Limits of Logic: The Gödel Legacy
ฝัง
- เผยแพร่เมื่อ 3 มิ.ย. 2024
- Kurt Gödel showed that mathematical thinking cannot be captured in a formal axiomatic reasoning system. What does this deep result mean in practice? What are the limits of computer thinking? Can beauty and creativity and a sense of humor be formalized?
Introduction by professor Douglas Hofstadter. - วิทยาศาสตร์และเทคโนโลยี
It reminds me of when I took a course in acoustics as a math major, half the class were musicians and half were engineers, the physicist who gave the course was a bassoon player of some accomplishment and he tried to make his course relevant to both groups - it didn't work. During one of the classes he read a physical description of what a violinist does while playing the violin, stopping the string at a precise length, the control of the sound through the speed and pressure of the bow on the string, etc. One of the engineering students asked, mystified, "how do they do all that?" To which one of the music majors called out "You practice."
lol Funny as heck and a great anecdote. Thank you!
Ron
play with mathematics, reason with music. peek into the edge of the disciplines and maybe you can reach beyond the guardians at the gates.
Formal logic professor here. This theorem is the limits of mathematics not logic. Gödel proved formal logic to be complete and mathematics to be incomplete. All logic rules are understood beyond the rule itself. Formal logic doesn't have the same problems mathematics do. He literally used logic is shown the hole in math.
Now here's where it gets super interesting is this belief of things being true or false is called the law of the excluded middle. But not all logic system abid by this the way math does. Fuzzy logic for example which is used mainly in AI rejects this premise and deals with degrees of truth. It's super super interesting when you start pinning axioms against each other in this system because there are degrees of truth but even in a system like this you'll have to pick one. Oh yeah logic is a shit ton of fun.
Came here to say this. Godel's incompleteness theorems only apply to formal systems that are capable of quantification, which classical/propositional logic is not.
Hats off to Prof Hofstadter for beautifully stringing ideas to illustrate Gödel's first incompleteness theorem. Not one boring second.
This is the best overview of Gödel's first incompleteness theorem that I've come across.
th-cam.com/video/U84nxCsQpus/w-d-xo.html
His book “Godel Escher And Bach” introduced me to these concepts.It is still an excellent book.Thank you Douglas
Hey Jim looks like there are 8 of us that have that book. Cheers
th-cam.com/video/SOWt2fBI1VI/w-d-xo.html
@@ouimetco never loan it to anyone!
@@illygah Nope, never. Buy them their own copy if you feel strongly they should read it.
I have had my copy since 1987.
In 1989 I bought another of his books, "The Mind's I".
“the exact reason why it’s not provable, is that it’s true.”
One of the deepest, most philosophical concepts in the universe…
Great lecture. OTOH somebody should tell the camera op what needs to be in frame.
a little hint for formal systems. the book with the same titel by luis Carrol. Yes the auther of Alice in Wonderland was a mathematical logician too. I learned from his book when I was a student. Thanks to prof. Douglas' Hofstatter's nice talk, maybe somebody wants know a little bit more, for these people I suggest my book ( together with John Casti) "Gödel: a life for logic", Cast+DePauli Perseus books,MA 2000).
*Lewis Carroll
The nom de plume of Charles Lutwidge Dodgson.
@29:30 This is a shorter sequence to 27:
1 -> 2 -> 4 -> 8 -> 16 \ 5 -> 10 -> 20 -> 40 \ 13 -> 26 -> 52 \ 17 -> 34 \ 11 -> 22 -> 44 -> 88 \ 29 -> 58 \ 19 -> 38 -> 76 -> 152 -> 304 \ 101 -> 202 -> 404 -> 808 \ 269 -> 538 \ 179 -> 358 \ 119 -> 238 -> 476 -> ***952*** \ 317 -> 634 \ 211 -> 422 -> 844 \ 281 -> 562 \ 187 \ 62 -> 124 \ 41 -> 82 \ 27
Tried to link to code which confirms this, but those comments are removed.
WOW!!! What a great lecture!! Hofstadter is imaginative and clear
Thanks for this. Read one of this distinguished academic's books, namely Godel Escher and Bach, long ago. It's taken a while for my understanding of this momentous result in mathematics to sink in. But in life the most difficult challenges are the always the most satisfying and long lasting.
th-cam.com/video/SOWt2fBI1VI/w-d-xo.html
Cool idea to draw lines between the edges of centered symmetrical lists of axioms and compare them -- to check whether or not (a particular kind of) visual aesthetics correlates with well-formed formulas.
