WEBVTT 00:00:00.000 --> 00:00:02.520 Picture this for a second. You've got this massive 00:00:02.520 --> 00:00:05.379 subway train, right, packed with hundreds of 00:00:05.379 --> 00:00:08.300 commuters, and it's hurtling through this dark 00:00:08.300 --> 00:00:12.279 underground tunnel at like 60 miles an hour. 00:00:12.580 --> 00:00:16.300 Right. High stakes. Exactly. Or, well, consider 00:00:16.300 --> 00:00:18.480 a totally different scenario. You have a medical 00:00:18.480 --> 00:00:21.539 algorithm scanning a super high resolution MRI, 00:00:22.339 --> 00:00:25.120 and it's trying to determine the exact microscopic 00:00:25.120 --> 00:00:28.480 boundary of a brain tumor. basically to guide 00:00:28.480 --> 00:00:31.519 a surgeon's scalpel. Oh wow, yeah. That is, I 00:00:31.519 --> 00:00:33.539 mean, that's literally life or death. It is. 00:00:33.799 --> 00:00:36.079 And in both of these crazy high -stakes situations, 00:00:36.119 --> 00:00:37.799 you'd probably assume the computers running the 00:00:37.799 --> 00:00:40.560 show are operating on, you know, absolute iron 00:00:40.560 --> 00:00:42.060 -clad certainty. Because that's how we think 00:00:42.060 --> 00:00:44.600 of computers, right? Rigidity. Yeah. We have 00:00:44.600 --> 00:00:46.679 this deep -seated expectation that a machine 00:00:46.679 --> 00:00:48.780 is just built on a foundation of microscopic 00:00:48.780 --> 00:00:51.060 switches that are either, like, completely on 00:00:51.060 --> 00:00:53.280 or completely off. The ultimate binary universe. 00:00:53.280 --> 00:00:55.600 Yeah. Just zero or one. True or false. Yeah, 00:00:55.600 --> 00:00:58.799 exactly. And... You know, I think there is a 00:00:58.799 --> 00:01:00.859 real comfort in thinking the digital world is 00:01:00.859 --> 00:01:04.859 purely black and white. But the reality is, well, 00:01:05.219 --> 00:01:07.519 it's that the most sophisticated systems we rely 00:01:07.519 --> 00:01:09.739 on today, they don't operate on that principle 00:01:09.739 --> 00:01:12.439 at all. Not even a little bit. No. They actually 00:01:12.439 --> 00:01:14.959 rely on the mathematics of vagueness. Yeah. The 00:01:14.959 --> 00:01:17.620 digital world is operating in millions of shades 00:01:17.620 --> 00:01:20.560 of gray. Which is wild. So if you're listening 00:01:20.560 --> 00:01:23.810 to this, our goal today for the learner is to 00:01:23.810 --> 00:01:26.469 kind of shortcut your path to understanding how 00:01:26.469 --> 00:01:29.129 machines are actually taught to process all this. 00:01:29.650 --> 00:01:32.049 Because human reality is messy, it's subjective, 00:01:32.450 --> 00:01:35.629 it's highly imprecise. It really is. So we're 00:01:35.629 --> 00:01:37.689 going on a deep dive today. We have this remarkably 00:01:37.689 --> 00:01:40.590 comprehensive Wikipedia article on our desk covering 00:01:40.590 --> 00:01:43.230 a topic called fuzzy logic. A great topic. It 00:01:43.230 --> 00:01:45.790 is. And we're going to explore this whole concept 00:01:45.790 --> 00:01:48.939 without like. drowning you in an overwhelming 00:01:48.939 --> 00:01:51.439 sea of mathematical equations. We promise. Yeah, 00:01:51.519 --> 00:01:52.879 we're getting straight to the core mechanics 00:01:52.879 --> 00:01:55.219 of how the world around you actually works. So, 00:01:55.519 --> 00:01:57.799 to really understand what fuzzy logic is, we 00:01:57.799 --> 00:01:59.799 kind of first have to understand what it's replacing, 00:01:59.879 --> 00:02:02.519 right? We have to escape the trap of classical 00:02:02.519 --> 00:02:05.260 logic. The Boolean trap. Exactly, the Boolean 00:02:05.260 --> 00:02:08.780 trap. Classical Boolean logic is the absolute 00:02:08.780 --> 00:02:11.240 bedrock of traditional computing. I mean, it 00:02:11.240 --> 00:02:13.659 relies entirely on absolute integers. Everything 00:02:13.659 --> 00:02:16.400 has a box. Right. Everything must be categorized 00:02:16.400 --> 00:02:20.000 as a zero, which means completely false, or a 00:02:20.000 --> 00:02:23.039 one, meaning completely true. There is just zero 00:02:23.039 --> 00:02:26.259 mathematical room for anything in between. statement 00:02:26.259 --> 00:02:29.020 is a fact or it is a lie okay let's unpack this 00:02:29.020 --> 00:02:31.680 because I think an analogy helps here if traditional 00:02:31.680 --> 00:02:35.120 Boolean logic is a standard lice which like you 00:02:35.120 --> 00:02:37.060 know it's either completely flipped on or completely 00:02:37.060 --> 00:02:39.539 flipped off yeah