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DEAL: For $25 - Add A Second Phone Number To Your Smartphone for life! Use promo code SLASHDOT25. Also, Slashdot's Facebook page has a chat bot now. Message it for stories and more. Check out the new SourceForge HTML5 internet speed test! × ## Is Mathematics Discovered Or Invented?798 An anonymous reader points out an article up at Science News on a question that, remarkably, is still being debated after a few thousand years: is mathematics discovered, or is it invented? Those who answer "discovered" are the intellectual descendants of Plato; their number includes Roger Penrose. The article notes that one difficulty with the Platonic view: if mathematical ideas exist in some way independent of humans or minds, then human minds engaged in doing mathematics must somehow be able to connect with this non-physical state. The European Mathematical Society recently devoted space to the debate. One of the papers, Let Platonism die, can be found on page 24 of this PDF. The author believes that Platonism "has more in common with mystical religions than with modern science." This discussion has been archived. No new comments can be posted. #### Is Mathematics Discovered Or Invented? More Login ## Is Mathematics Discovered Or Invented? Comments Filter: • #### Logical positivism to the rescue... (Score:5, Insightful) on Saturday April 26, 2008 @05:42PM (#23209122) Homepage When faced with an awkward question, logical positivism [wikipedia.org] asks: what would the answer tell me about the future? Suppose you had a definitive, 100% guaranteed answer to the "discovered vs invented" question. What would it allow you to do that you couldn't do before? What could you predict? What would you gain? Nothing, nothing and nothing. It's meaningless; merely a matter of perception, wordplay and people having too much time on their hands. Oh, and the correct answer is "discovered". • #### Re:Logical positivism to the rescue... (Score:5, Insightful) by Anonymous Coward on Saturday April 26, 2008 @05:50PM (#23209156) Oh, and the correct answer is "discovered". No, the correct answer is "both." The relationships and observations that we use mathematics to model are discovered. They are out there, we discover them, and then we model them. That should be obvious to all but the most die-hard of idealists. The language that we use to do this modeling is invented. It is also refined (i.e. slightly reinvented) over time to better fit our discoveries. That, too, should be obvious to all but the most die-hard of determinists. I know, this answer isn't very deep, but in my opinion the question isn't nearly as deep as it is being made out to be. • #### Re: (Score:2, Insightful) I'm just amazed when stuff can be worked down to an amazingly simple formula. Like e=mc^2 . I mean, why exactly ^2 and not ^2.14332544988? I think the correct answer is basically as you've described. Like most absurd debates where both sides are vehemently opposed, the answer actually lies in the middle. • #### Re:Logical positivism to the rescue... (Score:5, Insightful) by Anonymous Coward on Saturday April 26, 2008 @06:14PM (#23209344) Because squared gives you the right units. • #### Re:Logical positivism to the rescue... (Score:5, Informative) <nine.times@gmail.com> on Saturday April 26, 2008 @06:22PM (#23209406) Homepage Yes, it's also amazing that the equation isn't 2.14332544988e=2.14332544988mc^2. Yes, sorry, I'm being a smart-ass and it's not polite. But c^2 is just a constant. • #### Re:Logical positivism to the rescue... (Score:5, Insightful) on Saturday April 26, 2008 @07:08PM (#23209688) Homepage the reason that is it not (some value here)mc^2 is because c is a natural constant with a non-integer value, and all the "non-roundness" that seems to amaze you is contained in this constants. Another example of a fundamental constant is pi. Is it really so amazing that the ratio of circumference to diameter is exactly pi and not 2.143243*pi ? These numbers and constants are discovered, as they clearly exist whether or not we know what they are. Other parts of math do resemble invention more than discovery. E.g., the definition of mole being the number of atoms of carbon 12 needed to make exactly 12g and the Coulomb, both of which are numbers that are arbitrarily assigned to fit in with the system of measurements that has been devised over the years. All of these constants could easily be multiplied by any non-integer value and the whole system would still work. To answer the article's original question however, my answer would be: Who gives a toss? Math is useful. Whatever semantic definition we apply to the process by which we expand our mathematical capabilities