Slashdot Mirror
Is Mathematics Discovered Or Invented?
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."
When faced with an awkward question, logical positivism 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".
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.
It's intelligently designed.
"A Lisp programmer knows the value of everything, but the cost of nothing." - Alan Perlis
Of course the answer could lead to further locking up knowledge... You can't read my theorem until you pay the license type deal.
Shh.
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.
i\hbar\dot{\psi}=\hat{H}\psi
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
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?
09 F9 11 02 9D 74 E3 5B D8 41 56 C5 63 56 88 C0
The concept was invented.
What can be done with it is then discovered.
Paying taxes to buy civilization is like paying a hooker to buy love.
it's the name of a system designed to model reality
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?
God, I wish I could say things with such brevity. Bingo. You win the cupie doll.
--
Toro
We discover the already existing relationship between numbers and we invent the theory that describes this relationship.
What's up with this box everyone has to think inside of or outside of? Why does there have to be a box?
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?
Is Mathematics Discovered Or Invented?
Neither. It is defined.
It's a floor wax and a dessert topping.
...that an Intelligent Designer made Mathematics about 6,000 years ago.
Donte Alistair Anderson Roberts - hi son!
Karma: Chameleon
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.
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. 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. 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.
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.
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.
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.
Two wrongs don't make a right, but three lefts do.
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?
Are songs discovered or written?
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.
I will shred my adversaries. Pull their eyes out just enough to turn them towards their mewing, mutilated faces. Illyria
> 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.
Warning: this article may contain humor, sarcasm, parody, and perhaps even irony. Read at your own risk.
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.
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
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.
Shop as usual. And avoid panic buying.
Oh no, math and I have a far more... intimate relationship.
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.
Some drink at the fountain of knowledge. Others just gargle.
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!
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.
I work for the Department of Redundancy Department.
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).
And speaking of observation...Schrodinger's cat is dead.
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?
Intellectual Property is a monopolistic, selfish, and defective concept. It is "tyranny over the mind of man"
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.
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.
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.
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.
McCain/Palin '08. Now THAT's hope and change!
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.
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
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).
...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.
These posts express my own personal views, not those of my employer
That in itself is perhaps not really a good argument for Platonism being wrong. One could also question the infallibility of modern science.
What's purple and commutes? An Abelian grape.
I fell off my chairness.
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 is mostly misunderstood!
Further reading... http://en.wikipedia.org/wiki/Philosophy_of_mathematics
The distinction could have implications for mathematical IP and the validity of software patents, if software is a way of getting a computer to do certain math, and math is discovered, vice invented.
Of course, does that mean that the MP3 encoding algorithm was discovered? And that I'm anxiously awaiting the discovery of Jedi Knight: Force Unleashed?
-- My Sig is a P228.
It's not like Pi (the relationship between the radius and it's circumference) wouldn't exist if mankind didn't exist or know about it. It's not even rational. Nether is the square root of two. Make a square with length one whatever sizes, then measure the distance between two opposite corners. It's not like nature can't work with squares, check out copper crystals. Man certainly didn't "Invent" them.
> Integers were discovered. Beyond that, it's human invention.
Totally bogus as well. Whoever started that idea was an arrogant anthropocentric imbecile. Any alien civilization that had to build anything would run across these relationships. It's not a great leap to abstract them into numbers, symbols and rules (what we call Mathematics). There's a obvious and logical progression from counting numbers, zero, the integers (includes negative numbers), rational numbers, irrational numbers, infinity, greater order infinities. To say that they wouldn't exist if Man didn't shows a great lack of understanding.
Erdos spoke of a book that God possessed that was a collection of elegant proofs. Erdos said he didn't know if God existed or not, but was quite sure the book did, having discovered some of its pages.
This argument is completely silly. Of course mathematical results exist before someone thought of them. Were there any integer solutions to x^n + y^n = z^n for n > 2 before Wiles? Was there some weird rational polynomial with pi as a root before Lindemann?
Philosophers should stick to fluffy pointless subjects that no one cares about. When they start thinking about mathematics the results usually range from stupid to ridiculous.
It is sensed.
Is it Patentable?
Starbucks, Harbuckle of Breath.
What's purple and commutes? An Abelian grape.
Math is an objective way to express things we observe around us, from the strength of gravity's pull to the correlation between the angles in our knees and elbows. Math is a language we invented to show and to prove the things we've discovered.
If a tree falls in the woods and there is nobody there to hear it does it make a noise? The question of course to scientist is "of course it makes a nose, noise is the consequence of matter vibrating the air." Math is no more an invention than noise is an invention. Would you also declare the discovery of light beyond the visible spectrum an invention? If invisible light is a discovery, than transcendental numbers must be a discovery: they already exist in their entirety, only as our computing power increases can we calculate what the next number in the sequence is.
Do you think a circle cares what PI is? Do you think an object along a ballistic trajectory thinks of it's decent as a mathematical formula? Yet the functions exist, have always existed, and are only awaiting our observation and discovery.
I've saved the best point for last: if math was invented, how could be be "corrected" as new discoveries are made unless it's truly just our interpretation of what we think math is.
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.
But math *is* a mystical religion...
Mathematics has both invention and discovery parts to it.
Mathematical systems are invented. Non-Euclidean
Geometry is an invention as shown by the
varieties available.
Invention to me is an arbitrary choice made amidst
a multitude of choices that is often
made for artistic or stylistic reasons.
Discovery is what you find when you use a mathematical
system. Theorems are the discoveries.
Base 10 is an invention. Pi is a discovery.
The Euclidean geometric system with its set of
axioms, definitions, etc. is an invention. The
Pythagorean Theorem is a discovery.
In response to the original post.
"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.
(The Question is) meaningless; merely a matter of perception, wordplay and people having too much time on their hands."
I must firmly but respectfully disagree that this is the outlook we should have. First of all, we have no true way of knowing what what we could predict or what we could gain if we knew the anser. Even if we knew that right now we could not predict or gain anything, there is no way of knowing that future discoveries wont prove the answer useful. I for one, do not believe it is necessarily the scientist's (and I do mean a scientist of any kind) duty to discover anything useful, insofar as it is difficult to know the full ramifications of a discovery until after it is made. And even if you did have a goal there are always unintended consequences that may overshadow your original design. Just look at TNT.
Well one could argue this isn't a matter of science, it is a matter of philosophy. Well in that case I would say that even if we don't believe we can find an answer, or even if we don't believe an answer would be useful, it is still part of our human desire to seek these answers, and to deny this is in my estimation, foolish. There is a part of us that thrives on this type of thinking and we should not ignore it.
This is not to say that all people get any kind of pleasure from this sort of exercise. But all the same, I do not consider it a hollow pursuit, regardless of what the outcome may or may not be, and regardless of if an outcome can even be reached.
"Taboo, like anything else, goes in and out of style."
The problem with the current wide following of materialisim is that it somehow synthetically imposes a decoupling of the spiritual and the material, in order to then be able to fully dismiss the spiritual as non-sense. I that way todays abundant confession of pure materialisim is just as wacky as the exact opposite: spiritualisim. Which tries to project the laws and mechanisims of the material world on to the spiritual.
Of-f*cking-course does math have more in common with mysticisim that with natural science - there is no way you can measure or weigh Math. We can sense it in the reflection of our sentinent minds only. That doesn't make it less real. It's called the spiritual world, you twit!
I'd even go further: I say a modern Mathematician has *much* more in common with ancient religious prophets and leaders than todays blind followers and proponents of constructed confessions people call 'religion'. Be they some variant of the monotheistic theme or some degenerated kit of current liturgies modelled after pantheistic beliefs and traditions, carried out with no true sense of their initial meaning *and* - even worse - with no intent whatsoever to explore or discover, none-the-less discuss their origins.
In my point of view a bright and carefull thinking & observing philosopher or is far closer to God than some evangelistic priest. The big problem is however that these people often propone pure materialisim and neglegt all holisticisim as pure and utter nonsense. And yet again, with the islam fundamentalists gaining so much attention these days and the most powerfull country in the world run by a evangelistic loonie (both claiming to base their actions on a religious and spiritual inclination and insight) I somehow can't blame them.
We suffer more in our imagination than in reality. - Seneca
Congratulations, you've just invented the Projectively Extended Reals! 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.
What's purple and commutes? An Abelian grape.
And I came to the conclusion that everything is discovered. The example I used was Music. All beats, rythms, lyrics, etc, already exist and are waiting to be discovered. It may be a stretch, but I'm starting to believe more in fate than free will, and there is no such thing as Random.
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.
Definitions & Axioms: Invented.
Theorems: Discovered.
Proofs: Invented.
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
"Yes".
To be more specific, Mathematical rules are discovered, Mathematical techniques are invented; "Mathematics" consists of both.
The answer is very simple:
Anything that is universaly true cannot be invented, it can only be discovered because it always existed.
Mathematical truths always existed so they cannot
be invented.
To me, the argument has nothing at all to do with metaphysics; it's about knowledge, and the semantics of the argument are weak and kind of obvious. The "internalism versus externalism" debate from centuries of epistemological thought is probably wiser (as in wisdom of crowds) than a faddish notion posed by contemporary Euro-math-geek-elites.
I can accept the semantic difference between principles and observations, and I really don't understand why some people can't. This argument belongs to the same family of paradoxes like "proof of Divine existence," or "whether information can be destroyed." Arguing about the root of knowledge is like shouting to make the wind stop. It might be fun for some, but it just doesn't make sense.
I don't understand why it's hard to accept that certain points of knowledge are a priori factual, and others need to be supported by other facts. However, I can understand why some people could be anxious to discover (or invent) an ideological foothold that would allow for the irrelevancy of absolutes. To me, the "discovered or invented" argument resembles reconstructionist attempts to Inject connotations where none previously existed. The arrogance with which these reconstructionists would dispute their own contradictions makes me completely nauseous.
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.
in soviet russia, math discovers you!
what about "goodness" or "justice"?
That approach is not logical positivism. It's Pragmatism. Two completely different schools. Logical positivists regard all statements as meaningless that do not have a truth value determined by either the logical system itself (tautologies, contradictions) or by contingent empirical facts ascertainable through observation. Pragmatists, on the other hand, don't believe in truth-conditional semantics; the meaning of a linguistic expression is a function of the practical consequences of its use.
