—Stuart K. Hayashi
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| This is Bartolomeo Carduci's fresco of Zeno of Elea (he is the old man leading the young men). |
Chris McKenzie explains:
Zeno's Paradox is as follows. The ancient Greek philosopher Zeno stated that in order to cross a room, one must first cross half the room. In order to cross the remaining half, one must cross half the remaining distance, and so on -- infinitely. Zeno concluded that one can therefore never cross the room.
Let's change Zeno's numbers but keep his intent: In order to cross a room, one must be first cross 99% of the room. In order to cross the remaining 1%, one must cross 99% of the remaining 1%, infinitely.
Zeno has a problem though, and it's one he's smuggled in. Why does Zeno think you can cross 99% of the room? After all, if you treat 99% as a whole, you must first cross 99% of the new whole. The end result is that you can't move at all because in order to move you must first move through 99% of whatever infinitesimally small space you've chosen to move through -- say the first 1% of the distance.
Now to the claim that "nothing is certain" or we can only be 99% certain. It's easy to explode this claim with a single question: Are you sure? However, we can see that the 99% certainty claim suffers from the same problem as Zeno's room: In order to be 99% certain, you must first be 99% of 99% certain. Which is to say, you must be wholly certain of a certain whole (the first 99%).
Or, you could just walk across the room and proudly claim, with 100% certainty, "I did it."
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Stuart K. Hayashi adds:
As Chris said, we can turn this around on Zeno.
Every fraction is, in another context, one whole. That is, Y may be a fraction of X, but Y is still one whole of Y. If the room is 30 feet across, and I only walk "half that distance," that is 15 feet. But that "half distance" is actually "the entire distance" of another measurement: 15 feet. Therefore, every time you travel a distance, you travel the entirety of that distance. Every measurement is, in at least one context, "100 percent."
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UPDATE from the Same Day:
Robert Nasir wrote the following comments to me. I am quoting them here with his permission.
Similarly, if you can travel half the length of a given room, then you can travel what is half of twice the length of the room.
The real issue is the integration of the discrete and the continuous. Everything is discrete. Entities are discrete. The distances they travel (to the extent they're measurable) are discrete.
Mathematics treats space (and time) as continuous. That's fine, it's useful (and arguably necessary) to do so.
But to understand why apparent paradoxes arise, one must never lose sight that math is method, not metaphysics.
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UPDATE: Additional comment from Stuart on January 26, 2018:
I think I have a new way of phrasing this matter:
Zeno's Paradox is based on the false presumption that no measurement can ever be 100 percent. But the exact opposite is true: every measurement is 100 percent, at least in terms of each measurement being 100 percent of itself.
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UPDATE: Addendum from Stuart on February 9, 2020:
Only today did it occur to me that pointing out that every measurement is “100 percent of itself” is a restatement of “A is A” (here, “1 = 1”).
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After I first published this blog post, I learned that in his Physics, Aristotle himself provided a refutation of Zeno Paradox’s similar to the one I provided above. In fact, the explanation of Zeno’s Paradox in Physics has been, throughout the modern age, an important document through which archaeologists and historians have been aware of this idea having been circulated in ancient Greece.
I will say more explicitly why, years ago, Chris McKenzie started discussing this matter with me over Facebook. The reason is a bromide commonly repeated in the culture, even by scientists and science journalists influenced by Karl Popper and Thomas Kuhn in the philosophy of science.
The bromide starts by pointing out that when it comes to forming conclusions, we can err. A big reason for this is often that, in forming the conclusion, we had yet to come across a piece of information that would have provided a more-complete understanding. And because we are not omniscient and cannot know everything that can ever be known, it follows there will always be more learn. It then follows that will always be a risk of error in our conclusions. Therefore, concludes the bromide: no principle should be deemed to be absolute, and we can never be a certain of any conclusion. And if we can never be certain of any conclusion, any and every claim to any knowledge is subject to existential doubt, no better than tenuous.
That is what Karl Popper meant in saying, “...we never know what we are talking about.” In Forbes magazine, physics professor and science writer Ethan Siegel insists, “In fact, when it comes to science, proving anything is an impossibility.”
An interpretation such as Siegel’s concedes that scientists do gather evidence, and that, as evidence accumulates for a particular conclusion, we can legitimately grow more confident of that conclusion. However, goes that interpretation, as no one can gain all data ever to exist, complete confidence can never be achieved. In other words, we can never have 100-percent of all information, which means we can never have 100-percent confidence. Ergo, says the argument, even as we can cite additional evidence to become justly more confident in a proposition, actual complete proof of the conclusion will forever elude us.
Being influenced by Karl Popper and Thomas Kuhn, the argument above is often provided in introductory science courses in university. The Forbes writer is far from alone, even among science instructors. I know my introductory biology professor gave lip service to it before moving on to aspects of the lecture — about cells and respiration — that he found more interesting. But when people say, “We can never achieve certainty in the end,” the phrasing is terribly misleading.
