Is Mathematics Invented or Discovered?

>>8357080
Neither, it was a gift.

>>8357080
It depends on the type of maths.

For example, take an identity like
[eqn]\text{sin}^{2}\;\theta+\text{cos}^{2}\;\theta=1[/eqn]
This would have to be discovered, sin and cos always had this property, so it couldn't be invented.

However, take something like a complex numbers. These didn't necessarily exist in maths before, so they were invented.

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>>8357080
invented

>>8357080
Math has always existed, all statements in math have been true since at least the beginning of our universe. How we formulate it might be considered an invention, but the properties of the objects were certainly discovered.

>>8357080
What criterion do you use to distinguish between something which is invented vs something which is discovered?

>>8357225
whether some property has always been the case or not

>>8357232
Well, this is something you sort of have to take on faith, but if an entire mathematical theory was lost, then it would be possible to recreate its conclusions as long as someone had whatever axioms it was based on. And the axioms themselves can be formulated anywhere you have a Turing machine, so to that extent they are eternal. I suggest not thinking about it in these terms though, it's more that it has a different kind of existence than material existence. General vs particular, abstract vs concrete, etc.

Math is like a language, instead of letters you've got numbers.
I'm pretty sure we invented it to make living in bigger communities possible (like 7500 years ago or so)

>>8357104

nigga how the fuck u sayin' dis shit was invented?

You're telling me we invented a system that just happens to be self consistent AND is capable of modeling every physical phenomenon in the universe?

That's one fucking hell of a coincidence bro. Dat shit was discovered.

Assuming that there is any correlation between our observations and the universe we observe, then there is order in the universe. You could call this order "mathematics" or you could call our understanding and representation of it math. The former is discovered, the latter invented. The difference is semantic.

>>8357080
The rules/definitions are invented, the consequences of those rules are discovered

>>8357257

nigga how the fuck u sayin' dis shit was discovered?

You're telling me we discovered a system that just happens to be self consistent AND is capable of modeling every physical phenomenon in the universe?

That's one fucking hell of a coincidence bro. Dat shit was invented.

is pi a fundamental property of the universe or was the ratio of the circumference to the radius of a circle invented?

>>8357257
>You're telling me we invented a system that just happens to be self consistent AND is capable of modeling every physical phenomenon in the universe?
Well, I don't know if it is self-consistent and I don't know if it is capable of modeling every physical phenomenon, but otherwise, yes.

It isn't a coincidence. If our modeling doesn't work, we stop using it. If you're only willing to consider successes and disregard all failures, I have a great roulette strategy to sell you.

A better question is "Does it matter?"

Was the structure of the atom discovered or invented?

Was the mathematics (in the form of quantum physics) that perfectly describes and predicts the structure of the atom invented or discovered?

>>8357351

Meant to quote you.

>>8357363

>>8357363
The structure of the atom was also invented. It went through several iterations, which even a cursory glance at 20th century physics will tell you. It will likely go through some more over time, though the time gap between iterations is likely to increase considerably.

>>8357080
Yes.
The axioms of Math were invented. These axioms then provided boundaries through which the rest of math was discovered

>>8357080

I don't understand how anyone can even entertain the possibility of it being "invented." It's not a fucking opinion, mathematics has right and wrong answers which are discovered, not made up.

>>8357351

>If you're only willing to consider successes and disregard all failures

That's called "discovery." You're learning what works rather than making something up and proclaiming that it works.

>roulette

Obviously we get a shitload more mileage out of mathematics than we do from decision making based on random impulse. If it were just a few lucky guesses you wouldn't be reading this on your computer hooked up to a vast telecommunications network.

>>8357378
>You're learning what works rather than making something up and proclaiming that it works.
This is what you think "invention" is? Assertion?

>>8357392

http://www.merriam-webster.com/dictionary/inventing

>: to create or produce (something useful) for the first time
>: to create or make up (something, such as a story) in order to trick people

At best you can argue the trivial / superficial aspects to mathematics like notation are created, produced, or made up. The real thing is cross-cultural and has to be discovered by taking simple premises like the concept of repetition (e.g. the realization having "two" apples and "two" hands means your apples and hands share the abstract quality of twoism) and expanding on them in rational and consistent ways.