has absolutely zero impact upon that expansion. • #### Re: (Score:2, Interesting) Actually its not quite e=mc^2, thats just the first term in a taylor series for the actual answer. The first term is just close enough that it works for day to day use. The universe is incredibly elegant in its mathematics though. • #### No, mc^2 is exact for an object at rest (Score:5, Informative) on Saturday April 26, 2008 @06:39PM (#23209494) Actually its not quite e=mc^2, thats just the first term in a taylor series for the actual answer. No. For an object measured in its rest frame, the energy is possesses is exactly mc^2 (where m = m_0 = rest mass). The only situation where you're using a Taylor series approximation is when you approximate the energy of a moving object with speed v much less than c by mc^2 + (1/2)mv^2. But if you want the exact answer for a moving object it's easy enough to use E = \gamma mc^2 anyway. • #### Re:No, mc^2 is exact for an object at rest (Score:5, Insightful) on Saturday April 26, 2008 @07:53PM (#23209998) How would you suggest I measure an object in its rest frame? This may seem like a nitpicking question, but it brings us to the point that I really want to make: Mathematics is interesting because there are no ambiguities in a well described mathematical problem. There are many problems that have a finite set of solutions. However, every mathematical model we develop to describe our surroundings is only an approximation of our observations. With time, we can create more and more accurate models, but there will always be something about that model that is derived experimentally, and is therefor imperfect. This does, in fact, tell us something about the underlying nature of the universe. Either it was created with some arbitrary parameters, or it exists in a way such that there is no way to perfectly describe it. Or maybe there are other possibilities I have not considered. What philosophical meaning you derive from all this is up to your own reasoning. • #### Re:Logical positivism to the rescue... (Score:4, Funny) on Saturday April 26, 2008 @07:12PM (#23209728) Homepage You say "day to day use" as though I'd use e-mc^2 when working out value for money on the small vs large box of cereal in the supermarket or something like that. "Hmm... I wonder if the larger box would still be better value for money if I were eating it in a spaceship with a velocity approaching c" • #### Re:Logical positivism to the rescue... (Score:5, Funny) on Saturday April 26, 2008 @06:06PM (#23209294) No, it is not "both". Math exi Damn, I am too drunk to type. I have one eye closed as I type.... so you win :-) • #### Re:Logical positivism to the rescue... (Score:5, Informative) on Saturday April 26, 2008 @06:09PM (#23209322) Homepage Journal The language we use to describe mathematics is not the math itself. The math exists regardless of the symbolism used to describe it. Hence, you are incorrect. It is all discovered, but the means to describe it and put it to use is invented. • #### Re:Logical positivism to the rescue... (Score:5, Insightful) on Saturday April 26, 2008 @06:17PM (#23209362) The math exists regardless of the symbolism used to describe it. Depends what you mean by "exists". For example, mathematical concepts are not observable (which is the condition for existence in an empirical framework), but physical systems can be observed which implement the concept. One can observe one apple or one galaxy, but one cannot observe the number one. • #### Re:Logical positivism to the rescue... (Score:5, Insightful) by Anonymous Coward on Saturday April 26, 2008 @07:09PM (#23209708) math is truth truth is discovered truthiness is invented • #### Re:Logical positivism to the rescue... (Score:5, Insightful) on Saturday April 26, 2008 @07:04PM (#23209662) Journal The math exists regardless of the symbolism used to describe it. Math is the symbolism used to describe the universe. Physical reality does not need symbols or tools or sentience to function, we however need math to describe the functions of the universe in precise detail. Math is a tool and so is an invented thing where the ideas have come from observing the world around us, just like a knife or velcro are tools that where invented based off of ideas gleaned from observations of the world around us. • #### Re: (Score:3, Funny) Sad but true. Mathematicians are tools. • #### Re: (Score:2, Interesting) I agree with your opinion. My first reaction when reading the summary of the story is was to think that mathematics was "discovered" was utter bullshit (and may only make sense if you think that you heavenly$Deitity created it and we mere mortals are just obtaining whatever \$Deitity wants to give us).

