This one should be easy to answer. It was invented, like most (if not all) of mathematics. Various reasons:

1) Pi is a number; numbers don't "exist" in the physical world; they only exist in mathematics, which is an invention of humanity. Now, granted physics is also an invention of humanity, but physics at least is looking at the physical world and describing it. There is nothing in the physical world you can look at and say "that's a number 1" (though obviously you can use numbers when interpreting the real world..."there are 3 cows in the field", etc.)

2) Not only is Pi a number, it's an irrational number, which means it's infinitely long (among other things). You don't get infinities in the real world (as far as we know), so there is no way for Pi to exist in the real world. Even we can't calculate the true value of Pi, since we don't have enough material / energy in the universe to do it.

3) Slightly related to point 2. Pi is generated by taking the circumference of a circle and dividing it by it's diameter. Sounds easy to replicate in real life right? Well, it's not. Circles are "a shape of Euclidean geometry that is the set of all points in a plane that are a given distance from a given point, the centre." That means that every point on the circle is the same distance from the centre as every other point. If this doesn't hold true, then you haven't got a perfect circle; you've just got something that looks quite like a circle, without actually being one.

Given the limitations of the universe again, it is impossible to construct a perfect circle. If you have two points next to each other, no-matter how small the distance between them, you can always find a point in between. This effectively means that the number of points in a perfect circle tends to infinity. At some point you will either reach the minimum length in the universe, or run out of stuff to construct your circle with. Either way, you have not been able to construct a perfect circle, and thus are not able to calculate the true value of Pi.

As Vihart once put it, maths is about making up rules and seeing what happens. We didn't discover negative numbers, imaginary numbers, etc. We just wondered what would happen if you could subtract larger numbers from smaller numbers, or could multiply two equal numbers together to get a negative number, and created a way to make it work whilst being consistent with other parts of mathematics.

Pi was born

(March 13, 2013 at 8:36 pm)Tiberius Wrote: This one should be easy to answer. It was invented, like most (if not all) of mathematics. Various reasons:

1) Pi is a number; numbers don't "exist" in the physical world; they only exist in mathematics, which is an invention of humanity. Now, granted physics is also an invention of humanity, but physics at least is looking at the physical world and describing it. There is nothing in the physical world you can look at and say "that's a number 1" (though obviously you can use numbers when interpreting the real world..."there are 3 cows in the field", etc.)

2) Not only is Pi a number, it's an irrational number, which means it's infinitely long (among other things). You don't get infinities in the real world (as far as we know), so there is no way for Pi to exist in the real world. Even we can't calculate the true value of Pi, since we don't have enough material / energy in the universe to do it.

3) Slightly related to point 2. Pi is generated by taking the circumference of a circle and dividing it by it's diameter. Sounds easy to replicate in real life right? Well, it's not. Circles are "a shape of Euclidean geometry that is the set of all points in a plane that are a given distance from a given point, the centre." That means that every point on the circle is the same distance from the centre as every other point. If this doesn't hold true, then you haven't got a perfect circle; you've just got something that looks quite like a circle, without actually being one.

Given the limitations of the universe again, it is impossible to construct a perfect circle. If you have two points next to each other, no-matter how small the distance between them, you can always find a point in between. This effectively means that the number of points in a perfect circle tends to infinity. At some point you will either reach the minimum length in the universe, or run out of stuff to construct your circle with. Either way, you have not been able to construct a perfect circle, and thus are not able to calculate the true value of Pi.

This is about as absurd as saying given the limits of the universe, it is impossible to construct a perfect vacuum, therefore speed of light in vacuum is not a property of the physical universe, but a invented number.

Nor is the rest any less absurd than that finding natural occurrences of two related properties in the universe to seem to converge on a irrational dimensionless multiplier, and claiming by its irrationality the relationship itself is thus an invention.

All of mathematics invented, so before it was invented, what happened when you added x cows to x cows? Nothing? It's like asking if red was invented. Was red something else before someone named it red? Is your word red the actual thing we're referring to when we say red? We're obviously quite capable of producing (inventing) concepts for things which are not the things themselves but are nevertheless tied at the hip in some way (though we're clearly just as competent at divorcing the concept from the thing to which it refers).

