What makes a math proof true? There's logical reasoning with common proof methods but that logic is incapable of reconstructing math from the ground up. Is math some mystical entity in the aether that we access using our intellect? Math is very bizarre and I feel dumb for not realizing that sooner.

Bro, it's all a dream. We just accept some shit as facts and then go along, deeper into that specific belief until it becomes undeniable reality science tm. Now we await the good big brain posts that strictly adhere to the religion of science and mathematics.

Platinists are psychotic and mathematical fictionalism is the right answer.

Math doesn't have a solid foundation in first-order logic.

Oh no my thread has been invaded by a schizo

im afraid happens to often on IQfy, don't seat it

> What makes a math proof true?

Axioms bro. Axioms.

That works in a vacuum but I'm talking about the entirety of mathematics, every concept and theorem as a unified whole. Or is math not unified?

There is no area of math that is distinct and separate for all the rest. Every mathematical field is built upon another and at the bottom of the ladder are the axioms. It is a tree, not a set of islands.

So could somebody just put math in a taxonomic diagram and figure out how everything relates with enough branches on the tree? Is it that simple?

Yes. Pretty sure it has been done too.

But what about the axioms. Has anything ever justified adding/removing one? How is it decided? Aren't there people who reject lots of what people would say is "common sense" like the law of the excluded middle and still get results? Lots of the foundations were retrofitted to the math that was already there.

And there you get to the crux. By definition axioms cannot be proved. They are assumptions, taken to be either true or false. If you could (dis)prove an axiom give the rest it's not an axiom. If the self-consistency of the mathematics you create fails then you don't have enough. Math works because we have made it work, that it works so well has amazed people for centuries.

So is it a process of active creation by adding and/or removing axioms as new math is formulated, as opposed to something that can he made sense of ahead of time? I can understand that but it also worries me. This makes it seem too easy for math to go off course, and that doesn't fall in line with how firmly established many mathematical concepts are

People don't think about that anymore. The axioms have been fixed forever, they don't change. You could use ZFC on some field of math but there's little to no point. We *could*, but it would not accomplish anything.

Is this another of those threads were philosopher-wanna-bes want to feel smart?

Yes in order to make math works you need assumptions.

Literally anything we can conceive works the same way, even what we believe to be objective, more so to what we believe to be subjective.

Pointing that fact (assuming that you don't want to also relinquish your capacity to point facts, because of course, that also needs assumptions) is not very smart nor usefull in any way.

Saying that "under some assumptions we have a perfectly objective system" is better than saying "because I can't have objectivity without assumptions then I will reject any kind of objectivity". More so when our "system under assumptions" is capable of producing empirically observable results with high degree of determinism.

You can LARP that math doesn't hold up and by consequence that physics don't hold up.

But a russian can still use trigonometry and geolocalization to pin point your geographical location with high level of accuracy and fire a sarmat hypersonic missile that will use all sort of differential equation calculations solved by finite method algorithms in order to btfo your ass.

Common sense practical science triumph over linguistic schizo babble

If this is a thread to hate on "pure mathematicians".

Then you have my blessing of course.

The bullying shall continue until the mathematics improve.

I think

>who cares if it works it works

Is a lazy cop out and overgeneralization

Anon asked "what make math proof work".

The answer is that they work by definition because axioms could be thought as derivation rules.

A pawn in chess move forward and capture diagonally for no other reason than it is stated that way in the rules.

If you think of axioms as "valid moves of a game" then a mathematical proof is no other thing that a derived set of moves in your game.

The interesting thing about math is that humans were able to infer from their observation of reality axioms that are general and usefull enough to be able to derive mathematical truths that are actually usefull.

The rules of the game analogy sounds like formalism which from what I remember was kind of debunked.

>The interesting thing about math is that humans were able to infer from their observation of reality axioms that are general and usefull enough to be able to derive mathematical truths that are actually usefull.

This is a very bold claim. I would like for you to elaborate

>This is a very bold claim. I would like for you to elaborate

The two seeds of math were "computation" (arithmetic) and "geometry".

Both came to be from observation of physical reality.

So when the first thinkers attacked those problems they were both using axioms without knowing and formalizing the axioms into existance.

do you think that everything is empirical then? the axioms are based on our experience of the world?

>they were using axioms without knowing

but were they using the correct axioms? what about instances where rejection of what seemed to be common sense math was actually beneficial down the line, like in the case of non-euclidean geometry or imaginary numbers? It doesn't seem like what you're describing is how people actually think

>imaginary numbers

not him, but don't get filtered by moronic nomenclature

>do you think that everything is empirical then?

Reformulating your question.

> Do you think all math is empyrical?

Obviously no. You can literally describe things that don't exist. Even if something COULD potentially exist if it DOESN'T exist then it is not empyrical.

You could mathematically describe a system where earth gravity works differently. That would be mathematically correct but obviously not empirical.

Going back to my analogy game.

