00:00:00.000 Welcome to Topcast episode 6 about the problem of induction, the first podcast in a series
00:00:19.680 on the nature of science. Professor Russell owned a chicken. Each morning he would get up early
00:00:43.280 and feed his chicken. The chicken being an anthropomorphic chicken thought what a wonderful thing this
00:00:50.800 was and was excited each morning to see Professor Russell come into the coop with the feed. The
00:00:55.920 chicken went to sleep each night expecting that tomorrow the food would come and indeed the chicken
00:01:01.520 seemed to be right. The chicken predicted that each day he would be fed. He was sorely surprised then
00:01:08.400 when on Christmas Eve of all mornings, kind Professor Russell came in to decapitate him.
00:01:16.160 What does the story of Russell's chicken suggest? It suggests this,
00:01:21.600 no amount of observing a certain thing over and again is enough to entitle you to predict
00:01:27.200 that that thing will continue on again even one more time that alone 10 more times or forever.
00:01:32.960 No number of sunrises suggests that tomorrow the sun will rise. No number of breaths that you take
00:01:41.680 suggests that the next won't be your last. In other words, making predictions from the particular
00:01:48.400 to the general is simply not possible. Well, so what? For so many years it was unclear exactly
00:01:56.960 how science was justified in making the claims that it did. A group of philosophers and scientists
00:02:02.880 in the 1950s worked on this particular problem among many others. The main group of scientists
00:02:08.800 and philosophers worked in Vienna and came to be known as the Vienna Circle. The Vienna Circle
00:02:15.680 developed a style of philosophy called logical positivism. It was all about the idea that all
00:02:22.080 knowledge comes to us through our experiences and a purely logical process is the best way to solve
00:02:28.480 a philosophical problem insofar as there are philosophical problems. One consequence as they saw it
00:02:34.560 of logical positivism was the ability to justify scientific theories. Everyone knew by then
00:02:40.960 that mathematics was a deductive system and they believe mathematics proved its truths with certainty.
00:02:48.480 In the last few episodes, I've argued that this position turns out not to be correct.
00:02:52.640 You do not get certainty in mathematics. As for science, well, if it's not purely deductive,
00:02:59.840 then what is it? How could you justify your theory if it was not able to be purely deduced
00:03:06.240 from the evidence in the same way that mathematics proves its conclusions? The logical
00:03:10.960 positivist came up with the idea of verification. You verify your theory. How? Well, through observations,
00:03:20.480 repeated observations. So if you were Newton, presumably you watched an apple fall from a tree,
00:03:27.360 and then another, and then another. According to the logical positivists,
00:03:34.160 you then came up with a theory about gravity and your theory is verified through further observations.
00:03:41.120 Now, you cannot prove your theory. It's not deductive, but well, it's still pretty well justified,
00:03:49.280 isn't it? So the process of verifying a theory using a method of justification based upon repeated
00:03:56.320 observation became known as induction. Scientific induction. So scientific induction is supposed
00:04:05.200 to be a process of justification. You repeatedly observe a certain phenomena, like a falling apple,
00:04:11.840 and then come up with a theory, which is then verified by further observations.
00:04:21.920 You were then entitled, based on your theory, to make predictions like,
00:04:26.560 each time an apple falls from a tree, it'll hit the ground. Now, there's a real problem with this.
00:04:33.920 We saw that with Russell's chicken. Russell's chicken observed that each morning the farmer
00:04:39.120 would feed him. He predicted this would continue. He turned out to be wrong when he lost his
00:04:45.680 head and was put on the table that night for Christmas dinner. What's happened here?
00:05:01.680 But if induction can go badly wrong, doesn't that surely spell trouble for scientific theories
00:05:06.560 which are justified by induction? Does this bring all of science into question as unreliable
00:05:12.000 and poorly justified? No. No, it does not, because science does not justify its theories using
00:05:19.760 induction. Induction plays no part in the formation of scientific theories, and it never has.
00:05:26.320 In fact, no one ever justifies anything. Scientific theories are not using induction.
00:05:32.160 Okay, that's a big claim. It's a big claim to say that we never use induction.
