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/sci/ - Science & Math

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10130086 No.10130086 [Reply] [Original]

I don't know what the main idea behind formal logic is (think hilbert style logic, propositional logic, predicate logic, PL+natural deduction, etc., => vs <=>, modus ponens, etc.)
in math, are proofs made with formal logic? can you formalize any math argument? inb4 principia mathematica, I'm asking what the fuck is "iff" or "if... then..." in mathematical proofs. Can it be translated to propositional logic or is it informal logic, is it an informal argument that should convince the reader about the truth of the proved thing? is it informal like solving a law case is? proof elegance and etc. are just measures of how eloquently you defended your client?

>> No.10130095

>>10130086
I Think Susanne K Langer wrote some books on this shit. I read half of one and decided I was too stupid. She was a student of one of the authors of the book you refere to.

>> No.10130101

>>10130095
she was a student of Whitehead? do you remember anything from that book? If not, it's ok, I can do some research on it, thanks for the suggestion
*sucks a well-formed, 4-on-the-bristol-chart turd*

>> No.10130105

bump
*sucks a well-formed, 4-on-the-bristol-chart turd*

>> No.10130122

the second bump
bump
*sucks a well-formed, 4-on-the-bristol-chart turd*

>> No.10130128

>>10130086
yes math is made with formal logic, this was actually a problem in the 1800s because nobody could actually prove some things in math

>> No.10130133

>>10130128
please elaborate, or else you will have to suck on my well-formed crohn turd

>> No.10130145

>>10130128
please, more, turd-sucker! I was becoming interested and i briefly stopped sucking on my turds, but if you dont' continue I will suck everyting!

>> No.10130150

>>10130133
>>10130145
wait what do you want to know exactly

>> No.10130156

>>10130150
well i mentioned principia mathematica, check the OP text again
what's the connection between what you said, PA and godel's incompleteness theorems?

>> No.10130182

>>10130086
“Iff” is <=>
“If then” is =>

At the end of the day mathematical proofs are essentially just exploiting the principles of formal logic. Direct proofs, contrapositive proofs, proof by contradiction, proof by induction, etc. can all be represented through formal logic.

Complexity in mathematics typically follows an onion structure. The more complex stuff is usually that complex simply because it relies on some manifestation of the under layers, something like differential geometry is a perfect example. You would never want to go through the trouble of formulating a proposition in differential geometry from pure logic. The metric in general relativity is 10 (actually 16) separate differential equations just by itself.

To this end, this is the motivation behind axiomizing theories such that formal logic is better suited to tackle analysis of complex mathematics. Here is one such example (it’s been done numerous times) of an axiomization of special & general relativity. https://pdfs.semanticscholar.org/5362/cb94bcf48824782b176fbb2a1cc507f9a1fe.pdf

The length is no surprise.

>> No.10130186

>>10130156
All of logic, ie objective distinguished properties in the abstract, can be described in terms of set theory, set theory in the abstract, sometimes loosley called category theory, says that any abstract thing is simply a collection of of more abstract things ie more sets, ie the elements of that set. Elements in the set are compared with logical operators and conclusions can be reached. (whether two sets of something both contain or don't contain something)

Logicians had known this is the fundamental properties of how the mind uses logic for a while, but it got to the point about 200 years ago where people started questioning whether these properties actually applied in math, which was getting extremely complicated or was math not logical at it foundation. Principa mathemtica was written to try to try to give a logical foundation for mathematics, because if there wasnt one, you could quite literally prove 1+1=3 in terms of propositional logic. This book had axioms that were actually meant to be the foundation of the abstract logic for math.

However, russel proved there was a paradox in the institutional structure of the set theory proposed: you could use propositional logic to claim that there exists sets that contain all sets but them selves. Its really just an inconsistency that proved mathematics was not completely logically sound.

Enter Kurt Godel, this problem interested him so he tried to solve it, he actually ended up finding something profound: you cant actually prove everything that is propositionaly true in a logical system, even if they are necessarily true. His method is a bit convoluted but basically in every system built upon this foundational logic, including math, is NECESSARILY incomplete, meaning the problem we are facing in math right now could very

George cantor in the 1800s pointed out that there can be some infinities that are actually more than other ones and he proved this using set theory at the time, this proof still holds

>> No.10130197

>>10130186
including math, is NECESSARILY incomplete, meaning the problems we are facing in math right now, like the Riemann hypothesis could very well be true, but there is literally no way to prove it. This was a revolution in logic and is actually quite underrated

>> No.10130233

>>10130186
>>10130197
>>10130182
PUSSY GAVE ME LIFE!
PUSSY SHALL MAKE ME A MATHEMATICIAN!

I AM a stupid human, an evolutionary peon, I am the human equivalent of a bee drone or a worker bee, I am a slave, I have a role, a goal, I am manipulated, controlled, violated, destroyed, abused by biological imperatives

so what? with the little awareness and will I have I will beg the god of PUSSY to make me a mathematician! after all, being a mathematician is no different from being a bee drone, correct? you're both losers, virgins, slimeballs, turd-sucking fuckers waiting for a chance to have sex but in reality you're fucking useless to the hive and you can get kicked out by the bees at any moment, and remember, the queen bee can sting without killing herself, FUCKER!

My mantra: I am the homo sapiens equivalent to the bee drone. I was made to fuck, and maybe I will get lucky. If I don't, it doesn't matter, the worker bees will kick me out and I will just fuck off. My mantra: biological imperatives use and abuse me, but I am allowed to have a say in this because rationality is an evolutionary experiment. I can use pussy to get to mathematics and vice versa! PUSSY => MATHEMATICS /\ MATHEMATICS => PUSSY

the geometry of the pussy gives an incentive to practice mathematics, and this incentive is 100% instinctive, it's bullet proof, I will do math like the monkey fucks another monkey! repetitive, mechanistic, 100% simple!

>> No.10130270

>>10130086
The formal logic you describe is most commonly used for "real life" concepts that uses natural language like English. Concepts that are tied to "physical" things. The physical is logical, our bodies are logical in their structure, the "elements" of the earth interact logically (put water on fire, it goes out).

Mathematics (in terms of numbers and their operations) is also a language, but one that is much more precise, and doesn't need to be tied to the physical to work logically, it can operate separately to it as it is metaphysical in nature, it is not subject to physical "laws", but still subject to logical "laws".

>> No.10130311

>>10130270
causality is not logical, turdman, it's a category, and if you add the intuition of space-time you can start understanding shit, according to Kant, the number one authority on the philosophy of mind!

>> No.10130427

>>10130311
What is time made of?

>> No.10130845

>>10130427
no idea, but I just wanted to give a counter argument