These are slides for video of "Wittgenstein versus Gödel on the Foundations of Logic" Stanford Logic Colloquium on April 23, 2010.
Video can be viewed at:
http://wh-stream.stanford.edu/MediaX/CarlHewittEdit.mp4
Types of errors
Among the most frequent sources of errors Brown counts
(1) interlingual transfer,
(2) intralingual transfer,
(3) context of learning,
and (4) various communication strategies the learners use
Jean-Yves Béziau the metalogical hexagon of opposition
The difference between truth and logical truth is a fundamental distinction of modern logic promoted by Wittgenstein. We show here how this distinction leads to a metalogical triangle of contrariety which can be naturally extended into a metalogical hexagon of oppositions, representing in a direct and simple
way the articulation of the six positions of a proposition vis-à-vis a theory. A particular case of this hexagon is a metalogical hexagon of propositions which can be interpreted in a modal way. We end by a semiotic hexagon emphasizing the value of true symbols, in particular the logic hexagon itself.
Seven Ineffective Coding Habits of Many ProgrammersKevlin Henney
Presented at NDC 2014 in Oslo (4th June 2014)
Video available on Vimeo: https://vimeo.com/97329157
Habits help you manage the complexity of code. You apply existing skill and knowledge automatically to the detail while focusing on the bigger picture. But because you acquire habits largely by imitation, and rarely question them, how do you know your habits are effective? Many of the habits that programmers have for naming, formatting, commenting and unit testing do not stand up as rational and practical on closer inspection.
This talk examines seven coding habits that are not as effective as programmers believe, and suggests alternatives.
Seven Ineffective Coding Habits of Many ProgrammersKevlin Henney
Presented at NorDevCon (27th February 2015)
Habits help you manage the complexity of code. You apply existing skill and knowledge automatically to the detail while focusing on the bigger picture. But because you acquire habits largely by imitation, and rarely question them, how do you know your habits are effective? Many of the habits that programmers have for naming, formatting, commenting and unit testing do not stand up as rational and practical on closer inspection.
This talk examines seven coding habits that are not as effective as programmers believe, and suggests alternatives.
Types of errors
Among the most frequent sources of errors Brown counts
(1) interlingual transfer,
(2) intralingual transfer,
(3) context of learning,
and (4) various communication strategies the learners use
Jean-Yves Béziau the metalogical hexagon of opposition
The difference between truth and logical truth is a fundamental distinction of modern logic promoted by Wittgenstein. We show here how this distinction leads to a metalogical triangle of contrariety which can be naturally extended into a metalogical hexagon of oppositions, representing in a direct and simple
way the articulation of the six positions of a proposition vis-à-vis a theory. A particular case of this hexagon is a metalogical hexagon of propositions which can be interpreted in a modal way. We end by a semiotic hexagon emphasizing the value of true symbols, in particular the logic hexagon itself.
Seven Ineffective Coding Habits of Many ProgrammersKevlin Henney
Presented at NDC 2014 in Oslo (4th June 2014)
Video available on Vimeo: https://vimeo.com/97329157
Habits help you manage the complexity of code. You apply existing skill and knowledge automatically to the detail while focusing on the bigger picture. But because you acquire habits largely by imitation, and rarely question them, how do you know your habits are effective? Many of the habits that programmers have for naming, formatting, commenting and unit testing do not stand up as rational and practical on closer inspection.
This talk examines seven coding habits that are not as effective as programmers believe, and suggests alternatives.
Seven Ineffective Coding Habits of Many ProgrammersKevlin Henney
Presented at NorDevCon (27th February 2015)
Habits help you manage the complexity of code. You apply existing skill and knowledge automatically to the detail while focusing on the bigger picture. But because you acquire habits largely by imitation, and rarely question them, how do you know your habits are effective? Many of the habits that programmers have for naming, formatting, commenting and unit testing do not stand up as rational and practical on closer inspection.
This talk examines seven coding habits that are not as effective as programmers believe, and suggests alternatives.