then fuzzy logic is essentially 00:02:39.789 --> 00:02:42.289 The dimmer switch, like it can be set anywhere 00:02:42.289 --> 00:02:44.169 along a continuous spectrum. That's a perfect 00:02:44.169 --> 00:02:46.150 way to look at it, because in fuzzy logic, a 00:02:46.150 --> 00:02:48.590 variable can be literally any real number between 00:02:48.590 --> 00:02:50.969 zero and one. So like point two or point eight 00:02:50.969 --> 00:02:53.469 five. Exactly. It introduces this mathematical 00:02:53.469 --> 00:02:56.150 concept of partial truth. Now, the actual term 00:02:56.150 --> 00:02:59.370 fuzzy logic was famously coined in 1965 by this 00:02:59.370 --> 00:03:02.870 mathematician named Lotfi Zada. Right. But the 00:03:02.870 --> 00:03:05.069 seeds of this idea, they only go all the way 00:03:05.069 --> 00:03:07.370 back to the 1920s. Oh, really? I didn't realize 00:03:07.370 --> 00:03:09.150 it was that early. Yeah, there are these poles. 00:03:09.759 --> 00:03:12.659 mathematicians Jan Jukosevich and Alfred Tarski 00:03:12.659 --> 00:03:14.819 and they were exploring what they called infinite 00:03:14.819 --> 00:03:17.960 valued logic. Infinite values. So instead of 00:03:17.960 --> 00:03:21.860 just two rigid values you have infinite possibilities 00:03:21.860 --> 00:03:24.560 between zero and one. Exactly. But I guess what 00:03:24.560 --> 00:03:26.500 was the catalyst for this? Like what was the 00:03:26.500 --> 00:03:28.939 actual problem with a simple on and off switch 00:03:28.939 --> 00:03:31.879 that necessitated inventing an entire new branch 00:03:31.879 --> 00:03:34.360 of logic? Well the problem was trying to map 00:03:34.360 --> 00:03:38.080 human reality onto machine hardware. Zadeh and, 00:03:38.080 --> 00:03:40.659 you know, his predecessors, they basically observed 00:03:40.659 --> 00:03:43.060 that humans almost never make decisions based 00:03:43.060 --> 00:03:46.259 on crisp, precise numerical information. True. 00:03:46.659 --> 00:03:49.259 Like, consider asking a room full of people to 00:03:49.259 --> 00:03:51.879 identify a specific color. Say, a particular 00:03:51.879 --> 00:03:54.400 shade of teal. Yeah, that's a recipe for an argument. 00:03:54.620 --> 00:03:58.060 Right. You aren't going to get a unified binary 00:03:58.060 --> 00:04:00.539 answer where every single person points and says, 00:04:00.960 --> 00:04:03.939 that is 100 % blue or that is 100 % green. You 00:04:03.939 --> 00:04:06.719 get a spectrum of subjective answers. One person 00:04:06.719 --> 00:04:08.539 says, you know, it's mostly green. Another says 00:04:08.539 --> 00:04:10.860 it's slightly blue. And neither of them is technically 00:04:10.860 --> 00:04:14.400 wrong. Exactly. But classical logic completely 00:04:14.400 --> 00:04:16.860 breaks down in that scenario because it forces 00:04:16.860 --> 00:04:19.540 a true or false conclusion on a concept that 00:04:19.899 --> 00:04:23.019 inherently just lacks boundaries. Fuzzy logic 00:04:23.019 --> 00:04:25.079 provides the strict mathematical model for that 00:04:25.079 --> 00:04:28.300 exact human inexactness. Which sounds like a 00:04:28.300 --> 00:04:30.939 total oxymoron, like a strict mathematical model 00:04:30.939 --> 00:04:33.740 for inexactness. It does sound weird. I really 00:04:33.740 --> 00:04:36.199 have to push back here, though, because natural 00:04:36.199 --> 00:04:39.959 language is incredibly slippery. How do human 00:04:39.959 --> 00:04:42.600 words, which are inherently messy and totally 00:04:42.600 --> 00:04:46.000 subjective, literally become math? Because, like, 00:04:46.079 --> 00:04:47.620 old to a 10 -year -old is very different than 00:04:47.620 --> 00:04:49.540 old to an 80 -year -old. That's the core challenge, 00:04:49.860 --> 00:04:52.220 yeah. The translation from human language to 00:04:52.220 --> 00:04:54.439 machine code happens through what mathematicians 00:04:54.439 --> 00:04:57.079 call linguistic variables. Linguistic variables. 00:04:57.399 --> 00:04:59.959 Right, so instead of using pure numbers, fuzzy 00:04:59.959 --> 00:05:03.079 systems actually use words to express facts and 00:05:03.079 --> 00:05:06.100 rules. They take base words like young or old, 00:05:06.420 --> 00:05:08.800 and then they modify them with mathematical adverbs. 