But then, thinking a bit deeply, I agree that as you said, maths is both discovered and invented. There is no doubt that mathematical symbols were created by us humans. I just created some symbolisms while doi
• #### Re:Logical positivism to the rescue... (Score:5, Insightful)

<nine.times@gmail.com> on Saturday April 26, 2008 @06:53PM (#23209578) Homepage

No, the correct answer is "both."

No, I think the correct answer is, "What are you asking?"

The problem with questions like this is that it isn't clear what's in the mind of the person asking the question. What do you mean by "invented" and what do you mean by "discovered"? What difference do you see between the two?

For example, some people will think that "invented" means "made up". So in that person's mind, if math is "invented", then it's based only on human thought, and not on real principles of the universe itself. Of course, this line of thought makes me want to ask what it would mean to be a "real principle", and what is the "universe itself" when detached from human conception, but I'll leave that aside.

The problem I see immediately with this concept of "invented" is that real inventions don't exist independently of the universe. For example, was the wheel "invented", or did someone discover that rolling a circularly shaped object requires less energy than dragging an equally massive object? Was gunpowder "invented", or did someone discover than mixing certain chemicals together and setting fire to them caused an explosion? Was the telephone "invented", or did someone discover that you could convert sounds into electrical signals and back again by using magnets?

All inventions are a discovery of sorts, which makes this whole question a bit nonsensical.

• #### Re:Logical positivism to the rescue... (Score:4, Insightful)

on Saturday April 26, 2008 @07:26PM (#23209822)
No, the correct answer is "both."

No, I think the correct answer is, "What are you asking?"
.
.
For example, was the wheel "invented", or did someone discover that rolling a circularly shaped object requires less energy than dragging an equally massive object? Was gunpowder "invented", or did someone discover than mixing certain chemicals together and setting fire to them caused an explosion? Was the telephone "invented", or did someone discover that you could convert sounds into electrical signals and back again by using magnets?

Um, you have just given threee great examples supporting the original poster's answer of *both*...

In each case the basic scientific principle (mechanics, chemistry, elctricity & magnetism) was discovered (sometimes unwittingly) and then the knowledge of that discovery used to engineer an invention (wheel, gunpowder, telephone). The "discovery" was an observation of a natural phenomenom, etc, and the "invention" was creating something that otherwise did not exist in nature that took advantage of those phenomena. If you wanted to be pedantic you might argue the first "wheel" could have been discovered ("hey, look at how that round rock rolls!") but please don't try to claim that set of 18" forged alloy wheels with vulcanized radials was "discovered".

This is exactly the same argument the OP was making. Mathematics clearly involves the invention of a language to express discoveries (or assist in making those discoveries).

• #### Re: (Score:3, Funny)

No, I think the correct answer is, "What are you asking?"

I don't know what I am asking, but the answer is definitely 42
• #### Re:Logical positivism to the rescue... (Score:4, Interesting)

on Saturday April 26, 2008 @08:57PM (#23210400) Homepage Journal

What do you mean by "invented" and what do you mean by "discovered"? What difference do you see between the two?
Yes, the question asked is a question of semantics and philosophy.

Semantics:

In old French, both were essentially synonymous.
• -You can find mentions of Christopher Columbus having "invented" America(s).
• -"dÃ©couvrir" both means uncover and discover in French.
• -From my harrap's shorter french and english dictionnary of 1962 :
inventer v.tr. To invent. (a) To find out, discover. [...]

Philosophy :

Under platonism, there's actually no distinction (see allegory of the cave).
By suggesting to let platonism die, the anonymous reader seems to want us answer "invent"...
• #### Score +1 (Score:5, Insightful)

on Saturday April 26, 2008 @07:14PM (#23209742)
The question itself, as you pointed out but in a different way, is a false dichotomy (is it this or that??). There are a number of explanations that might be found in a mix of the two camps, or somewhere else altogether. As such, the question is pretty much meaningless, really.
• #### Re:Logical positivism to the rescue... (Score:5, Insightful)

on Saturday April 26, 2008 @06:01PM (#23209256) Homepage

Suppose you had a definitive, 100% guaranteed answer to the "discovered vs invented" question. What would it allow you to do that you couldn't do before? What could you predict? What would you gain?

I tend to agree. I'm reminded of the Dutch computer scientist, Dijkstra, who said that ""The question of whether a computer can think is no more interesting than the question of whether a submarine can swim." Some questions are just meaningless.

I think the thing to learn here is that language isn't reality, it merely describes reality.

Oh, and the correct answer is "discovered"

No, I think the correct answer is "Why are you asking the question?" There might be a more interesting (and perhaps answerable) question that underlies it.
• #### Re:Logical positivism to the rescue... (Score:5, Funny)

on Saturday April 26, 2008 @06:19PM (#23209378)

Oh, and the correct answer is "discovered"

No, I think the correct answer is "Why are you asking the question?" There might be a more interesting (and perhaps answerable) question that underlies it.
And how does that make you feel?

• #### Re: (Score:2, Insightful)

Suppose you had a definitive, 100% guaranteed answer to the "discovered vs invented" question. What would it allow you to do that you couldn't do before? What could you predict? What would you gain?
You can own an invention, but not a discovery.

So the answers to your questions aren't nothing x 3, but rather in lines with patenting and making money.

//T

• #### It is indeed discovered (Score:2, Funny)

Obviously! I mean, look : one apple, two apples, three apples. There. Numbers. See that funny relation between the diameter and circumference/area of a circle? There's pi. And so on...

I mean, it's a bit like asking wether a tree falling really makes a sound if nobody's there to hear it. Of course it bloody well does!

• #### Re: (Score:2)

Obviously! I mean, look : one apple, two apples, three apples. There. Numbers. See that funny relation between the diameter and circumference/area of a circle? There's pi. And so on...