The Gödel incompleteness can be modeled on the alleged incompleteness of quantum mechanics
Then the proved and even experimentally confirmed completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics
Infinity is equivalent to a second and independent finiteness
Two independent Peano arithmetics as well as one single Hilbert space as an unification of geometry and arithmetic are sufficient to the self-foundation of mathematics
Quantum mechanics is inseparable from the foundation of mathematics and thus from set theory particularly
Chapter 1 Logic and ProofPropositional Logic SemanticsPropo.docxcravennichole326
Chapter 1: Logic and Proof
Propositional Logic Semantics
Propositional variables: p, q, r, s, ... (stand for simple sentences)
T: any proposition that is always true
F: any proposition that is always false
Compound propositions: formed from propositional variables and logical operators (all binary except negation):
Negation ¬
Conjunction ∧
Disjunction ∨
Implication →
Biconditional ↔
Exclusive Or ⊕
Truth Tables: assign all possible T, F to all possible variables, and determines all possible T, F of compound propositions; with n variables there are 2n rows in the table
Negation changes T to F and vice versa
Conjunction is only T if both conjuncts are T
Disjunction is only F is both disjuncts are F
Implication is only F is the antecedent is T and the consequent is F
Biconditional is only true if they have the same tvalue
Exclusive Or is only T if they differ in tvalue
Two (compound) propositions are equivalent (≡) iff they always have the same tvalue (see also below)
English translations:
Conjunction: and, but, although, yet, still, ...
Disjunctions: or, unless
Implication: if, if ... then, only if, when, implies, entails, follows from, is sufficient, is necessary, when, whenever
Biconditional: if and only if, just in case, is necessary and sufficient
A set of propositions is consistent iff there is some assignment of tvalue that makes all T
A set of propositions is inconsistent iff there is no assignment of tvalue that makes all T
A tautology is a compound propositions that is always T
A contradiction is a compound propositions that is always F
A contingency is a compound propositions that is sometimes T, sometimes F
A compound proposition is satisfiable iff some assignment of tvalues make it T
A compound proposition is unsatisfiable iff no assignment of tvalues make it T
Two compound propositions p and q are logically equivalent iff p ↔ q is a tautology
Common equivalences:
DeMorgan’s Laws (Dem)
¬(p ∨ q)≡¬p ∧ ¬q
¬(p ∧ q)≡¬p ∨ ¬q
Identity Laws (Id)
p ∧ T ≡p
p ∨ F ≡p
Domination Laws (Dom)
p ∨ T ≡T
p ∧ F ≡F
Idempotent Laws (Idem)
p ∨ T ≡T
p ∧ p ≡p
Double Negation Law (DN)
¬(¬p) ≡ p
Negation Laws (Neg)
p ∨ ¬p ≡T
p ∧ ¬p ≡F
Commutative Laws (Comm)
p ∨ q ≡q ∨ p
p ∧ q ≡q ∧ p
Associative Laws (Assoc)
(p ∨ q) ∨ r ≡p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡p ∧ (q ∧ r)
Distributive Laws (Dist)
p ∨ (q ∧ r) ≡
(p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡
(p ∧ q) ∨ (p ∧ r)
Absorption Laws (Abs)
p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p
Conditional Laws (Cond)
p →q≡ ¬p ∨ q
¬(p →q)≡ p ∧ ¬q
Biconditional Law (Bicond)
p ↔ q ≡ (p →q) ∧ (q →p)
Quantifier Negation (QNeg)
¬ ∀x P ( x ) ≡ ∃x ¬ P ( x )
¬ ∃x P ( x ) ≡ ∀x ¬ P ( x )
Predicate and Relational Logic (Quantificational Logic, First Order Logic): Semantics
Variables: x, y, z, ...
Predicates/Relations, Propositional Functions: P(x), M(x), Q(x,y), S(x,y,z), ...
Constants: a, b, c, 0, -1, 4, Socrates, ...
Domain (U): set of things the variables range over
Propositional functions are neither T nor F; however, if all the variables are re ...
In this contribution the philosphical consequences of the theorem of Goedel are studied. It is shown that with formal systems, like mathematics or physical science only part of the reality can be described.
