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# Is Mathematics Discovered Or Invented?

#### kdawson posted more than 6 years ago | from the plato-says-yeah-but dept.

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."

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### Logical positivism to the rescue... (5, Insightful)

#### 26199 (577806) | more than 6 years ago | (#23209122)

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... (5, Insightful)

#### Anonymous Coward | more than 6 years ago | (#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:Logical positivism to the rescue... (2, Insightful)

#### somersault (912633) | more than 6 years ago | (#23209228)

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... (5, Insightful)

#### Anonymous Coward | more than 6 years ago | (#23209344)

Because squared gives you the right units.

### Re:Logical positivism to the rescue... (0)

#### Anonymous Coward | more than 6 years ago | (#23209372)

I think that math is invented, physics ( e=mc^2 ) are discovered. also it's excactly 2 because you hace to also conserve the units, you can have (in the physic world ) square foots, but what's the physics meaning of foot^2.14332544988?

### Re:Logical positivism to the rescue... (5, Informative)

#### nine-times (778537) | more than 6 years ago | (#23209406)

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... (2, Interesting)

#### fredcai (1015417) | more than 6 years ago | (#23209418)

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 (4, Informative)

#### SEMW (967629) | more than 6 years ago | (#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:Logical positivism to the rescue... (4, Funny)

#### JimDaGeek (983925) | more than 6 years ago | (#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... (4, Informative)

#### dreamchaser (49529) | more than 6 years ago | (#23209322)

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... (4, Insightful)

#### khallow (566160) | more than 6 years ago | (#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.

### No, the answer is CLEARLY invented... (1)

#### rtilghman (736281) | more than 6 years ago | (#23209466)

The "relationships and observations" you mention have NOTHING to do with the methods and language we have created for understanding and analyzing them. Mathmatics is, by definition, the system we use, not the phenomena themselves.

You can say that nature lends itself to analysis via mathmatics, but clearly mathmatics doesn't exist as some kind of absolute form.

Why is this even debated? Sounds like another stupid chicken/egg topic thrown around by people who can't manage deductive reasoning (the egg comes first folks since the first instance of what we define as "a chicken" was a genetic aberration born to a non-chicken).

-rt

### Re:No, the answer is CLEARLY invented... (1)

#### Eighty7 (1130057) | more than 6 years ago | (#23209588)

It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention. (Bridgman, P. W. The Logic of Modern Physics)

### Re:Logical positivism to the rescue... (2, Interesting)

#### xtracto (837672) | more than 6 years ago | (#23209538)

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 doing my thesis. However, those symbols are used to classify or "label" different patterns that *happen* in our universe and that we "perceive". It is then when we use such mathematical symbols to establish a classification of such patters (for example, we know that Weight = Mass Ã-- Gravity, because of experimentation, however we did not created such relationship or pattern. We just labeled it "\times" (or sometimes *).

### Re:Logical positivism to the rescue... (1)

#### HungSoLow (809760) | more than 6 years ago | (#23209568)

I completely agree. I think we need to consider an alien species that is advanced as we are, and what their mathematics would look like...

I think it's obvious that there would be similarities. Any similarities would probably be due to the ubiquitous nature of mathematics and hence lend credence to the 'discovery' mantra. But as far as I can see, concepts such as matrices, vector spaces and the like are merely tools we've developed and are therefore invented. An alien species would in no way need to have a theory of matrices. It's a tool mathematicians have invented to make their work a hell of a lot easier.

There are clearly both concepts discovered and invented. A prime number is not something someone has invented. Their distribution is not something someone has invented - both are discovered. How we compute them, on the other hand, is definitely invented.

### Re:Logical positivism to the rescue... (5, Insightful)

#### nine-times (778537) | more than 6 years ago | (#23209578)

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... (1, Insightful)

#### Anonymous Coward | more than 6 years ago | (#23209244)

What would it allow you to do that you couldn't do before? What could you predict? What would you gain?

