SCIE1000_workbook.pdf

SCIE1000/1100 Theory and Practice in Science Scientific Modelling thinking and analysis Programming Communication N...

SCIE1000/1100 Theory and Practice in Science Scientific Modelling thinking and analysis Programming Communication Name: _____________________________________ Contact details: _____________________________________ Twenty-ninth edition, Semester 2, 2024 © School of Mathematics and Physics, The University of Queensland, Brisbane QLD 4072, Australia  SCIE1000/1100 The core content falls into four key areas: scientific thinking, modelling, programming and communication. The content is taught in a highly ``interleaved'' manner rather than in separate blocks, but the figure below may help you to appreciate the course goals and content. Thinking Modelling Programming Communication scientific units linear equations software design units in dimensional quadratics errors communication analysis power functions input and output communicating mechanistic with graphs sine functions variables modelling science in the exponentials calling functions phenomeno- media logarithms conditionals logical modelling how to be precise, average rates of loops clear, and concise hypotheses change arrays history of scientific understand your meaning of plotting purpose thinking derivatives inductive writing functions know your Newton’s method audience reasoning areas under curves convey your key Popperian science differential messages equations quantitative reasoning Euler’s method systems of DEs Predator-prey models SIR models Scientific Contexts blood alcohol daytime hours contraception Mars Climate Orbiter migration of terns exposure to nicotine cancer radioactive decay glucose thrombosis carbon dating bioavailability of a drug HIV screening pressure bacteria growth atmospheric temperature atmospheric CO2 oyster populations Bicknell’s thrush pharmocokinetics endangered turtles biodiversity heart disease lynx and snowshoe hares sun protection apparent temperature disease models seasons metabolism of alcohol vaccination targets 2  SCIE1000 Modelling and Modelling with functions analysis Linear Binary classification Power Periodic Exponential & Logarithmic Rates of change Area under curve Combining Functions Differential equations Scientific Communication thinking Systems of DEs Life cycle diagrams Predator-prey models Disease models Newton’s method Euler’s method Trapezoidal rule Programming 3 Table of Contents 1 A short discussion of nearly everything 9 1.1 Workshop Preparation............................. 9 1.2 About this course................................ 12 1.3 Science and philosophy............................. 16 1.4 Science and modelling............................. 18 1.5 Science and Communication.......................... 24 Case Study 1: Please raise your right hand................ 24 1.6 Science and Programming........................... 26 Program 1.1: Blood Alcohol Concentration................. 26 2 Thinking and communicating 28 2.1 Workshop Preparation............................. 28 2.2 Losing patients with mathematics?...................... 30 Case Study 2: Cancer............................ 32 2.3 Binary classification............................... 35 3 Scientific reasoning - Part 1 44 3.1 Introduction: science and the assumption of rationality........... 45 3.2 Getting Philosophical.............................. 46 3.3 Some Preliminaries............................... 46 3.4 Science and Inductive Reasoning........................ 50 3.5 The Renaissance: experimentation and mathematics................................ 52 3.6 A Common View of Science.......................... 54 4 Scientific reasoning - Part 2 63 4.1 Popperian Science................................ 63 4.2 Kuhn’s View of Science and its Challenges.................. 70 4.3 Exercise..................................... 75 5 Modelling - power functions 76 5.1 Workshop Preparation............................. 76 5.2 Temperature.................................. 77 Case Study 3: Higher than a kite...................... 77 4  5.3 Bend it!..................................... 81 Case Study 4: Climate change and Bicknell’s thrush........... 81 5.4 (Super) powers................................. 84 Case Study 5: Species-area curves and biodiversity............ 84 Case Study 6: Ban the tan, man...................... 90 6 Modelling - periodic functions 92 6.1 Workshop Preparation............................. 92 6.2 Seasons, daytimes and cycles.......................... 95 Case Study 7: Modelling daytimes..................... 98 Case Study 8: To every thing, there is a season, tern, tern, tern..... 104 7 Modelling - exponential functions 107 7.1 Workshop Preparation............................. 107 7.2 Exponentials in action............................. 