WEBVTT

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sleep at time facts, paradoxes, find a comfortable place where the day's edges can begin to soften,

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let your breathing settle into its own slow rhythm. No effort needed, your body already knows

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how to rest. It simply needs your permission. This is a book about paradoxes. Those quiet puzzles,

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that philosophers have turned over for centuries, they sit at the boundary of clear thinking,

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little knots that don't easily untie tonight, though. They are not problems to solve,

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or arguments to win. They are more like patterns, on a ceiling that you notice just before sleep,

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almost familiar, slightly strange, entirely peaceful. They ask nothing of you in the chapters

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ahead. We'll meet Senos endless rays, Russell's careful set, and the ship that changes every

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plank, but right now, they are nothing but gentle companions. Ideas that invite your mind to

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wander, without gripping too tightly, let your thoughts drift like a leaf, unstill water, the words

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are here, they will wait, and there is no hurry, consider a single sentence, printed plainly on a page.

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This statement is false. You can hold it in your mind for a moment, let it settle, and then

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watch what happens if the sentence is true. Then what it says must be the case, namely,

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that it is false, so if it is true, it must be false, but if it is false, then what it says is not

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the case, meaning it is not false, so it must be true, the sentence curls back on itself.

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A small, quiet loop, with no resting point, there is no resolution. Only the gentle back and forth,

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that keeps going as long as you look at it, this little knot of language is ancient, one early shape

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of it appears in a story about epomenides. A Cretan philosopher from the 6th or 7th century BC,

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he is supposed to have said that all Cretans are liars, if he was telling the truth, then as a Cretan,

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he must have been lying, making the statement false, and if he was lying, then not all Cretans are

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liars, which leaves room for the statement to be false, without immediate contradiction, so epomenides

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version is not quite the tight loop we have today. It can be untangled by supposing that some

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Cretan's tell the truth and epomenides was simply wrong on that occasion, still the feeling of

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self-reference was there, and it intrigued later thinkers, a sharper form, came from ubulides

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of my leadus, in the fourth century BC, he asked simply, a man says he is lying, is he telling the truth,

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this stripped away the escape route, when a speaker asserts his own falsehood. The two possibilities

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chase each other around, in a circle, the ancient logicians, particularly the stoics,

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took the problem seriously, Cricypus, is said to have written several books on the liar paradox.

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Though none survive, we know they saw it as a challenge to the idea that every statement must be

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either true or false, a principle that seemed natural and firm the paradox points to a fault line,

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in our intuitive picture of truth every day thinking treats truth as a property. A sentence

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has when it matches the world, but here the sentence refers only to itself, and that self-involvement

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creates a kind of short circuit, it is not that the sentence is vague or ambiguous.

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It's meaning is perfectly clear, the trouble is that the standard rules for assigning truth

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break down, when the assignment loops back on itself this reveals that our ordinary concept of truth

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may not be as seamless as we assume, legicians have responded in several careful ways,

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one approach, developed most influential by Alfred Tarsky in the 20th century,

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holds that a language cannot consistently contain its own truth-predicate, that is,

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a sentence in a given language cannot talk about the truth of sentences in that same language,

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without risking paradox Tarsky's solution builds a hierarchy, an object language talks about the

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world, a metal language talks about truth in the object language, a metal language talks about truth

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in the metal language, and so on the liar sentence, which tries to speak about its own truth,

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becomes ill-formed because it straddles levels it should not. The loop is broken by denying the sentence,

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a place in the hierarchy this is elegant, but it restricts the expressive power of language

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more than many philosophers find comfortable, another response, explores the idea of a truth-value

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gap, perhaps the liar sentence, is neither true nor false, it falls into a third category,

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like a statement whose presupposition fails, if truth requires a grounding in something beyond the

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sentence itself, then a statement that refers only to its own truth might simply lack a footing,

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leaving it unassigned this avoids contradiction, but it forces us to give up the principle that every

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well-formed declarative sentence must be true or false, which is a concession sum are reluctant to make

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a third direction, rethink the logic itself in most logical systems,

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a statement and its negation cannot both be true, and a contradiction implies any conclusion,

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periconsistent logics, relax that rule they allow that the liar sentence might be both true and false,

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treating the contradiction as contained rather than explosive, this keeps the language whole

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and self-referential, but at the cost of revising deep assumptions about what a contradiction means,

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there are other paths to some philosophers' focus on the way context shifts, as we evaluate the sentence,

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others notice that the paradox depends on the sentence, referring successfully to itself,

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and they question whether self-reference of this sort is always achievable, each response illuminates

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something, but none has settled the matter completely, the liar paradox remains an open country,

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in logic and philosophy, a place where a simple sentence undoes a seemingly simple idea as you let

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the sentence circle once more, this statement is false, notice how little strain it takes to feel the

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loop, the mind can rest there, not solving it, just watching the movement, the sentence does not

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demand you fix it, it just sits there, turning quietly, a small reminder that some puzzles are

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more about the asking than the answering, a set in the usual understanding, is just a collection of

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things you can gather numbers, chairs, ideas, or even other sets, the rule for membership,

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seems open-ended, specify a condition, and the set contains exactly those items that satisfy

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it, the set of all red things, the set of all even numbers, the set of all sets that have more than three

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members for decades, this generous principle guided the early work in set theory, and it felt natural

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a set is defined by its condition, and any well-formed condition ought to pick out a set,

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then in 1991, Bertrand Russell asked what happens when you let the condition be this,

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the set of all sets that do not contain themselves, most sets, do not contain themselves, the set of all

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teaspoons is not a teaspoon, so it does not belong to itself, the set of all even numbers

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is not an even number, so it stays outside, but now imagine collecting all such well-behaved sets

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into one new set, call it R, the membership question for R, is whether it contains itself,

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if it does, then it must meet the condition, meaning it does not contain itself,

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if it does not, then it fails the condition, meaning it must contain itself, the contradiction

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folds neatly in both directions, there is no escape, R belongs to itself exactly when it doesn't

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Russell came upon this while studying the logical foundations of mathematics, the trouble,

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was not a distant curiosity, got love for each, one of the most careful thinkers of the time,

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had spent years building a formal system meant to ground arithmetic in pure logic, his basic laws

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of arithmetic relied precisely on the unrestricted set-forming rule that Russell's construction