Are you adequate?
Let's see if this spelling gets through Slashcode: Erdös
You can't even round, so spare us the philosophy.
who on earth seriously believes that all mathematics exists in some universal psychic brain?*
Is the Psychic Math Brain Theory (sorry, I mean Platonic Theory) really a serious contender among mathematicians as stated?
*As opposed, that is, to the idea that physical relationships that exist in our universe are sussed out through experimentation and logic and expressed in mathematical language.
Our current mathematical framework still carries the stigma of the evil of zero and infinity.
The idea that zero and infinity were to be feared held us back for centuries, and is so deeply ingrained in our thinking that it has quite possibly prevented us from discovering some quite interesting mathematics. It is for this reason that we cannot yet fully realise the potential of the work of Georg Cantor.
That aside, invention of mathematics really depends on what viewpoint you take.
As part of my doctoral thesis I describe a new mathematical technique. Did I invent it or discover it?
Beats me...
Personally, I think I invented my method. Mostly because discovery is to my mind reserved for the likes of Pythagoras and his ilk.
Invention could be discarded, but mathematical discoveries like the properties of triangles are, as far as I am concerned, fundamental statements about the workings of the universe.
Just a little beyond my league.
A learning experience is one of those things that say, 'You know that thing you just did? Don't do that.' - D. Adams
interestingly the word invent comes from the latin inventus: to come upon, to find. "you lucky bastard: you invented world domination by chance. Keep up the good work or we'll throw you to the lions.
This is perhaps the most vacuous debate I have ever seen /. post an article about. But it's a very good discrimintation tool for intelligence:
If you believe mathematical facts exist and operate whether they are known or not and require discovery by analysis, experimentation and critical thinking in order to further our understanding of our universe then you're right. (Yes, I agree the language and symbol used to describe them is invented) In this case please continue to exist.
If, on the other hand, you believe that you have the power to somehow magically create or invent a mathematical fact simply because you thought of it then please take your psychotic god-complex elsewhere and let the rest of us get on with the business of figuring out how things really work and relieve us from having to be frustrated by your delusions of grandeur. In this case please create a cult for you and your followers based on soul-transport and "sacred" kool-aid.
I will never live for sake of another man, nor ask another man to live for mine.
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?
this nation, under God, shall have a new birth of freedom. -- Lincoln, Gettysburg Address
Discoveries are when you find out about things that have already existed. If you travel to a place where no one has ever been before, you discover it. But it would still exist without anyone knowing about it. Gravity is still gravity, inertia is still inertia, and two plus two is still four. Whether or not we understand it has no bearing on the subject at all. (And don't give me that "high values of two" bunk. Yes, you're very clever, now shut up.)
Hearing well-educated people say "just let it die" is somewhat alarming to me.
Yeah, of course they're entitled to their oppinion on this, but I sure hope they let their rationality do the ruling on this, and not as some knee-jerk reaction to the less-than-bright creationists who seem to enjoy the spotlight nowadays.
If they let creationists influence their inclination towards the opposite side of the pond, the damage has already been done.
You don't need any mystical/religious crap to deal with mathematics being discovered. The ratio between a circle of *any* size and it's diameter is always the same, regardless of whether we have given it a name, or figured out an approximation. Long before this ratio was called Pi, and before we discovered that it was approximately 3.14159, it was still true that this ratio held. Same thing with the ratio of sides of a right triangle.
Now, one could argue that man invented triangles, and circles, except that, particularly circles, do occur in nature (the earth, the orbits of the celestial bodies, etc), triangles, somewhat less so, though *angles* do exist naturally, and a triangle is just closing the two 'primary sides' of an angle with a third side at some arbitrary distance along one of the sides of the angle. But, with things like shadows, angles occur naturally.
I think a great example of how mathematics is 'discovered', as opposed to invented, is something I heard about how illiterate shepherds have often used pebbles (or other physical objects) to track that they have the right number of sheep - for each sheep they start out with at the beginning of a day, they put one pebble in a bag, or pile, or pot, whatever. Then they check to make sure they have the same number of sheep as pebbles at the end of the day - without necessarily knowing anything about 'numbers' or how to count. They just remove one pebble from the collection for each sheep at the end. They just better end up with 0 pebbles or else they lost a sheep (or somehow gained one - e.g. from another herd that wandered into theirs). So, addition and subtraction work even when you don't know numbers, or how to count. I think that strongly shows that math is 'inherent' in nature.
Almost all the rest of math just flows from addition and subtraction.
Has to be discovered really. However, there are various degrees to how arbitrary or how 'beautiful' mathematical concepts can be.
;)
An alien race is likely to also use the majority of our 'foundation' concepts. As the theory gets more fuzzy and more complicated, there may be various other ways, and perhaps some of these ways are less elegant and more arbitrary than other ways. However, it's still possible to get elegant AND complicated mathematicss.
A good example is the Mandelbrot set. I don't we think we invented that shape
Why OpalCalc is the best Windows calc
I guess I fall in the discovered / defined category and yet the math we use day to day is invented. Our brains are born with the abilities to do complex math 'on the fly' doing physics geometry and who knows what else, simply to catch a ball in mid-flight. Yet if I took pencil and paper it would take me several minutes if not hours to figure out where the ball will be at point x on it's path.
To me this says that math is in our nature, there to be discovered. However, as we define it with pencil and paper we created [invented] the methods to find the answers we needed. And in my opinion not very efficient ones. When we actually find [discover] the math that's done in our head to catch the ball, I think we'll make some earth-shattering discoveries about well, everything, really.
I've done very little research on this, so forgive my ignorance if I'm missing something basic or obvious.
Everyone knows you have to discover mathematics before you can build catapults
Anyway, we built the representation of natural numbers. Plus, minus, and multiplication are invented transforms on natural numbers that were invented. as is the concept of exponentiation. These representations are useful.. but often have shortcomings.. naturals cant represent all decimal values.. fixed length decimals cant represent irrationals.
Anyway we invent all sorts of representations, from cartesian coordinates to Hilbert space.. but the relations of the various transforms are discovered... That the hypotenuse of a right triangle squared is equal to the sum of the sides squared is discovered... That 360 degrees are in a circle is invented.. The concept of a right triangle is invented. But the trig that transforms angles to sin/cos is discovered.. Anyway if a representation is wrong, then we can create a new representation, and start discovery from scratch.. IE roman numerals for modern math, horrible for exponents, fantastic for adding numbers together.... however bad discoveries point to a bad model, or to poor logic.
Storm
So does this debate have a root in the fact that one cannot patent discoveries but can patent inventions?
is this debate idiotic or retarded ? Jeez who cares . Please get back to doing some work (math, cs, physics) that might be good rather than wasting time discussing this crap.
Can we not agree that if a mathematical proof is postulated and proved - that it must be by definition, have always been true? Can we agree that the techniques through which we 'discover' these pre-existing mathematical proofs, are indeed, created by men and women?
It's a perfectly well defined question with a concrete answer. Mathematical symbols and techniques are invented by us, just like tools like hammers and microwave ovens. And just like hammers and microwave ovens we can discover interesting, novel and surprising things to do with them. That's it. No mystery.
Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.
Isn't this discussion purely academic?
To say we "discover" something with math assumes that the math is "real", and if so, then how we acquire information from such a realm (that clearly doesn't exist in the physical realm) becomes a very significant and serious issue regarding the composition of the universe - far more disturbing than the "missing mass of the universe."
If we deny that we discover things in math, but instead create things in math, then we are in a different quandary, because then mathematics takes on a certain arbitrariness, and investigations in maths becomes something more like butterfly or stamp collecting than the Fearless and Intrepid Discovery that gives us a reason to go on discovering things.
If it's invention, then the process is more like Sodoku - filling in the blanks of the universe, as opposed to finding New and Important Things about the world.
My personal guess is that it is neither - there are things about the universe we can find that is mathematic - specifically, numeracy, which permits addition and subtraction. (Some more things, some fewer things), but the rest of it we make up, as "higher mathematics" (anything above counting, addition and subtraction) is our language center's abstraction capacity playing games with our inherent numerical sensibilities. If i had to come down on one side or the other, i'd side with invention over discovery. I don't think humans are all that special, and I don't think we were built to figure out how the universe works.
RS
Shoes for Industry. Shoes for the Dead.
those who see it as being invented are nihilists who cannot see that there is great order to the universe. We've applied math to achieve feats of engineering. Even nature has employed technical mechanisms in organisms that existed long before human- kind came into the picture. To suggest that we invent math is pompous at best.
Yes we do invent impressive but often inadequate math to simulate or understand the natural world. It is from first realizing that there are simple and then increasingly more complicated mathematical observations that we know there is possibly an ultimate mathematical description of the universe that exists outside of human thought.
The nihilists believe we are the only intelligent life forms in the entire universe. It is from this viewpoint that they base their argument. Who's the mystic?
E=mc^2 is a special case.
http://en.wikipedia.org/wiki/Mass-energy_equivalence#Background
The underlying question, or question that may come out of this many years later maybe something as outrageous and ambitious as;
Since the theories of mathematics are models of real world observations, by definition we would consider it a discovery. However, we must acknowledge the underlying language and symbolism which is core to our mathematics is invented. Then the question would be, is this language of maths correct or can it be improved and hence remove some of the anomalies we have in mathematics today. Is our "elegant" language elegant enough?
If you have enery in kg.m2/s2 and you round up the useful variables, that's rest mass, velocity v and lightspeed c.
Then you already know you can write E as m0.f(v/c).c^2 because otherwise how are you going to get the dimensions right. f can still be any weird thing at that moment, but it's dimensionless. And surely you can set m = m0.f(v/c)/f(0) before starting to find out what the subject is you're looking at. Don't mind me though. streamofconsciousness.
The author of the EMS article spends most of it talking about how brains function in understanding mathematics, a different form of Platonism. That mathematical truths would exist without something to understand them is akin to saying A implies B, and A, thus B if and only if something thought about it. As far as the utility of if they are one, the other, or something totally different, if it's more akin to discovery, is there an easier way to survey the land?