Seven Ineffective Coding Habits of Many Java ProgrammersKevlin Henney
Presented at GeeCON (15th May 2014)
Video available on Vimeo: https://vimeo.com/101084305
Habits help you manage the complexity of code. You apply existing skill and knowledge automatically to the detail while focusing on the bigger picture. But because you acquire habits largely by imitation, and rarely question them, how do you know your habits are effective? Many of the habits and conventions Java programmers have for naming, formatting, commenting and unit testing do not stand up as rational and practical on closer inspection.
This session examines seven coding habits that are not as effective as many Java programmers believe, and suggests alternatives.
66 C O M M U N I C AT I O N S O F T H E A C M J A.docxblondellchancy
66 C O M M U N I C AT I O N S O F T H E A C M | J A N U A R Y 2 0 1 9 | V O L . 6 2 | N O . 1
contributed articles
T H E C H U R C H - T U R I N G T H E S I S (CTT) underlies tantalizing
open questions concerning the fundamental place
of computing in the physical universe. For example,
is every physical system computable? Is the universe
essentially computational in nature? What are the
implications for computer science of recent speculation
about physical uncomputability? Does CTT place a
fundamental logical limit on what can be computed,
a computational “barrier” that cannot be broken, no
matter how far and in what multitude of ways computers
develop? Or could new types of hardware, based perhaps
on quantum or relativistic phenomena, lead to radically
new computing paradigms that do
breach the Church-Turing barrier, in
which the uncomputable becomes com-
putable, in an upgraded sense of “com-
putable”? Before addressing these ques-
tions, we first look back to the 1930s to
consider how Alonzo Church and Alan
Turing formulated, and sought to jus-
tify, their versions of CTT. With this nec-
essary history under our belts, we then
turn to today’s dramatically more pow-
erful versions of CTT.
History of the Thesis
Turing stated what we will call “Turing’s
thesis” in various places and with vary-
ing degrees of rigor. The following for-
mulation is one of his most accessible.
Turing’s thesis. “L.C.M.s [logical com-
puting machines, Turing’s expression
for Turing machines] can do anything
that could be described as … ‘purely me-
chanical’.”38
Turing also formulated his thesis
in terms of numbers. For example, he
said, “It is my contention that these op-
erations [the operations of an L.C.M.]
include all those which are used in
the computation of a number.”36 and
“[T]he ‘computable numbers’ include
all numbers which would naturally be
regarded as computable.”36
Church (who, like Turing, was work-
ing on the German mathematician
David Hilbert’s Entscheidungsproblem)
advanced “Church’s thesis,” which he
expressed in terms of definability in his
lambda calculus.
Church’s thesis. “We now define the
notion … of an effectively calculable
The Church-
Turing Thesis:
Logical Limit
or Breachable
Barrier?
D O I : 1 0 . 1 1 4 5 / 3 1 9 8 4 4 8
In its original form, the Church-Turing thesis
concerned computation as Alan Turing
and Alonzo Church used the term in 1936—
human computation.
BY B. JACK COPELAND AND ORON SHAGRIR
key insights
˽ The term “Church-Turing thesis” is used
today for numerous theses that diverge
significantly from the one Alonzo Church
and Alan Turing conceived in 1936.
˽ The range of algorithmic processes
studied in modern computer science
far transcends the range of processes a
“human computer” could possibly carry out.
˽ There are at least three forms of
the “physical Church-Turing thesis”—
modest, bold, and super-bold—though,
at th ...
