Speaker 1: Welcome to the deep Dive today. Our mission is to
Speaker 1: basically decode the invisible architecture that's operating inside your computer
Speaker 1: right now.
Speaker 2: Yeah, we are taking the physical breaks of reality, you know, steel, glass, concrete,
Speaker 2: and we're just trading them completely for pure thought exactly.
Speaker 1: We are going to explore some pretty mind bending stuff
Speaker 1: like infinite sets, digital logic, and will end with this
Speaker 1: wild mathematical puzzle about thirty six military officers.
Speaker 2: Oh, the Eiler one. Yeah, that is such a good.
Speaker 1: Story, right, It famously proved a mathematical genius wrong and
Speaker 1: actually made the front page of the New York Times.
Speaker 2: It really is an incredible journey. And you know, while
Speaker 2: the term discrete mathematics sounds like a well, like a
Speaker 2: purely academic, intimidating concept, it is actually the ultimate toolkit
Speaker 2: for clear thinking.
Speaker 1: Which is something I think everyone could use a bit
Speaker 1: more of.
Speaker 2: Oh, definitely. It's essentially the discipline of breaking down these massive,
Speaker 2: chaotic problems into solvable, totally logical steps, which is a
Speaker 2: highly practical skill for you the listener, whether you are
Speaker 2: writing complex software code or honestly just trying to organize
Speaker 2: the logistics of your daily life.
Speaker 1: Yeah, Okay, so let's unpack this. We are pulling all
Speaker 1: of this from Scheriman, Shridharan and ar Bellachristian's textbook on
Speaker 1: the Foundations of Discrete Math, right, and what's really fascinating
Speaker 1: about their approach is how well we have to start
Speaker 1: by defining the absolute baselines before we can build anything else.
Speaker 1: Like we all generally know the basics of set theory, right,
Speaker 1: just grouping unique elements together, sure, like a basket of
Speaker 1: fruit exactly. But where this deep dive gets interesting is
Speaker 1: how rigorously the text defines the boundaries of those sets.
Speaker 2: Yeah, because of the mathematical reality is that you just
Speaker 2: cannot manipulate objects until you know exactly where the borders are.
Speaker 2: And this is where concepts like de Morgan's laws come
Speaker 2: into play.
Speaker 1: Right, which sounds intense.
Speaker 2: It sounds worse than it is. A fundamental rule from
Speaker 2: the source material states that the complement of a union
Speaker 2: is the intersection of the compliments.
Speaker 1: Okay, wait, let's translate that. So, rather than looking at
Speaker 1: sets as just buckets of items, de Morgan's laws look
Speaker 1: at the logical boundaries. So if we have say set
Speaker 1: A and Set B, the union is everything inside both
Speaker 1: sets combined, and the complement of that union is just
Speaker 1: the entire rest of the universe, Like everything that is
Speaker 1: strictly outside of both A and B.
Speaker 2: Exactly, which de Morgan mathematically proved is completely identical to
Speaker 2: taking everything outside of A and then finding the exact
Speaker 2: places where it overlaps with everything outside of B.
Speaker 1: Wow.
Speaker 2: Okay, yeah, you are literally defining what a thing is
Speaker 2: by mathematically proving what it isn't You're establishing this perfect
Speaker 2: logical symmetry.
Speaker 1: And once we establish those boundaries, then we can start
Speaker 1: organizing the chaos. Right, Because the text has this whole
Speaker 1: section on equivalence relations.
Speaker 2: Right, which is just a fancy way of saying we
Speaker 2: need a mechanism for grouping things based on a specific,
Speaker 2: totally unbending rule.
Speaker 1: So the example they use is grouping integers by whether
Speaker 1: their difference is a multiple of five.
Speaker 2: Yeah, and think about the mechanics of what that rule
Speaker 2: actually accomplishes. If you group integers modulo five, you take
Speaker 2: the infinite, literally never ending sea of whole numbers, just
Speaker 2: endless numbers, right, and you perfectly partition them into exactly
Speaker 2: five neat, non overlapping equivalence classes. Everything finds an exact
Speaker 2: non contradictory place. You are imposing a finite order onto
Speaker 2: an infinite universe.
Speaker 1: Which is awesome until human intuition completely breaks down, because
Speaker 1: then we get into cardinality.
Speaker 2: Ah yeah, the infinities, right.
