Nurturing Science Talent K. P. Mohanan and Tara Mohanan

Contents 1. The Role of Learning Resources 2. Letters to Mishti

2.1 Investigating Questions 2.2. Congruent Shapes

3. Inquiry in the Kitchen 3. 1. Statistical Thinking in the Kitchen 3.2. Theory Building in the Kitchen: A Scientific Theory of Banana Chips

4 Mother-Son Dialogues

This article is a companion piece to our article “Assessing Science Talent” which discusses questions such as “What is science talent?” and “How do we assess it?” In what follows, we assume the general discussion in part I of that paper, and proceed directly to the question nurturing the capacity for scientific thinking and inquiry. 1. The Role of Learning Resources Given that most science programs at the primary, secondary, and tertiary levels of education focus almost exclusively on knowledge and (mechanical) application of knowledge, identifying the potential for the thinking and inquiry abilities outlined in part I becomes a difficult task. How do we assess a student’s potential for swimming if (s)he has never had the prior experience of splashing about in water?

To identify the potential for scientific thinking and inquiry, we would need to develop learning resources outside of traditional curricula to help the young acquire the relevant thinking and inquiry abilities. These resources will have to be made available to them such that they are at least exposed to scientific thinking and inquiry, sufficiently in advance of the exams that would probe into their talent.

In nurturing science talent, it is important to go back to an even earlier stage, make available learning resources that will expose children to the excitement of science, help them figure out for themselves if they are genuinely interested in “science”, and then make them want to pursue studies/career in science. For this, making the resources available just to “candidates” who have decided to do the entrance exams is not enough: this will not serve the purpose of nurturing. It is important to capture the attention of those students who are put off by the traditional modes of teaching science — of meaningless memorization and mechanical routines, but have the potential for scientific inquiry. Chances are that among the children who have developed an intense dislike of science, we will find many who have the potential to be outstanding scientists.

In what follows, we offer a sample of such self-learning materials to help fifteen year olds (and above) acquire the ability to engage in scientific thinking and inquiry. It is meant as an illustration of some of the strands of scientific thinking and inquiry that can be incorporated into science education – going beyond the traditional instructional goals of understanding and application – and how students can be initially exposed to these strands. We hope that it would encourage other science educators to produce similar learning resources to help the young develop the mental abilities prototypical of scientific inquiry.

A program of tertiary education that aims to produce future scientists needs to identify science talent that calls for a selection procedure based on the potential of the candidates for scientific research, as opposed to the information and problem solving skills they have managed to acquire through coaching and slogging. Given the current forms of primary and secondary

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education that restrict themselves to the dissemination of scientific knowledge and the training in its application, however, it is not feasible to test the applicants’ potential to acquire the ability to think like scientists: that would be like testing the swimming abilities of those who have never seen swimming, let alone have the experience of swimming.

Self-learning resources of the kind illustrated in the following pages might offer a way out of this dilemma. If the applicants are supplied with learning materials sufficiently in advance, we may expect highly motivated candidates to use those materials to develop their thinking abilities. It would then make sense to measure the degree to which they attain the relevant abilities, and design selection criteria that include these abilities. The combination of learning materials and test would yield a more reliable assessment of the candidate’s potential for further learning to become a researcher, as opposed the quantum of scientific knowledge already acquired.

This is a draft, not the final product. It will have to go through several rounds of revision on the basis of comments from scientists, teachers, and students before it can be used as preparatory material for science aptitude assessment. As stated above, it will also have to be supplemented by additional material, produced by others.

2. Letters to Mishti 2.1 Investigating Questions

Dear Mishti, When I visited you and your parents in Melbourne, I was asking you all kinds of questions, and I had a feeling that you were getting overwhelmed by them. So I think I owe you an explanation.

The reason for my badgering you with questions was not to find out what you have learnt in your school, or to judge your knowledge, but to help you develop the capacity to inquire and think for yourself. This ability is essential if you want to be a scientist. You told me that you want to become a scientist. So I am trying to help you achieve that goal.

This might sound strange to you. How can students become scientists by answering questions? Don’t they have to learn science first, by attending classes, reading textbooks, and learning what the answers are? Aren’t questions what students have to answer in their exams?

Well, the questions that Bua and I ask are different. There are certain kinds of questions that you can answer simply by consulting your memory. If I were to ask you, “What is the formula for the circumference of a circle?” all you need to do is access from your memory the statement that the circumference of a circle is its diameter multiplied by π. Questions in your class tests and exams are typically of this kind: they are designed to assess what you have learnt. Our questions are not of this kind, and I had no intention of assessing you, so don’t worry.

Unlike exam questions, the questions we ask you are ones for which you don’t already have an answer. What do you do when faced with such questions? If you consulted your memory you would find no answer there. You can’t say, “I don’t know,” because we already know that, which is why we are asking you in the first place. To make you look for answers.

When faced with such a question, you might think of asking someone who knows the answer or consult a document that carries it. If I were to ask you, “What is the area of a sphere?” and you don’t know the answer, you could ask your teacher, or look it up on the web. But this kind of consulting isn’t going to help you develop the capacity for independent inquiry. So the questions we ask you are not of this kind either.

Now you might be stuck. If you don’t already know the answers, but we don’t want you to ask someone or look it up in a book or on the web, how are you supposed to answer the questions? Aha, that is precisely where independent inquiry comes in. A scientist is an inquirer. The

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purpose of our badgering you with questions is to help you learn the strategies of inquiring so that you can become a good scientist before you finish your school.

Let us take the questions I asked you about clouds. Some clouds are like huge chunks of cotton. On a bright sunny day, you might find that certain parts of a cloud are white while other parts of the same cloud are dark. What makes some parts of a cloud white and other parts dark? Go out and look at clouds, think about what you see, and see if you can come up with an answer.

Another question. Is it possible to look at clouds in the sky and figure out where the sun is even when you can’t see the sun? Go out, look at the clouds, and see for yourself if you can figure out where the sun is. If you can figure out the position of the sun by looking at the clouds, what allowed you to infer that?

If you find answers to these questions, you will realize that it is easy for you to answer the other questions I asked you. When you have both white clouds and dark clouds in the sky, it is quite often the case that the dark clouds cover the white ones. But it is unlikely that you will ever find white clouds covering the dark ones. Is this true? Go out and look, and find out if it is true. If it is true, can you think of an explanation on your own? (Invent one.)

Mohanmama

2.2. Congruent Shapes

Dear Mishti,

If you cut out the shapes in figure 1 and figure 2 below, you will find that they fit one on top of the other exactly. You can do this with figures 3 and 4 as well: figure 1 figure 2 figure 3 figure 4

Even without actually cutting out the shapes and putting them one on top of the other, it is easy to simply look at figures 1 and 2 and see that what I am saying is correct. It might be a little more difficult to do that with figures 3 and 4. If you can’t tell by looking, then do try cutting them out and putting one on top of the other.

If one shape can be put on top of the other exactly, we say that they are congruent. Shapes 1 and 2 are congruent. So are shapes 3 and 4. It is fairly easy to see right away that the following pairs are not congruent.

Shapes 1 and 3 Shapes 1 and 4 Shapes 2 and 3 Shapes 2 and 4.

We have seen two ways of telling whether two shapes are congruent or not: by simply looking, and by cutting them out and putting them one on top of the other. Suppose I now told you that shapes 4 and 5 are congruent. I haven’t shown you shape 5: it is on a piece of paper hidden inside a book. Can you tell me if the following pairs of shapes are congruent? Circle the answer that you think is correct.

Shapes 1 and 5: YES NO Shapes 2 and 5: YES NO Shapes 3 and 5: YES NO

This must have been easy. But how did you arrive at the answer? By putting some pieces together and making an inference. What you did was to arrive at a conclusion on the basis of

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what you already knew about shapes (1)-(4), and what I told you about shape 5.

Suppose I asked you, “How many legs does Zoumi have?” you will not be able to answer. If I point Zoumi out to you, you will be able to count its/his/her legs and give me an answer. Suppose instead that I tell you that Zoumi is an insect. You will immediately say that Zoumi has six legs, even though you have never seen her. How come? You are able to make that inference from what I told you:

Given: Zoumi is an insect. you infer: Zoumi has six legs.

Connecting the two statements is a general statement that you are implicitly appealing to:

All insects have six legs.

So the inference is:

Given: Zoumi is an insect; and All insects havesix legs; you infer: Zoumi has six legs.

Your inference will not be the same if I told you that Zoumi is a bird, or that she is a spider. If Zoumi is a spider, given the general principle that all spiders have eight legs, you would infer that Zoumi has eight legs.

To arrive at a conclusion about the congruence of shapes 1-3 with 5, you crucially needed a general principle about congruence that applies to all shapes. Such a principle would be analogous to: “All dogs have four legs,” and “All spiders have eight legs.” Can you state that principle? After formulating the general principle needed to answer the question about shape 5, I want you to answer another question. Are shapes 1 and 6 congruent? I can see you rolling your eyes and saying, what kind of question is that? How can you answer this question without knowing anything about shape 6?

