On the eve of the computer age, philosophers and mathematicians are looking for the possibility of basing mathematics on the principles of logic.
In the previous lecture, we saw what motivated such an undertaking. Russell and Whitehead's "Principia Mathematica" (1910 - 1913) was a masterpiece in this endeavor.
I'm no expert in this kind of mathematical stuff, but if you can reduce mathematics to the principles of logic, maybe you could solve all the unsolvable problems in mathematics.
The basic principle of logic is simple: you have statements that are true or false and if you put them into a valid argument that leads to a conclusion, then the conclusion MUST be true if those statements are true.
For example: When it rains (p) I get wet (q), but I don't get wet (~q) because I use an umbrella (r)...
((p --> q) & (r -->~ q)) --> ~q
Translated into the formalization of propositional logic, the sentence could look like this. I cannot guarantee that it is 100% correct. It has been half a century since I was intensively involved in studying proposition logic.
If the argument is valid and all premises are true, then the conclusion is also true. Making this applicable to all of the mathematics seems to me to be the best thing you can imagine as a mathematician.
The mathematician who built on the ideal of the "Principia Mathematica" is the German mathematician David Hilbert (1862 - 1943).
This ambition comes to an abrupt end when Kurt Gödel demonstrates the incompleteness of the "Principia Mathematica".
He shows that there are mathematical statements that the "Principia Mathematica" can express, without, however, being able to prove their truth or falsity.
A variation on Epimenides' statement about the lying Cretans is central. Gödel formulates in the "Principia Mathematica" language the sentence "this sentence is unprovable", also called the Gödel sentence.
The Gödel sentence refers to itself being unprovable, which proves to be an insurmountable problem for Hilbert's plans. The Gödel sentence is both true and unprovable, with which Gödel shows that the "Principia Mathematica" is incomplete.
I will spare you all the complicated reasoning regarding this sentence and the "Principia Mathematica", but the main point that Gödel showed was that there are inherent limitations to what a formal mathematical system can prove.
But of course, this does not mean the end of the story. There is always someone who can come up with something new that will help us further.
Ok, the ideal of the "Principia Mathematica" seems to be out of the question, but Hilbert also worked on another issue, which was a source of inspiration for Alan Turing (1912 - 1954), one of the first computer scientists.
Thank you for your attention again...
Main Sources:
MacMillan The Encyclopedia of Philosophy, 2nd edition
TABLE OF CONTENT -----------------------------------------------------------------
1 - 100 Philosophers 9 May 2009 Start of
2 - 25+ Women Philosophers 10 May 2009 this blog
3 - 25 Adventures in Thinking 10 May 2009
4 - Modern Theories of Ethics 29 Oct 2009
5 - The Ideal State 24 Febr 2010 / 234
6 - The Mystery of the Brain 3 Sept 2010 / 266
7 - The Utopia of the Free Market 16 Febr 2012 / 383
8. - The Aftermath of Neo-liberalism 5 Sept 2012 / 413
9. - The Art Not to Be an Egoist 6 Nov 2012 / 426
10 - Non-Western Philosophy 29 May 2013 / 477
11 - Why Science is Right 2 Sept 2014 / 534
12 - A Philosopher looks at Atheism 1 Jan 2015 / 557
13 - EVIL, a philosophical investigation 17 Apr 2015 / 580
14 - Existentialism and Free Will 2 Sept 2015 / 586
15 - Spinoza 2 Sept 2016 / 615
16 - The Meaning of Life 13 Febr 2017 / 637
17 - In Search of my Self 6 Sept 2017 / 670
18 - The 20th Century Revisited 3 Apr 2018 / 706
19 - The Pessimist 11 Jan 2020 / 819
20 - The Optimist 9 Febr 2020 / 824
21 - Awakening from a Neoliberal Dream 8 Oct 2020 / 872
22 - A World Full of Patterns 1 Apr 2021 / 912
23 - The Concept of Freedom 8 Jan 2022 / 965
24 - Materialism 7 Sept 2022 / 1011
25 - Historical Materialism 5 Oct 2023 / 1088
26 - The Bonobo and the Atheist 9 Jan 2024 / 1102
27 - Artificial Intelligence 9 Feb 2024 / 1108
The Discussion
[13:13] .: Beertje :. (beertje.beaumont): Thank you Herman
[13:13] Max Chatnoir: Thanks, Herman.
[13:13] Particle Physicist Bejiita (bejiita.imako): Well that Principia Mathematica sounds just how every programming language works
[13:13] Particle Physicist Bejiita (bejiita.imako): you check statements and alter execution flow
[13:13] Daruma Boa: Thank u
[13:13] Gemma (gemma.cleanslate): nice
[13:14] Max Chatnoir: I'm wondering if the Goedel sentence is a limitation of mathematics or a limitation of language?
