Topic Index: P

P equal to NP

“for which you can easily recognize solutions, generically, there's an algorithmic shortcut, akin to what we saw in shortest paths. Even though there's an astronomical number of possibilities, by…” (Tim Roughgarden)

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P not equal to NP

“that would be P not equal to NP, or whether we live in World Number II. That would be P equal to NP. So I mentioned earlier, John von Neumann's, slightly cynical quote about how you don't really…” (Tim Roughgarden)

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P versus NP

“in all of mathematics, certainly the deepest open question in computer science, P versus NP, which will be the subject of the next two episodes. Hope to see you there.” (Tim Roughgarden)

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P versus NP question

“So, episode four, we're gonna do a deep dive on the most important open question in all of computer science, one of the most important open questions in really all of mathematics, something which is…” (Tim Roughgarden)

Computation as a Universal and Fundamental Concept · Intro: Computation and Its Limits▶︎ watch 4:33 · read ↗︎

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paint-based theory of surface area

“So we discussed the idea, look, maybe it's a sort of paint-based theory of surface area. I wanna say, look, if I have a geometric object, a container, then what does it mean to” (Joel David Hamkins)

Lectures on Infinity · Lecture 3: The Paradox of Giants: Strange Consequences in High Dimension▶︎ watch 25:07 · read ↗︎

painter's paradox

“Okay. So I call that the painter's paradox. So let's now do some other paradoxes of higher dimension.” (Joel David Hamkins)

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pairing function

“So it's a sort of nicer formula. This is called the Cantor pairing function. A pairing function is a function that takes pairs of objects and gives you back individual objects in a way that is a…” (Joel David Hamkins)

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paradox of giants

“I'm Joel David Hankins. Welcome to these lectures on infinity, and today, I want to tell you about the paradox of giants. So according to legend, giants once roamed the Earth. Everyone knows Odysseus…” (Joel David Hamkins)

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paradox of supertasks

“We're gonna see Zeno's paradox, and the paradox of supertasks, Galileo's paradox, the distinction between the potentially infinite and the actually infinite. Eventually, we'll find the distinction…” (Joel David Hamkins)

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paradox of the miniature human

“So there's a similar thing with the, not just giants, but the paradox of the miniature human. Maybe you've seen some of these Hollywood films, Downsizing or Ant Man or Honey, I Shrunk the Kids. It's…” (Joel David Hamkins)

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paradoxes of infinity

“to this series of lectures on infinity. I wanna show you all my favorite conundrums and paradoxes of infinity. We're gonna see Zeno's paradox, and the paradox of supertasks, Galileo's paradox, the…” (Joel David Hamkins)

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Paradoxes of the Infinite

“So, this is in Paradoxes of the Infinite, 1851. And what Bolzano observed is the following. Bolzano observed that, well, look, if you have a” (Joel David Hamkins)

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parallel postulate

“that which occurs in geometry. When, for example, you consider the parallel postulate in geometry and the question of whether it's possible to prove it from the other axioms of set theory, this was…” (Joel David Hamkins)

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parallelist

“It's not an epiphenomenalist position, it's not a parallelist position. It's what's called an interactive dualist position. It's the kind of dualist position that Descartes had, okay? And that not…” (David Albert)

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Parmenides

“And I'll give you a little digression on Parmenides. Since today is the day for digressions.” (Richard Polt)

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partial products

“then one times the top, okay? So I'll spare you all of the, all of the details, but basically you wind up with the following, the first partial product's 22,712. Again, then you do a shift, the next…” (Tim Roughgarden)

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parts and wholes

“So, we're thinking about parts and wholes, because we're breaking a whole down into the parts. Now here's a question that you've probably heard. Is the whole more than the sum of its parts? I think…” (Richard Polt)

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passive observation

“up until quantum mechanics of what you might call passive observation, okay? Everybody knew, of course, that in order to observe anything, in order to get any information about any physical system,…” (David Albert)

