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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."
Is Mathematics Discovered Or Invented?
Neither. It is defined.
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.
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
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.
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!
The problem with the question is that the answer is predicated on interchangable assumptions. To discover something it has to apriori exist. Inventing something requires that it not. So the fundamental question is:
Does mathematics which no one knows about exist?
Well it is obvious that on some level it should. It is likely that whatever new field of mathematics we invent, it will (eventually) be described using axiomatic set theory. But does the fact that we already have the language we need to describe a theorem mean that the theorem already exists? Does a sonnet exist before I write it? All the words I'm likely to use will be in some version of the Oxford English Dictionary. I can symbolically write down the abstract idea of every sonnet imaginable in only a few lines of mathematics. It would seem clear that mathematics, like poetry and prose, is invented then.
But then mathematics is different from prose, because mathematics can be used to make quantitative predictions about the world around us. It would seem that independent of human being nature itself 'knows' about mathematics. Before we invented calculus the acceleration on an electric charge due to a electromagnetic field could still be found using Maxwell's equations. The Falkland Islands were still there before the Spanish arrived right?
So now it would seem that at least some mathematics is discovered, at least as to how it relates to nature. Of course the mathematics we use to describe nature is just an approximation. Maybe nature doesn't know about math, maybe we just got luck.
Then there is another problem, whose to say that just because we think of prose as invented it really is. That might just be our sloppy use of language. I said earlier I could, at least in the abstract, write down every possible sonnet in the English language. That at least implies that those sonnets exist in some way before I write any of them, even if it is as an abstract sonnet.
Bottom line, it all comes down to what you think exists. If under your philosophy mathematical theorems can be shown to exist independent of if someone knows about them or not, then they are discovered. It is likely sonnet are discovered under that philosophy as well. If on the other hand theorems only exist after someone has conceived of them then they are invented. Now you have to be careful that at least some part of the Falklands weren't invented by the Spanish as well.
I'm going to have to go with discovered. To me Euler's equation is, was and ever shall be true and there isn't a darn thing anyone in this universe can do about it.
Of course the discussion doesn't really yield any useful results, so I would like to propose the Dirac interpretation for the uncovering of mathematical knowledge:
Shut up and calculate.
It comes with a corollary of my own devising:
No you cant patent it.
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.
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.
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.
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
those who see it as being invented are nihilists who cannot see that there is great order to the universe.
I may be the nihilist, but you're the egotist - the one who believes that the order he sees in the universe is really there, not simply the result of his choice to define "order" in such a way that some parts of the universe seem to fit.
To suggest that we invent math is pompous at best.
To suggest that we discover it - that our brains, somehow, are able to tune in to an entire dimension of mystical mathematical truths - is arrogant.
And I have to ask you the question that completely dispels mathematical platonism - where do the wrong ideas come from? If they come from a special universe for wrong ideas, then discerning the difference is the same thing as inventing them. If they come from human imagination, if humans can invent wrong ideas, then surely they can invent right ideas too, and again, it's all invention.
I never have frustrations, the reason is, to wit:
If at first I don't succeed, I quit!
Semantics:
In old French, both were essentially synonymous.
inventer v.tr. To invent. (a) To find out, discover. [...]
Philosophy :
Under platonism, there's actually no distinction (see allegory of the cave).
By suggesting to let platonism die, the anonymous reader seems to want us answer "invent"...
I have discovered a truly marvelous proof of killer sig, which this margin is too narrow to contain.
I deviated from the profession of mathematics long ago, but as far as I'm concerned, the question of invented/discovered was adequately retired by Kolmogorov-Chaitin complexity theory. For some reason, most mathematicians and most physicists seem determined to ignore this.
The formal system you begin with has an arbitrary beginning: the nature of the universal computer used to measure sequence length. In practice, the arbitrary starting point rarely makes a whiff of difference. The maximum disagreement on sequence length is bounded by the complexity of the program by which one machine is able to simulate the other. Since it is possible to construct universal computers of startlingly low complexity (you could easily write out the rules on the back of a business card with a blunt pencil), this bound tends to be minuscule for most universal machines we might choose to adopt for serious purposes.
I recall reading an article, by Putnam I think, where he talks about two different axiomatic formulations of the integers. Both formalisms agree on all the properties of the integers we regard as essential. However, in one system it is always true that if n < m then the set used to represent n id a subset of the set used to represent m. It the other axiomatic foundation, this is not true.
Some foundation points can introduce some strange discrepancies, but rarely anything we regard as material. This could probably be stated as an theorem in complexity theory. You'd have to put some elbow grease into the project to come up with a universal machine which can't compute pi using a "short" program where short is less than say Ackermann(4,4) and more likely, within a golf score of Ackermann(3,4).
Strange fact I didn't know:
http://en.wikipedia.org/wiki/Ackermann_function
Perhaps this is why KC theory is so often ignored. People can't wrap their minds around A(4,4) as an example of an extremely small number. The problem is, the philosophical question of invented/discovered demands this cognitive shift. A(4,4) is *not* a large number on the *philosophical* landscape.
Chaitin's omega, however, is the total perspective vortex of theoretical mathematics.
There seems to be a small number of special constants, such as e and pi, that any universal computer anyone has ever found a use for can obtain from a short program. Within this nucleus, a nanoscopically small filagree in the multidimensional fractal of all possible mathematics, the balance shifts toward "discovered". The further one departs from this minute filigree of felicity and virtue, the more the scale tips toward "invented".
If that sounds like fluff, answer this: what is the shortest number one can copyright?
Due to subitization it has never been possible to copyright the integers 1..4. The copyright on 5 probably expired 50,000 years ago. In modern society, there is evidence that 128-bit numbers remain fair game, though the difficulties of enforcing this are notorious. Clearly, five was discovered, the AACS constant was invented.
Not everyone agrees with Chaitin. This post makes a coherent statement of what he might be presuming:
http://coding.derkeiler.com/A