Is Math an Invention or a Natural Phenomenon?

[zoobyshoe]

This is essentially what I've been saying about the third option, but to me it appears to emerge from the interaction of external and internal, not solely internal or external. We don't have a circle clump of neurons, per say, we have a more generalized visual system and circular things are common enough that we lump them into one category (like we do with so many other things).

I'm really not sure how much weight to put on the external input. But for anyone still in the "math is out there in the universe" camp, it is important to emphasize how much of math is actually "in here".

The brain seems to spontaneously generate geometric images under certain circumstances:

http://plus.maths.org/content/uncoiling-spiral-maths-and-hallucinations

Less formal blog:

http://disregardeverythingisay.com/post/9331287956/the-visual-components-of-a-psychedelic-experience.

The same thing happens to some people experiencing visual migraine "aura".

These experiences seem to "drop out" of the way neurons interact when under the influence of a variety of "toxins" or pathological situations, or they may "drop out" of the cytoarchitecture of the brain, itself, under those circumstances.

This should cast doubt on whether "invented or discovered" is a proper question, and get people thinking about the "third option."

edit:

There must be experiments on this - does a baby distinguish a circle from an oval "at birth"?

See the above. You can't find the answers to some questions by experimenting on babies.

Edit: Article from Scientific American:

http://blogs.scientificamerican.com...ts-nature-turing-patterns-and-form-constants/

Edit: Wikipedia article on Form Constants:

http://en.wikipedia.org/wiki/Form_constant

 

[Medicol]

I think Math is Expanded, not invented or discovered.

 

[BobG]

What does discovering a constant (Pi) have to do with whether math was discovered or invented?

Math is invented. Hopefully, the invention is useful for modeling things that occur naturally in the real world (vs the imaginary world of mathematicians ).

Take the invention of quaternions. Hamilton spent a long time trying to invent trinarians, but it was virtually impossible to develop a mathematical system that would work with one real and two imaginary components. He invented quaternions because it was actually possible, most importantly, and if you just set the real part to zero and used three imaginary components, his quaternion math could accomplish all the things Hamilton hoped his trinarian system would accomplish.

And, like a true mathematician, the new math he invented didn't actually solve any problems that were very pressing during his time (although over a few hundred years, other people did actually find some problems that quaternions were good for). Quaternions were just something that gave him cool things to carve into bridges while on moonlight strolls with his wife (See what I mean about mathematicians?)

 

[homeomorphic]

And, like a true mathematician, the new math he invented didn't actually solve any problems that were very pressing during his time (although over a few hundred years, other people did actually find some problems that quaternions were good for). Quaternions were just something that gave him cool things to carve into bridges while on moonlight strolls with his wife (See what I mean about mathematicians?)

Disagree. The main reason why quaternions are useful was already apparent back then. That is that they can be used to calculate rotations. That might not be all the applications, but the use in computer graphics today, for example, is basically not anything new, other than how to get a computer to do it and make the computer. A lot of the things mathematicians do today are much more removed from reality than quaternions are. What could be more real than 3D rotations? Of course, then Hamilton went and tried to do all sorts of things with quaternions that weren't so successful, but I think one of his main motivations was to describe 3D rotation.

Some of the semantic difficulties are also evident in your post. By math, do we mean math as a whole subject or do we mean specific math? Also, using one example is not really sufficient. If you look at a wide variety of examples, you see that it becomes awkward to always insist on either "invention" or "discovery", and that it is semantically much more natural to use both words, depending on the math in question.

 

[BobG]

My comments about the usefulness of quaternions were slightly tongue in cheek (but vector analysis enthusiasts sure spent a lot of time bashing quaternions, as this quote shows: "Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way" - Lord Kelvin, 1892).

They're actually pretty useful in satellite attitude control, as well.

But they are a good example of "discovered" vs "invented". I could easily understand a person saying the commutative property of multiplication was discovered, as any person looking at a group of chairs assembled in five rows of 10 could easily discover that property themselves. But that doesn't mean multiplication has to be commutative. A person could design a non-commutative version of multiplication - and Hamilton did with his quaternions.