I would try to explain it by saying that all equations are out there already in the same way that all integers are out there already. Once we had the idea for the infinite sequence of integers, we knew of every one of them. Even though mankind will never see most of the set. It's the same kind of relationship (a simpler one, but the same.) Cheers
I have a theory that the truth is never told during the nine-to-five hours. - Hunter S. Thompson
Mathematics is a field with enormous breadth and depth. I think it highly implausible that given the two possibilities, all of the various areas fall under only one origin. Some math is created to explain observed phenomena, and so "found". Some is created in the purely theoretical sense and only later found to have real world application and so "made". To muddy the waters, in all cases I can think of, those things developed under one of these inevitably are found to have connection with other things developed under the other, meaning neither can lay claim to being the original source.
I tend to think that not all areas of math that were created theoretically find a real world connection, making these at least purely "made", with no natural frame of reference. However, it can't be said they will never have such a connection until the time constraint for the definition of "never" has been exhausted, "Which is, of course, impossible."
The main problem I have arguments in the subject of "found" vs. "made" is that both presume to say the human mind is sufficiently advanced enough to be able to recognize true connections with Platonic ideals, or to show that they do not or cannot exist in any sense. The human mind is not terribly advanced beyond the realm of other animals', and in any case is advanced only along the path it has evolved, where as an entirely different kind of mind might only be able to operate under one context or the other, may see no contradiction in holding to both simultaneously. or may hold to a similar but orthagonal set of assumptions that would create a third option, although we might never be able to grasp the thought processes involved; our brain wouldn't be able to think along lines it wasn't evolved to think along (the "the problem with aliens is, they're alien" axiom).
I do tend to think that the answer, if such exists, will be something like "both, and then some" because absolutist either/or thinking such as the argument presented usually ends up being found to be a naive viewpoint from underdeveloped minds. In any case, from what I can tell, the vast majority of makers and users of mathematics carry on marvelously without tripping themselves up on the question, because the making and using does not require an answer to the question in order for these to happen.
"I may be synthetic, but I'm not stupid." -- Bishop 341-B
... when you're not anywhere at all?
I have started to wonder about Pi. I mean, isn't it all in the definitions? What's a circle -- it's the set of all points equidistant from a center point, What's "equal" when applied to "distance"? As soon as the distances are of a magnitude where relativity applies -- non-Newtonian space - then, whether two distances are equal depends on the velocity of the observer, on the frame of reference. A circle is only a circle if it's holding still -- but in reality, it's not holding still. So no circle is real; so in reality, there is no Pi. There is only Pi in mathematics.
Pi is not invented or discovered. It is imagined.
Then I say, hey, just take this circle and move toward it, or away from it, until you get an ellipse you like -- one where the radii have a nice whole-number relationship to each other and to the circumference -- and throw all those extra digits away. Problem solved.
That's my story and I'm sticking to it. Ulekewan Manatepe.
I get the car.
:)
Seriously, this seems silly to me. Math is a descriptive method of explaining calculation. It seems more akin to a language. Language explains things, unless it's computer docs, then it might explain things, unless it's man pages and then it only explains what you currently know or less(often lessing what you do know). But, basically math explains calculations dealing with numbers or number like things. So it's an expression, and as such could express a discovery, invention, or something else. As a language-like system though it's more line a story than an invention. But, as i type I find myself agreeing more with people say it just explains a natural law. And since often there is only one method for making that observation correctly then it would be dumb as hell to put it under copyright or patents. Could you imagine how that would retard the world of math as it has computers. Kids would not be able to learn in schools that could not afford the 2+2=4 tax, and calculating the taxes on 2+2=4 would also generate a fee.
-- Prepared at the direction of, or to be sent to Legal Counsel, in anticipation of litigation. Attorney Client Pri
Math is perfection embodied and completely logical. Humans are incapable of both perfection and pure logic. We can understand the concepts but we cannot actually achieve them. Conclusion, math is discovered. My two cents anyway and you know what they say about opinions, so the debate shall rage on.
"I reject your reality and substitue my own." ~ Adam Savage, Mythbuster extraordinaire.
You can only discover them. Inventing rules of reality is a pipe dream of the current presidential administration.
You CAN, however, invent applications of the rules of reality.
I'm not a mathematician. My experience with mathematics is limited to my college education (electrical engineering and computer science).
Mathematics, in my use, is nothing more than language to describe the natural world. In that sense, it is both an invention and a discovery. The invention part is the actual mechanics of the language, and the stuff it is trying to describe is the discovery.
-ted
It's a nonsensical question, all you are doing is arguing semantics.
For example consider how Socrates and his buddies argued about "good". There is no such thing as good existing outside of a "good meal", "good deed", or "good idea". It just modifies the word that follows.
The ONLY thing this argument is about is the difference between the words "invented" and "discovered". Can you REALLY invent something, or do you just put together previous "discoveries" in a novel way?
Math is math, good is good.
Arguing about math on slashdot is like arguing about pussy...some people think it is good, and easy and feels really good, and yet others having the same math or pussy will not like it, and say it is rough and feels bad. The one constant here is that you will really never get either.
"My immediate reaction is "WTF? What kind of moron doesn't make things 64-bit safe to begin with?" Linus
I'm certainly no mathematician, and this questions does seem absolutely pointless, but I'm going to bet on "invented". Sure, quantities of objects (Which could be added, subtracted, etc) would still exist even had we not invented mathematics. But with that in mind, having not invented language wouldn't stop objects with assigned names from existing.
"He who can destroy a thing, controls a thing." --Paul Atreides, Dune
The hard part in mathematics are not theorems and proofs; the hard part are elegant (or useful) theorems and proofs. You can generate as many theorems and proofs you want using automated computer (they are, after, recursively enumerable). But then you won't recognize the really interesting ones.
The point is, the really interesting theorems (and proofs) require good definitions. And the definitions are invented. Consider for example the progress that was made by replacing Riemann's integral by Lebesgue's integral (which is just a different definition of the same concept), or using distributions instead of real functions (again, different definition of similar concept). There are also many examples where good definition simplified existing proof considerably. And don't me even let to start on algorithms... Thus the main progress in mathematics is done by good definitions more than anything else.
So mathematics, as a usable science (as opposed just to a bunch of tautologies valid within a given axiomatics), is invented, not discovered.
P.S. When I realized this I also realized that the school of formalism in mathematics is missing the most interesting part.
THe figure resides in the block of marble before he began sculpting. he just chipped away what wasn't the sculpture.
I cannot see anything that would come of being able to say one way or the other if Math is discovered or invented. So why would the distinction matter?
END COMMUNICATION
Mathematics was invented. Take for example Euclidean geometry which requires perfectly straight lines with no thickness. In the real world, nothing makes a perfectly straight line - not even light. There are no true circles, triangles, parabolas, squares, rectangles. Not to mention planes - which span an unbounded space.
this is a no brainer
You're confusing Platonism with dualism. In Platonic principle, the idea is manifested in reality, so we can discover its form and from that pattern understand its design principle. Ever read the Pattern Language books by Christopher Alexander? Yep, that idea. Dualism says there's another world, and we must touch that world to be able to understand it. It's crazy talk for Semitic shepherds on fly agaric.
Anti-Globalism, Traditionalism, and FreeBSD.
travel to Logopolis and find out..
mathematics is simply the science that maps reason to specific; mostly numeric problems. Rudimentary mathematics are simply rudimentary logical operations. and advanced mathematics are advanced logical operations. When a given proof correctly identifies the facts of reality, it becomes the proof or law. The world learned at Hiroshima that Einstein's mathematics were not just academic, but that they correctly identified the facts of atomic reality. Finally, the idea that platonism must de because of its associaltion with mysticism is to toss the baby out with the bath water. The mysticism associated with Platonic theory has indeed died its own death. This despite its absorption by the medieval Catholic church. Do we see any "platonists" as such expressing the same or for that matter any mysticism today?
So, yes, it is invented.
I do not believe in karma. "Funny"=-6. Do good and forbid evil. Yours, Oft-Offtopic Flamebaiting Troll.
The newsletter of the EMS the Find announcement talks about is dated June 2007, so this no recent debate :-)
Besides, I believe a lot of maths is invented, but simultaneously the vast majority of it is discovered : Invention first (maybe), then lots of discoveries pertaining to this invention. Sometime the invention is simply an idealisation of the known world.
For instance, in the same issue of the EMS is the announcement that Stephen Smale was a co-recipient of the 2007 Wolf Prize (major prize in math). He is famous for many things, one of which was a proof that the standard sphere can be pulled inside out while remaining in standard space, without tearing or creasing.
There are many illustrations of the phenomenon, one of which is called "outside in" found on google video.
Inasmuch as a sphere is an abstract, "invented objet", the process of eversion, as it is called, a feature of 3-space topology, is definitely a discovered phenomenon.
In another instance, if one will, one could call natural numbers an invention, however a pretty natural one (eh). All the number theory results are however discoveries of properties of the natural numbers.
[ ]Half Empty [ ]Half Full [x]Twice as big as it needs to be
How does mathematics related to logic, to reason? Is the potential truth of a verbal statement invented or discovered? If I say "Maths is discovered (or maths in invented)" is that statement discovered or invented? Does some maths not make observations of purely numerical phenomena? Is all maths applicable to the universe? Does some maths exceed potential limitations in the universe, for example perhaps maths dealing with infinity? Is perhaps the universe, perhaps ultimately purely mathematical? Will one day in thousands of years a formula be dis coved which explains the whole universe?
I'm going to have to say that it was discovered, to me this is like asking "did we invent the earth as a spherical object or did we discover it was a spherical object?"...well we invented the term "spherical" and the language to allow that word, but it has almost always been spherical (I know it's not a perfect sphere). The order in the universe which is measured by mathematics have existed before humans...we simply discovered its order and called the order mathematics. Everything we do is in observation and evaluation, we simply shape our society on the order of everything and observe ways to make it work for us.
It seems pretty clear to me. All mathematical "knowledge" that can be proved using prior mathematical work and ultimately based on a set of axioms, was implicit in the original axioms. When we do mathematics, all we are doing is "discovering" the ramifications of our axioms - but that doesn't mean that mathematics are discovered from some platonic realm.