109 Case Study 9: Radioactive decay...................... 109 Case Study 10: Hot stuff, cold stuff..................... 113 Program 7.1: Temperatures......................... 114 7.3 Logarithms in action.............................. 116 8 Modelling - combined functions 121 8.1 Workshop Preparation............................. 121 8.2 The Keeling curve................................ 123 Case Study 11: Atmospheric CO2...................... 123 8.3 Drugs in the blood and surge functions.................... 128 Case Study 12: Zoloft and depression................... 128 8.4 Predicting heart disease............................ 133 Case Study 13: To the heart of the matter................. 133 8.5 Apparent temperature............................. 137 Case Study 14: Apparent temperature for Aussies............. 137 9 Rates of change 140 9.1 Workshop Preparation............................. 140 9.2 Sobering rates of change............................ 142 Case Study 15: Whisky (back to BAC)................... 142 Case Study 16: Drink deriving....................... 144 Program 9.1: BACs and food consumption................. 149 5  9.3 Derivatives and Newton’s method....................... 151 Case Study 17: Contraceptive calculations................. 151 Program 9.2: Using Newton’s method for contraception.......... 158 10 Area under the curve 159 10.1 Workshop Preparation............................. 159 10.2 Areas under curves............................... 161 Case Study 18: Dying for a drink...................... 162 Program 10.1: Wilful exposure (to alcohol)................. 167 Case Study 19: Smoking areas....................... 168 Program 10.2: AUC for nicotine...................... 170 10.3 All in the blood................................. 172 Case Study 20: Sweet P’s.......................... 172 Case Study 21: Bioavailability of drugs................... 176 11 Differential equations and populations 180 11.1 Workshop Preparation............................. 180 11.2 Writing differential equations.......................... 182 11.3 The exponential DE.............................. 183 Case Study 22: Poo............................. 184 11.4 Euler’s method................................. 187 Program 11.1: E. coli............................ 190 11.5 Limited scope for growth............................ 192 11.6 Oysters in Chesapeake Bay........................... 194 Case Study 23: Overfishing annoys an oyster............... 194 12 Systems of DEs 201 12.1 Workshop Preparation............................. 201 12.2 Going through a difficult stage......................... 204 Case Study 24: Total turtle turmoil..................... 204 Program 12.1: Turtles............................ 207 12.3 Eat or be eaten................................. 209 Case Study 25: Snowshoe hares and Canadian lynx............ 210 Program 12.2: Lotka-Volterra model of hares and lynx........... 213 12.4 Epidemics and SIR models........................... 216 Case Study 26: Rubella........................... 219 Program 12.3: SIR model of rubella..................... 220 6  Case Study 27: Vaccinations........................ 222 A Programming manual 225 A.1 Getting started................................. 225 A.2 Basic use of Python............................... 227 Program A.1: Printing things........................ 229 Program A.2: Simple calculations...................... 230 Program A.3: Spacing inside Python programs............... 230 A.3 Variables, functions and input......................... 231 Program A.4: Variables........................... 232 Program A.5: Functions........................... 233 Program A.6: Input............................. 235 A.4 Software errors and bugs............................ 235 Program A.7: Multiple errors........................ 236 A.5 Conditionals................................... 238 Program A.8: The basic if......................... 238 Program A.9: An if-else......................... 240 Program A.10: An if-elif-else..................... 240 A.6 Loops...................................... 241 Program A.11: Simple For Loop...................... 241 Program A.12: Nested For Loop...................... 242 Program A.13: While Loop Prompt..................... 243 Program A.14: While Loop......................... 243 Program A.15: Infinite loop......................... 244 A.7 Arrays...................................... 245 Program A.16: Our first array....................... 245 Program A.17: Creating arrays....................... 247 Program A.18: Creating new arrays from old............... 247 Program A.19: Accessing individual array elements............ 248 Program A.20: Arrays and loops...................... 249 A.8 Writing functions................................ 249 Program A.21: Degrees to radians, 1.................... 