00:05:40.080 Never? No. It was the great Scottish philosopher, David Hume,
00:05:47.680 who, around 1740, clearly stated that induction cannot be logically justified,
00:05:54.400 and that this seemed to be a problem for science. But it's not, because science does not rely on
00:06:00.080 induction and never has. But Hume was one of the first to most clearly express the problem
00:06:06.800 of how to explain just why it is we are justified in believing our scientific theories
00:06:11.920 and believing that they're reliable. He was a smart man who wondered about the nature of cause
00:06:17.440 and effect, and how science worked. Hume was right in believing that we cannot be certain
00:06:22.400 of our scientific theories, and he also knew that induction was flawed. So his problem was to wonder
00:06:28.720 how, then, are we justified in believing our scientific theories? He wasn't the first,
00:06:34.800 the Greek philosopher Sextos Impiricus, in around 200 AD, mentioned the problem of induction
00:06:40.800 and his writings, and we're going to find that so many discussions in analytical philosophy
00:06:46.080 can trace their roots to one or more of the ancient Greeks. Since Hume, people have still
00:06:52.000 wondered how it is that we can justify our scientific theories. People who still wonder about that
00:06:56.960 today do not realise that the question has indeed been answered.
00:07:10.800 Now sure, this philosophical question had to wait around two more centuries for a solution,
00:07:16.880 but the solution was found. It turned out that science simply does not proceed by induction.
00:07:24.640 Science does not work that way. So how does science work then? Well I'm glad you asked.
00:07:31.600 Forget induction. Well don't forget it. It's useful to know the misconceptions that many,
00:07:37.040 especially sociologists of science, and those who write some science textbooks have about science.
00:07:44.880 Induction is certainly a big concept, and it's still debated today, but most of us interested
00:07:50.720 in the philosophy of science, and scientists themselves know that induction is dead. The problem with
00:07:57.280 induction is that it is simply not a way of justifying things. Induction not only cannot justify
00:08:03.440 scientific theories, it cannot justify anything. It's not a method of justification.
00:08:10.080 You simply are not entitled to reason from the particular to the general.
00:08:15.040 Russell's chicken shows you that. Given a set of observations, the farmer fed me today,
00:08:21.440 the farmer fed me yesterday, the farmer fed me today before that, and so on. The prediction,
00:08:27.440 the farmer will feed me tomorrow, does not follow. Not only does it not follow, but something even
00:08:35.600 worse than that, a general prediction like this is simply not a theory. A theory explains something.
00:08:44.720 In this case, the thing that requires explaining is the farmer's behavior. Now given one theory about
00:08:50.960 the farmer's behavior, namely that the chicken is a loved pet, then the chicken might very well
00:08:56.960 derive a certain prediction from this theory. One prediction that he might derive would be,
00:09:03.360 well, the farmer will continue to feed me, and he believes that this is justified because he
00:09:07.920 believes in the theory that the farmer loves him because he's a pet. But given another theory to
00:09:14.000 begin with, another as it turns out, more accurate theory, namely that the chicken is being
00:09:19.920 fattened up for Christmas dinner, then another quite different prediction would be derived.
00:09:25.600 One day this feeding will stop when the farmer wants to cook me.
00:09:29.600 So given nothing but a set of observations, namely the feeding of the chicken repeatedly,
00:09:35.760 we are not entitled to predict anything just yet. We first need a theory.
00:09:53.600 So how do we reason in science? The scientific method is not about verification. It's about
00:10:00.880 falsification. We'll get to precisely what that means in a moment. For now, let's also look at
00:10:07.600 how the inductiveists get science wrong. The idea that induction works at all,
00:10:13.520 forms a significant part of how some talk texts and the talk guide explain how science works.
00:10:20.640 The inductiveist idea suggests that scientists make observations, and on the basis of those
00:10:26.480 observations, generalize a theory. We have seen this does not work, and it cannot work.
00:10:32.960 So what does scientists do then? What does scientists actually do is solve a problem?
00:10:39.920 They're problem solvers. Currently, for example, the world's cosmologists have a problem.
00:10:46.000 The rate of the expansion of the universe is greater than expected according to general relativity,
00:10:51.600 given the amount of mass and energy in the universe. The rate of expansion is accelerating.
00:10:57.040 That's a problem. One proposed solution is what is called dark energy.
