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Zeno's Paradox, Aporia: Achilles Can't Outrun the Tortoise?!
The First-Ever Metaphysical Solution to Zeno's Paradoxes

Truth is no longer, if just one of its elements is amputated
By Ion Saliu, Anti-Sophist At-Large

Zeno or Zenon Sophist is famous for his aporias or riddles, paradoxes, puzzles.

  • The final version published in October 2005; first captured by the WayBack Machine ( on October 13, 2005.

    Zenon made an extremely attractive art to prove that motion was not possible. He came up with a riddle that used Achilles as the main character. We know that the legendary hero was also known as the Quick-Foot Achilles, or Speedy Achilles. Really? Zeno questioned the speed of Achilles. I, Zeno, will prove that your speedy Achilles will never beat a tortoise in a running race!

    Okay, your Achilles is faster than a tortoise, admitted Zeno. Therefore we must necessarily give the tortoise a long start. But here is the kicker, puzzled Zeno his audience. Achilles does reach the starting point of the tortoise. But when Achilles reaches that space, the tortoise would reach a farther position in space. Let's name it position 2. When Achilles reaches position 2, the tortoise would have reached a new position; let's name it position 3. When Achilles reaches position 3, the tortoise would have reached a new position; let's name it position 4. When Achilles reaches position 4, the tortoise would have reached a new position; let's name it position 5. And so on ad infinitum. Achilles can never pass the tortoise...

    Many thinkers have been impressed by that logical construct. We are talking here over two thousand years and millions of admirers of a paradox! Is it really a paradox? Is it really an impeccably logical argument that leads to a false conclusion? NOT!

    The argument is NOT logical. Logic assumes that ALL the elements of a system are present. Leaving out just one element makes the system not only ILLOGICAL, but, worse, ABSURD. Absurdity is the opposite of logic.

    Motion assumes two and exactly two elements: Space and Time. Motion is represented relationally by Speed. Speed equals Space over Time (v = s / t). Zeno eliminates one of the sine-qua-non elements: Time. Zeno equation of Speed (or velocity, hence v) leads to the absurdity: v = s / s. Metaphorically speaking, Zenon amputates one of Achilles' legs! Amputate one of anybody's legs — and the person would fall down motionless! It is not a paradox!

    Zeno's paradox takes into account Space only. Time is completely ignored. Achilles will cover one unit of space in less time than the tortoise. Equivalently, Achilles will cover a longer distance than the tortoise in the same time. By the time Achilles reaches the starting point of the tortoise, the tortoise would have moved a shorter distance. The distance could be so short that Achilles could surpass it in a very short time. Generalizing, the gap gets shorter and shorter as the time progresses. At some point in time — NOT point in space — the gap reverses. The faster competitor surpasses the slower competitor who had an early start.

    It's a no-brainer! It all depends on the speeds of the two competitors. At equal speeds, the competitor with an early start will be always in front, keeping the same gap between her and the competition. It is not the case here. Zeno admits that Achilles is faster than the tortoise. Let's say Achilles is 10 times faster (in reality, a human is much faster than that compared to a tortoise). Let's say the tortoise has a ten-meter early start. The tortoise is 10 meters ahead of Achilles. Achilles walks, instead of running. He walks ten meters in ten seconds. When Achilles reaches the starting point of the tortoise, the Sophist's favorite will be exactly one meter ahead. It took 10 seconds to reach the new point in space. In the next 10 seconds, Achilles will be 8 meters ahead of the tortoise. The tortoise is no longer ahead. Motion does not stop at so-called “points in space”. Motion is fluid in Space AND Time.

    What Zeno's paradox considers new position is a chimera. Such new position is an absurdity because it can't be determined by considering the space only. A new position in space can be determined only IF the speed is considered; the speed can be determined only if both Space and Time are known. The Achilles and the tortoise puzzle relies on a chimera. In order to move, you must reach a new position regardless of speed! says that highly paid tricky guy Zeno of Elea! There is no such a thing as a new position IF excluding the Time!

    • • There is a similar Zeno riddle: The arrow and the bow paradox. Zeno had to create this new one because he realized the fundamental flaw of Achilles and the tortoise. Probably many contemporaries posed Zeno with the inherent absurdity of his riddle. Zenon himself admits that Achilles is faster than the tortoise. Therefore, the tortoise must be placed ahead of Achilles. By admitting that an element is faster than the other, the implication of motion is mandatory!

    The arrow will never leave the bow, because it must surpass first half the distance between the bow and the target, reckons Zenon of Elea. Before that, the arrow must surpass half of the half the distance. Further backwards, half of the half of the distance; and so on ad infinitum.

