Why Computers Can't Generate True Random Numbers
43sReveals a surprising limitation of everyday technology, sparking curiosity about how randomness actually works.
▶ Play ClipAI Summary
This video explores the philosophical and mathematical challenge of selecting a number at random, highlighting why true randomness is impossible for computers and why ordering the real numbers is a fundamental problem. It traces the historical mission beginning in 1870 to resolve this paradox, setting the stage for a deeper discussion.
Chapters
[00:02]
Computers lack true randomness
Computers cannot generate true random numbers because formulas always produce the same result; they rely on algorithms seeded by local time to simulate randomness.
[00:16]
Rules for selection
In mathematics, selection requires following a rule, such as choosing the smallest element, which works for sets like positive integers (1) or primes (2).
[00:31]
The real numbers problem
The real numbers include all positives, negatives, fractions, and irrationals like pi, making it impossible to define a smallest element because they extend to negative infinity.
[00:44]
Specific rules fail
Even a specific rule like 'choose the smallest number after one' fails because there is always a smaller number (1.01, 1.0001, etc.), leading to an infinite regress.
[01:13]
The paradox of infinite options
Despite knowing there are infinite real numbers, we cannot specify an order (next, previous, first, last), leaving us stuck on how to pick one.
[01:27]
Historical mission begins
In 1870, a man took on the task of ordering the real numbers definitively, risking his life—and nearly dying—in the process.
The video underscores a deep mathematical paradox: true randomness is unattainable, and even with infinite choices, we cannot define a selection rule for the real numbers. This sets the stage for exploring historical attempts to resolve the ordering problem.
Study Flashcards (6)
Why can't computers generate true random numbers?
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Why can't computers generate true random numbers?
easy
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Because formulas always produce the same result; they use algorithms seeded by local time to simulate randomness.
00:02
What is the smallest positive integer?
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What is the smallest positive integer?
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1
00:31
What is the smallest prime number?
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What is the smallest prime number?
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2
00:31
Why is it impossible to choose the smallest real number?
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Why is it impossible to choose the smallest real number?
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Because the real numbers extend to negative infinity, so there is no smallest element.
00:44
What happens when you try to specify the smallest number after one?
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What happens when you try to specify the smallest number after one?
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You get stuck in an infinite regress: 1.01, 1.0001, 1.00000001, and so on.
00:58
In what year did the mission to order the real numbers begin?
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In what year did the mission to order the real numbers begin?
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1870
01:27
💡 Key Takeaways
📊
Computers simulate randomness
Clarifies a common misconception about computer randomness, grounding the discussion in practical reality.
00:02
💡
The real numbers are unorderable
Highlights a fundamental mathematical limitation that challenges intuitive notions of selection.
00:44
📊
Historical quest begins
Introduces a dramatic narrative hook that connects abstract math to human endeavor.
01:27Full Transcript
[00:02] random because formulas always give the same result, which is why computers don't have true random number generators. Instead, they usually run an algorithm on your current local time to generate numbers that appear random. So,
[00:16] if we can't pick randomly, how do we select anything in math? Well, the only way is to follow a rule of some sort. So, a rule could be always choose the smallest thing. For example, if we're looking at whole positive integers, the
[00:31] smallest is one. For prime numbers, it would be two. Easy. But, what about the real numbers? That's any number, positive, negative, whole, fraction, even irrational, like pi or the square root of two.
[00:44] Now, try to choose the smallest one. It's impossible. The real numbers stretch off to negative infinity. Even if we try to fix our rule by making it super specific, like choose the smallest number after one, we still get
[00:58] stuck. There's 1.01, then 1.0001, then 1.00000001, and so on. and so on. So, really, what number comes after one?
[01:13] If we can't begin to specify the order of the real numbers, next and previous, first and last, we're stuck. The ridiculous part is we know we have infinite options, but despite that, we can't figure out how to just pick one.
[01:27] The mission to resolve this began with one man in 1870. He took on the task of putting the real numbers in a definitive order, even if it killed him. And, it nearly did.
