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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."
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 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.
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.
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
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!
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.
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!
What's purple and commutes? An Abelian grape.
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.
> If it's discovered, we can patent it. If it's invented we can copyright it.
No. If it is invented it can be patented. If it is created it can be copyrighted. If it is discovered it can be neither patented nor copyrighted.
Warning: this article may contain humor, sarcasm, parody, and perhaps even irony. Read at your own risk.
You got it all wrong. You can not patent a discovery, and you have no copyright to an invention. You can, however, patent an invention.
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.
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.
"Yes".
To be more specific, Mathematical rules are discovered, Mathematical techniques are invented; "Mathematics" consists of both.
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.
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?
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
Everyone knows you have to discover mathematics before you can build catapults
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!
If mathematics is invented it can be patented.
HTH.
Deleted
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.
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.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.