66 C O M M U N I C AT I O N S O F T H E A C M J A.docxfredharris32
66 C O M M U N I C AT I O N S O F T H E A C M | J A N U A R Y 2 0 1 9 | V O L . 6 2 | N O . 1
contributed articles
T H E C H U R C H - T U R I N G T H E S I S (CTT) underlies tantalizing
open questions concerning the fundamental place
of computing in the physical universe. For example,
is every physical system computable? Is the universe
essentially computational in nature? What are the
implications for computer science of recent speculation
about physical uncomputability? Does CTT place a
fundamental logical limit on what can be computed,
a computational “barrier” that cannot be broken, no
matter how far and in what multitude of ways computers
develop? Or could new types of hardware, based perhaps
on quantum or relativistic phenomena, lead to radically
new computing paradigms that do
breach the Church-Turing barrier, in
which the uncomputable becomes com-
putable, in an upgraded sense of “com-
putable”? Before addressing these ques-
tions, we first look back to the 1930s to
consider how Alonzo Church and Alan
Turing formulated, and sought to jus-
tify, their versions of CTT. With this nec-
essary history under our belts, we then
turn to today’s dramatically more pow-
erful versions of CTT.
History of the Thesis
Turing stated what we will call “Turing’s
thesis” in various places and with vary-
ing degrees of rigor. The following for-
mulation is one of his most accessible.
Turing’s thesis. “L.C.M.s [logical com-
puting machines, Turing’s expression
for Turing machines] can do anything
that could be described as … ‘purely me-
chanical’.”38
Turing also formulated his thesis
in terms of numbers. For example, he
said, “It is my contention that these op-
erations [the operations of an L.C.M.]
include all those which are used in
the computation of a number.”36 and
“[T]he ‘computable numbers’ include
all numbers which would naturally be
regarded as computable.”36
Church (who, like Turing, was work-
ing on the German mathematician
David Hilbert’s Entscheidungsproblem)
advanced “Church’s thesis,” which he
expressed in terms of definability in his
lambda calculus.
Church’s thesis. “We now define the
notion … of an effectively calculable
The Church-
Turing Thesis:
Logical Limit
or Breachable
Barrier?
D O I : 1 0 . 1 1 4 5 / 3 1 9 8 4 4 8
In its original form, the Church-Turing thesis
concerned computation as Alan Turing
and Alonzo Church used the term in 1936—
human computation.
BY B. JACK COPELAND AND ORON SHAGRIR
key insights
˽ The term “Church-Turing thesis” is used
today for numerous theses that diverge
significantly from the one Alonzo Church
and Alan Turing conceived in 1936.
˽ The range of algorithmic processes
studied in modern computer science
far transcends the range of processes a
“human computer” could possibly carry out.
˽ There are at least three forms of
the “physical Church-Turing thesis”—
modest, bold, and super-bold—though,
at th.
Category theory is general abolute nonsensPawel Szulc
Happened to be part of conversation recently that you did not get or the library which README.md was to obscure because it consistently kept on using those weird phrases? "Typeclasses", "semigroups", "monoids", "applicatives" - they all seem so weird, so academic, so pointlessly detached from the real-world problems. But then again, they pop up here and there in conversations (especially if talk to functional programmers). So are they really irrelevant. Is knowing them is just an academic exercise or do they have a real applications? Well, you have to join this talk to see and judge it for yourself.
Intention of this talk is not to give just raw definitions of such terms as 'monoid' or 'functor'. The more important question (rarely explained in other talks & tutorials) is what are the motivations behind those concepts. And this is exactly what we want to discover.
Preferable audience for this talk are people who:
- want to stay relevant to the continuously changing industry
- confuse their wives/husbands/boyfriends/girlfriends ("You know honey, a monad is just a monoid in the category of endofunctors")
- sound smart on the next job interview
A quantum framework for likelihood ratiosRachael Bond
The ability to calculate precise likelihood ratios is fundamental to science, from Quantum Information Theory through to Quantum State Estimation. However, there is no assumption-free statistical methodology to achieve this. For instance, in the absence of data relating to covariate overlap, the widely used Bayes’ theorem either defaults to the marginal probability driven “naive Bayes’ classifier”, or requires the use of compensatory expectation-maximization techniques. This paper takes an information-theoretic approach in developing a new statistical formula for the calculation of likelihood ratios based on the principles of quantum entanglement, and demonstrates that Bayes’ theorem is a special case of a more general quantum mechanical expression.