00:05:09.579 --> 00:05:11.939 The mathematicians actually call these modifiers 00:05:11.939 --> 00:05:15.699 hedges. Hedges, like the word somewhat, or rather, 00:05:16.100 --> 00:05:18.019 or very. Exactly. But wait, how can a totally 00:05:18.019 --> 00:05:21.639 subjective word like... somewhat be turned into 00:05:21.639 --> 00:05:24.079 a mathematical formula. It feels like trying 00:05:24.079 --> 00:05:26.879 to nail jelly to the wall. It does, but the process 00:05:26.879 --> 00:05:29.579 of pinning that jelly to the wall is most famously 00:05:29.579 --> 00:05:32.060 encapsulated in what's called the Mamdani system. 00:05:32.160 --> 00:05:34.860 The Mamdani system. Yeah, and it works in three 00:05:34.860 --> 00:05:38.199 distinct highly logical steps. The very first 00:05:38.199 --> 00:05:40.860 step is called fuzzification. Fuzzification? 00:05:41.019 --> 00:05:42.459 I kind of love that word. It's great, right. 00:05:42.800 --> 00:05:45.220 So imagine you are programming the climate control 00:05:45.220 --> 00:05:47.879 system for a massive office building. You define 00:05:47.879 --> 00:05:51.139 fuzzy sets for the temperature. So cold, warm, 00:05:51.240 --> 00:05:54.000 and hot. Okay, simple enough. Mathematically, 00:05:54.360 --> 00:05:56.500 these sets are drawn on a graph as overlapping 00:05:56.500 --> 00:05:58.959 shapes. They might be triangles or trapezoids 00:05:58.959 --> 00:06:02.100 or even like bell curves. And because these shapes 00:06:02.100 --> 00:06:04.939 overlap a specific room temperature, say at 68 00:06:04.939 --> 00:06:07.519 degrees, it doesn't just fall into one isolated 00:06:07.519 --> 00:06:10.259 category. Oh, I see. Because looking at the temperature 00:06:10.259 --> 00:06:12.660 example in our source material, there's a vertical 00:06:12.660 --> 00:06:15.060 line representing a specific temperature reading, 00:06:15.480 --> 00:06:17.459 and it intersects these overlapping triangles 00:06:17.459 --> 00:06:19.920 on the graph. Exactly. So the red arrow points 00:06:19.920 --> 00:06:23.199 to zero for hot. So the machine knows it's definitely 00:06:23.199 --> 00:06:26.339 not hot, but the orange arrow points to 0 .2 00:06:26.339 --> 00:06:29.220 for warm, and the blue arrow points to 0 .8 for 00:06:29.220 --> 00:06:31.420 cold. There you go. The system has just assigned 00:06:31.420 --> 00:06:33.680 degrees of membership. The room is 20 % warm 00:06:33.680 --> 00:06:36.720 and 80 % cold at the exact same time. That's 00:06:36.720 --> 00:06:38.600 crazy. It is all of those things at once to different 00:06:38.600 --> 00:06:40.839 degrees. That is fuzzification. OK, so that's 00:06:40.839 --> 00:06:42.879 step one. Right. The next step is applying the 00:06:42.879 --> 00:06:46.629 rules using fuzzy logic operators. Now, in classical 00:06:46.629 --> 00:06:49.129 Boolean logic, you have logic gates like AND, 00:06:49.629 --> 00:06:52.649 OR, and not to combine rules together. Fuzzy 00:06:52.649 --> 00:06:54.870 logic mimics these using what are called zeta 00:06:54.870 --> 00:06:58.310 operators. So wait, if Boolean logic uses rigid 00:06:58.310 --> 00:07:02.069 rules, how do zeta operators handle these overlapping 00:07:02.069 --> 00:07:04.490 percentages? Like, how does it deal with 0 .2 00:07:04.490 --> 00:07:07.420 and 0 .8? They use a brilliantly simple comparative 00:07:07.420 --> 00:07:10.879 method. The fuzzy AND operator looks at two overlapping 00:07:10.879 --> 00:07:13.660 values and simply takes the minimum of the two. 00:07:13.740 --> 00:07:15.920 The minimum, OK. Yeah. So if a rule requires 00:07:15.920 --> 00:07:19.240 the room to be both warm and humid, and it's 00:07:19.240 --> 00:07:23.079 0 .2 warm but 0 .6 humid, the AND operator just 00:07:23.079 --> 00:07:26.180 outputs 0 .2. It takes the lowest common denominator 00:07:26.180 --> 00:07:28.860 of truth. Oh, interesting. Conversely, the OR 00:07:28.860 --> 00:07:31.120 operator takes the maximum of the two values. 00:07:31.139 --> 00:07:33.439 Makes sense. And the NOT operator, which handles 00:07:33.439 --> 00:07:35.800 the negative, it just simply takes 1 minus the 00:07:35.800 --> 00:07:39.800 value. So if cold is 0 .8, then not cold is mathematically 00:07:39.800 --> 00:07:42.519 0 .2. That is remarkably elegant. I mean, it 00:07:42.519 --> 00:07:44.579 preserves the nuance of the gray area through 00:07:44.579 --> 00:07:47.079 the entire calculation. It really does. But eventually, 00:07:47.220 --> 00:07:49.100 this machine has to actually do something. Right. 00:07:49.220 --> 00:07:50.939 Right. It can't just sit there being nuanced. 00:07:51.079 --> 00:07:53.120 Right. And what's fascinating here is the final 00:07:53.120 --> 00:07:56.120 step, which is called defuzzification. Defuzzification. 