(March 13, 2013 at 9:21 am)Rhythm Wrote: Yet we have no problem constructing them. Come now, a vast number of perfect circles exists anywhere we care to place them, as both the circle and pi are descriptions of a relationship between dimensions that we found it useful to define in a certain way. I'm not entirely sure what there would be to disagree with in the sentence you quoted, as this is how pi is defined. If pi is not a description of the relationship between the circumference and diameter of a circle please alert the mathematical community at your next convenience Aractus.

I already defined Pi earlier in the thread for you:

Pi is simply the answer to the question what is the ratio of the length of the sides of a regular polygon to the average diameter "as the number of sides approach infinity"?

We have no problem imagining perfect circles, but we can't construct something perfect if it requires a transcendental number - can we?

Also, this is exactly - verbatim - how Pi is calculated to an infinite number of decimals:



That's Francois Vieta's calculation - prior to that, he calculated Pi to nine decimal places, and the way he did that was by calculating the area of a regular polygon with 393,216 (393,216 = 3 x 217) sides.

(March 13, 2013 at 10:36 pm)Rhythm Wrote: All of mathematics invented, so before it was invented, what happened when you added x cows to x cows? Nothing?

Yes! Addition is part of mathematics. You cannot add things without it. Certainly, taking x cows and putting them with x other cows will get you more cows than you had before, but here's the point: the amount (number) of cows is not an intrinsic physical property of the cows. It's merely a way for humans to describe them.

Quote:It's like asking if red was invented. Was red something else before someone named it red? Is your word red the actual thing we're referring to when we say red?

Red is the name of a color, which exist in the physical universe. Red has always been there, but the name for it is a human invention.

Circles however do not exist in the physical universe. They are theoretical but useful for making calculations in the real world. If circles do not exist in the physical world, pi cannot exist either. Likewise, since circles were invented, and pi is a property of circles, pi was invented as well.

Colours don't "exist" per se... the frequency of the light entering our eyes gets interpreted by the brain and in turn we have the experience of colour from a physical phenomena (the frequency of the light) that has a huuuge range, not just the range where colours "exist".

Another observation that sort of ties in with the above is that the colour "red" isn't an objective thing. In the colour spectrum, where does red begin and end?

I've kind of gone off topic, but meh.. thought it might be an interesting insight.

I'm unfamiliar with philosophy of mathematics, and also the current set theoretical underpinnings of mathematics, but it's not clear that we're even talking about things at the right level. If these things are analytical consequences of much simpler things, like a set, identity, non-contradiction, and so on, the same questions recur, but seem a much clearer consequence of how the mind makes sense of the world. In other words, simply having a mind that is human may mean automatically having knowledge of the concepts upon which things like pi and circles depend, without doing any real thought to "construct" them; they're a priori analytical, but not things which are obvious analytical consequences of the a priori. Indeed, it might even be impossible to think in the human way without already being possessed of conceptual truths whose consequences are these derivative concepts. While, as noted, I think some of these higher level concepts exist as nodes in our neural nets, of which we have privileged access, even if not, it's not clear we created these truths. (That's one of the reasons I think it's somewhat misleading to refer to them as "inventions." Seeing the world as three dimensional is an automatic consequence of the way our minds work, it's not a necessary consequence of the objective world that we should think in these terms, nor is it necessarily true that there are three dimensions. Yet I'd look querulously at anyone who claimed that we "invented" three-dimensionality.)

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(March 14, 2013 at 5:57 am)FallentoReason Wrote: Colours don't "exist" per se... the frequency of the light entering our eyes gets interpreted by the brain and in turn we have the experience of colour from a physical phenomena (the frequency of the light) that has a huuuge range, not just the range where colours "exist".

Another observation that sort of ties in with the above is that the colour "red" isn't an objective thing. In the colour spectrum, where does red begin and end?

And the existence of tetrachromatics blows that even further out of the water.

Many common things that seem to defy explanation seem to hide themselves in plain sight. Music consists of a series of notes played in sequence. Where is melody? Is it contained in a particular set of those notes, all of them, or none of them? Where is the melody?

apophenia Wrote:I'm unfamiliar with philosophy of mathematics, and also the current set theoretical underpinnings of mathematics, but it's not clear that we're even talking about things at the right level.

apophenia, dear... maths is something that some people get and others just don't. I saw many of my fellow classmates fall away and quit the highest level of maths in highschool. Are you sure you even know what you're saying?

/mirror

All in good spirit bud

Apophenia,

You slay me!!!!

Fucking funny shit...