Fair enough. So let me say something about shape 6. The circumference of shape 6 is twice as much as the circumference of shape 1. Now can you answer the question whether shape 1 and shape 6 are congruent? This should be easy for you. But, I want you to give me your reasons for arriving at that conclusion. Can you show that shapes 1 and 6 cannot be congruent, using the strategy we used to show that Zoumi the dog has four legs? Make sure that you don’t forget to include the general principle.

Mohanamama

3. Inquiry in the Kitchen 3. 1. Statistical Thinking in the Kitchen1

Some of my biology friends were meeting at my place for potluck dinner and a movie that evening. I was preparing vegetables for a vegetable-cheese dish. Alka, my sister-in-law, who was visiting, walked into the kitchen, saw me cutting the vegetables, and offering to help, picked up the potatoes.

“How do you want these cut?” she said.

“Oh, into about half-centimeter cubes,” I replied.

“Not half-inch, huh?” She laughed. “Half-centimeter! Finicky, aren’t we?” 1 This is a mildly embellished true story. Though the story is told in first person, narrating an experience of KPM’s, it has two authors; a rough draft made by KPM was significantly modified by TM.

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“Yes, I want them very small.”

As I diced the celery, capsicum, and carrots, I kept thinking of the importance of the way vegetables are cut, how it affects the taste, texture, and soul of a dish, and of the professional pride in cutting vegetables just right. I am not a professional cook, but the way vegetables are cut, I certainly am finicky about.

When Alka finished, she asked, “Does this qualify?” I looked at the bowlful of potato cubes, and said, “It doesn’t really matter, but those look more like three-fourth centimeter cubes to me.”

“No, I think they are half-centimeter cubes,” Alka said emphatically.

Mishti, my fifteen-year-old niece (Alka’s daughter), was watching a TV commercial in the living room. “Mishti, come over here,” I called out. She turned off the TV set, ambled to the kitchen, and stood at the door looking at me.

“Your mother says these are half-centimeter cubes,” I said. “I think they are three-fourths centimeter cubes. What do YOU think?”

She looked at the pile of potato cubes in the bowl, and shrugged. ”I don’t know, I can’t tell whether they are half or three fourths. That’s such a small difference.”

“How will you find out?” I said.

“I guess I’ll take a ruler and measure.”

“Go get a ruler, then.”

She got a ruler, picked up a piece of the chopped potato, and measured it.

“It’s half a centimeter.”

“So?”

“So Mom is right. It is half a centimeter, not three-fourths.”

“Pick another piece, Mishti, and measure it.”

The one she measured this time was three-fourths centimeter. “Oh, so you are both right. Some are half and some are three-fourths.”

“Can I ask you a question, Mishti?”

“Yup?”

“There must be more than a couple of hundred cubes in that bowl, right?”

“Right.”

“What if just one piece is half a centimeter and the rest are all three fourths, and you happened to pick, just by sheer chance, that single half-centimeter one the first time. Who would be right if that were the case?”

“You would be.”

“And what if just one piece is three-fourths and the rest are half, and you just happened to pick, just by sheer chance, that single three-fourths one the second time. Who would be right?”

“Mom would be.”

“And how do you know which is the case?”

“Oh!” Mishti frowned for a moment, lost in thought, then said, ”I think I’ll have to measure all the pieces.”

“Suppose you find they are all different sizes. Some are 0.4 centimeters, some are 0.5, some are 0.8, and so on. Then what will you do.”

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“I’ll put them in different piles.”

“What kind of piles?”

“I will make a pile of pieces with 0.1 to 0.5, from 0.6 to 1.0. and 1.1 to 1.5.”

“And suppose you found each pile to be equal?”

“Then you will both be equally right, or both equally wrong.” She was beginning to get into the spirit of the question.

“Suppose you found that out of two hundred pieces, a hundred and eighty are in the 0.6 to 1.0 centimeter pile, and ten each in the other two piles.”

“Then Mom would be right. Oh, wait, no, then you would be right. Oh no, … well, … I don’t know.”

“So making piles won’t work?”

“Er… I guess not.”

“So what are you going to do?”

“I don’t know.”

“Take your time, Mishti. You can give me an answer when you’re ready.”

“Wait, I know what I will do. I’ll take the average.”

“What is average?”

“Jeez, don’t they teach these professors anything? Average is this number.”

“What number?”

“Well, you measure all the cubes, then you add up all the numbers and then you divide them by the total number of pieces. That would be the average.”

“Very good, Mishti. The kind of average you are talking about is called ‘mean’.”

“Oh yeah! Some mean people like you must have invented it.”

“Ha ha! You can see me cracking up.”

“Okay, bye, Mohanmama.”

“Mishti, wait. Don’t go yet. To do this you said you have to measure all the pieces in the bowl?”

“Yeah, YOU are the one pestering me with questions. How else will I find out who is right, Mom or you?”

“Suppose we’re cutting vegetables for a party of about a hundred people. Instead of a small heap of potato cubes in a bowl, there’s a huge pot of them. More than ten thousand pieces. Are you going to measure ALL of them? Your hair will turn gray by the time you’re done. Is there a more practical way to find a reasonably approximate answer? Not necessarily an exact one.”

“What other way is there?”

“Think about it. Let me know when you come up with a solution.”

“Mohanmama?”

“Yes?”

“My brain is all fried up. Can I go back to my TV?”

“Sorry, go watch your commercials.”

* * *

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Ten minutes later, I found Mishti sitting in front of the TV but not looking at it. She was lost in thought. And then she jumped up. “Mohanmama, may be I can take a sample and measure the pieces in the sample.”

“What is a sample?”

“A sample is what you take from the pot. Like a handful.”

“What you take from the pot? A handful of peanuts?”

“No. A handful of potato pieces,” she said patiently, in all earnestness. “For a pot of potato pieces a sample is a few pieces from the pot.”

“A sample from the population. Okay.”

“No. Not from a population. We’re talking about potatoes, not people, Mohanmama. Jeez, these profs.”

“Populations are not necessarily populations of people, Mishti. Have you heard of the subject called statistics?”

“Yeah. You showed me a video lecture on the statistics thingie yesterday, remember?”

“Oh yes, I did. Okay, in statistics, a population is a collection of any thing. It can be people, rabbits, tables, marks, prices, just any collection. A sample is a part of the population. So if you pick a handful of potatoes from a pot, the potatoes in the pot is the population and the handful is the sample.”

“Okay, so I pick a sample from the population of potato pieces.”

“Yes. Go ahead, pick your sample from the bowl.”

Mishti picked about ten pieces from the top of the pile. She was going to measure them when I stopped her.

“Can I ask you something, Mishti?”

“I am beginning to get scared about your can-I-ask-you-something questions. Every time you ask that question, you get me into deep trouble.”

“That’s my job, Mishti. Getting kids into trouble.”

“What were you going to ask?”

“You picked all your ten pieces from the top, right?”

“Yes.”

“Suppose you found that those in the sample are mostly half a centimeter cubes. But is it possible that when your Mom started cutting the potatoes, she was cutting them bigger, and when she was about to finish she cut them smaller? She is not a machine, after all, she is a human being. Isn’t that possible?”

“Yes.”

“If that’s what happened, how would you know if what your sample is telling you what is true of the population, or if it’s giving you distorted information?”

“Oh… I think my sample would be distorting. It would say one thing, but the population would be something else.”

“In statistics, they would say that the sample doesn’t adequately represent the population, or is not representative of the population.”

“What’s that?”

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“A sample should be representative of the population. Whatever is true of the population should be true of the sample as well. The sample should reflect the properties of the population.”

“I see. So if Mom changed her cutting style towards the end, my sample from the top of the pile would not be representative of the population.”

“That is not unlikely, right? How would you make sure that your sample is representative? At least increase the probability that the sample represents the population?”

Mishti thought for a moment. She dug her fingers into the bowl of potatoes, and did a thorough job of mixing the pieces. Then she closed her eyes, chanted some children verse like “eeni-meeni-maini-mo”, thrust her hand into the pile, and with her eyes still closed, gathered a handful of pieces.

There were eighteen pieces in her sample. Their measurements were:

0.4; 0.7; 0.9; 0.5; 0.4; 0.7; 1.1; 0.7; 0.8; 1.0; 0.6; 0.7; 0.8; 0.8; 0.9; 0.5; 0.6; 0.8.

“Mohanmama, the mean is a little more than 0.7,” she said after her calculations. “So it looks like you were right.”

* * * A few minutes later, when Mishti came back to the kitchen to invade the fridge, I said, “Mishti, I want you to take a few potato cubes, say around ten, and find their mean size.” She picked ten pieces randomly, and measured them. The measurements were:

0.4; 0.7; 0.5; 0.5; 0.4; 0.6; 1.1; 0.3; 0.8; 0.4.