Patents, trademarks, copyrights, and with that, giant gobs upon gobs of money.

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

#### Vellmont (569020) | more than 6 years ago | (#23209256)

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... (5, Funny)

#### goombah99 (560566) | more than 6 years ago | (#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:Logical positivism to the rescue... (1)

#### zoomshorts (137587) | more than 6 years ago | (#23209274)

The correct answer is DEVELOPED :P

### Re:Logical positivism to the rescue... (0)

#### Anonymous Coward | more than 6 years ago | (#23209330)

Oh, and the correct answer is "discovered".

Yeah, I don't even know how this is up for debate. It's like those weirdos who think Lisp was invented.

### Re:Logical positivism to the rescue... (2, Insightful)

#### TheTapani (1050518) | more than 6 years ago | (#23209334)

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 (2, Funny)

#### g253 (855070) | more than 6 years ago | (#23209350)

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:It is indeed discovered (1)

#### azaris (699901) | more than 6 years ago | (#23209428)

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 the rest are deduced based on those things.

### Re:Logical positivism to the rescue... (1)

#### bipbop (1144919) | more than 6 years ago | (#23209358)

For a science fictional reason why these answers might not be "nothing", take a look at Greg Egan's short stories _Luminous_ and _Dark Integers_. If nothing else, they're pretty entertaining :-)

### Re:Logical positivism to the rescue... (1)

#### nine-times (778537) | more than 6 years ago | (#23209460)

Yes, and if science fiction were real, I could fix every one of my problems by realigning the deflector array.

### Re:Logical positivism to the rescue... (1)

#### xTantrum (919048) | more than 6 years ago | (#23209424)

actually the concept of quantity was discovered, and its applications in the various fields were invented. this really isn't that hard.

### Re:Logical positivism to the rescue... (3, Interesting)

#### professionalfurryele (877225) | more than 6 years ago | (#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:Logical positivism to the rescue... (1)

#### telso (924323) | more than 6 years ago | (#23209462)

It's meaningless; merely a matter of perception, wordplay and people having too much time on their hands.
Philosophers to the rescue!

### Re:Logical positivism to the rescue... (0)

#### Anonymous Coward | more than 6 years ago | (#23209502)

Obviously the answer is 42.

### Re:Logical positivism to the rescue... (0)

#### Anonymous Coward | more than 6 years ago | (#23209508)

If I may gently ask: What makes you think the positivist approach is the correct one?

Logical positivism has fallen out of favor in contemporary academic philosophy because it suffered from one critical flaw: self-referential inconsistency. (discussed in the Wikipedia article). I would be loath to put forward any sort of pronouncement based on a positivist premise, even if I thought I was correct.

I personally think mathematics is discovered too, and that the formal system we call mathematics is a mapping to real phenomena.

But that's only my guess; I can't prove it.... for all the corner points.

### Mathematics in the forms of human intuition (4, Insightful)

#### traindirector (1001483) | more than 6 years ago | (#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.

### Re:Mathematics in the forms of human intuition (1)

#### Samgilljoy (1147203) | more than 6 years ago | (#23209332)

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.

As soon as I saw "Kantian," I knew what my answer would be to the question, and I am very happy to see that someone articulated it breve et clare much better than I could have.

.

Sadly, I'm without mod points today, but I'm tipping my virtual hat.

### Re:Mathematics in the forms of human intuition (1)

#### Reality Master 201 (578873) | more than 6 years ago | (#23209582)

One could have summed up ever more succinctly:

Both; see the Critique of Pure Reason, synthetic a priori judgements.

### My inner electron (1)

#### goombah99 (560566) | more than 6 years ago | (#23209340)

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 program it is a left brained experience. I'm fully aware of the rest of the programs existence even though my right-brained eyes see but a few lines at a time. I feel it's flow though the coed itself never moves.

