Is Mathematics Discovered Or Invented? 798
An anonymous reader points out an article up at Science News on a question that, remarkably, is still being debated after a few thousand years: is mathematics discovered, or is it invented? Those who answer "discovered" are the intellectual descendants of Plato; their number includes Roger Penrose. The article notes that one difficulty with the Platonic view: if mathematical ideas exist in some way independent of humans or minds, then human minds engaged in doing mathematics must somehow be able to connect with this non-physical state. The European Mathematical Society recently devoted space to the debate. One of the papers, Let Platonism die, can be found on page 24 of this PDF. The author believes that Platonism "has more in common with mystical religions than with modern science."
Logical positivism to the rescue... (Score:5, Insightful)
When faced with an awkward question, logical positivism [wikipedia.org] asks: what would the answer tell me about the future?
Suppose you had a definitive, 100% guaranteed answer to the "discovered vs invented" question. What would it allow you to do that you couldn't do before? What could you predict? What would you gain?
Nothing, nothing and nothing.
It's meaningless; merely a matter of perception, wordplay and people having too much time on their hands.
Oh, and the correct answer is "discovered".
Mathematics in the forms of human intuition (Score:5, Insightful)
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.
Patently Obvious.... (Score:4, Insightful)
Re:Logical positivism to the rescue... (Score:5, Insightful)
No, the correct answer is "both."
The relationships and observations that we use mathematics to model are discovered. They are out there, we discover them, and then we model them. That should be obvious to all but the most die-hard of idealists.
The language that we use to do this modeling is invented. It is also refined (i.e. slightly reinvented) over time to better fit our discoveries. That, too, should be obvious to all but the most die-hard of determinists.
I know, this answer isn't very deep, but in my opinion the question isn't nearly as deep as it is being made out to be.
How is this a debate? It's both. (Score:4, Insightful)
What can be done with it is then discovered.
Re:Logical positivism to the rescue... (Score:2, Insightful)
Re:Logical positivism to the rescue... (Score:1, Insightful)
What would it allow you to do that you couldn't do before? What could you predict? What would you gain?
Axioms vs. theorems (Score:5, Insightful)
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.
Re:Logical positivism to the rescue... (Score:5, Insightful)
Suppose you had a definitive, 100% guaranteed answer to the "discovered vs invented" question. What would it allow you to do that you couldn't do before? What could you predict? What would you gain?
I tend to agree. I'm reminded of the Dutch computer scientist, Dijkstra, who said that ""The question of whether a computer can think is no more interesting than the question of whether a submarine can swim." Some questions are just meaningless.
I think the thing to learn here is that language isn't reality, it merely describes reality.
Oh, and the correct answer is "discovered"
No, I think the correct answer is "Why are you asking the question?" There might be a more interesting (and perhaps answerable) question that underlies it.
Re:How is this a debate? It's both. (Score:4, Insightful)
The concept was discovered, then we invented new methods of math based upon the discovery of Math
Re:Logical positivism to the rescue... (Score:2, Insightful)
So the answers to your questions aren't nothing x 3, but rather in lines with patenting and making money.
Re:Logical positivism to the rescue... (Score:5, Insightful)
Re:Logical positivism to the rescue... (Score:5, Insightful)
Depends what you mean by "exists". For example, mathematical concepts are not observable (which is the condition for existence in an empirical framework), but physical systems can be observed which implement the concept. One can observe one apple or one galaxy, but one cannot observe the number one.
ideas possess a location? (Score:2, Insightful)
I thought the article was weak. It asked:
Where is the edge of the world? Where is the center of the universe?
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?
Re:Logical positivism to the rescue... (Score:5, Insightful)
No, the correct answer is "both."
No, I think the correct answer is, "What are you asking?"
The problem with questions like this is that it isn't clear what's in the mind of the person asking the question. What do you mean by "invented" and what do you mean by "discovered"? What difference do you see between the two?
For example, some people will think that "invented" means "made up". So in that person's mind, if math is "invented", then it's based only on human thought, and not on real principles of the universe itself. Of course, this line of thought makes me want to ask what it would mean to be a "real principle", and what is the "universe itself" when detached from human conception, but I'll leave that aside.
The problem I see immediately with this concept of "invented" is that real inventions don't exist independently of the universe. For example, was the wheel "invented", or did someone discover that rolling a circularly shaped object requires less energy than dragging an equally massive object? Was gunpowder "invented", or did someone discover than mixing certain chemicals together and setting fire to them caused an explosion? Was the telephone "invented", or did someone discover that you could convert sounds into electrical signals and back again by using magnets?
All inventions are a discovery of sorts, which makes this whole question a bit nonsensical.