I think most people interpret Goedel too harshly. What we can learn from his insights is that even the most rigid systems, and formal systems are certainly extremely rigid in their construction, can allow for surprises. They are rich in features and that is what makes them worth exploring in the first place. Isn't that beautiful?
Wow. Brilliant lecture.
Great talk that cleared some ideas for me. Also great pronunciation, especially of: Goldbach, and Gödel. It's a shame how so many English speakers butcher these words.
Itube "kuhhrt guhhrrdl" and variants. I know it's so inaccurate haha
th-cam.com/video/U84nxCsQpus/w-d-xo.html
Nicely done -- thanks! Or, should say, "This is a great lecture." is a decidedly true.
Beyond great lecture. (I love his GEB book too). Thank you for the video!
This presentation helps to show that Immanuel Kant was right about the limits of Reason as described in his "Critique of Pure Reason". Kant's central claim is that Reason is a biological function and has no pedigree. Apriori truths are those truths we can't deny and are not dependent upon any observation of the physical world. However, the aprioriness of such truths does not grant them a pedigree of perfection, rather they are the limits and measuring stick of Reason. Thus that which does not comport with Reason will not make sense to us.
Isn't that last sentence a tautology? There's also the interesting ambiguity of the word "sense", which of course refers to a feeling. (Same thing in German, "Sinn".)
being a tautology is not always a bad thing. Here is another. Anything (transformations) that does not break the laws of physics (or more broadly natural law) is possible, if the required knowledge is available
Thanks. Haven't read Kant, but I was called Kantian the other day.
Great presentation, thanks.
Gödel Escher Bach was recommended to me, when, during a long walk I was looking at the stars, thinking about the universe. I thought, “what if the universe at its core is illogical? 🤔 How can that be proven? Well in order to prove it, I’d need to use logic. So if I was successful in the proof, the proof would negate itself!” And so I texted a colleague of mine who’s one of the few who’s brain I respect. And he recommended this book to me and even loaned me his copy.
This book is the best book I have ever read. It opened up my mind to new levels of though and how the universe and logic at its core works.
@29:30: The lowest number that includes 9232 in its shortest sequence is 159: 1 -> 2 -> 4 -> 8 -> 16 -> 5 -> 10 -> 20 -> 40 -> 80 -> 160 -> 53 -> 106 -> 35 -> 70 -> 23 -> 46 -> 92 -> 184 -> 61 -> 122 -> 244 -> 488 -> 976 -> 325 -> 650 -> 1300 -> 433 -> 866 -> 1732 -> 577 -> 1154 -> 2308 -> 4616 -> *9232* -> 3077 -> 6154 -> 2051 -> 4102 -> 1367 -> 2734 -> 911 -> 1822 -> 607 -> 1214 -> 2428 -> 809 -> 1618 -> 539 -> 1078 -> 359 -> 718 -> 239 -> 478 -> 159.
Gödel's insight by his two incompleteness theorems is fitting beautifully to Karl Popper's insight that mankind can know truths about nature, but can never ever be sure about it ... ;-)
Amazing lecture
Spectacular! More please...
th-cam.com/video/U84nxCsQpus/w-d-xo.html
In the late 13th century, Arab algebraists were the first to use only symbols to formulate equations. Historians often see this time as the turn to standard symbolic notation.
@29:30: The lowest number that has a shortest path of the same length (112) as in the video is 1951: 1 -> 2 -> 4 -> 8 -> 16 -> 5 -> 10 -> 20 -> 40 -> 80 -> 160 -> 53 -> 106 -> 212 -> 424 -> 848 -> 1696 -> 3392 -> 6784 -> 2261 -> 4522 -> 1507 -> 3014 -> 6028 -> 2009 -> 4018 -> 1339 -> 2678 -> 5356 -> 10712 -> 21424 -> 7141 -> 14282 -> 28564 -> 9521 -> 19042 -> 6347 -> 12694 -> 4231 -> 8462 -> 16924 -> 5641 -> 1880 -> 3760 -> 1253 -> 2506 -> 5012 -> 10024 -> 3341 -> 6682 -> 13364 -> 26728 -> 8909 -> 17818 -> 5939 -> 11878 -> 3959 -> 7918 -> 2639 -> 5278 -> 10556 -> 21112 -> 7037 -> 14074 -> 4691 -> 9382 -> 3127 -> 6254 -> 12508 -> 4169 -> 8338 -> 16676 -> 33352 -> 11117 -> 22234 -> 7411 -> 14822 -> 29644 -> 9881 -> 19762 -> 6587 -> 13174 -> 4391 -> 8782 -> 2927 -> 5854 -> 1951.
