐹‘𝐴))

Hyp. ⊢ (𝜑 → (𝐴(tc‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(tc‘𝐹)𝐴))

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐵(tc‘𝐹)𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶(tc‘𝐹)𝐵))

———

Proposition 131, p. 85: If 𝐹 is a function and 𝐴 contains all elements of 𝑈 and all elements before or after those elements of 𝑈 in the transitive closure of 𝐹, then the image under 𝐹 of 𝐴 is a subclass of 𝐴.

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((◡(tc‘𝐹) “ 𝑈) ∪ ((tc‘𝐹) “ 𝑈))))

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

———

Proposition 133, p. 86: If 𝐹 is a function and 𝐴 and 𝐵 both follow 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹 (or both if it loops).

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝑋(tc‘𝐹)𝐴)

Hyp. ⊢ (𝜑 → 𝑋(tc‘𝐹)𝐵)

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐴(tc‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(tc‘𝐹)𝐴))

———

So what's nice about the transitive closure that #Frege felt compelled to invent a new language in which to present mathematical arguments? When 𝑅 is a function, two sets being related by the transitive closure of 𝑅 is much like induction. When 𝑅 is a more general relation, we have a more general form of induction, that is truly #ancestral in the language of #Whitehead and #Russell.

Proposition 129, p. 83: If 𝐹 is a function and (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹, the successor of 𝐴 is either 𝐵 or it follows 𝐵 or it comes before 𝐵 in the #TransitiveClosure of 𝐹.

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝐴 ∈ dom 𝐹)

Hyp. ⊢ (𝜑 → 𝐶 = (𝐹‘𝐴))

Hyp. ⊢ (𝜑 → (𝐴(tc‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(tc‘𝐹)𝐴))

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐵(tc‘𝐹)𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶(tc‘𝐹)𝐵))

———

Proposition 131, p. 85: If 𝐹 is a function and 𝐴 contains all elements of 𝑈 and all elements before or after those elements of 𝑈 in the transitive closure of 𝐹, then the image under 𝐹 of 𝐴 is a subclass of 𝐴.

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((◡(tc‘𝐹) “ 𝑈) ∪ ((tc‘𝐹) “ 𝑈))))

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

———

Proposition 133, p. 86: If 𝐹 is a function and 𝐴 and 𝐵 both follow 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹 (or both if it loops).

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝑋(tc‘𝐹)𝐴)

Hyp. ⊢ (𝜑 → 𝑋(tc‘𝐹)𝐵)

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐴(tc‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(tc‘𝐹)𝐴))

———

So what's nice about the transitive closure that #Frege felt compelled to invent a new language in which to present mathematical arguments? When 𝑅 is a function, two sets being related by the transitive closure of 𝑅 is much like induction. When 𝑅 is a more general relation, we have a more general form of induction, that is truly #ancestral in the language of #Whitehead and #Russell.

Proposition 129, p. 83: If 𝐹 is a function and (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹, the successor of 𝐴 is either 𝐵 or it follows 𝐵 or it comes before 𝐵 in the #TransitiveClosure of 𝐹.

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝐴 ∈ dom 𝐹)

Hyp. ⊢ (𝜑 → 𝐶 = (𝐹‘𝐴))

Hyp. ⊢ (𝜑 → (𝐴(tc‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(tc‘𝐹)𝐴))

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐵(tc‘𝐹)𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶(tc‘𝐹)𝐵))

———

Proposition 131, p. 85: If 𝐹 is a function and 𝐴 contains all elements of 𝑈 and all elements before or after those elements of 𝑈 in the transitive closure of 𝐹, then the image under 𝐹 of 𝐴 is a subclass of 𝐴.

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((◡(tc‘𝐹) “ 𝑈) ∪ ((tc‘𝐹) “ 𝑈))))

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

———

Proposition 133, p. 86: If 𝐹 is a function and 𝐴 and 𝐵 both follow 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹 (or both if it loops).

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝑋(tc‘𝐹)𝐴)

Hyp. ⊢ (𝜑 → 𝑋(tc‘𝐹)𝐵)

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐴(tc‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(tc‘𝐹)𝐴))

———

So what's nice about the transitive closure that #Frege felt compelled to invent a new language in which to present mathematical arguments? When 𝑅 is a function, two sets being related by the transitive closure of 𝑅 is much like induction. When 𝑅 is a more general relation, we have a more general form of induction, that is truly #ancestral in the language of #Whitehead and #Russell.