Speaker 1: Cardinality is the mathematical measure of the size of a set.
Speaker 1: For a finite set, you just you know, count the
Speaker 1: items one, two, three.
Speaker 2: But when we get to infinite sets, you obviously can't
Speaker 2: count them. You have to compare them logically, which leads
Speaker 2: us straight to the Shorter Bernstein theorem.
Speaker 1: And Shorter Burnstein is just a brilliant solution to this
Speaker 1: exact problem of infinity. The theorem basically states that if
Speaker 1: set X is smaller than or equal to set why,
Speaker 1: and set why is smaller than or equal to set ACT,
Speaker 1: then they.
Speaker 2: Must be the exact same size.
Speaker 1: One incredibly obvious if we're talking about physical objects, like
Speaker 1: if I have equal or fewer books than you, and
Speaker 1: you have equal or fewer books than me, we obviously
Speaker 1: have the exact same number.
Speaker 2: Of books, right. But proving that holds true when the
Speaker 2: sets are infinite, when you literally cannot count the books,
Speaker 2: that requires a very specific mechanism.
Speaker 1: Because you can't just tally them up exactly.
Speaker 2: It requires mapping. So since set X is smaller than
Speaker 2: or equal to HY, we know every item in X
Speaker 2: can find a unique partner and why even if Y
Speaker 2: has some items left over, And because why is smaller
Speaker 2: than or equal to X, every item and Y can
Speaker 2: map back to a unique partner in X.
Speaker 1: So you basically play ping pong between the two infinities.
Speaker 2: You do, you map an element from X to Y,
Speaker 2: and then map that new element back to X, and
Speaker 2: so on. This creates these chains of connections.
Speaker 1: Like invisible threads between them.
Speaker 2: Yeah, some chains might loop in a circle, some might
Speaker 2: just stop, and some might bounce back and forth infinitely.
Speaker 2: But because every single element in both sets belongs to
Speaker 2: exactly one of these chains, you can use to perfectly
Speaker 2: pair up every item and X with an item and
Speaker 2: Y with zero leftovers.
Speaker 1: Oh wow, Yeah.
Speaker 2: Guarantees a one to one correspondence without ever needing to
Speaker 2: actually count a single item.
Speaker 1: So you use the structure of the chains to prove
Speaker 1: the sizes are identical. But I feel like the real
Speaker 1: twist on infinity comes when we look at the power set.
Speaker 2: Oh Canter's proof?
Speaker 1: Yes, yes, So for the listener, the power set of
Speaker 1: a set is the set of all its possible subsets.
Speaker 1: So if your original set is a list of three ingredients,
Speaker 1: the power set is every possible recipe you can make
Speaker 1: by combining them, including, you know, making nothing at all, right,
Speaker 1: the empty set exactly. But the text details Canter's proof,
Speaker 1: which demonstrates that the power set of any set is
Speaker 1: always strictly larger than the original set itself. Wait, so
Speaker 1: if a set is already infinite, its power set is
Speaker 1: a bigger infinity. How can something be bigger than endless?
Speaker 2: I know it hurts your brain a little, Yeah, but
Speaker 2: the mechanism can't use for this proof relies on a
Speaker 2: logical contradiction involving what we call ordinary and extraordinary elements. Okay,
Speaker 2: Imagine you try to pair every single item in an
Speaker 2: infinite set X to a unique subset in its power set.
Speaker 2: You assume you can match them one to one perfectly.
Speaker 1: So we are setting up our ping pong match again,
Speaker 1: hoping to prove they are the same size exactly.
Speaker 2: Now, some elements might happen to map to a subset
Speaker 2: that actually contains that element. The proof labels these elements
Speaker 2: as ordinary okay, But other elements might map to a
Speaker 2: subset that does not contain the element itself. These are
Speaker 2: labeled extraordinary.
Speaker 1: Okay, let's unpack this. Let's gather all those extraordinary elements,
Speaker 1: the ones that don't belong in their partner subsets, and
Speaker 1: put them into a brand new subset. Let's call it
Speaker 1: the misfit subset.
Speaker 2: I like that. So, since it is a combination of elements,
Speaker 2: this misfit subset must exist within the power set somewhere right.
Speaker 2: And because we assumed every subset has a unique partner
Speaker 2: element from our original set, this misfit subset must have
Speaker 2: a partner item. Let's call that partner item Bob.