00:11:04.160 This solution itself raises questions because when the mathematics of quantum theory is applied
00:11:09.680 to dark energy, we get a value for the equivalent zero point energy as it's called to be 10
00:11:15.680 to the power of 120 times greater than expected. This is another problem observed. In fact,
00:11:22.960 it has been said that this is the worst theoretical prediction in the history of physics. We need
00:11:27.680 a new solution. So this is how science works. Problems require solving and phenomena require
00:11:37.120 explanations. Here's a historical problem from science. Why is it that all the objects in the sky
00:11:46.080 seem to rise in the east and set in the west? The sun, the moon, the stars, and the planets?
00:11:51.360 One solution is that the Earth is a motionless sphere in the center of the universe
00:11:56.400 surrounded by a bigger sphere called the celestial sphere, which itself rotates.
00:12:01.920 Now holes in the celestial sphere give us the appearance of stars in the sky.
00:12:08.080 But now there's a problem. This particular theory can't account for the fact that the
00:12:13.600 planets appear not to be fixed to the celestial sphere like the stars are. They appear to wonder
00:12:19.040 about the sky and rather random ways. That's a problem. Now one solution was to give up the
00:12:27.440 idea of the celestial sphere, but keep the Earth at the center. In general,
00:12:33.120 theories with the Earth at the center are called geocentric theories.
00:12:37.200 And you can have a geocentric theory in which all the planets are objects that are in orbit
00:12:42.320 around the Earth. However, these planets don't go around the Earth in simple circles,
00:12:47.600 but rather in complex orbits called epicycles. The notes to accompany this topcast have a
00:12:54.000 diagram explaining what epicycles are. The purpose of the geocentric theory and epicycles
00:13:00.240 was to explain the problem of why it is that the planets appear to wander across the sky.
00:13:05.280 The astronomer, told me, in 300 BC, developed this theory to such sophistication that it was able
00:13:12.080 to predict the motion of planets to a high degree of accuracy. This theory was terribly
00:13:17.200 complicated though, and as time went on, it was obvious that Ptolemy's predictions were not as
00:13:21.760 accurate as they first seemed to be. The theory required ad hoc revision. The theory was kept,
00:13:27.920 but it required the revision of putting epicycles onto epicycles onto epicycles.
00:13:34.240 Every time a planet deviated from where it was predicted to be, astronomers just added another
00:13:39.280 epicycle. Clearly, this was unsatisfactory. It was Copernicus who suggested that, really,
00:13:47.760 things would be much simpler with the Sun at the Center and the Earth and other planets going
00:13:51.920 around it. That requires fewer assumptions, and you do not need all these complications of epicycles.
00:13:59.040 Later, Galileo and Kepler would further improve the theory with precise mathematics and the
00:14:05.120 idea that the orbits were ellipses, not perfect circles. Now, perhaps you've heard of Occam's
00:14:11.840 razor. I'll have more to say about that in the next episode on science, but let's just say for now
00:14:17.520 that even in the absence of better predictions, the heliocentric theory, the theory that the Sun
00:14:22.800 is at the Center, beats Ptolemy's geocentric theory, not necessarily because it makes better
00:14:27.920 predictions, indeed at the time it couldn't, but because it relies on far fewer assumptions
00:14:33.680 about the existence of all these complicated epicycles. The Galilean worldview that put the Sun
00:14:40.080 at the Center explained in a simple way, why it is that celestial bodies appear to rise in the
00:14:45.280 east and set in the west, and why it is that the planets follow more complicated paths across the
00:14:50.720 sky. Further, it made predictions about where any particular celestial object would be at
00:14:57.360 any time on any night of the year with a high degree of accuracy. All of this with far fewer
00:15:02.400 assumptions, unlike the raft of assumptions that Ptolemy needed epicycles after assumed epicycles
00:15:08.000 after assumed epicycles. Galileo was able to solve this problem with an elegant solution.
00:15:15.200 His heliocentric theory predicted a number of things, including the paths planets followed,
00:15:20.240 and the fact that because not everything had to go around the Sun, the moons of Jupiter could be
00:15:24.720 accounted for. Galileo himself was the first one to discover moons going around Jupiter,
00:15:30.160 the first four are named after him, they're called the Galilean moons.