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Paul Feyerabend

“When I first read the Theaetetus, it was taught by Paul Feyerabend, who is a brilliant philosopher of science at UC-Berkeley, and the main thing that he focused on was some uncanny parallels between…” (Richard Polt)

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Pavlov

“B.F. Skinner is probably its most famous practitioner. It was created by Pavlov. Pavlov was a Russian psychologist, very famous for his experiments with dogs. He had a lot of dogs, and what he would…” (Lee Braver)

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Pavlov's dogs

“Science is never about one thing or two things. When the lessons of Pavlov's dogs were not about Pavlov's dogs, they were not that particular group of canines. They're about animal minds, about dogs…” (Lee Braver)

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Peano

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Pentium chip

“When the first Pentium chip came out, an early computer chip, it had to be recalled at huge, huge expense because it made this one mathematical error, very, very complex error, and they, the” (Lee Braver)

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perfect set

“which is a closed set with no isolated points. And Cantor knew that every perfect set, every non-empty perfect set, was equinumerous with the whole real line. So therefore, this shows that every…” (Joel David Hamkins)

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perfect squares

“For example, zero is zero squared and one is one squared, two squared is four and three squared is nine and so on. Four squared is 16 and so on. And if you notice, the squares, the perfect squares…” (Joel David Hamkins)

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Peter Shor

“to 2 times 2 times 5 times 5, okay? So it turns out, and this was a big, big deal when this was first proved by Peter Shor in 1994, it turns out that if you can build large-scale quantum computers,…” (Tim Roughgarden)

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Phaedo

“And in Theto, which is about Socrates' death, the narrator says, "Oh, Plato couldn't be there. He was sick." Which is crazy. How could he miss the death of Socrates? So he's deliberately taking…” (Richard Polt)

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Phaedrus

“There's also a dialogue named Phaedrus... Where Socrates talks about writing. He says that, "Writing was invented in” (Richard Polt)

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Phaenarete

“extended metaphor in which Socrates compares himself to his mother, who apparently was a midwife with the wonderful name Phainorete, meaning she who brings virtue to light or virtue shower, which…” (Richard Polt)

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phenomenology

“is less, well, you suspect there's something sophistical about this whole thing. The next one is less sophistical. It's more based on a phenomenology, a description of the actual experience of sense…” (Richard Polt)

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philosopher kings

“And in fact, he goes on to describe philosopher kings or philosopher rulers who would necessarily have to have both kinds of wisdom. So okay, I think we have a rough idea of what's going on in the…” (Richard Polt)

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Phoenicians

“and analysis that you get in a literate culture. They owed their alphabet to the Phoenicians, by the way, who had a, they had a near alphabet. They didn't have letters for vowels, but everything…” (Richard Polt)

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photons

“are easier to do with photons. That's not gonna affect the sort of logical structure of the situation, which is mainly what I wanna get across. And I'm also going to say that these stories are gonna…” (David Albert)

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picture of the universe

“The reason why it became so crucial at that point was because this was the beginning of the scientific revolution. For a long time, there was one picture of the universe, one sense of how things…” (Lee Braver)

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pig and the baboon

“And in particular, he brings up the pig and the baboon, they're not smarter than the pig or a baboon. Yeah. So, it's not only people who all already have knowledge, but other animals too, right? They…” (Richard Polt)

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pigeonhole principle

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pineapples

“And his example is pineapples. Pineapples were this new thing that they had. They, they, some explorer had found pineapples and brought them back to, to, to, to England, and people were eating them…” (Lee Braver)

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pink sunglasses

“we believe are actually out there in the world. It is the person with the pink sunglasses looking at a white wall, thinking that the wall is pink, not that their glasses are making it pink. We take…” (Lee Braver)

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placebo effect

“something that would fit right here. something that would fit right here. Placebo effect is very good medicine. Nothing wrong with it. If he can make you feel healthy and feel good, fine. Great…” (Richard Polt)

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Plato

“Doesn't matter. Plato said it doesn't matter. You can say they're wrong. But you have to have a reason. You cannot just say, "It's wrong. Obviously it's wrong. Can't be right." That's not a” (Lee Braver)