 

[homeomorphic]

But they are a good example of "discovered" vs "invented". I could easily understand a person saying the commutative property of multiplication was discovered, as any person looking at a group of chairs assembled in five rows of 10 could easily discover that property themselves. But that doesn't mean multiplication has to be commutative. A person could design a non-commutative version of multiplication - and Hamilton did with his quaternions.

Well, I'm not sure if I agree that Hamilton "invented" that feature of the quaternions. The fact is that 3D rotations don't commute, so that is probably why he realized he needed non-commutative multiplication, I'm guessing. You could say he invented it in order to model the 3D rotations, which are out there in reality. But he discovered that he needed to invent because we aren't born knowing that 3D rotations don't commute, but instead, figure it out ("discover"). In fact, to poke some fun at engineers, I once heard of a mathematician whose whole job was to explain to engineers that 3D rotations don't commute.

 

[BobG]

I once heard of a mathematician whose whole job was to explain to engineers that 3D rotations don't commute.

Probably a bunch of mechanical engineers.

 

[Domenico94]

The reason mathematics is admirably suited to describing the physical world is that we invented it to do just that. We make mathematics up as we go along to suit our purposes. Mathematicians agree on the same subject partly because there is continuity of knowledge across time and space. The Arabs learned from the Greeks, and the West learned form the Arabs.

In cases of similar developments, but no contact, there is "one obvious invention". For example, integers up to about seven are gifted to us by evolution, and it seems a very obvious step to make up more integers by "adding 1", just as we make 2 from 1 by adding 1. One can imagine many cultures inventing this "adding 1" process to "invent new integers" without learning from another culture. Of course, this process would be useless if "adding 1 object" led to to arbitrary number of objects. If that had been the case the invention of integers would have died at birth, like a tissue paper steam engine.

Yes...but thanks to math Dirac found the evidence for his theories (Spooky action at distance), scientists made other previsions about black holes..and so on..that was my point.
When we solve physical problems at school or university, we never go by intuition..We always use a more or less sophisticated mathematical model which always points to the real solution we're actually finding. In addiction, many mathematical topics (like the "potential function " or the integral or a curve, or the derivative, are themselves strictly connected to physics).

If mathematics wasn't perfect why would many mathematicians at different places and different times come up with the same soolution for a theorem?

 

[mal4mac]

Yes...but thanks to math Dirac found the evidence for his theories...

No! Experimental physicists found the evidence for his theories. A prediction for anti-matter might have emerged form his math, but the evidence for anti-matter needed to be found through experimental observation.

If mathematics wasn't perfect why would many mathematicians at different places and different times come up with the same solution for a theorem?

Do they? Can you give examples of the many times this has happened?

 

[Domenico94]

Theorem of Banach-Cacciopoli, in analisis, theorem of Bolzano weierstrass...Theorem of Ascoli-Arzolla (in numeric series), and so on...The euler's identity was discovered , with 200 years difference, by euler and an indian mathematician...I think it's enough :D

 

[mal4mac]

Theorem of Banach-Cacciopoli, in analisis, theorem of Bolzano weierstrass...Theorem of Ascoli-Arzolla (in numeric series), ...

These guys were all very close in space, time & influence.

 

[zoobyshoe]

Theorem of Banach-Cacciopoli, in analisis, theorem of Bolzano weierstrass...Theorem of Ascoli-Arzolla (in numeric series), and so on...The euler's identity was discovered , with 200 years difference, by euler and an indian mathematician...I think it's enough :D

I'm not sure you have a good sense of what constitutes an invention. You should agree that a machine for recording sound is an invention, right? At the same time, it only works by virtue of the fact it takes advantage of principles found in nature. What was man's input into the sound recording machine such that we call it an "invention" and distinguish it from a discovery?

