The question was posed to me:
Is pi – the ratio between the circles circumference and its diameter – a constant approximating 3.14159? How can we be certain that math is ever truly “right”?
As it just so happens, we can be certain that it isn’t. Most of math is in fact wrong.
Everything from graph theory, shapes, areas… the very definition of pi itself, which have significant basis on all of mathematics… all of it and everything that sits on top of it… is based on Euclidean geometry.
Euclid, and most people until Einstein, all believed that space was flat. Hence the notion of Euclidean space. Most of mathematics is based on the assumption. A line is the shortest distance between two points… in Euclidean space. Pi is the ratio between a circles circumference and its diameter… in Euclidean Space.
Einstein showed that space was curved.
Either you believe that Einstein proved space was curved, conclusively, and space really is curved, in which case most all of mathematics rests on false presumption. Pi, and all other geometrical arguments and designs, have different conclusions due to their application in a space they cannot be applied in. We have assigned a false value to pi for the space we apply it in.
Or, even if space is still flat and Euclidean, and Einstein was wrong, Einstein still showed curvature in the very mathematics he used… in which case, since Einsteins proof holds to mathematical scrutiny, we would be able to deduce that mathematics is self-inconsistent.
Either way, the conclusion is the same: math is wrong.
And if I’m wrong about this logical argument/observation then logic itself might very well be wrong as well, which only serves my argument on yet another count. Since the foundations of mathematics, of proof, and of our perceptions about reality are all subject to logical scrutiny and everything from mathematical rigor to empirical science to philosophy has fundamental roots in logic. What happens to it all if logic itself is wrong?
Most all of math is proven, inasmuch as anything mathematical can be proven. Each proof for each theorem is based on premises which comprise of theorems, which are themselves proven in turn, and axioms which are not – one so-called “truth” leading into the next subsequent one – as is the nature of the accumulation of all of human knowledge.
Axioms, and “Principles”, are where it all falls apart. They are taken on first principle – as “intuitive knowledge” – which is exempt from being proven and which is based on human perception and cognition. Human imagination. They are “proven” by demonstration and observation – of a single, or of few, yet limited account of this so-called “truth”. Human perception and cognition is limited, restricted and compromised by the same reality we purport to perceive and reason about.
In the case of Euclidean geometry and Einsteinian relativity, we have a contradiction – a result which contradicts the fundamental precepts it is based on – proving that it all is garbage to begin with. It is a rare occurrence, as even fallacious reasoning can be self-consistent. And we all “know” that self-consistency is not proof of validity while contradiction is proof of invalidity.
Empirical proof of the fallacies of human reasoning which extend to the fundamental precepts of all of our tools with which we reason is outrageously unheard of. The advent of quantum physics, and the nature of the quantum world, to demonstrate observable self-contradiction hasn’t forced us to question our notions of logic much. We make an arbitrary exception for it, saying that it makes sense only on a quantum level, pursuing the futile endeavor to logically understand something so far outside our logical capacity, which contradicts the very tools of reason we utilize to reason about it. Instead of seeing the quantum as proof of the invalidity of our own rules, we see it instead as an exception, with some arbitrary and magical threshold barring one world from another.
Let me not even touch on the shortcomings of logic at resolving pretty much all human controversy. One can only wonder about logic. We hold it in such high esteem… but then only on paper and only in pre-constructed realities and word-problems we designed ourselves. In the real world, logic fails to hold up to people, to statistics, to the quantum, to the real world phenomenon.
Is quantum uncertainty proof of randomness? Or is it proof of the limitations of science – not just our modern science now, but science forever? Could there be a deterministic explanation for quantum uncertainty which exists beyond all empirical measurements, beyond sciences capacity to detect?
The quantum world is one example where self-contradiction is permitted – albeit with a sense of curiosity and awe – but permitted nonetheless. We question the reality we perceive, but not the precepts of science which made those observations, nor do we question the perceptions we make with our senses, nor do we question the validity of “our” notions of logic or reason with which we reason about those observations. We presume that a healthily functioning brain accurately reasons about the world it perceives. We say the quantum world is uncertain, but do we know where the uncertainty really is?
Of course I say all of this with confidence in my own notion of logic. Am I right? Does this post hold up to logical scrutiny? Does it fail? Does that even matter? Would a valid argument necessarily prove my case if my case about logic is right? Would a logical counter-argument necessarily “prove” my case wrong if my case about logic is right?
I have always wondered if, given the right position, the right angle, the right perception, if all of reality could make sense. I call it a “God logic” or a “divine perspective”. Could there be a place, inside our universe or outside, in which a perception on our world made perfect predictable sense?
What I mean is this: We make rules for the Newtonian and for Einsteinian relativity, and for the quantum. Different rules for different worlds. There may be a transition. It may be predictable. But is there a perspective anywhere in conceivable existence, not barring the divine, in which all the rules are the same? Where there is no contradiction, no inconsistency, no unpredictability?
We claim the the quantum is uncertain, but could it be the Newtonian that is? Could it be that, if we knew the mechanisms of the quantum, it would be far more predictable and deterministic than even the Newtonian? And if so, the predictability and comprehension of the Newtonian is nothing more than a consequence of our intellectual and biological evolution within a Newtonian environment.
We only live in one small corner of the universe. I wonder if, in the neighboring galaxies or star systems, if pi is the same as it is here. If the speed of light is. If the gravitational constant is. If Euler’s number is. Is that an absurd inquiry? I don’t see why. Unless we have been everywhere, and have proven it everywhere, we cannot say with certainty. The empirical evidence we used was gathered here. The logic we used was reasoned here, evolved in our brains here. And all observations we perceive here are subject to the space in which we live, affected by its influences prior to our perceiving it. And after.
Science has a lot to say, I’m sure, about the homogeneity of our universe. But how do we really know? I don’t know. It seems to me that we cannot truly be certain. Space is curved. That much we know. The value of the constant pi is affected by the type of non-Euclidean space a circle is in, and thus by the degree of curvature. Given the right cosmic circumstance, why must pi necessarily be a constant?
We know from the quantum and the relativity worlds that the rules are not always applicable in the same ways. Could variations in these constants help to account for the universe? Many physicists already believe so, citing the first moments of the big bang in which the constants were different or otherwise variable. What is it about the universe we exist in that necessitates their constancy, and more significantly their universality.
If the constants are indeed variable then all of theoretical science is… all the more in doubt than its ever been. The precepts we hold to the very reality we live in and perceive ought be doubted as well.