    Again, the Time is completely eliminated from the equation. The arrow will reach the half of the half of the half…of the distance to the target in half of the half of the half…of the time, approximately. There is another important parameter regarding motion: Acceleration. There was an excuse for the Ancient Greeks: At that time, calculus was unknown. Calculus makes clear such ideas as acceleration, or approximating Infinity, or approximating Nothingness (Zero). Nevertheless, the arrow and the bow aporia can be solved simply by considering both Space and Time in representing Motion.

    Where is the logical brilliance that has impressed millions of humans for hundreds of years? Zeno says that Speed is just a matter of Space! And the audience applauds in ecstasy! No, fools, Socrates said. Speed is a matter of Space AND Time! Look at the truth in its fullness. Don't discard some elements as a convenient means to avoid logical difficulty.

    Socrates started his philosophical career as his stated mission to expose the Sophists. As brilliant as they seemed, Socrates considered that the Sophists corrupted reasoning. Unluckily for Socrates, the Sophists represented the establishment. They became the first educators-for-hire. They charged very high fees. Only the rich and powerful were able to hire Sophists as tutors. By attacking the Sophists, Socrates collided with a rich and powerful segment of the Athenian society. It was one of the major charges against Socrates at his trial.

    Zeno's aporia resembles also a magic show. The magician wants the audience to watch those beautiful women, rather than stepping up on stage and checking that mysterious box. Otherwise, the magician would cut in two the beautiful woman in the box. Then, the magician would put back together the woman! That's so because the magician eliminates important elements from view!

    Impeccably logical? Not at all. It's impeccably tricky! It is not an aporia, it is not a paradox.

    If only Space exists, but no Time, Motion does not exist. It would be all frozen. Humans understand it more easily. Space is more intuitive than Time. What if only Time existed, but no Space? That would be harder to represent mentally. I was in an earthquake (Romania, 1977). To me, the most intuitive representation of lack of Space is an earthquake. I had the strong impression that the quake lasted forever. There was only Time present. There was no Space. I had the impression that there was nowhere to go. I felt Time was infinite, while Space was zero. An Achilles versus the Tortoise aporia seems impossible to construct IF Space is left out while only Time is considered.

    Mathematics always comes to the rescue. Put a construct in mathematical expression — and see if it's logical or illogical. Like in the case of the absurd relation v = s/s; instead of the correct formula v = s/t.
    It is not that Mathematics creates Reality. Mathematics, and Logic, and Philosophy are creations of the mind. They represent attempts of the humans to understand Reality — to find the Truth. In the case of the aporiae, humans attempt to create reality — their personal reality. It is absurd right from the start. Zeno knows that objects move. In fact, he admits that Achilles is faster than the tortoise. Thus he starts with a premise based in reality. But, then, he attempts to create a new reality where the motion is impossible. And, finally, he forces upon us the idea that his reality is the true reality! If we follow his distortion, Zeno himself did not exist. Motion was impossible, therefore Zeno's parents could not have moved close enough to each other to create Zeno!

    Logic outside the Reality — the only Reality — is just fiction, or religion, or magic show. Mathematics is the most objective instrument of verifying if mental constructs comply with Reality.

    • • • Summary of solving Zeno's (Zenon) aporias: The paradox of Achilles and the tortoise
    We demonstrated that an aporia is an absurdity. That is the end of an aporia. An absurdity is absurd; therefore no argument is necessary. The fanatics of aporiae, however, insist on giving a solution in the same field of knowledge. Zenon The Eleatic's aporiae are metaphysical in nature; therefore they require a metaphysical solution. Others consider Zeno's paradoxes to be logic in nature; therefore they require logical arguing. The only solutions - up until the writing of this very page - were mathematical. But instead of applying the most intuitive form of mathematics, such solutions resorted to most difficult math: Mathematics of Infinity.

    1. The philosophical or metaphysical solution

    Motion consists of exactly two and always two elements: Space AND Time. The objects Move from Here to There in various lengths of Time. Zenon of Elea only considers Space as defining motion. Zenon moves from here to there in NO time! Because there is NO time, Zenon's false reasoning eliminates the possibility of motion. That would be correct only and only if Mind could recreate reality. If a human, or mind, or any other object could recreate a "reality" where Time does not exist but only Space, Motion would not exist.

    No single object, no single force can create or recreate reality. Reality is the result of random interaction of forces. Mind or reason cannot create reality; it is the other way around. Mind is part of reality alright; reality comes first, mind comes afterwards. The Sophists reacted to Heraclitus' ideas of change. Change meant beginning and ending, therefore death. The Sophists served the needs of the religion of their time - and all other times (eras). Humans don't want to die - period. Let us live forever by finding arguments proving no-change, therefore eternity, therefore immortality. Such metaphysics or philosophy wins in most popularity contests anywhere on Terra (or anywhere in the Universe, for that matter).