00:07:56.560 --> 00:07:59.240 Yeah. This is where the system brings all that 00:07:59.240 --> 00:08:02.329 abstract math. back into physical reality. Because 00:08:02.329 --> 00:08:04.569 a climate control system cannot tell a mechanical 00:08:04.569 --> 00:08:06.870 fan motor to run somewhat fast. Oh right, the 00:08:06.870 --> 00:08:09.589 motor needs a number. Exactly. The motor requires 00:08:09.589 --> 00:08:13.589 a crisp specific electrical voltage. Defuzzification 00:08:13.589 --> 00:08:16.009 is the process of getting a single actionable 00:08:16.009 --> 00:08:18.730 variable out of those overlapping fuzzy truths. 00:08:18.889 --> 00:08:21.350 So how does it pick the final number? Well, the 00:08:21.350 --> 00:08:23.930 machine takes all those partial truths, so the 00:08:23.930 --> 00:08:26.870 slightly warm and the fairly cold, and it maps 00:08:26.870 --> 00:08:30.709 them onto a final output curve. Then it calculates 00:08:30.709 --> 00:08:33.370 the exact mathematical center of weight of the 00:08:33.370 --> 00:08:35.509 area under that curve. Under of weight? Yeah, 00:08:35.590 --> 00:08:37.590 think of it like balancing a weirdly shaped piece 00:08:37.590 --> 00:08:40.110 of cardboard on the tip of your finger. The machine 00:08:40.110 --> 00:08:43.669 finds that exact balancing point and the X coordinate 00:08:43.669 --> 00:08:46.269 of that specific center point becomes the final 00:08:46.269 --> 00:08:49.320 single output command sent to the motor. Wow! 00:08:49.700 --> 00:08:51.720 So the machine isn't just taking a wild guess. 00:08:52.259 --> 00:08:55.019 It is literally calculating the exact geometric 00:08:55.019 --> 00:08:58.120 center of gravity of our vague human intentions 00:08:58.120 --> 00:09:01.409 to produce one perfect concrete action. It's 00:09:01.409 --> 00:09:04.129 beautiful, really. The Mamdani system is deeply 00:09:04.129 --> 00:09:07.309 intuitive for that very reason. Now, our text 00:09:07.309 --> 00:09:10.029 does detail an alternative to Mamdani called 00:09:10.029 --> 00:09:13.490 the TSK system, which stands for Takagi -Sujino 00:09:13.490 --> 00:09:16.889 -Kan. TSK is actually much more computationally 00:09:16.889 --> 00:09:19.690 efficient because it skips that visual center 00:09:19.690 --> 00:09:22.909 of gravity step entirely. Defuzzification is 00:09:22.909 --> 00:09:25.429 built directly into the rules using polynomial 00:09:25.429 --> 00:09:28.049 or linear functions. Meaning, instead of the 00:09:28.049 --> 00:09:30.690 computer having to draw a graph plot over overlapping 00:09:30.690 --> 00:09:32.929 triangles and calculate a physical balancing 00:09:32.929 --> 00:09:36.190 point, it just runs a direct algebraic math formula 00:09:36.190 --> 00:09:38.950 to spit out the final number. Precisely. Which 00:09:38.950 --> 00:09:41.970 is why it is fantastic for algorithmic optimization 00:09:41.970 --> 00:09:44.210 and machine learning. I mean, it saves massive 00:09:44.210 --> 00:09:47.850 amounts of computing power, but it is much less 00:09:47.850 --> 00:09:50.269 intuitive for a human programmer to visualize 00:09:50.269 --> 00:09:52.789 or troubleshoot. Mamdani remains popular because 00:09:52.789 --> 00:09:55.389 it translates human thought into machine action 00:09:55.389 --> 00:09:57.509 in a way we can actually picture in our heads. 00:09:57.809 --> 00:09:59.750 Okay, I have to imagine you listening to this. 00:09:59.559 --> 00:10:01.700 Might be having the exact same thought I had 00:10:01.700 --> 00:10:03.879 when I first started reading these sources We 00:10:03.879 --> 00:10:06.039 are talking about assigning values between 0 00:10:06.039 --> 00:10:09.919 and 1 and a 20 % chance of rain is 0 .2 So what 00:10:09.919 --> 00:10:12.460 is the actual difference between fuzzy logic 00:10:12.460 --> 00:10:14.820 and standard probability theory? Aren't they 00:10:14.820 --> 00:10:16.960 the exact same thing? It's a completely natural 00:10:16.960 --> 00:10:19.179 assumption, and honestly, it's one that actually 00:10:19.179 --> 00:10:21.799 sparked a massive academic turf war. Really? 00:10:22.019 --> 00:10:24.740 A turf war over math? Oh, absolutely. While both 00:10:24.740 --> 00:10:27.759 fields use the exact same numerical range, so 00:10:27.759 --> 00:10:31.159 between 0 and 1, they model fundamentally different 00:10:31.159 --> 00:10:34.299 concepts of reality, because probability models 00:10:34.299 --> 00:10:38.059 ignorance. but fuzzy logic models vagueness. 