so lut ions in the or ig ins of math T h e G o l d b a c h C o n j e c t u r e prime i define as Goldbach did what Euler & our recent math deny: prime def 1 = a product of only 1 and itself. the a, only, 1 and itself each imply primacy. the four refract prime into unity. notice that by prime def 1, 1 is prime. 1 is the exemplar prime. all agree 1 is unique: ▪ Euclid agreed by denying it numberhood and calling it the Unit: that which any Number is a multitude of. ▪ Euler agreed by denying it primacy, and thereby spoke a paradox: " I deny the primacy of 1. " if rather defined as prime def 2 = a number with only two distinct divisors then 1 is not prime, since 1 has only one distinct divisor: 1. i'd amend prime def 2 to: prime def 1.i = a number whose only divisors are 1 and itself by prime def 1 or prime def 1.i, the Goldbach Conjecture resolves to: every even number is a sum of two primes. if a Goldbach number is a sum of two primes, then the Goldbach conjecture is these seven words: every even number is a Goldbach number. 1 is the common factor, the unit all numbers are built from. the Fundamental Theorem of Arithmetic is odd without it. if primes are "the basic building blocks of all numbers"1 , then 1 is the exemplar prime: the primes are all like 1. 1 mathisfun.com : Fundamental Theorem of Arithmetic if 1 is prime, we simplify the FTA, from every integer greater than 1 is either a prime or the product of primes to every natural number is a product of primes i.e. every non-prime is a product of primes, and every prime is a product of itself & its exemplar. 2 2 what about the FTA's unique factorization claim? on granting 1 primacy, we add two symbols to the FTA corollary [ 1< ] : which i rather like. see Chris K. Caldwell & Yeng Xiong, What is the Smallest Prime? Journal of Integer Sequences, 2012 [Vol 15] Recent espousers of 1's primacy include G.H. Hardy and the aliens in Carl Sagan's Contact. it's the first thing the aliens say: their opening VHOOMP is a correction of our Math. odd how Goldbach, Gödel/Bach, and Goldberg-Bach imply the other aurally & in my limited list of Aufklärer how music & a kinderlogic affirm the self-reference i'd restore the first word of Math to; and how lovely Rebecca Goldstein is, who elicits by a footnote in her Gödel book this Sunday morning exercise.3 3 Rebecca Goldstein, Incompleteness: The Proof and Paradox of Kurt Gödel. Norton, 2006 she lets me know the Vienna Circle's favorite word, sachlichkeit, means "fact-mindedness". her beauty demands we let in song, when analyzing. 1 is prime's exemplar. 1 is prime's namesake. i assert the primacy of 1. i sum again to solve as Goldbach intended; and up the reductivist structure, levels 4 thru 20 of the Sick Kids research tower, i'd free the mouse from rack & decapitator. Grothendieck had a flair for choosing striking, evocative names for new concepts; indeed, he saw the act of naming mathematical objects as an integral part of their discovery, as a way to grasp them even before they have been entirely understood.4 4 Allyn Jackson, Comme Appelé du Néant-As If Summoned from the Void: The Life of Alexandre Grothendieck. Notices of the AMS 51:10, p 1197 for Bob Palais, it's not that the circumference/diameter ratio isn't 3.14 it's that we should have saved the special name for 6.28, for the circumference/radius ratio. Palais complains of "the litany of important theorems and formulas into which a ubiquitous factor of 2 has crept and propagated". he advocates "good notational conventions."5 radius gives the circle simply: diameter reduces to two continuous radii. a point on a plane, evenly radiating: this is what a circle is ! 5 Bob Palais, π Is Wrong! The Mathematical Intelligencer, 2001 [vol 23:3] T h e L i a r ' s P a r a d o x One of themselves, even a prophet of their own, said, the Cretans are always liars, evil beasts, slow bellies. This witness is true. Wherefore rebuke them sharply, that they may be sound in the faith [Titus 1:12-13] solution 1a the prophet disclaims from the vantage of banishment. when Paul/pseudo-Paul [P/p-P] comes upon him, he's exiting Crete, with the weight of the Cretans' judgement on him. some of that weight he offloads to P/p-P, and P/p-P onto Logical history. solution 1b they've just killed their Socrates, and the prophet escapes with his life he shall not let them sin against Philosophy twice. solution 1c the prophet speaks from the standpoint of their future Redemption: all Cretans at Paul's present are liars. : in solution 1, the prophet isn't strictly "one of their own". solution 2 P/p-P insists that the Cretan "is true". perhaps P/p-P means that the flip-flopping Implication [if he's true then he lies; if he lies then he's true, and so on] sums into he's true, in logical space: it's a rounded sum of competing infinities. solution 3 are you a true witness, P/p-P? that depends on who you are, Paul or pseudo- : for it's Paul you claim in the byline: Paul, a servant of God, and an apostle of Jesus Christ, according to the faith of God's elect, and the acknowledging of the truth which is after godliness [Titus 1:1] scholars approach consensus that the Pastoral Epistles are pseudonymous. i for one am wary of a writer who begins with Trust me; so the Paradox displaces to Titus 1:1 itself, to the document's authenticity. this Paradox is nested in witnesses. in orbitals our answers displace to. the [putative] Cretan's [putative] words we circle in our recursion. we widen from the site of speech, we de-indent from the Dialogue's inner column to interrogate its Witness, then the Citer, and so on. The speaker, Epimenides, is a Cretan. who says this, and why do we believe them? R u s s e l l ' s P a r a d o x does the set of all self-exclusional sets exclude itself? it seems to me that Russell's decree Whatever involves all of a collection mustn't be part of the collection is not wholly arbitrary. a self-inclusional set is quasi-contradictory: "a kind of vicious circle", he says. the quasi is his opening: it frees him from perplexity to resolution, action. his Theory of Types is folly, some said. an ad hoc pencil trick to save all Arithmetic. yet something must be done to stave off logical collapse! the lemmas must be pried apart & quarantined: as Yahweh parted Sea from earthly Promontory. Yahweh tells Job: your pious thought, your Math alone, cannot make a World. a World must be willed into coherence. it's a lonely achievement, a terrible secret, that Yahweh confesses. the people sleep easy, unknowing their existence is deeply contingent: that consistency itself is contingent on a primal volition! A.J. Ayer briefly died and found himself squinting at a dire red light, a siren impelling him to fix some kind of rift in the structure of Time. in a ground-level lab where cyberspace gestates, a toggle-switch is stuck with gum to perpetual ON with "MAGIC" written on the masking tape label. they've let in something senseless. their need is what demands it. their irony a leavening in their Naturalism allows it. you're at the gates of Night & Day, the Paradox all & anything stream from. & what did Earl Russell do? a Hierarchy, he decreed: a ranking of Types, to let in the light. he decreed, not quite deduced. yet it's sensible, in its way: for by it we ascend thru levels of increasing generality.