When I wrote about this on Facebook in 2016, Chris McKenzie replied to me in a comment post that “We can get more confident, but can never achieve complete confidence” is a variant on Zeno’s Paradox. The same applies to “Something can improve but never reach perfection.” These are variation’s on Zeno’s Paradox because the main message of Zeno’s Paradox is, You can gain increments bringing you closer to 100 percent, but no form of 100 percent shall ever be reached.
Here is how “We can’t be certain” is a variation on Zeno’s Paradox. Isaac Asimov, the famed science-fiction author who was also a trained biochemist, told PBS talk-show host Bill Moyers, “...we can’t be absolutely certain. Science doesn’t purvey absolute truth. Science is a mechanism. It’s a way of trying to improve your knowledge of nature.”
Note how Asimov concedes that through science we keep getting closer to the absolute truth. But he warns us that absolute truth itself cannot be reached. Asimov’s statement is a rephrasing of Zeno’s idea that we can get closer to 100 percent but not 100 percent itself. One is “A principle might apply much of the time, but it will never absolute.” Another is “Something can improve, but it will never be perfect.”
Zeno’s Paradox is a fallacy because it is inverse to the truth. The argument is sloppy in terms of how it defines 100 percent, largely conflating 100 percent and completion with infinity. This Paradox is also sloppy in how it treats the meaning of the concept of context.
In reality: in at least one context, every increment is 100 percent. This is because, in at least one context, every quantity is 100 percent. It is 100 percent in that it is 100 percent of itself. That is, “1 = 1.” Phrased differently, “A is A.” That is, every distance traveled is 100 percent of itself. By that same token, a fraction of a Particular Distance C is itself 100 percent of yet another distance, Distance D.
Let us say we are in a room that is 18 feet long (5.4864 meters). And let us revisit Zeno’s Paradox. We want to cross the room from one end to the other. We travel half the distance (9 feet; 2.7432 meters). We then travel half the remaining distance: 4 feet and 6 inches, or 1.3716 meters. Then we travel half the remaining distance again, and again, and so on. We keep inching closer to the desired destination, and yet if in every interval we travel half the remaining distance, we will never reach the desired destination. How, Zeno asked, can we possibly reach the desired end of the room? Again, the point is that we can never reach 100 percent; every distance traveled is a fraction of a distance and not a complete distance. This means we can never attain something that is one whole; we can never attain something that is a single unit; we can only obtain a fraction of that unit.
By now, we see the fallacy. Half the distance of an 18-foot-long room is itself a distance. The nine feet in our first increment (2.7432 meters) may be a mere fraction of the length of the room, but, in another context, it is indeed 100 percent, one whole, and a complete distance — as is any and every quantity. That is why, as quoted above, Robert Nasir cheekily said that if we are allowed only to walk “half a distance,” then he can offer a work-around. It is that we say we will travel half of twice the length of the room. By walking half that distance, we end up walking the entire length of the room.
In Physics, Book 6, Aristotle already spotted this. As he said it, there is no distance traveled “less than itself for it to traverse first...” Hence, any time we travel a distance that is a fraction of another, the distance we have traveled is already a “distance equal to itself.” Yes, “everything that is in motion ... traverses a distance as great as itself.” Hence, every fractional distance we have traveled is, in at least one context, already one “whole.”
The same principle applies to the concepts of certainty, perfection, and absolute. All of those are contextual. By that, I mean that any time we talk rationally of certainty, perfection, and absolute, those are parameters set by a pertinent context. And, within that pertinent context, every one of those can indeed be achieved.
It is a false philosophic presumption that any claim to true knowledge requires infallibility and omniscience. It is the same false philosophic presumption that perfection and absolute are achieved only when not limited to any parameters, so much so that they transcend context. That false presumption can largely be attributed to Plato, who propounded that certainty, perfection, and absolutes could exist only as ideals beyond human capability, with everything observable in human experience necessarily being of inferior stature. In the modern age, Immanuel Kant popularized the presumption that human fallibility precludes certainty and perfection.
Once you toss away the Platonic misconceptions about them, it happens that certainty, absoluteness, and perfection all exist. They exist within their respective proper contexts, and each are defined by that context.
Whether you reach “100 percent” — whether you have a complete unit, as opposed to a fraction of that unit — is determined by the quantity you defined, in the first place, as 100 percent and a complete unit. In his Paradox, Zeno was equivocal about how he defined 100 percent of a particular length. The Paradox’s upshot is that you can never reach 100 percent of any important quantity, but, as we have identified, the half-the-room’s-length you have walked is itself 100 percent of an important quantity.
Zeno’s sloppiness when it comes to his parameters on exactly what quantity is needed to achieve a goal, is mirrored in platitudes about the impossibility of perfection, absoluteness, and certainty. When people say perfection is unachievable, I reply that their interpretation of it is unachievable for the reason that they are talking of a perfection that can be called such only if it transcends all context. Hence, I call this invalid idea one of Platonic perfection, in contrast to what is actually legitimate: contextual perfection. By that, I mean “perfection within a particular practical context” — which is indubitably achievable.