Speaker 1: Bob the misfit Bob.
Speaker 2: So does Bob belong inside the misfit subset?
Speaker 1: Wait, hold on, walk me through that one more time.
Speaker 1: I'm losing the threat of the paradox here. If Bob
Speaker 1: is the partner to the misfit subset, where does he
Speaker 1: logically go.
Speaker 2: Let's just test both options. Option one. Bob is inside
Speaker 2: the misfit subset. But remember the rule for being a misfit.
Speaker 2: You are only extraordinary if you do not belong to
Speaker 2: your partnered subset.
Speaker 1: Oh, so, if Bob is inside his partnered subset. He
Speaker 1: is ordinary exactly, and.
Speaker 2: If he's ordinary, he cannot be allowed inside a subset
Speaker 2: made exclusively of misfits.
Speaker 1: Right, so we move to option two. Bob is not
Speaker 1: inside the misfit subset, but if he is.
Speaker 2: Not in his partnered subset, that makes him a misfit
Speaker 2: by definition, which means he should be in it.
Speaker 1: Oh, that's wild. It's an impossible loop. He can only
Speaker 1: be in the subset if he isn't in it, and
Speaker 1: he can only be kept out if he belongs in it.
Speaker 2: Right, the logical rule literally breaks its own definition. That
Speaker 2: contradiction proves our initial assumption was wrong. A perfect one
Speaker 2: to one mapping is impossible. There are always more subsets
Speaker 2: than elements, so the power set is a fundamentally larger
Speaker 2: tier of infinity.
Speaker 1: So there are sizes of infinity that literally dwarf other infinities.
Speaker 1: That's okay, we have our sets, We have these different
Speaker 1: infinite piers, but we can't just leave them sitting in
Speaker 1: an abstract pile.
Speaker 2: Now we have to organize them, right, which brings us.
Speaker 1: To the next structural block, partially ordered sets or posits.
Speaker 2: Yeah, puzzits introduce hierarchical structure, we start looking at how
Speaker 2: things relate to one another. Visually, you can map a
Speaker 2: posit using a Hassei diagram, where larger items sit physically
Speaker 2: above smaller ones, connected by vertical lines.
Speaker 1: It looks a lot like a corporate or chart.
Speaker 2: Honestly, it really does.
Speaker 1: But the partially impartially ordered is the crucial distinction here,
Speaker 1: because in a total order, like numbers on a number line,
Speaker 1: everything can be compared. Three is less than four, four
Speaker 1: is less than five, But in a partial order, some
Speaker 1: elements are completely incomparable.
Speaker 2: The Org chart analogy is perfect for this. Think about
Speaker 2: two mid level managers who both report to the same
Speaker 2: vice president and they manage the same entry level employees
Speaker 2: beneath them. Okay, they share upper bounds and lower bounds,
Speaker 2: but they have no direct authority over each other. They
Speaker 2: sit at the exact same level, entirely unconnected. You cannot
Speaker 2: say manager A is greater than manager B.
Speaker 1: Right, So posits give us structure, but it can be
Speaker 1: a bit loose and disconnected. To get closer to the
Speaker 1: actual machinery of computer science, we have to tighten the rules.
Speaker 1: We move from posits to lattices.
Speaker 2: Right, and lattices are a highly specific, honestly beautiful refinement
Speaker 2: of a posit. In a lattice, any two elements you
Speaker 2: choose must have two things, a meat and a join.
Speaker 1: Okay, define those for me.
Speaker 2: The meat or infamum is their greatest lower bound, the
Speaker 2: join or supremum is their least upper bound.
Speaker 1: So no matter which two items you pick on the chart,
Speaker 1: even if they are unconnected to each other, you can
Speaker 1: always trace their lines down to a shared foundation the meat,
Speaker 1: and trace their lines up to a shared ceiling of
Speaker 1: the joint.
Speaker 2: Exactly, and that rigid structure yields the duality principle. The
Speaker 2: symmetry of a lattice dictates that any valid logical formula
Speaker 2: remains completely valid if you just swap every meat with
Speaker 2: a join and every join with a meat. It's a
Speaker 2: mathematical mirror image.
Speaker 1: But of course, not all lattice is play nice. There
Speaker 1: are two specific formations that totally ruin the structural symmetry,
Speaker 1: the diamond lattice and the pentagonal lattice. Ah.