    2. The logical solution

    When you argue in one field of knowledge, you always stay in the respective field of arguing. That's one of the fundamental requirements of logic. The Sophists had bona-fide supporters in the modern era (post Renaissance). Moving the argument to mathematics did violate the Same-Field requirement of Logic. Also, moving the arguing to Plane Reality seemed to violate the Same-Field requirement of Logic. They did it first in the ancient times. Okay, Zeno, some said. Just put your tortoise ahead of me. I don't need the Quick-Feet Achilles run for me. I'll run and beat your tortoise myself - by an incredible margin of victory, I might add, o wise son of Elea, o exquisitely idiosyncratic disciple of Parmenides!

    But the sin of plane changing belongs to Zeno first. The premise of his aporia is this very reality, as boring and non-exciting, and non-hopeful it may be. Zenon admits that Achilles is faster than the tortoise. In the very next step, Zenon creates a new reality, in which he eliminates Time. Zeno breaks a fundamental rule of logic two times.

    First, he recognizes the True Reality, where Motion does exist. So far, so good. But Zenon commits the first logical sin when he recreates a reality where Time does not exist; therefore Motion cannot exist. He already moved to a different field of arguing. The aporia ends right there. Not! Zenon goes a step further. He tries to impose upon us, faithfully reasoning humans, that the reality he single-handedly created is the true reality! Now, that we cannot take, Zeno! So, Zeno moved from one field (plane) of arguing (logic) to another that served his purpose. Nothing, therefore nobody can change reality to serve a particular goal. The Everything goes from almost zero to almost one - randomly - and back, and forth, and back, and forth - ad infinitum... The work of Randomness Almighty...

    3. The mathematical solution

    As I said previously, the mathematical solutions resorted to most difficult math: Mathematics of Infinity. The Greeks were brilliant mathematicians. They set the foundation of mathematics. But it was only the beginning. They hadn't reached the level of the mathematics of infinity, specifically differentials and integrals.

    Many argue that Archimedes had already discovered the mathematics of infinity. In his mostly destroyed work named The Method, Archimedes appeared to have discovered the famed constant e - the base of the natural logarithm (2.718…..). But let's be frank about it. The Ancient Greek luminaries were not aware of e, or any other infinity concept. Modern attempts to solve Zeno's paradoxes employed the mathematics of infinity. Zenon presumed dividing Space in half ad infinitum: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, etc. That resembles the mathematical construction of e. e = lim(1+1/1+1/2+1/3+1/4…+1/n)n= lim(1 + 1/n)n. Well, it was too complicated. The Greeks of that era didn't comprehend that kind of mathematics.

    Even when mathematics is considered, the philosophical (metaphysical) solution takes priority. It all started with the philosophical concept of motion. We solved it philosophically by employing that Motion always assumes two and exactly two elements: Space and Time.

    This is undoubtedly the first ever solution to Zeno's Paradox also Zenon's Paradox.

    Achilles and the Tortoise in Zeno's paradox keeps Space, leaves Time out.

    Indeed, it might appear that I play with updating this page! I discovered it is an efficient method of fighting plagiarism and/or piracy. Up until I published this page (in 2005 ... yes, that recently!) the only idea was that Zeno's paradoxes (aporiae) had had no solutions. They had been considered unsolved mysteries. No human could solve a Zenon paradox or aporia or riddle!

    I quote from a reputable book: The Story of Thought by Bryan Magee (published in 1998):

    “People have found [Zeno's paradox] terribly disconcerting. There must be a fault in the logic, they have said. But no one has yet been wholly successful in demonstrating what it is.” (page 19). I tell you what it is not: There is no Time in Zenon's paradoxes. All objects change position in Space in NO Time — therefore with a velocity equal to Infinity! And that's a blunt absurdity!

    I published this page — and all of a sudden people started to tell me that others had solved the paradoxes before me! Do the Web log files or archives show such solutions prior to my writing? NOT! NON! Quod erat demonstrandum.

    You must fight for what belongs to you. It is a virtue. The real sin is to allow others to rob you. The suckers might shout in public that you are not modest, or that you are bragging, or that you are narcissistic! One must make sure that what is his/hers always belongs to him/her. Muhammad Ali, the greatest champion of them all, was absolutely right. “It ain't bragging, if you back it!”


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    Ion Saliu: The very first to solve Zeno's Paradox of Achilles and Tortoise, plus Paradox of Arrow.