Rather, we have deliberately chosen the axioms, the underlying assumptions of our game. These axioms, and the rules of the game, have been invented to be consistent with one another (Godel aside). All other "knowledge" that comes out of consequences of the rules and axioms isn't "new", as such, but merely an implication of our chosen axioms.
Axioms are invented. Theorems are discovered.
150 Opening BINARY mode data connection for slashdot.sig (129323052 bytes).
Invented...
atleast it is the way I do it.
Blazing Spiders
Why would I need to connect with mathematics to understand it? That's like saying well we're not sure if botany exists because no one connects with plants... When we breed do we not start out at one cell then double and double and double? Isn't that a "connection" with mathematics? The math is there and was there and always will be there. The fact that we understand it doesn't give it being, but rather proves that we are truly special animals in that we can comprehend that which is there. A rabbit doesn't understand exponents, but is able to breed using them...
Defined consistently and in a closed system way? Wouldn't that violate Godel's incompleteness theorem?
http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/godel.html
http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
Serious question BTW, non expert here.
p.s. Godel's theorem would seem to imply mathematics is discovered, why would we invent something fragmentary?
Tired of all the isms, don't exploit people as an employer, or a government, mmmmK?
Like all phenomena that abides a rigid law, mathematics can't be "invented", nor can it be "discovered".
:-)
To be invented, it must first not exist. But since the rules are there, even tho the practicioners may not comprehend them at some point in time, everything still follows these rules and once the rules are understood the notation is invented. Yes, the notation. The way of putting the rules down on paper. That's an invention, as may well be seen with 18th century mathematicians, who many of which managed to understand the same things at similar times at different places in the world, putting them down on paper for future generations. Many of these "moments of enlightenment" are put down so differently that f.ex. when studying discrete mathematics, you generally learn several different notations for the same ground law.
But to be discovered, there's no requirement of an understanding. It's simply that you stumble upon something possibly great.
Take f.ex. gravity. We can with farly good certainty say that humans don't know how gravity works. We know that Newtons law og acceleration is affected by gravity, but we don't comprehend the laws and physics behind gravitational forces. Not enough to be able to say with any certainty what the laws of gravitation are.
But the gravity is still there. Ask anyone over the age of sixty. They'll show you proof.
Now, if someone really bright would think long and hard about gravity and how it works, he might finally understand the inner workings of gravity itself, and the laws that control it.
Understand.
He wouldn't discover anything, since what he was "searching" for was already there. He just didn't comprehend it.
Much like a native of the Maku tribe of South America wouldn't comprehend a fission reactor, although he might understand that it's a dangerous object, and he would certainly understand that it's there.
And however much you'd want to be able to claim to have invented the laws of gravity - I think that "doubtful honour" belongs to Al Gore, inventor of the internet, universe and everything.
This signature is DRM protected. By the DMCA, you are not allowed to counteract or oppose to it.
who gives a shit. Get a girlfriend.
Plain and simple.
The things that math describes are already there.
We invent a way to frame the concepts that we discover.
Nothing was nothing before we thought of zero.
We just invented a way to explain it.
The discovery is that it actually makes sense.
09 F9 11 02 9D 74 E3 5B D8 41 56 C5 63 56 88 C0
If its a discovery, who gets the claim to fame?
But if its just a human created abstraction set created for communication as most all other human created abstractions are for, then the copyright has long expired.
According to what I've learned from video games, it is discovered, and you need to know "Writing" to be able to do so.
Yes.
stupid fucking question
Debian FTW
We invented mathematics. But we keep only the ideas that we can reuse and adapt in a structured way to a variety of situations. The resulting "harmony", structure, "beauty", etc., we see in the resulting mathematical system are there because we selected the concepts that had those attributes. It's sort of like natural language; different cultures invent different languages to describe the same things, but in essence we all arrive at nouns, verbs, and so on. A hypothetical extraterrestrial race would invent their own mathematical system, which likely would look different from ours, but it would still describe integers, ratios, arithmetic, ...
I suspect this has to do with the problem solving bias one has - if you are a theoretician, you are trying to prove theorems. They are either true or false (yes, I'm ignoring the posibility they are indeterminate), and the fact that you haven't figured out which doesn't mean the answer is some wishy-washy thing - the theorem is TRUE or it is FALSE, we just don't know which yet. This sense of the answer being there but not being found yet is like a process of discovery, seeking something that already exists, but we just don't quite know where.
Applied people, OTOH, are rarely interested in proving a theorem for it's own sake, but usually are hunting down algorithims for determining answers to well modeled problems, perhaps proving some bounds on how rapidly their solution converges. There are frequently many different ways of calculating something, and when we come up with a faster converging, more computationally convenient method, it seems a clever invention, something new created. We don't think of methods and algorithims as being discovered, because they seem more like plans of action rather than a claim about a static thing.
People who have no feel for numbers tend to be in the 'invented' camp, but that's because they view mathematics as a bunch of arbitrary rules - the connections and underlying consistency is invisible to them, so they conclude it's all just made up - i.e. invented.
I personally like Ivar Peterson's interpetation; mathematics is like a vast jungle, of which we have only cleared out the tiniest areas. We have made many forays into the jungle, cleared a few spots, blazed many trails, and quite frequently find that many different trails wind up at the same place; the peaks of mountains, the waterfalls at the edge of a cliff.... Those who focus on the meeting spots and large cleared areas see them as being there already, and think of mathematics as a process of discovery of these common meeting points. Those who focus on the multiple trails, all the random and winding ways they go, watching how old, treacherous ones get replaced by shorter and eaiser ones, in a somewhat arbitrary fashion and with an apparent degree of luck, see mathematics as being invented.
But it is discovered, you know.... ;-)
Otherwise, if the math was invented as a tool that is not necessary (i.e. there are other ways to do it), then it is invented.
I would argue that some maths is discovered and some is invented. A lot of ideas used in mathematics can be found in nature and hence were discovered, the sine and exponential functions are good examples of this. Other maths is simple created either for it's own sake or to help us solve a given problem, examples of this can be found by studying abstract algebraic structures which are often used as tools for solving complex problems.
Hmm, seems pretty obvious to me. It's both.
Mathematics is about identifying patterns that exist in the real world. No matter how abstract it becomes later, the topic at hand WILL be derived from past "reality" based observations. This is why math is partly about discovery.
The notations of mathematics, however, is invented. FTA they make the cliche argument of 2 + 2 = 5 and embellish the conundrum mathematicians have with such an invalid statement.
But it really isn't much of a problem.
2 + 2 = 5 is perfectly valid if it has been DEFINED that way. It is fine if we have 2 white dots and 2 black dots to conclude we have "5" black dots if we agree to denote 5 as a symbol for the meaning for four. It is, on the other hand, idiotic to conclude there aren't any dots or infinite dots based on what's observed. The denotation of symbols is the invention side of math.
It just so happens that, with enough definitions, remarkable conclusions can be made about things that we can't hope to perceive in the real world because our senses aren't capable. These conclusions are related to patterns so qualify as discoveries.
It's been discovered that it's invented.
Humans invented what they discovered.
Logical positivism doesn't ask whether an answer will tell us something about the future. Instead, it has a verificationist theory of meaning which states that a sentence is meaningful if (a) it is true by virtue of the meaning of its terms (analytic), or (b) it is empirically verifiable. Besides this, a statement has to be well formed, which means that it conforms to the rules of syntax for that particular language, otherwise it is pure nonsense right out of the gate.
The meaning/meaningless distinction was intended by the logical positivists to distinguish between scientific statements --statements we can say involve legitimate knowledge claims-- and non-scientific statements. Carnap had an exterminationist mission against metaphysics, and sent as many statements to his guiliotine as possible. Eventually, however, the verificationist theory of meaning had to be executed, and the whole project fell apart.
This particular question would indeed be tossed by Carnap, on the grounds that statements like, "Mathematics involves Platonic forms," is neither analytic, nor empirically verifiable. The Platonic realm is by definition outside of the realm of experience.
I imagine that anyone saying it's both is on the right track, but not in the way most people are thinking. You see in Nature, long before Man, long before our Imagination to Invent Math, there was nothing and then something. From that something became something else and thus a great pattern formed into a complex creature born out of energy from the Universe. Sounds diluted at the scientific basic level I know. Then came Man who knew nothing and then was cast out of Eden having taken the fruit to know good and evil. Now that sounds biblical basic I know. Man with his fresh cleaned slate discovered a great many things in the World and with those discoveries his Imagination grew to Invent and discover more things and so on and so forth this continued till todays question. So do we Discover Math or Invent it? Do other creatures use Geometry to figure the nuts are too many for the capacity for the old tree, No. Does a God need to know Math to the number of floods, No. So then Math is an Invented Tool Created by Man for Man to Aid Himself in Exploring and Discovery. So Math is Invented, but we are Discovering a great many things with it.
The act of counting is a human behavior. Counting relies on our ability to differentiate the units of stuff to be counted. Where does one unit begin and end? This is a human notion. Counting gives rise to mathematical operations and also allow us to conceptualize ways to measure the universe (volume, speed, groups and sets, angles, etc..) Math is a model for a human perception of reality and is therefore an invention.
And they are discovered, not created.
I don't think they should be patentable either, but I'm not in charge.
Build a man a fire, he's warm for one night. Set him on fire, and he's warm for the rest of his life.
Some math is discovered. Like pi for example. You can say pi is the ration of circumference to diameter in a circle, or also a number that comes up in probability theory, or several other ways you could define it - but they are all the same actual value. Would anyone claim that Euclid invented the value of pi? No, he discoverd it (to the degree he was capable of anyway). But some other aspects of math are invented - say, number systems. Base ten or base two (or for that matter Roman numerals) were invented. Some people have invented number systems based on factorials, or negative or even complex bases - or my favorite, numbers based on the Fibonacci sequence. I suppose that goes back to the second answer above then. Number systems are part of the "language" of math - how we express mathematical statements. Terms like triangle and isometry and the symbol pi are invented, formulas and geometry proofs and finite simple groups are discovered
If Math is invented, then how can mathematical proofs exist? One can't base a logical proof on a theory or invention - a proof is either true, or it isn't.