Speaker 2: Yes, the troublemakers of discrete math. We call them that
Speaker 2: because they break two highly desirable mathematical rules, distributivity and modularity.
Speaker 2: A distributive lattice behaves like regular algebra, where you can
Speaker 2: distribute terms across an equation. The diamond and pentagonal lattices
Speaker 2: warp the math, so distributivity just fails.
Speaker 1: How does a pentagon shape actually warp the math? Though? Like,
Speaker 1: what does it conceptually mean for the lattice to fail?
Speaker 2: Okay, go back to your org chart. Imagine a mid
Speaker 2: level manager who reports to the VP, but somehow simultaneously
Speaker 2: sidesteps the director who is supposedly on on the chain
Speaker 2: of command directly above them.
Speaker 1: Oh, so they have a shortcut exactly.
Speaker 2: The pentagonal lattice creates a structural bypass. It breaks the
Speaker 2: logical chain of command. If you try to distribute a
Speaker 2: mathematical operation across that lattice, the bypass causes the equation
Speaker 2: to resolve to two completely different answers depending on the
Speaker 2: path you take. It ruins the predictability of algebraic distributivity and.
Speaker 1: The mathematical reality detailed in the source material is that
Speaker 1: any non modular lattice secretly contains this pentagonal shape hiding
Speaker 1: inside it.
Speaker 2: Yeah, it's wild.
Speaker 1: So if a logical structure is fundamentally flawed in that
Speaker 1: specific way, that five sided bypass is buried in the
Speaker 1: architecture somewhere, just causing trouble.
Speaker 2: Which leads us to the ultimate structural question. What happens
Speaker 2: when you build a lattice with zero bypasses, a perfectly
Speaker 2: distributive lattice where every single element has a unique exact opposite,
Speaker 2: a complement, and the entire structure is bounded by an
Speaker 2: ultimate ceiling of a universal one and a basement floor
Speaker 2: of a universal zero.
Speaker 1: Here's where it gets really interesting. You get Boolean algebra.
Speaker 1: George Bull developed this in eighteen fifty. He was literally
Speaker 1: just trying to mathematically manipulate logical statements true or false,
Speaker 1: A and D or not t pure abstract mass exactly.
Speaker 1: But by proving that logical thought could be manipulated algebraically,
Speaker 1: he inadvertently laid the absolute groundwork for the digital computer,
Speaker 1: decades before anyone even dreamed of building physical hardware.
Speaker 2: Boolean algebra is the invisible architecture of the digital age.
Speaker 2: It's a complemented distributive lattice. Every element has a unique opposite,
Speaker 2: everything operates strictly between zero and one, and there are
Speaker 2: absolutely no pentagonal bypasses to warp the logic.
Speaker 1: Okay, So looking at the representation theorem for finite boolean algebras,
Speaker 1: my takeaway is that it's basically squashing all this complex
Speaker 1: logic into simple binary strings. Is it really just saying
Speaker 1: everything reduces down to ones and l's.
Speaker 2: It is not just reducing them, it is preserving them perfectly.
Speaker 2: The representation theorem guarantees a lossless translation.
Speaker 1: Lossless Okay, you weren't.
Speaker 2: Losing the complexity of the algebra. You are mapping it.
Speaker 2: Think of the atoms of the lattice. These are the
Speaker 2: smallest indivisible elements, sitting just above that universal zero floor.
Speaker 2: The theorem proves that any element in the entire infinite
Speaker 2: algebra can be constructed by a unique combination of these
Speaker 2: foundational atoms.
Speaker 1: So if you're lettice relies on say five specific atoms,
Speaker 1: you just need a five digit sequence of ones and
Speaker 1: nails exactly.
Speaker 2: The sequence serves as a characteristic vector. It tells you
Speaker 2: exactly which atoms are on and which are off to
Speaker 2: build any specific logical argument in that system.
Speaker 1: So you are perfectly translating abstract logical thought into binary code.
Speaker 2: Yes, that mathematical mapping is the exact reason we can
Speaker 2: take pure logic and run it through the physical electrical
Speaker 2: switches of a computer processor.
Speaker 1: Okay, so we've established this perfect binary Boolean universe where
Speaker 1: everything is one or zero. But abstract algebraic rules aren't enough, right,
Speaker 1: We need to know if specific structures can actually exist
Speaker 1: with in those rules in the real world, which pulls
Speaker 1: us out of pure algebra and write into combinatoric commonataorics.