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undermined when Russell wrote to him in June 1902, Fridge had just finished the second volume,

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he added a short appendix, acknowledging the difficulty, beginning with the line, a scientist

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can hardly meet with anything more undesirable than to have the foundations give way just as

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the work is finished, the simplicity of Russell's question had caused a genuine crack, the problem

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lives in that unrestricted comprehension principle by allowing any condition to define a set, we invited

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conditions that refer to sets themselves in a looping way, self-reference, when coupled with negation,

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so often produces this kind of spiral in daily life, sentences like this statement is false,

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twist in a similar knot, but here the twist was inside the formal apparatus mathematicians

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used to talk about everything if a system admits a contradiction, it becomes trivial in classical

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logic, anything follows from a contradiction, so the discovery threatened to pull the whole structure

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apart, the response was not despair, but careful rebuilding, Russell himself worked on a theory of

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types, a hierarchy that forbids a set from being a member of itself by organizing objects into levels,

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a set can only contain items from a lower level, so a condition like does not contain itself

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becomes ill-formed, others notably Ernst Sermilo and later Abraham Frinkle,

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built axiomatic set theory, unrestricted principles, they kept the useful operations of pairing,

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union, and power sets, while replacing unrestricted comprehension with separation,

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given a set you already have, you can carve out the subset of its members that satisfy a condition

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that way, a new set never reaches out to gather members from the entire universe of all possible things,

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in a single reckless gesture, it stays local, the paradox, also prompted a philosophical shift,

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we learned to distinguish between a collection and a condition, noticing that not every well-fraised

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description, points to a completed hole, there is a gentleness in this insight, a language that felt

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perfectly general, actually contained quiet limits, the move from intuitive gathering, to careful

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axioms, changed how logicians think about existence to say a set exists, you need a bit more

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than a crisp definition, you need a construction that follows the rules, Russell's paradox,

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indoors, as a kind of guardrail, it reminds anyone who works with formal systems that self-application

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can be delicate, and that the very tools we used to organize thought occasionally need their own

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boundaries drawn, the set of all sets that do not contain themselves does not exist in the rebuilt

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theory, and its absence is a small quiet landmark along the road of modern logic, it points to the fact

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that even the most rigorous frameworks carry small seams, places where the fabric must be stitched with

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care that care ended up producing a richer understanding of order, one that accommodates infinite

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collections and precise structure, without giving way under a single curling question, stand at one end

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of a room, and pick a point on the far wall before you can arrive there, you must first cross

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half the distance, that part is easy to picture, now from that half way mark, you must cross half

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of what remains, then half of that remainder, then half of that, the sequence never announces a last step,

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because after any fraction, no matter how small, there is still a remaining half waiting,

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scene of a Leah, reasoning this way in the 5th century BCE, concluded that motion could never

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begin, the dichotomy, is not a puzzle, about running out of time, or energy, it is a puzzle

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about the sheer number of tasks that seem to be packed into any short glide across a room, walk through

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the arguments slowly, and it feels almost hypnotic, imagine the distance, as a line of unit length,

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the first milestone sits at the midpoint, one half, the next sits at three quarters, then seven eighths,

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then fifteen sixteenths, the markers crowd together, near the far end, each step you might try to take,

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brings you only to the next fraction, in an unending list, because there are infinitely many such

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points between you and the door, seeing how claimed that covering them one by one would require

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an infinite number of distinct motions, an infinite number of motions, he thought,

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cannot be completed in a finite stretch of time, so the arrow never leaves the bow,

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and the foot never lifts from the floor, what makes the dichotomy linger, is that the math and

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medical machinery we now bring to it seems, at first, to dissolve the problem cleanly, the distances

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see no, asks us to traverse add up like this, one half plus one quarter plus one eighth plus one sixteenth,

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and so on without end, this is a geometric series, and its behavior was understood by later Greek

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mathematicians, and formalized fully in the calculus, the partial sums approach one,

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one slash two, three slash four, seven slash eight, fifteen slash sixteen, they creep arbitrarily close to

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the whole distance, the limit of the infinite sum is exactly one, so far all is smooth, the total distance

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is finite, a finite distance you might say, requires only a finite time at a steady walking pace,

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the infinity of subdivisions seems tamed, yet that summation hides a subtle shift,

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summing an infinite series in mathematics, is a single conceptual operation, it does not instruct you

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to actually perform infinitely, many separate acts of adding, when you walk, you do not consciously

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tick off the halfway point, then the three quarter point, then the seven eighths point, the motion happens

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all at once, Zino's challenge though, was aimed at the very idea that you could finish a process,

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whose parts never stop proliferating, even if the hole is finite, can you really complete, a journey made of endlessly,

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many distinct sub-journey's, the mathematical formula, gives the sum, it does not provide a step-by-step method

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for physically visiting every point in order Aristotle, who preserved Zino's arguments for us,

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tried to untie the knot by distinguishing between a line and the points on it, he said that a continuous

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magnitude is not composed of indivisible points, the points exist only potentially,

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not as a completed infinity that you must exhaust, you can keep dividing a distance in thought,

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but the distance itself does not contain an actual infinite set of halfway markers,

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waiting to block your way, later thinkers, refined this line of response in modern terms,

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the real numbers between zero and one form, a dense order, but traversing a continuum,

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does not require enumerating them one after another, the motion is continuous,

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not a succession of discrete jumps to each midpoint, even with these replies. The dichotomy refuses

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to become a mere, solved, exercise, it nudges us toward a peculiar fact about the everyday world,

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a single step, one that feels immediate and whole, contains within it an unbounded cascade of smaller

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intervals, no matter how close you are to the door, there is always a distance smaller than the one

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remaining, a fraction needing to be crossed before the next, the infinite does not shout here,

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it sits quietly beneath the floorboards, harmless and strange as you turn the light out and let the

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thought drift. What remains is the image of that soft, uncompleted line of markers,

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endlessly fine, waiting in a space that takes no effort at all to cross, a swift runner,

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and a slow tortoise, side by side at the start of a dusty track, Achilles, the great hero of the Greeks,

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offers the tortoise, a friendly head start, the tortoise accepts, and begins to crawl forward, Achilles

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waits, patient and confident when the tortoise has moved a hundred paces, the race begins, Achilles runs