There IS a reason to ask the question; the answer matters, be it to a few. It has nothing to do with physical applications -- more than half the posts here are irrelevant, because Math not only is not physics; it is not a physical science at all. If Math is "discovered", i.e. if mathematical objects lead a charmed life of their own and are not artifacts of ours, if the notions of "real numbers", "set of real numbers", "the set of sets of real numbers" are meaningful and well-defined, then it DOES maker sense, for example, to look for an answer to the Continuum problem (Cantor's Continuum Hypothesis, "CH")...beyond the current axiomatic framework (ZFC), which is known not to imply CH, nor not-CH. If Math is not, one might retreat into formalism, or formalism of sorts, where CH (among other things) is a consequence of this and its negation is a consequence of that, and so on and so forth...without commitment to, or maybe even a notion of, what is ultimately true. About the real numbers, if you will -- this is no abstruse question regarding large cardinals or something. Some mathematicians are formalists on Sundays; not always, though, when they are grappling with a problem :-P
[ Compare: "there are few atheists in the trenches". ]
My answer, if you haven't guessed yet: Math is discovered.
Having said that, the counter argument against 'math is discovery' (which is that, if math is discovery, then it must be Platonic, but Platonism is bullshit, so math is not discovery) is that nothing points towards 'math is platonic' more so than other theories, like 'math is a set of ideas that we must inevitably reason towards'(the nomological view)
Note that the 'everyone knows math' argument doesn't support Platonism; anything that is even meaningful is shareable. (yes, even 'internal' emotions)23413241324014151381098383771346 17236763476002374693746767349165862485 77264391763900632417623 9487162394871632948 7162394876193248 716239847169 23874169328746198 237461982374619823746 19283 741928374 1923847619
did I invent this number, or did I discover it? Nobody has written it down already, so I invented it. However, the number system exists already as a definite point on the number line which we all think to know, so I discovered it.
Mathematics is a word with many meanings, it can mean the product of mathematicians researches, what they supposedly refer to, or the human activity of thinking about those stuff.
Considering mathematics as the set of all possible axiomatic theories, I believe it's discovered, because of its very complex structure, which is what makes mathematics interesting. Each new detail of this intricate structure (proven proposition, theorem) is something no one knew (discovered), and was not under someone's control to make it true or false (invented).
Axiomatic theories can be developed mechanically, if there were physical systems that acted as theorem-proving computers on some random place in the universe, we could say that mathematics is really about the behavior of those computers (just like biology). There is no need of thinking beings or abstractions.
The real question is, where does the mathematics structure comes from? Does it depends on computational paradigms or notational systems? Is it specified by some more fundamental system?
If the answer is that it's somehow specified by aspects we invented or control, it might be considered an invention as well, otherwise, say, "it's a consequence of some physical law concerning information and computation", it might be considered a discovery, and a very special one.
Mathematics exists only as a combination of BOTH the human mind and external reality. Without the human mind there is no mathematics, without the external reality there is no mathematics.
It is analogous to the colour "blue", which is a combination of the human perceptual system combined with the external electromagnetic radiation at certain frequencies. Remove either and blue no longer exists.
Does a falling tree create a sound if no one is there to hear it?
How can you invent something that already exists?
I consider math to be software for your brain. The Romans had "Roman numerals I", but most people today have "Decimals 2.0" and "Algebra XP" installed.
Were Roman numerals invented or discovered? Were decimals invented or discovered? Both were invented, but they point at the underlying concepts that were discovered.
"e" is an human invention that points at the discovery that all growth rates share a fundamental base. "pi" is a human label for the discovery that all circles share a common ratio.
We invent the language of math (mental software) to describe relationships that already exist in the wild. We write physical equations (F=ma) to describe relationships that already exist in nature. So I think it's a little of both.
Does it matter? Yes! Not in the "practical" sense, just like art, literature, music, rainbows and *gasp* Star Wars don't matter in the practical sense. But they are still interesting in their own right, and we are curious creatures. Life would be a miserable place if it was only about making better widgets.
The feeling of invention or discovery can lead us to a better understanding of knowledge and our relationship with the world. Does Godel's incompleteness theorem change anything in your daily life? Not really, math keeps on trucking along, and 1 + 1 = 2 still works. But it's quite interesting philosophically, that some truths will always be unprovable.
no, one and one make two, two and one make three. it was destiny.
i've had just about enough of your vassar bashing.
human invented the way of looking at the universe, and discovered many insightful findings.
1
there, fixed that for ya
Was time discovered or invented?
I think it is discovered, but I think this goes for all inventions. It also goes for art. If you believe in God, then you believe that all ideas have already been thought before, before the creation of the world.
Consider the game of Chess. It is a deterministic game, with a huge but finite space of games that exist. Any game could then be considered to take a one of the fixed and limited number of paths through through this space. As such, it could be said that a chess game is not "played", it is only "discovered".
However, the experience of actually playing or watching chess results in an entirely different feeling -- so maybe it is not such a useful way to define it.
WHO is observing, physically? An abstract-entity?
Why can't this abstract-entity observe/perceive abstractions?
The Buddhists have held this to be possible for ages ( http://www.amazon.com/Understanding-Mind-Geshe-Kelsang-Gyatso/dp/8120818911/ )
The Mental Sense Consciousness is what they call it.
This question simply is false. Math is neither discovered nor invented: It is applied. What is invented is the structure. The symbol + means something, and that was invented. Digits as well are an invention, but there are uses for these structures. What is discovered is the uses. People don't just make things like integral mathematics up off the top of their head. It is a slow and often painful discovery. Math itself is the constructive use of numbers to describe things. It is the application of equality, functions, and funny letters to explain numeric principals.
I've already cleared the block out with just one echo. It is sensed. All this jibber jabber book talk is useless.
haven't these guys read "Zen and the Art of Motorcycle Maintenance?"
sheesh..
Mathematicians should know better than to have arguments about terms that haven't been defined. "Invented" and "discovered" aren't well-defined terms, and debating about whether mathematics is one or the other is pointless until one has defined them.
As for Roger Penrose, despite his mathematical insights, he seems to be prone to ill-founded leaps of logic and unfounded generalizations. Being able to reason logically in one domain apparently doesn't generalize to the consistent application of logic in other areas.
[eom]
In the same way that the other sciences are reproducible physical experiments mathematics consists of reproducible mental experiments.
Proofs give you the steps to reproduce the experiment.
To that end math tells us as much about the human mind (what is considered beautifull or elegant in mathematics?) as it does about the universe.
http://www.maths.manchester.ac.uk/~avb/micromathematics/downloads
It was discovered. The universe is made up of 2 fundimental areas. Those that fall into order(structure, logic, positive) and those that fall into chaos(non-structured, random, negitive). Mathematics can be said to be the method used to understand those orders. The connection from man to the order makes it discovered as it existed before man, just that man didn't understand it. Calling it mathematics, is invented.
If mathematics is invented it can be patented.
HTH.
Deleted
We do invent the artifact we call pure mathematics. However, the fundamental truths of mathematics are not created just because we 'discover' them. Nor will those truths cease to exist just because we will eventually be extinct. Existence is not an attribute of the concept of a triangle. The confusion of the scope of 'existence' was at the kernel of Plato's problematic regressions.
In contrast, mathematical physics is purely discovered. That's just figuring out how things are. Applied mathematics is sort of stuck in the crack.
(While one of my degrees includes philosophy. However, I think the only important thing I learned studying philosophy was that you don't learn anything important from studying philosophy. The important philosophy is how you live.)
Freedom = (Meaningful - Coerced) Choice != (Speech | Beer^2), and sad sock puppets' bad mods avail them naught.
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.
And that is a weakness of your theorem prover. The nq in NQTHM aka Boyer-Moore theorem prover stands for non-quantified. Quoting from the manual:
Nqthm does not support, especially well, attacks on theorems in set theory or about such nonconstructive entities as the real numbers.
Set theoretic formulations for integers exist and no theorem prover is ever going to give you any insight into what is discoverable and what is not. so, please stop spreading that BS around.
There exists a distinction between invention and discovery only because we create one. Any given mathematical technique that we "invent" could also be considered a "discovery", because we simply discovered that this particular technique works correctly within the framework of mathematics that we have already defined.
Let's take an example: calculus. Newton and Leibniz both invented calculus simultaneously. It could be said, then, that they both simply discovered the same thing!
This is a question of linguistic semantics.
I thought this had been settled long ago. Axioms are invented, theorems are discovered.
It depends very strongly on each persons style how he or she performs mathematics. I have met some people who are extremely good in developing a certain technique until the point when it makes a new abstraction possible. And then other people may use these tools and maybe can do some discovery. It'like in physics (the subject i have a PhD in): for sure it need a lot of "invention" to build an Atomic Force Microscope. To invest these "invention" is a matter of intuition, even if the process itself is very technical. When the AFM is working "discoveries" can be made. Some of them rqeuire more intution, some less. Some are expected before the AFM is build, others not. Non-Scientists, and some scientists, too, would put the expected discoveries into the "invention" class. However, from a scientific viewpoint the results have the same quality as the unexpected ones. In math it's the same. There are very constructive tehings, like exploring finite fields. In the poor framework of the original question asked, this would be something like an invention. However, sometime things like Fermats conjecture appear out of nothing. The really interesting question is: finally a combination of highly developed abstract methods and mathematical intuition very far beyond my personal horizon did the proof.
So was this a result of discovery or an invention? I would say it doesnt matter to me!
If there were no humans, would 2+2 still equal 4? The answer is yes, so people merely discovered this fact. Now, arithmetic is only a branch of math, but the theory applies to the whole (otherwise I'm wrong, and that can't be).
If there were no humans, would there still be cell phones? The answer is no, so we can say that cell phones were invented. Don't give me any crap about alien cell phones -- you get my point.
For those that want to get hung up on the first question, it's not as difficult as you're making it out. Of course the symbols would be different if another group of discoverers were to define the language of math, but the fundamental premise would be unchanged.
If you have any good reason why I'm wrong, I'd appreciate you not letting me know. Adapting to new ideas is a real bitch.
Kevin
Also significant in this debate is Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," which observes the myriad of ways in which mathematics predicts rather than describes phenomena.