250 Program A.22: Degrees to radians, 2.................... 250 Program A.23: Converting to radians................... 250 Program A.24: Using a new function.................... 252 A.9 Graphs...................................... 253 7  Program A.25: Using plot(...) and show(...)............. 253 Program A.26: Plotting graphs, 2...................... 255 Program A.27: Some customised plots................... 255 A.10 Python summary................................ 257 B Communication in Science 260 B.1 Principle 1: Being clear............................. 262 B.2 Principle 2: Knowing your purpose...................... 267 B.3 Principle 3: Knowing your audience...................... 271 B.4 Principle 4: Identifying key messages..................... 273 C Prerequisite maths review 278 C.1 Linear, quadratic, and power functions.................... 278 C.2 Periodic functions............................... 280 C.3 Exponential and logarithmic functions.................... 282 C.4 Rates of change................................ 284 C.5 Area under the curve............................. 286 D The use of units in Science 287 D.1 SI Units and prefixes.............................. 287 D.2 Derived units.................................. 288 D.3 Algebra for quantities and units........................ 288 D.4 Unit conversions................................ 289 List of Tables 290 List of Figures 291 List of Images 293 List of Original Photographs 294 List of Programs 295 Bibliography 296 Index 303 8  Chapter 1: A short discussion of nearly everything Workshop 1: Quantitative Modelling Learning objectives ✓ Course rationale ✓ Introduce modelling, communication, philosophy and programming ✓ Develop critical evaluation skills Scientific examples ✓ Blood alcohol concentration ✓ Mars Climate Orbiter Maths skills ✓ Developing a plausible mathematical model Preparation ✓ Read through the course profile ✓ Look through the preparation section below 1.1 Workshop Preparation At the start of each chapter of this workbook you will find a preparation section. Please read through this material before attending the associated workshop. The reading will provide context and background information which will be built upon during classes. Course rationale The purpose of science is to understand, explain, predict and influence phenomena. SCIE1000/1100 aim to help you to develop a range of relevant skills, integrating aspects of science, philosophy, mathematics and computing. SCIE1000/1100 use a unique delivery style at UQ, as they are taught in teams of two. Your teaching staff are various discipline experts, including practising scientists, mathematicians and philosophers. Each brings something different to the discussion, and this is intentional. In class you will see that the study 9  §1.1. WORKSHOP PREPARATION Workshop 1: Quantitative Modelling of any one topic is, quite naturally, interdisciplinary. A scientist’s toolkit is never limited to “biology skills” or “chemistry skills.” Consideration of a single topic may, at different times, require us to think like a biologist, a chemist, a philosopher, a mathematician and a computer programmer, in order to achieve our aims. The scientific contexts we examine are real and important. Almost every example and case study is based on a research paper, or is a reasonably accurate model of a real situation. For example, when we derive an equation to model blood alcohol content, the equation genuinely models real data. You can estimate your own blood alcohol content using the equation. What could be more topical than rising CO2 levels in the atmosphere? Or an understanding of pandemic modelling? As compelling as the contexts are, memorising the specific details of any particular context is not your goal as a student. Instead, your focus will be on applying the underlying concepts to novel situations, understanding and practicing how to think critically, developing and applying models, interpret- ing your results, and communicating with clarity and precision. We encourage you to engage fully with the course material. The development of your quantitative skills requires regular practice! The content covered in this course is all within this course book. We will introduce the material in class and then you will have the opportunity in a second session to apply these skills. A separate contact session will assist you in developing your programming capabilities (a skill which is important for almost any scientific discipline). We will regularly be referring to journal publications throughout this course. We recommend that you obtain copies of these articles and glance through them prior to each class. The UQ library has free access to many journals and you can thus search online and obtain free copies if you are logged on in the UQ domain (on campus or UQ VPN). If this seems daunting, don’t worry. Almost all of the mathematics we use will be familiar and you will be provided with resources to build your con- 10  §1.1. WORKSHOP PREPARATION Workshop 1: Quantitative Modelling fidence; we will build your computer programming skills from scratch; and any scientific principles we need will be introduced as if you have never seen them before. You need only bring curiosity, diligence, and a determination to critically inquire. 