Read More: http://www.worldscientific.com/doi/abs/10.1142/S0219749918500028
Similar to Incompleteness Theorems: Logical Necessity of Inconsistency (20)
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Incompleteness Theorems: Logical Necessity of Inconsistency
1. WittgensteinvsGödelonFoundations of Logic Carl Hewitt Stanford Logic Seminar: April 23, 2010 Slides updated: May 8, 2011 Video available at: http://wh-stream.stanford.edu/MediaX/CarlHewittEdit.mp4
2. Abstract This talk (in four parts) explains how some of the writings of Ludwig Wittgenstein can be interpreted as precursors of important developments in the foundations of mathematical logic for information systems applications. These Wittgenstein writings stand in almost exact opposition to the views of Kurt Gödel. First part: the current state of foundations of mathematical logic for information systems applications is overviewed with regard to issues of expressiblity, incompleteness, and inconsistency tolerance. Second part: The above developments have precursors in the following writings of Wittgenstein: *There can’t in any fundamental sense be such a thing as meta-mathematics. . . . Thus, it isn’t enough to say that p is provable, what we must say is: provable according to a particular system.True in Russell’s system” means, as we have said, proved in Russell's system; and “false in Russell's system” means that the opposite has been proved in Russell's system.Have said-with pride in a mathematical discovery [e.g., inconsistency of Russell’s system because incompleteness is self-proved]: “Look, this is how we produce a contradiction.”Indeed, even at this stage, I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from consistency. Third part:the above views are contrasted with the almost opposite ones of Gödel.The situation between Gödel and Wittgenstein can be summarized as follows: Gödel Mathematics is based on objective truth. Roundtripping proves incompleteness but (hopefully) not inconsistency. Theories should be proved consistent. Wittgenstein Mathematics is based on communities of practice. Self-proof of incompleteness leads to inconsistency. Theories should use inconsistency tolerant reasoning. Fourth part:How do the above provide framework and guidance for the further development of logic for information systems applications? *The (posthumously edited) writings of Wittgenstein are idiosyncratic and fragmentary. Interpretations in this talk are directed to putting them in their best possible light as precursors.
3. Further reading Common sense for concurrency and inconsistency tolerance using Direct Logic™ ArXiv:0812.4852
4. Overview State of the art Cherry-picking Wittgenstein Gödel on Wittgenstein Inventing the future
5. Pervasive Inconsistency “find bugs faster than developers can fix them and each fix leads to another bug”--Cusumano & Selby 1995
6. Wittgenstein 1930 Indeed, even at this stage, I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from consistency.
7. Predicalc Inconsistency Tolerant Spreadsheets Michael Kassoff, Lee-Ming Zen, Ankit Garg, and Michael Genesereth. PrediCalc: A Logical Spreadsheet Management System 31st International Conference on Very Large Databases (VLDB). 2005.
11. Transitivity in Argumentation Scientist and engineers speak in the name of new allies that they have shaped and enrolled; representatives among other representatives, they add these unexpected resources to tip the balance of force in their favor.Bruno Latour (├T) , (├T)⇨ (├T)
12. if an argument holds and furthermore if the antecedent of the argument holds, then the consequence of the argument holds. Soundness in Argumentation (├T)⇨((├T)⇨(├T)) an argument holdsimplies the antecedent of theargument holds implies thatthe consequence of the argument holds
13. if an argument holds and furthermore if the antecedent of the argument holds, then the consequence of the argument holds. Adequacyin Argumentation (├T)⇨ (├T(├T)) an inference holdsimplies it holds that the inference holds
14. if an argument holds and furthermore if the antecedent of the argument holds, then the consequence of the argument holds. Faithfulness in Argumentation (├T(├T))⇨ (├T) it holds that an argument holdsimplies the argument holds.
15. Reification & Abstraction The execution of code can be dynamically checked against its documentation. Also Web Services can be dynamically searched for and invoked on the basis of their documentation. Use cases can be inferred by specialization of documentation and from code by automatic test generators and by model checking. Code can be generated by inference from documentation and by generalization from use cases.