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smoothly, his strides eating up the ground in a moment, he reaches the spot where the tortoise started,

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but the tortoise is no longer there, in the small time it took Achilles to cover those hundred

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paces, the tortoise has moved another ten, so Achilles must now run those ten paces, he does

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effortlessly when he arrives at the new point, the tortoise has advanced a little farther,

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perhaps a single pace this time again Achilles covers that distance, and again the tortoise

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has crept ahead by a small amount this continues, the gap shrinking but never quite vanishing,

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the argument says that Achilles can never overtake the tortoise each time he arrives at a place where

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the tortoise has been, the creature has already departed for somewhere new, there will always be another

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small distance to cover, another interval of time before the next arrival, the chase breaks into an

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unending sequence of stages, and completing an endless sequence seems to require an endless

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duration, on its face, this is absurd, we know a fast runner, overtakes a slow one, every time,

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yet the logic feels tight, the structure closely resembles Xeno's dichotomy, we're crossing a room

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requires first crossing half, then half of the remaining half, and so on, here instead of having

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a fixed distance, the steps shrink according to the relative speeds of the two racers, if Achilles

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runs ten times as fast as the tortoise, the first leg is a hundred paces, the next ten,

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the next one, and on it goes, the mathematical reply, emerged in clearer form, long after Xeno,

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the distances Achilles must run form a geometric series, 110, 110, 100th, and so forth,

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add them all up, and the total is not infinite, it is exactly 111, and one night the paces,

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at that spot, Achilles finally draws level with the tortoise, the times needed for each segment,

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form a similar shrinking series, an infinite number of time, intervals, sums to a finite duration,

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so the chase does have an end, and it arrives within a finite stretch of time, once this is pointed out,

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the paradox can feel almost like a trick of faulty bookkeeping, Xeno asked us to picture,

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an endless list of tasks, but an endless list can have a finite sum, just as the fraction one-third

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unfolds, as 0.3 repeated forever, without passing a fixed boundary, the mathematics of limits,

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developed centuries later by Newton and Labeness, gave precise tools for handling such infinite

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sequences, yet for sum, the deeper philosophical tangle remains, the mathematical sum describes a limit,

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but does it describe the actual completion of a supertask, an infinite number of separate actions,

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one after another, if each stage of the chase is a distinct event, then the runner would have to

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perform an unlimited number of catch-up acts, whether physical motion can be carved into infinitely

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many discrete parts, is a question that continues to draw careful argument, one response holds

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that motion is fundamentally continuous, not chopped into separate little advances,

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another suggests that at the limit, the infinite sequence collapses into a single smooth passage,

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these finer debates need not disturb the gentle image of the race, picture the scene softening now,

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the bright track fades at the edges, Achilles' strides blur into a steady rhythm,

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and the tortoises slow shuffle becomes a quiet, regular pulse, the gap between them shrinks to a

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sliver, then a spec, then something too small to hold in the mind's eye, the infinite steps,

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fold themselves into a single fleeting moment, the race does not reach its finish here,

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the finish is simply allowed to dissolve, the runner and the tortoise drifting out of the frame,

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before the overtaking ever occurs, an arrow is released from the bowstring, and lifts into a quiet

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arc, for a moment, if you could freeze the world, you would see it suspended against the sky,

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perfectly still in that single slice of time, Xenoe of Alia, writing in the fifth century before the

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common era, asked us to consider that frozen moment, and then to think about what it really contains,

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the arrow paradox runs like this at any indivisible instant of its flight, the arrow

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occupies a space, exactly equal to its own length, it is not larger than that space, nor smaller,

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it fits it precisely if something occupies a space equal to itself, it is what we call it rest,

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at that one instant, nothing about the arrow's position is changing, now time Xenoe supposed,

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is composed entirely of such indivisible instance, arranged one after another, if at every instant,

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the arrow is at rest, and time is nothing but instance, then the arrow is always at rest, motion

00:30:32.720 --> 00:30:42.000
he concluded cannot happen, the logical steps are gentle, but they tighten around a genuine difficulty,

00:30:42.000 --> 00:30:49.520
the word instant, does the heavy lifting, an instant is supposed to have no duration,

00:30:49.520 --> 00:30:58.880
it is a point in time, not a stretch of it, in a durationless point, no change can occur because change

00:30:58.960 --> 00:31:06.960
requires a before and an after, so if you try to catch the arrow, in the act of moving inside an

00:31:06.960 --> 00:31:16.720
instant, you fail, the instant is too narrow to hold motion, all it can hold is a location, the arrow

00:31:16.720 --> 00:31:26.560
paradox becomes a meditation, on what time must be like, if it can contain movement at all, one long

00:31:26.560 --> 00:31:36.320
standing reply, is that motion is not the sort of thing that exists inside an instant, an instant,

00:31:36.320 --> 00:31:45.200
is merely a boundary, like the edge of a ruler marking, where one centimeter ends and the next

00:31:45.200 --> 00:31:52.560
begins, you would not say that length lives at the boundary, motion lives across intervals,

00:31:52.560 --> 00:32:00.160
the arrow moves, when at one moment it is here, and at a later moment it is there,

00:32:00.160 --> 00:32:09.600
the motion itself is a fact about the relationship between those two moments, not a property that any

00:32:09.600 --> 00:32:18.320
single moment possesses in this view, being at rest is also a property that needs an interval,

00:32:18.320 --> 00:32:27.040
something is at rest, if over a stretch of time its position does not alter, so senos claim that the

00:32:27.040 --> 00:32:38.160
arrow is at rest at each instant might already be mistaken, because rest, like motion, requires duration,

00:32:38.160 --> 00:32:47.120
mathematicians, and physicists later, offered a more refined version of this idea, using the tools

00:32:47.120 --> 00:32:56.240
of calculus, they defined instantaneous velocity, not as something that occurs at a literal

00:32:56.240 --> 00:33:06.400
isolated point, but as the limit of average velocities over smaller and smaller intervals around

00:33:06.400 --> 00:33:16.000
that point, the arrow speed at an instant is a number that summarizes how its position changes when

00:33:16.080 --> 00:33:25.040
you zoom in on the timeline, it exists as a mathematical convenience, a description of a tendency,

