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I don't like the argument about pi. I've mentioned this a couple of times above and nobody seems to want to address my points about the fact that we do not live in Euclidean space. Real circles with the property that ratio of circumference to diameter is a constant and that that constant is exactly equal to the transcendental number we call pi is a complete fairy story. It's just not true physics. It's approximately true physics for sure, but "approximately true" is infinitely far from "perfectly true" when it comes to mathematics. Perhaps this point is too subtle for anyone to notice, but I believe it is the crux of the entire question.
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I'm really not understanding your point about Pi. The concept of Pi is quite basic and does not rely on a perfect physical model for the concept to be discovered and examined!

In the "early science" days of a civilisation they must examine geometric shapes and their characteristics. A square has a circumference of 4 times it's diameter, this does not need a perfect physical model as it can be easily understood. Sure the early scientists would theorise the relationship for a circle, it's a very important relationship. Pi exists and would be discovered and refined in accuracy as the science develops.

Are you really saying you think Pi is "invented"? If so how do you justify that?

 

Although....

Mathematics is a wholly human construct inspired by the nature of our universe.

Just an opinion now, not the gospel.

The finest thing about a scientist is that he is not prejudice. He draws his conclusions upon observation, data collection, finding empirical relation in data, reaching a general mathematical formula, totally unbiased from his own choices and likings. When we examine nature, we observe that some mysterious force is controlling every thing with an unmatched precision and by following certain laws which are never compromised. Now, we don't know that force yet but we are trying to unravel its secrets by trying to unify four fundamental forces of nature by a single mathematical equation.
As soon as some religious claims come in the form of scripture, defining about same controlling force, most of the so called unbiased scientist lose their temper and start prejudicing the same. Whereas, in my humble opinion a real scientist would take these claims at least as an hypothesis and would try to prove or reject them axiomatically. Even the meta physics statements are no exception since if we delve in human history, we will find so many meta physical believes then, now are proved scientific facts.

 

I'm really not understanding your point about Pi. The concept of Pi is quite basic and does not rely on a perfect physical model for the concept to be discovered and examined!

It all depends on your definitions of "discovered" and "invented". I agree with you in the sense that I think we can discover and examine pi within the context of our invented mathematical landscape, and in fact, historically we have done that. However, this landscape is not reality. In this discussion, we are bound by the OPs definition of "discovered". He said ...

There are some fixed mathematical relations in nature which exist on their own such as ratio of circumference over diameter, ... These 'natural' relations constitute part of discovered mathematics. ... I believe the part which includes natural relations (such as ratio of circumference and diameter) is universal and will be true and applicable everywhere.

He is clearly defining "discovered" to refer to that which is real and exists as truth in our physical world "everywhere", not an abstract world that we created.

Are you really saying you think Pi is "invented"? If so how do you justify that?

Yes, that is the position I'm arguing in this thread, and I provided my justification for this above. But, I can reword it a little because it is a subtle point.

In order to discover pi in the mathematical sense (not the real sense), you first have to develop a mathematical framework (what I figuratively called "landscape"). In modern times we call this framework Euclidian geometry, this geometry relies on primitive notions of a point, line and plane which are not clearly definable, but we accept. There are then 5 important postulates put in place to make a logical framework. The 5'th postulate amounts to the statement that you can have parallel lines that never intersect, and this postulate has been the object of study and debate for centuries. Eventually mathematicians, like Gauss and Riemann, discovered that this 5'th postulate could be removed and still allow a logical framework. Einstein made use of a modified form of this Riemannian geometery to develop his General Relativity. The end result is that we now know that we do not live in a Euclidean universe, and any two lines in a plane (even parallel lines) will always intersect in this more general geometry.

So, my argument is simple. We invented this concept of Euclidean geometery. Why? Because it is so very close to what we experience that we intuitively feel that it is right and true and an exact representation of reality, even though it is not any of that. The concept of pi as a ratio of circumference of a circle to diamter exists only in this invented framework. So, can we invent the Euclidean framework abstactly and then discover things (like pi) within that framework? Yes, i think so. However, this does not fit PG's definition of discovery.

This is more than just the issue that we can't do measurements to prove pi is the ratio of circumference to diameter. It is much more. We have actually scientifically proved that this notion of Euclidean geometry (and hence all that follows from it) is false reality. It's not a limit of measurement precision, but an accepted law of nature that contradicts what the simple math is saying.

 

I think math is discovered as well as invented. There are some fixed mathematical relations in nature which exist on their own ...

Since no one else has bothered to say it, I'll say it.

The mathematical notion of "one" or "unity" is perhaps the simplest example to consider. You could also claim that the simplest idea in physics is the idea of the existence of a "universe". If we group everything that we know and everything that exists, or has ever existed, or will ever exist, into one thing, and call it "the universe", perhaps we have an example of a mathematical concept that is discovered in the sense you mean.

One "universe" exists, which is easy to prove because if you find anything else, you just throw it back into the definition of universe. The idea, and the ideal, of unity, oneness and completness describes that reality.

 

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In order to discover pi in the mathematical sense (not the real sense), you first have to develop a mathematical framework (what I figuratively called "landscape"). In modern times we call this framework Euclidian geometry,
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If I understand your point correctly you claim that it cannot be "discovered" as the "discovery" requires an invented framework?

I don't see that as true. The island of Hawaii was discovered, even though it required an invented framework of ocean going boats. The moons around planets were discovered, although needing the invented technology framework of telescopes.