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Duality between invention and discovery. Subjectively we invent and experience free will. Reality is likely deterministic though and thus we discover, it's just that because of the principle of computational irreducibility and our limitations of analysis that we have the experience of invention and free will. Assuming the probabilistic aspect of QM is ultimately incorrect.

>>8357080
Invented. A/x/ioms belong to >>>/x/ btw

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>>8357420

>Invented

LOLOL 5 + 5 = 10 today but I'll make it equal 7 tomorrow because it's all just invention and not based on objective reality independent of us!

>>8357427
The axioms that allow that arithmethic to be 'objectively' correct were invented.

>>8357437

Go make up your own arithmetic at a bank where you receive millions of dollars more than what you would under the existing "made up" system and tell me how that works out for you.

>>8357102
This comment is a bit ironic considering the connection between complex numbers and trig functions.

>>8357439
Just because our system is made up doesn't mean it is bad or easily replaceable.

It just means it was invented. Why do you have to put so much baggage onto that word? It means what it means, nothing more.

>>8357427
How is it independent of us? How did math interfered in life before we use it?

>>8357444

>Just because our system is made up doesn't mean it is bad or easily replaceable.

Why isn't it bad or easily replaceable?
Answer: Because it was discovered.

>it was invented

No, reality existed before humans did. Four monkeys and four bananas had something in common with each other before humans showed up to give it the label "four."

>>8357446

>How did math interfered in life before we use it?

Numbers of things obviously impacted animals long before humans were around.

>>8357454
>No, reality existed before humans did.

Please read again the first post I made and notice the word objectively.

If you have 5 bananas and then you get 5 additional bananas then you will always have 10 bananas. Why? Well, you count them obviously.

But count? What do you mean by counting? Well, to get the amount obviously.

Amount? Isn't that what you determine by counting?

See how even the idea of number seems really subjective. Who says that 5 + 5 = 10

Intuitively you can see it, like in my example, but many intuitive things are also not true.

So how do we allow objectivity to take the place of subjectivity? By inventing axioms that mimick our initial intuition. That is, we make axioms presupposing that they will alow 5+5=10 to be true and then we test them against our intuition to see if they match and if they do then now we can throw away intuition and subjectivity because now we have objective logical statements.

Like geometry, everything euclid did was known before his publication but it was all really intuitive and really subjective work. How would you settle an argument in geometry before the axioms? That is something I would have loved to see.

Euclid saw that problem and thought to replace this subjectivity and intuition with axioms and then he tested those axioms against his intuition, showing that his axioms imply the theorems known before (like pythagoras theorem) and then after he concluded that indeed his axioms were equivalent to the previous intuition he published his work and proclaimed that now no more would geometry need intuition, just the objective axioms.

>>8357464

You're taking the difficulty of providing a rigorous mathematical proof of mathematics as a discovered thing as reason to believe it isn't a discovered thing. There's no reason to make that leap. You can just as easily suppose it's difficult for any system natural or otherwise to prove basic things about itself without appealing to other systems.

>>8357080
Both.
From a human perspective however, there's little difference.

>>8357473
What I am saying is that

5+5=10 was discovered

The axioms that generate that result in a more general and better system were invented by mathematicians.

>>8357481

I'd argue any aspect of mathematics that can be demonstrated to have held true for reality before our species was walking the Earth ought to be considered "discovered." If you can make predictions about the positions of celestial bodies at earlier points in history for example and then verify those predictions with different forms of observation and observation based inference, then the maths you're using (after you set aside the superficial implementation details) are truths about the world that were discovered.

>>8357403
So you want to turn an invention/discovery discussion into a platonism vs the world discussion? No thanks.

America was discovered, but "America" was invented. Atoms were discovered, but our models of them—ALL of the models—are invented. And combinations were discovered but describing them by addition is invented.

There is no closed-form solution to the differential equation modeling the motion of a pendulum but pendulums move all the same.