 

[Domenico94]

These guys were all very close in space, time & influence.

but they didn't work toghether..that's the point..and came to the same conclusion...Euler and the indian mathematician were not close...Euler was Swiss. Srinivasa ramanujan was Indian and he lived in the 19th century..

 

[Domenico94]

I'm not sure you have a good sense of what constitutes an invention. You should agree that a machine for recording sound is an invention, right? At the same time, it only works by virtue of the fact it takes advantage of principles found in nature. What was man's input into the sound recording machine such that we call it an "invention" and distinguish it from a discovery?

I didn't talk about inventions...I'm just saying people that never met themselves, came up with the same conclusion..

 

[zoobyshoe]

I didn't talk about inventions...I'm just saying people that never met themselves, came up with the same conclusion..

Which you offer as evidence of what?

The list of disputed credit for inventions is about long enough to wind around the Earth three times. You name the invention and there is a dispute somewhere about who actually invented it first. This is because, at any given time, there are huge numbers of inventors working, and the chance of two or more accidentally being at work on the same invention completely unbeknownst to each other is quite high. The same goes for math. This is not evidence that either activity is "perfect."

 

[Domenico94]

If it wasn't..they would have come with different theories for a theorem...but they didn't..that's my point

 

[zoobyshoe]

If it wasn't..they would have come with different theories for a theorem...but they didn't..that's my point

OK. I understand your point now. The point you're missing, though, is that the whole activity of "proving" something in math is something humans invented for the benefit of other humans. We did not learn it from nature.

 

[Domenico94]

yes...maybe that happened too ( Einstein too used saying that). But, take for example, Euler's formula :
$$ e^(i*pi) + 1 = 0 $$

That was a completely abstract formula, with numbers we couldn't imagine (the complex ones). BUt that led to the introduction of the Complex analysis, the Fourier transform and all the applications of this abstract thing into real life.
We could say the same about differential equations :)

 

[zoobyshoe]

yes...maybe that happened too ( Einstein too used saying that). But, take for example, Euler's formula :
$$ e^(i*pi) + 1 = 0 $$

That was a completely abstract formula, with numbers we couldn't imagine (the complex ones). BUt that led to the introduction of the Complex analysis, the Fourier transform and all the applications of this abstract thing into real life.
We could say the same about differential equations :)

I'm not following your logic, or, at least, I don't have a clear sense of what statement you're making here regarding invention vs discovery. Euler invented a formula that later had applications to real life problems proving...? As evidence for what?

 

[collinsmark]

yes...maybe that happened too ( Einstein too used saying that). But, take for example, Euler's formula :
$$ e^(i*pi) + 1 = 0 $$

That was a completely abstract formula, with numbers we couldn't imagine (the complex ones). BUt that led to the introduction of the Complex analysis, the Fourier transform and all the applications of this abstract thing into real life.
We could say the same about differential equations :)

I'm not following your logic, or, at least, I don't have a clear sense of what statement you're making here regarding invention vs discovery. Euler invented a formula that later had applications to real life problems proving...? As evidence for what?

Actually, that particular formula, e^{i \pi} + 1 = 0 comes directly from a more general equation of Euler's,

e^{i \theta} = \cos \theta + i \sin \theta.

While that might at first glance seem like an equation that he merely pulled out of his buttocks, it is not. You can prove it to be a true, mathematical relationship using rigorous deduction (within the confines of our basic axioms of course, which I won't list here).

(There are several ways to derive this formula. One method is to start with f(x) = e^{ix}, and then state f(x) = - f''(x). Solving that differential equation produces f(x) = A \cos x + B \sin x, where you can invoke the uniqueness theorem along the way (twice actually), showing the solution to be unique, and apply boundary conditions to show A = 1 and B = i. Other methods are also possible: http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs)

But now I fear I'm getting off topic.

I'll follow up with a more on-topic post, but I'm not sure if I'll make it in before the PF 4.0 migration.