That reasoning fails when you realize mathematics is not the same as arithmetic. To claim that purely abstract concepts that have no representation in the physical world are discovered pretty much necessitates belief in a Platonic 'world of ideas'.

I would say a few things are invented (like axioms of plane geometry), then

• #### Re:Logical positivism to the rescue... (Score:4, Interesting)

on Saturday April 26, 2008 @06:29PM (#23209454)
The problem with the question is that the answer is predicated on interchangable assumptions. To discover something it has to apriori exist. Inventing something requires that it not. So the fundamental question is:

Does mathematics which no one knows about exist?

Well it is obvious that on some level it should. It is likely that whatever new field of mathematics we invent, it will (eventually) be described using axiomatic set theory. But does the fact that we already have the language we need to describe a theorem mean that the theorem already exists? Does a sonnet exist before I write it? All the words I'm likely to use will be in some version of the Oxford English Dictionary. I can symbolically write down the abstract idea of every sonnet imaginable in only a few lines of mathematics. It would seem clear that mathematics, like poetry and prose, is invented then.

But then mathematics is different from prose, because mathematics can be used to make quantitative predictions about the world around us. It would seem that independent of human being nature itself 'knows' about mathematics. Before we invented calculus the acceleration on an electric charge due to a electromagnetic field could still be found using Maxwell's equations. The Falkland Islands were still there before the Spanish arrived right?

So now it would seem that at least some mathematics is discovered, at least as to how it relates to nature. Of course the mathematics we use to describe nature is just an approximation. Maybe nature doesn't know about math, maybe we just got luck.

Then there is another problem, whose to say that just because we think of prose as invented it really is. That might just be our sloppy use of language. I said earlier I could, at least in the abstract, write down every possible sonnet in the English language. That at least implies that those sonnets exist in some way before I write any of them, even if it is as an abstract sonnet.

Bottom line, it all comes down to what you think exists. If under your philosophy mathematical theorems can be shown to exist independent of if someone knows about them or not, then they are discovered. It is likely sonnet are discovered under that philosophy as well. If on the other hand theorems only exist after someone has conceived of them then they are invented. Now you have to be careful that at least some part of the Falklands weren't invented by the Spanish as well.

I'm going to have to go with discovered. To me Euler's equation is, was and ever shall be true and there isn't a darn thing anyone in this universe can do about it.

Of course the discussion doesn't really yield any useful results, so I would like to propose the Dirac interpretation for the uncovering of mathematical knowledge:
Shut up and calculate.
It comes with a corollary of my own devising:
No you cant patent it.

• #### Re: (Score:2)

It's meaningless; merely a matter of perception, wordplay and people having too much time on their hands.
Philosophers to the rescue!
• #### Re: (Score:3, Insightful)

I'll agree that it's meaningless in the sense you describe, but I'll also add that it's a relatively easily answered question as long as one keeps superstition at bay.

Mathematics is a language, one intentionally of the most precision we can manage. This language is very well able to, and intended to, describe many of the methods and mechanisms of the universe we live in, and is additionally capable of describing things that are abstract and/or impossible.

As a language, it evolves with use, and it mai

• #### Wrong (Score:3, Insightful)

If mathematics is invented it can be patented.

HTH.

• #### Mathematics in the forms of human intuition (Score:5, Insightful)

on Saturday April 26, 2008 @05:43PM (#23209126)

I much prefer the Kantian approach, which, simplified, is that space and time are the forms of human intuition, and it is these forms of intuition that lead to us understanding things the way we do (spacially and temporally, whose relationships are mathematical). "Things in themselves" are unknowable, and can only be approached through some set of references, whether it be through the space and time we perceive, other possible ways time and space could work (non-Euclidian geometries?), or ways we can't even imagine. Unlike Plato's idea, which is that mathematics involves universal truths we discover, Kant's "Copernican turn" puts the subject as the one who projects mathematics onto everything it experiences. Arguably, this is the idea that has lead to the "modern era".

This makes mathematics the study of these forms of intuition, so unlike Plato's approach, we're not "discovering" universal ideas, but rather coming to understand the way we interpret the world (and by "we", I mean me, the beings who do science that makes sense to me, and probably most beings on earth whose methods of sensation resemble that of humans).

To answer the question of discovery or invention from this perspective, we can invent ways to do mathematics, but the relationships themselves are a discovery of the way we intuit anything we can sense.

• #### My inner electron (Score:2)

I must be imbued with electron-ness, I'm made of electrons so by this logic I must have to connect with my inner electron to manifest my existence.

No, wait, that's all negative. Let me connect with my inner proton, that's a positive outlook. Me thinks that these people who require one to "be the thing you think about" are searching for more than is there. One might say they are "more"-ons.

Why do I have to have a carrier signal to planet math to use math?