>>8357427
>smugsadfrog.jpg
opinion discarded

>>8357366
>The structure of the atom was also invented


wew lad


So you're saying the atom exists the way it does because we fucking said so?

>>8357574
Which way?

>>8357080
The main reason people disagree on this question is between how people define what is math.

There are the people who say math is the weird stuff

Then there are the people who say math is just the notation used to describe the weird stuff.

>>8357375
If axioms are invented, why were they chosen? Math is arbitrary and/or narrowly defined according to you I guess.

>>8357257
You mean the same coincidence that happens for Moon to be exactly 400 hundred times away from Sun and 400 times smaller than Sun so it can perfectly cover it on our sky?

>>8357590

No, brainlet. I mean in the way that we can mathematically demonstrate the quantization of energy and make useful predictions from it. For example we can predict the shape of electron clouds of atoms, and when we observe them, they match. You think that shit's a fucking coincidence?

>>8357413
This ^ saged

Maths is a model for things... we invent the axioms and then discover facts that emerge from the systems we create.
It's entirely artificial. It just works really, really, really well in some cases.

>>8357080
We invent systems of axioms which we use to discover theorems.

>>8357315
2pi is the period of the complex exponential function.

sidenote but is anyone else fucking amazed that we can create models of reality, do mathematics on the models, and somehow it works out and we get accurate predictions?

>>8357618

>It just works really, really, really well

How do you not come to the conclusion something real corresponds to your system when it works well enough to allow instant intercontinental communication, transportation off of our planet to the Moon, nuclear bombs capable of destroying entire cities, etc.? At best I can see how you could insist we don't know with 100% that something real corresponds to the system, but not being certain of its existence shouldn't lead you to be certain of its nonexistence. There's a lot more evidence for its reality than against it as far as I can tell. And it would be pretty absurd for nature to just play along with your system those billions of times where it made perfect sense for everyone and everything it was tested with / against all around the world only to suddenly stop working that way. Even then I'd argue there was a temporary reality corresponding to our systems in that case.

>>8357498

The part that you're missing is that the models we use to describe those processes are models that we actually invented. We created a set of rules (axioms) then tried to describe what is already happening with the rules that those axioms provide.

When the model fails, we make a new model. Sometimes this involves the creation of a new field.

This is exactly what happened when Einstein came about. Before him, it was largely decided that Physics had been "solved" and was a dying science. But then he came up with a new model that retired the old models, relativity.

As we find new phenomena I the universe that cannot be explained with our current models of physics, we will quite literally throw everything in the trash and write a new model to try and explain the phenomena.

The hang up "discovery" people have is that they think that the universe was written in code. This is not true. The universe exists, and we invented code to describe it.

>>8357514

This guy gets it.

>>8357705

I'm not amazed because I've always been a Platonist even before I took a philosophy class and learned that label existed. I also program for a living and the abstract world of Forms constantly strikes me as more real and fundamental than the trivial / culture specific ways we participate in them.

>>8357716

>retired the old models

We still use Newton's work for a solid majority of today's practical physics applications. It's not like Newton had some made up model that was completely wrong and had to go in the trash. His work clearly matched reality in most of the cases creatures of our scale deal with on a daily basis. There was both truth and non-truth in his work, and the truth part is what he discovered about reality.

>>8357713

The reason it works that way is not because math already existed, but that when we made math to describe what is happening with a model, we remade the model until it described what was already happening. That's the "coincidence", except it's not. We made it to be true.

But if we had a new observation that defeated the model, we would then make a new model.

This isn't that hard to understand guys. Mathematics models are basically words in a textbook. They accurately describe the thing that is already happening in nature. But sometimes textbooks get outdated, and we write new words (math models) to make it accurate again.

This will happen until we find the so-called "Theory of Everything", the model in which all known models are derived. But it will not be discovered, we will invent it after reinventing all existing models to the point that they can be all described with a unifying theory

>>8357732

I wasn't talking about Newton specifically.

He ivented a model to solve problems in physics. The parts that weren't true... We're reinvented.