I see a couple of fundamental characteristics of the concept "discovered";
1. The Alien rule; If "discovered" by two different researchers from different backgrounds etc the discovery would be the same. Like the "discovery" that water could be separated into hydrogen and oxygen gas.
2. The Existed rule; If something is real and exists, it will likely be discovered. A circle is a real thing, as are characteristics like diameter and circumference. So are the planets' moons and H20, and volts and amps, etc. They can be demonstrated and examined as real things and the math relationship of the real things can be compared to a purely theoretical concept.

So to answer your point I don't see the requirement of science or technology as negating the concept of "discovered".

 

If I understand your point correctly you claim that it cannot be "discovered" as the "discovery" requires an invented framework?

Not at all. I'm saying that it is not discovered if we use the OP's definition of what discovered means. It's important not to argue about semantics. If the OP asks a question, then let's understand what he is asking and try to stick to that.

As I said above, "I agree with you in the sense that I think we can discover and examine pi within the context of our invented mathematical landscape, and in fact, historically we have done that."

I don't see that as true. The island of Hawaii was discovered, even though it required an invented framework of ocean going boats. The moons around planets were discovered, although needing the invented technology framework of telescopes.

I see a couple of fundamental characteristics of the concept "discovered";
1. The Alien rule; If "discovered" by two different researchers from different backgrounds etc the discovery would be the same. Like the "discovery" that water could be separated into hydrogen and oxygen gas.
2. The Existed rule; If something is real and exists, it will likely be discovered. A circle is a real thing, as are characteristics like diameter and circumference. So are the planets' moons and H20, and volts and amps, etc. They can be demonstrated and examined as real things and the math relationship of the real things can be compared to a purely theoretical concept.

So to answer your point I don't see the requirement of science or technology as negating the concept of "discovered".

That's fine, but as I said you are using a different definition of "discovery" than the OP. I feel bound by the OP's established constraints when debating in this thread. Outside this thread I might agree with many of your points. Even within the constraints, I like some of your arguments, but I disagree that the example of pi meets the OP's definition of math that is discovered.

He used the example of pi as a ratio of circumference to diameter of a circle and claimed that this ratio called pi would be valid everywhere. Yet, it is not generally valid in our world which rules out ALL of everywhere, and only leaves the abstract place of Euclidean space without time, as a "where" where the thing exists and could be discovered in his sense of the word.

So your examples of using tools to discover something is fine under the constraint of the OP's definition of "discover". Things like Hawaii and moons are real, while pi is not real but an abstract construct. Even the ideal circle as defined by Euclidean geometry is not real as you claim. Approximate circles exist in the real world, but perfect circles do not exist in the real world. In fact, the concept of a circle is one of the 5 postulates of Euclidean geometry I mentioned above, so it is not even discovered in the abstract world of Euclidean geometry, but it is postulated. So approximate circles were discovered in the real world. They were also invented (the wheel comes to mind). The essential ingredients of "circle" were then abstracted in the human mind and then used as a postulate to invent the framework of Euclidean geometry.

As I said above, mathematics is a great tool of discovery, which compares well with your boats and telescopes, but the things that the tool helps you discover are not themselves mathematical concepts if we use the OP's definition of discover. For example, when Maxwell used mathematics to modify Amperes law and then derive the wave equation to discover that light is electromagnetic waves, he wasn't discovering an abstract feature of math, but he was discovering a true feature of reality.

 

Thank you very much, Steve, The RB. I really appreciate your help, and through your productive discussions I have learned how mathematics really maps the world around us.

Regards
PG

 

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He used the example of pi as a ratio of circumference to diameter of a circle and claimed that this ratio called pi would be valid everywhere. Yet, it is not generally valid in our world which rules out ALL of everywhere, and only leaves the abstract place of Euclidean space without time, as a "where" where the thing exists and could be discovered in his sense of the word.
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It sounds to me like you used a lot of words to basically re-affirm your earlier point that Pi is not real or "valid in the real world" because there "is no such thing as a perfect circle in the real world". (I apologise if I have paraphrased you in such a way as to misquote you.)

I don't think the concept of Pi requires perfection in any way, or can be excluded from the real world due to imperfection.

You can make any number of real world circle models, internal, external, averaged etc and measure the relationship and circumference to diameter, and although all real world models will have some small error the RELATIONSHIP that is Pi will be quite apparent as a real thing, a ratio relationship that exists between all real diameters and real circumferences.

So say this Pi exists ONLY in "Euclidian space" is a poor argument. The relationship (as provable to a couple of decimal places) clearly exists even in the simplest real world measurable models that can be examined by emerging civilistions, and that relationship is a "real thing" that will be discovered as more or less the *same value* by EVERYONE, allowing for the precision inherent in whatever models they possess.

Pi is real, and is entirely "discovered" and the limitation of technology simply limits the precision of the value discovered, to the abilities of the particular technology used.

 

Hi The RB

You and Steve are both full of ideas and I appreciate it. I think we are now going around in circles. So, I think it's better to start doing something else. You can help me with queries in some other threads, if you like to. I would be thankful. But let's be done with it. Many thanks.

Regards
PG

 

I agree with PG, that we are going around in circles. I don't expect anyone to agree with me, and there is no point in me repeating the same points over and over. However, it's interesting that the most important points I made are not addressed at all.

So, anyone who wants to understand the subtleties in what I'm saying can go back and reread my posts and note all the points. Since this is an unprovable thing, all any of us can do is make a series of plausibility arguments. In the end, one either finds value in those arguments, or one doesn't. I entered this discussion saying "I'll be a devil". Hopefully, the reference is clear. Arguing as a "Devil's Advocate" is more useful to the OP than just saying, "hey, sounds good to me!".