16. Propositionsversus Sentences Sentence “Gallia estomnisdivisa in partestres.” starts with “Gallia” Proposition All of Gaul is divided into three parts. believed by Caesar
17. Reification and Abstraction Reification Gallia estomnisdivisa in partestres. “All of Gaul is divided into three parts.” Gallia estomnisdivisa in partestres. “Todos Galia está dividida en tres partes.” Abstraction “Gallia estomnisdivisa in partestres.”⇔All of Gaul is divided into three parts.
20. Hilbert on Incompleteness that within us we always hear the call: here is the problem, search for the solution: you can find it by pure thought, for in mathematics there is no ignorabimus.
31. Wittgenstein on “meta-theory” There can’t in any fundamental sense be such a thing as meta-mathematics. . . . Thus, it isn’t enough to say that p is provable, what we must say is: provable according to a particular system.
32. Wittgenstein onTruth in Russell “True in Russell’s system” [Russell] means, as we have said, proved in Russell's system; and “false in Russell's system” means that the opposite has been proved in Russell's system.
33. Wittgenstein onIncompleteness and Inconsistency Let us suppose I prove the unprovability (in Russell’s system[Russell )] ) of P [⊢Russell⊬RussellPwhereP⇔⊬RussellP,i.e. P⇔⊢Russell P];then by this proof I have proved P[⊢RussellP]. Now if this proof were one in Russell’s system[⊢Russell⊢RussellP]—I should in this case have proved at once that it belonged [⊢RussellP] and did not belong [⊢RussellP]to Russell’s system.—That is what comes of making up such sentences. But there is a contradiction here!—Well, then there is a contradiction here[in Russell]. Does it do any harm here?
34. Wittgenstein onInconsistency Tolerance Can we say: ‘Contradiction is harmless if it can be sealed off’? But what prevents us from sealing it off? Let us imagine having been taught Frege’s calculus, contradiction and all. But the contradiction is not presented as a disease. It is, rather, an accepted part of the calculus, and we calculate with it. Have said-with pride in a mathematical discovery: “Look, this is how we produce a contradiction.”
35. Overview State of the art Cherry-picking Wittgenstein Gödel on Wittgenstein Inventing the future
37. Gödel onWittgenstein It is clear from the passages you cite that Wittgenstein did ''not'' understand it [1st incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox*, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics). *in Principia Mathematica
38. More Gödel onWittgenstein He [Wittgenstein] has to take a position when he has no business to do so. For example, “you can’t derive everything from a contradiction.” He should try to develop a system of logic in which that is true.It’s amazing that Turing could get anything out of discussions with somebody like Wittgenstein.
39. Gödel versusWittgenstein Gödel Mathematics has objective truth Continuum hypothesis? Roundtripping proves incompleteness but (hopefully) not inconsistency Theories should be proved consistent Wittgenstein Mathematics is a community of practice Proof of incompleteness leads to inconsistency Theories should use inconsistency tolerant reasoning
40. Tarski onInconsistency I believe everybody agrees that one of the reasons which may compel us to reject an empirical theory is the proof of its inconsistency. . . . It seems to me that the real reason of our attitude is...: We know (if only intuitively) that an inconsistent theory must contain false sentences.
41. Frege onTruth when we say that it is true that seawater is salty, we don’t add anything to what we say when we say simply that seawater is salty, so the notion of truth, in spite of being the central notion of [classical]logic, is a singularly ineffectual notion. It is surprising that we would have occasion to use such an impotent notion, nevermind that we would regard it as valuable and important.
42. Overview State of the art Cherry-picking Wittgenstein Gödel on Wittgenstein Inventing the future
43. Building “a new box” Direct Logic DirectArgumentation (argumentation directly expressed) Direct Inference (no contrapositive bug for inference) Self-refutation Incompleteness self-provable Inconsistency Tolerance Two-way Deduction Theorem for natural deduction Boolean Equivalences hold Concurrency Actor Model iScript TM scriptJ TM
44. Boltzman onBloodandTreasure What the poet laments holds for the mathematician. That he writes his works with the blood of his heart.