00:15:33.920 Galileo's heliocentric theory could also account for the phases of Venus,
00:15:38.800 and he could explain precisely when they were going to occur. Eventually, all of these things
00:15:45.280 were observed and explained and predicted. So there we have it. Problem, proposed solution,
00:15:53.920 prediction, observation. Now the thing is, Galileo's predictions about the motions of planets
00:16:02.480 was falsifiable. That means it was possible for Galileo to have been wrong,
00:16:08.800 and this is one of the big keys about scientific theories, falsificationism,
00:16:17.040 Karl Popper is perhaps the most famous philosopher of science to have ever lived.
00:16:32.320 He was born in Austria in 1902, and is well known not only as someone who developed a theory
00:16:37.440 of how science works and how scientific theories are justified, but also as a political philosopher,
00:16:43.120 who produced some of the most advanced and creative theories about how an open and flourishing
00:16:47.760 society should be structured. His ideas are some of the strongest and most articulate
00:16:52.800 against authoritarianism and totalitarianism of all sorts.
00:17:02.320 Now Popper, despite being from Austria, was not a member of the Vienna Circle,
00:17:07.360 although he was in contact with them. He was a critic of their logical positivist philosophical
00:17:12.960 movement, and he rejected the idea that scientific theories could ever be verified.
00:17:17.840 He also rejected their idea that there was no such thing as a philosophical problem.
00:17:21.520 This was a theory put forward by Wittgenstein. He found that what scientists actually do
00:17:27.920 is propose a bold conjecture that seeks to explain phenomena.
00:17:32.400 Scientists then attempt to refute the conjecture. If the conjecture, or theory,
00:17:36.800 is not refuted, then it is conditionally accepted.
00:17:40.160 Now Popper was a genius, and he did most of his work at the prestigious London School
00:17:45.040 of Economics before going to New Zealand to take up a position at the Canterbury University
00:17:49.760 College in Christchurch. He was rejected for a job in Australia due to antisemitism in the late 1930s.
00:18:00.080 The view of science that Popper has is really a view of how knowledge in general is created.
00:18:05.920 For this reason, his view is more broadly called Popperian epistemology.
00:18:12.400 He saw the acquisition of knowledge as a problem-solving process.
00:18:16.960 At schools around the world now, this underpins the way students learn,
00:18:20.800 at least it is directing the way students are educated more than in the past.
00:18:25.040 There is less emphasis on wrote learning and more emphasis on discovery.
00:18:29.600 Students tend to construct their own theories with guidance.
00:18:32.960 The students then test these theories about the world.
00:18:37.840 In science, we call those tests, or attempts to refute the theory, experiments.
00:18:44.480 Those which are refuted are dismissed or altered, while those which are not are conditionally accepted.
00:18:51.200 All knowledge therefore is conditional. All knowledge consists of theories about the world.
00:18:57.600 These theories are explanations, and explanations can be good or bad.
00:19:04.160 Your knowledge of science is made up of scientific theories.
00:19:07.440 Your knowledge of history, historical theories, politics, political theories,
00:19:11.280 your beliefs about ethics constitute an ethical theory about the world.
00:19:15.280 Your knowledge of what your best friend is doing right now consists of a theory about her.
00:19:22.640 Or far from it. It may be based upon much convincing evidence, or it may not be.
00:19:29.280 In other words, your theory may be good or bad.
00:19:35.760 The criteria for good is that a theory explains what it sets out to and explains a broad range
00:19:42.240 of phenomena. The broader range of phenomena the theory explains than the deeper that theory must be.
00:19:48.240 And importantly, a good explanation also makes precise predictions.
00:19:54.720 Consider the scientific explanation that is called quantum theory.
00:19:58.880 Quantum theory is the theory about matter and energy at the subatomic scale.
00:20:03.280 In fact, it explains the behavior of all subatomic particles.
00:20:06.720 It explains superconductivity and how computer chips work.
00:20:10.320 It explains how fluorescent tubes produce light, and it is quantum theory that enables us to
00:20:15.360 have knowledge of extremely distant galaxies in our universe. The disparate phenomena
00:20:20.560 that quantum theory links is so broad that no other theory comes close.
00:20:24.640 And as for predictions, well, the degree of precision that quantum theory can obtain
00:20:31.520 is like specifying the width of Australia to an accuracy of the width of one human hair.