Proposition 129, p. 83: If 𝐹 is a function and (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹, the successor of 𝐴 is either 𝐵 or it follows 𝐵 or it comes before 𝐵 in the #TransitiveClosure of 𝐹.

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝐴 ∈ dom 𝐹)

Hyp. ⊢ (𝜑 → 𝐶 = (𝐹‘𝐴))

Hyp. ⊢ (𝜑 → (𝐴(tc‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(tc‘𝐹)𝐴))

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐵(tc‘𝐹)𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶(tc‘𝐹)𝐵))

———

Proposition 131, p. 85: If 𝐹 is a function and 𝐴 contains all elements of 𝑈 and all elements before or after those elements of 𝑈 in the transitive closure of 𝐹, then the image under 𝐹 of 𝐴 is a subclass of 𝐴.

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((◡(tc‘𝐹) “ 𝑈) ∪ ((tc‘𝐹) “ 𝑈))))

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

———

Proposition 133, p. 86: If 𝐹 is a function and 𝐴 and 𝐵 both follow 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹 (or both if it loops).

Hyp. ⊢ (𝜑 → 𝐹 ∈ V)

Hyp. ⊢ (𝜑 → 𝑋(tc‘𝐹)𝐴)

Hyp. ⊢ (𝜑 → 𝑋(tc‘𝐹)𝐵)

Hyp. ⊢ (𝜑 → Fun 𝐹)

Therefore ⊢ (𝜑 → (𝐴(tc‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(tc‘𝐹)𝐴))

———

So what's nice about the transitive closure that #Frege felt compelled to invent a new language in which to present mathematical arguments? When 𝑅 is a function, two sets being related by the transitive closure of 𝑅 is much like induction. When 𝑅 is a more general relation, we have a more general form of induction, that is truly #ancestral in the language of #Whitehead and #Russell.

Notation guide (adapted from Metamath):

• 𝜑, a metavariable standing for any logical formula, abbreviates the conjunction of all hypotheses listed for a given proposition; writing each line as ⊢ (𝜑 → …) puts the theorem in "deduction form," which can be easier to apply in #Metamath.

• ⊢ 𝜑 asserts that 𝜑 is true; the turnstile is descended from #Frege's own Urteilsstrich (judgment stroke).

• 𝐴𝑅𝐵 means the ordered pair ⟨𝐴, 𝐵⟩ is an element of 𝑅, or we could say 𝐵 immediately follows 𝐴

• 𝐵 = (𝑅‘𝐴) means 𝐵 is the unique set such that 𝐴𝑅𝐵 is true (when such a 𝐵 exists) which means 𝑅 is function-like when restricted to operating on the singleton {𝐴}

• (𝑅”𝐴) is the image of 𝐴

• dom 𝑅 is the domain of 𝑅, the class of all sets 𝑥 such that there is a set 𝑦 that would make 𝑥𝑅𝑦 true.

• Fun 𝑅 is true when 𝑅 is function-like for all sets in its domain.

• ◡𝑅 is the converse of 𝑅 so 𝐴◡𝑅𝐵 iff 𝐵𝑅𝐴

• (tc‘𝑅) is the #TransitiveClosure of 𝑅 (Metamath uses (t+‘𝑅) which can be awkward.) Whitehead and Russell use the term ancestral to describe how 𝐴(tc‘𝑅)𝐵 means 𝐴 is some “ancestor” of 𝐵. Alternately, we can say 𝐵 eventually follows 𝐴.

• V is the universal class, every set is a member, and only sets may be members of any class. After Frege’s later work ran into Russell’s Paradox, it was discovered that not every class {𝑥 | 𝜑} makes sense as a set and so we need the hypothesis ⊢ (𝜑 → 𝐴 ∈ V) before we can talk about the function value of 𝐴 or the ordered pair ⟨𝐴, 𝐵⟩ being an element of 𝑅. V is not italic because it is a constant symbol, like tc, dom, and Fun.

Begriffsschrift (1879), is one of the first manuscripts on #SymbolicLogic. As such, it literally invents a new language to describe the subjects the author, #GottlobFrege, wants to introduce. And this notation is very unlike what we see in math before or after this.