00:10:38.259 --> 00:10:39.940 Here's where it gets really interesting. I want 00:10:39.940 --> 00:10:42.480 you to imagine you are holding a completely opaque 00:10:42.480 --> 00:10:45.139 bottle of water. You can't see inside it at all. 00:10:45.379 --> 00:10:47.679 Okay, an opaque bottle. Someone tells you there's 00:10:47.679 --> 00:10:50.259 a 20 % chance the water inside is totally toxic, 00:10:50.720 --> 00:10:52.799 and an 80 % chance it's pure mountain spring 00:10:52.799 --> 00:10:55.659 water. That is probability. The bottle is full. 00:10:55.820 --> 00:10:58.779 The water has an absolute binary state -like. 00:10:58.960 --> 00:11:02.000 It is either pure or it is poison, but you are 00:11:02.000 --> 00:11:04.059 entirely ignorant of which one it is until you 00:11:04.059 --> 00:11:05.990 drink it. And because you are ignorant of the 00:11:05.990 --> 00:11:08.049 state, probability measures the frequency or 00:11:08.049 --> 00:11:09.950 likelihood of an event occurring in the future. 00:11:10.409 --> 00:11:12.710 Exactly. Now contrast that opaque bottle with 00:11:12.710 --> 00:11:15.529 fuzzy logic. Right. Now imagine a clear glass 00:11:15.529 --> 00:11:18.149 bottle. You can see perfectly inside, you know, 00:11:18.190 --> 00:11:20.830 with 100 % certainty that it contains pure, safe 00:11:20.830 --> 00:11:24.250 drinking water. But the bottle is only 20 % full. 00:11:24.909 --> 00:11:27.210 There is no ignorance there. You know exactly 00:11:27.210 --> 00:11:29.470 what it is. The state of the bottle's fullness 00:11:29.470 --> 00:11:32.659 is simply a matter of vagueness or degree. It 00:11:32.659 --> 00:11:35.500 is a partial truth. And because you can clearly 00:11:35.500 --> 00:11:38.539 see that it's 20 % full, the mathematical tools 00:11:38.539 --> 00:11:41.039 you use to interact with it completely change. 00:11:41.720 --> 00:11:44.019 This distinction is exactly why fuzzy sets were 00:11:44.019 --> 00:11:46.039 developed at Berkeley in the mid -20th century. 00:11:46.580 --> 00:11:48.960 Traditional probability theory simply could not 00:11:48.960 --> 00:11:51.399 model uncertainty and vagueness jointly. It just 00:11:51.399 --> 00:11:54.149 didn't work. Right. Probability forces you to 00:11:54.149 --> 00:11:56.470 guess the state of a hidden system, while fuzzy 00:11:56.470 --> 00:11:58.629 logic allows you to measure the state of a vaguely 00:11:58.629 --> 00:12:01.690 defined system. But the text mentions this massive 00:12:01.690 --> 00:12:04.789 academic debate over this exact distinction. 00:12:05.370 --> 00:12:07.529 There was a prominent researcher named Bart Kosko, 00:12:07.929 --> 00:12:09.909 and he argued that probability is actually just 00:12:09.909 --> 00:12:12.509 a sub -theory of fuzzy logic. He went so far 00:12:12.509 --> 00:12:15.970 as to mathematically derive Bayes' theorem, which, 00:12:16.129 --> 00:12:18.289 you know, is basically the holy grail formula 00:12:18.289 --> 00:12:21.409 of probability. Yeah. Entirely from the concept 00:12:21.409 --> 00:12:24.200 of fuzzy subsea -hood. He did, essentially arguing 00:12:24.200 --> 00:12:26.100 that the likelihood of something happening is 00:12:26.100 --> 00:12:28.679 just a specific slice of the broader mathematics 00:12:28.679 --> 00:12:31.600 of vagueness. But on the other side of the aisle, 00:12:31.980 --> 00:12:34.399 Lot Fisate, you know, the father of fuzzy logic, 00:12:34.899 --> 00:12:38.259 he argued the exact opposite. He maintained they 00:12:38.259 --> 00:12:41.340 are entirely different in character. Zada even 00:12:41.340 --> 00:12:43.620 created something called possibility theory as 00:12:43.620 --> 00:12:46.700 a generalization to handle the types of uncertainty 00:12:46.700 --> 00:12:49.080 that probability mathematically failed to capture. 00:12:49.519 --> 00:12:51.539 Possibility theory? Yeah. It really forces us 00:12:51.539 --> 00:12:53.419 to ask a profound question about how we view 00:12:53.419 --> 00:12:56.100 the world. Like, how often in our daily lives 00:12:56.100 --> 00:12:58.840 do we confuse our own ignorance of a situation 00:12:58.840 --> 00:13:01.259 with the inherent vagueness of the situation 00:13:01.259 --> 00:13:03.519 itself? That is so true. We try to put betting 00:13:03.519 --> 00:13:05.700 odds on things that aren't actually events waiting 00:13:05.700 --> 00:13:08.080 to happen. They're just realities that exist 00:13:08.080 --> 00:13:10.289 perpetually in the gray area. So we have this 00:13:10.289 --> 00:13:12.870 profound philosophical distinction and we have 00:13:12.870 --> 00:13:15.309 the clever math of fuzzification