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."
Connection to math = The Universe (Score:2, Interesting)
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?
Is Mathematics Discovered Or Invented? (Score:5, Interesting)
Neither. It is defined.
Only the integers (Score:5, Interesting)
Integers were discovered. Beyond that, it's human invention.
I used to do work on mechanical theorem proving, and spent quite a bit of time using the Boyer-Moore theorem prover [utexas.edu]. When you try to mechanize the process, it's clearer what is discovered (and can be found by search algorithms) and what is made up. Boyer-Moore theory builds up mathematics from something close to the Peano axioms. [wikipedia.org] But it's a purely constructive system. There are no quantifiers, only recursive functions. It's possible to start with a minimal set of definitions and build up number theory and set theory. The system is initialized with a few definitions, and, one at a time, theorems are fed in. Each theorem, once proved, can be used in other theorems. After a few hundred theorems, most of number theory is defined.
But you never get real numbers that way. Integer, yes. Fractions, yes. Floating point numbers, representation limits and all, yes. But no reals. Reals require additional axioms.
both? (Score:4, Interesting)
Parallel (Score:5, Interesting)
All the same? (Score:4, Interesting)
Why so human-centric? (Score:3, Interesting)
So, regardless of the whole platonic debate, basic mathematics definitely exist independently of humans.
Re:Logical positivism to the rescue... (Score:2, Interesting)
But did God invent or discover it? (Score:3, Interesting)
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.
The super-imaginary number, j. (Score:3, Interesting)
Re:Logical positivism to the rescue... (Score:4, Interesting)
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.
Are jokes discovered or invented? (Score:2, Interesting)
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).
Re:Only the integers (Score:4, Interesting)
Integers were discovered. Beyond that, it's human invention.
I can make a strong case that negative integers are invented. Because you can't have -3 apples. We invented the negative numbers to indicate a loss of positive numbers. We also invented fractions to stand in for ratios.
Sort of.
Re:Logical positivism to the rescue... (Score:2, Interesting)
But then, thinking a bit deeply, I agree that as you said, maths is both discovered and invented. There is no doubt that mathematical symbols were created by us humans. I just created some symbolisms while doing my thesis. However, those symbols are used to classify or "label" different patterns that *happen* in our universe and that we "perceive". It is then when we use such mathematical symbols to establish a classification of such patters (for example, we know that Weight = Mass Ã-- Gravity, because of experimentation, however we did not created such relationship or pattern. We just labeled it "\times" (or sometimes *).
Re:Is Mathematics Discovered Or Invented? (Score:3, Interesting)
Applied to math, you could say mathematics is a series of definitions we've created to describe an observed phenomenon or hypothesize the existence of an as yet unobserved phenomenon.
But what the hell do I know? I'm neither a philosopher or a mathematician.
Bah (Score:2, Interesting)
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's neither (Score:3, Interesting)
You've just reinvented Projectively Extended Reals (Score:5, Interesting)
Just reading about this... (Score:5, Interesting)
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.
Re:Logical positivism to the rescue... (Score:3, Interesting)
I vote "invented" because.... (Score:3, Interesting)
Re:Well it's obviously discovered (Score:4, Interesting)
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.
Re:Logical positivism to the rescue... (Score:4, Interesting)
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"...
Re:Logical positivism to the rescue... (Score:4, Interesting)
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 [wikipedia.org]
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 [wikipedia.org] 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 [derkeiler.com]
Re:Logical positivism to the rescue... (Score:2, Interesting)
Re:Logical positivism to the rescue... (Score:3, Interesting)
One adds those things together that can be separated. Mass and velocity are intrinsic to an object, so they can't be separated. In other words, if I accelerate one part of an [inelastic] object, I accelerate all parts that same amount. For example, if I have a 12 kg bowling ball moving 10 km/hr, all 12 kg have to go the 10 km/hr. So, the simplest way to do the accounting is 10*12.
It depends on what you mean by those words. Did we invent the wheel or simply discover it? Its a very philosophical indeed, but I'm not so sure we achieve anything by trying to answer it, especially since, in the end, it boils down to semantics. In such cases, I usually refer to American Heritage as that is what came with my Mac.
Re:Logical positivism to the rescue... (Score:2, Interesting)
However, if you're referring to "unit dimensions" in physics specifically, there's a simple reason for that. And it doesn't have to do with the structure of the world. Quite simply, the fact follows from our use of integer derivatives to study change, as opposed to fractional derivatives. Units are syntactically variables, and must be treated as such during computation. Of course, this is why 3m x 5m is 15m^2. But it is also why the 3/4th derivative of position in time would end up with whacky unit exponents.
http://mathworld.wolfram.com/FractionalCalculus.html [wolfram.com]
The relations derived using the fractional calculus are just as true as the standard treatment. The integral formulation is merely computationally simpler.
Re:Logical positivism to the rescue... (Score:2, Interesting)
To put it another way, if mathematics were not discovered, we need a pretty good explanation for simultaneous and independent discovery (calculus by Newton and Leibniz, zero by Indians, Mayans, and Chinese, many theorems with a hyphenated name like Schur-Zassenhaus or Cauchy-Kovalevkskaya, gauge or Henstock-Kurzweil integrals). Independent discovery, whether simultaneous or not, is a pretty good argument in favour of the discovery portion of mathematics, that mathematics has an intrinsic content for us to discover that does not depend at all on the formalism of symbols we use to describe that content.
The day we run into alien civilisation, the first thing I'm gonna ask is to see their mathematical books. I expect to find a lot of familiar things in there.
Re:Logical positivism to the rescue... (Score:3, Interesting)
"is philosophy discovered or invented"
Philosophy is just repeatedly applying rules like mathematics, and I could always play the same trick I did with the poems to concieve a very large number of philosophical system. I would argue all philosophy is discovered.
A Question of Semantics (Score:2, Interesting)
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.
Re:Logical positivism to the rescue... (Score:2, Interesting)
This is why different cultures tend to agree on scientific grounds much more than they do in governance, trade, etc.
As for Einstein, what he taught us was that relativistic effects can be explained consistently from different points of view. We can factor-in those POVs without reducing our predictions and observations to mere subjectivity.
We humans (and even some animals) evolved with more than our five senses and intuition. We have the ability to calculate, or deduce, truths that cannot be intuitively grasped.
Math is discovered (Score:2, Interesting)
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.