So I will list some theorems adapted (by me, circa 2020) from #Frege with proper set-theoretical bounds.

#SetTheory #Logic #Metamath

Die Geburtsstadt #Wismar begeht das #Frege-Jubiläum mit vielen Veranstaltungenund einer Ausstellung im Museum Schabell
» https://www.wismar.de/B%C3%BCrger/Aktuelles/Pressemitteilungen/Frege-Jubil%C3%A4um-Viele-Veranstaltungen-in-der-Hansestadt-Wismar.php?object=tx,2634.5.1&ModID=7&FID=2634.47323.1&NavID=2634.13&La=1
» https://www.wismar.de/Tourismus/Veranstaltungen/H%C3%B6hepunkte-des-Jahres/Frege-Jubil%C3%A4um/Gottlob-Frege-und-die-Macht-der-Logik-Ein-Wismarer-pr%C3%A4gt-die-moderne-Welt-.php?object=tx,2634.4&ModID=11&FID=2634.19216.1&NavID=2634.4719

Vor 100 Jahren starb der "Aristoteles der Neuzeit": Am 26.7. jährt sich der Todestag des #Mathematik​ers, Logikers und Philosophen Gottlob Friedrich Ludwig #Frege. Er gilt als Begründer der modernen mathematischen #Logik, schuf eine Definition des #Zahlbegriff​s und publizierte wichtige Beiträge zur #Sprachphilosophie z.B. zur Trennung von Syntax und Semantik.
Die ULB besitzt einige Dokumente aus dem #Nachlass, die Teil des Nachlasses von Heinrich Scholz sind: https://www.ulb.uni-muenster.de/ULB/sammlungen/nachlaesse/sammlung-frege.html

You almost can't tell that we were experiencing technical difficulties this AM. Today on #IanAndJaySpaceOut, @jay & I get into "Hubris Maximus", #Signal shenanigans at the Pentagon, Gottlob #Frege, #guillotines, and more!

If recordings aren't your thing you can follow @live and tune in to the next show.

https://archive.org/details/ian-and-jay-space-out-2025-04-27

Bertrand Russell: Why I Am Not a Christian (1927)
https://users.drew.edu/~jlenz/whynot.html
#ycombinator #Bertrand_Russell #philosophy #essays #articles #books #Russell #Bertrand_Russell_Society #Wittgenstein #Carnap #Frege #G_E_Moore #Gottlob_Frege #Ludwig_Wittgenstein #Rudolf_Carnap #Bertrand_Russell_Society_Quarterly #Analytic_Philosophy #Early_Analytic_Philosophy #History_of_Analytic_Philosophy #History_of_Early_Analytic_Philosophy

Bertrand Russell: Why I Am Not a Christian (1927)
https://users.drew.edu/~jlenz/whynot.html
#ycombinator #Bertrand_Russell #philosophy #essays #articles #books #Russell #Bertrand_Russell_Society #Wittgenstein #Carnap #Frege #G_E_Moore #Gottlob_Frege #Ludwig_Wittgenstein #Rudolf_Carnap #Bertrand_Russell_Society_Quarterly #Analytic_Philosophy #Early_Analytic_Philosophy #History_of_Analytic_Philosophy #History_of_Early_Analytic_Philosophy

Bertrand Russell: Why I Am Not a Christian (1927)
https://users.drew.edu/~jlenz/whynot.html
#ycombinator #Bertrand_Russell #philosophy #essays #articles #books #Russell #Bertrand_Russell_Society #Wittgenstein #Carnap #Frege #G_E_Moore #Gottlob_Frege #Ludwig_Wittgenstein #Rudolf_Carnap #Bertrand_Russell_Society_Quarterly #Analytic_Philosophy #Early_Analytic_Philosophy #History_of_Analytic_Philosophy #History_of_Early_Analytic_Philosophy

Movie idea: an action flick where #Frege and #Russell realize their theories are inconsistent and they have to stop the publishing of their works. They have two hours to get to the printing house.
#math #mathjoke

"If you study philosophy at a British or American #university, your #education in the history of the subject will likely be modest. Most universities teach #Plato and #Aristotle, skip about two millennia to #Descartes, zip through the highlights of #Empiricism and #Rationalism to #Kant, and then drop things again until the 20th Century, where #Frege and #Russell arise from the mists of the previous centuries’ Idealism ...”