 

[collinsmark]

Here's my 2 cents on the subject.

Is math invented or discovered? Let's get to the root of the question and make it even more basic.

Is the concept of prime numbers discovered or invented?

A mathematician -- a good mathematician -- might make the claim that the set of prime numbers is invented. This in turn might bring about a typical reaction from a person less well versed in mathematics,

Typical reaction to prime numbers being invented:

You got to be kiddin' me. Invented? That's about the most arrogant statement I can imagine. Do you honestly think that if we ever made contact with another species of intelligent life, with advanced technology, that they wouldn't have a concept of prime numbers? And if they did, which they assuredly would, what would you do? Throw your hands up and yell, "Thieves! Thieves! Prime numbers are man-made! The Zorgon empire stole them from us! Prime numbers were invented by humans so the Zorgon empire is just a bunch of thieves! Just like those damn, thieving cicadas! Thieves!"​

But that's not what the mathematician means by inventing prime numbers. As a matter of fact, most mathematicians actually like cicadas. More to the point, the invention of prime numbers is very different than inventing a novel recipe for a tuna sandwich on rye bread with some mayonnaise.

The mathematician is purposefully being inconclusive -- agnostic if you will -- on the subject of whether prime numbers are a crucial, fundamental part of our universe. Rather the mathematician is saying that prime numbers were at least invented by humans. If in the future we make contact with the Zorgon empire, and they also happen to have a concept of prime numbers, the mathematician would say the set of prime numbers was independently invented by both humans and Zorgons. The mathematician may very well suspect that prime numbers are in fact a fundamental part of our universe -- but he just cannot prove it. And I don't mean that we just haven't found a proof yet, but rather it seems as though the concept may not even by provable. The good mathematician, not wanting to make claims of what has not been proven, or make conjectures to what is unprovable, stays quiet on the subject.

The set of prime numbers fall out from our created axioms, such as Peano axioms which go without saying that they are obviously invented.

Then, the real crux of the issue comes from Gödel's incompleteness theorems, which when applied here, means that we can't prove anything outside our axioms. We cannot definitively prove that another technologically advanced, intelligent species would have made the same axioms.

I fully expect that if you were to take that mathematician -- or any good mathematician -- aside in private and ask him or her,

Question: If we were to encounter another form of intelligent life, of considerable technological advancement, do you think they would have a concept of prime numbers? The same set of prime numbers that we have?
Answer: Well, yeah. Of course. I mean I would really be surprised if they didn't! Well, the notation would obviously be different. Any they might include the number '1' in the set (we do not anymore). But the rest of them sure. Can you imagine! A technologically advanced society without prime numbers! That's not going to happen. So yeah, I'm pretty sure they would. But I cannot prove it.​

And similarly, although I cannot prove it, I fully suspect that with the Zorgon empire's own equivalents to Gödel's incompleteness theorems, if the zorgons were confronted with the same question, they would throw their arms up in the air and run around chanting "Qz! Qz!" where the word "qz" from the zorgon native tongue is translated as:

qz
/ kwǝz /

interjection

The affirmation that the set of prime numbers was invented by the ancient zorgonians and was independently invented by the despicable kladies of the planet Kladbor, as well as being independently invented by a species of the planet Earth know as cicadas. The term continues the affirmation that there is also another, very tasty species of Earth beings called humans that claim to have independently invented the set of prime numbers as well, but it is equally likely that the delicious humans stole the idea from the cicadas. The term has a connotation that it is welcomed and encouraged to sit down during tea and discuss the origin of prime numbers with humans over rye bread and some mayonnaise.
​

noun

A sandwich consisting of humans, rye bread and mayonnaise. Sometimes served with a light spread of cicada sauce.

​

​

 

[phion]

Here's an insightful clip about what Stephen Wolfram thinks:

 

[zoobyshoe]

Very interesting video, Phion. He doesn't exactly say our math is invented, but that it is one of many possible mathematics, an "artifact" of the particular route taken by human history.