00:33:26.080 --> 00:33:32.800
not a snapshot of activity, the arrow is moving, because its trajectory is smooth,

00:33:34.160 --> 00:33:42.880
and its position function has a derivative, even with these clarifications, the paradox retains a stubborn

00:33:42.960 --> 00:33:52.160
appeal, some philosophers point out that the language of limits and derivatives explains how we can

00:33:52.160 --> 00:34:00.640
model motion successfully, but it does not fully dissolve the conceptual puzzle, the feeling that there

00:34:00.640 --> 00:34:10.000
is a jump between a sequence of stationary positions, and genuine movement is hard to shake, a film

00:34:10.080 --> 00:34:19.840
strip is a collection of still frames, yet when projected it creates the illusion of motion is reality

00:34:19.840 --> 00:34:28.320
like that if time really were built from durationless slices, with the arrows flight be any more than a

00:34:28.400 --> 00:34:37.360
succession of quiet, motionless tabloes, these are open questions, and they lie at the edge

00:34:37.920 --> 00:34:46.240
of what physics and logic can comfortably say there is a serene image that emerges from thinking

00:34:46.240 --> 00:34:54.320
about senos arrow, picture the arrow halfway along its path, held in a moment of thought,

00:34:54.400 --> 00:35:01.040
the arrow around it is still, the fletching does not quiver, the point is utterly fixed,

00:35:01.040 --> 00:35:08.400
this single instant, if it could be examined, would contain almost nothing of the flight,

00:35:09.360 --> 00:35:18.720
except a silent, perfect location, the paradox invites you to rest with that image not to solve it urgently,

00:35:19.680 --> 00:35:27.440
but to notice how strange it is that so many of these quiet points, placed side by side,

00:35:28.640 --> 00:35:37.440
add up to the rushing arc, we see when we let go of the bow, the arrow paradox, does not threaten

00:35:37.440 --> 00:35:44.880
our daily experience of motion, arrows still reach their targets, yet it leaves behind a gentle

00:35:44.880 --> 00:35:51.520
puzzlement, about the structure of time, it asks whether the flowing river of moments

00:35:52.000 --> 00:36:00.000
can be made from drops, that taken alone, our entirely dry, you can put the question down

00:36:00.880 --> 00:36:08.640
and pick it up again whenever you like, the still arrow will be there, waiting in the same instant,

00:36:09.360 --> 00:36:17.120
inviting you to wonder, start with a single grain of sand, resting in the palm of an imagined

00:36:17.120 --> 00:36:25.440
hand, no one would look at that one small grain and call it a heap, a heap of sand is something larger,

00:36:26.480 --> 00:36:34.800
something substantial, add a second grain, still not a heap, a third, no chain, the transition

00:36:34.800 --> 00:36:44.800
from not heap to heap never arrives in a single step, because one grain seems powerless to cross

00:36:44.800 --> 00:36:52.640
that line if one grain cannot create a heap and you begin with something that is definitely not a heap,

00:36:53.840 --> 00:37:01.760
then no matter how many grains you add one by one, you never reach a heap, that is the current

00:37:01.840 --> 00:37:10.080
of the Sarate's paradox, flowing quietly from a few innocent looking premises to a conclusion

00:37:10.720 --> 00:37:17.760
that cannot be right, the word Sarate's comes from the Greek word for a pile or heap,

00:37:19.040 --> 00:37:26.800
and the paradox was known to the logicians of antiquity, the puzzle is not really about sand,

00:37:26.800 --> 00:37:34.160
it is about concepts whose boundaries have no crisp edge once you state the reasoning clearly,

00:37:35.280 --> 00:37:42.080
it applies to any gradual change, a man with a full head of hair, is not bald,

00:37:42.080 --> 00:37:51.040
pluck one hair, and he still is not bald if no single hair loss transforms the non-bould into the bald,

00:37:51.040 --> 00:38:02.080
then by this step-by-step reasoning a man will never become bald, even after every hair is gone,

00:38:02.080 --> 00:38:10.160
the same pattern repeats for tallness, someone who is five feet tall is not tall, adding a millimeter

00:38:10.800 --> 00:38:18.480
cannot make the difference between short and tall, in one instant yet we are perfectly comfortable,

00:38:19.360 --> 00:38:25.600
calling some people tall, and others not, the paradox pushes us to ask what we mean,

00:38:26.560 --> 00:38:35.200
when we use these everyday words, the argument works because of a logical principle that seems

00:38:35.200 --> 00:38:42.480
hard to deny, the principle says that if a small change does not alter the application of a word,

00:38:42.480 --> 00:38:53.120
then repeated small changes, also cannot alter it step by tiny step, you move from a clear case

00:38:53.840 --> 00:39:02.480
where the word does not apply to a clear case where it does, and the logical structure forces you to say

00:39:02.480 --> 00:39:09.760
the word never applies, but the real world clearly contains heaps of sand, bald men,

00:39:10.640 --> 00:39:18.560
and tall people, so something must give philosophers, have offered three broad kinds of reply,

00:39:18.560 --> 00:39:26.480
the first accepts that there is somewhere, a sharp boundary, one grain does make the difference between

00:39:26.480 --> 00:39:34.400
a heap and a non-heap, but we simply cannot know which grain it is, the boundary exists,

00:39:35.280 --> 00:39:41.360
hidden from our ordinary awareness, this answer preserves the neatness of classical logic,

00:39:42.560 --> 00:39:51.840
because the statement, this is a heap, can be sharply true or false for every number of grains,

00:39:51.840 --> 00:40:00.000
the trouble is that it leaves us with a fact that feels both arbitrary and unknowable if the

00:40:00.000 --> 00:40:10.240
thousandth grain may the heap, why not the 999th and how could our word for heap latch onto a precise

00:40:10.240 --> 00:40:19.360
number that no language user can identify a second response, softens the idea of truth instead of

00:40:19.360 --> 00:40:27.920
true and false as an all or nothing switch. We can speak of degrees of truth, a collection of two

00:40:27.920 --> 00:40:38.160
grains is hardly a heap at all, so the proposition, this is a heap, is almost entirely false,

00:40:38.160 --> 00:40:47.600
a collection of 200 grains is more heap-like, the proposition is true to a high degree as the grain