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Plato's Theaetetus

“Welcome everybody to my little course on Plato's Theaetetus. My name is Richard Polt. I've been teaching philosophy at Xavier University for over 30 years, and this is a platonic dialogue that I've…” (Richard Polt)

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playing the game and making up the rules at the same time

“So, there were a number of figures, like Descartes, uh, Bacon, Galileo, Newton, Leibniz, who were all doing science and philosophizing about the nature of science simultaneously. They were playing…” (Lee Braver)

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Poincaré conjecture

“Like, you might have heard of some of the other ones, like the Riemann hypothesis, or Navier–Stokes equations, or the one that has been solved actually, the Poincaré conjecture. So those are other…” (Tim Roughgarden)

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Poincaré disk model

“was eventually observed not to be the case, by doing a similar kind of model construction method. If you say, take the Poincaré disk model of hyperbolic space, you know, of hyperbolic geometry, then…” (Joel David Hamkins)

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polar coordinates

“If you're familiar with polar coordinates, where you specify a point not by a point in the plane, not by specifying the x and y coordinates, but rather by specifying the radial and angular…” (Joel David Hamkins)

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polynomial

“algorithm, and the P, that stands for polynomial. Okay, what's a polynomial? You know, polynomials are things like, you know, X plus 3X plus 2, or, you know, X to the tenth minus” (Tim Roughgarden)

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polynomial time

“with you-know-it-when-you-see-it solutions also solvable by a fast algorithm? And remember, we're identifying fast algorithms with algorithms that run in polynomial time, algorithms that when you…” (Tim Roughgarden)

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polynomial-time algorithm

“So we're going to identify fast algorithms with what's known as a polynomial-time algorithm, and I'll tell you how to think about that in concrete terms. But just for now, the terminology is…” (Tim Roughgarden)

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position

“But this is an example of exactly the same thing. We call observable properties like color and hardness incompatible with one another because our experience is that there is no procedure by means of…” (David Albert)

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positional number system

“8547, 8,547, what it means is, this is a positional number system, so it means that we have eight thousands, and five hundreds, four tens, and seven ones.” (Joel David Hamkins)

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positive rational numbers

“And so these runners all wanted to check into the hotel. They're running the half marathon, not the full marathon, because they really like fractions. So, can the manager accommodate all of the…” (Joel David Hamkins)

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possible

“there is a larger V that thinks phi is true. So, maybe there's some V up here where phi becomes true. So, something is possible at a world if it becomes true in a larger world. And U thinks that phi…” (Joel David Hamkins)

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post-quantum cryptography

“post-quantum cryptography, which means cryptographic systems which remain secure even when we have available to us large-scale quantum computers. And a lot of super smart cryptographers have come up…” (Tim Roughgarden)

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Postulate of Collapse

“of particle number one. This is the postulate of collapse, von Neumann's rule number two. This looks strikingly non-local, and Einstein noticed this. Okay? I don't know the history” (David Albert)

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potential infinity

“infinite sum which has a kind of potentialist character. There's this distinction between potential infinity and actual infinity. Potential infinity is the idea that you never complete the infinite…” (Joel David Hamkins)

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potentialism

“and philosophers sometimes disagree about what exactly that means because, according to the philosophy of potentialism, yes, there are infinitely many numbers. You can have as many as you like, more…” (Joel David Hamkins)

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potentialist character

“you never complete the infinite task. You only take more and more, finitely much of it, at a time. So the very meaning of an infinite sum has a kind of potentialist character because we look at what…” (Joel David Hamkins)

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potentialist translation

“So, let me talk a little bit about the what's called the potentialist translation. So we were already doing the potentialist translation before when I was translating assertions of mathematics into…” (Joel David Hamkins)

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potentially infinite

“paradox of supertasks, Galileo's paradox, the distinction between the potentially infinite and the actually infinite. Eventually, we'll find the distinction between the countably infinite and the…” (Joel David Hamkins)

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potion for wisdom and memory