[13:14] Particle Physicist Bejiita (bejiita.imako): if this bool == true then do this and this integer == 42 then do that and so and so
[13:14] herman Bergson: I find this relation between logic and maths fascinating...
[13:14] Particle Physicist Bejiita (bejiita.imako): thats how all programming works in the ground
[13:14] herman Bergson: and the fact that within a mathematical system you still can not proof every statement of the system
[13:14] Gemma (gemma.cleanslate): it has come upi before with some of the other projects
[13:14] Gemma (gemma.cleanslate): yes
[13:14] Particle Physicist Bejiita (bejiita.imako): so here i say the principle holds, if i now grasped it all correct
[13:15] herman Bergson: What is even more fascinating to me is that all this is produced by our brain
[13:15] Particle Physicist Bejiita (bejiita.imako): hmm
[13:16] herman Bergson: and what consequences does it have for computers, the pure logic machines and Artificial Intelligence?
[13:16] herman Bergson: That is what I am after to learn in this project
[13:16] Gemma (gemma.cleanslate): I am going to have to log out before class ends
[13:16] Gemma (gemma.cleanslate): I will just poof
[13:17] .: Beertje :. (beertje.beaumont): ok
[13:17] Max Chatnoir: Good to see you, Gemma. :-)
[13:17] herman Bergson: As long as you stay healthy everything is allowed here Gemma :-)
[13:17] Particle Physicist Bejiita (bejiita.imako): well in programming everything boils down to true or false no matter if it is a number, a bool foat ect, but thats only with programming and not general math
[13:17] Gemma (gemma.cleanslate) GIGGLES!!
[13:17] Gemma (gemma.cleanslate): ...LOL...
[13:17] Particle Physicist Bejiita (bejiita.imako): in programming u check statements, in general math you do a bit more
[13:17] Particle Physicist Bejiita (bejiita.imako): i guess thats the difference
[13:17] herman Bergson: That is the point Bejiita.....this black and white world computers live in....
[13:18] Gemma (gemma.cleanslate): i like the idea of combining it all since it all does go together
[13:18] Particle Physicist Bejiita (bejiita.imako): yes
[13:19] herman Bergson: Computers are better than our brain within well defined areas like chess rules or Go rules.....
[13:19] Gemma (gemma.cleanslate): sure thinks faster than I do
[13:19] herman Bergson: But that is just related to their unnatural calculation powers
[13:19] Max Chatnoir: Programming language has to be more precise than "regular" language.
[13:19] Gemma (gemma.cleanslate): right
[13:19] Particle Physicist Bejiita (bejiita.imako): aaa
[13:19] herman Bergson: It is Max.... otherwise you have bugs
[13:20] Max Chatnoir: So does that restrict the kind of problems that artificial intelligence can productively address
[13:20] herman Bergson: By the way....funny story...do you no why a programming error is called a BUG?
[13:20] Max Chatnoir: ?
[13:20] .: Beertje :. (beertje.beaumont): but the world isn't only black and white, it has a lot of greyscales
[13:20] Gemma (gemma.cleanslate): It was a bug that took down ATT last week
[13:20] Max Chatnoir: Yes.
[13:20] Max Chatnoir: It was a real bug
[13:20] herman Bergson: So funny
[13:20] Max Chatnoir: In a very large computer.
[13:20] herman Bergson: It was Gemma???????
[13:20] Gemma (gemma.cleanslate): yes
[13:21] herman Bergson: THAT is hiostorical!
[13:21] Particle Physicist Bejiita (bejiita.imako): for ex when i work in Unreal Engine i see this all time. i have branch for boolean but i also have the switch on string , int, float statements and they all boil down to if a comparison is true or false
[13:21] Gemma (gemma.cleanslate): script error
[13:21] herman Bergson: ohhh...not a real bug?
[13:21] Particle Physicist Bejiita (bejiita.imako): aaa yes that is where the word bug come from
[13:21] Max Chatnoir: No, I meant the original "bug"
[13:21] Max Chatnoir: was a real insect
[13:21] Max Chatnoir: moth?
[13:22] Gemma (gemma.cleanslate) GIGGLES!!
[13:22] Gemma (gemma.cleanslate): ...LOL...
[13:22] Particle Physicist Bejiita (bejiita.imako): a cockroach in a relay, and the one fixing it also coined the name debugging writing "i debugged the relay" or something similar
[13:22] herman Bergson: Yes it was, Max....blocked a relais somewhere in 1948 or so :-))
[13:22] herman Bergson: I love that story :-)
[13:22] Max Chatnoir: Humans and insects!