00:40:47.600 --> 00:40:56.640
secumulate, the truth value slides upward, like a dimmer switch rather than a click, this fuzzy

00:40:56.640 --> 00:41:05.600
logic approach captures the gradual feel of vague language, it explains why we never find a single

00:41:05.600 --> 00:41:14.640
grain that makes the difference no grain does, because the transition is spread out across many grains,

00:41:14.640 --> 00:41:23.600
critics worry though that the machinery of degrees gets complicated quickly when we combine statements,

00:41:23.600 --> 00:41:33.040
if this is a heap, is 0.7 true, what is the truth of this is a heap, and that is a heap,

00:41:34.080 --> 00:41:43.600
when the second pile is 0.6 true, the exact numbers feel as artificial as the sharp boundary,

00:41:43.600 --> 00:41:51.760
the third kind of reply, challenges the logical step itself, it denies that if one grain cannot

00:41:51.760 --> 00:42:00.960
change a non-heap into a heap, then an endless series of such grain additions also cannot do so,

00:42:00.960 --> 00:42:10.400
this reply uses a form of reasoning, called super-valuationism, on this view, a statement is true,

00:42:11.120 --> 00:42:19.520
if it comes out true, on every acceptable way of making the vague term precise for borderline cases,

00:42:20.400 --> 00:42:28.480
the statement is neither true nor false, it is indeterminate, one grain cannot be the turning point,

00:42:29.520 --> 00:42:36.800
is true for each particular grain you check, but it does not follow that there is no heap,

00:42:38.000 --> 00:42:46.480
because the phrase turning point has different reference, depending on how you sharpen the word,

00:42:46.480 --> 00:42:54.880
the logic of fakeness on this view demands a departure from simple, two-valued reasoning,

00:42:54.880 --> 00:43:04.000
the appeal is that it keeps the ordinary truth of clear cases, while allowing a genuine fuzziness

00:43:04.000 --> 00:43:12.240
in the middle one can get a feel for the paradox by watching a mental image, a flat surface,

00:43:13.200 --> 00:43:22.320
and onto it grains of sand descend slowly, one by one at first, a few scattered specs,

00:43:22.320 --> 00:43:29.840
then a light dusting, then a small mound begins to lift from the flatness, the concept of a heap,

00:43:30.560 --> 00:43:39.120
does not snap into focus, at some countable instant, it coalesces, like the edge of twilight,

00:43:40.080 --> 00:43:46.480
or like the borderland between being awake, and being somewhere else entirely,

00:43:47.600 --> 00:43:53.600
or the contours of awareness soften, the heap was never in the grains themselves,

00:43:54.800 --> 00:44:03.600
it was a pattern our mind traced across them, the sorry it's paradox, invites us to notice the gentle blur

00:44:04.400 --> 00:44:12.720
at the edges of our own categories, and to sit with that blur, without demanding a sharper line,

00:44:13.520 --> 00:44:20.800
then the world provides, blue dark, told the story first, theses, the mythical king of Athens,

00:44:21.840 --> 00:44:28.800
sailed a ship back from creed after defeating the Minotaur, the people of Athens, kept that ship

00:44:28.800 --> 00:44:37.200
in their harbor as a memorial, and as the years passed, they would replace its planks one by one,

00:44:38.560 --> 00:44:47.760
each time a piece of wood weakened, eventually every plank, every nail, every length of rope

00:44:47.760 --> 00:44:54.480
had been swapped for new material, the question arose quietly, almost as a pastime,

00:44:55.360 --> 00:45:05.520
after all those replacements, was the vessel still the ship of theses, the puzzle feels gentle at first, a wooden ship,

00:45:05.920 --> 00:45:15.680
is a physical thing, so if the physical parts change gradually, maybe the thing itself remains, that is how we talk

00:45:15.680 --> 00:45:23.520
about ordinary objects, a bicycle with a new wheel, is still your bicycle, but push a little further,

00:45:23.520 --> 00:45:31.520
if all the matter is different, what is it that persists, is it the shape, the function, the name,

00:45:32.560 --> 00:45:40.080
or the continuous history of replacement, the old planks did not vanish, imagine that a careful ship

00:45:40.960 --> 00:45:51.280
gathered every discarded piece and reassembled them exactly as they had been, restoring the original ship

00:45:52.000 --> 00:46:00.000
with its weathered timber and faded paint, now there are two candidates, one has the continuous history,

00:46:01.040 --> 00:46:07.120
the other has the original material, the problem sharpens into a single question,

00:46:07.120 --> 00:46:16.480
which one is the ship of theses, Thomas Hobbes, tightened the dilemma, in the 17th century,

00:46:16.480 --> 00:46:25.200
he imagined a second ship, built from the original planks, and then asked whether the original ship

00:46:25.920 --> 00:46:34.640
had persisted in the continuously maintained vessel, or in the reassembled planks if you believe the answer

00:46:34.640 --> 00:46:44.240
cannot be both, you face a choice, neither choice is obviously absurd, and neither satisfies completely,

00:46:44.240 --> 00:46:53.200
the continuously maintained ship went through an uninterrupted life of repairs, but at last,

00:46:53.200 --> 00:47:02.240
its original substance, the reassembled ship regained its substance, but its history was interrupted,

00:47:02.320 --> 00:47:10.880
philosophers have responded in several broadways, one approach says that objects are four-dimensional

00:47:10.880 --> 00:47:20.000
things that extend across time, as well as space, on this view, the ship is not a three-dimensional

00:47:20.000 --> 00:47:29.120
object that lasts through time, but a four-dimensional entity that has temporal parts, the continuously

00:47:29.200 --> 00:47:39.120
maintained ship, and the reassembled ship are two different temporal slices that share an earlier moment,

00:47:39.120 --> 00:47:48.400
the original ship, just is that earlier stretch of the four-dimensional worm, and the later candidates

00:47:48.400 --> 00:47:56.640
branch off from it, there is no deep puzzle. Only a question about which branch we choose to call

00:47:56.720 --> 00:48:05.680
by the old name, a contrasting view, is that identity for material objects is not a precise fact of the

00:48:05.680 --> 00:48:15.520
world, but a matter of how we decide to use words, we sort things into kinds, and for each kind,

00:48:16.240 --> 00:48:24.720
we have fuzzy rules about what counts as the same thing over time for a ship, practical continuity,