Mishti did the calculation on a piece of paper. “Oh, oh!”

“What happened?”

“The mean is 0.57. That means Mom was right. Not you. This is driving me crazy. What is it with these idiotic potatoes any way. Why do they keep switching sides?”

“Let us take a look. Your first sample of eighteen was, where is it, … okay, the measurements were: 0.4; 0.7; 0.9; 0.5; 0.4; 0.7; 1.1; 0.7; 0.8; 1.0; 0.6; 0.7; 0.8; 0.8; 0.9; 0.5; 0.6; 0.8. And the mean was 0.7.”

“Yes.”

“Let’s pick five from this sample. I am going to pick the first one, 0.4, the fourth, 0.5, the fifth, 0.4, the eleventh, 0.6, and the seventeenth, 0.6. What is the mean?”

“It is 0.5.”

“So even from the first sample, if you took a smaller subset, your conclusion can be different, right?”

“Yeah, the numbers are all over the place.”

“Which sample would you trust more to give you a more accurate conclusion?”

“I don’t know. May be if we took a very large sample, say fifty pieces or so, we can get a more trustable mean. If we took two samples of five pieces each, the mean can fluctuate like crazy, but if we took two samples of fifty pieces each, I have a feeling that they will be much closer.”

“So what you’re saying is that when the sample size increases, the sample become more representative of the population.”

“Am I saying that? I didn’t think of it that way, actually, but now that you have put it that way, yes, that’s reasonable. As the sample size increases, the sample becomes more representative of the population.”

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“And the more representative a sample is, the more we can trust our conclusion.”

“That looks like it.”

“A little while ago, we figured out that if you pick all your pieces from the top, you won’t get a representative sample: it will be a biased sample. So we made it random to make it representative. But now we find that random selection is not enough. The size of the sample should also be large enough. So the sample has to be both large and random to make it representative.”

“You sound like a teacher, Mohanmaama.”

“I am a teacher. Remember, I teach in a university. So here is my question for you. You want to arrive at a reasonably certain conclusion about a population on the basis of a sample. What makes a sample large enough? What would your decision on the sample size depend upon?”

“Erm, … that depends on the population, right? The sample size should be proportionate to the population.”

“If you take a sample of 10 for a population is 100, you should take a sample of 200 for a population of 2,000, and a sample of 5000 for a population of 50, 000?”

“I guess.”

“Let us see. Suppose you see a cup of soup on the table. You want to check if it has enough salt, so you take a teaspoonful and taste it. If the salt is enough, you will conclude that there is enough salt in the soup in the cup, right?”

“Right”

“So a sample of one teaspoon is sufficient for a cup of soup. But what if it is not a cup of soup, it is a big bowl of soup. Do you need to taste more teaspoons of soup to find out if there is enough salt?”

“No, one teaspoon from the bowl is enough.”

“I see. What if it is a huge pot of soup. Or a barrel of soup?”

“It is all the same soup, right? One teaspoon is still enough.”

“But you said a little while ago that the size of the sample should be proportionate to the size of the population. Won’t that mean that you take a small sample for a cup of soup and a big sample for a big pot of soup?”

“No… Yes… I mean, I did say that, but that doesn’t work here for the soup.”

“So what should the choice of the sample size depend on?”

“I don’t know. Mohanmaama, why are you asking me? I am just fifteen, you are the professor, you should know.”

“I am trying to help you to figure it out for yourself, Mishti. It is more fun that way. Don’t you feel good when you find out something on your own?”

“Yes, I suppose so.”

“Alright, then. Let us try another example. We were looking at the potatoes that your Mom cut. Suppose the potatoes were cut not by her but a very precise machine. Would you still need a large sample size? When should we use a larger sample size, if something is cut by a machine or by a person?”

“Er… when cut by a person?”

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“And suppose you have another population of potatoes cut not by one person but by many different persons. Which one would need a larger sample, when cut by one person or when cut by many?”

“When cut by many people.”

“So when cut by a machine, we use a small sample, when cut by a person we use a medium size sample, and when cut by many different people, we use a large sample. What does this depend on?

“A machine would cut vegetables uniformly. All pieces would be the same. But when a person cuts it, it will be different. And when there are many people, it will be still more different.”

“What you are pointing to, Mishti, is uniformity and variability. If the property we are investigating is uniform across the population, like in soup, a small sample will do. But if the population is not uniform, if it exhibits variations, then we should take larger samples. The more variability we expect, the larger our sample. Are you falling asleep?”

“No, I am not asleep. I am thinking about what you are saying.”

“Did you understand what I said?”

“I think so, kind of.”

“Alright, tell me then, why is it that in the population of diced potatoes we were talking about, the machine-cut potatoes call for the smallest sample, and the potatoes cut by different people calls for the largest sample?”

“Because machines are mechanical and they cut all the potatoes the same way, so a smaller sample is enough. But humans don’t do everything so uniformly, so human-cut potatoes would be more variable, so we need a larger sample. And if they are cut by different humans, they will be still more variable. So that needs the largest sample.”

“Excellent.”

“Do I pass the test, Sir Professor?”

“You do, Mishti. With distinction.”

“Now can I watch TV to throw out all the potatoes from my brain?”

“I will watch with you.”

“No, you won’t. If you do, you are going to ask me another question in ten minutes. Go away, sit in front of your beloved computer, not in front of the TV.”

“Okay, bye, Mishti.”

“Bye, Mohanmama.’

3.2. Theory Building in the Kitchen: A Scientific Theory of Banana Chips2

One rainy morning after breakfast, my wife Tara, our daughter Ammu, and I decided to taste some fancy banana chips someone had given us. I bit into a piece and said that the chips were crisp. Tara and Ammu disagreed; they thought the chips hard, not crisp. This was puzzling, because we were making different assertions about banana chips from the same bag.

“Looks like we have different sensory experiences of the chips — different ‘qualia’, as the ancient Greek philosophers would say,” I commented.

“Huh? What’s that?” Ammu said.

2 This incident happened when our daughter Ammu was nine years old. The first person narrative was drafted by KPM, and significantly modified by TM.

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“You and I eat the same thing, but you experience it to be hard, and I experience it to be crisp. Remember when we eat the same chutney, you find it hot and I don’t? Our experiences of the same chutney, or the same banana chip, can be different.”

“It may not be a matter of subjective experiences,” said Tara. “It could be that we have different meanings for the words ‘crisp’ and ‘hard’.”

“What do you mean, different meanings? We’re all speaking English, aren’t we?” Ammu said.

“Try this,” said Tara, holding up her right hand. “How many fingers do I have on my right hand?”

“Five.” Ammu obviously thought this was a silly question.

“If I said that I have exactly four fingers on my right hand, no more, would that be true?”

“Of course not, it would be wrong. You have five fingers, not four.”

“Is a thumb a finger?”

“Of course, it is.”

“But you know, Ammu, the word ‘finger’ has two meanings. One meaning is that it is a digit on the hand. That includes the thumb. That’s your meaning. But there is another meaning: a finger is a digit on the hand other than the thumb. So the thumb is not a finger. Look it up in a dictionary if you want. So if my meaning of the word ‘finger’ excludes the thumb, and I say that I have exactly four fingers on my right hand, not five, is that true or false?”

“Oh! Oh, I see. Yeah, that would be true.”

“So could we be disagreeing about the chips because our meanings of ‘crisp’ and ‘hard’ are different?”

“Let’s taste some more and find out,” I said.

As we continued helping ourselves to more chips, Ammu said, “Hey, I just got a crisp one. So they’re not all hard.” Tara had also found some crisp ones. And I had found a few hard ones.

“So this means we came to our conclusions about hard and crisp based on just one chip. That’s why we were all wrong,” Tara said.

“We relied on anecdotal evidence,” I added.

Anecdotal evidence is an extreme case of defective sampling. Such hasty conclusions based on bad sampling are quite common both in both ordinary life and in scientific research.

Our curiosity was aroused; we had to make sense of why some chips were crisp and others hard.

Mo: So is there a pattern in the difference in crispness? Is it a random difference, or a systematic one?

Tara: Looks like the ones with more sugar coating on them are the hard ones, and those with less sugar are crisp.

Mo: That sounds like an intuitively satisfying explanation: what causes the difference in hardness and crispness is the amount of sugar coating. But how do we measure the amount of sugar coating on the chips? There’s no observable difference.

Tara: Yes, without an independent test, whenever we find a hard chip, we can claim that it has a lot of sugar coating, and whenever we find a crisp chip, we can claim that it has less sugar coating.

Mo: That makes the claim untestable.

Ammu: What does that mean?

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Mo: It means there is no way to find out if it is true or false. Suppose I tell you that headache is caused by headache demons. Every time someone has a headache, I say, “Ah, he’s possessed by a headache demon, that’s why he has a headache. And when headache is cured, I say, “Oh, the headache demon has left him.”

Ammu: But what if the headache goes because of taking aspirin?

