(March 14, 2013 at 11:47 pm)FallentoReason Wrote:

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?

I was distinctly saying that there were bounds to my ability to speculate deeply about the matter. I understand you're making a joke, but I'm not altogether sure of what is being said. I was a terrible math student, and had to abandon mathematics in college when I abandoned my college career due to depression, so there are certainly valid qualifications to be made. (I certainly didn't get as far as I would have liked, and as noted, missed important areas. I more or less fell away from my college career about the time that I was struggling with abstract algebra and differential equations, so the bulk of my knowledge comes from the practical side having to do with the calculus. So no, I don't have any real experience with the more theoretical disciplines of mathematics such as set theory and number theory and such.)

(And I'm told by younguns that nowadays maths are taught at a higher level in high school than they were in the Cretaceous period, so things have changed.)

Don't worry, apo, you probably didn't miss any philosophical insight of maths.... just calculations and more calculations.

On to a bit more of philosobabble...
To me, it's always been easier to imagine numbers as lying on a straight line... like an axis.
The kind of line that has all the numbers in it. Infinite numbers just between 1 and 2, and pi, somewhere between 3 and 4.
pro tip: if you're doing mental calculations that require pi, approximate it to 3

That said, I've always imagined that numbers just exist. All of them. Not as physical entities, of course, but as abstract ones. Sure they require an intelligence to become aware of them, but they're just there, regardless of the physical world... Just like theists would like their god to be.
You probably can tell that I'm not a big fan of the uncertainty in the Schrodinger's cat dilemma.
The cat is either dead or alive, regardless of anyone's awareness of it. Real numbers just exist, regardless of people's awareness of them.

The significance of the particular number someone defined as pi as the "ratio of a circle's circumference to its diameter" was manifestly discovered, inferred, determined by an intelligence capable of performing that ratio.

I'm not sure if this goes against anything I've written in this thread previously, but, if it does, it just comes to show that you guys managed to change my mind.

(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".

Colors exist due to that interpretation though. We can get all philosophical and say that colors only exist if there is something to view them, which is true enough, but when we are talking about discovery / invention, it's quite obvious to me that colors were discovered rather than invented.

Names of colours, well yes, they were invented, but the colors they actually represent were not.

Quote: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?

True, but that doesn't negate colors existing prior to humans, it only demonstrates a limitation on human descriptions of colors.

Having thought through this a bit more, it really all depends on what you define "discovery" and "invention" as. Pi is a property of perfect circles, which we definitely did invent (they do not make an appearance in the physical world, nor can they). Now, one could say that since Pi was worked out by taking a perfect circle and dividing the circumference by the diameter, it counts as a discovery (or a discovery of an invention). If you want to look at it that way, fine, I have no objection. However, the point is that the "discovery" of Pi was based on an invention of humans, and nothing else. I still think it's easier to think of Pi as having been invented, given that perfect circles were also invented.

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(March 13, 2013 at 9:54 pm)Chuck Wrote: 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.

There is a difference between the actual speed of light in a vacuum, and c, which is an estimation of that speed. The actual speed of light is clearly a property of the physical universe; light exists and it moves. The value we have assigned c is an invented number, as accurate as we can currently get it (and it's currently based off a 2009 calculation as far as I can tell). We are trying to get more accurate, but due to the limitations you noted above, as well as computational limitations, it's unlikely we'll ever get the true value (though perhaps we may be able to infer it).

In terms of Pi, the analogy here would be that however hard you try to make a perfect circle in the physical world, if you take the circumference of that circle and divide it by the diameter, you will never end up with the actual value of Pi.

Quote: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.

The "related properties" I assume you are talking about here are the circumference and the diameter of a perfect circle? If so, then what you've said is categorically wrong. Those properties do not make any "natural occurrences" in the universe, because it is impossible to find (or construct) a perfect circle in the universe. You can only get the irrational number Pi by creating a perfect circle. Any imperfect circle will get you a number that does not equal Pi (though it may be very close).

(March 15, 2013 at 7:47 am)Tiberius Wrote: There is a difference between the actual speed of light in a vacuum, and c, which is an estimation of that speed. The actual speed of light is clearly a property of the physical universe; light exists and it moves. The value we have assigned c is an invented number, as accurate as we can currently get it (and it's currently based off a 2009 calculation as far as I can tell). We are trying to get more accurate, but due to the limitations you noted above, as well as computational limitations, it's unlikely we'll ever get the true value (though perhaps we may be able to infer it).

In terms of Pi, the analogy here would be that however hard you try to make a perfect circle in the physical world, if you take the circumference of that circle and divide it by the diameter, you will never end up with the actual value of Pi.