“Egypt by a god named Famous, who brought this wonderful new invention to the King of the gods, Thoth, and said, 'Look, I've discovered a potion for wisdom and memory.'" And Thoth says, "No, you've…” (Richard Polt)

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power of computation

“efficient computation, and then, on the other hand, the limitations of computation, okay? So both of those themes are obviously already there in Turing's 1936 paper, the introduction of Turing…” (Tim Roughgarden)

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power of pre-processing

“And let me call it the power of pre-processing. Okay?” (Tim Roughgarden)

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power set

“is the power set of the natural numbers. You know, the power set of a set is the collection of all subsets of the set. So I could write it like this. It's the set of all sets A such that A is a…” (Joel David Hamkins)

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power set of the natural numbers

“So, for example, we can see that the power set of the natural numbers is equinumerous with Cantor space, which is the set of binary sequences, and that's equinumerous with the real line. So these…” (Joel David Hamkins)

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power set of the real numbers

“And so therefore, this set here, Z to the R, the functions from the reals to the integers, it's at least as big as the power set of the real numbers, which is strictly bigger than the real numbers.…” (Joel David Hamkins)

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prediction

“And so if we're allowed, which maybe is a big if, to look beyond the present moment of perception and allow for predictions, you know, the normal human being in real life is going to believe that…” (Richard Polt)

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prime factorization

“We would have to say, for every prime except one, and so on, or for example, in the uniqueness of prime factorization, every number can be factored uniquely into primes. It wouldn't be true anymore…” (Joel David Hamkins)

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Principle of Locality

“So, I want to talk about a certain very deep intuition, very primordial intuition that we have about the structure of the world, a very deep conviction” (David Albert)

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probabilistic law

“So- But there, but there is predictability. Not predictability on any individual outcome- Yes, there's statistical predictability. ... but there is. And that's what differs this from magic. Or, or…” (David Albert)

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probability claims

“Of course, you're absolutely right that the business of being fully clear and explicit about what we mean by probability claims is a highly non-trivial business, philosophically. Well, maybe that was…” (David Albert)

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probability problem in Many-Worlds

“And there's just prima facie a puzzle about a theory like this, okay? If the world is just deterministically splitting all the time and there is for sure, at the end of the experiment,” (David Albert)

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probability zero

“So, what that means is that the chance that the ball survives for infinitely many steps is less than this number for every n, which means it must have probability zero. So therefore, what I'm saying…” (Joel David Hamkins)

Lectures on Infinity · Lecture 2: Supertasks: Doing Infinitely Many Things▶︎ watch 21:01 · read ↗︎

problem of induction

“It is difficult, but it's really, extraordinary in its conclusion. It's come to be called the problem of induction. He doesn't call it that, and I'm not gonna get into any real details about what…” (Lee Braver)

From Descartes to Nietzsche: The Birth of Science and the Death of God · Lecture 4: Why You Can't Prove Tomorrow▶︎ watch 0:11 · read ↗︎

process philosophy

“So, there are no things. There are only motions, intersecting, interacting. You can picture it this way. There's an active motion and a passive motion reacting to it.” (Richard Polt)

Plato’s Theaetetus: An Inquiry into Knowledge · Lecture 2: When Every Perception is True▶︎ watch 20:23 · read ↗︎

Prodicus

“So if philosopher means lover of wisdom, sophist means wise guy. And these wise guys would travel around Greece and they would teach on all sorts of topics, but especially rhetoric, the art of…” (Richard Polt)

Plato’s Theaetetus: An Inquiry into Knowledge · Lecture 1: Is Knowledge Just Perception?▶︎ watch 28:40 · read ↗︎

programs as data

“fundamental concepts in computer architecture, which is programs as data. So you can have value stored in the memory of your computer. It might represent something like your bank balance, you know,…” (Tim Roughgarden)

Computation as a Universal and Fundamental Concept · Lecture 1: Is There Anything Computers Can't Do?▶︎ watch 32:43 · read ↗︎

projective sets

“One wants to maybe make the move to the projective sets, which are the sets of reals that you can define by a property that quantifies over integers and also real numbers. So there's a, a rich…” (Joel David Hamkins)