Mo: I would say that some headache demons don’t like the taste of aspirin, so they go away. But there are other headache demons who don’t mind aspirin, so even if you take it, they’ll still be inside your head.

Ammu: That’s a silly theory, Acchaa.

Mo: Yes, I know. It’s silly because it is untestable. No amount of evidence can tell us whether the theory is true or false.

Ammu: So, saying that sugar coating is what causes crispness is like saying that headache demons are what causes headaches, unless there is some other way also to show that it is the sugar coating.

Mo: Exactly.

Scientists don’t like untestable claims in their theories because they are unfalsifiable. So we were unhappy. But that didn’t stop us from eating more chips. As we did this, Ammu examined the chips in the bag.

Ammu: Amme, Acchaa, look. Some chips are darker than others.

Tara: I wonder if that has something to do with the amount of sugar.

We tasted more chips, both dark and light ones, and as it turned out, the dark ones were systematically hard, and the light ones were crisp.

Tara: Let’s see if the harder ones have more sugar on them. I think thicker sugar coating tends to make fried stuff darker.

Ammu: Taste them. The harder chips, which are darker, are also sweeter. That means they have a thicker coating of sugar.

Mo: So now we have an independent correlation. Though the amount of sugar can’t be observed, colour and taste can. So there is an observable correlate.

Tara: This is beginning to look like a theory of crispness in banana chips!

Mo: We should now crystallize the intuitions into an explicit testable theory. Ammu, what are the patterns we’ve found?

Ammu: The harder chips are darker.

Tara: What else?

Ammu: The darker chips have more sugar coating.

Tara: Anything else?

Ammu: The sweeter chips have more sugar coating.

Mo: Good. So shall we write this down?

I got a sheet of paper, and wrote on it:

(1) Theory a. The darker the chip, the greater the amount of sugar coating. b. The more the sugar coating, the harder the chip. c. The more the sugar coating, the sweeter the chip. d. The less the hardness, the greater the crispness of the chip.

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Mo: The advantage of stating an intuition as a set of precise and explicit propositions is that

we can test our theory by checking the predictions that follow from it. Tara: Then we can check if our intuitive understanding is on the right track, or if we are

completely wrong.

Ammu: What do you mean, check the predictions? You mean the theory can foretell the future?

Tara: No, Ammu, the predictions of a theory are not the same as predictions about the future.

Ammu: But when you say, “I predict that it will rain today,” and it actually rains, you say that the prediction came true. It’s like a prophecy, right?

Tara: Yes, sweetie pie, but the word ‘prediction’ has a different meaning here. It doesn’t mean ‘to foretell’ the future.

Mo: In scientific theories, a prediction is a proposition about the observable state of affairs that follows as a logical consequence from the hypotheses of the theory or analysis.

Ammu: Acchaa!!! I don’t have a clue what you’re saying!

Mo: See if an example helps. Remember when were talking about why water is called H2O, we mentioned valency? We said that: (a) hydrogen has a valency of one, (b) oxygen has a valency of two, (c) the atomic weight of hydrogen is one, and (d) the atomic weight of oxygen is sixteen. With these four statements, or propositions, can you tell the ratio of hydrogen to oxygen when they combine to form water?

Ammu: Yeah. The ratio is 1:8.

Tara: So the four propositions ‘predicted’ that ratio. You were able to deduce the ratio on the basis of those statements. That is what prediction means in physics, in linguistics…

Mo: In scientific theories in general. So when we say, “The theory predicts X,” we mean “Given the hypotheses of the theory, we derive the consequence X as the outcome of a logical derivation, and we can check whether X is observationally correct or not.”

Ammu: Okay.

Mo: So now, if we put together two of the statements in our theory, that the darker the chip, the greater the amount of sugar coating, and that the more the sugar coating, the harder the chip, we get a prediction. Can you state it?

Ammu: The darker the chip, the harder it will be.

Tara: Excellent. And we found this prediction to be correct. All the dark chips were also hard.

Mo: Now, take the two statements that the more the sugar coating, the harder the chip, and that the less the hardness, the more crisp the chip, what result would you expect?

Ammu: That the lighter a chip is, the more crisp it will be.

Tara: Let’s check if this is true.

Mo: That would need further observation. You know, Ammu, one of the qualities of a good theory is that it points us in the direction of new and interesting observations.

Ammu: Oh, oh, I just had a chip that light but not crisp at all, it was actually hard. So the-lighter-the-crisper thing is not true.

Mo: Oh well, may be the dark chips are systematically hard, but the light ones show no pattern, they’re randomly hard or crisp.

Tara: M-hm, that can’t be. It can’t just be random like that. Let’s see. What about the thickness of the banana slices themselves? Oh, no. I was beginning to think we’re done, we’ve already eaten too many chips this morning. Now we’ll have to eat some more.

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Ammu: You’re right, Amma, I just ate a thick one and a thin one, both light ones. The thick one was hard, and the thin one crisp.

Mo: Is that so? Let me try some.

After a few more chips, we were now ready to modify our theory by adding a new statement.

(1) Theory (expanded) a. The darker the chip, the greater the amount of sugar coating. b. The more the sugar coating, the harder the chip. c. The more the sugar coating, the sweeter the chip. d. The less the hardness, the greater the crispness of the chip. e. The thicker a chip, the greater its hardness.

Mo: So Ammu, given the new statement, what result should we expect?

Ammu: That a light chip that is thin will be the crispest.

Further observation — more eating of chips — confirmed this expectation.

Tara: If we find this result in the lighter chips, shouldn’t we expect to find the same difference among the dark ones? The thicker the chip, the harder it will be, and the less crisp?

When we checked this prediction, we found it to be correct. Among the dark chips, the thick ones were extra hard. The verification of the prediction from the new data, which was revealed only after the theory was formed, increased our confidence in the theory, and we were happy.

Tara: Nice! So now we have a theory that makes falsifiable predictions. Thick and dark but crisp chips, or thin and light but hard ones, would be counterexamples. And if we find such chips, we’ll have to either modify or abandon the theory.

Mo: If we hadn’t formulated our intuitions as (1a-d), we wouldn’t have realized that our initial theory didn’t match our observations, and we wouldn’t have thought of looking for the relation between thickness and crispness, and arrived at (1e). It’s only because of stating our understanding in terms of explicit, logically connected, testable hypotheses that we explore factors and domains that we wouldn’t have considered otherwise.

4 Mother-Son Dialogues 4.1 Dialogue 1

S: Mom, do you believe in God?

M: Why do you want to know, Rafa?

S: I want to know if God exists.

M: Okay, but tell me why you’re interested in God all of a sudden.

S: Well, Sandy was asking me if I was an atheist. I asked her what an atheist was, and she said one who doesn’t believe that God exists. Do you believe that God exists?

M: If I told you that I believe that God exists, would you believe that God exists?

S: Yes.

M: And if I told you that God doesn’t exist, would you believe that God doesn’t exist?

S: Yes.

M: Then you would be very stupid, my darling son.

S: Huh? Why should that make me stupid? You’re not a liar, I believe you.

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M: I may not be a liar, but I might be mistaken. If you believe everything I say without questioning, you would be simply absorbing my beliefs, or taking my word for it, without thinking about it and making up your own mind. What would you do if I told you that God exists and your father told you that God doesn’t exist?

S: Oh! Er… I don’t know. I’d get very confused. How does one figure out for oneself if God exists?

M: Well, first you need to learn how to figure things out for yourself. Not just about God, but about everything in general. Then you can investigate the God question for yourself. Are you ready for that? It’s not going to be easy.

S: I’m ready.

4.2 Dialogue 2

M: Tell me what objects there are in front of you.

S: There’s you in front of me, of course, there’s a table, there’s a book on the table, there’s a bookshelf over there, …

M: Okay, how do you know there’s a bookshelf in front of you?

S: Huh? What do you mean?

M: You said there’s a bookshelf in front of you. How do you know that there’s a bookshelf in front of you?

S: I can see it. There, right in front of me. You’re weird, Mom, you know that?

M: Yes, I know I’m weird. But I’m making a point, and you’ll soon see what it is. You said you know that there is a bookshelf in front of you because you see a bookshelf in front of you.

S: How smart!

M: No wise cracks, junior! I’m going to put it a bit differently: you conclude that there is a bookshelf in front of you because you see a bookshelf in front of you. Better still, you see a bookshelf in front of you, and on the basis of what you see, you conclude that there is a bookshelf in front of you.

S: Are we going to do this hair splitting all day?

M: Just one more question. Is it possible that your conclusion based on what you see could be mistaken?

S: Er… I guess it’s possible, like sometimes I see something in front of me, but then I realize that it’s a reflection in the mirror.

M: Right. It could be an illusion, so you might be mistaken.

S: But that bookshelf is not an illusion.