and the center 00:13:15.309 --> 00:13:18.330 of weight calculation. But, I mean, what does 00:13:18.330 --> 00:13:21.129 this actually do in the real world? Oh, it fundamentally 00:13:21.129 --> 00:13:24.659 runs the modern world. Many of the earliest and 00:13:24.659 --> 00:13:27.879 most spectacular successes of fuzzy logic actually 00:13:27.879 --> 00:13:31.159 happened in Japan during the 1980s. Yes, the 00:13:31.159 --> 00:13:34.220 Sendai subway system. The Sendai subway 1000 00:13:34.220 --> 00:13:37.539 series. So before fuzzy logic, automated trains 00:13:37.539 --> 00:13:40.700 used really rigid binary rules. A computer would 00:13:40.700 --> 00:13:44.159 literally say, if speed is over 50, apply 10 00:13:44.159 --> 00:13:47.299 % break. Which sounds awful. It was. The result 00:13:47.299 --> 00:13:49.580 was a jerky, uncomfortable ride as the train 00:13:49.580 --> 00:13:52.980 constantly toggled between absolute states. Sendai 00:13:52.980 --> 00:13:55.639 engineers, they implemented a fuzzy system. Using 00:13:55.639 --> 00:13:57.860 those hedges. Exactly. It used those vague rules 00:13:57.860 --> 00:14:00.000 we talked about. Like, if you're close to the 00:14:00.000 --> 00:14:02.379 destination station and moving fairly fast, increase 00:14:02.379 --> 00:14:04.490 the brake pressure somewhat. Wow. The system 00:14:04.490 --> 00:14:07.129 processed all those overlapping conditions continuously 00:14:07.129 --> 00:14:09.830 and the result was a train that accelerated and 00:14:09.830 --> 00:14:12.090 starved so smoothly and with such microscopic 00:14:12.090 --> 00:14:14.549 precision that it vastly improved ride comfort 00:14:14.549 --> 00:14:18.210 and fuel economy. And the source text lists so 00:14:18.210 --> 00:14:20.250 many everyday applications that followed that, 00:14:20.289 --> 00:14:22.470 like the anti -lock brakes on your car. They 00:14:22.470 --> 00:14:24.909 map the temperature ranges and speed of the brake 00:14:24.909 --> 00:14:27.590 pads to fuzzy truth values to constantly adjust 00:14:27.590 --> 00:14:30.009 the braking force so you don't skid. Yeah. Or 00:14:30.009 --> 00:14:31.990 washing machines that use a single button to 00:14:31.990 --> 00:14:34.090 figure out exactly how much water. are an agitation 00:14:34.090 --> 00:14:36.950 you need based on like fuzzy readings of the 00:14:36.950 --> 00:14:39.769 load weight and dirtiness. It's everywhere. Sony 00:14:39.769 --> 00:14:42.149 even used it for handwriting recognition in early 00:14:42.149 --> 00:14:44.870 pocket computers because human handwriting is 00:14:44.870 --> 00:14:48.350 basically the ultimate vague data set. It really 00:14:48.350 --> 00:14:51.169 is. And if we connect this to the bigger picture 00:14:51.169 --> 00:14:53.730 beyond just watching machines and subways, the 00:14:53.730 --> 00:14:56.549 most vital application of fuzzy logic today is 00:14:56.549 --> 00:14:59.340 the foundation of artificial intelligence. Wait, 00:14:59.399 --> 00:15:02.519 really? Because isn't AI all about massive databases 00:15:02.519 --> 00:15:04.860 and binary code? Well, at the hardware level, 00:15:05.039 --> 00:15:07.580 yes, computers process ones and zeros. But at 00:15:07.580 --> 00:15:09.779 the architectural level, the underlying logic 00:15:09.779 --> 00:15:12.200 of modern neural networks, which are the systems 00:15:12.200 --> 00:15:15.899 actually driving the current AI boom, that logic 00:15:15.899 --> 00:15:18.500 is fundamentally fuzzy. The text actually mentions 00:15:18.500 --> 00:15:21.240 a really pivotal moment in the 1980s for machine 00:15:21.240 --> 00:15:24.299 learning. Researchers were divided into two main 00:15:24.299 --> 00:15:27.360 camps. One camp was using what's called decision 00:15:27.360 --> 00:15:30.759 tree learning. Right. And decision trees use 00:15:30.759 --> 00:15:34.200 strict binary logic. Imagine a flow chart. The 00:15:34.200 --> 00:15:36.440 computer looks at a pixel and asks, is this pixel 00:15:36.440 --> 00:15:39.080 dark? If yes, go down the left branch. If no, 00:15:39.200 --> 00:15:41.200 go down the right branch. Careful enough. It 00:15:41.200 --> 00:15:43.679 forces a rigid either -or choice at every single 00:15:43.679 --> 00:15:46.580 step, and it perfectly matched the computer hardware 00:15:46.580 --> 00:15:49.320 of the time. But the other can't use neural networks. 