Lectures on Infinity · Lecture 10: Set Theory's Deepest Mystery▶︎ watch 28:36 · read ↗︎

Prolegomena to Any Future Metaphysics

“the first Critique himself. So he wrote this book, this Prolegomena to Any Future Metaphysics, as a sort of a shorter, easier version of his philosophy. Easier, it's still not easy. But it's…” (Lee Braver)

From Descartes to Nietzsche: The Birth of Science and the Death of God · Lecture 5: Kant Wakes from His Dogmatic Sleep▶︎ watch 2:10 · read ↗︎

proof by contradiction

“So if we wanna show that having such a correspondence is impossible, then we can do so by using proof by contradiction. So what we're gonna do for the first step of the proof is suppose toward…” (Joel David Hamkins)

Lectures on Infinity · Lecture 9: Beyond Countable Infinity▶︎ watch 3:15 · read ↗︎

Computation as a Universal and Fundamental Concept · Lecture 1: Is There Anything Computers Can't Do?▶︎ watch 37:32 · read ↗︎

propagation

“a sort of process of propagation of the effect between adjacent spatio-temporal points stretching from the initial cause to the final effectSomething we do know about quantum mechanics now is that any” (David Albert)

Quantum Mechanics and the Crisis of Scientific Realism · Intro: The Crisis In Physics▶︎ watch 12:05 · read ↗︎

proper part

“According to Euclid's principle, the whole of the natural numbers is greater than the proper part consisting of the perfect squares. And yet, Galileo observed, nevertheless, we can make a one-to-one…” (Joel David Hamkins)

Lectures on Infinity · Lecture 6: Galileo's Paradox of Infinity▶︎ watch 19:20 · read ↗︎

Protagoras

“a nothing, a nothing burger, we would say. So Socrates praises this, and he immediately connects it to the view of Protagoras. Protagoras was another Sophist, maybe the most famous of all the…” (Richard Polt)

Plato’s Theaetetus: An Inquiry into Knowledge · Lecture 1: Is Knowledge Just Perception?▶︎ watch 35:07 · read ↗︎

Plato’s Theaetetus: An Inquiry into Knowledge · Lecture 2: When Every Perception is True▶︎ watch 2:56 · read ↗︎

Plato’s Theaetetus: An Inquiry into Knowledge · Lecture 3: Does Relativism Refute Itself?▶︎ watch 1:03 · read ↗︎

Plato’s Theaetetus: An Inquiry into Knowledge · Lecture 4: The Soul Behind the Senses▶︎ watch 19:54 · read ↗︎

psyche

“So, he uses this word, psyche, traditionally translated as soul, but he says, "Call it a psyche or whatever you want to call it." It's a... So he's not dogmatic about this. There's certainly no…” (Richard Polt)

Plato’s Theaetetus: An Inquiry into Knowledge · Lecture 4: The Soul Behind the Senses▶︎ watch 3:02 · read ↗︎

public key encryption

“credit card number when you do that is because your credit card number is hidden from eavesdroppers through encryption, public key encryption. And the most, one of the most common forms of public key…” (Tim Roughgarden)

Computation as a Universal and Fundamental Concept · Lecture 5: AI, Quantum Computing, and Beyond▶︎ watch 7:46 · read ↗︎

purely rational thinking things

“is enabling, because it's what enables us to make use of our past experience when reason could not. If we were to become purely rational thinking things as Descartes wanted, Hume says we would die…” (Lee Braver)

From Descartes to Nietzsche: The Birth of Science and the Death of God · Recap: Making Your Beliefs Your Own▶︎ watch 3:50 · read ↗︎

putting your judgment into words

“So, the second one is putting your judgment into words. Well, why isn't that enough to be an account?” (Richard Polt)

Plato’s Theaetetus: An Inquiry into Knowledge · Lecture 5: Defeat as Philosophical Victory▶︎ watch 14:35 · read ↗︎