Re:Logical positivism to the rescue... (Score:5, Insightful)
Math is the symbolism used to describe the universe. Physical reality does not need symbols or tools or sentience to function, we however need math to describe the functions of the universe in precise detail. Math is a tool and so is an invented thing where the ideas have come from observing the world around us, just like a knife or velcro are tools that where invented based off of ideas gleaned from observations of the world around us.
Re:Logical positivism to the rescue... (Score:5, Insightful)
Other parts of math do resemble invention more than discovery. E.g., the definition of mole being the number of atoms of carbon 12 needed to make exactly 12g and the Coulomb, both of which are numbers that are arbitrarily assigned to fit in with the system of measurements that has been devised over the years. All of these constants could easily be multiplied by any non-integer value and the whole system would still work.
To answer the article's original question however, my answer would be: Who gives a toss? Math is useful. Whatever semantic definition we apply to the process by which we expand our mathematical capabilities has absolutely zero impact upon that expansion.
Re:Logical positivism to the rescue... (Score:5, Insightful)
truth is discovered
truthiness is invented
Score +1 (Score:5, Insightful)
Re:It's neither (Score:5, Insightful)
So you invent/assume your choice of axioms, and everything else follows from them and can be discovered at leisure.
My Take (Score:2, Insightful)
Theorems: Discovered.
Proofs: Invented.
Glib answer... (Score:4, Insightful)
To be more specific, Mathematical rules are discovered, Mathematical techniques are invented; "Mathematics" consists of both.
Re:Logical positivism to the rescue... (Score:4, Insightful)
No, I think the correct answer is, "What are you asking?"
.
.
For example, was the wheel "invented", or did someone discover that rolling a circularly shaped object requires less energy than dragging an equally massive object? Was gunpowder "invented", or did someone discover than mixing certain chemicals together and setting fire to them caused an explosion? Was the telephone "invented", or did someone discover that you could convert sounds into electrical signals and back again by using magnets?
Um, you have just given threee great examples supporting the original poster's answer of *both*...
In each case the basic scientific principle (mechanics, chemistry, elctricity & magnetism) was discovered (sometimes unwittingly) and then the knowledge of that discovery used to engineer an invention (wheel, gunpowder, telephone). The "discovery" was an observation of a natural phenomenom, etc, and the "invention" was creating something that otherwise did not exist in nature that took advantage of those phenomena. If you wanted to be pedantic you might argue the first "wheel" could have been discovered ("hey, look at how that round rock rolls!") but please don't try to claim that set of 18" forged alloy wheels with vulcanized radials was "discovered".
This is exactly the same argument the OP was making. Mathematics clearly involves the invention of a language to express discoveries (or assist in making those discoveries).
Re:No, mc^2 is exact for an object at rest (Score:5, Insightful)
This may seem like a nitpicking question, but it brings us to the point that I really want to make:
Mathematics is interesting because there are no ambiguities in a well described mathematical problem. There are many problems that have a finite set of solutions. However, every mathematical model we develop to describe our surroundings is only an approximation of our observations. With time, we can create more and more accurate models, but there will always be something about that model that is derived experimentally, and is therefor imperfect.
This does, in fact, tell us something about the underlying nature of the universe. Either it was created with some arbitrary parameters, or it exists in a way such that there is no way to perfectly describe it. Or maybe there are other possibilities I have not considered. What philosophical meaning you derive from all this is up to your own reasoning.
Re:Logical positivism to the rescue... (Score:2, Insightful)
Re:Logical positivism to the rescue... (Score:5, Insightful)
Momentum scales linearly with both mass and velocity, fields fall off with inverse square relations, and so on. You cannot change the equations describing them away from these truths in any meaningful fashion without making the equations wrong - this is not human convention or definition, it is how the universe works.
Re:Logical positivism to the rescue... (Score:3, Insightful)
Sure E=MC^2 is very nice looking, but if it wasn't, it would not be famous, take a look at some of the other equations involved in relativity, they are not so pretty.
Re:Logical positivism to the rescue... (Score:3, Insightful)
I'll agree that it's meaningless in the sense you describe, but I'll also add that it's a relatively easily answered question as long as one keeps superstition at bay.
Mathematics is a language, one intentionally of the most precision we can manage. This language is very well able to, and intended to, describe many of the methods and mechanisms of the universe we live in, and is additionally capable of describing things that are abstract and/or impossible.
As a language, it evolves with use, and it maintains consistency with use. It also can lose ideas and dialects.
To get all happy-assed because one has a technically specialized language available is akin to a programmer thinking he is discovering a hitherto unknown facet of the universe because he just learned Python. Fun, interesting, mind-expanding, all that... but not a "connection to a non-physical state." Just a new language.