The "related properties" I assume you are talking about here are the circumference and the diameter of a perfect? If so, then what you've said is categorically wrong. Those properties do not make any "natural occurrences" in the universe, because it is impossible to find (or construct) a perfect circle in the universe. You can only get the irrational number Pi by creating a perfect circle. Any imperfect circle will get you a number that does not equal Pi (though it may be very close).

"perfectness" of the circle totally misses the point.

The understanding of a property is not the mere cataloging of its manifested impact on the measurements actually made. It includes the capacity to make predictions on what the measurement would turn out to be if the circumstances of observation were to be changed from those of the actual observation. This includes predicting behavior even in notional circumstances marginally beyond physical capacity to actually achieve. So the fact that notionally perfect circle is impossible to achieve is irrelevant. What is relevant is actual instance of roundedness can be conceptually changed to reflect this notional perfect circle and predictions can be made upon it.

So through observation of the behavior of actual ratios between measurable circumference and dimension through centroid of real and quantifiable shapes, we developed an understanding of the property of this relationship such that we could predict how the ratio would change if a theoretical arbitrary shape were to deviate from actual observed shape. We so happen to find one instance of this predicted ratio to be the most practically encapsulating and useful. This so happen to be a shape requiring a minimum of fuss to describe, and yet adaquately approximating a large and useful collection of actual applications. This we call Pi.

We didn't invent Pi. We dicovered how to predict a specific, theoretical but encapsulating instance of the manifestation of a property.

This is all about perfectness of a circle.

That the circumference and diameter (and radius) ratios are common to all circles is a discovery if you want to call it that, but this is about Pi itself, which is an exact number, and depends a perfect circle. One can easily calculate an estimate of the circle ratio; 3 is used a lot, but if you want more accuracy, 3.1, or 3.14, etc. All of these are observable and measurable in real world circles, but none of them are Pi.

Pi is what happens when you take our observation / discovery and apply it to an invented perfection: the perfect circle. If we invent the perfect circle, what is the value of the circle ratio? The answer is Pi. This is an invented answer; you can't find an instance of Pi in any real circle.

(March 15, 2013 at 9:41 am)Tiberius Wrote: This is all about perfectness of a circle.

That the circumference and diameter (and radius) ratios are common to all circles is a discovery if you want to call it that, but this is about Pi itself, which is an exact number, and depends a perfect circle. One can easily calculate an estimate of the circle ratio; 3 is used a lot, but if you want more accuracy, 3.1, or 3.14, etc. All of these are observable and measurable in real world circles, but none of them are Pi.

Pi is what happens when you take our observation / discovery and apply it to an invented perfection: the perfect circle. If we invent the perfect circle, what is the value of the circle ratio? The answer is Pi. This is an invented answer; you can't find an instance of Pi in any real circle.

That's like saying we discovered the acceleration of gravity on earth but invented, rather than predicted, the acceleration of gravity on a notional planet of a different but specified size and mass because you can't find an actual instance of this particular value of acceleration of gravity.

No it isn't.

Both your examples use real-world scenarios. The construction of a perfect circle (and the calculation of Pi) cannot be done in any real-world measurement. It's entirely invented. Pi does not exist in the real world. It does not exist in any circle you find in the universe. All circles in the universe have ratios which look similar to Pi, but once measured, turn out to not be Pi.

Pi only exists in mathematics, as an invented number for the ratio of a perfect circle.

The acceleration of gravity on a planet of arbitrarily imagined mass and density could also exist nowhere in the real world. No real acceleration of gravity would turn out exactly like those of the imaginary planet. So what?

The key is whether predicted for the perfect circle or the fictional planet, or measured around a pizza on on earth, they reflect the case specific menifestations of the same underlying properties.

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(March 15, 2013 at 9:49 am)Tiberius Wrote: out to not be Pi.

Pi only exists in mathematics, as an invented number for the ratio of a perfect circle.

So caprice of the inventor has room to make pi a different number?

No. Perfect circle, like the imaginary planet, is invented. Pi, like the acceleration of gravity on the imaginary planet, is the predicted case specific menifestation of a discovered property in a invented case.

On a lighter note, have you seen/heard this Pi song? Kinda cute.

http://youtu.be/hJJJmQojcLM

(March 14, 2013 at 5:50 am)Tiberius Wrote: 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.

"Red" is not an intrinsic property of light, it's merely a way for humans to describe how our eyes perceive certain wavelengths.

Quote: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.

"Circle" is the name of a defined surface area, which exists in the physical world. Circles have always been there, but the name for them is a human invention.