28:30 I don't really get why and how do we do that jump from 16 to 5 (aswell as logic behind jumps, what's going on lol?)
The equals sign was invented in 1557 by a Scottish mathematician, Robert Recorde.
Dear Dr Hofstadter, At the core of Gödel’s proof of his famous incompleteness theorem there is a sentence. I have been told that the mechanism of reasoning of that sentence is reminiscent of Jules Richard’s paradox. Can you please comment on that or perhaps even introduce that sentence to me and other readers.
My objective for asking this is to investigate some reservations I have about logical reasoning, specifically the admissibility of self-reference in theorems. The famous example that sensitized me to this issue is the following sentence:
“This statement is false.”
The quoted statement purports to be a false theorem. For clarity let me adapt this statement:
“This theorem is false.”
The quoted theorem does two things, it claims of itself it is a theorem and it claims to be false. The quoted theorem certainly is a grammatically correct sentence, so we will call it the sentence, as that is not in doubt. The sentence contains a reference to itself. We can always replace the reference to something by the entirety of the thing referred to. When we do that we get:
““This theorem is false.” is false.”
But still this sentence contains a reference, we substitute and get:
“““This theorem is false.” is false.” is false.”
The structure of this sentence is:
(((()is false)is false)is false)
At the center of the successive false-claims being made, however, nothing is claimed. This sentence interpreted as a theorem is an empty construct. However, I posit the obvious by claiming that a theorem must claim something to be true, or false. Our empty construct makes falsehood claims, but not about anything; at the heart of the construct there is nothing that the falsehood claim is about. As a consequence I claim that the sentence is NOT a theorem. Because it is not a theorem it cannot be said to be false or true, just like a question cannot be false or true, it’s just a question. The sentence, although it claims to be a theorem, cannot be a theorem, because on inspection it is an empty (infinitely recursive) self-referencing construct, not actually claiming anything.
I suspect that the Gödel sentence that Gödel used in his proof of his incompleteness theorem, is in much the same way a sentence that can be discerned to be a self-referencing construct and NOT a theorem. If that Gödel sentence is not a theorem, the operator “Beweisbar” or in English “Provable” cannot be applied to it, just like that operator cannot be applied to a question. As a result Gödel’s proof would fail. That of course would not mean that his incompleteness theorem is false, but unless there exists at least one different and correct proof, it would stand unproven as a possible false conjecture.
Dr Hofstadter, I read your book, big fan! Please shine some light on this issue!
"We can always replace the reference to something by the entirety of the thing referred to. When we do that we get:
““This theorem is false.” is false.”" is false.
See what I did there?
I'm not an expert but here is my understanding: I think your statement is dissimilar to Godel's in an important way. His statement is about provability, not about its own truthfulness. This is why it's such an elegant/beautiful device for proving his big idea: that truthfulness and provability are in fact separate. Its not self-referential in the same way as your example statement - which can be thought of more as a sort of linguistic paradox.
you need Tarski, not Godel.
@@rg3788 No I don’t see what you did there. Please explain, Sir/Madam.
@@100macaco3 I need an explanation, not a mere remark, Sir.
What happened to the discussion at the end?
It wasn’t included in the video, to state the obvious. The lack of inclusion fits the topic, don’t you think?
A litmus test of any exposition. As while building up a literary scenario. Thanks.
Robert Recorde invented the equals sign in 1557, and he was Welsh
That is if we don't look for it in India, China, .....
Did he also invent the not equals sign ?
Are you sure it wasn't 1337?
