how do proofs in math work if there's no empirical basis for them, just consensus by peers who agree to certain assumptions/axioms and linguistic framework, it's basically like religion.
also how does proof in science work if it's all inductive, tentative to change and based on peers "repeating" your results. is that really knowledge? No.
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Tongue punch your fart box to increase your low IQ OP.
Go look at the ZFC axioms. Do you really disagree with them on an empricial basis?
You are a retard.
First of all: There are lots of logics that have been developed over the years. Each logic contains (among other things) a "proof system" that outlines how one can deduce if a statement is true (or other things for more exotic logics). A deduction (informally called an argument) within the proof system is called a proof and a true statement for which there exists a proof is called a theorem. It is a purely formal thing with its own semantics and limitations.
A logic by itself has theorems but they aren't super interesting from a semantic perspective. So, from there one typically constructs an "axiomatic system" by attaching to the logic a number of sentences that are true by assumption. These sentences are called axioms. Also it is important to point out that the axioms are not sentences in natural language they are sentences in formal logic and they don't inherently have any semantic meaning. Applying the proof system to these axioms yields more interesting theorems.
Since an axiomatic system by itself doesn't have any meaning/semantics one can try to give it some by associating to it an interpretation of the system called a "model". The model is like an analogy for the axiomatic system where one creates a mental picture (in natural language) about what the system could mean. Then theorems in the system can be interpreted as assertions/insights in the model.
In mathematics people create axiomatic systems where the models deal with abstract concepts like Euclidean Geometry, Set Theory, Number systems, etc... In science people create axiomatic systems where the models are theoretical models about the real world.
Mathematics is not empirical since it doesn't care about the real world. If the axiomatic system diverges from the model then a new axiomatic system is created and both are studied in parallel (eg. all the set theories). Science is empirical and if the model diverges from observations then the theory is wrong.
fedoras belong on lelddit, fuck off
I should add that formal logic wasn't a thing until the early 1900s and around the same time mathematics and science were both put on a logical foundation (to resolve paradoxes and shit).
The point of using logics is that they're super precise and you can be extremely clear about what you mean and extremely precise about your assertions. A drawback of using them is that they are ultimately a formal languages and some things can be difficult to express in them.
As long as people agree on what a triangle and an angle is. The sum of all angles within a triangle will always be equal to 180 degrees, and though this can be verified empirically to an extant, it's still preferable to prove it mathematically.
Besides, being able to find a single counter-exemple is enough to refute a theorem.
Literally everything about this post is wrong. Congrats on being dumber than OP.
>math is all logical with perfect proofs that cover everything and it all makes sense
>can't divide by zero
>can't divide zero by anything
Special case allowance(s): 0/0 OR 0/1 -> Ultimately non-constructive (destructive if followed rigorously)
>that are true by assumption.
The entire field of maths is literally imagination.
"If we assume these things are true then what then must follow?"
>lol muh imagination
Your right. There is no empiricism, because its not empirical.
Mathematicians agree on definitions (eg, lines). And then use logical rules (P -> Q. P is true so Q is true) to determine what follows from the definitions.
Whether those definitions correspond to the world is absolutely open to discussion, and yes proofs usually aren't extremely formal, but the follow the same patterns. Most proofs certainly aren't unquestionable certainties, when they are first presented, but they can usually be translated into more logical forms and analyzed if one is skeptical.
>Most proofs certainly aren't unquestionable certainties, when they are first presented, but they can usually be translated into more logical forms and analyzed if one is skeptical.
This is an important point to emphasize. Most modern mathematicians work at a semi-formal level where the proofs they write aren't in formal logic directly. Rather the proofs are written in natural language with the intention of being clear and rigorous enough that it can convince other mathematicians that a formal logic proof exists and is obtainable.
This is both because formal logic is tedious to work in directly and because it is difficult to read.
An analogy can be drawn to programmers. A programmer who is trying to convince someone that a program exists to perform some task will likely give a general outline using simple abstractions and general ideas. It would be possible for the programmer to actually write the program and present that as proof but that would be tedious to do and difficult to read. Along the same lines a programmer is typically tasked with commenting their code in natural language that is easy to understand (typically giving main ideas and explaining why they're doing what they're doing). This is similar to the work of mathematicians.
What's the highest level math course you've ever taken?
...
A universe in which 2 things make 12 when you put them together would like a word with you.
>how do proofs in math work if there's no empirical basis for them, just consensus by peers who agree to certain assumptions/axioms and linguistic framework, it's basically like religion.
You state what you're assuming and work out the logic.
>just consensus by peers who agree to certain assumptions/axioms and linguistic framework
Well, yes. No one in math is hiding this. You have to make assumptions if you're going to come up with anything. That is how things work.
OP, I suggest you go study philosophy a bit before coming back.
Nobody in the sciences actually believes that what they are doing is absolute- unless of course they aren't real scientists or mathematicians. The core of any field of knowledge or system of understanding is the idea that these are systems that are only supposed to be internally consistent, without relationship to what we might call 'reality'. Science, in the more narrow sense is to take phenomenon that occur in 'reality', and build a system of understanding around those phenomena, in hopes that practical application of the consistencies of the system may be analogous to what reality has to offer.
what is a cyclic group of order 2?
proofs are true because they are deductive arguments
>people are rude to OP but fundamentally agree with him
why do you guys sound so bitter?