00:15:50.179 --> 00:15:52.779 And despite binary logic seeming like the obvious 00:15:52.779 --> 00:15:55.360 choice for a computer, the neural networks won 00:15:55.360 --> 00:15:57.820 out. They did. Why? Because the real world doesn't 00:15:57.820 --> 00:16:00.879 operate in zeros and ones. Exactly. The mechanism 00:16:00.879 --> 00:16:02.980 behind a neural network's success is that it 00:16:02.980 --> 00:16:06.000 doesn't force an immediate binary choice. A neural 00:16:06.000 --> 00:16:08.539 network takes a huge variety of inputs. If a 00:16:08.539 --> 00:16:11.200 pixel is 40 % dark, it doesn't force it to be 00:16:11.200 --> 00:16:14.019 a one or a zero. It just accepts it as 40%. Right. 00:16:14.240 --> 00:16:17.259 It takes that .4 value. assigns it a numerical 00:16:17.259 --> 00:16:20.500 weight, and passes that intermediate value forward 00:16:20.500 --> 00:16:22.960 to the next layer of the network. It carries 00:16:22.960 --> 00:16:25.639 the nuance of the gray area through the entire 00:16:25.639 --> 00:16:27.980 system without ever dropping it. That's brilliant. 00:16:28.200 --> 00:16:30.480 Neural networks resulted in much more accurate 00:16:30.480 --> 00:16:33.500 models of complex situations precisely because 00:16:33.500 --> 00:16:36.379 they embraced the fuzzy nature of the data. Which 00:16:36.379 --> 00:16:38.860 totally brings us back to that MRI example from 00:16:38.860 --> 00:16:41.659 the beginning. Medical decision -making is a 00:16:41.659 --> 00:16:44.779 massive frontier for fuzzy logic because medical 00:16:44.779 --> 00:16:47.700 data is incredibly subjective. Like a patient 00:16:47.700 --> 00:16:50.600 says their pain is a 7 out of 10. Or a doctor 00:16:50.600 --> 00:16:52.220 is looking at an image trying to find the edge 00:16:52.220 --> 00:16:54.899 of a brain tumor. An image -based computer -aided 00:16:54.899 --> 00:16:57.740 diagnosis relies heavily on fuzzy sets. When 00:16:57.740 --> 00:17:00.200 you look at an MRI of a tumor or a photograph 00:17:00.200 --> 00:17:02.679 of an infected wound after surgery, the pixels 00:17:02.679 --> 00:17:05.259 don't suddenly turn from healthy tissue to tumor 00:17:05.259 --> 00:17:07.859 in a perfectly crisp, mathematically straight 00:17:07.859 --> 00:17:10.279 line. It's a gradient. It is a gradient. The 00:17:10.279 --> 00:17:13.740 boundary is fuzzy. By using fuzzy logic, the 00:17:13.740 --> 00:17:16.599 computer can analyze the biomedical signals and 00:17:16.599 --> 00:17:19.279 segment those images by recognizing degrees of 00:17:19.279 --> 00:17:21.960 membership. It says, you know, this pixel is 00:17:21.960 --> 00:17:24.299 80 percent tumor, 20 percent healthy tissue. 00:17:24.619 --> 00:17:27.450 But the text points out a beautiful almost tragic 00:17:27.450 --> 00:17:30.490 irony in this pursuit. It notes that the envelope 00:17:30.490 --> 00:17:33.230 of what can be achieved and what cannot be achieved 00:17:33.230 --> 00:17:36.250 in medical diagnosis, ironically, is itself a 00:17:36.250 --> 00:17:38.609 fuzzy one. Yeah. The underlying challenge is 00:17:38.609 --> 00:17:40.990 that we are trying to mathematically map human 00:17:40.990 --> 00:17:44.029 subjectivity. We rely on humans to define what 00:17:44.029 --> 00:17:45.910 healthy or sick looks like in the first place, 00:17:45.910 --> 00:17:48.049 and there is an inherent mathematical limit to 00:17:48.049 --> 00:17:50.630 how perfect that translation can ever be. And 00:17:50.630 --> 00:17:53.450 this limitation actually extends far beyond medicine. 00:17:53.930 --> 00:17:55.940 The computational theorist Leslie Valiant coined 00:17:55.940 --> 00:17:58.039 a brilliant term for this. He called them ecorythms. 00:17:58.119 --> 00:18:00.640 Ecorythms. Algorithms that learn from their complex 00:18:00.640 --> 00:18:02.759 environments, like your ecosystems. Exactly. 00:18:03.220 --> 00:18:05.619 Valiant argues that machine learning is essentially 00:18:05.619 --> 00:18:08.700 evolutionary biology playing out in code. Ecorhygums 00:18:08.700 --> 00:18:11.220 share fuzzy logic's core philosophy. They focus 00:18:11.220 --> 00:18:13.960 on possibilities rather than exact probabilities. 00:18:14.079 --> 00:18:17.019 Wow. They are designed to generalize, to approximate, 00:18:17.240 --> 00:18:19.740 and simplify logic in environments that are simply 00:18:19.740 --> 00:18:23.039 too chaotic. too noisy, and too complex to understand 00:18:23.039 --> 00:18:25.720 perfectly or discreetly. They survive by adapting 00:18:25.720 --> 00:18:28.420 to the gray area. So what does this all mean? 00:18:28.759 --> 00:18:30.400 When you step back and look at the whole picture, 00:18:30.839 --> 00:18:34.099 there is this incredible, almost poetic irony. 