00:48:25.600 --> 00:48:33.040
may weigh more heavily than a original matter for a painting, the original canvas and pigment

00:48:33.040 --> 00:48:40.720
might matter more on this account, the ship of the easiest riddle is not a mystery waiting to be solved,

00:48:42.000 --> 00:48:51.280
but a sign that the concept of sameness has slack in it, slack that ordinary talk leaves untied

00:48:51.280 --> 00:48:58.240
when the puzzle is shifted to persons, the same questions reappear in softer form,

00:48:58.240 --> 00:49:06.640
your body replaces itself, gradually over years, your memory's shift, your habits evolve,

00:49:06.640 --> 00:49:15.040
if every part of you is eventually replaced, are you the same person you once were, hear the discussion,

00:49:15.120 --> 00:49:24.880
often turns to psychological continuity, the thread of memory and personality that links one moment

00:49:24.880 --> 00:49:33.120
to the next that thread rather than the material is what many philosophers have placed at the center

00:49:33.840 --> 00:49:42.320
of personal identity yet the ship of the easiest reminds us that continuous psychological change is still

00:49:42.320 --> 00:49:51.600
change and the self you were ten years ago might be connected to you now by a chain of small adjustments

00:49:52.960 --> 00:50:02.320
each one unremarkable, together a quiet transformation there is also the simple thought defended by

00:50:02.480 --> 00:50:10.160
some that the question has no fact of the matter answer it is not that the answer is hidden

00:50:11.360 --> 00:50:19.360
but rather that the situation is under-described the ship after repair and the ship from old

00:50:19.360 --> 00:50:29.200
planks each have a claim and nothing in the world decides between them besides the conventions we bring

00:50:29.200 --> 00:50:37.840
on still nights that may be the most calming response of all some riddles are not storms to be mastered

00:50:38.880 --> 00:50:47.600
but patterns to be observed and then set aside the ship sits in the harbor half old half new

00:50:48.800 --> 00:50:56.240
never quite the same from one wave to the next the planks that were replaced yesterday are already

00:50:56.240 --> 00:51:04.400
drying in the sun ready to be fitted again imagine a room with two boxes one box is clear

00:51:05.440 --> 00:51:14.400
and inside it you can see a thousand dollars the other box is opaque and you cannot see what it holds it might

00:51:14.400 --> 00:51:21.920
contain a million dollars or it might contain nothing at all a being with a remarkable talent for

00:51:21.920 --> 00:51:31.120
prediction has already made a choice about what to place in the opaque box this being often called

00:51:31.120 --> 00:51:40.080
the predictor has played this game with many people before its record is nearly perfect when a person

00:51:40.080 --> 00:51:48.720
chose both boxes the predictor nearly always left the opaque box empty when a person chose only the

00:51:48.720 --> 00:51:57.440
opaque box the predictor nearly always filled it with a million dollars the prediction was made before

00:51:57.440 --> 00:52:06.400
you enter the room and the contents of the boxes are now fixed you know all this now you must decide

00:52:07.440 --> 00:52:15.440
do you take both boxes or do you take only the opaque one the puzzle was introduced by the physicist

00:52:15.440 --> 00:52:25.280
William Newcomb and later brought into wider discussion by the philosopher Robert Nasik it is since

00:52:25.280 --> 00:52:33.360
become a quiet fault line in decision theory two lines of reasoning lead to opposite conclusions

00:52:34.560 --> 00:52:43.040
and each one feels like simple common sense the first line says you should take only the opaque box

00:52:43.120 --> 00:52:50.960
the reasoning runs like this the predictor is almost always right people who take only the opaque

00:52:50.960 --> 00:53:00.400
box walk away millionaires people who take both boxes almost always get just the thousand dollars

00:53:00.400 --> 00:53:08.160
from the clear box if you want the million you should act like the kind of person the predictor for

00:53:08.160 --> 00:53:16.960
sees choosing one box more formally expected utility theory tells you to weigh the possible

00:53:16.960 --> 00:53:26.000
outcomes by how likely they are the probability that the opaque box is full given that you take only it

00:53:27.040 --> 00:53:36.240
is very high the probability that it is empty given that you take both boxes is also very high

00:53:36.240 --> 00:53:44.560
so the expected value of one boxing is close to a million dollars while the expected value of two

00:53:44.560 --> 00:53:54.320
boxing is close to a thousand even though the money is already in the boxes or not your decision

00:53:54.320 --> 00:54:03.280
seems to indicate what is already there many decision theorists find this reasoning persuasive then

00:54:03.360 --> 00:54:10.720
there is the second line which says you should take both boxes its argument is just as clear

00:54:11.920 --> 00:54:19.520
and just as hard to dismiss the predictor has already acted the money is either sitting in the opaque

00:54:19.520 --> 00:54:29.200
box or it is not nothing you do now can change that if the opaque box is full then taking both boxes

00:54:29.840 --> 00:54:37.600
gets you the million plus a thousand if it is empty taking both boxes gets you a thousand

00:54:37.600 --> 00:54:45.040
instead of nothing either way taking both boxes leaves you a thousand dollars richer

00:54:45.840 --> 00:54:53.360
then taking only the opaque box this is the dominance principle when one option is better no matter

00:54:53.360 --> 00:55:02.000
how the world turns out rationality seems to demand it two boxing dominates one boxing the

00:55:02.000 --> 00:55:11.440
past is fixed and so is the contents of the boxes so the choice should be for both every time here

00:55:11.440 --> 00:55:20.240
the puzzle splits the mind both arguments cannot be right yet both appear unassailable from within their

00:55:20.240 --> 00:55:28.080
own perspective the one boxer sees a clear connection between your choice and the likely state

00:55:28.080 --> 00:55:36.720
of the world a connection that acts through the predictors for knowledge the two boxer sees a straight

00:55:36.720 --> 00:55:46.000
forward independence between a choice made now and an event already settled in the past much of the

00:55:46.000 --> 00:55:54.160
quiet debate around this problem turns on how we think about free will and for knowledge if the

00:55:54.160 --> 00:56:03.440
predictor can know your choice in advance with near certainty does that mean your choice was already

00:56:03.440 --> 00:56:11.360
determined and if it was does that alter what it means to choose rationally some philosophers see the