00:20:37.120 Such bold predictions are just asking for the theory to be falsified,
00:20:40.960 and yet it has not been. As we will see, some of the conclusions of quantum theory are very
00:20:45.680 counterintuitive, but we are compelled to accept them because of the fact that quantum theory
00:20:50.160 puts itself out there to be falsified with highly accurate predictions, and yet it has not been
00:20:54.880 falsified. Sure, one day it might be, but this is not to suggest that we cannot trust quantum theory
00:21:00.640 now. For example, we explain genetic inheritance now through the theory of DNA.
00:21:06.320 We explain that genes are passed from one generation to another via the DNA.
00:21:12.560 The DNA contains a code for producing all of your features under the right conditions.
00:21:17.760 This whole field of genetics, like quantum theory, constitutes a theory.
00:21:22.080 Now both may be wrong, but what are the chances that DNA has absolutely nothing to do with
00:21:27.520 genetics? What are the chances that everything we know about quantum theory is false?
00:21:32.240 Well, the chances are effectively zero. We may change these theories, but because the
00:21:38.640 predictions they make are so precise, a wholesale rejection of our best theories today is going
00:21:43.360 to be very difficult. But we will have more to say about this in a different podcast.
00:21:48.320 We will also find that it is falsificationism that enables us to distinguish between what
00:21:52.800 cancer science and what does not. Popper's idea of falsifiable theories enable us to explain how Galileo
00:22:02.080 can say things like, on the night of October 27, 1640, Jupiter should be right overhead with
00:22:07.520 Mars at 45 degrees declination, or Venus should be rising just in the east, years before any of
00:22:14.160 these events happened. If he turned out to be wrong, then he would know that his theory was flawed.
00:22:20.080 So that's the method. It has nothing whatever to do with the so-called induction scheme.
00:22:26.880 At no point in the story, the Galileo sit night after night watching planets and generalizing
00:22:31.280 a theory from those observations alone. No, there was a problem. The Tolemaic theory was
00:22:37.520 unwieldy and imprecise. A simpler theory, with fewer assumptions, could solve the problems
00:22:44.240 with the Tolemaic theory. And this simpler theory is closer to the truth. Now it's worth noting
00:22:51.440 that a Tolemaic theory, although way off, could have been worse, had at least had had the
00:22:56.000 planets moving in orbit. Sure, they were the wrong type of orbits, but orbit still. The fact that
00:23:01.600 planets moving in orbit was something that was true, that was subsumed by the first heliocentric
00:23:07.120 theory, which got us closer to the truth. Copernicus heliocentric theory suggested the orbits
00:23:13.280 were perfect circles. Now this was better, but it took a Galileo to improve it further still.
00:23:19.440 Later, the astronomer Kepler, and also especially Newton, improved things still further with
00:23:24.880 elegant laws of motion, which linked the orbits of the planets, with the tides on the earth,
00:23:29.760 and apples falling to the ground. Newton's theory rested on very few assumptions,
00:23:35.920 made accurate predictions, and unified diverse phenomena solving many problems in the process.
00:23:41.120 This is the mark of a good scientific theory. We can never be sure that we have certain
00:23:46.560 truth in science, but this is not a problem for a scientist who is not after certain truth.
00:23:52.320 Scientific theories are explanations, and the best explanations survive.
00:23:57.200 So what about Professor Russell's chicken? Well, the chicken had actually formed a
00:24:01.280 theory in his head to explain the behavior of the professor before or the moment when he started
00:24:06.160 making predictions. He theorized that the professor was keeping him as a treasured and much loved
00:24:12.160 pet. A prediction which comes logically from such a theory is that the professor will therefore
00:24:17.360 continue to feed the chicken, but the theory was wrong, and it was falsified in a dramatic
00:24:22.800 fashion on Christmas Day. As I explained earlier, if the chicken had developed a theory which
00:24:30.640 correctly explained the farmer's behavior, namely, why is it that he is being fed each day
00:24:35.520 by the farmer, then the chicken would have formulated the following theory.
00:24:40.160 The farmer is feeding me because he wants to fatten me up for Christmas dinner when he will eat me.
00:24:45.040 The point is that any number of theories can explain a set of observations.
00:24:49.760 In order to make a prediction that a certain phenomena will continue to occur,
00:24:53.760 you first need a theory in mind that explains what's going on.