Which has always seemed to me the problem of non-Platonic approaches to mathematics - mathematics is a system more precise than reality that nonetheless approximates and describes it. Atoms do not neatly form right angles, and yet we know many properties of right triangles that are repeatedly borne out in the physical world. Mathematics, in this regard, seems like it must have a metaphysical component. Which would require "discovered" to be the answer.
Philip Sandifer's academic website
That which needs to be explained is the power that mathematics provides vis a vis practical life. Mathematical forms in this regard are said to be normative. So if bald platonism is rejected then the question remains: what explains the normative power of mathematics? The nominalists raised these objections centuries ago - then the post-modernists: these forms are just 'flatus vocis,' empty sounds, symbols without that which is symbolized. Needless to say these two groups can't explain the normative power of mathematics either. Extreme Platonism may be unfashionable, but at least it offers some sort of explaination that the 'invented' camp cannot provide.
The problem is that 'mystic' has a negative co-notation of 'un-scientific'. Which is wrong. Just because a scientist precisely predicting some phenomenon or a programmer doing seemingly decoupled things to suddenly make an automated process work might appear mystic/magical to non-scientists and non-programmers doesn't mean there is nothing but simple reasoning based on true insight into the matter involved.
The problem with the current wide following of materialisim is that it somehow synthetically imposes a decoupling of the spiritual and the material, in order to then be able to fully dismiss the spiritual as non-sense. In that way todays abundant confession of pure materialisim is just as off the mark as the exact opposite: spiritualisim. Which tries to project the laws and mechanisims of the material world on to the spiritual.
Of course does math have more in common with mysticisim that with natural science - there is no way you can measure or weigh Math. We can sense it in the reflection of our sentinent minds only. That doesn't make it less real. I'd even go further: I say a modern Mathematician or even a natural scientist has *much* more in common with ancient religious prophets and leaders than somebody simply performing rituals with no understanding of their origins or intent.
Modern belief seperates *knowledge* of the spritual and *knowledge* of the material just as it would seperate the material and the spiritual itself - despite any kind of knowledge or consciousness allways being spiritual. And very real indeed. Not seeing that clearly imho is the single largest misconception of todays materialisim. Or as a famous philosopher once pointed out: Todays thinkers use all the might of their spirit for the strangest of tasks: To prove that it itself doesn't exist.
We suffer more in our imagination than in reality. - Seneca
The so-called difficulty regarding the non-physical state is only a difficulty when the narrow-minded (and haughty) assumption is made that the physical is all there is.
The word "science" originally meant "knowledge" or "to know". i.e. follow the truth wherever it leads, even if it suggests that there must be a non-physical state able to interact with the physical. There is no proof that knowledge is restricted only to the physical dimension.
It is often because men are afraid to give any leeway to the possibility of God existing that they outright deny the non-physical.
I think that this is one of those things that will never really be sorted out until we finally make contact with Aliens (if we ever do). The real question in my opinion is : is there something about our brains that makes our mathematics come out the way that it does ? Or would any intelligent creature come to the same conclusion ? I don't see how you can answer that without seeing a few independent realizations.
I don't think whether mathematics is discovered or invented plays a role in patents - genes are also discovered, not invented (usually). Still they can be patented, simply because it makes economic sense to let the people who invested lots of money into finding out about the build up and role of a particular gene allow to publish their findings and still reap the benefits of them.
The same could (similar to well/conservatively applied software patents) also be true for "pure" mathematical "ideas". If "discovering" or "inventing" them was done with a lot of effort (think of using super computers), and they have lots of useful commercial applications, why not make sure the people profiting from the research pay a little to the ones who spent so much time developing it (often at the expense of their social life)?
As to the actual subject of this threat:
I think some people here are confusing completely different things with the actual question. We are not talking about language, nor about subjective feelings towards mathematical research, we are (as far as I understand) talking about whether there is any "true mathematical entity/concept" underlying this world - whether mathematics exists in some way (discovery-side), or whether it's just a phantasy made up by us to describe what we observe or what we think about (invention-side).
On a purely practical level, it's obvious that mathematics only exist in our brains, books, and bytes, because it's only a way of modelling/describing things - even the 2+2=4 part. Still it is apparently so that what we find out with our "invented tool" of mathematics often holds true for anything to which it can be applied - a=b, b=c => a=c is basically true all over the universe (within the usual constraints). Obviously, the universe has an "innate" logic or mathematically coherent structure.
From that, one can safely assume that "logic" is something everything in this universe is subject of. This logic is everywhere, it's not the mathematicians who discover that, but the physicists (and other scientists). The mathematicians basically only develop the "tools" used by all the others, sometimes before, sometimes after the according real world discovery.
I also see no real difference between discovering or inventing, except in more trivial matters. Both use similar creative processes from a subjective view, so the difference gets less and less the more complex concepts mathematicians deal with. It appears to be only a simplifying distinction between differing degrees of creativity. For things for which there will never be a real life equivalent or which only got created after mathematicians came up with them, like some geometric shapes or encryption methods, I think invention is the more correct word, though. The proof of validity for any of those "inventions" would be closer to discovery, though.
Still, as the whole logic behind mathematics is innate to our universe, especially the descriptive parts of mathematics can be considered even less - an exercise of "mapping", as someone stated some time ago in this threat. But this map is at least in some parts something which doesn't exist in nature and which may never have come into existence without a sentient being thinking about it. Similar to a differential as it is used in a car. Even in mapping, though, words like discovery (that two objects are close to each other, for instance) or invention (that a certain method easily shows all objects with certain characteristics, for instance) can make sense.
Congratulations, you've just "discovered" how to hide exponents.
Observe and weep: 1
n/t
Well, okay, I'll rant a bit anyway.
There are absolutes. We can't approach them very closely as long as we are limited by being mortal. Sometimes that makes us think the whole universe is relative (to us).
But we can sure try to approach the absolutes, and we can sure learn a lot from our efforts. And if we are careful, the things we can learn can be useful.
A tree falling requires something for it to fall towards. The full version of the question hypothesizes a forest. Forests necessarily contain trees. Trees have means to detect and measure sound waves, thus there is no way for a tree to fall in a forest without a "listener" present.
I mean, yes. We have other geometries, some of which turn out to have direct application to parts of the known universe.
But a non-plane geometry rarely makes a good fit for problem areas where a plane geometry fits well.
How about this? One winesap apple, one crabapple, one Fuji apple, one Golden Delicious apple. Do I have four apples? Three?
Where does the abstract concept of unity come from?
I guess she's bored and gone to sleep? ;-)
For my purposes, I can verify the existence of God. I can also recognize that what I understand of "good" is tightly bound with what I understand of God.
I can also recognize that what I understand is tightly bound to things I am and to things I need.
I can trace this line of reasoning further, but I'm going to state categorically that the conclusion that I must have created God would require that I had created myself, all the way back to either Adam & Eve or to the primordial swamp.
Yes, I have created part of my current state of existence. No, neither my conceptions of good nor of God are free from the biases in my brain. But there is something there that is antecedent to both my choices and my biases.
n/t
At least, the integers beyond one.
The integers are an invention that closely (relative to us) models deeper truths about the physical universe.
Woops.
While this may mean nothing to your argument (which I don't understand), it mischaracterizes Einstein's work to say he "makes some new observations." Einstein's key work was based on thought experiments - on visualizations - not observations. These led him to postulate a symmetry, an invariance in the speed of light. He was in a long tradition in both math and physics of exploring an intuition of symmetry, and then finding real-world evidence to fit after, not before, working out a system based on that intuition.
Good popular books on this are Ian Stewart's Why Beauty Is Truth and A. Zee's classic Fearful Symmetry, giving the accounts from math and physics respectively.
The story largely goes: if you invent a symmetrical system - even just the abstract mathematics of one - sooner or later there's a high likelihood of it being discovered that some aspect of nature can be very well modeled by the symmetrical system you've invented. How far this works is open to dispute. Currently the string theorists are working hard to make up stuff that fits observation, while the loop quantum gravity theorists are betting on continuing to make progress on the model of Einstein-type application of intuitions regarding symmetries. If the string theory people are right, the run of luck with symmetries has reached its limit. But if the loop quantum gravity people are right, then the symmetry between human mathematical invention and natural discovery may continue to be fruitfully delved into for many centuries to come.
"with their freedom lost all virtue lose" - Milton
This is like asking whether or not Time exists or if we simply created it by comparing one repetitive event to another. So long as a machine can use the concept to make our lives easier, it's origins really don't matter to anyone other than scholars or historians.
Besides, concepts like math and time are driven by human greed more than anything. We always want to know who has more than the next guy as a means of quantifying our own existence.
8==8 Bones 8==8
but it doesn't alter the fact that setting C to one may hide the exponent, but doesn't make it disappear, doesn't make it anything other than 2.
Nor does the series that more closely models things that have relativistic momentum relative to use make the power any less integral.
It may be true that the beauty one person perceives in physical realities may be proof to that person about the existence of God (positive or negative, depending on the definition that said person has attached to the concept of God, among other things). That may, indeed be true.
It may also be true that said proof is applicable only to the person who perceives it so. But the limit of applicability is not a universal proof of God, either, whether negative or positive.
But to answer your question, making the unit of fuel for your guess tank useful when you set C=1 should point you in the right direction, if it isn't a red herring for you.
This is of course a problem of semantics. The terms discovered and invented are posed as distinctly different, when they are not. You pose the question as an âeither/orâ(TM) problem. Discovered: is universal Invented: is cultural. Things can be both. Therefore âfireâ(TM) (or mathematics) as a cultural application can be invented, and at the same time there natural universal constants are discovered.
The question keeps coming up, if math is "out there" already, where is "there"? I don't have an answer, but I do have an observation. When I am having a "math" problem (writing/debugging a program), I often experience going to bed not knowing the solution, dreaming about it, and waking up knowing the answer. So, wherever "there" is, it's where I go when I dream.
What would it allow you to do that you couldn't do before?
If it is invented, you can patent it. If it is discovered, you can not.Assume we determine the answer is "discovered". Then if we could extend this line of reasoning to inventions in general, we could reform our patent system doing everyone alot of good.
So, the general public would have much to gain as patent abuse affects every level of society.