To there is a sensory nature to programming that is like being able touch things that are abstractions.

But so what I can use it without being it.

### Re:Mathematics in the forms of human intuition (1)

#### nine-times (778537) | more than 6 years ago | (#23209346)

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) through empirical means, but rather through intuition. Whether those things exist "in themselves" depends on whether our intuition is actually accessing some source of real knowledge/understanding/whateveryouwanttocallit. I don't remember him saying asserting that this intuition is not coming from some sort of "world as it is in itself". Instead it seems to me that he indicates that we should trust intuition because we must trust it-- we have no other choice.

Sorry to be pedantic, but it seems worth noting that Plato and Kant don't really differ that much on this subject. Both of them would say that mathematics are not created by a person, but derived from principles that we know a priori. Of course they might use different words.

### Re:Mathematics in the forms of human intuition (0)

#### Anonymous Coward | more than 6 years ago | (#23209480)

Precisely. Also, mathematicians should stick to math. This question has absolutely nothing to do with mathematics, theoretical or otherwise. It is a question of epistemology. The particular subject is irrelevant.

### I know this! (5, Funny)

#### ForumTroll (900233) | more than 6 years ago | (#23209130)

It's intelligently designed.

### Re:I know this! (1)

#### nabil2199 (1142085) | more than 6 years ago | (#23209200)

It's intelligently designed.
6000 years ago like everything else

### Re:I know this! (1)

#### goombah99 (560566) | more than 6 years ago | (#23209398)

It's intelligently designed.
by turtles!

### But did God invent or discover it? (2, Interesting)

#### CustomDesigned (250089) | more than 6 years ago | (#23209434)

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 called Nominalism - "good" is just a label for what God does. Or is what God does "good" in some objective sense? (Realism.) But that would give "goodness" an existence independent of God.

The answer to that question actual *does* affect future decisions. Unfortunately, it is hard/impossible to *verify* the answer, which is what I though Logical Positivism was about. "Statements which cannot, in principle, be verified, are meaningless." Of course this self refuting formulation would not be popular with adherents.

### Re:But did God invent or discover it? (0)

#### Anonymous Coward | more than 6 years ago | (#23209520)

Umm... God doesn't exist.

### Patently Obvious.... (3, Insightful)

#### headkase (533448) | more than 6 years ago | (#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 (1)

#### xPsi (851544) | more than 6 years ago | (#23209146)

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 a state space exists containing all outputs of the rules; indeed also containing all the possible rules themselves. If that state space is big enough, the process of discovery becomes indistinguishable from creativity, since finding non-trivial points at random in that space becomes very improbable.

### Invented and Discovered (1)

#### Torodung (31985) | more than 6 years ago | (#23209150)

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 one, which has no real mathematical meaning. We just set a value because we needed to.

It is also discovered in that we discover how our invented system relates to the real world, the non-finite system, by which all of "nature" operates. Discovering this relationship between our invention, mathematics, and the universe at large, is what drives mathematics. Discovering the point at which they interface is a profound experience.

So I'd have to say: it is both an invented and discovered system, and the two forces (reality vs. theory) are what drive new mathematical concepts, and most of the natural sciences.

It's a false dichotomy. Have fun assuming you can't have it both ways, folks. ;^)

--
Toro

### Connection to math = The Universe (2, Interesting)

#### Shatrat (855151) | more than 6 years ago | (#23209152)

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. (3, Insightful)

#### ArsonSmith (13997) | more than 6 years ago | (#23209158)

The concept was invented.

What can be done with it is then discovered.

### MOD PARENT UP (1)

#### Torodung (31985) | more than 6 years ago | (#23209178)

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

--
Toro

"kewpie".

### Re:How is this a debate? It's both. (3, Insightful)

#### Bill, Shooter of Bul (629286) | more than 6 years ago | (#23209304)

Thats absurd.

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

### Re:How is this a debate? It's both. (1)

#### SpinyNorman (33776) | more than 6 years ago | (#23209420)

That's deep.