That said, I do know that when I'm deep inside a p
• #### Re: (Score:2)

Unlike Plato's idea, which is that mathematics involves universal truths we discover, Kant's "Copernican turn" puts the subject as the one who projects mathematics onto everything it experiences.

Does Kant actually discuss the source of intuition? I don't remember him doing so in anything I've read. In that case, then Kant is not saying that we project math onto anything. What he's really pointing out there is that we don't learn about space (and therefore arguably geometry, and therefore arguably math)

• #### I know this! (Score:5, Funny)

on Saturday April 26, 2008 @05:44PM (#23209130)
It's intelligently designed.
• #### Re: (Score:2)

It's intelligently designed.
by turtles!

• #### But did God invent or discover it? (Score:3, Interesting)

Did God invent mathematics, or simply make use of it (being omniscient, there is no need to discover)?

That would be a good question for Theists. The origin of the Universe poses few logical problems for a Theist (thousands of years ago thinkers realized the universe was a sub-reality like a story - or in modern computer terms, a virtual machine). But the origin of things like logic or justice are trickier. For instance, is everything God happens to do "good" because He is God and says so? That view is ca

• #### Patently Obvious.... (Score:4, Insightful)

on Saturday April 26, 2008 @05:47PM (#23209140)
Of course the answer could lead to further locking up knowledge... You can't read my theorem until you pay the license type deal.
• #### discover? create? same difference (Score:2)

I think the characterization of "discovered" in this context has been somewhat mischaracterized. Mathematics, the study of generalized rule sets using logic, the language of algorithm, is "discovered" in the same way creative works of literature or music are "discovered." With some generalized state space of rules, every possible output, idea, or concept constrained by those rules can be indexed in this space. This can be in "in principle" thing -- i.e. you don't have to know all the rules to acknowledge
• #### Invented and Discovered (Score:2)

It is invented, in that we have set the rules of logic, and other rules and therefore it is one of the few disciplines where there is a "correct" answer, and all other answers are demonstrably wrong. That's because we set the rules, and it is therefore a finite system that we can fully understand.

It is discovered in that when we set new rules, we have yet to discover all the implications of that new rule. Such as chaos mathematics being a natural implication of setting a value to the square root of negative
• #### Connection to math = The Universe (Score:2, Interesting)

human minds engaged in doing mathematics must somehow be able to connect with this non-physical state.

Wouldn't this just be our observations of the world around us?
I haven't read TFA yet but it sounds like a troll written by someone who doesn't really grok math and physics (not that I completely do either).
Take addition for example.
Did some balding Greek define addition, or did he have 1 apple in one hand, 1 apple in the other hand, and discover that he had 2 apples total?

• #### How is this a debate? It's both. (Score:4, Insightful)

on Saturday April 26, 2008 @05:50PM (#23209158) Journal
The concept was invented.

What can be done with it is then discovered.
• #### MOD PARENT UP (Score:2)

God, I wish I could say things with such brevity. Bingo. You win the cupie doll.

--
Toro
• #### Re:How is this a debate? It's both. (Score:4, Insightful)

on Saturday April 26, 2008 @06:07PM (#23209304) Journal
Thats absurd.

The concept was discovered, then we invented new methods of math based upon the discovery of Math ;)

• #### Re: (Score:2)

I'm not entirely following, are you saying that mankind, or some mathematically non-contingent entity invented Mathematics?

cheers
-b
• #### Is Mathematics Discovered Or Invented? (Score:5, Interesting)

on Saturday April 26, 2008 @05:53PM (#23209186)
Is Mathematics Discovered Or Invented?

Neither. It is defined.
• #### It's both (Score:2)

It's a floor wax and a dessert topping.
• #### The short answer: (Score:2)

Neither.

Mathematics is just another rock to the sculptor: It's in plain sight for all to see but it takes skilled artisans to give it life and make sense of it.
• #### Only the integers (Score:5, Interesting)

on Saturday April 26, 2008 @05:56PM (#23209214) Homepage

Integers were discovered. Beyond that, it's human invention.

I used to do work on mechanical theorem proving, and spent quite a bit of time using the Boyer-Moore theorem prover [utexas.edu]. When you try to mechanize the process, it's clearer what is discovered (and can be found by search algorithms) and what is made up. Boyer-Moore theory builds up mathematics from something close to the Peano axioms. [wikipedia.org] But it's a purely constructive system. There are no quantifiers, only recursive functions. It's possible to start with a minimal set of definitions and build up number theory and set theory. The system is initialized with a few definitions, and, one at a time, theorems are fed in. Each theorem, once proved, can be used in other theorems. After a few hundred theorems, most of number theory is defined.