M: I can grant you that. All I’m saying is, our eyes can deceive us, so we shouldn’t trust everything our eyes tell us. A reflection in a mirror is only one kind of illusion. If you put a straight rod in water, you see it as bent. You know it’s not bent but you can’t help seeing that it’s bent. That’s also an illusion. Can you think of other optical illusions?

S: Like the sky being blue?

M: Excellent. You’re catching on. Let’s take another one. Which is bigger, the moon or the stars?

S: The stars.

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M: But do you see the stars as bigger?

S: No, what I see is that the moon is bigger than the stars. Oh boy! If I had been born a few thousand years ago, I would have been totally certain that the moon is bigger than the stars.

M: Very good. So my point is this. To develop knowledge of the world we live in, we do have to rely on our sensory experience: what we see, what we touch, what we hear, taste, smell, and so on; but we have to be aware that our sensory experience is not totally reliable: it can deceive us. The conclusions that we draw from sensory experience are fallible, prone to error.

S: Mom?

M: Can we stop here? I’ve got to pee.

4.3 Dialogue 3

S: Mom?

M: Yes?

S: You said that all sensory perceptions are fallible. Including touch?

M: Including touch.

S: But there aren’t any illusions of touch. What we touch has to be real.

M: You think so? Let’s try something. Which of these two paperweights on the table has greater weight?

S: I don’t know.

M: Why don’t you find out?

S: Well, to find out how much each one weighs I’ll have to weigh them on a weighing machine. But when I lift them both, I can feel that the blue one is heavier.

M: So your conclusion is based on the sensation of strain you feel on your muscles when you lift them, right? You are saying, I experience greater strain on the muscles when I lift the blue paperweight, so I conclude that it has greater weight.

S: Yep. But you’re still talking weird, Mom. Normal people don’t say, “I conclude that it has greater weight.” We just say, “It’s heavier.”

M: Normal academics do say such things. And we are engaged in academic inquiry, aren’t we?So you better learn to talk weird if you want to learn this kind of thing. Now, suppose you find a red suitcase and a blue suitcase on the floor. You lift the red one. It’s heavy, but you can lift it. But when you try to lift the blue one, try as hard as you may, it doesn’t come off the floor. You will conclude that the blue suitcase is heavier, right?

S: Right…

M: But you might be wrong. It is possible that the suitcase is empty, but unknown to you, someone has screwed it to the floor, which is why it doesn’t come off the floor. The sensation of strain on your muscles is the same as that for lifting a heavy suitcase.

S: Wow! I hadn’t thought of that.

M: That means what our sense of touch tells us is also prone to error.

S: Is the muscular strain when lifting something part of the sense of touch?

M: Strictly speaking, no. It is felt more by the muscles than by the skin. Let me give you an example of actual sense of touch. This one comes from the ancient Greeks. They thought up this beautiful experiment in which you immerse one of your hands, say the left hand, in

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a bucket of very hot water for a minute or so, and your right hand in a bucket of very cold water. Now you take out both hands and immerse them in a bucket of water at room temperature. Your left hand will feel the water to be cold, but the right hand will feel the same water to be warm. It couldn’t be the case that both hands are telling you the truth, right? So at least one of them is mistaken.

S: What does all this have to do with the question I asked you about God?

M: Remember we said you have to figure that out for yourself? Right now we’re talking about how to investigate any question. And we’re beginning with questions having to do with sensory perception. The point I was making was, we need to rely on our sense perception as reliable grounds for our knowledge of the world, but our senses are fallible. They might tell us the truth, but they might also deceive us, so we have to examine critically whatever they are telling us.

S: Oh, okay… Mom, I just thought of something.

M: What?

S: You asked me which paperweight has greater weight?

M: Yes?

S: Paperweights don’t have weight.

M: What?

S: Yes, they have mass, they don’t have weight. What we call weight is the effect of the earth’s gravity acting on something that has mass, and the greater the mass, the greater the pull.

M: Okay, I see what you’re saying…

S: You were talking about the strain on the muscles, right? The strain that we experience on our muscles is not an inherent property of the paperweights, but the force that the earth exerts on the paperweights. Had we been on the moon, we would experience less strain on the muscles with the same paperweights, and in outer space, none at all. When we attribute that strain to the paperweights themselves, we’re mistaken. Another wrong inference of the senses. Yes?

M: Absolutely! That’s a pretty good discovery for a fuzzyheaded teen! I am impressed.

S: So you shouldn’t have asked me, “Which paperweight has greater weight?” Your question is misguided.

M: Go away! Go do your homework!

4.4 Dialogue 4

S: Mom, you pointed out yesterday that our senses are only partly reliable. We build our knowledge on conclusions on the basis of sensory experience, but those conclusions can be mistaken, so we have to think critically before we accept what our senses tell us.

M: You got it.

S: But to figure out whether some particular sense perception is an illusion or a true perception, you have to go to something more dependable. What is that more dependable source?

M: There is no such completely dependable source. We match inferences based on sense experiences with other inferences based on other sense experiences. If they are in conflict, choose the best one.

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S: I don’t understand.

M: If you see something in front of you, how would you make sure that it is not an illusion created by a mirror?

S: I’ll look behind me to check if I can see it behind me too. I might also check if I can see in front of me an image of me touching that object.

M: So you take a conclusion from one set of visual experiences and check it against the one from another set of visual experiences.

S: Yes.

M: Let’s take another example. Suppose you walk into a room and you see a statue on a stand. You walk around the statue to make sure that it’s not a reflection in a mirror. You would conclude that there’s a statue in front of you, right?

S: Yes.

M: Now imagine that you try to pick up the statue, and your hands go right through it. They don’t encounter anything solid that you can feel with them. What would your conclusion be, that there is a statue in front of you, or there’s no statue in front of you?

S: That there’s no statue in front of me. What I am seeing may be a hologram of a statue, not a real statue.

M: So given the conflict between the visual information that your eyes are giving you, and the tactile information that your hands are giving you, you decide to accept the tactile.

S: Hm! … That’s right, but … I don’t understand why I prioritize the tactile over the visual. Why do I accept the testimony of my hands and reject the testimony of my eyes when they are in conflict?

M: We’ll come to that after we do some more basic stuff. For know, I want you to think about sources of knowledge other than that of sensory perception.

S: Right now?

M: No. Keep thinking, and we’ll talk about it tomorrow.

S: Mom, there’s something that’s bugging me.

M: What’s bugging you, son?

S: Sensory experience is internal to us, right? It’s like other experiences, like when I experience a stomachache, nausea, or anger, it’s something that exists inside my consciousness.

M: That’s right. Experiences exist in our subjective world.

S: But when we say something is bigger, something is heavier, something is hot, we’re talking about what exists outside our consciousness.

M: That’s right. Those properties exist in the objective world. I know what you’re driving at. So our sensory experience is subjective, but our inferences about the world based on thesensory experience are ‘projections’ from the subjective to the objective world. The grounds we rely on are indeed subjective, but the conclusions are objective.

S: Mom?

M: Yes?

S: You’re cool, you know that?

M: Oh, really? So I am not weird any more?

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S: NO! This is awesome stuff we’ve been doing.

M: Thank you, my dear fan. That makes my day.

4.5 Dialogue 5

S: Mom, can we talk about this sensory perception thing?

M: Wait, let me finish washing the dishes first.

S: But I thought it was Dad’s turn to do that this week.

M: No, he’s cooking this week. He did the dishes last week and I do the dishes this week. It’s your turn next week.

S: Okay, but can we talk while you’re doing the dishes?

M: Alrighty, pull up a chair and sit down.

S: Well, I was thinking, …

M: That’s a pleasant surprise! So you have a new pastime now!

S: Oh, stop it, Mom. When I asked you about God you got me thinking about how we arrive at conclusions based on sensory experience. So I was wondering how I can find out if God exists through sense perception. I haven’t seen God, heard god, touched God and so on, so given my sensory experience, there’s no basis for concluding that God exists. But I can’t conclude that God doesn’t exist either, because there are plenty of things that exist that I haven’t seen, heard, smelled, touched or tasted. So does that mean I can never know?

M: I wouldn’t say that.

S: That’s what I thought. Can I ask you something different, Mom? I remember your telling me that your grandmother had six fingers on her left hand.

M: Right, and she wore black metal rings on five of those fingers whenever she went out…

S: She died before I was born, so I’ve never seen her. But I know that she had six fingers on her left hand. And when I ask myself how I know this, the only possible answer is your testimony. You told me that she had six fingers, and I believe you.

M: That’s correct.

S: That means that one of the sources of my knowledge is what other people tell me.

M: Right. Sense perception is not a person’s only source of knowledge.

S: Yet when I asked you about God, instead of giving me a straight answer, you made me jump through all these hoops of sensory experience and knowledge. If it’s okay for me to believe that my grandmother had six fingers just because you told me so, why isn’t it okay for me to believe that God exists – or doesn’t exist – because you told me so?

M: Excellent question! Let’s try this. Suppose I tell you that my grandmother had six fingers. And you doubt and question me, asking, “How do you know that?” What do you think I would say?