The same applies to absolutes. People say that a principle cannot be absolute because, for it to be absolute, it must apply in any and every context uniformly. That would mean that the principle applies in such a manner that it transcends all context. I call that sort of unachievable absolute the fallacious idea of Platonic absolute. An actual absolute principle is a contextual absolute. It recognizes that a principle applies consistently within its own pertinent context.
Within a particular purview with clearly-defined goals, perfection is attainable. If “perfectly beautiful” and “perfect morally” are defined no more than nebulously, of course those misconceptions of Platonic perfection cannot manifest. As so many people are vague, at best, on what they interpret perfection to be, it is wholly unsurprising that they repeat clichés about how nothing and no one but Jesus is perfect. But suppose your child in school is given an examination of ten math problems. And suppose your child has gotten the correct answer on all ten problems. In that purview, it is entirely legitimate to say that on that exam your child has gotten a score that is perfect.
That every quantity is 100-percent, by the way, is also why people contradict themselves in their proclamation, with implicit certainty — and with the subtext of that this applies as an absolute — that no proclamation should ever be taken as absolutely certain. (Any science-fiction character who shouts, “Only a Sith deals in absolutes!” must, by his own definition, be a Sith.)
The fallacious idea that we have certainty only through infallibility and omniscience, I identify as the fallacious idea of a Platonic certainty. The foil to that is actual certainty, which is contextual certainty. To achieve a particular life-affirming goal — the pertinent context — we need to have a quantity of knowledge that is sufficient to achieve it. Insofar as applying the pertinent knowledge is consistent in achieving the goal satisfactorily, we are indeed rationally certain of that knowledge. Attaining such rational certainty does not require knowing everything there is to know. It requires only knowing enough that is pertinent to the context and scope in question.
Isaac Asimov elsewhere tells Bill Moyers, “That is really the glory of science. That science is tentative, that it is not certain, that it is subject to change.” Note the conflation of science being “subject to change” with science being uncertain. In light of new information, aspects of scientists’ interpretations are properly subject to change and revision. But that does not preclude contextual certainty. The aspects of conclusions which we gained through rational observation, minding pertinent context — and which continue to withstand challenge repeatedly — are indeed certain.
In the future, we may gain some additional knowledge that provides insight into additional facets of, and nuances regarding, whatever it is we are studying and doing. This new knowledge adds to the knowledge we had already acted upon successfully in the past, and it may give us a new perspective whereby we revise aspects of our prior interpretation. But it does not render, as uncertain, the aspects of our old ideas that we had acted upon successfully in the past and which continue to produce the same results.
Every fraction of Quantity C, is itself one-whole of itself, Quantity D. Perhaps we can imagine a quantity of knowledge that consists of All-Knowledge-That-Can-Be-Had-and-Ever-Can-Be-Had — omniscience — and call that Quantity-of-Knowledge C. And the quantity of knowledge we will accumulate in our lifetimes is only a fraction of that. Ours is Quantity-of-Knowledge D. That Quantity-of-Knowledge D is less than C is not sufficient to render Quantity-of-Knowledge D uncertain. That consideration is the same, in principle, as the fact that our indoor stroll measuring 9 feet, though less than 100 percent of the length of our 18-foot-long room, does not mean that our 9-foot stroll is not 100 percent of any quantity.
Even if, when it comes to what we think we know at present, in the future we will find aspects mistaken or at least in need of further clarification, that consideration still is not enough to render our Quantity-of-Knowledge D uncertain. The aspects of our Quantity-of-Knowledge D that hold up today, and which continue to hold up in the future, are rationally certain. And insofar as the principles within that knowledge apply consistently within the purview and scope of what we are studying and doing, those principles are absolute.
Thus, the fact that every quantity is 100 percent is something that can be applied to our philosophic guidance over everyday life. For a principle to be absolute in any manner is not contingent upon it applying in every context. It need only be consistent in its own proper context, in its own proper scope. The more dissimilar a new context is from the original, the more it may be that the original principle does not apply to the new context. That does not change the absoluteness of the principle to the original context. Likewise, for you to be legitimately certain in an instance is not contingent upon infallibility of thought and reasoning. Nor is it dependent upon having all knowledge of every pedantic nuance, including knowledge of everything that ever can be discovered. You can legitimately call yourself certain in an instance insofar as your proposition of thought reliably demonstrates itself satisfactorily within, once again, the pertinent context and scope.
Thus, the fact that every quantity is 100 percent is something that can be applied to our philosophic guidance over everyday life. For a principle to be absolute in any manner is not contingent upon it applying in every context. It need only be consistent in its own proper context, in its own proper scope. The more dissimilar a new context is from the original, the more it may be that the original principle does not apply to the new context. That does not change the absoluteness of the principle to the original context. Likewise, for you to be legitimately certain in an instance is not contingent upon infallibility of thought and reasoning. Nor is it dependent upon having all knowledge of every pedantic nuance, including knowledge of everything that ever can be discovered. You can legitimately call yourself certain in an instance insofar as your proposition of thought reliably demonstrates itself satisfactorily within, once again, the pertinent context and scope.
