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Entrance : Philosophy as the origin of all science
Parmenides |
It considered that the interview that been made 450 B.C between Socrates & the famous philosopher Parmenides is one of the greatest interviews ever made in the ancient history, Parmenides proposed in this interview some ideas that changed the history and was an entrance for an argues that didn’t finished till present days, the idea of this proposal was that the universe is a single union that can’t be parted, in another word, space and time (and those also movement itself) is just un illusion! |
The coincidence of this proposal was a great amount of argues that can’t be mentioned here, but the actual reason that made this proposal alive till current day, was a different philosopher , and he was the Greek philosopher Zeno, he presented in the mentioned interview, this owner of an extremely powerful and abstract view of the world, proposed 8 logical paradoxes (they was reduced lately to only 3 due to it similarity) as a proof for Parmenides proposal, they confirmed that space, time and movement has no real existence.
The contradiction of this result with our real life experience from a side, and the power and deepness of the logic behind Zeno paradoxes, despite of their simplicity, was the reason for a lot of philosophical argues and even a scientific one later, because nobody could provide a really complete explanation to this magic paradoxes.
The Paradoxes
1- Spatial Paradoxes
a. Two moving objects, never will be able to meet (the paradox of the runner and the tortoise)
A race happened once between a runner and a tortoise , and because the runner was very sure of his winning, he allowed the tortoise, to start in a some distance in front of himself, in some point of time, the race is began, in a some latter point of time , the racer will reach the place that the tortoise started the race from , but till that point of time, the tortoise will be covered some distance too, and so on, and this mean that the runner will never become ahead the tortoise! Whatever he tries!
b. You can’t ever reach any place! (The paradox of the apple or Dichotomy)
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If we hold an apple, and twisted our hand down, then we released it, the apple will start falling down, but before the apple reaches the ground surface, it should cover half distance between your hand and the floor, after that it will need to cover half of the rest of the distance again, and so on, thus the apple will need always to cover some distance as following :
1\2, 1\4, 1\8 ......
And as it’s clear, this distance will never become zero, because there is always some “half of a half of …” that needs to be covered, this mean that the apple will never reach the ground, even more, it never can even start to fall! |
2-Time Paradox (the arrow paradox)
Let us suppose that there is some flying arrow, in each point of time; the arrow will occupy some space in the air, so the question is when this arrow will move? If the arrow will move, it needs to change it position in the space during time, but it’s fixed in each point of the time, so the arrow not moving at all!
As it is clear, Zeno depends on dividing the object (space & time) to infinitely small parts, and because those parts are very tiny, movement in those parts are impossible, in the same time, because the amount of this parts are infinity, summarizing all this parts in order to reproduce space or distance is nonsense, those both space & time are out of the meaning.
Zeno Paradoxes through History
Summarizing 2500 years of history full with argues and different points of view is difficult a task, so I will try to mention only the break through ideas that proposed to solve those paradoxes.
1- Philosophically:
As I already mentioned, Zeno paradoxes produced a lot of hot argues between philosophers, even some new ideologies based on this paradoxes appeared, because nobody could provide a satisfactory explanation to the nature of this paradoxes.
The reaction of most philosophers was one of two:
1-Denying the problem about the paradoxes by not admitting these paradoxes or that they have a real meaning or that they treating our space – time understandings.
2-or that this paradoxes actually very simple and clear to solve them.
But some philosopher that owned a higher level of abstraction realized that there is something missed and that these paradoxes really deserve to be think about! Thus none of the philosophers could ever put a widely accepted solution during all this ages, and that because the solution as it will be figured later, was deeper & smarter than what they ever thought!
2- Physically:
Taking in account that those paradoxes are related directly to the space & time, which are the core concepts that physics studies, a lot of physicians supposed that this paradoxes are basically a physical paradoxes, and they took on their behalf to solve them, thus, and because this paradoxes blowing up all foundation of the physics, most physicians was bypassing to argue on them.
With the beginning of 20th century, physics was rapidly developing in the small scale world, and secrets of the atom and nuclear become more and more accessible by developing foundation of the quantum physics & theory of relativity.
Max Plank |
One of the basic ideas that quantum physics built on, was the idea the proposed by the famous scientist, Max Plank, and it stated that, energy flow is not continues! This mean that it always produced in separate parts, (each part called "Quant") and that will be despite of the energy source, but this quants are extremely small (actually, they are proportional to the object mass, so the quant of the our everyday life objects will be very small and almost impossible to be detected using current technology, but for the very tiny objects like an electron, this quants become quit detectable , and of course this idea had been approved by the experiments) |
Werner Heisenberg |
The probabilistic nature of the quantum theory that raised due to the dual Material-Wave nature of the objects (and it means that everything in the universe behaves as a material and wave in the same time!) was always a very rich source for scientists imagination and also a source for objections (at the beginning of theory life) lately, and as a prime result of Schrödinger famous equation in the quantum physics, the also very known scientist, Werner Heisenberg, concluded his famous Uncertainty principle that been named with his name. |
This principle itself raised a huge amount of philosophical argues regarding the nature of the universe, and it states that it is impossible to measure different related properties of the elementary particles and atoms in the same time infinitely accurate, only till a specified accuracy is allowed (and it is also proportional to the mass of the measured object) and that despite of how much our measure devices and technology are advanced and complicated!