S: You would say that you have seen and counted your grandmothers fingers, and touched them too. And there is no reason to suggest that your sensory experience was mistaken in this case.

M: Excellent, Now suppose I tell you that God exists. And you doubt and question me, asking, “How do you know that?” What do you think I would say?

S: Hm! I see what you mean. You couldn’t have seen, heard, smelled, touched or tasted God, so how did you conclude that God exists? May be someone reliable told you.

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M: And how did that person come to conclude that God exists? Someone else told him?

S: At some point, there must have been someone who saw and heard God.

M: And how do you know that this person who claims to have seen and heard God wasn’t having a hallucination? What if he was suffering from left temporal lobe epilepsy?

S: What’s left temporal lobe epilepsy?

M: It’s a form of epilepsy caused by high electrical activity in the brain around the region of the left ear. People who have left temporal lobe epilepsy have visual and auditory hallucinations, typically of some power greater than themselves, and they take this as experiencing God.

S: Hmm! That’s a tough one. How can we distinguish hallucinations from what could be an authentic experience of God? I don’t know.

M: Think about it, but in the mean time, let me ask you a different question. Suppose someone tells you that God appeared to him and said that you should give him all your savings. If you don’t, God says He will send you to Hell. Would you give this person all your savings?

S: No way.

M: You’re not afraid of going to Hell?

S: I won’t go to Hell for not giving him my money.

M: So you won’t believe him when he says God wants you to give him your money?

S: No.

M: You think he must have been hallucinating?

S: Or simply telling a lie to get my money.

M: Let’s look at another example. Suppose someone tells you that Capitatians are dishonest people. Are you going to believe him?

S: No. I’ll ask him why he thinks Capitatians are dishonest.

M: And he says he knows someone who’s a Capitatian, and that person is dishonest.

S: That’s ridiculous. May be that particular Capitatian he knows is dishonest, but that doesn’t allow him to conclude that all Capitatians are dishonest, not even that most Capitatians are dishonest.

M: Very good. That’s called anecdotal evidence.

S: Anec what?

M: Anecdotal evidence. It’s a flawed way of arriving at an inference on a population, based on just one or two convenient examples. If, instead of a particular example, we took a different one, our conclusion would be quite different.

S: Anecdotal evidence. Hm! I am going to use that word on Sally tomorrow.

M: Don’t you bully that poor girl!

S: No I won’t. Sorry, I forgot what we were saying.

M: I was pointing out that, when someone tells you that something is true, you need to check the grounds for that conclusion, and check the reasoning from the grounds to the conclusion. You accept the conclusion only if you’re satisfied with the reliability of the grounds and the validity of the reasoning.

S: That’s a lot of words, and a lot of work.

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M: Becoming intelligent doesn’t come free: you need to work your mind to become intelligent. Having a strong, fast, agile mind is like having a strong, fast, agile body: you have to exercise your mind just like you have to exercise your body.

S: Can’t I just accept as true what I’m told by people who know more than me, like you and Dad, my teachers, the textbooks, and so on?

M: You can, but only if you’re okay with being stupid.

S: Okay, okay, I’ll exercise the biceps and triceps and other muscles of my mind. Oh, no, I am late for my ballet lesson. Bye, Mom!

M: Bye, sweetie. Don’t be late for dinner. Your Dad is making a great eggplant and broccoli curry.

S: Yuck, I’ll eat at my friends place today. Eggplant and broccoli indeed. Bye Mom.

4.6 Dialogue 6

S: Mom, I still don’t know how to wrap my head around the does-God-exist question.

M: You’re not ready to do that yet, Rafa. You need to learn to walk before you can run. Tell me what you learnt about epistemology from our kitchen conversations so far.

S: Apiscomology? Why do you use such big words?

M: Epistemology, not apiscomology. Some important concepts sometimes are carried by big words. Part of being educated is becoming familiar with those words. Epistemology is the study of knowledge, how we discover, justify, and evaluate knowledge claims.

S: Is a knowledge claim something that is put forward as knowledge, but has not been established as knowledge?

M: Yes. You wanted to know if God exists. To investigate that question, you need to learn how knowledge is arrived at and established, and how it is evaluated.

S: What did I learn about epistemology? Let me see. Well, I learnt that I can figure out things for myself, just by exercising my mind. I also learnt that it is important to doubt and question what people tell me, not blindly accept what others say, even if they are authorities or experts. That was an eye opener.

M: Very good. What else did you learn?

S: Looks like it’s a good idea to think of knowledge as a body of conclusions. If we do that, we can ask what are the grounds for those conclusions, and what reasoning derives those conclusions from the grounds. Then we can check if the grounds are reliable and the reasoning is valid.

M: Excellent. Spoken like a budding epistemologist. Sometimes you surprise me, Rafa. What else?

S: There are two kinds of grounds, sensory experience and the testimonies of others.

M: There are other kinds of grounds too, but go on.

S: Neither of these grounds is foolproof. They are fallible, so we need to check them carefully. Our own sensory experiences can be illusions or hallucinations, or just plain errors. And when other people report their sensory experiences, they too can be illusions, hallucinations or plain errors. And their reasoning might be flawed too.

M: That pretty much summarizes everything we’ve said.

S: It sounds like a philosophy of distrust, Mom.

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M: I hadn’t thought of it that way, but now that you mention it, I agree, it is indeed a philosophy of distrust, distrust of oneself and distrust of others, a healthy kind of distrust. And from that sense of fallibility comes the habit of doubting and questioning; also the readiness to correct our existing beliefs, and accept new beliefs if they are supported by sound justification.

S: But all that makes knowledge very tentative, doesn’t it?

M: Yes, it does.

S: It makes me uneasy when you say that anything I know can turn out to be false.

M: What you think you know might turn out to be false, what you judge as knowledge can turn out to be false. But when you discover it’s false, you can’t say that you knew it, can you?

S: Can we really know anything at all, then?

M: That was a big question in epistemology. Some ancient Greek philosophers, and even people more recently like Descartes, defined knowledge as “justified true beliefs”. If knowledge is indeed true beliefs, it can’t have even the slightest possibility of being false. That leads to the conclusion that we cannot know anything at all. But if we relax our demand for total certainty, and accept the idea of fallible knowledge, then there are certain things that we can know and do know. What’s important is being as rigorous as we can in our pursuit of knowledge.

S: Why don’t they teach us any of this in school?

M: That you should ask the people in your school, not me.

S: Schools are supposed to give us knowledge, but they force us to swallow everything that textbooks and teachers say, without questioning. So by your criteria, they make us more and more stupid.

M: Well, may be…

S: Can I stop going to school, Mom? I don’t want to become stupider than I already am.

M: Shut up and do your homework, junior!

S: You parental authority! You matriarch!

M: Scoot.

4.7 Dialogue 7

M: You’re early from school, Rafa. How come you didn’t go loafing with your friends today?

S: Well, I was thinking of something that happened in school. I couldn’t figure it out, so I thought I might as well come home and discuss it with you.

M: That’s flattering, you prefer discussions with me to chatting with your friends? What happened in school?

S: My math teacher was talking about pentagons and hexagons today and I asked him how he knew that pentagons have five sides and hexagons have six sides. He said haven’t you read the chapter on polygons in the textbook.

M: And what did you say?

S: I said yes, I have, but how does the textbook writer know that pentagons have five sides and hexagons have six sides. And he gave me one look, and said, Rafa, please don’t ask stupid questions in my class. Was that a stupid question, Mom?

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M: No, that was actually a very good question, especially given our conversations on sensory perception and verbal testimonies as sources of knowledge. But you can’t blame your teacher for getting annoyed, because he couldn’t have known why you asked that question.

S: Here’s what I’m confused about. Suppose a Martian comes to the earth, and I tell him that cows have four legs. If he asks me how I know that, I can show him a cow, and ask him to count the number of legs, and verify for himself that the number of legs is four. I can take a large random sample of cows, and show him that every cow in my sample has four legs. And I can tell him, you can go look for a cow which has more than four legs or less than four legs, but you’re not going to find any. So until you find a cow that shows that my statement is wrong, you have to accept my claim.

M: You constantly amaze me, Rafa. That’s fairly sophisticated reasoning using radical induction.

S: Radical induction? What’s that?

M: It’s a way of arriving at generalizations. If you observe some property P in every member of a sample, it’s reasonable to conclude that every member of the population has property P until you find evidence to the contrary. That means we acknowledge that our conclusion can be wrong, but until it is demonstrated to be wrong, it is legitimate to take it to be correct.

S: Yeah, that’s kind of what I was saying. So that’s called radical induction? Wow! So I am a logician?

M: Don’t let that get to your tiny head, boy.