[13:22] .: Beertje :. (beertje.beaumont): the power of a moth:)
[13:22] herman Bergson: It teaches us modesty
[13:22] Particle Physicist Bejiita (bejiita.imako): its quite charming
[13:23] Gemma (gemma.cleanslate): goodness
[13:23] Gemma (gemma.cleanslate) GIGGLES!!
[13:23] Gemma (gemma.cleanslate): ...LOL...
[13:23] Max Chatnoir: Yes it is.
[13:23] Particle Physicist Bejiita (bejiita.imako): in Swedish we also use the term avlusning = delicing
[13:23] Particle Physicist Bejiita (bejiita.imako): but i think thats an older term here
[13:24] Max Chatnoir: You use it for computers?
[13:24] Particle Physicist Bejiita (bejiita.imako): yes
[13:24] herman Bergson: But now we are at the verge of the computer era.....Alan Turing....
[13:24] Particle Physicist Bejiita (bejiita.imako): aaa
[13:25] herman Bergson: But that is for next Tuesday :-)
[13:25] Max Chatnoir: Yum!
[13:25] Daruma Boa: ;-)
[13:25] herman Bergson: Can't spill all my beans now :-)
[13:26] Particle Physicist Bejiita (bejiita.imako): hehe no we have to have some good moments with this project indeed
[13:26] herman Bergson: YEs we aren't done yet, Bejiita....much more to come
[13:26] Max Chatnoir: An excellent project!
[13:26] Particle Physicist Bejiita (bejiita.imako): ㋡
[13:26] Daruma Boa: Good!
[13:27] .: Beertje :. (beertje.beaumont): interesting:)
[13:27] herman Bergson: And as I said in the introduction...AI looked at from a philosophical perspective
[13:27] herman Bergson: You can have your fun with ChatGPT and BARD ....:-)
[13:28] herman Bergson: Put them to the test
[13:28] Particle Physicist Bejiita (bejiita.imako): aaa
[13:28] herman Bergson: So...
[13:28] Max Chatnoir: I have a new project for Chat GPT. I want it to construct a phylogeny for a mythical creature.
[13:29] Daruma Boa: sounds interesting
[13:29] herman Bergson: a what, Max??? :-))
[13:29] Max Chatnoir: Maybe a dragon?
[13:29] Max Chatnoir: or a unicorn?
[13:29] Max Chatnoir: any other suggestions?
[13:29] herman Bergson: what is a phylogeny?
[13:29] herman Bergson: they never told me that in highschool
[13:29] Max Chatnoir: a family tree for different species
[13:30] herman Bergson: ahha...ok ...a family tree
[13:30] Max Chatnoir: showing the relationships between then.
[13:30] Particle Physicist Bejiita (bejiita.imako): that could be an idea
[13:30] herman Bergson: yes, I get it
[13:30] Max Chatnoir: Like bonobos separated from chimps after we did.
[13:31] .: Beertje :. (beertje.beaumont): but a family tree for an unicorn?
[13:31] herman Bergson: but how do you find out such relations between non existing veatures?
[13:31] Max Chatnoir: Mostly these days based on DNA.
[13:31] Max Chatnoir: That's why I want to see what Chat GPT will do with it.
[13:31] .: Beertje :. (beertje.beaumont): ツ
[13:31] Particle Physicist Bejiita (bejiita.imako): new mythical creatures can at least go into my Unreal 5 projects. I can need some for my backrooms project that im working on at moment ㋡
[13:32] Particle Physicist Bejiita (bejiita.imako): need new entities
[13:32] Particle Physicist Bejiita (bejiita.imako): hehe
[13:32] Particle Physicist Bejiita (bejiita.imako): i havent even started that part
[13:32] Particle Physicist Bejiita (bejiita.imako): need to get better at character making
[13:32] herman Bergson: OK...nice of topic....so time to say
[13:32] Max Chatnoir: character in what sense?
[13:32] Particle Physicist Bejiita (bejiita.imako): but I've seen some awesome ones
[13:33] herman Bergson: Thank you all again...
[13:33] Particle Physicist Bejiita (bejiita.imako): well monsters, creatures, all kind of things
[13:33] herman Bergson: Class dismissed ....enjoy your weekend :-)
[13:33] .: Beertje :. (beertje.beaumont): Thank you Herman
[13:33] Max Chatnoir: Thanks, again Herman, and all the rest of you.
[13:34] bergfrau Apfelbaum: thank you again Herman and class!
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