But you never get real numbers that way. Integer, yes. Fractions, yes. Floating point numbers, representation limits and all, yes. But no reals. Reals require additional axioms.

• #### Re: (Score:2)

Correct. Quantities do exist in the universe - otherwise, the universe wouldn't have things in it, it would just be one big undifferentiated mass.

But the rest of it? Nope. It's our language center tapping in to our numeracy center, and then confabulating all this "math".

The Platonists are funny - they get all worked up about this "NO!!! WE ARE DISCOVERING TRUTHS!!!" and well, they're not, but it's nice that they think they are - gives them and their pointy little heads something to thump their chests ov

• #### Re: (Score:2)

Those axioms being the Archimedical one and the Cauchy/Dirichlet/WeierstraÃY one.
• #### Re:Only the integers (Score:4, Interesting)

<nine.times@gmail.com> on Saturday April 26, 2008 @06:38PM (#23209488) Homepage

Integers were discovered. Beyond that, it's human invention.

I can make a strong case that negative integers are invented. Because you can't have -3 apples. We invented the negative numbers to indicate a loss of positive numbers. We also invented fractions to stand in for ratios.

Sort of.

• #### Discovered (Score:2)

In the first few chapters of Roger Penrose's "The Road To Reality", he convincingly puts forth the argument that mathematics have been discovered by human intellect. More so, mathematical discoveries have been explored long before their physical manifestations were understood, such as hyperbolic geometry, imaginary numbers and the complex plane.

It's more than a line from the movie "Pi", it's the plain truth; "Mathematics is the language of nature". Too bad we remain collectively illiterate.

• #### Axioms vs. theorems (Score:5, Insightful)

on Saturday April 26, 2008 @06:01PM (#23209254)
I'd say that one "invents" a set of axioms and "discovers" the inevitable logical consequences of those axioms. For example, one might invent a negation of Euclid's 5th postulate and discover non-Euclidean geometry. In the process, one might "invent" a proof which is a path that leads from axioms to theorems.

The point is that the axioms don't exist until we create them. But once we create a set of axioms, then the results are an inevitable (if arduous) journey of discovery which might require clever inventions to reach the destination of mathematical knowledge.
• #### both? (Score:4, Interesting)

on Saturday April 26, 2008 @06:02PM (#23209268) Journal
Geometry and number theory can be derived from a few axioms. These axioms are chosen to give geometries and/or numbers which are useful for describing nature, but you could also generate other geometries by using a different starting point. Since the starting axioms are ultimately arbitrary, everything constructed from them is just an invention. However, at some level, the proofs fall back on pure logic and set theory. Is logic invented? I don't know. There are forms of logic with different rules, but there's seems to be something fundamental about the basic logic of sets. So some of math might be called discovered?
• #### Parallel (Score:5, Interesting)

on Saturday April 26, 2008 @06:05PM (#23209288)
Are songs discovered or written?
• #### Re:Parallel (Score:5, Funny)

<andrey.petrov@shazow . n et> on Saturday April 26, 2008 @08:26PM (#23210200) Homepage
You mean... is the music on my hard drive stolen or was it collaborated on by hundreds of peers, each telling my box of one small chunk from it?

- shazow
• #### All the same? (Score:4, Interesting)

<thyamine@@@ofdragons...com> on Saturday April 26, 2008 @06:05PM (#23209290) Homepage Journal
Isn't it very close to being the same thing. It seems to me that you could argue that anything invented is really just being discovered. Someone can invent carbon steel, but aren't they just discovering the formula that nature says will work? Even complex systems that are invented (machines, computers, etc) are really just taking simple discoveries and weaving them together to discover something new and more complicated.
• #### Discovery Need Not Imply Metaphysics (Score:2)

> The article notes that one difficulty pointed out with the Platonic view is that, if
> mathematical ideas exist in some way independent of humans or minds, then human minds
> engaged in doing mathematics must somehow be able to connect with this non-physical
> state.

That doesn't follow. The math may be embodied in the physical universe in which the human brains are embedded. One need not postulate a non-physical state. The convergence of math and physics tends to support this.
• #### They assume (Score:2)

....that the abstract must be connected to, as per the Platonic ideas of, say, circleness or triangularness. This is fundamentally flawed, in that it assumes a property is inherited from somewhere. This defies common experience, where things can be approximate, or can even hold multiple characteristics. In other words, things are far more fluid than either pure Platonists or these anti-Platonists would have you believe. Things do not have a single, fixed, unit, discrete value. Well, some things do, and it's
• #### "Remarkably"? No. (Score:2)

It's not remarkable that such a thing is still being discussed. It is not a question that mathematics, however advanced, can resolve, since it is not a mathematical question. It's a question about mathematics itself.

