/sci/ - Science & Math

Mathematics isn't based on real world but axioms which means there is no gravity where mathematics exists so there is no need for foundations.

>>7366785
The point is to build on top of simpler foundations than the thing of interest.

All human reasoning has this sort of problem actually. You can't prove logic works through logic.

>>7366800
What mathematics does is explicitly state that logic is based on set theory, and that set theory is based on logic. This is obviously flawed. It isn't the same as using the clarity of Euclidian reasoning, and simply posing some definitions and postulates as being self-evident.

>>7366806
My point being that Hilbert's, Russel's, etc. mental masturbation is fucking retarded. Trying to make mathematics self-contained and based on solid foundations won't fucking work. Poincare was right.

>>7366806
>>7366811
The point of studying mathematical logic from the perspective of set theory is to understand more general structures than the usual logical system which might also be called logical systems. Set theory isn't meant to replace logic. That would be silly.

>>7366785
are you past middle school yet?

>>7366785

njwildberger pls go

>>7366785
Wrong. Mathematics is based on:

1) Definitions such as numbers and how to ad/subtract these numbers.
2) Axioms which are thing assumed to be true even if they can not be proven. Eg. axiom of choice (a personal favorite)
3) Theorems which are proven from the definitions and axioms with a few conditions thrown in for good fun.

If math were circular it would not yield any useful results. Unfortunately for you math is really useful in the real world.

>>7366811
well no set can be "self contained", godels incompleteness theorem is pretty much correct

>>7366963
And what do you think, say, the axiom of choice is based on? It's expressed using first-order logic, which is a formal system which is expressed using (partly, of course) set theory (think of the truth functions, etc. which make up this formal system) - which is expressed (partly) using the axiom of choice.

>>7366963
>If math were circular it would not yield any useful results.

What's the reasoning behind this? It can, if we take an intuitionist stance. And most mathematicians are implicitly intuitionist.

>>7366978
Ok you really are an idiot.

>And what do you think, say, the axiom of choice is based on? It's expressed using first-order logic, which is a formal system which is expressed using (partly, of course) set theory (think of the truth functions, etc. which make up this formal system) - which is expressed (partly) using the axiom of choice.

Axioms are not provable. If it were provable it would be a Theorem not a axiom. Anyone who claims to prove them is either right or wrong. If they are right they become theorems.

I am a mathematician, and there is a concept of APPLIED mathematics. You are painting yourself into a corner with the institutionalist angle. There are more of us than institutional mathematicians.

>>7367013
>Axioms are not provable.

Nobody claimed that they are. The question was of them being expressed using first-order logic. Which is, in turn, expressed using set theory (which is expressed using first-order logic).

You seem to mix up "proving" and "expressing". Not only that, but you call someone who maxed out the performance score on a standardized IQ tests "an idiot".

>>7367013
>Institutional
>Same thing as intuitional

Nope

>>7367027
>he brings up his IQ in an online argument
I'm not that guy but this just makes you look like a tool because you're not providing any proof for it and it doesn't add to the discussion in any useful way anyway. People will start taking you seriously when you stop doing this.

No one disputes math and logic are circular. They essentially "mean" nothing. That's why Wittgenstein concludes his Tractatus with

"7. Whereof man cannot speak, thereof he must remain silent."

If you did not realize this and still see the usefulness of logic and math you are a retard. If you did, then you are a shitlord.

>>7367077
>claiming logic is useless
>claiming that math is useless
Can you guess why we love the "can't 'no nothin' " caricature of philosophers here?
Anyway philosophy ought to suffer the same difficulties. It claims dependence on logic. And while many philosophers claim dependence on only logic more philosophers will say it is constructed from observations of the world like you claim axiomatic set theory to be.

>>7366785
I can accept incompleteness, for example OP is a FA_GOT

>2015

Well, there's a lot of different interpretations of what the formalization of mathematics is and I think this is one of the most reasonable ones, but all this is philosophy and you really can't be sure about anything...
First you have your naive set theory, the objects live in your head and satisfies those basic principles that were formalized in ZF(C, or maybe not C).
Now you can collect a number of "meta"-objects, those sets (which are really meta-sets) and form a language and a formal logic based on them and this is a formal theory, with which you can axiomatize set theory, like ZFC, and a model of this axiomatic theory is the universe of actual sets.

read principia mathematica

I came here hoping for a good answer to OP's question but instead I just had a bunch of highschoolers trying to sound smart but missing the point, and the other ones trying to have a point, while nobody actually tried to adress the question.