@@faustdownunder what about India and China? Did they use a equal sign like our modern one? Not the concept of equality and algebraic manipulation, but the symbol itself
The Welsh always longed for equality 😎
For those unfamiliar with predicate logic the upside down A is called universal. There's a rule in logic called Quantifier Negation which allows us to say Universal is equivalent to Not Existential (the backwards E). There's 8 forms of this and they're all proven on predicate logic. Mathematics only uses 2/8 of these forms.
An excellent lecture!
The word 'self' is self evidently very interesting to me as
its meaning is also the very essence of me in
its instantiation as neural vibrations in my vocabulary,
the thing that is in fact the very thing the word refers to, me.
Back in grade school ...
Teacher: "Johnny - what's five add five?"
Johnny: "Err... that's ten Miss"
Teacher: "That's very good Johnny!
Johnny: "Very good Miss? It's bloody perfect"
Adam Mangler Depends what she is commenting on
I dont get it
Donlans Donlans 5+5 is necessarily 10. There is no subjective extent to which it is true which would be implied if the teacher literally meant what she said in terms of the question
When and where was this lecture given?
"Pi-symposium", Sunday 2016.06.12, 1700-1900 at Pop House Hotel, Djurgårdsvägen 68, Stockholm.
A statement that is true and not provable. Wow!
A lovely vid sir quite inspirational
I suppose another place this shows up relates to Turing's Halting Theorem. Since you cannot in the general case decide if a given algorithm will halt, you run into the situation where there are necessarily algorithms that do not halt, but you cannot prove they do not halt.
I couldn't tell you how that relates to Godel's work in any meaningful way, but that is a problem that's obviously more useful to consider than Godel's example.
Transformers And Power Rangers Toy Reviews the commonality is what Bertrand Russel called the vicious cycle principle, where a paradox is formed by either defining a set or a function or whatever in terms of itself, or giving a function itself as an input, or giving it an input which is defined in terms of the function to which it is an imput. Russel and others discovered these types of paradoxes in naive set theories such as Cantor's, and Russel set out to relinquish the vicious cycle paradox with type theory. He failed because the formal system he invented can itself be looked at as a function which relates whether or not a string of symbols is a theorem of the system, in the same exact way that the hypothetical program in the halting problem tells whether or not an input is acceptable to a program. When you feed the formal system a function defined in terms of the formal system itself, you get an ultimate form of the Vicious Cycle paradox which can never be put down by virtue of the essence of formalism in general.
Transformers And Power Rangers Toy Reviews the halting problem is just a special case of the incompleteness theorem.
Just thumbed this up to 1000..feels good
awesome!!!!!
Is the mind capable of reasoning about things in more flexible ways than a formal system allows?
If not, does this mean there are truths that just can't be reached by any reasoning we do?
Well… ya? We live in a formal system
@@alexbuckley4378 How can you prove the statement “we live in a formal system” to be true or false? I don’t believe you can.
@@MarcosElMalo2 I infer the statement. I observe that many other systems are run off of formalisms. I believe to be living in a system therefore Formalisms run my life
Hofstadter: explains how Godel encoded one formal system into another to prove incompleteness
Also Hofstadter: this is how you're conscious
me: wait, wat?
@Simeon A. Scott th-cam.com/video/SOWt2fBI1VI/w-d-xo.html
@Simeon A. Scott
"It remains a theorem, and not a proof, because it only shows that there is no way to establish a proof."
That is categorically not true. You don't understand the theorems at all so try doing some more research. And you clearly don't even know what a theorem is either because a theorem by definition IS a mathematical proof lol!
Go back to school.
@Simeon A. Scott
Sure, if you look at the full sentence which gives it context it means that as long as you accept the axioms of the formal system being used, proving a theorem means it's true, ie a theorem is a proof.
The only way you can deny the truth of a theorem is to deny the axioms of the system which no rational person would do as axioms are self-evidently true.
However I'm curious as to why you ignored the first sentence in that Wiki entry which confirms what I said.
I'm not sure if you are cherry picking, or just genuiney curious to learn but were confused by the sentence you quoted.
@@Oners82you can have theorems that are not mathematically proven. Scientific theorems for example. Which is probably what he meant.
@@Hvantmiki
There is no such thing as a scientific theorem. Theorems are mathematical proofs by their very nature, and are necessarily true given the truth of the assumptions going into them and the validity of the logical inferences inferences used.
Science gives us laws and theories, NOT theorems. Theorems are mathematical, period.