00:18:34.319 --> 00:18:37.839 the crispest, most precise, most reliable systems 00:18:37.839 --> 00:18:41.019 we rely on every single day. From the anti -lock 00:18:41.019 --> 00:18:43.319 brakes keeping us from crashing, to the medical 00:18:43.319 --> 00:18:46.000 imaging saving lives, to the AI recognizing our 00:18:46.000 --> 00:18:48.640 voices, they actually require fuzzy thinking 00:18:48.640 --> 00:18:51.220 to function flawlessly. It's a huge paradox. 00:18:51.359 --> 00:18:54.410 In order to be exact in the real world, The machine 00:18:54.410 --> 00:18:56.690 must first learn how to be vague. It represents 00:18:56.690 --> 00:18:59.170 a profound paradigm shift in how we approach 00:18:59.170 --> 00:19:01.930 problem -solving. We literally had to teach silicon 00:19:01.930 --> 00:19:03.990 to understand the spectrum of human intention. 00:19:04.569 --> 00:19:06.349 So to wrap this up, we've gone on quite a journey 00:19:06.349 --> 00:19:09.549 today. We started with Lafizadeh's 1965 rebellion 00:19:09.549 --> 00:19:13.130 against the rigid binary Boolean logic that defined 00:19:13.130 --> 00:19:16.170 early computers. We did. We walked through Mamdani's 00:19:16.170 --> 00:19:18.900 rule books. learning how overlapping triangles 00:19:18.900 --> 00:19:21.140 and center -of -weight calculations turned words 00:19:21.140 --> 00:19:23.940 like somewhat, and rather, into precise mechanical 00:19:23.940 --> 00:19:27.579 voltages. We rode the buttery smooth Sendai subway 00:19:27.579 --> 00:19:30.500 in Japan, and we peered under the hood of the 00:19:30.500 --> 00:19:33.480 neural networks, carrying partial truths through 00:19:33.480 --> 00:19:36.170 the modern AI revolution. It's all built on the 00:19:36.170 --> 00:19:38.970 math of the gray area. And as we close, there's 00:19:38.970 --> 00:19:42.829 one final lingering idea drawn from the decidability 00:19:42.829 --> 00:19:45.309 issues section of the source text that I really 00:19:45.309 --> 00:19:47.009 think is worth reflecting on. OK, let's hear 00:19:47.009 --> 00:19:49.670 it. The text notes an unresolved open question 00:19:49.670 --> 00:19:52.329 in mathematics regarding whether Kurt Gödel's 00:19:52.329 --> 00:19:55.490 famous incompleteness theorems can be extended 00:19:55.490 --> 00:19:59.009 to fuzzy logic. Ah, Gödel. So for those who might 00:19:59.009 --> 00:20:00.690 not spend their weekends reading theoretical 00:20:00.690 --> 00:20:03.730 math, Gödel basically proved that in any complex 00:20:03.730 --> 00:20:05.849 set of mathematical rules, you will eventually 00:20:05.849 --> 00:20:08.910 find statements that are 100 % true, but the 00:20:08.910 --> 00:20:11.269 rules themselves are fundamentally incapable 00:20:11.269 --> 00:20:14.109 of proving them. The system is always technically 00:20:14.109 --> 00:20:16.890 incomplete. Which is just a staggering reality 00:20:16.890 --> 00:20:19.650 for mathematicians to accept. Now, apply that 00:20:19.650 --> 00:20:21.210 to what we've learned today. I want to pose this 00:20:21.210 --> 00:20:24.849 directly to you, listening right now. If human 00:20:24.849 --> 00:20:27.150 reasoning is fundamentally fuzzy, as we've discussed, 00:20:27.670 --> 00:20:30.210 and if fuzzy logic itself is eventually proven 00:20:30.210 --> 00:20:32.630 to be bound by those Godelian limits of what 00:20:32.630 --> 00:20:35.069 can ever be mathematically proven, are there 00:20:35.069 --> 00:20:37.170 fundamental truths about your own life, about 00:20:37.170 --> 00:20:39.109 human nature, and about our universe that aren't 00:20:39.109 --> 00:20:42.190 just currently unknown to us, but are mathematically 00:20:42.190 --> 00:20:44.990 destined to remain in the gray area forever? 00:20:45.049 --> 00:20:48.200 Wow. testing to be gray. Even the math admits 00:20:48.200 --> 00:20:50.380 they can't know everything. Well, on that note, 00:20:50.480 --> 00:20:52.119 we want to thank you for joining us on this deep 00:20:52.119 --> 00:20:54.500 dive. The next time you sit down at your computer 00:20:54.500 --> 00:20:56.680 or you hit the brakes in your car or you just 00:20:56.680 --> 00:20:59.279 try to describe the color of the sky, remember 00:20:59.279 --> 00:21:01.980 that the world isn't a light switch. Look at 00:21:01.980 --> 00:21:03.680 the some -whence and the rathers of your own 00:21:03.680 --> 00:21:06.400 day in a whole new mathematically brilliant light. 00:21:07.059 --> 00:21:09.500 Embrace the dimmer switch. Thanks for listening.