You must be the next fuckin' Plato.

### Re:How is this a debate? It's both. (1)

#### PenguinX (18932) | more than 6 years ago | (#23209452)

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

cheers
-b

### invented (1)

#### kennylogins (1092227) | more than 6 years ago | (#23209170)

it's the name of a system designed to model reality

### So why do we care? (0)

#### Anonymous Coward | more than 6 years ago | (#23209174)

The terms 'discovered' and 'invented' are only ever approximations to what is really going on in someone's head any given situation. They are just words. Why would mathematicians (or non-mathematicians for that matter) care?

### Twofo Ghey Niggers (-1, Offtopic)

#### Anonymous Coward | more than 6 years ago | (#23209176)

Eating my goatse'd penis! [twofo.co.uk]

That's right, you cock-smoking tea-baggers!

### Discovering the relationship-Inventing the Theory (1)

#### thinktech (1278026) | more than 6 years ago | (#23209180)

We discover the already existing relationship between numbers and we invent the theory that describes this relationship.

### isn't this more a question of philosophy (0)

#### Anonymous Coward | more than 6 years ago | (#23209184)

and the history of philosophy?

or am i not supposed to bring up the 'fluffy liberal arts major' stuff here on the great engineering round table discussion board?

### Re:isn't this more a question of philosophy (1)

#### PenguinX (18932) | more than 6 years ago | (#23209522)

Yes, perhaps and it would be accurate to say that mathematics, logic, science, and philosophy have more to do with each other than meet the eye.

-b

### Is Mathematics Discovered Or Invented? (5, Interesting)

#### SamP2 (1097897) | more than 6 years ago | (#23209186)

Is Mathematics Discovered Or Invented?

Neither. It is defined.

### Re:Is Mathematics Discovered Or Invented? (0)

#### Anonymous Coward | more than 6 years ago | (#23209360)

That's beside the point. The question is what the nature of a definition is: Does a definition describe or does it create? The object of a definition either existed before the definition or it didn't. It's like asking whether a statue exists "inside" a block of marble before the artist removes the other pieces of marble. Yes, certainly everything that comprises the statue already exists, but the artist's selection didn't. In this case we're not interested in the marble, but in the artist's separation of statue and non-statue.

The philosophical question behind all that is whether math is "natural", if it is the only "logical" choice of axioms because the set of axioms is inherent to the universe or because it is built into us, who observe the universe. Are we the source of math or do we discover something which exists beyond us?

### Re:Is Mathematics Discovered Or Invented? (2, Interesting)

#### aztektum (170569) | more than 6 years ago | (#23209576)

Too paraphrase some Buddhist thoughts on the idea of value, the only value anything has is that which we create for it. Nothing has intrinsic value until individuals express value for it.

Applied to math, you could say mathematics is a series of definitions we've created to describe an observed phenomenon or hypothesize the existence of an as yet unobserved phenomenon.

But what the hell do I know? I'm neither a philosopher or a mathematician.

### Re:Is Mathematics Discovered Or Invented? (1)

#### gmuslera (3436) | more than 6 years ago | (#23209590)

Kinda like when we ask ourselves if a tree falls it make sound if noone is there to observe that. Im more akin to the "define" choice... we just put a name (numbers, operations, etc) of what is already there.

Dont think in something like a big rock from where we take from it an statue, think in sand, water drops, etc that adds, substracts, etc. At least, for the very basic of mathematics.

Now. mathematics is too big for a word, probably part can be counted as defined, another part is discovery, and another, invention (maybe non-euclidian geometries, or imaginary numbers fall in this category).

### It's both (1)

#### Eevee (535658) | more than 6 years ago | (#23209192)

It's a floor wax and a dessert topping.

