Is Math an Invention or a Natural Phenomenon?

[zoobyshoe]

What about the number of days in a year? That's not rational.

I think that fact supports my argument, doesn't it? It's another example of Nature not working out to perfectly dovetail into clean, whole numbers, or whole number fractions of 1, the way the Pythagoreans thought it would at first.

 

[Pythagorean]

What I'm asserting is that the Euclidian definition of a circle, which is a concept, pretty much drops in your lap during the act of drawing a physical approximation of a circle.

Only after you've invented the concepts of points, centers, and planes with which to construct the Euclidean definition.

Still, the circle was never "out there" waiting to be discovered. Circularity is a property we can discover in the natural world (only because it's something we've classified as "this property of things we find in nature that looks like this", but the noun, "circle", is rather abstract and meaningless (with respect to natural phenomena) when used alone.

 

[zoobyshoe]

Only after you've invented the concepts of points, centers, and planes with which to construct the Euclidean definition.

OK, I shouldn't have said "the Euclidian definition," but, "the essence of the Euclidian definition." The concept of an ideal flat figure distinct from other flat figures, and characterized by the equality of any measurement made from center to circumference drops into your lap from the act of drawing the circle, without you even needing to have a name for any of it. Given time you could work out how to articulate that in words. (The concept usually precedes the ability to articulate it.)

Still, the circle was never "out there" waiting to be discovered.

It was not invented, and if it wasn't invented and it wasn't "out there" waiting to be discovered, it must have been "in here" waiting to be discovered. By which I mean our ability to conceive of the 'mathematical object' (I think that's the term I want), the circle, emerged along with our ability to perform the varied logical processes concerning abstract quantities that constitute math. When we discover things in math, I think they're discovered in an interior conceptual arena constructed to try out model trains of logic.

 

[zoobyshoe]

Incidentally, it occurred to me that any math must, by necessity, be built on the assumption of conservation of quantity. If that assumption isn't made, there's pretty much no point to the math. It's possible those primitive tribes who don't count higher than 10 haven't made that assumption, and may unconsciously suppose quantities might spontaneously change.

 

[mal4mac]

It [the circle] was not invented ... and it wasn't "out there" waiting to be discovered, it must have been "in here" waiting to be discovered.

Are babies born knowing what a circle is?

This is open to empirical investigation. here's some background:

http://www.psychologytoday.com/blog/babies-do-the-math/201101/brainy-babies

"babies appear to be born knowing that objects cannot magically appear or disappear, that they cannot pass through each other, and that they cannot move unless contacted by another object. These expectations hold for objects, but not for non-object entities like substances (e.g., liquid, sand)."

So babies are born knowing quite a lot of physics .

"by their first birthday (and long before they can talk), babies exhibit quite sophisticated number knowledge. They can enumerate visual and auditory items, items presented sequentially and items presented simultaneously."

Nothing there about a circle being "in there". "Numbers" and "assuming continuing existence" are in there because they are useful to evolution. But is spotting a circle useful to evolution? Being able to choose two apples, rather than one, is obviously useful; but taking the more circular apple doesn't seem useful. My guess is that the circle was invented, and has to be explained to children - it's not "in there", and no perfect circle is "out there", so it had to be invented.

 

[zoobyshoe]

Are babies born knowing what a circle is?

This is open to empirical investigation. here's some background:

http://www.psychologytoday.com/blog/babies-do-the-math/201101/brainy-babies

"babies appear to be born knowing that objects cannot magically appear or disappear, that they cannot pass through each other, and that they cannot move unless contacted by another object. These expectations hold for objects, but not for non-object entities like substances (e.g., liquid, sand)."

So babies are born knowing quite a lot of physics .

"by their first birthday (and long before they can talk), babies exhibit quite sophisticated number knowledge. They can enumerate visual and auditory items, items presented sequentially and items presented simultaneously."

Nothing there about a circle being "in there". "Numbers" and "assuming continuing existence" are in there because they are useful to evolution. But is spotting a circle useful to evolution? Being able to choose two apples, rather than one, is obviously useful; but taking the more circular apple doesn't seem useful. My guess is that the circle was invented, and has to be explained to children - it's not "in there", and no perfect circle is "out there", so it had to be invented.

Due to (PF member) Pythagorean's earlier criteria about what constitutes a circle:

An approximation of a circle is not a circle. The circle is the idealization; the simplification.

the capacity to idealize a circle from experience of the approximation should be "in there" with numbers. We perform math on idealized internal models. We can account for a large herd of sheep by idealizing the sheep as all equal, when they're actually all different weights, and colors, etc.

 

[mal4mac]

the capacity to idealize a circle from experience of the approximation should be "in there" with numbers.

Idealizing a few sheep by making them "equal and simple" in the process of "number discrimination" seems an easy thing for the brain to do - they are obviously separate white bits on a field of green. But idealising objects as circles seems more difficult - is an apple circular, is an orange? Does the baby know a circle at birth? Or does the baby learn to distinguish circles later on? I suspect a long time "later on" - after learning about apples, oranges, balls... and why one is more circular than another. I might, of course, be wrong. There must be experiments on this - does a baby distinguish a circle from an oval "at birth"?

 

[Pythagorean]

OK, I shouldn't have said "the Euclidian definition," but, "the essence of the Euclidian definition." The concept of an ideal flat figure distinct from other flat figures, and characterized by the equality of any measurement made from center to circumference drops into your lap from the act of drawing the circle, without you even needing to have a name for any of it. Given time you could work out how to articulate that in words. (The concept usually precedes the ability to articulate it.)

It was not invented, and if it wasn't invented and it wasn't "out there" waiting to be discovered, it must have been "in here" waiting to be discovered. By which I mean our ability to conceive of the 'mathematical object' (I think that's the term I want), the circle, emerged along with our ability to perform the varied logical processes concerning abstract quantities that constitute math. When we discover things in math, I think they're discovered in an interior conceptual arena constructed to try out model trains of logic.

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).

 

[iDimension]

So can't we just say that some areas of mathematics are invented and some are discovered? The perfect circle doesn't exist in nature anywhere we know of which in turn would mean that Pi can't exist until humans or some other intelligent life draw the perfect circle.

It's a hard one but after reading some of the other posts I think it's fair to say that some math is invented and some math is discovered. The concept or mathematics must surely be discovered and not invented as any intelligent life form must surely use mathematics too.

The symbols and notation is invented but the idea and concept is discovered.

 

[mal4mac]

The concept or mathematics must surely be discovered and not invented as any intelligent life form must surely use mathematics too.

Maybe. But they might invent different mathematics, just as they would invent different steam engines (or different engines if they skip steam!) The Greeks got a long way without algebra. Calculus was developed using limits, but now calculus using infinitesimals is acceptable. Maybe the aliens would invent mathematics based on infinitesimals from the get go - or maybe just have a discrete mathematics rather than messing with all those nasty infinities.

 

[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.