Excellent take down of Integers and Symbols. Do we also have similar problems with self-referential movies (possibly video feedback, as proposed in I am a Strange Loop), and self-referential geometry (moebius strip, Klein bottle). What is the most complex self-referential geometry (DNA?). What if we're a bunch of bottles in a bottle universe? I'd rather have a bottle in front of me, than have a prefrontal lobotomy! Cheers!
Good lecture. I like Godel, and also the lecturer here. But, I laughed at the sound of "this twiddle means not", or one could say, "This twittle means naught". I'll take humor anywhere I find it (even if it isn't there).
"Would you answer "no" to this question?"
- Thank you!
I would not.
@@plaguedpixel39 😀
maybe
No, because I would answer yes 😊
Absolutely.
At about 14:00 he mentions the unpredictable nature of the length of
formal proofs. I'm not so sure about the universality of that, as at
least from my read of them, some protothetical systems have much more
regular properties in terms of how proofs can work.
Consider this system (the axioms aren't independent, but say we
ignore the requirement of independence of axioms), under substitution
and detachment for the conditional C, and with D as a functorial
variable (it's a variable for any of the unary functions in {0, 1}):
1. 1
2. CD1DC00
3. CD1DC01
4. CD0DC10
5. CD1DC11
6. CD0CD1Dp
Now say we want to prove a formula with n variables. We need to prove at least (n2)
formulas to use axiom 6 to generalize to the intended formula. We also
have that any formula detached from a substitution instance of 2., 3.,
4., or 5. used in the proof of the sought after formula can come as
shorter or equivalent in length to the sought after formula.
Also, consider the system with CpCqp as it's sole axiom under
detachment. How many steps it takes to prove a formula, or a
substitution instance of a formula, ends up as completely predictable.
Additionally, if we instead consider CpCqp with only condensed
detachment as the rule of inference, then every formula derived after
the axiom is longer than the axiom and is longer than every other
formula previously proved. The formulas derivable in that system go
(CpCqp, CpCqCrq, CpCqCrCsr, CpCqCrCsCts, ...)
Edit:
As another example, consider an equivalential calculus under these three axioms:
1. Epp
2. CDEpqDEqp
3. CDEEpEqrDEpEqr
We can immediately take a substitution instance of Epp which will
have the same length as the sought after formula, and the same number of
variables as the sought after formula. Then we use instances of 2. and
3. (or 3.' CDEpEqrDEEpqr) to transform the substitution instance of Epp
into the sought after formula. All instances of 2. can have the same
length, and all instances of 3. can have the same length.
When he mentions the universality of Goedel's theorem, I'm sure he was considering only the systems that contain arithmetic.
@Simeon A. Scott "The fact there are short theorems does not mean we can predict the length of any theorems." Yes it does. There exist perfectly regularly ways of deriving theorems. There exists theorems of a *known* finite length. Since some theorems, along with some rule of inference, always allow for a longer theorem to get derived of a fixed greater length, it follows that the length of some theorems is predictable.
@@Spoonwood Some, but not all.
@36:11 "A picture of Gödel with a (relatively) unidentified peasant"
Mr Stone!
That's Godel standing with Einstein.
Xam Malist Is it tho?
Fibonachos the guy who married his cousin?
Unrelated to the lecture: this channel has an awesome logo.
50:00
How is it possible to "encode" a statement that contains number g AND some other stuff into a number thats not larger than g
nvm I got it
@@desambio can you explain
29:28 Over 9000 you say?!?!
Thank you for Godel Escher and Bach
I think Dr. Hofstadter knew Mark Twain.
All explanations of Godel fall short of explaining how the actual construction looks like.
His proofs are printed in plenty of books on mathematical logic. What stops you from reading the real thing?
Yes, even this lecture involved hand waving and glossing over. So? What’s your point?
What about tuppers self referential formula?
what about tuppers? I dunno never heard of it
As inventor of the "=" sign counts Robert Recorde (en.wikipedia.org/wiki/Robert_Recorde). He was imprisoned for debt.
Limits of Mathematics
Fabulous ❤
Is this a well formed formula:
0 => 2+2=4. (Read if 0 then 2+2=4)
No.
What's wrong with this cameraman?Doesn't show the white board...eh...
Is 1 above similar to the well-foundedness principle?? Are they "equivalent"? And what is that called in Formal Category Theory?? Yeah I know it is transferability, I said formal.