My point was that the seeds of math, the first development of math, the first questions over math develop axioms that came from physical reality, giving us an axiomatic system whose derivations gave actual usefull results.

> but were they using the correct axioms?

Yes and no?

When you develop mathematic by intuition you are not using any axiomatic system in particular.

1+1=2 works both in an axiomatic system were parallels never intersect or in one when parallels intersect of some point.

Also 1+1=2 could work in different axiomatic systems.

That's why I said

> They were using axioms without knowing and formalizing the axioms into existance

What they were doing without knowing was recursively iterating over different axioms that were compatible with their observations of reality.

The result of that is they set the basis for math to aftweards develop modern calculus and modern physics.

The fact that our mathematical results "magically" align with our results in physics is not magic at all.

We selected axioms that served us to model physical reality.

This is exactly the opposite scenario compared with was happening today in academia in the ghettos filled with pure mathematicians.

Were they create piles upon piles of interesting (for them) bullshit that goes nowhere.

To fix math we need to go back to the basis

> geometry

> arithmetic (computation)

Mathematicians should be working in better ways to describe the real numbers as processes with infinite arbitrary precision, so we can develop new tools and language to describe orders of complexity that don't fall for the

> INFINITES BIGGER THAN OTHER INFINITES LMAO!

Bullshit.

So we can finally answer the P=/=NP.

Instead of that they are creating mathematical bullshit in non-existant geometry or whatever.

Mathematicians today prefer to prove properties for empty spaces than to do something actually usefull.

Bad faith NJW fanatic. Get a job

Numerics already exists

Removing the 5th axiom of Euclid's geometry has actually led to the development of non-euclidean geometries. On spheres, there are no parallel lines and on saddles there are multiple parallel lines (in Euclidean geometry there is only one). Changing the axioms can result in new math, so long as the axioms do not contradict. If your axioms contradict, then you will have a nonsensical system (an example would be adding an axiom to Euclidean geometry that no parallel lines can exist. This contradicts the fourth axiom, and so results in a system which does not make logical sense); however, this is the only restriction on changing or even making an entirely new set of axioms so anything other than that is fine. The main reason we don't do it as often is that it isn't the easiest to prove for certain that a set of axioms make sense, and ours are useful enough to keep going along with them in most context (an example of the axioms we are used to not being enough which as also commonly used is the sphere, with the axiom change mentioned above. In astronomy, for example, it is common to have to do geometry on sphere and so the 5th axiom of Euclidean geometry is changed to allow for this).

look up reverse mathematics

also, there's a general "canon" that most people build off of. this canon is historically developed and often parallels developments in the sciences and applications, or can lead to new ones. you could do your own thing, but you shouldn't expect many people to care.

More like a forest of disconnected trees.

Where you can technically think of new trees but most of the time will be shitty trees already included in other trees. So your tree is suboptimal and doesn't add anything new.

But there is a chance that you could think about a new tree that includes an already existing tree but also adds new mathematical objects with interesting properties. So your tree would actually add something usefull.

In other cases you could have an equivalent tree but that allows you to say the same thing in a different way, which in some cases could be usefull.

I'm 100% layman on the subject of P=NP problem.

But for those who actually study it the more common argument is that our current math is incapable of describing the problem or even the solution.

Is like wanting to say "my house is on fire" without using the word fire, combustion, exchange, particles, heat, alteration, etc.

If the only thing you can say is "this is a physical phenomenon" then you can't differentiate it from others physical phenomenons.

The only way to give this phenomenon its own identity is to create new words to describe the phenomenon.

When I grade for my friend's real analysis class, there comes a point when he has the students do that exercise of constructing a Lebesgue nonmeasurable set on the line. When they come to me for help, I tell them that they can readily find the solution online. I propose that they instead do the exercise on the circle.

Convince yourself, if you haven't yet learned Haar measure, that there is a rotation-invariant measure on the circle, whose outer measure is just given by arc length covering. Consider the countable subgroup of elements with finite order, choose a representative from each coset, assume that the set of these representatives is measurable and produce a contradiction.

If you have a certain presence of mind, you'll appreciate why the axiom of choice is controversial: you have to make an uncountable number of choices. You can choose anything you want, but I think you do have to choose, since there is no formulaically tractable alternative. Do you have that kind of time and resources? Obviously not. The real question is, does even God have that kind of time and resources? Pure mathematics gives you the power to legitimately dare to surpass God.

Truth is a subset of math, it's unfortunate but true.

True/Not True is boolean logic rather than arithmetic ie mathematical logic.

And boolean logic falls under a field of math, hence a subset of math...

No, it falls under logic next to math.

That's like saying that geometry isn't math because it's shapes. It falls under drawing next to math

No. Geometry is not just drawing shapes its the numbers and calculations that correspond to the shapes. True/Not true doesn't require calculation or quantization, it requires only rational observation and the resulting logic.