00:24:57.840 That theory would explain that the phenomena is going to continue,
00:25:01.040 or perhaps it will explain that the phenomena is going to stop. You see the sun rise each day.
00:25:07.680 You don't simply conclude on that basis alone that therefore the sun will continue to rise
00:25:13.360 every other day. In fact, that's not a scientific theory, that's just a prediction,
00:25:20.400 and science is not primarily about predictions. It's about explanations.
00:25:25.520 The explanations will allow you to derive predictions. That the sun has risen every day might,
00:25:32.480 conceivably, give rise to a problem. A problem like, why does it seem to be the case that the sun
00:25:38.400 rises in the eastern sets in the west each day? Or what is the sun at all? To solve this problem,
00:25:46.400 you need to use a theory. Now at the moment, the best available to us is the theory from
00:25:53.040 astronomy and astrophysics and physics and a whole raft of sciences that go on to explain
00:25:59.920 what the sun is and how it works and why it is that it appears to rise in the eastern set in the west.
00:26:05.840 In this case, the full armamentarium of science would explain to us that the sun appears to rise
00:26:10.960 due to the motion of the earth, and this motion is not expected to cease for billions of years.
00:26:16.960 Well before this, but still billions of years from now, the sun will see shining.
00:26:21.840 However, this will not occur in our lifetime, and all of this allows us to predict that the sun
00:26:27.040 is going to continue to rise each and every day of our lives.
00:26:32.480 So if you read some text on the subject of how science works, including various resources about
00:26:38.800 talk, you will find an erroneous scheme. The claim is that scientific theories are produced
00:26:44.480 in the following way. You make observations, you generalize these observations into a theory,
00:26:50.320 you then make more observations, and these justify the theory. And it is this scheme I have argued,
00:26:58.640 along with Popper, is clearly false. The way science actually works is in the following way,
00:27:05.360 and I have taken almost everything that I've said here directly from a scientist I have mentioned
00:27:10.080 before, David Deutsch. Science works by first identifying a problem. It then suggests solutions,
00:27:18.000 which possibly solves a problem. These solutions are criticized, now in science we include among
00:27:24.560 our methods of criticism a special type called experimental tests. These experimental tests
00:27:31.440 then enable us possibly to replace our erroneous theories with the ones that solve the original
00:27:36.640 problem. We may then encounter new problems to solve and the cycle begins again.
00:27:41.680 At this point, I want to highlight a link that exists here between science and mathematics.
00:27:49.360 We have seen that induction in science, indeed induction as a means of justification in general,
00:27:54.480 simply as not a way that we reason, not a way we can reason or do reason. The logical
00:27:59.760 positivists, textbooks and common sense are all wrong. But in mathematics there is a procedure
00:28:06.000 called mathematical induction. What about it? Mathematical induction is a process senior
00:28:11.920 students of mathematics learn in order to prove certain theorems. It is a very valuable and useful
00:28:17.600 means of proof, and it is deductive. Let me say that again, mathematical induction is a deductive
00:28:25.600 process. Unfortunately, it shares the term induction with that other induction we just
00:28:31.680 discredited, but the two are entirely different. Some talk texts, and even some maths texts,
00:28:40.240 are lured to the idea that mathematics uses induction, that naive induction, to establish
00:28:46.000 certain mathematical formulas true. This is incorrect. For example, consider the following
00:28:53.120 surprising fact about odd numbers. The number one is equal to one squared. Okay, big deal.
00:28:59.600 Stay with me on this. One plus three has two terms, and it's also equal to two squared.
00:29:07.840 All right, one plus three plus five has three terms, and it's equal to three squared.