Speaker 2: Yes, this is the study of finite structures. We transition
Speaker 2: from theory into the actual mechanics of existence. The field
Speaker 2: generally splits into two categories, existence problems, which ask if
Speaker 2: a structure can even exist, and enumeration problems, which asks
Speaker 2: how many of them exist?
Speaker 1: So, as an expert, is the real thrill of combinatorics
Speaker 1: actually counting all the variations? Or is it the existence
Speaker 1: problem like the Aha moment of proving something is fundamentally
Speaker 1: possible or impossible without having to physically test every single variation.
Speaker 2: Oh, finding a structural impossibility through an existence problem is
Speaker 2: often much more elegant. You don't need to exhaustively count
Speaker 2: the solutions if you can prove the architecture itself forbids
Speaker 2: any solution from existing in the first place. The mutilated
Speaker 2: checkerboard problem is the classic example of this logical shortcut.
Speaker 1: Okay, picture a standard checkerboard for a second, alternating black
Speaker 1: and white squares sixty four squares total. Now, take a
Speaker 1: pair of scissors and cut off the top right corner
Speaker 1: square and the bottom left corner square. Okay, because they
Speaker 1: are diagonally opposite, they are the exact same color. Let's
Speaker 1: say we removed two white squares, so we are left
Speaker 1: with a mutilated sixty two square board thirty two black
Speaker 1: squares and thirty white squares, right, And.
Speaker 2: The challenge here is to cover this entire sixty two
Speaker 2: square board perfectly with thirty one dominoes. A domino is
Speaker 2: exactly the size of two adjacent squares. If you treat
Speaker 2: this like an enumeration problem, you might try to physically
Speaker 2: test thousands of domino arrangements, and.
Speaker 1: You'd get frustrated every single time you end up with
Speaker 1: two empty squares that aren't touching exactly.
Speaker 2: But the existence problem approach looks for the structural mandate.
Speaker 1: It looks for the parity argument.
Speaker 2: Yes, every single domino, because it covers two adjacent squares,
Speaker 2: must cover exactly one black square and exactly one white square,
Speaker 2: There is no other physical way to place it on
Speaker 2: the grid. Right Therefore, thirty one dominos will perfectly cover
Speaker 2: thirty one black squares and thirty one white squares.
Speaker 1: But our mutilated board has thirty two black squares and
Speaker 1: thirty white squares. It literally doesn't matter how clever you
Speaker 1: are with your arrangement. The fundamental parity of the board
Speaker 1: mathematically clashes with the fundamental parity of the dominos.
Speaker 2: Is it structural impossibility, no trial and error required.
Speaker 1: You locate the invisible wall that prevents a solution which
Speaker 1: transitions us perfectly to the grand finale puzzle of the
Speaker 1: source material. Euler's thirty six officers problem a.
Speaker 2: Leonard Euler is one of the greatest mathematical minds in
Speaker 2: human history, and he encountered an existence problem that really
Speaker 2: challenged his assumptions.
Speaker 1: You have thirty six military officers. They come from six
Speaker 1: different regiments, and they hold six different ranks. Euler wanted
Speaker 1: to arrange them in a six by six grid, sort
Speaker 1: of like a complex Sudoku board, so that no rank
Speaker 1: or regiment repeats in any row or column.
Speaker 2: Every row must have one of each rank, in one
Speaker 2: of each regiment, and the exact same requirement applies to the.
Speaker 1: Columns right an Eiler tried to find a solution and
Speaker 1: just couldn't. Another mathematician named Terry eventually proved by exhaustive
Speaker 1: enumeration that the six by six grid is indeed impossible
Speaker 1: to construct. The structural wall is totally real.
Speaker 2: But Eiler didn't just stop at the six by six grid.
Speaker 2: He looked at the math and noticed a pattern. He
Speaker 2: knew a two by two grid was impossible, he knew
Speaker 2: six by six was impossible, so he famously conjectured that
Speaker 2: this specific kind of grid was impossible for any size
Speaker 2: that fit the mathematical formula four and plus two.
Speaker 1: Meaning a ten by ten grid, a fourteen by fourteen grid,
Speaker 1: and eighteen by eighteen grid, all of them fundamentally impossible.
Speaker 2: Yep.