S: Never mind. Here’s what I was driving up to. Suppose the Martian asks me how I know that a rectangle has four sides, and I use your radical induction and say, look, here is a rectangle. You can count the number of sides, and you can see for yourself that it is four. And I could take a thousand rectangles and show him that every rectangle has four sides. And I can tell him, you can go look for a rectangle which has more than four sides or less than four sides, but you’re not going to find any. So until you find a rectangle that shows that my statement is wrong, you have to accept my claim. I am doing exactly the same thing, but I have a funny feeling that there’s something wrong here.

M: You’re right. Let’s take a closer look to see why radical induction doesn’t work in this case. Suppose your Martian shows you a three-sided polygon and says, here is a rectangle with three sides, not four. Won’t you say, yes, but that’s not a rectangle, that’s a triangle?

S: Yes.

M: Suppose he shows you a five-sided polygon and says, here is one with five sides, not four. Won’t you say, yes but that’s not a rectangle, that’s a pentagon?

S: Yep.

M: And suppose he says, look, every time I show you what I think is evidence that contradicts your claim, you say that’s not a rectangle, so it doesn’t count. You’ve got to tell me exactly what a rectangle is so that I can look for contrary evidence. So tell me, what is a rectangle?

S: A rectangle is a four-sided polygon in which each angle is a right angle.

M: So what you’re saying is, “I choose to define rectangle as a four-sided polygon in which each angle is a right angle.”

S: Yes.

M: If you define a rectangle as a four-sided polygon in which each angle is a right angle, it follows from your very definition that a rectangle has four sides, doesn’t it? That is simply a logical consequence of your definition. You don’t need to do any counting of sides once

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you define it this way.

S: Hm! I kind of see what you’re saying, but not quite. Can you give me another example?

M: Take spiders. If you choose to define a spider as an eight-legged arthropod, …

S: What’s an arthropod?

M: Arthropods are invertebrates with external skeletons, segmented bodies, and jointed limbs.

S: I thought spiders were insects.

M: No. Insects have antennae. Spiders don’t. As I was saying, if you choose to define a spider as an eight-legged arthropod, it follows logically from the definition that a spider has eight legs. Any creature that has less than eight legs or more than eight legs by definition does not qualify as a spider. It also follows logically from the definition that a spider is an arthropod. An eight-legged insect by definition does not qualify as a spider.

S: Hm! That’s right. … But how come radical induction works for the number of legs cows have, but not the number of sides rectangles have?

M: If you defined a cow as a four-legged mammal, that would be true of cows as well, but I am assuming that you’re not going to define cows as four-legged mammals, because there are many four-legged mammals that are not cows. So if you have an independent way of defining cows, then whether or not a cow has four legs can be established through observation and radical induction.

S: Oh! Suppose I define a spider as an arthropod that spins webs, and then say that a spider has eight legs. That won’t be true by definition, right?

M: Right. That would be fine. I can look for arthropods that spin webs and yet have six legs or ten legs, and if I can show you that such creatures exist, you would be wrong.

S: Which of them is the correct definition, then? That a spider is an arthropod that has eight legs, or that a spider is arthropod that spins webs?

M: Definitions are not true or false. You define something in a certain way because it’s useful for some purpose. There’s no question of being correct or incorrect in your definition.

S: Mom, that doesn’t work. If my teacher asked me what is a pentagon, and I said a pentagon is a polygon with eight sides, he’s going to flunk me. I can’t choose to define a pentagon as an eight-sided polygon.

M: That’s because when a teacher asks you how many sides does a pentagon have, he’s not asking you for your own definition. He’s asking you how a pentagon is conventionally defined in the field of mathematics. So you need to know how mathematicians define it. It would definitely be wrong to say that mathematicians define a pentagon as an eight-sided polygon.

S: My head is all heated up, Mom. Before it starts smoking, can I have some ice cream.

M: Sorry, kiddo, there’s no ice cream in the fridge.

S: What? There was a whole quart of chocolate ice cream in the fridge when I looked last night.

M: Yea. But your father had a mid-night snack, and he polished it off.

S: Gee-whiz, Mom, if I become the president of this country I am going to banish all ice cream-eating fathers from this land.

M: In that case you have my vote.

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4.8 Dialogue 8

S: Mom, Mom, Mom!!!

M: What happened, Rafa?

S: Mom, this is awesome.

M: Stop jumping, Rafa. Sit here and calm down. Okay, you want to tell me what got you so excited?

S: I can define a spider as an arthropod with eight legs, or I can define it as an arthropod that spins webs, right?

M: Okay. So?

S: Suppose I define a spider as an arthropod that spins webs. I can now claim that spiders have eight legs. To test if my claim is true, I take a large random sample of arthropods that spin webs, and count their number of legs. If all of them have eight legs, I can take my claim as correct, until I find evidence to the contrary. I can do the opposite if I define a spider as an eight-legged arthropod. Right?

M: Yes, sir.

S: But that is not what I would do in the case of triangles. I can define a triangle either as a three-sided polygon or as a polygon whose angles add up to 180 degrees, right?

M: Right.

S: Suppose I define a triangle as a polygon whose angles add up to a hundred and eighty degrees. If I want to prove that a triangle has three sides, I won’t take a large random sample of polygons whose angles add up to a hundred and eight degrees, and count their sides. And if I define a triangle as a three-sided polygon, and I want to prove that its angles add up to 180 degrees, I am not going to take a large random sample of three-sided polygons and measure their angles. Right?

M: That’s correct.

S: Instead, I am going to prove it mathematically. I won’t make observations, and I won’t use radical induction.

M: That’s an excellent insight Rafa. I can see why you were jumping up and down with excitement.

S: A few days ago, we were talking about sensory perceptions and verbal testimonies as sources of knowledge. Then you asked me if there were other sources of knowledge.

M: Yes.

S: Okay, here’s another source of knowledge: what we already know. We can infer new knowledge from what we already know. Isn’t that what we do in mathematics.

M: That really really deserves a reward, Rafa. Come into the kitchen for ice cream. There’s strawberry, and there’s chocolate. Which of them do you want?

S: Can I have both?

M: A scoop each?

S: Can I have two scoops of each?

M: You’re stretching it, junior. Oh, okay. Just for today, you deserve four scoops. Get me that bowl on the table.

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4.9 Dialogue 9

S: Mom? This thing that you call radical induction, that’s subject to error, right?

M: Yes.

S: So your grounds can be correct, and the reasoning can be valid radical induction, and yet the conclusion can be false.

M: That’s right. Radical induction is a form of defeasible reasoning.

S: You’re into your big words trip again, Mom. What’s defeasible reasoning?

M: In defeasible reasoning, the grounds can be correct, and the reasoning can be valid, and yet something else can demonstrate that the conclusion is incorrect, so we accept the conclusion only tentatively, until we find reasons to reject the conclusion.

S: Does mathematics use defeasible reasoning?

M: Mathematical reasoning uses classical deductive reasoning. If the grounds are correct, and the reasoning is valid, the conclusion can never be false.

S: I don’t get it.

M: Okay, suppose I tell you that the Mr. Jenkins works in a rectangular office whose sides are 10 feet and 20 feet. One day, Jenkins decided to re-do the office floor with new tiles. Each tile is a six-inch square. How many tiles did he use?

S: What do you think I am, a third-grader?

M: Answer the question, Rafa.

S: 800 tiles.

M: Do you go to Mr Jenkin’s office and count the number of tiles?

S: No.

M: Well, how do you know that it has 800 tiles then?

S: You told me, remember?

M: I didn’t tell you that it had 800 tiles.

S: Well, you didn’t exactly tell me that it had 800 tiles, but you did say that it is a rectangular office with sides of 20 feet and 10 feet. You also told me that each tile was a six-inch square. So there must be 800 tiles to fill its area. You’re treating me like a third-grader, Mom.

M: Please humour me, Rafa.

S: Okay. Here it is, with every gory detail. The information given to me: Mr Jenkins has a rectangular office, 20 feet long and 10 feet wide. Each tile is a six-inch square. What I already know: the area of a rectangle is the product of its length and width. The product of 20 and 10 is 200. So the area of Mr Jenkins’ office is 200 square feet. You also told me that the tiles are six inch squares. One square foot has four six inch squares, so 200 square feet has 200 multiplied by four, that is 800 six inch squares. So you need 800 tiles. Satisfied?

M: Good. So, to answer the question I asked you, you used the information I gave you, and using what you already know about rectangles and squares, you inferred the answer. The mode of reasoning you used when you inferred the answer is classical deductive reasoning. If the information you have been given is correct, and if the knowledge you appeal to is also correct, and your reasoning is valid, your conclusion can never be wrong.

S: Can you give me an example outside math?

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M: Try this. Suppose I tell you that all gleeks are slobins, and that Mr Brigost is a gleek. Then you can conclude that Mr Brigost is a slobin. If it is true that all gleeks are slobins and if it is true that Mr Brigost is a gleek, it can never be false that Mr Brigost is a slobin. That reasoning is classical deduction.

S: What about mathematical theorems?

M: What about them?

S: Are theorems proved using classical deduction?

M: Yes.