00:56:11.360 --> 00:56:20.320
problem as a gentle way to test our intuitions about causal loops and backward influence

00:56:20.320 --> 00:56:29.040
the predictors foresight does not cause your choice but it correlates so perfectly that the usual

00:56:29.040 --> 00:56:38.320
boundaries between past and future seem to soften there is no settled resolution decision theorists

00:56:38.400 --> 00:56:47.200
remain divided some have tried to break the puzzle by proposing that the predictor is logically

00:56:47.200 --> 00:56:56.400
impossible under certain views of free will others have argued that the mere act of describing the problem

00:56:57.200 --> 00:57:05.200
commits us to one answer or the other but for most people who encounter it the problem lingers

00:57:05.840 --> 00:57:14.640
as a kind of mental lullaby the two arguments sit side by side equally calm equally reasonable

00:57:15.680 --> 00:57:24.720
each as inviting as the other and in that quiet space the idea of a simple choice one box or two

00:57:25.760 --> 00:57:33.760
expands into something vast and unresolved the way a dream can feel perfectly logical

00:57:34.720 --> 00:57:42.480
until you wake and try to hold it a teacher stands before a class and makes a simple announcement

00:57:42.480 --> 00:57:50.800
there will be an examination next week she says on one of the five weekdays you will not know which day

00:57:51.760 --> 00:57:57.280
until you arrive that morning the exam will be a surprise the students listen

00:57:58.240 --> 00:58:06.400
and after the teacher leaves a quiet conversation begins someone points out that the exam cannot

00:58:06.400 --> 00:58:15.680
possibly be on Friday the reasoning is straightforward if Thursday night arrives and the exam has not yet

00:58:15.680 --> 00:58:25.520
happened then only Friday remains at that point the day would be known in advance and the surprise

00:58:25.600 --> 00:58:34.800
would vanish so Friday is impossible with Friday ruled out the same logic slides backward consider

00:58:34.800 --> 00:58:44.240
Thursday if Wednesday night passes with no exam then only Thursday and Friday are left but Friday

00:58:44.800 --> 00:58:53.040
has already been shown to be impossible that leaves only Thursday once again knowing it must be

00:58:53.040 --> 00:59:00.400
Thursday removes the surprise so Thursday cannot work either the students continue

00:59:01.520 --> 00:59:11.520
step by step eliminating Wednesday then Tuesday then Monday by the end no day remains the exam

00:59:11.520 --> 00:59:19.360
cannot happen at all the announcement appears to be a contradiction a few days pass on Wednesday morning

00:59:20.240 --> 00:59:27.840
the teacher walks in with a stack of papers the students blink they did not expect the exam that day

00:59:27.840 --> 00:59:34.720
it is in fact a complete surprise the paradox is that their careful reasoning

00:59:35.760 --> 00:59:44.560
which seemed to prove no surprise was possible has just been defeated by a real event the logic was

00:59:44.560 --> 00:59:53.440
persuasive yet the world paid it no mind this puzzle has been discussed since at least the 1940s

00:59:54.880 --> 01:00:03.440
often called the unexpected hanging paradox or the surprise examination it belongs to a family of

01:00:03.440 --> 01:00:11.440
problems about self-referential knowledge and the effort to reason about what other people know

01:00:12.080 --> 01:00:19.040
and what they know that you know the student's argument depends on a chain of deductions

01:00:20.240 --> 01:00:28.400
that assumes each link holds with the same certainty but the later links especially those about

01:00:28.400 --> 01:00:38.400
Thursday and Friday rest on a particular assumption that the students will still believe the exam is

01:00:38.400 --> 01:00:46.160
coming on those later days and that they will still accept their own earlier reasoning as the week

01:00:46.160 --> 01:00:55.280
unfolds their epistemic state changes on Monday morning the students do not know the day even if

01:00:55.280 --> 01:01:04.240
their deductions about Friday is sound in some abstract sense it is a deduction made from a vantage point

01:01:05.120 --> 01:01:12.960
they have not yet reached the world on Monday contains four remaining possibilities the surprise

01:01:12.960 --> 01:01:22.720
condition only requires that on the evening before the exam the students cannot be certain it will come

01:01:22.720 --> 01:01:30.160
the next day as long as they cannot uniquely pick out the correct day from the set of remaining days

01:01:31.120 --> 01:01:38.320
the condition holds the backward induction tries to collapse that set prematurely

01:01:39.360 --> 01:01:48.240
by importing a certainty that only exists at the end of the week into the beginning some philosophers have

01:01:48.240 --> 01:01:58.560
suggested that the paradox exposes a hidden tension in how we handle iterated knowledge for the

01:01:58.560 --> 01:02:06.880
Friday step to work the students must know on Thursday night that they have survived the week so far

01:02:08.320 --> 01:02:15.680
and they must know that the teacher's announcement was true but is their own memory of the announcement

01:02:16.640 --> 01:02:24.320
something they can trust with absolute certainty might they wonder even slightly whether they

01:02:24.320 --> 01:02:32.160
misheard or whether the exam was canceled real reasoning includes such tiny grains of doubt

01:02:33.360 --> 01:02:41.360
and the formal deduction often ignores them another angle turns on what the teacher means by surprise

01:02:43.120 --> 01:02:51.280
if she simply means that the students will not be able to deduce the day in advance using formal

01:02:51.600 --> 01:02:58.960
logic then their failed deduction might be part of the point their reasoning is self-defeating

01:03:00.320 --> 01:03:09.280
by concluding that no surprise is possible they ensure that the exam when it comes will indeed be

01:03:09.280 --> 01:03:17.680
unexpected the student's argument also relies on common knowledge they assume that everyone in the class

01:03:17.680 --> 01:03:28.560
accepts the rules that everyone knows everyone else accepts them and so on without any fuzzy edge

01:03:28.560 --> 01:03:36.880
but common knowledge is a fragile thing a single student who harbors a private doubt about the whole chain

01:03:37.760 --> 01:03:46.240
can break the reasoning and the collective certainty may not be as solid as it seems in the end

01:03:47.120 --> 01:03:55.360
the surprise examination paradox does not announce a verdict it simply lays out a sequence of thoughts

01:03:56.720 --> 01:04:04.160
and lets you notice where the gears begin to slip you can sit with it letting the logic unspool