The truthfulness of the statement "1 + 1 = 2" is not absolute. For example, take this math test:
1. 1 + 1 = ?
A. 2
B. 10
C. both of the above
D. none of the above
2. 1 + 1 = ?
A. 1
B. 9
C. both of the above
D. none of the above
For those uncertain, the answers are 1.C. and 2.A. So, even in math, there are multiple right answers. Of course, there are always mulitple wrong answers.
For once, what we know as addition (and numbers, and equality) was defined that way. If everybody else defined addition on a different way, 2 + 2 could be different from 4 for them. But our definition of addition would still make sense and 2 + 2 = 4 would still hold for it.
Now, if everybody used a definition where 2 + 2 != 4, and you used our current one, you'd have a severe communication problem, and math would be almost useless for you.
Math is only usefull if everybody agrees, but it doesn't makes 2 + 2 = 4 less a convention.
Rethinking email
"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"
?!
Human minds are indeed able do that. The "state" is called reality (which is more than the static observable physical realm), and the "connection" is called thinking.
No big mystery, sorry.
Vasco Figueira
But is mathematics the study of generalized rule sets using logic?
In the socratic dialogue Meno, Socrates publicly gives an uneducated boy a math puzzle involving 2-dimensional area. This demonstration serves to show that the boy
That part of the dialogue is summed up with the following exchange:
I've experienced (roughly) two different kinds of teachers in my life, those who acted like they were pouring knowledge into my mind and those who acted like they were leading me on a journey of discovery. Invariably the latter kind proved more interesting and helpful to me.
Can you remember back to when you learned the "multiplication tables"? I was initially taught them by rote from 1x1 to 12x12. Then we'd be given speed tests with random two-number problems in that domain. But the rote method didn't work well for me. I can still remember the glorious moment when it dawned on me that multiplication was just a shorthand version of addition and a whole world of beautiful, numeric patterns opened up for me.
(Somehow I managed not to fall off of John Nash's deep end, but I digress)
Imagination is more important than knowledge -Einstien
I meant, "knowledge in himself of how to answer correctly"
Imagination is more important than knowledge -Einstien
The answer to this question is important because if we agree math is inventable it also becomes patentable.
There seems to be a confusion in many posts between pure and applied mathematics. Applied mathematics is used to describe the physical world, and thus successful mathematical descriptions are discoveries. But mathematics is also used to describe mathematics. This is pure maths.
In pure maths, you can invent mathematical objects, and define their properties and relationships in a logical sense. These definitions are axioms. Then you can discover facts about this invented world. These facts are theorems. This has already been said more briefly in previous posts.
Disclaimer: I am an applied mathematician, but I appreciate the value of pure mathematical theorems related to the maths I use.
We invent assumptions, aka axioms, and discover interesting consequences of those assumptions. Those assumptions are often born of our own interaction with the world, such as the idea that the shortest distance between two points is a line, or that two parallel lines never intersect. Whole new mathematical realms are discovered when we relax these rules, but they're not necessarily out there in nature. And there's certainly no rule that nature must follow these rules, everywhere or anywhere.
I Browse at +4 Flamebait
Open Source Sysadmin
Nothing, nothing and nothing.
Sounds like Logical Negativism to me.Oh, and the correct answer is "discovered".
Think a little deeper. What follows if you are correct; what if the correct answer is "discovered"?If it is discovered:
- Where is it?
- How do we discover it?
- Is there a better way to discover it?
- etc...
These are questions tackled by truly deep thinkers like Roger Penrose. He actually follows through with his beliefs. He's championing your "meaningless" answer.
Penrose's conclusion is wrong (IMHO -- way wrong). But at least he understands why it is important.
Oh, and the correct answer is "invented" not "discovered". Discovery implies Absolute Truth -- which is a philosophical quagmire.
Mathematics is a man made tool. And, like all tools, it is shaped (not discovered) by nature.
Quite right; though so many people confuse the models with reality itself. We build models that map onto the real world, and we consider them more or less "true" as our predictions match up with observations.
...and they're trivial to dismantle. You simply find the flaw in the model - you don't assume that reality much be wrong (e.g., concluding "all movement is illusory").
We structure math with rules to make it *internally* consistent, but the mathematical models we build of the world do not always work.
Another example of models that don't match correctly: "paradoxes", such as the well-known Zeno's.
Try analyzing them with the understanding that:
* our words and numbers are NOT reality; they are a model of it
* the model is often limited or flawed
There's a lot of confusion about what "discovered" or "invented" means. The real question is not about the details of discovery or inventiveness, it's about truth -- absolute truth.
Is mathematics Truth?
Is mathematics "written in the fabric of the universe" or should we wonder about its "unreasonable effectiveness"?
Is mathematics a "sure path to understanding nature" or is it inherently subject to human error.
Discovery implies that the Truth is "out there" and math is how we find it.
Invention implies that the Truth is not (yet?) knowable and math is just a tool we use to get a glimpse at it.
"Empirically verifiable" is not identical with "tells us something about the future." A statement could be empirically verifiable without actually ever being empirically verified, thus telling us nothing about the future. For example, I might declare that inside my grandmother's body is a digital watch. This is a meaningful statement because it is theoretically empirically verifiable. I'm not going to cut my grandmother open to find out, but it is in principle discoverable through experience, the realm in which logical positivists believe proper verification can occur. (As opposed to mystical insights, visions, whatever.) Another example is one in which we evaluate a theory on the basis of past observations recorded in a database; here, again, the meaningfulness of statements do not involve futurity. We could make no further observations, and still hold the theory to be meaningful according to the logical positivists.
Inductive laws are supposed to tell us something about the future. We posit laws based on limited experience with the hope that those laws will tell us something about the future. Clearly statements that make no claim about the future are not inductive laws. But that is not precisely what is at issue with the logical positivists' verification theory of meaning. To say we cannot develop inductive laws from this debate is to say nothing about how a logical positivist would evaluate it as meaningful or meaningless. They do not distinguish between meaningful and meaningless debates on the basis of induction, but the possibility of verification. This possibility does not have to be actualized because the status of meaningful comes prior to a statement's being accepted or discarded as true.
Having not read through all of the posts yet, I do not know if this idea has been mentioned.
I would like to pose a question related to physics first. Are the laws of physics discovered or invented? Gravity, as everyone knows of course, exists independent of whether or not we know about it. The laws of the universe exist and we discover them.
As for math, say you have two apples, one in one hand the other in the other hand. One apple with another apple makes two apples. Did someone invent this idea, or discover it? This seems like something that would exist regardless of what we think, which probably is how things should be looked at in the first place - how things really are.
Do these mathematical principles exist regardless of whether or not we do? Would these ideas still exist in our world if we did not know about them? I would have to say yes.
On another side, taken from a religious point of view, God is the source of all knowledge and truth. Therefore, we do not "invent" anything. Quite simply, God reveals it to us when He deems it necessary for us to have the knowledge.
Regardless of everything said, I think it should be looked at from a point of view of how things are. It should not be based off opinion, but off of truth and fact. We can debate as much as we want about truth and fact, but that will never change it. (i.e., it doesn't matter how much say, debate or whatever that gravity doesn't exist, it's still there. It's a fact.)
Do hole-diggers really create holes? Or do they merely displace the dirt that obscures them?
Note to mods: The best post in this thread is the one by alexhs.
Note to anyone who cringed in anticipation of a strong underwhelming feeling when they read Slashdot's summary of this story, here is a much more interesting question: What does it mean if I build a computer that successfully files 1000 contiguous US patents?
My position (as an undergraduate math major) has always been simple:
The laws of the universe are governed by mathematics. In physics, F=MA regardless of what I wished F equaled. In pure math, d(5)/dx = 0 regardless of what I wished the slope of 5 were with regard to x. Math exists, indeed, it *is* the study of pure logic with an extension into the world of computation. Math can no more be invented than you can choose yourself to be born. It is, and why it is how it is is beyond us.
At the heart of this discussion, I believe, lies a misconception about what math is. There is a difference between math and our representation of math. We make up all of our math symbols, but math is not symbols. We make up our number system, but math is not numbers. We make up all of our vocabulary, but math is not vocabulary. We discover inefficient ways of doing things before we discover more efficient ways of doing things, but math is itself not efficiency.
And -- this is the one that trips most people up, especially amongst the replies I'm seeing in this thread -- we make up representations and models for the universe and concepts in the universe, but mathematics is not a representation or a model of the universe. It is what allows us to create/make-up a representation or model of the universe. Most people, I believe, err in recognizing the distinction here. They argue that models are not absolute and mere representations. They are correct, but see a limited picture.
Math is an existence, not a process or a tool. Math is logic. The absence of an absolute mathematics is the absence of logic.
Yes, this does require the fact that there exists something beyond our physical world. But to any mathematician, this is not a hard concept to grasp. Many do not think of it as an inconvenience, but as a requirement.
Without making personal attacks, I would like to point out that the majority of people who claim mathematics is "invented" are themselves not mathematicians -- and I do not count amature hobbyists as mathematicians. It strikes me as the naive and/or ego-centric viewpoint, these people either cannot see that there exists something greater than themselves, or they cannot bring themselves to acknowledge the fact that they study something greater and more fundamental to the universe than they can even understand, let alone that they are themselves.
Now, on the speculative side, here's some flame-bait: I believe that an understanding of math is, in part, dictated by how one is born. It is so abstract that hopes of communicating it to someone without that understanding or changing another's view of math is close to impossible. One is either born understanding it or one is not. If one is not, age and time may help them understand the inherent existence of math better, but they cannot be persuaded by anyone else.
I agree, it's a mistake to accord any special quality to our brains as ultimate observers. An electron detector is an observer too. It can capture and record an event, and in doing so it collapses the wave function of the electron, creating an absolute where only a potential had existed a blind moment before. As Schrodinger points out, the event is still only a wave function until your brain takes in the measurement.
... it is not merely an abstract link!
Did the wave function collapse for the detector at the moment that the electron hit it, or did the whole shebang collapse when you looked at the screen? Actually, the question is meaningless because the result is the same. Regardless of when it actually happened, for the detector too, the time of the event will be retrospectively consistent. That is, the detector, if it could talk, would say it felt the electron hit it at 1:32 even though it didn't tell you until 1:35.