As example, we can’t measure the position & speed of the electron intently accurately at the same time, we can measure only on of this two properties accurately! If we measured it speed, we will be not able to know exactly it location in the same time!
This results of Heisenberg uncertainty principle are also applies on anything measurable, speed, mass, time, and even electron diameter and much more can’t be measured absolutely accurately!
Some scientist supposed that these results combined with some additional results of theory of relativity can explain Zeno paradoxes, but for most, this explanation was not satisfactory.
Later, new extensions of quantum physics began to rise, especially what is called "Quantum Field theory", and which proposed a much farther idea of the one Plank gave, it considered that not only energy is constructed of quants, but also space and time are! This means that time and space are not continuous (but this pieces are ultra small , about 10^-32 of the meter ! so it is impossible to measure it!) and here again a new sounds raised calming that this theory explains Zeno paradoxes even better as it restricts dividing space and time to intently small parts as Zeno did.
Unfortunately, and despite that Zeno paradoxes seems to be solved Physically, this scientist missed a very important point: the problems that this paradoxes proposed are much deeper than what they expected, and that they not only affect physical reality, but the real problem is in the abstract mathematics! because denying the existence of space denying existence of the Geometry at whole! Also it denying existence of the axis’s of real numbers for example, thus, the physics became itself nonsense as it depends on this concepts as a basics!
3- Mathematically:
Once a rule of the sum of the geometric progression had been found, it was supposed by some scientist that at least Spatial paradoxes had been solved, because actually it is very easy to get the sum of the progression we get from apple’s free fall using that rule:
And this obviously means that, the sum of all pieces that the apple should go though before striking the floor is 1! And that despite that there is an intently many pieces to be taken in count, even so the result will be a finite number which is 1!
Actually, the proof of this sum is very easy, but the problem that it is not showing that this result is really true ! because there is some progressions that can’t have a sum ! for instance :
Because by changing the way of grouping this ones and calculating there sum we can get different answers!
This was the beginning of the progression Algebra and Calculus, and despite that the beginning of this branches of math was by Newton and Libenz in the 18 century, this aspects was depending totally on the progressions as the above mentioned, plus a new obscure concepts like infinitely large and small, and despite that it been found an enormous applications for it, the concepts and basics of this theory wasn’t quit strict and powerful from a math point of view!
Lewis Cauchy |
Later, in the 19th century, other scientists started to rebuild it basis independently, and what called "Weierstrass-Cauchy" theorem appeared and it was about converge of infinite progressions, it showed for example that summarizing of the above mentioned Zeno pieces is indeed right, and that it has really a finite sum.
But lately again this theorem had been criticized, then mathematicians modified it do become as it known "Epsilon-Delta" theorem, and you can find this theorem now in any school book, and which supposed to be a very strong fundamental basis for Calculus and progressions, and by this it had been announced that Calculus reached the "perfection" stage only after a long time of it using in the practical life. |
Even so, infinitely small & infinitely large values , and despite that it had by this time a very strict definitions, this concepts stayed so ambiguous, very strange, Hard to imagine, and it always seemed as it is out of the general Math stream, and most Mathematicians and even Physicians tried to not discuss this concepts because they was always leading to a strange results that not obviously fits real life (for example that the sum of infinity with any number stays the infinity itself !), and it was the same situation that Physician had in other concepts such as the meaning of the mass, cause this concepts are always needed a higher degree of awareness to be realized, and it will not be overstate that mental health of some scientist become under suspension when they tried to do some research in this fields (in the same way that been in medieval, even in our modern world!) and some of them become psychosis due to reasons related directly to their researches!
This status of confusion and ambiguity made some mathematician and physicians with recognition at a glance suspicious in the validity, strength and abstraction level of Calculus fundaments and if Epsilon-Delta theorem is really enough as a basis for it!
In the mean time, another branch of Mathematics called "Set Theory" was intensely developing, and this Brach is the deepest & the most abstracted branch of mathematics, this theory had proposed a very strong basis for the whole math even for the abstract concept such a number itself ! And for the first time it had been proved that 1+1=2!! (By the way, these proof not a small or an easy one!) By using a much deeper and more abstract concept than a number, this made math even more close to the philosophy itself!
Later, a more powerful version of this theory had been proposed and it called "Axiomatic Set Theory" , and which is differs from the first one, that is depends in whole on only 5 basic axioms starting from which you can build all our current knowledge about math, this axioms called ZFC and it considered as a fundaments of math till current day (by the way, most people suppose wrongly that axioms are simple because it should be very clear and acceptable without a proof, but if you will take a look on ZFC axioms, you will see that they are far away from being simple !)even though this axioms putted the basics of all math, and it was totally compatible with Epsilon-Delta theorem, Infinitely small and big concepts stayed out of this axioms possibilities to explained them !
Abraham Robinson |
This enigma stayed in this ambiguous stage till 1966, when and after about 2500 years of Zeno paradoxes birth, and only after about 250 years from Calculus birth, a new genus mathematician : "Abraham Robinson" dared finally to dig in this dark world of infinitely small structures, and he putted the basics of totally new branch in mathematics called now Non-Standard Analysis by proposing a new revolutionary ideas that provided and for the first time a direct solution for the infinitely small & large numbers , and lately this concepts been used to solve finally Zeno paradoxes (as the main goal for the theorem wasn’t Zeno), and I will talk about this new theorem in the next Article (The Smallest thing in Universe, is so small that you even can't discover it!). |
References from Internet:
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