(I need to simplify my last post - too complex)
Pi is heavily used in all geometries. You may want to google around to find the formula for areas of a triangles on a sphere. Even simple items like surface area of a sphere use Pi (Area =4(Pi)(R squared) ).
Einstein's equation for space-time curvature uses Pi also.
So - All Geometry uses Pi.
Irrational numbers exist and without them the Real numbers are incomplete. i.e. the axioms of Real Numbers will be violated. The vast majority of real numbers are irrational. (Cantor proved this)
In any case. Pi is a constant without flat space. It has several definitions, all give the same number, and also shows up outside of Geometry. examples can be given if asked. (i.e. Fourier transforms)
I am not sure of your use of Fractals. So I'm confused there.
I agree that Math expands as we "discover" new things. However - Math remains the same e.g. The Pythagorean Formula. It is originally defined on flat space, However if you look at other parts of Math it will show up in strange places.
In any case -
We all get to believe on either side of the issue.
I appreciate your comments. I need to think about your fractals. Seems I have yet to Grok it :)
Thanks, John. Your opinion is well respected. However, I have a different notion on somethings you have raised. Pi is constant only on a flat surface. Like square roots, whose decimal expansion goes on forever, they are irrational constants. As I understand this, the word irrational states, "incomprehensibility", maybe because of the absence of an exact answer. Notwithstanding, there will always be variants/variables that underlies a perceived knowledge. In Fractals, wikipedia has this say:
The word "fractal" often has different connotations for laypeople than mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception. The mathematical concept is difficult to formally define even for mathematicians, but key features can be understood with little mathematical background.
The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in the infinite regress in parallel mirrors or the homunculus, the little man inside the head of the little man inside the head...).
The fundamental way and mechanism to calculate things may be the same in a finite or fixed sample set and approximation. In an infinite world, our present knowledge is still elementary. Math, for me, will change and evolve with our consciousness. I prefer to subscribe to Einstein's mantra: " Not everything that can be counted counts and not everything that counts can be counted."
there are two groups in Math - one is Platonists (Math is real - unchangeable), the other are in the view that it is created by humans. Of course, some are undecided.
I am a Platonist
I fail to see the Human-only viewpoint as very valid. The above book (kid used in High School) was wrong. Books at different levels will take positions that makes the subject easier.
Part of the problem is that the exponent is a short-hand. e.g. y times y times y is y cubed, yyy. if one writes yyy all the time the formulae get unbearable.
It works because y squared times y cubed is y to the 5th power (power is 2+3).
at that point, exponents with fractions, negative numbers, negative fractions, even irrationals can be defined rigorously. Indeed the notation can be expanded (and has to be) using the axiomatic rules of the real number system. There is essentially only one way to do it as well.
This shorthand then becomes incredible useful and powerful to expressing these higher concepts.
Mathematics deals with things that are not defined "only" by humans. A circle, square etc exist without us. Pi is a constant . it is the same constant anywhere you go.
Does a casino have different odds if it is moved to another planet?
The way and mechanism to calculate things may look different, but it is fundamentally the same.
So that is how a Platonist thinks. There is no choice in Math. Group Theory examines symmetries, and those symmetries will not change regarding where you are, or who can recognize them.
Math NOT being real is rather ego-centric by a human.
The people on both sides can get their own way, After all -- the math does not change so there is no way to test the mathematical philosophy.
That which does not change is Real and True. That is only my opinion <G>
Thanks for the video. It sorts of validates Robert Lanza's, " Biocentrism theory", which I think is also a very good topic coz it claims to be , "The Theory of Everything."
Oops forgot to post the video.
Thanks for sharing that Rolly, the comment at the end there is true, whether it's Science, Math, Spirituality or even History we have to have prudence in matters of where we are gaining knowledge. They say that unlike most sciences, math lacks an empirical component, where you can't see math happening.. I watch this video a couple of days ago that goes into that theory, it's pretty entertaining and poses some really good questions, Like do people Discover math? Or do people create it? Almost leaves you wondering too, do people discover God or the Divine or do we create it?
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