11  §1.2. ABOUT THIS COURSE Workshop 1: Quantitative Modelling 1.2 About this course Course staff: Dr Mel Robertson-Dean Dr Ashleigh Richardson Mathematics Computer Science Course Coordinator Dr Tatsuya Amano Dr Ben Roberts Environmental Science Physics A/Prof Joel Katzav Dr Peter Evans Philosophy Philosophy 12  §1.2. ABOUT THIS COURSE Workshop 1: Quantitative Modelling Course aims: In this course we aim to develop your capabilities as a scientist. This will include elements of quantitative modelling and analysis (building on your prior mathematics) communication the philosophy of science programming Course assessment: Practical exercises Philosophy assignment Python and communication assignment Final exam You should read the Course Profile in detail, where you will find the weighting of the assessment items as well as course policies. Learning resources: This course book, together with annotations Course Blackboard site (announcements, access to other material) Workshop recordings Practical tasks and solutions SCIE1000/1100 Online Modules for Support and Enrichment (SOMSE) Selected past exams and solutions ShiFoo for extra programming practice Getting help: Ask questions during the workshops Talk to your tutors and peers during your scheduled practicals Attend teaching staff office hours 13  §1.2. ABOUT THIS COURSE Workshop 1: Quantitative Modelling Visit the online course discussion board Check out “Course Help” on Blackboard for FAQs and support Science Science aims to understand, explain, predict and influence phenom- ena. Understanding science, and thinking in a ‘scientific manner’, requires: discipline knowledge and content – the language, information, knowl- edge and skills specific to a discipline; scientific thinking and logic – the conceptual process of performing systematic investigations, hypothesising, thinking critically and de- fensibly, and making valid deductions and inferences; communication and collaboration – the process of working with others, sharing information and resources; curiosity, creativity and persistence – the relatively intangible charac- teristics that include the ability to ask and answer ‘interesting’ ques- tions, and solve difficult problems in novel ways; observation and data collection – the processes and techniques used to collect useful data about particular phenomena; modelling and analysis – the process of developing conceptual rep- resentations of phenomena, then using approximation, mathematics, statistics and computation in order to allow predictions to be made. Most science is intrinsically quantitative, because quantifying phenomena allows us to measure, describe and compare them efficiently and precisely. Science often proceeds by: observing phenomena and measuring values, such as the amount, fre- quency, magnitude, duration or rate using these data and scientific thinking to develop a scientific model applying this model to answer predictive questions, such as 14  §1.2. ABOUT THIS COURSE Workshop 1: Quantitative Modelling − “What will happen if... ?” − “What causes... ?” − “How can... ?” − “Why does... ?” To learn the science in this course, we encourage you to: Prepare for workshops by looking through the listed pre-reading prior to class Attend all learning activities and actively participate Make use of the extra resources available to aid your learning Ask questions - often! Direct questions to the teaching staff, to tutors and to your peers Appreciate that learning is about understanding, not memorising Note that AI can be a useful resource in the learning process. However, be aware that generative AI programs can provide outputs that are entirely convincing but factually incorrect. You should thus critically evaluate any material that you access (this includes AI, the internet and publications). Also note that generative AI must not be used in the development of so- lutions to assessment items in this course. Such use will be considered as academic misconduct. The potential penalties of academic misconduct at this university are available through the UQ Policy and Procedures Library (PPL). 