@philosophy
#philosophy
https://www.prospectmagazine.co.uk/ideas/philosophy/46795/does-the-history-of-philosophy-matter

Ich war beim #Javaland und hab da gelernt, dass #Haskell als #Frege inzwischen angeblich mit #Java interoperabel/kompatibel sei. (mein letzter Versuch vor Jahren war mehr Krampf als brauchbar).

Dadurch spiele ich gerade wieder mit Haskell rum und freue mich. z.B. über Literate Haskell. Ein Dokument, wo man gleichzeitig Doku und Code schreibt. Wenn da steht "in diesem Block *someBlock* wird xyz gemacht", dann ist *someBlock* der reale Code, der compiliert wird.

Buroker likes to cite #Frege's essay 'Negation' as showing that the theory of propositions as acts of affirmation/denial (& of affirmation & denial as separate acts) in Port-Royal & #Locke is confused, but arguably this theory captures something Frege misses. /1

#philosophy #logic #language #histodons

It also the first three axioms of explains #Frege's logical calculus (https://en.wikipedia.org/wiki/Frege%27s_propositional_calculus).

I. The first axiom, 𝐴→(𝐵→𝐴), means that if a statement 𝐴 is true, it is also true under an assumption 𝐵. One can therefore “move” statements under an assumption.

II. The complex second axiom, (𝐴→(𝐵→𝐶))→((𝐴→𝐵)→(𝐴→𝐶)), means that logical conclusions work under an assumption the same way as outside: If 𝐵→𝐶 and 𝐵 are true under assumption 𝐴, then 𝐶 is true under 𝐴.

-->

I'm currently writing about #Locke, & I'm worried that my draft sounds like I'm simultaneously grumpy at Locke's interpreters for failing to take his ideas seriously & grumpy at Locke for not being as good at philosophy as #Arnauld & #Berkeley.

Trouble is, poor Locke was very popular among 'analytic' historians of #philosophy in the 20th century & those folks think every time Locke agrees w/ #Frege he's confused, but if they had understood the Port-Royal Logic they wouldn't've thought that.

I'm currently writing about #Locke, & I'm worried that my draft sounds like I'm simultaneously grumpy at Locke's interpreters for failing to take his ideas seriously & grumpy at Locke for not being as good at philosophy as #Arnauld & #Berkeley.

Trouble is, poor Locke was very popular among 'analytic' historians of #philosophy in the 20th century & those folks think every time Locke agrees w/ #Frege he's confused, but if they had understood the Port-Royal Logic they wouldn't've thought that.

I'm currently writing about #Locke, & I'm worried that my draft sounds like I'm simultaneously grumpy at Locke's interpreters for failing to take his ideas seriously & grumpy at Locke for not being as good at philosophy as #Arnauld & #Berkeley.

Trouble is, poor Locke was very popular among 'analytic' historians of #philosophy in the 20th century & those folks think every time Locke agrees w/ #Frege he's confused, but if they had understood the Port-Royal Logic they wouldn't've thought that.

I'm currently writing about #Locke, & I'm worried that my draft sounds like I'm simultaneously grumpy at Locke's interpreters for failing to take his ideas seriously & grumpy at Locke for not being as good at philosophy as #Arnauld & #Berkeley.

Trouble is, poor Locke was very popular among 'analytic' historians of #philosophy in the 20th century & those folks think every time Locke agrees w/ #Frege he's confused, but if they had understood the Port-Royal Logic they wouldn't've thought that.

@ereliuer_eteer

@philosophy
@philosophie
#philosophy

It was #Kant suggesting that, where the #mathematician starts, #philosopher 's work is already done [Über die Deutlichkeit der Grundsätze... §1...]

Of course in a formal system, say #FOL, #identity is simply the strictest equivalence relation, the unit classes of the domain being its equivalence classes. And yes, this is really elementary stuff, high school level, if you like. And no, that's not, why #philosophers at least since #Leibniz (and in a sense since #Parmenides), and more recently #Frege made a case for it.

Typically math philosophical issues with identity from my point of view do include the question of indiscernibles, the sense of an identity statement wrt. redundance, the relation status of identity, ... open list