Also, the answer should be "discovered," but some things that people do look more like invention. I think they're fooling themselves, but try telling them that.
• #### Platonic? (Score:2)

Oh no, math and I have a far more... intimate relationship.
• #### what can be discovered? (Score:2)

Starting with the most basic maths like multiplying something by 2, that looks like something you could discover. When you get into something like calculus or trig, this is not an intuitive process anymore, and has to be invented, and taught to the next generation. We went for centuries not knowing calculus, but how long have we as a people known addition? We teach our children how to add and multiply in school yes, but isn't that something that they could eventually figure out themselves?

It's a muddy li
• #### Re: (Score:2)

We went for centuries without knowing there is America, too.

Trig is very intuitive, and so is calculus. What is not intuitive is the way we do calculus, i.e. limits and stuff. That was definitely invented.
• #### i'd say "discovered" (Score:2)

the study of math is the *natural* relationship of numbers, so it should be classified discovered.

we might *invent* theories to deduce the relationships if they're complex, but it's possible that we just haven't *discovered* the true path from A to B.

for example, we technically haven't fully *discovered* pi or e, since they're transcendental, but we have *invented* easier ways to approximate them in order to simplify our lives (~3.14 and ~2.72).
• #### ideas possess a location? (Score:2, Insightful)

I thought the article was weak. It asked:

Where, exactly, do these mathematical truths exist?

Where is the edge of the world? Where is the center of the universe?

Can a mathematical truth really exist before anyone has ever imagined it?

Of course it can! For instance, 3 has always been a prime number. There have always been prime numbers. Doesn't matter that the ideas weren't conceptualized and expressed in prehistoric times. This is the same question as the previous, with "when" substituted for "where".

As to inventions, the almighty lever would have worked the same before our solar system had formed as it does today

• #### Why so human-centric? (Score:3, Interesting)

on Saturday April 26, 2008 @06:23PM (#23209410)
Believe it or not, it has recently been discovered that dogs can count. I wouldn't be surprised if apes (other than us) or parrots could do this too.

So, regardless of the whole platonic debate, basic mathematics definitely exist independently of humans.
• #### The super-imaginary number, j. (Score:3, Interesting)

on Saturday April 26, 2008 @06:28PM (#23209448)
If mathematics is invented, then let's invent some right now. First, let's set the scene: Mathematicians ran into this annoying problem that you can't take the square root of a negative number, so they invented this number, i, that is defined as the square root of -1. Then, by using this i in your answer, any root can be expressed. Ok, now that the scene is set, I find it incredibly annoying that you cannot divide by zero. Therefore, I am hereby inventing a number, j, that is defined as one divided by zero. Henceforth, you can express any number divided by zero by using this j in your answer. Who knows, such a thing might actually be useful.
• #### You've just reinvented Projectively Extended Reals (Score:5, Interesting)

on Saturday April 26, 2008 @07:09PM (#23209702)
Congratulations, you've just invented the Projectively Extended Reals [wikipedia.org]! Yes, it is certainly possible to get a consistent system with 'a point at infinity'. Trouble is, it isn't very useful. Why not? A lot of things that make the Reals useful come from the fact that they're a field. The projectively extended reals aren't a field, so you lose a lot of useful theorems. And there really isn't very much you can do with them that you can't do with the normal reals or that wouldn't be better done in a Riemann sphere anyway. The complex numbers as an extension to the reals, by contrast, are enormously useful, not only in Mathematics (complex numbers are a field) but also in Physics and Engineering.
• #### Are jokes discovered or invented? (Score:2, Interesting)

Substitute "jokes" in place of "mathematics, and the question becomes both stupid AND enlightening.

"Are jokes discovered or invented?" Obviously, jokes are invented. Also almost as obvious, more than one person can invent the same joke at around the same time.

Just as obvious, nobody "invents" the ratio between the circumference and the diameter of a circle - so this mathematical truth is discovered, not invented. It (pi) existed before anyone discovered it and named it.

Or to put it another way, truth

• #### Lawyers are circling again I see (Score:2)

...like hungry sharks. If it's discovered, we can patent it. If it's invented we can copyright it. Imagine something as fundamental as Pi falling under copyright. I think that'd be bad. Imagine a cease and desist for reproducing Pi to 10 digits and publishing the forumla for a circle. Then again perhaps patents would be worse. With copyright not requiring registration, if it's obviously frivolous and you can show prior art it's out the door. However I wouldn't put it past certain patent offices to grant a
• #### Re:Lawyers are circling again I see (Score:4, Informative)

on Saturday April 26, 2008 @06:55PM (#23209592) Homepage
> If it's discovered, we can patent it. If it's invented we can copyright it.