15  §1.3. SCIENCE AND PHILOSOPHY Workshop 1: Quantitative Modelling 1.3 Science and philosophy The School of Athens (left) depicts some famous scientists, mathematicians and philosophers, including Plato, Aristotle, Euclid, Socrates and Pythagoras. Science and knowledge play fundamental roles in human history, culture and society. In order to understand phenomena in the world around us, we use a variety of math- Image 1.1: The School of Athens (1510 – ematical and scientific techniques, real- 1511), Raphael (1483 – 1520), Stanze di world data, as well as “common-sense” in Raffaello, Apostolic Palace, Vatican. order to develop models. Whatever model we are considering, as scientists, we must compare its predictions to real-world data. Hypotheses, laws, Mathematical theories techniques Policy Common-sense Model Etc. Data Figure 1.1: Philosophy of Models: Models as Mediators 16  §1.3. SCIENCE AND PHILOSOPHY Workshop 1: Quantitative Modelling Question 1.3.1 The figure below shows climate model projections according to various modelling scenarios. Figure 1.2: Climate model predictions for global warming relative to global average temperatures in 2000, according to model scenarios considered by the IPCC Third Assessment Report As we have just noted, a variety of considerations are required when de- veloping a model. How is the fact that models are mediators illustrated in the example of the climate models? 17  §1.4. SCIENCE AND MODELLING Workshop 1: Quantitative Modelling 1.4 Science and modelling Models Models are simplifications and approximations of reality, usually based on measured data, that allow us to understand phenomena, make predictions and evaluate possible impacts of interventions. All models need to strike an appropriate balance between accuracy and complexity. Models can be physical or conceptual. Examples of physical models include building scale models of bridges or dams, and subjecting the model to tests. In this course, we will focus on conceptual models. Some common ways of developing and presenting (conceptual) quantitative models are: words (descriptive text); values (for example, weight / height / age tables); pictures (such as graphs or flow diagrams); mathematics (using equations); computer programs. Note that there is nothing “right” or “wrong” about each approach – each is suited to different uses and/or target audiences. Most models can be developed and presented in all of these ways. Mechanistic Models vs Phenomenological Models A mechanistic model is a model that is derived theoretically by exam- ining the individual physical components of a system. A phenomenological model is a model that is derived empirically by fitting a hypothesised relationship to the observed data. When developing a mechanistic model, it is important to ask yourself what are the key factors that could influence the phenomenon in question. 18  §1.4. SCIENCE AND MODELLING Workshop 1: Quantitative Modelling Phenomenological models are usually determined using statistical software. In this course, you will not be required to run such software; instead we will discuss the basic features of mathematical functions that can be used to fit various sets of observed data. Critical thinking will be valued more than precision. Gender and sex In this course, and often in science more generally, one thing we do is to aim to keep models as simple as possible. When modelling medical phenomena, it can be useful to have a variable that considers the influence of a person’s sex. First, it is important to recognise the distinction between a person’s sex and a person’s gender. The World Health Organisation says: “Gender interacts with but is different from sex, which refers to the different biological and physiological characteristics of females, males and intersex persons, such as chromosomes, hormones and reproductive organs. Gender and sex are related to but different from gender identity. Gender identity refers to a person’s deeply felt, internal and individual experience of gender, which may or may not correspond to the person’s physiology or designated sex at birth.” Often when we consider the influence of sex in scientific modelling, the relevant factors are related to body fat percentage and hormones. Under the hormonal effects of oestrogen, which is typically higher for females, comparatively larger amounts of fat tissue are deposited around the body. Within any category used to describe a person’s sex, there still may also be quite a bit of natural variation around what are considered to be “standard” levels of fatty tissue and hormones. However, tracking these levels for individuals is quite a complex task. Thus, for the purpose of making simple models, we will sometimes use a binary choice for “sex”; for example, when modelling blood alcohol concentration. We hope that you recognise the limitations that come with making such an assumption in a model. 19  §1.4. SCIENCE AND MODELLING Workshop 1: Quantitative Modelling Question 1.4.1 Develop a mathematical model of the blood alcohol content of a person after they consume a quantity of alcohol. 