>>7367027
It was the OP who said "based on" not "expressed in"

Who cares what something is expressed in?
All of mathematics is just ideas or sentences expressible in language.

>>7366926
Fucking this.

How did you advance so far into mathematics and never learn that the theorems of mathematics are logical statements, not assertions or beliefs.

<span class="math"> P \Rightarrow Q [\math] being true does not imply that either P or Q is true.

Fuck off back to youtube Ausberger.[/spoiler]

>>7369782

<span class="math"> P \Rightarrow Q [/spoiler] does not imply that either P or Q is true.

Fuck off back to youtube Ausberger.

>>7366978
the part of logic using truth functions is its semantics, not its syntax. by stipulation, people are usually interested in extensional systems, typically in the category of sets, so of course the semantics will involve set-functions. however, they are distinct and separable components.

for example, if you work in a grothendieck topos the semantics will correspond not to functions, but natural transformations between sheaves. if you work in other topoi, there may not even be mention of set functions, but simply the morphisms of the category where the semantics is carried out.

p much you are equivocating things which are not equivalent. you can't simply say "logic IS truth functions, etc" because you are assuming a set-based semantics, where of course the interpretation of truth will be a set-function, but that's by YOUR choice, and not logic per se. there is nothing circular there. often, the language of the logic is required to be a set, for reasons of computation an manipulatability, which is totally appropriate for most uses.

the results of set theory, as they pertain to logic, are usually about the collections of their models, which are assumed to be sets, but not the working of logic itself. ie no syntactic inference ever says "assuming the axiom of choice". in fact, the same logic can have extensions in different topoi, which may or may not have a certain property, eg AC. but you are not strictly required to view the collection of models in a particular category as a set. in fact, every logical theory encodes a classification in a particular category abt which objects are models. these could be proper classes, not sets, and logic can still handle them. for example, people reason about the class of ordinals all the time.

>>7366926
>>7369785
Maybe you should watch the videos before pretending to critique them?

>>7367027
in short
>first-order logic. Which is, in turn, expressed using set theory

Is simply a gross oversimplification that is wrong. the semantics of a first order logic with set-valued models is expressed using set theory, definitionally, but the workings of logic do not depend on set theory itself.

>>7369981
Norman Wildberger is an amazing teacher.
I would love to attend one of his lectures one day if it were possible.

To study mathematical logic, formal logic needs to be developed twice. First you develop formal logic in a very rudimentary way. In this step you develop the notion of formula, the notion of formal proof. This is done in a finitistic way so there is no chance of criticism. Now that you have a metatheory ready, you can develop ZFC. After that you can use ZFC to develop mathematical logic.

>>7370033
basically sums it up

>>7366785
This is my understanding of the process.

>First you only construct the formal language used as a basis for predicate logic.
>Next you construct primitive notions of truth, false, truth functional, and a very primitive version of set theory. Note that this set theory is extremely limited and really only used as more of a "conceptual shorthand" than a fully fleshed set theory.
>Next combine the two steps above to construct a crude but fully functional version of predicate logic.
>Use this crude predicate logic as a basis for set theory.
>Now that you have set theory go back to the beginning.
>Use set theory and logic to develop the theory of formal languages.
>Once again construct the formal language used for predicate logic but now do so in the context of formal language theory.
>Once again define all the notions of predicate logic but this time do it in the context of set theory (with proper set theoretical functions and everything).
>Now go on and develop the rest of the theory surrounding predicate logic.
>Finally, in the context of your fully fleshed out predicate logic theory, go ahead and reconstruct set theory.

In this way we bootstrap the construction of set theory and logic. The second pass gives us a much more streamlined versions of our theories that are much easier to work in. One can also now investigate other logics within this framework.

Man, its like OP got BTFO.

>>7366785
Mathematics is based off nothing more than the natural numbers, everything else is just abstraction and manipulation.

>>7373327
This, its the study of quantity and space.

>>7373327
you forgot the well-ordering axiom

mathematics isn't based on reality, which causes some weird loopholes like .9999... = 1 and imaginary numbers. Basically math nothing in math is actually true, it only appears true because of the axioms we have chosen are arbitrarily close to the real world. For example, in math a = a, but in real life a can equal a + .00000000...1 due to quantum fluctuations in spacetime.

>>7373795
nice
niiice