### Re:It's both (1)

#### wbaxter1 (1240732) | more than 6 years ago | (#23209456)

Thanks, that line was going around in my brain when I saw your response freakin' awsome :D

### I think (0, Redundant)

#### maroberts (15852) | more than 6 years ago | (#23209204)

...that an Intelligent Designer made Mathematics about 6,000 years ago.

### The short answer: (1)

#### Ethanol-fueled (1125189) | more than 6 years ago | (#23209206)

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 (4, Interesting)

#### Animats (122034) | more than 6 years ago | (#23209214)

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:Only the integers (1)

#### Ralph Spoilsport (673134) | more than 6 years ago | (#23209282)

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 over.

A good book on a similar topic is Stanislaus Dehaene's "The Number Sense".

RS

### Re:Only the integers (1)

#### Sique (173459) | more than 6 years ago | (#23209422)

Those axioms being the Archimedical one and the Cauchy/Dirichlet/WeierstraÃY one.

### Re:Only the integers (3, Interesting)

#### nine-times (778537) | more than 6 years ago | (#23209488)

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 (1)

#### sbillard (568017) | more than 6 years ago | (#23209220)

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.

### Sorry, It's Invented by Discovery (0)

#### Anonymous Coward | more than 6 years ago | (#23209232)

Here's how it works-- a person discovers how nature works, and then invents math to describe it.
Most people use the word 'math' to describe both the underlying function of nature and the language we use to describe such events, which is the only reason the debate exists.

Basically this is just a bunch of people with nothing better to do than argue about terminology that nobody else cares about.

### Axioms vs. theorems (4, Insightful)

#### G4from128k (686170) | more than 6 years ago | (#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? (3, Interesting)

#### Khashishi (775369) | more than 6 years ago | (#23209268)

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 (5, Interesting)

#### blaster151 (874280) | more than 6 years ago | (#23209288)

Are songs discovered or written?

### All the same? (3, Interesting)

#### Thyamine (531612) | more than 6 years ago | (#23209290)

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 (1)

#### John Hasler (414242) | more than 6 years ago | (#23209298)

> 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.

### Re:Discovery Need Not Imply Metaphysics (0)

#### Anonymous Coward | more than 6 years ago | (#23209356)

My answer would be that mathssss is a representation of the fundamental symmetries in nature. What explains the fundamental symmetries? Well if they weren't there we wouldn't be here to ask the question.

### They assume (1)

#### jd (1658) | more than 6 years ago | (#23209310)

....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 nothing to do with humans ascribing the value. Pi would hold the value of Pi had humans never evolved.

They also assume mathemtics is uniquely human. We know of animals that can perform basic arithmetic and even have a notion of zero. Ergo, mathematics is not uniquely human. Coincidental inventions happen, usually when the basic idea has been around for a while and the invention is "ready" to be invented, but this clearly does not apply to cross-species discoveries of things like zero, as there is no connection whatsoever between those discoveries.

### Re:They assume (0)

#### Anonymous Coward | more than 6 years ago | (#23209396)

You say that "we know of animals that can perform basic arithmetic and even have a notion of zero". Being a bear of very little brain indeed, I am intrigued by this, at the very least from a methodological point of view. Would you care to give references backing up this statement?
Or do I find myself yet again in the "goldfish have an attention span of seven seconds" territory so dearly held by the tabloid press?

### Re:They assume (0)

#### Anonymous Coward | more than 6 years ago | (#23209554)

Pi would hold the value of Pi had humans never evolved.
Since you state that so nonchalantly, I suspect that you don't understand the problem. The concepts which result in Pi and its value may well only exist in us. The fact that we recognize mathematical concepts everywhere, for example in the behavior of animals, does not contradict that.

### "Remarkably"? No. (1)

#### BorgCopyeditor (590345) | more than 6 years ago | (#23209324)

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? (1)

#### DrEldarion (114072) | more than 6 years ago | (#23209326)

Oh no, math and I have a far more... intimate relationship.