0*0=0

0*1=0

1*0=0

1*1=1

Same holds for logic. Sorry logic is simple enough that you feel you don't need the math and that geometry is too hard that you feel you do need the math. Your preferences and what you call "difficult" don't dictate what is and is not math

counting is real and your thread is a fricking troll bait

Truth comes from the platonic realm. Noesis is the epistemological faculty of consciousness to connect to the platonic realm.

>but that logic is incapable of reconstructing math from the ground up.

No, it isn't.

Yes it is. Formalism failed (Gödel). ZFC is the compromise that *seems* to work

An interesting concept would be.

a) Is a diagonal argument bullshit because it would imply a set being a member of itself violating ZFC?

b) Or is a diagonal argument valid precisely because for the recursive enumeration of the dagional elements to exist you should be able to make a set be member of itself violating ZFC thus being a proof by contradiction?

My guess is that b) is the correct way of looking at it.

>Is a diagonal argument bullshit

it is not, educate yourself and stop talking through your ass

You didn't understand what I said.

I gave two options dummy.

The second one explicitly state that the diagonal argument IS NOT bullshit and explain the core of the argument.

This is your brain in symbolic schizo babble.

>The second one

is literally link related

https://en.wikipedia.org/wiki/All_horses_are_the_same_color

>Formalism failed (Gödel).

It hasn't failed.

(OP)

If you don't know formal logic it's pointless trying to explain this, and if you do know formal logic then you understand how proofs work. Throw in a lesson on inequalities and you are good to go.

But there is zero sense trying to break it down if you haven't studied formal logic, because you will still just ask "but how do we KNOW?" like a moron.

Mostly, it's just that things are defined a certain way, and if you try to define things a different way, it's no longer useful. Like if I said the word "word" now can also refer to apples and cats, this exact sentence becomes ambigious as shit. What is the worth of this sentence?

Some "I fricking love science" goypop channels on YouTube maybe have attempted.

if math is nothing more than an ad hoc tool that's defined as long as it "just works" then it wouldn't ever be the case that math seems ahead of the curve in many ways compared to empirical observation?

If I understand you correctly (I find your post a bit hard to parse), then most of 20th century physics was actually very much calculated before it was discovered. The same is even true for things like Neptune's orbit, which we knew before we knew Neptune.

>because you will still just ask "but how do we KNOW?" like a moron.

all you're doing is saying the tool justifies the construction, which doesn't make any sense

>the tool justifies the construction,

To a degree, yes.

You start with some axioms. The thing about axioms is that they can't be broken down any more. Luckily, logical axioms (upon which math relies) are unassailable. A logical axiom is the law of identity:

>thing A is thing A

There is not a possible universe in which that is not true. The other axioms are similar. You just have to take them for given.

We combine these into some higher-order sentences. We shape these towards a goal where the tool (which is our desired endresult which we determine before we map out its actual construction) justifies the construction.

That does not make the tool invalid, since it still relies on the axioms and the logically valid combination thereof (logically valid as outlined. There is no possible universe in which they are not valid).

Tools are inherently arbitrary. This can be shown the following way: suppose you, some scholar, have some tool, and in it you can make the statement

>cats are red

But I, another scholar waltz by and say "Wait till you hear my tool! It's even better than yours, because I have included additional features. Actually, just one additional feature: you can now say

>cats are red, yes, really red

I have added the +[yes, really red] clause to the tool!"

So, my tool has n + 1 additional rules, but does precisely the same thing. Obviously, both tools are equally valid, and both tools are also useful in precisely the same situations. But certainly you see how my tool is just a meaningless extension of your tool with that one additional rule, no?

This is true for all logical tools and frameworks. Some frameworks, however, have the interesting property that they are "minimal". That if you take a rule away, you suddenly can't talk about cats being red anymore. That is a meaningful distinction.

>true

implying anything can be "true"

true doesn't exist

you only have grades of certainity

truth just like god are philosophical terms

now we cannot proceed with the rest of your post

A proof is a set of sound logic from a set of axioms. You can't prove the axioms so they are usually either something that is commonly agreed to be true or based on the results of another proof. Your proof means, as long as my axioms are true, this other thing must also be true. What makes a proof "true" is that the logic is sound so what you are saying from the axioms will necessarily be true as long as the axioms are true.

In other words a proof says [math] p implies q[/math] where p is your axioms, q is the resulting statement, and the proof is [math] Implies [/math]

The answer is that it is up to the standard a mathematician applies. Math isn’t rigorous because most of Math isn’t formalized, that takes way too long. Math proofs leave out a ton of steps because it’s assumed you’re smart enough to fill them in as you read it. In theory math could be formalized entirely and checked by a computer but the only reason to believe that is that trained mathematicians are relatively reliable in recognizing when a proof is wrong.

God knows all the math, it's cool that he's been giving me trillions of dollars. He's discovered how to rape a israeli man from thousands of miles away, that's so cool because I love getting fricking raped in my butthole by apparitions. This likely happens using math, I'm not sure the specifics. I guess it's not so hard but god has humanity, or at least understands it as well as I do. He seems to care a lot about the world, not something I share.