As nouns the difference between formula and theorem is that formula is (mathematics) any mathematical rule expressed symbolically while theorem is (mathematics) a mathematical statement of some importance that has been proven to be true minor theorems are often called propositions'' theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called ''lemmas . "wik Diff"
Is the incompleteness theorem itself incomplete ?
Let's hope so !
Footnote:- I wonder if the universe is incomplete.
No.
@@Oners82
No what ?
@@tedgrant2
No the theorem is not incomplete.
@@Oners82
Newton thought that his theorem was complete.
Then along came Einstein.
And I say, his isn't complete either.
Our puny mathematics can never completely describe everything.
@@tedgrant2
Newton's theory was not incomplete, it was just fundamentally an incorrect description of spacetime. You are confusing completeness for accuracy.
"And I say, his isn't complete either."
Read your post again dude. You ASKED if it is incomplete in your very first sentence so I answered you.
"Our puny mathematics can never completely describe everything."
Sorry but that's just not true. I fear you do not understand what the incompleteness theorems actually say about mathematics at all.
thank you.
hashtags are the thing now.
Magic thinking about the psychology of condensation is an exercise is discovered quantization, ie is Mathemagical Thought Experimentalist's practical Intuition confirmation of the Universe of Logarithmic Time Duration Timing modulation cause-effect mechanism to discover coherence-cohesion objectives in orthogonality, thoughts uncertain because unlimited content is another format of i-reflection containment => inclusion-exclusion timing-phase proportioning.
That’s over 9000!
El limite es la locura.
Didn't khwarizmi the founder of algebra work with symbols before the time he says?
No
Nope. Al-Khwarizmi used written out words.
58:16 sometimes smaller???
1:58 Robert Recorde. He was Welsh, not Scottish.
The man's hip.
Fri tanke=Free thinker
The undecidable is liberating.
th-cam.com/video/U84nxCsQpus/w-d-xo.html
Robert Record invented the equal sign, and he was Welsh.
Celestial Teapot stfu we know already
is that on the record?...=P
*Recorde
Perceptions of scaling/proportioning density-intensity, alignment with line-of-sight holographic unity is the intellectual process of positing-> positioning a coherence-cohesion sum-of-all-histories resonance state, ..posturing limited content in Perspective context and logical i-reflection superposition containment-condensation coordination. Perfect Gas Thermodynamics alternative path terminology.. (Not recommended for beginners?)
'How can 'This sentance is false' be a paradox when it is the unequivocal fact of an utterance.
Wasn't Ludwig Wittgenstein part of this dialogue: between Kurt Gödel & Bertrand Russell?
I really don't know why anyone imagines that "this sentence is false" is a paradox or any sort of challenge to logic. This is simply a category error. A 'sentence' purely as a 'sentence' - i.e. a vehicle to communicate content, but without any reference to any content - cannot be true or false. The adjective is irrelevant. It's as absurd as saying "this signpost, which does not have any information on it, and which does not have any reference to any location, is pointing in the wrong direction". The signpost, purely as a signpost, cannot be right or wrong. All such paradoxes contain a fallacy. Logic cannot be relative or limited, otherwise it cannot function as logic. The principle of explosion (ex falso quodlibet) proves this.
@K F So if logic can be relative or limited, then please explain to me what OTHER method I am using - or appealing to - in my statement, which you quoted.
But why are you reducing sentences to the words they're composed of when they can be clearly interpreted. What we're really evaluating is the statement that's conveyed through the sentence, not the sentence itself, but the way it's usually phrased is a useful shortcut. What if I said "this statement is false"?
I would say that your critique is rather analogous to saying "this signpost, which has a left pointing arrow on it doesn't really point anywhere because it's just metal and paint".
@@julija5949🙏🏼
10:25 But if you remove what it is the symbols represent then you don't have anything...
You have a string of symbols, sure - but it's now just comparable to nothing.
Albeit_Jordan adding this as a "disclaimer" would then solve the issue, right? as in, we attribute this and this meaning to these and these symbols
That's the whole point. You just manipulate the string of symbols according to the rules and at the end you bring the meaning back onto the string and you have something that has to be true.