From that it should be clear that an observer is just another name for a physical system, so an observation is the same thing as an interaction. Since every observation involves a material interaction of some kind, it is safe to assume that what's true for detectors is true for ourselves. Every time a photon strikes your retina a physical event has taken place, something has been determined, an effect has followed a cause.
It's important to screen out the false abstraction about human observers being disconnected from the chain of events. If I tell you by email that my detector registered 6.2 you don't have to come touch the detector to make it so. If you theorize that it was still a wave until I told you, that says there's a physical link developed through my act of telling you
For as it happens, if you trace the photon from my email back to its physical source, then follow the electrons in your screen's transistors back to their source, and so on, you will find that there's been a direct physical chain set up between my finger hitting the "A" key just now and the expelled CO2 in your amazed exclamation about this fact.
That puts a new spin on things!
Philosophically speaking, yes, it is impossible to determine what - if anything - exists in between measured physical interactions. Time could stop while a huge team of quantum elves set up all the dominos for the next moment of the universe, and we would never know the difference.
So does "reality exist, then we observe it" ?
I would say, no. Existence, interaction, and observation are all the same thing.
1. Interaction of energy/matter gives rise to existence in the present.
2. Observation by an instrument is the state of the instrument, and observation by a mind is merely the conditioned state of that mind.
3. Knowledge is in part a pure conditioned state, but in effect it only arises when engaged in a process of comparison, and that comparison process only gives a significant effect in the world if it in turn moves a body.
I want to reiterate one of the most important but subtle points that came up in this rumination, which is that reality only exists right now. And as for that 'now' reality at any given time... how long is it? If the present is itself a phantom, in a sense we cannot really observe it either. We can only see it in any meaningful way, in fact, when we have a previous measurement to take it against. So all observing devices depend on some form of stateful comparison.
What does this say for "acting" devices, like bodies attached to brains? Can the brain can get the same effect - a useful action to follow a coherent stimulus - without recourse to any form of stateful comparison? I don't think so. A neuron whose resistance increases with current isn't a stateful comparison per se, but could give rise to comparable effects.
Regardless of the way states are modeled, I would bet that all special observers in nature (nervous systems) evolved from the earliest stages to perform temporal-spatial comparisons. This would be strong evidence that change is a very real phenomenon in the world even when no special observers exist, and that time can make an adaptational mark without itself having an independent existence apart from change.
-- thinkyhead software and media
You make the assumption that the universe is not a simulation (i.e. programmed according to a specific set of rules), or that it does not follow a specific set of rules without having a programmer as such*. If that is true, then all those rules are essentially sitting there waiting to be discovered by whatever emergent intelligent life inhabits such a system.
In that situation, saying that we "invent" mathematics makes as much sense as saying an ant "invents" a spill of soda somewhere on the ground. Even a mathematical technique is in a way a discovery, because it allows our brains to work a certain way with the wiring that is common to enough people. I.e. it is an emergent property of the system and our brains, which are in turn an emergent phenomenon of the universe. The brains were wired the same way all along, it just took someone to explore enough search space to find the new efficiency.
Of course, this is one of those things that cannot really be "known" as such by our sensory apparatus. It's a lot like God. I don't believe the existence of God can be proven conclusively one way or another. However... if the universe was not a system that worked according to a specific set of rules, then there'd be a heck of a lot of engineers and scientists out of work. Put my neck under a guillotine pending on the outcome? No way. Put my money on the outcome? You bet.
The reason why science progresses the way it does is because simpler models provide some benefit. Complex models provide increasing benefit, but the returns are diminishing. For example, I might play a FPS. I make a rough mental model - learn what weapons work best against what targets, learn the map by playing it, etc. It stands to reason that anyone who studies the source code can make a better model... by investing a lot of work. They can see exact damage counts for each weapon, know exact times and locations for respawning, etc. It is a better model and given the same skill level will make them a superior player most probably. But... by no means did they invent anything, they merely discovered what was already in the source code.
*Note that the existence of quantum mechanical "God playing dice" type effects does not mean that we aren't in such a system. I don't see why God can't make calls to a random number generator. Or why his random number generator can't be perfect, since he operates outside this universe. Or maybe everything does operate according to mechanistic rules, and it is impossible to directly observe such things from within the system.
If I have seen further it is by stealing the Intellectual Property of giants.
http://en.wikipedia.org/wiki/False_dilemma. Learn to spot one.
This post expresses my opinion, not that of my employer. And yes, IAAL.
I wondered this a long time ago, but in the more general sense. If I find an elegant solution to a programming problem, I feel satisfied. But did I discover it or invent it? It's not like an evolutionary algorithm might not have come up with it, given enough time.
How is evolution a learning process? Is evolution merely cells "learning" about the world? Our existence is in essence a continual discovery... Or were we there all along, just waiting to be discovered and made flesh?
is under the tent. We had better hope that the eventual conclusion is that "discovery" wins. If mathematics is determined to be "invented" then there will a mad rush to the already screwed up patent offices around the world.
....
Beware
> Is Mathematics Discovered Or Invented?
Neither. It is taught. Like language.
Now, if I discovered your wallet in your back pocket logically fallaciously I would have discovered North America in 1492.
18th century mathematician Leopold Kronecker gets credit for saying it.
Bill Stewart
New Fast-Compression-only CPR http://preview.tinyurl.com/dy575ks
I must confess to not remembering much of the class I took from him 30 years ago (:-), since it's not material I ended up using much over the years, but it did contribute a lot to my mathematical maturity and appreciation of elegance, and had I done more research rather than engineering the matroid theory would have been an especially useful place to work from.
Bill Stewart
New Fast-Compression-only CPR http://preview.tinyurl.com/dy575ks
As interesting as I find the discussion, and by the way I believe Mathematics to be "discovered", the real question is, "why is it important?" It comes down to two truths. If Mathematics is invented, then the basis that it can be patented is rock-solid. Yikes! If Mathematics is discovered, then the basis for Intelligent Design is rock-solid. Yikes! Either way, the current chaos continues unabated. I'd like these 3 questions answered together: Invented or Discovered? Patentable or No? Intelligent Design or No? I'll bet the subsequent statistics would be fascinating and enlightening.
The invented or discovered dilemma is easily solved by this argument: any basic FACT (not ARTIFACT) that other (possibly imaginary) intelligent alien life discovers (or invents), and it happens to be the same as our discovery (or invention), then it was really *discovered*. So, this applies to Math and to Physics. If they happen to discover the fire, then we both *discovered* the fire. If they invent the wheel, then nothing can be said about it through this argument (because the wheel is an artifact).
But I have a better question. Is Mathematics Transuniversal? What I mean is:
Do mathematics (or at least a subset of it) belong only to *our* universe (or some other look-alike universes)? Or can it be said to belong to ALL POSSIBLE UNIVERSES?
I disagree.
The saddest poem
Offtopic but somewhat related: Cheeseburger Brown (author of Simon of Space and The Darth Side: Memoirs of a Monster) is writing a story right now titled The Secret Mathematic. It's about the discovery of a new kind of math that can be used to effect (no, it's not a typo) the universe. It's up to chapter 20 so far. http://cheeseburgerbrown.blogspot.com/
Burn the land and boil the sea, you can't take the sky from me
When a mathematician writes down an subtly, but entirely, erroneous piece of mathematical reasoning, this question gets much easier. Clearly in that case it's "invented", because it can't be preexisting or "real" in any way that makes any kind of sense.
The question is: why would this process be fundamentally different from writing down a valid piece of mathematical reasoning? One could argue that that validity makes it a "discovery", but the process is exactly the same, and in fact it's often the case that something that appeared valid before appears invalid later or vice versa.
Does the math switch from "invented" to "discovered" and back to "invented" in that case? Does every bit of discovered/invented math exist in some kind of superposition of those two states?
Does the concept of "valid" even exist outside of human perception and understanding? Can we know whether there's a real difference between these? Metaphysical, yes, but I think the only valid answer is that what we *perceive* as reality is created by perception as much as it is by anything objective.
I vote for invented.
Besides, that will piss off the people arguing against software patents, which this whole discussion is a very thinly veiled reference to anyway.
Dang. I wish I could.
/.
But either she is gone or she has gone on.
(Probably the latter, and sometimes watching me from over there and trying still to tell me to do something more useful with my time than post to
This assumes that they start with the same axioms that we do. Some of the axioms are provided by common sense from the universe, one apple plus one apple equals two apples. Your definition of math doesn't have to define 1+1 as 2, and if it doesn't you will end up discovering different things then me.
We invent the basic principles (axioms) of math, hopefully so they share the same properties of the universe. We discover the relationships from then on.
Really there is nothing saying that the universe follows math. We think that the universe follows one giant equation, but that is only because the properties we see have the same properties that our math does.
If I neglect air resistance, -9.8m/s/s*t*t describes the motion of a falling object quite well. I might even think that the ball knows mathematics! But then I introduce air resistance... and then I have to add more things to describe its motion. What if I move it to another planet, or even far enough away from the earth... all of a sudden -9.8 doesn't apply anymore. So I have to add more properties.
I think that the fact math describes the universe well is a coincidence, we invented so we could describe natural properties of the universe, those properties we discover. The description of it (mathematics) we invented.
As for the universe thing, oh, wouldn't it be so cool if there was a universe where pi is 1? -1? i?
I don't know what this proves or not, but if, literally, there was a universe for every possibility, then at least one universe should be able to teleport into our universe. Thus, universes must be limited in some way less then inter-universe teleportation.
This is all assuming that the parallel universe theory is true. Keep in mind it was theorized as a way to explain the randomness of the quantum world, and as far as I know, the state of a given particle will not decide the value of pi, so I think that the constants stay the same, only the outcomes of chance are different.
Wonder what the public key field is for?
"well, so you say that 1/sqrt(1-vÂ/cÂ) is not pretty ?"
Use the fucking "Preview" button, and you will (may) look like less of an idiot.
On the other hand once you fix your axioms (and the flavor of logic used in some extreme cases) all derivative propositions are either provable or not (even if we cannot know which, even in principle in nontrivial systems). So we have to discover theorems one by one.
pfft, tell that to tenacious d
i've had just about enough of your vassar bashing.