00:29:15.040 Okay, let's keep going. One plus three plus five plus seven. Notice what I'm doing here. I'm just
00:29:20.800 adding up all the odd numbers. One plus three plus five plus seven, that's got four terms,
00:29:27.600 and you guessed it, it's equal to four squared. So the pattern seems to be that for however many
00:29:33.520 odd numbers you add up, then the sum of those numbers is equal to the square of the number of terms
00:29:38.880 that you've added together. Now, naive induction would say, well, you can conclude on the basis
00:29:44.960 what seems to be going on, that that's actually the case. So an inductive proof would be that
00:29:51.600 any sequence of odd numbers definitely adds up to a square of the number of odd numbers that you
00:29:56.320 've added up. Proof done. No, no, no. I presume you'd get your backside kicked pretty hard in
00:30:03.600 mathematics class if you tried to prove anything this way. No one in mathematics ever tries to
00:30:08.480 prove a theorem in that way, certainly not a university and not at school. That sort of induction
00:30:13.200 just doesn't have a place in mathematics. So why would some text that explain the way that
00:30:19.920 mathematics works? Even mention that this is a valid method of justification. I'm not too sure.
00:30:27.120 I know that what such an argument would do is that it leaves the door open for anyone not fully
00:30:32.000 trained in mathematics to be able to say, you can never be sure that your proof will not fail for
00:30:36.960 a certain number of odd numbers. So for our example, how do you know that the formula doesn't fail
00:30:42.080 once you get to, let's say, 1 million, 350, 4,291? You can't be sure. So mathematics is unreliable.
00:30:50.480 Well, if indeed we were proving our theorems in the way that I've just described, then yes,
00:30:55.120 that'd be a valid criticism. But no one in mathematics proves things the way we just did there.
00:31:01.200 But you can use a technique called mathematical induction to prove it.
00:31:04.960 Now I know, these uses of the word induction are confusing. Induction does have a lot of
00:31:13.200 different meanings. If you look it up in the dictionary, there are quite a few there that haven't
00:31:17.280 even been covered by what I've been talking about so far. But suffice it to say, in real
00:31:23.040 mathematical induction, the method of proof known as mathematical induction, what you do is
00:31:28.240 you check that the formula works in a particular case. So usually for the case of the number
00:31:32.960 one, in our example of figuring out what the sum of a certain sequence of odd numbers adds up to,
00:31:40.640 we can take the number one. One's an odd number and one happens to equal one squared, plain and
00:31:45.680 simple. Well, then you assume that the formula works for a certain general case. So we could assume
00:31:52.000 that it works for one plus three plus five plus all the way up to k where k is some odd number.
00:31:59.120 And you assume that that equals to n squared where n is the number of odd numbers that you've just
00:32:04.640 added up. Then you take this assumption, you show that if it holds, then it's the case that it works
00:32:10.160 for the next number up to all the numbers k plus two. This would be the next odd number.
00:32:16.000 For those taking higher levels of mathematics, the proof of what I'm talking about right now,
00:32:20.880 accompanies the notes to this podcast. Now I'm not going to attempt to go any further with the
00:32:25.680 proof. It's almost impossible to teach mathematics purely via audio. Suffice it to say that
00:32:32.240 mathematical induction can best be compared to dominoes. You push one over and the rest as
00:32:36.640 certain to go over as well. It's a rigorous deductive mathematical proof. As certain as anything
00:32:42.560 else in mathematics, it's not naive induction. It has none of the flaws of naive induction.
00:32:48.320 Its only flaw is that it's called mathematical induction. It should be called proof by iteration
00:32:53.760 or something. It seems to me that this unfortunate turn of phrase gives much fuel to some who want
00:32:59.600 to suggest flaws with mathematics. Now if you've heard the other top casts on the nature of
00:33:04.720 mathematics, you know that there are sensible and very interesting discussions to be heard about
00:33:08.800 the reliability of mathematical knowledge and its limits. But mathematical induction and
00:33:15.040 induction in general are not included as part of these limitations. The limitations and reliability
00:33:20.240 of mathematics are actually far more interesting than this stuff about naive induction.
00:33:25.120 Now finally for this episode, I want to emphasize that my criticism here of the entire idea of
00:33:31.360 induction is not shared by all. I said in the very first episode of top cast that I would be
00:33:38.480 presenting at times a decidedly one-sided view of reality. We don't have time to go into all the
00:33:44.160 details about the debate here. Indeed, some people still believe in induction, and they believe
00:33:49.520 in a certain type of induction called Bayesianism. It's quite popular among some philosophers,
00:33:54.880 but I think that that's induction in a cheap tuxedo. As always, these alternative views are
00:34:00.320 definitely worth researching, and we can continue the discussion online.
00:34:06.080 Now having spent an entire episode talking about what science is not, next time I'm going to talk