### what can be discovered? (1)

#### v1 (525388) | more than 6 years ago | (#23209368)

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 line, but I'd speculate that simpler maths cannot be claimed to be invented, while more complex maths cannot be claimed to be merely discovered. Obvious = discovered. Unintuitive = invented.

### Re:what can be discovered? (1)

#### lahvak (69490) | more than 6 years ago | (#23209518)

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" (1)

#### PureCreditor (300490) | more than 6 years ago | (#23209374)

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).

### Thought Game (1)

#### Nautical Insanity (1190003) | more than 6 years ago | (#23209388)

One unit together with one unit yields units that are two in number. This happens regardless of human observation, thought, categorization, what-have-you. Everything else follows.

And speaking of observation...Schrodinger's cat is dead.

### Re:Thought Game (0)

#### Anonymous Coward | more than 6 years ago | (#23209504)

actually 1 + 1 yields 10.

### ideas possess a location? (1, Insightful)

#### bzipitidoo (647217) | more than 6 years ago | (#23209408)

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.

The article takes a turn to the weird when it suggests that if these concepts already existed and we merely discovered them, then we somehow obtained this information-- from somewhere. From reading the inherent properties of the universe, perhaps. Except I don't see why this "obtaining" should follow. That's rather like saying we couldn't think of things on our own. The article begins to seem like a troll of the same sort as the Intelligent Design and the "God of the gaps" arguments. I also wonder if this is a devious argument meant to justify Intellectual Property laws.

Perhaps I have it wrong and someone could better express what the author means?

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

#### clichescreenname (1220316) | more than 6 years ago | (#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.

### Semantic pointlessness: why distinguish maths? (1)

#### gedhrel (241953) | more than 6 years ago | (#23209416)

So the question is: were mathematical truths invented by mathematicians, or were they always, platonically true, awaiting discovery?

Why distinguish mathematics in this question? Take any other field of invention. Is it the case that physical principles that a particular realised invention uses were not true prior to their "discovery"? Or was the operation of the invention always so?

Platonism would have all invention as merely "discovery". At that point, the word becomes distorted and devalued. It's pretty much a pointless debate.

Except where Penrose is concerned. As far as his opinion goes, it's extreme mysticism, and he invents (discovers?) an awful lot of hoops to justify his rather odd religion.

### The super-imaginary number, j. (3, Interesting)

#### suck_burners_rice (1258684) | more than 6 years ago | (#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.

### The Economic Factor (1)

#### SnarlSlayer (1279872) | more than 6 years ago | (#23209458)

I am not a mathematician, however I do know that the question has profound ramifications-- that which has been discovered usually cannot be patented; that which is invented can be patented. Additionally, once the question is answered (if it can be answered) it's only a matter of time before it's applied to code.

### Are jokes discovered or invented? (1, Interesting)

#### tomhudson (43916) | more than 6 years ago | (#23209470)

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, truths are discovered, not invented (and that REALLY pisses off the politicians).

### Lawyers are circling again I see (1)

#### syousef (465911) | more than 6 years ago | (#23209472)

...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 patent for Pi, which would require more effort to fight as you'd need to invalidate a registered patent. Standard IANAL disclaimer, so if I have something wrong, please feel free to correct.

### Jumping to conclusions (0)

#### Anonymous Coward | more than 6 years ago | (#23209482)

Platonism "has more in common with mystical religions than with modern science."

That in itself is perhaps not really a good argument for Platonism being wrong. One could also question the infallibility of modern science.

### Ha! I laughed so hard (0)

#### Anonymous Coward | more than 6 years ago | (#23209512)

I fell off my chairness.

### What Erds and Feynman believed about this (4, Informative)

#### Beryllium Sphere(tm) (193358) | more than 6 years ago | (#23209530)

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.

### Is Mathematics Discovered Or Invented? (0)

#### Anonymous Coward | more than 6 years ago | (#23209562)

It is mostly misunderstood!

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