The universe works in certain ways. Most languages exist to describe those ways, and how we interact with them. Math is one of a very few languages that attempt to do that precisely, and this is both notable and useful, but it isn't magical. Because the goal is to discover how the universe works, and how abstracts perhaps not in the universe might be expressed, we are driven to extend the language. We don't "discover" it, we invent it.
To the extent that we meet our goals -- that is in particular, successfully describe how the universe works -- we can expect that someone or something working elsewhere would come to the same or equivalent conclusions. Because knowing how things work seems to be is so fundamental to our industry and technology, we presume that it would be similarly fundamental to the industry and technology of those "others." If that presumption is correct, then thinking of math as a "universal language" is an idea that has legs; but again, there's nothing magical about it. We can be pretty certain that while another race might know what the idea of "pi" represents, they're not going to call it "pi", it's not that kind of universal.
Math can be expected to be a common ground just as other types of communications and specifications based upon communications are likely to be common ground. Also like metallurgy; chemistry; physics; etc. Not because it's magic, but because the universe offers only certain things to its inhabitants, and as we work with them and extend our knowledge about them, we're going to need very specific ways to describe and represent and work with them. So would anyone else, if they're even remotely similar to us.
So it's invented. But it is invented to describe something that already exists, in many cases, as well as imaginary things conceived by minds conditioned by experience with other things as they exist. That's why some people think math can be described as "discovered"; but they're simply confusing the universe being described with the description. We might discover how orbits work and very precisely describe that orbit with math; but the math for the orbit is not the orbit itself. It's a description; it's language.
That's my take, anyway.
Re:Logical positivism to the rescue... (Score:1, Insightful)
I'm going to have to go with invented. Euler's equation is true within a particular mathematical system. There is no unique system of mathematics. Do some reading on the old controversy surrounding the Axiom of Choice, and come back and talk about how math is discovered. Math is a game. We arbitrarily decide which rules we play by.
Wrong (Score:3, Insightful)
HTH.
Re:Logical positivism to the rescue... (Score:3, Insightful)
The problem is that you both are arguing about 'ideas' (physical laws) and 'reality' (observed nature). Both concepts share a single source, that's human perception, (cultural stimulus processing) but in our days those term have a very different scope. That's where the problem arises, and dicothomy between 'laws' and 'reality' becomes something 'real'. A little philosofy could help to understand that we build 'world images', 'mental representations', 'imago mundis' to talk about things, these images, no matter how exact we think they are, are not the thing being observed or perceived, just our representations.
Take the concept 'time'. Current interpretation is a somewhat lineal property of experience, not space bounded. Does we 'know' what time is? Clearly not. Does we know time exists? We suppouse the anwser to be yes, but that's just a cultural construct, inherited from our ancestors, our culture. Now we build upon such an undefined concept, the net result is that our knowldege, our science is more about how we perceive the world to be, than how we know the world to be. Pretending both to be the same, equating perception/stimulus and reality/representation, that's the problem.
Re:Logical positivism to the rescue... (Score:3, Insightful)
As an aside:
I'd have to wonder if there is a difference between our "observation" and our "description" of the universe. How do we differentiate between describing a thing as having eerie symmetrical properties and our need to observe said properties? Are we even able to observe that which our minds can't fathom?
To cut a long story short, the universe pretty much exists in our minds' eye and any statement we make about its nature will invariably be subjective. Even if we choose to use the language of mathematics linked to empirical data.
Perhaps these geniuses that don't come along too often marvel at the beauty of the human mind and its constructs rather than the universe, and the tricky bit is that neither they nor anyone else can prove any of this either which way.
This whole discussion is one of semantics, and the original poster is right in that the answers that might or might not be forthcoming are not likely to change our existence overnight.
Re:Logical positivism to the rescue... (Score:2, Insightful)
All statements about reality are statements about human observation; all physical laws are just patterns seen in observations. What is "real" outside of human observation is not answerable.
The idea that reality continues when no one is looking is a convenient simplifying assumption, but is ultimately, by definition, not determinable - you can't do the experiment to check.
It's a matter of patents ... (Score:2, Insightful)
What would it allow you to do that you couldn't do before?
Re:Logical positivism to the rescue... (Score:3, Insightful)
I was looking for an appropriate place to say exactly this - and it is the reason why a debate as to whether mathematics is invented or discovered is so important, and should not be ignored as merely frivolous. If we allow enough groups to declare that mathematics is invented, we will soon see patents allowable on mathematics, and any future resistance to expansion of the patent system into both mathematics and other pure sciences will rapidly fall. Mathematics must not be allowed to be seen as being "invented" if we want to still be able to build mathematics knowledge on the foundation of previous efforts unhindered by patents and "intellectual property" claims.