Speaker 1: And for a very long time the mathematical world just
Speaker 1: accepted it, because you know, if Euler says there's a
Speaker 1: structural impossibility, you'd tend to believe him.
Speaker 2: Right. It really serves as a cautionary tale about mathematical assumptions.
Speaker 2: Even a genius of Euler's caliber can spot a pattern
Speaker 2: that turns out to be a fantom. He saw the
Speaker 2: impossibilities at two x two and six by six and
Speaker 2: assumed a structural parity issue that extended infinitely through the numbers.
Speaker 2: He assumed an invisible wall was there, but he was wrong.
Speaker 1: The textbook highlights the incredible twist that occurred in nineteen
Speaker 1: fifty nine three mathematicians Bow's Shrekandan Parker. They earned literally
Speaker 1: the greatest nickname.
Speaker 2: In math history, boiler spoilers.
Speaker 1: Boiler spoilers. They proved that the ten by ten matrix
Speaker 1: is actually possible.
Speaker 2: And the textbook literally prints their ten by ten matrix
Speaker 2: counter example. You can look at the grid and verify
Speaker 2: the ranks and regiments yourself, and.
Speaker 1: Disproving Oiler on a ten by ten grid was such
Speaker 1: a massive shock to the system that it made the
Speaker 1: front page of the New York Times. It was headline
Speaker 1: news that a mathematical wall had been broken.
Speaker 2: It was a triumph of the existence problem. They didn't
Speaker 2: just theoretically argue against Eiler. They manifested the structure. They
Speaker 2: proved that the invisible architecture Oiler thought was blocking the
Speaker 2: ten by ten grid simply didn't exist. The math totally
Speaker 2: allowed for a solution.
Speaker 1: Wow, so we've covered a massive amount of invisible architecture today.
Speaker 1: We started by looking at the infinite resiicans of the
Speaker 1: power set, establishing that some infinities are mathematically low larger
Speaker 1: than others. We take those sets and built the hierarchical
Speaker 1: ladders of puffits and lattices. We saw how the strict
Speaker 1: distributive rules of Boolean algebra converted those ladders into the
Speaker 1: lossless ones and zeros of digital computer code. And finally,
Speaker 1: we saw how applying those discrete rules to combinations of
Speaker 1: items allowed us to see the actual structural walls of
Speaker 1: the mutilated checkerboard.
Speaker 2: And break through the assumed walls of Euler's thirty six
Speaker 2: officers exactly. You know, the true value of understanding discrete
Speaker 2: mathematics is that it fundamentally changes how you look at
Speaker 2: the world around you.
Speaker 1: Yeah.
Speaker 2: You start to spot the hidden parity arguments in your
Speaker 2: daily life.
Speaker 1: Yeah, that makes sense.
Speaker 2: You recognize when a project at work is failing, not
Speaker 2: because you aren't trying hard enough, or because your team
Speaker 2: lacks talent, but because of a structural impossibility. You are
Speaker 2: trying to cover thirty two black squares and thirty white
Speaker 2: squares with dominoes. You learn to stop frantically testing different
Speaker 2: domino arrangements, and you start analyzing the actual structure of
Speaker 2: the board.
Speaker 1: It gives you the X ray glasses to see the
Speaker 1: blueprints of logic. And I want to leave you the listener,
Speaker 1: with a final, somewhat provocative thought. Inspired by the representation
Speaker 1: theorem we discussed earlier, We learned that any finite Boolean algebra,
Speaker 1: no matter how massively complex, can be modeled perfectly by
Speaker 1: its smallest parts, its foundational binary atoms. Well, in modern physics,
Speaker 1: we are increasingly understanding that the physical universe we inhabit
Speaker 1: is fundamentally discreete. At the deepest level. Reality doesn't seem
Speaker 1: to be a smooth, continuous line, but rather a grid
Speaker 1: of finite quantum states, discrete particles, and plank lengths.
Speaker 2: Wow.
Speaker 1: So if the universe is completely discreete and governed by
Speaker 1: finite physical interactions, does that mean the entire complexity of
Speaker 1: our reality? Every thought, you have, every star on this guy,
Speaker 1: every physical law is at its core, just one massive
Speaker 1: calculable Boolean algebra. Are we just living inside the ultimate
Speaker 1: set of atomic ones and eighties? Something for you to
Speaker 1: ponder