Looked like Douglas Hofstadter was wearing a tube top & tie in the thumbnail lol 😆
Coat Girdle*
If there are mathematical truths that cannot be derived, then you essentially have to remain a platonist
How so? It basically invalidates Platonism because it tells us that we have free choice for every single unprovable theorem. We can assume that it's true and see where that leads us or we assume that it's false and check out that path. Whatever is prettier we keep.
@@schmetterling4477 That’s not what we did though in the historic case of the parallel axiom. We kept both Euclidean and Noneuclidean geometries, as both describe objects of interest. They both exist in parallel in the platonic sphere.
My point is: in order to even discover that something is logically independent of the axioms that you have, you have to look at concrete models, and those models themselves only exist in the platonic sphere. But you have to keep them around ontologically because what Gödel actually proved is that Hilbert’s empty formalism doesn’t work. Also, if we ever discover inconsistencies, that can only be so because we misunderstood something that is out there. If math was just something that we make up or that existed only in our language, we could essentially prove anything we want.
@@schmetterling4477 The major misconception many people have and why they think that Gödel invalidates Platonism (although he himself was an outspoken Platonist) is that the platonic sphere of ideas is limited by some fixed model of set theory because this is the foundation of large parts of contemporary mathematics. But the mere fact that logicians constantly look at different models of set theory and that category theory ditches sets altogether (it’s just one category among many) should really give you some pause. I mean, no linear algebra person would talk about *the* vector space. They derive a lot of theorems about things satisfying the axioms of vector spaces, but you can construct lots of very different instances - all of which co*exist* peacefully outside material reality in the realm of logically possible universes.
@@markuspfeifer8473 True, we can keep both options. Mathematics is an infinity of crossroads, it never ends and nobody can tell just how long a particular dead end will be, should we have the misfortune to enter one. I am pretty sure that Plato would have been horrified by that thought and he would probably have rejected it. Just my guess... the guys in antiquity were not that flexible in their thinking. They liked things simple, predictable and finite.
@@markuspfeifer8473 Logic has not much to do with set theory, though. If anything, it has something to do with finite automata and one could ask if that isn't way too restrictive a point of view, either. I do agree with the category theory folks: we have to stop thinking about "stuff". We need to learn to think in terms of properties and equivalences, only. Physicists have ditched "stuff" a hundred years ago, when quantum mechanics was discovered. I wonder what will happen to mathematics once mathematicians will catch up to that paradigm.
Kurt Goedel (I have no easy umlaut) did not show that mathematical thinking could not "be captured in a formal axiomatic reasoning system." A better expression might be that it could not be captured in a single, formal axiomatic reasoning system. Indeed, he showed the power of formal axiomatic reasoning and its limitation: that within any such system will be be meaningful statements that are true (or cannot be demonstrated to be false or leading to a contradiction) that are not provable. One may broaden the subject system with further axioms, but then there would be other true statements that are not there provable. I don't find this displeasing, as a personal matter.
Poor camera work .. absolutely butchered the slides
What is he talking about expressing the axioms of Principia Mathematica as an integer? He didn't write an integer and neither did Goedel. He wrote a numeral. Numerals aren't integers.
What's the diff between numerals and integers?
A numeral is used to represent an integer, but if you want to go further, he did not write a numeral, he just made some mark on a board
Thanks
If I understood you correctly, you are being extremely strict about the choice of vocabulary. 40:40 don't you think most English speaker would agree that that's an integer on the screen? The meaning is there and clear. Also, Goedel did map formulas into integers. In fact, Goedel numbering maps formulas into integers not numerals. (I know that doesn't mean he WROTE an integer according to your convention). I think you are pushing it too far. If you are going to be that strict you might as well say by me writing down "George", I didn't write a person's name, I wrote a string of letters.
I’m certainly no mathematician, but it seems Dr. Hofstadter mischaracterizes the nature of Godel’s paradoxical statement when he says, of any formal system, that “there are an infinite number of holes” (~55 min mark). It seems, rather, that there is but one hole that grows infinitely larger with the size of the formal system upon which it is applied.
If I’m mistaken I’d welcome correction so long as the response be made in good faith.
@Simeon A. Scott
And you are certainly no expert in writing coherent sentences, lol.
You’re confusing the English language with topology.
when fun part begins?