>it's basically like religion.
Oh fuck off
...so, highschool.
not what you think it is.
math.stackexchange.com
>level 3 in math studies
Why the fuck are you here?
t. got a level 7 in hl math
That's congruence in the integers. The cyclic group of order 2 actually has only two objects so we're not just dealing with congruence but actual identity (ie. =).
In that case you have to reduce 2 and 12 first, which he didn't do. Neither is even an element of the group then
Yes, I don't see how anyone could possibly agree with the Axiom of infinity.
In additive notation (used with additive groups which is commonly used for commutative groups) one uses multiplication to mean a thing added to itself n times.
So 2=2(1):=1+1.
Doesn't exactly seem like you proved him wrong if you change the notation he was using to prove your point
That is to say, I can define 2 to be equal to 12 in an unlimited number of ways if I change integer literals to no longer mean the integer I'm writing down as a value
one is equality the other is identity.
In logic integer literals don't exist. Instead things like "2" and "12" are just names. In the interpretation an object can have multiple names and saying 2=12 ("2 is identical to 12") means that the object named by 2 is the same object named by 12.
Even in peano arithmetic this is the case. Things only get weird when you start defining everything within set theory where the only objects are sets. In that case you encode "2" and "12" as specific sets and say that that = *("equals", not "identity") is a special thing to deal with your numbers encoded as sets.
To put it in simpler terms. Identity is to Equals as Equals is to Congruence.
>Instead things like "2" and "12" are just names. In the interpretation
And you think this is the interpretation he was using?
>Even in peano arithmetic this is the case.
And do you think he was using Peano arithmetic? He never even restricted what he said to the integers (or a subset), that was an invention from I don't get why you put other words, surrounding context, and notational systems into this guy's mouth when he literally said one sentence and everyone knew what he was talking about.
>how do proofs in math work if there's no empirical basis for them, just consensus by peers who agree to certain assumptions/axioms and linguistic framework, it's basically like religion.
informal proofs (i.e. the common style of writing proofs) always have room for error, which is why formal proofs are becoming a trendy thing
they are...axiomatic...
>when he literally said one sentence and everyone knew what he was talking about.
your thoughts are antithesis to rational discourse
Hi, how's highschool going for you? It'll be better once you're not freshman.
>The sum of all angles within a triangle will always be equal to 180 degrees
>draws a triangle on a globe
You can, but once you do, you're no longer working with CRing/Field and it becomes too much of a pain to work with.
>>can't divide by zero
yes you can
Think of the biggest set you can think of. Now just add one. Keep doing that until you are happy.
>how does proof in science work
science doesnt prove anything
you may only prove mathematics, in it's abstract purity
This
>math is all logical with perfect proofs that cover everything and it all makes sense
Nobody with a good background in mathematics claims this. There are limitations mathematics, as there are in science
>can't divide by zero
Yes you can, but in the sort of math usually dealt with, division by zero isn't defined.
>can't divide zero by anything
Zero divided by any number (other than zero) will equal zero, retard
>I don't see how anyone could possibly agree with the Axiom of infinity.
On an empirical basis? No. The axiom of infinity isn't an axiom of the real world; it is purely abstract. The constraints of the universe aren't needed to do math that's useful and aesthetic. Can I agree on the axiom of infinity on an empirical basis? No. But the notion of an infinite set, in which any element is bound to be followed by one different from the rest, is enough for me.
Shit man, Algebra 2 is more advanced than IB Math Studies.
all mathematicians know this and agree with you; we just don't care. math is still fun.
You can't deny it works
I guess you've always been curious as to how Von Neumann would talk to 'you'. Superior ab origine phenotype represent!
stop embarrassing yourself by posting here, seriously you're shitting up the board
Ever stopped to consider that all the peers and 'real' mathematicians are on this Veeky Forums board and not in ivory towers somewhere?
Or does my mathematica make you feel inferior somehow? It's just math dude, AND I am showing you my working out like a good boy.
i think you have brain damage
*sigh* christ, do people need me to get an MRI before they'll accept this concept called faith? A psychiatrist's report isn't enough apparently.
Should perhaps point out you are referring to 'brain damage', but you present no baseline of health. Median/mode/Medium.
So either 'humans cannot exist without some form of brain damage', or you have access to some chart nobody else does and aren't willing to share it.
This has to be the most cancerous thread on Veeky Forums right now
are you canadian?
Jesus fucking Christ man fuck off
...
>nothing works until there is empirical basis for it
you have to invent something before you can get empirical data from it
Only works with measurements OP.
Shit like "mass" is a joke.
>literally imagination
no, it's figuratively imagination, you retard
>just consensus by peers who agree to certain assumptions/axioms
Yes.
> it's basically like religion
I do not think you know what a "religion" is.
>also how does proof in science work
it does not
I'm starting to understand your posts. This scares me...
Fuck off stalker