No. If it is invented it can be patented. If it is created it can be copyrighted. If it is discovered it can be neither patented nor copyrighted.
• #### Re: (Score:3, Informative)

You got it all wrong. You can not patent a discovery, and you have no copyright to an invention. You can, however, patent an invention.
• #### What Erds and Feynman believed about this (Score:5, Informative)

on Saturday April 26, 2008 @06:45PM (#23209530) Homepage Journal
The late mathematician Paul Erds used to say, perhaps metaphorically, that the most elegant proof of every mathematical theorem was written in a great book in God's library. When he came up with a beautiful proof, he would say it was one from the book.

Feynman also felt like coming up with a proof was more discovery than invention. He said that the proof felt like it was already there all along, raising the question of where "there" is.
• #### It's neither (Score:3, Interesting)

on Saturday April 26, 2008 @07:03PM (#23209652)
It all starts with 1+1=2, and that's neither a discovery nor an invention, it's an assumption. The rest is just semantics.
• #### Re:It's neither (Score:5, Insightful)

on Saturday April 26, 2008 @07:15PM (#23209744)
You can go a lot more basic than 1+1=2. Go back to the Peano axioms and you'll find that all you have to assume is the existance of "0", a "successor" function, induction, and a few trivial things like the properties of equality and addition, and you get the whole of arithmetic -- including 1+1=2.

So you invent/assume your choice of axioms, and everything else follows from them and can be discovered at leisure.
• #### Glib answer... (Score:4, Insightful)

<Peet42.Netscape@net> on Saturday April 26, 2008 @07:17PM (#23209752)
"Yes".

To be more specific, Mathematical rules are discovered, Mathematical techniques are invented; "Mathematics" consists of both.

on Saturday April 26, 2008 @07:21PM (#23209782)
It is coincidental that I was just reading about this in Paul Davies' book "The Mind of God". My opinion on the matter is fairly simple. Mathematics are invented. Period. The reason is simple... all of mathematics is an abstraction. There is no "real" thing called 1 or 2 or 3. In fact, the "integers" we use for counting things is only allowed because of the way we abstract the thing which we count. If we really defined whatever we were counting (say, coins for instance), then we could not count more than one of them.

Here's a thought problem for you.

You have the following in your hand:

A one-cent piece from 1978
A one-cent piece from 1986
A one-cent piece from 2004

I could have said you have 3 cents. But there is no such thing as 3 cents. 3 cents is an idea, an abstraction. It is not a concrete thing in the real world.

So, despite all that we appear to discover about the world through mathematics, we cannot really say that math is "out there" somewhere waiting for our discovery. Rather, mathematics is our projection onto the universe. It it because of the shortcomings of our abstractions and models that our science must be continuously revised.

For example, Newton did not discover anything about the universe. He made observations and rationalized (projected?) an abstract model which works very similarly to the observations. It's repeatable and consistent, so we call it a theory.

But then along comes Einstein. He makes some new observations, some new hypothesis, and voila, a new theory. Even if you argue that Einstein, or anyone else for that matter, has made such discoveries through mathematical observation, that doesn't discount the fact that the observation in that case is made upon the abstraction of the universe, not the universe itself.

In summary, mathematics is a simulation of the universe. It's an abstraction. One we humans invent. The fact that our model is observable, predictable, and so on in no way justifies the position that we are discovering some thing which pre-existed. Here's a final analogy - a computer model can be created to simulate the design of a car. We can study, observe, made predictions, corrections, and so on with the model. Yet, despite how relevant those observations, predictions, corrections, and so on are to the real car, they are still NOT the real car. The model is our interpretation, our abstraction of the car. We invent it. We make it. We project our ideas about the car into it. We do not "discover" it. The model does not exist without us.
• #### I vote "invented" because.... (Score:3, Interesting)

on Saturday April 26, 2008 @07:38PM (#23209900)
mathematics is an abstract concept similar to language. In fact, mathematics should be considered just another language because of the symbols (numerals) used. We use various languages (English, Spanish, etc.) to describe our world in words. We use mathematics to describe the world around us but in a numerical manner. Obviously our world exists without mathematics but we can use various components of mathematics to describe the world and the universe. We have differing numbering systems as well. They all can be used to describe the world around us. An interesting question is if an alien race (which I don't believe in but this is hypothetical) created something similar to mathematics, would it be proper to say that they also invented something and if they did should it be considered mathematics? Or would it be more proper to say they discovered the same thing we did if their mathematics turned out to describe the universe the same way our mathematics does?
• #### Civ (Score:3, Funny)

on Saturday April 26, 2008 @07:54PM (#23210000)
Everyone knows you have to discover mathematics before you can build catapults

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