Idiot
"This Sentence is False" because => Duality. A ONE-INFINITY Singularity Eternity-now Interval Conception identification and e-Pi-i sync-duration positioning integration relative-timing ratio-rates superposition identification time-timing substantiation. Pure-math Fluxion-Integral Temporal superposition Calculus.
So it's the equivalent in Math-Physics to the Measurement Problem in Physics-Math.., floating in No-thing at the Centre of Time Duration Timing Origin, in 0-1-2-ness GD&P line-of-sight Superposition-point projection-drawing of symmetrical equation Sentencing, ie expressing Judgement in abstract reasoning.., which occurs naturally in probabilistic numberness dominance sequences of AM-FM time-timing quantization differentiates, condensation modulation of Ideal Gas Thermodynamics for Math-Phys-Chem and Geometrical phase-locked empirical laws of log-antilog Quantum-fields Mechanism Mathematical Form following Reciproction-recirculation e-Pi-i infinitesimal shaping phenomena.
"Logic" is minimal action positioning terminology, another aspect of wave-particle coordination-identification and Calculus. Sentencing is the combination of leading, lagging and synchronous spin-spiral quantization.
The logic of QM-TIME Completeness, or infinitesimal connection e-Pi-i cause-effect-> functional roots 1-0-infinity and therefore transverse trancendental zero-infinity range probability.., in superposition, self-defines the wave-particle "paradox of dualistic orthogonal-normal universality of axial-tangential No-thing => this particular Sentencing statement in/of ONE-INFINITY Singularity, a Fractal Function Flash floating Be-cause-effect Actuality, of Eternity-now timing spacing modulation, Math-Phys-Chem and Geometrical Interval Conception.
42
Roger Penrose would do well to explain Gödel like this and not by talking about taking a class from Dirac and Stein telling him it's about understanding.
He claims 1^2 = 1`
(Nah!)
1 ^2 1 (a number cannot both multiply and not multiply itself. Russell's Paradox.
B^2 B (A barber cannot both shave and not shave himself, in a village containing only one villager, and that is all you need; seets have nothing to do with it).
To multiply two elements requires they both exist 2+0 = 2:
# = 1 + 1 = 2
#^2 = (1 + 1) [1^2 + 1^2] + [2(1)(1)] = 4(1^2) binomial expansion
4(1^2) (# - 1) = 1
I call bull puckey
It seems to me that Godel can provide a number for the statement "x = 3" but cannot prove it, for x a variable)... so "x = 3" is "unnproveable" but 3=3 is "proveable", because proveablity requires tautologies? (x is not a symbol of the meta-language).
Duh!
Robert Recorde invented = and he was Welsh
What's this about quantum logic?
i think it's more like the limit of langauage than limit of logic.
It’s the limit of formal systems.
Or maybe Godel is fallowing in the steps of all who are fascinated by what is called, "understood", and recognised as logic, (logos), or not.
And in the process is warning, and proving, simultaneously, either way.
As a warning, as a proof, as warning of the proof, as the proof of the warning!!!
That you cannot used code, any type of code to represent logic, (logos).
If you do!!! Then any possible Godel will use the same code, to not prove the very code used.
Because code, any code can only prove or not prove itself.
Unless anyone says to any possible Godel, that anything to this, and this to anything, is a point, as is a line, as is a point, as is a line.
Oh wait, it has already been said, known but not fully understood as geometry.
Unless anyone calls it (wrongly) measurements, (the geometry that is) to buy- pass the confusion, and (vua la) anyone gets science, perfectly making sense, perfectly not making sense.
wff: write a compiler.....
21:00 Why would you keep Hofstadter's butt in frame instead of the dense formal notation he's pointing to?
Writing a math textbook is usually a route to financial success in both short and long terms, but Russell & Whitehead managed not to do that. They had poured years of work into their "Principia", but it took about 30 years to earn back its advance, having been proven incomplete almost as soon as it was published.
I opened a new chapter in memetics and with a new name. Psychovirology. Using this methodology, I also managed to demonstrate the invalidity of Kurt Gödel's first incompleteness theorem, because it only spread memetically and not scientifically. Douglas Richard Hofstadter was also wrong about this in one of his books.
Do you have any publications, reviewed papers, books? Thank you.
@@MuratIsikHome H: Trükkös Mém - Túl Richard Dawkinson e- könyv GB: Tricky Meme - Beyond Richard Dawkinson e-book