S: Can you give me an example?

M: Draw a rectangle on a piece of paper.

S: Done.

M: Now draw a line from one angle to the diagonally opposite one.

S: Like this?

M: Yes. You now have two triangles. Are their areas equal?

S: Yes.

M: How do you know that?

S: That’s simple. Let’s label the corners of the rectangle as ABCD, like this:

A B

D C

Well, we know that the area of triangle is half the product of its length and height. The base of ADC is DC and its height is AD, so its area is 1/2 DC x AD. Let us take the base of ABC as AB, so its height is BC. So its area is 1/2 AB x BC. But I know that AB = DC and AD = BC, so 1/2 DC x AD is equal to 1/2 AB x BC. The two triangles have the same area.

M: How do you know that AB = DC and AD = BC?

S: We know that the opposite sides of a rectangle are equal. AB is the opposite side of DC so they are equal. AD is the opposite side of BC and so they are equal.

M: How do you know that the opposite sides of a rectangle are equal?

S: Is this how-do-you-know-that going to go on forever, Mom?

M: It will end soon. Answer me now.

S: Well, I know that the opposite sides of a rectangle are equal because a rectangle is a parallelogram whose internal angles are right angles, and the opposite sides of a parallelogram are equal, so the opposite sides of a rectangle are equal.

M: How do you know that a rectangle is a parallelogram whose internal angles are right angles?

S: That’s how a rectangle is defined.

M: Okay, how do you know that the opposite sides of a parallelogram are equal?

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S: That too is a definition. A parallelogram is defined as a quadrilateral whose opposite sides are equal.

M: Good. Once we reach a definition or axiom we can stop asking, “how do you know that?” When I said that mathematical theorems are proved using classical deduction, you asked me for an example. Your proof that the diagonal of a rectangle divides it into two triangles of equal area is an example of mathematical proof. You began with two premises that you took to be true, one, that the area of a triangle is half the product of its base and height, and two, the base and height of one of them are equal to the base and height of the other. From these, it deductively follows that the areas of the two triangles are equal. But when I asked you to prove the statement that the base and height of one of them are equal to the base and height of the other, you deduced it from another premise, that the opposite sides of a rectangle are equal. I asked you to prove that premise, and you deduced it from the premises that a rectangle is a parallelogram whose internal angles are right angles, and the opposite sides of a parallelogram are equal. And these two premises are both definitions. Every step in a mathematical proof is a deduction from a set of premises you take to be true, and if you keep questioning the premises for each proof, you end up with a set of definitions and axioms.

S: Axioms? We didn’t have any axioms in our proof.

M: One of the premises that you took to be true was that the area of a triangle is half the product of its base and height. Right? If you start questioning that premise, you’ll reach Euclid’s parallel axiom.

S: Euclid’s parallel axiom? What’s that?

M: I am tired, Rafa, let’s stop now. We’ll talk about it tomorrow.

S: Just one question, Mom. You said legitimate scientific conclusions that use radical induction can turn out to be wrong.

M: Yes.

S: But not legitimate mathematical conclusions, because they use only classical deduction?

M: That’s right. But the operative word here is “legitimate”. That means that the grounds are correct and the reasoning is also correct. If there is an undetected flaw in the reasoning, the reasoning is incorrect, and that would make the conclusion illegitimate. If so, it might even be the case that the conclusion is false. But such things happen only if the proof is flawed, unlike in science.

S: Can someone have a high aptitude for mathematical thinking but not for scientific thinking?

M: Yes.

S: And the reverse? Can someone have a high aptitude for scientific thinking but not for mathematical thinking?

M: That too.

S: Does that mean that mathematics is not a science?

M: I would say that math is not science.

S: So why is math taught in the Faculty of Science? Isn’t that irrational?

M: Who says everything in the world is rational?

S: But Mom, educational institutions are supposed to be rational. They’re not like other things in the world.

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M: They’re supposed to be, I agree, but are they actually rational? I don’t think so. In actual practice they’re as irrational as any other form of human activity.

S: If educational institutions can be irrational, isn’t there a danger that I will be learning irrational things from them?

M: Yes, that’s exactly why we’re having these conversations. I want you to develop the capacity to decide for yourself what to believe and what to do, instead of blindly accepting what your teachers and textbooks tell you.

S: One more question, Mom. Aren’t you then being irrational in insisting that I should still learn from irrational institutions?

M: Rafa?

S: Yes Mom.

M: Don’t you have homework to do, instead of asking silly questions just to get me all riled up?

S: Yes, Madam Hitler. I’ll go and do my homework.

4.10 Dialogue 10

S: Mom, you were saying yesterday that in science you use defeasible reasoning to arrive at conclusions on the basis of observations, but in math you use non-defeasible deductive reasoning to arrive at conclusion on the basis of what we already know.

M: Well, kind of. That could be a misleading statement, but it’s a good starting point.

S: But we use non-defeasible deductive reasoning in science too. And we can use that to arrive at a conclusion from what we already know.

M: For example?

S: Let us suppose that I go to the moon, and make a big hole, say about 10 feet in diameter, all the way from one side to the other, right through the center.

M: You think the moon is made out of cheese to make a hole through the center?

S: Doesn’t matter. Imagine that situation. I haven’t been to the moon, and even if I do manage to go the moon at some point in my life, I am not going to make such a hole.

M: Small mercies!

S: Now, if I stand at the edge of the whole, and drop a coin, what would happen to the coin? Would it stay floating in the air, go up, go down and stop at the centre, go down all the way to the other edge, go down all the way to the other edge and come back to oscillate back and forth, or go down beyond the edge and away from the moon?

M: Interesting question!

S: That is a question to inquire into, right? Of course someone can go to the moon, and if the technology were available, make a hole and find out from actual observation what would happen to the coin. But we don’t have to do that, right? We can actually deduce the answer from what we already know.

M: What’s your answer?

S: It will go down all the way to the other edge, come back to where we’re standing, and keep oscillating back and forth.

M: Excellent. And how do you know that?

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S: Well, given Newton’s theory of gravity, when we drop the coin just above the surface of the moon, the gravitational force of the entire moon would be pulling it downward. But after it goes down, say a few feet, then the mass of the moon on one side of the coin would exert force in one direction, but the mass of the moon on the other side would exert force in the opposite direction. Like this:

moon: blue circle coin: orange circle gravitational force: purple arrows

When it begins to move, the gravitational force on the other side — the one with the longer arrow — is stronger, so its velocity increases until it reaches the centre where the gravitational pulls neutralize each other, and there is zero gravity. But given the second law, we know that it would keep moving, but it begins to slow down because the gravitational force from the opposite side is stronger, until it gets to the other edge where the velocity becomes zero. The coin stops and starts moving in the opposite direction.

M: Good reasoning, Rafa. S: Thanks Mom. But here’s my point. I arrived at this conclusion without making any

observations. All I needed was Newton’s theory of gravity and motion and the information that the moon has no atmosphere. I deduced the answer from what I already know.

M: You’re right. What you did is exactly what is done in mathematical inquiry. Actually, this is mathematical calculation, except that you did it qualitatively, without using numbers. But notice that your inference was from the scientific theory plus a piece of information about the moon. What would happen if I asked you how do you know that Newton’s theory of gravity and motion is correct? Can you use deductive reasoning to answer that?

S: Erm… Well, I guess this time I’ll have to appeal to observations. M: Right. And you’ll be using a form of defeasible reasoning. S: Radical induction? M: No. A form of reasoning called speculative deduction. S: What’s that? M: I’ll explain that later, but right now I want you to understand that there’s a difference

between justifying a theory, and justifying a conclusion arising from that theory. S: Come again? M: Suppose I am giving a talk on a scientific theory that I have constructed. Someone in the

audience asks, how do you know that your theory is correct? Why should I accept your theory? To respond to that question, I appeal to observations, and use speculative deduction to show that the theory should be accepted. Now imagine that my theory is already accepted. Assuming that the theory is correct, I arrive at a conclusion on what I would observe if such and such situations existed. Let us call it a prediction. To show that the prediction does follow from the theory, I use deductive reasoning.

S: So…. If I need to show that Newton’s theory is correct, I use one form of reasoning, but if I want to show that certain predictions logically follow from Newton’s theory, I use another form of reasoning.

M: Yes. S: Phew! This stuff is harder than I thought. M: Don’t worry about the niceties of reasoning for now. It will become easier to understand

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once we work through a few examples. The point I was making is that if you keep asking, “how do you know that,” at some point you get to observations in science. The ultimate grounds in science are observations.

S: And in math? M: In math, it is the combination of axioms and definitions, not observations. If someone asks

how you know that your theorem is correct, you begin with the theorems that you already know are correct, and use deductive reasoning to deduce the new theorem from the old ones. But someone might ask how you know those theorems, and you appeal to some other known theorems. But if you keep asking, “how do you know that,” for every theorem that’s given as the ground, you finally get to axioms and definitions.