01:04:04.160 --> 01:04:14.000
forward and backward watching how a faultless looking proof can collide with an ordinary Wednesday morning

01:04:14.000 --> 01:04:21.840
where exactly the proof goes wrong remains a question with more than one gentle answer

01:04:21.840 --> 01:04:30.720
imagine a hotel with no last room the hallway stretches on and on door after identical door

01:04:31.840 --> 01:04:40.080
and wherever you stand the corridor does not end David Hilbert the mathematician who proposed this

01:04:40.720 --> 01:04:49.360
thought experiment in a 1924 lecture used it to show how strangely the infinite behaves

01:04:50.400 --> 01:04:58.560
when we tried to treat it like an ordinary number the hotel is full every room has a guest a weary

01:04:58.560 --> 01:05:07.440
traveler arrives at the front desk and asks for a room in a finite hotel the answer would be a gentle apology

01:05:08.160 --> 01:05:17.840
but here the manager simply nods picks up the telephone and asks the guest in room 1 to move to room 2

01:05:19.120 --> 01:05:29.680
the guest in room 2 to move to room 3 and so on each occupant shifts over by one room 1 stands empty

01:05:30.640 --> 01:05:38.320
and the newcomer settles in the hotel is still full but a new person has been accommodated

01:05:38.320 --> 01:05:47.040
infinity plus one is still infinity the trek deepens a bus pulls up carrying not one extra person

01:05:48.160 --> 01:05:57.680
but a countably infinite number of them as many as there are natural numbers the manager does not panic

01:05:57.760 --> 01:06:06.240
he asks each existing guest to move from room n to room 2 n the person in room 1 goes to room 2

01:06:07.520 --> 01:06:16.640
room 2 goes to room 4 room 3 to room 6 all the even numbered rooms are now occupied

01:06:17.840 --> 01:06:25.200
and all the odd numbered rooms stand open the manager can then assign each new arrival to an odd room

01:06:25.520 --> 01:06:38.160
first person from the bus to room 1 second to room 3 third to room 5 and onward the hotel absorbs them all

01:06:39.360 --> 01:06:47.360
even though it had no vacancies a moment ago then the situation becomes even more layered and endless

01:06:47.360 --> 01:06:56.720
fleet of buses arrives each bus carrying its own infinite set of passengers the buses are numbered

01:06:57.760 --> 01:07:07.280
and the seats inside each bus are number 2 it seems as if the hotel must finally run out of room

01:07:07.280 --> 01:07:14.560
but the manager spreads a chart across the desk a grid where every bus forms a row

01:07:15.520 --> 01:07:23.520
and every seat a column he draws a path that visits each pair exactly once a careful zigzag

01:07:24.880 --> 01:07:32.880
first bus first seat first bus second seat second bus first seat first bus

01:07:33.920 --> 01:07:42.960
third seat second bus second seat third bus first seat and so on the path works its way

01:07:43.520 --> 01:07:51.760
through the entire grid without missing anyone he assigns each visited pair to a room number in the

01:07:51.760 --> 01:08:02.560
order he reaches them and once again the hotel swallows up all the newcomers the key is that every passenger

01:08:02.560 --> 01:08:10.480
can be mapped to a unique natural number the manager has proved that an infinite collection of infinite

01:08:10.480 --> 01:08:21.360
groups is still in a precise sense the same size as the natural numbers themselves this is where the

01:08:21.360 --> 01:08:31.360
idea of cardinality enters soft as a footnote two sets have the same cardinality if their elements

01:08:31.360 --> 01:08:38.160
can be paired off one to one the natural numbers have the smallest infinite cardinality

01:08:39.360 --> 01:08:46.400
usually called alifinal Hilbert's hotel shows that adding one or a finite number

01:08:47.520 --> 01:08:57.120
or even an infinite number of new guests to a set of size alifinal does not produce a larger set

01:08:57.120 --> 01:09:05.680
a part of an infinite set can be exactly as large as the whole which is a departure from our

01:09:05.680 --> 01:09:14.320
everyday experience of finite collections the paradox is not a contradiction in mathematics it is a

01:09:14.320 --> 01:09:23.520
demonstration that the rules for finite quantities do not carry over into the infinite once you accept

01:09:23.520 --> 01:09:32.880
the definition of cardinality the strange behavior becomes a natural consequence something to smile at

01:09:32.880 --> 01:09:41.120
rather than worry over the image that stays is a soft one an endless hallway of closed doors

01:09:42.400 --> 01:09:52.000
each shift of a guest a quiet step deeper into an idea not a problem to solve but a landscape to

01:09:52.000 --> 01:10:00.960
wonder the doors never stop and the mind relaxing does not need to count them it's simply notices

01:10:00.960 --> 01:10:10.880
that the corridor extends that a new room is always waiting and that the notion of full is not as fixed

01:10:10.880 --> 01:10:19.600
as it first seemed the numbers go on and so does the quiet the night itself has a riddling shape

01:10:19.600 --> 01:10:28.880
the mind that followed a sentence looping back on itself can now let that same loop become a cradle

01:10:28.880 --> 01:10:37.280
the set of all sets that do not contain themselves no longer talks at attention it merely hums

01:10:38.400 --> 01:10:46.560
a distant tidy puzzle that can wait until morning even the hotel with its endless corridors of

01:10:46.560 --> 01:10:56.720
even and odd numbers softens into the image of a door that opens on to sleep and then another

01:10:58.080 --> 01:11:07.440
each one quieter than the last there is a kind of sorry in the way wakefulness fades no single grain

01:11:07.440 --> 01:11:16.080
of thought marking the shift from one state to the next the ship of the self reassembled

01:11:16.080 --> 01:11:25.600
plank by plank with every exhale drifts without needing to ask whether it is still the same vessel

01:11:25.600 --> 01:11:33.920
and time which seen oh divided into halves that never quite arrived now does the gentlest thing

01:11:34.640 --> 01:11:42.000
it suspends its chase the breath draws a slow arc that is enough the paradoxes stay

01:11:42.800 --> 01:11:52.000
but they ask nothing they simply keep watch light as logic can be while the last thought dissolves

01:11:52.000 --> 01:11:55.440
into the pleasant blank were all of it rests