If aliens exist, it's possible they use a different kind of system to describe the universe. And it's possible that this system is both more elegant and incompatible with our understanding of math

The Universe operates in a manner that is often perfectly modeled and expressed through pure mathematics. Universal standards such as the rotation of planets and irrational numbers like PI are things that existed before and will exist long after us. How we explain express them is what most people see as Math, numbers and equations where technically created by humans though the properties they posses and express where discovered. If we ever meet another intelligent species Mathematics is our best bet for communication as it is Universal, and its properties are necessary for advancing ones own civilization, regardless of how one expresses it.

For how much you guys shit on philosophy, a little philosophy would actually answer these stupid questions of "was math invented or discovered" "what can we do with consciousness"

There are questions science alone can't answer you faggots

>>8357733

>They accurately describe the thing that is already happening in nature. But sometimes textbooks get outdated, and we write new words (math models) to make it accurate again.

Suppose there's this aspect of reality which makes our models correct when they match it and incorrect when they don't match it. Suppose this aspect of reality is something called "mathematics."

You're already talking about it, you're just not using the same name for it.

>>8357739

>more elegant

Specific means of expressing truth are irrelevant as long as they're still true. We've already had experiences with aliens from other countries on the same planet. The main concern in these cases are whether or not the different culture specific approaches to the discovery of mathematics are true.

>>8357705
Of course we'd choose the models that worked!

>>8357758

I think he's talking more about the times when a mathematical idea with seemingly no practical applications ends up being the perfect tool for understanding some physical phenomenon we hadn't figured out until recently. As opposed to a case where you chose the model because it worked on some problem and then were amazed it works with the problem.

By analogy, the former would be getting predictive results with your machine learning system on a test population you haven't seen before running it through a training population while the latter would be just getting predictive results from the same population you trained on.

The axioms of math were invented, but the applications were discovered.

>>8357769
Well I think being a "perfect" tool is "perfectly" overstated.

One falls in love with "I know," and forgets all the times "I thought I knew..."

>>8357769
Yes this exactly.

>>8357769
You can model SAT with ladders in go. Is it amazing that the game of go is the perfect tool for understanding NP complete problems?

But who discovered discovery anon

>>8357080
Define "invent" and "discover" first and then we talk.

>>8357785
>One falls in love with "I know," and forgets all the times "I thought I knew..."
That's because they're fucking morons immersed in an environment of trash feedback loops, potentiating poor behaviors and mindsets.

Fuck them.

Invented the moment you make axioms. Discovered for everything else.

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>he thinks acausal atemporal abstract entities exist outside of space and time

(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.

(P2) Mathematical entities are indispensable to our best scientific theories.

(C) We ought to have ontological commitment to mathematical entities.

What's so hard for you fucking faggoty antirealists to understand? You believe quarks exist, don't you?

>>8358105
I can prove mathematically that everything you think doesn't exists outside of space and time not only exists but is actually laughing at you

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>>8358110
>We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
For what purpose?

maths is literally nothing but the process of arbitrarily inventing rules (axioms) then deriving results (theorems) from those rules.

anyone who doesn't understand this doesn't know what maths is

>>8358068
correct, therefore overall mathematics is invented.

>>8357080
Maths exicted. What we invented is a lenguage (numbers and stuff) to decode it.

>>8357257
>Human mode of thought invented to describe complex patterns and relationships in human sensory experience is able to do so
Wow

>>8357375
t.guy who thinks math began with zfc

>>8357080
Properties of lines and geometry were discovered. In order to these properties, proofs and 'axioms' were formalized.

Take for example the number 2, 2 didn't mean anything in the past. Yet, in order to idenfity multiple pieces of something, a language was constructed.


Thus, math itself is an invented language, whose purpose is to explain certain properties of the universe.

>>8359290
in order to explain these properties*

>>8359290
identify* fuck, I gotta sleep

>Look at two objects. Count the objects you looked at - two.
>Add two more objects. Count the objects you looked at - four.

There are a few facts that apply to the above:

>1) It existed before our capacity to invent did
>2) It exists equally throughout the entire Universe
>3) It will continue to exist as it is up until the "end" of the Universe

Therefor, we can conclude that it is a mere observation of something that has always been and will always be, or in other words, a discovery.