20  §1.4. SCIENCE AND MODELLING Workshop 1: Quantitative Modelling A unit of measurement in science is an agreed upon quantity of something; any other quantity of the same kind can be described by giving the ratio between that quantity and the unit. For example, when we write that the minimum length of a long jump pit is 2.75 metres, we are saying that 2.75 is the ratio between the minimum length of the long jump pit and the length that we have agreed to call a metre. The consequences of the poor use of units can be severe. Example 1.4.2 The Mars Climate Orbiter was launched in 1998 as part of a $USD330 million project, but in September 1999 it crashed into Mars. Here is an extract from the report into the accident : “During the 9-month journey from Earth to Mars, propulsion ma- noeuvres were periodically performed... coupled with the fact that the angular momentum (impulse) data was in English, rather than metric, units, resulted in small errors being introduced in the tra- jectory estimate over the course of the 9-month journey. At the time of Mars insertion, the spacecraft trajectory was approximately 170 km lower than planned... it was discovered that the small forces ∆V s reported by the spacecraft engineers for use in orbit de- termination solutions was low by a factor of 4.45 (1 pound force = 4.45 Newtons) because the impulse bit data contained in the AMD file was delivered in lb-sec instead of the specified and ex- pected units of Newton-sec.” Photo 1.1: Mars Lander (proof test model) from the Viking program, launched 1975. (Source: PA.) 21  §1.4. SCIENCE AND MODELLING Workshop 1: Quantitative Modelling In this course, SI units will generally be used although we will encounter situations where other units have been introduced. Refer to Appendix D “The use of units in Science” for a review of scientific units. About the use of units in this course We will regularly highlight the units of numerical quantities that we calculate. We will use only scientific units (SI base units and combinations). A unit must always be included with the final values calculated (unless a value is unit-less). However, it is not necessary to include units in every intermediary step. We will generally work with two or three significant figures in our calculations (noting that uncertainties in values are not covered in this course). Dimensional analysis is an important technique in science. It involves applying the following principle: an equation describing a physical situation can be true only if it is dimensionally homogeneous; that is, both sides of the equation have the same units. Dimensional analysis allows a quick check of whether a calculation is ‘plausible’. If the dimensions do not match, then there must be an error. Question 1.4.3 Use dimensional analysis to evaluate if our equation modelling blood alco- hol concentration is plausible according to the dimensions. 22  §1.4. SCIENCE AND MODELLING Workshop 1: Quantitative Modelling The model we developed is quite similar to a model which is often used in courts to determine blood alcohol levels; this is called the Widmark formula and it was developed in 1932. The equation is: A B= × 100% − V t, rM where B is the blood alcohol concentration (as a %) at time t (where t is measured in hours since drinking commenced), A is the amount of alcohol consumed (in grams), M is the body mass (in grams) and r is an estimate of the proportion of body mass that is water. V is a constant with an accepted numerical value of 0.015. Question 1.4.4 Consider the Widmark formula. What are the units of the constant V. What does this constant represent physically? Question 1.4.5 Is the equation we developed for BAC mechanistic or phenomenological? 23  §1.5. SCIENCE AND COMMUNICATION Case Study 1: Please raise your right hand 1.5 Science and Communication Over the duration of this course you will develop habits of mind and good practice that help ensure your communication is effective. These will set you up to communicate effectively in your specific, precise science discipline, and more broadly as a scientifically informed member of society. Communication Communication involves writing, reading, listening, speaking. It also in- volves words, numbers, code, graphics and images. Effective communica- tion is essential for successful human interactions and scientific endeavours, from study to the workplace, from research to impacts in society. Today we begin by considering with whom you communicate: your audience. The study is from a Court of Appeals case in Texas, USA. Case Study 1: Please raise your right hand In 1993, Raul Mata was charged for driving while intoxicated. An expert witness (a scientist) testified that Mata’s blood alcohol concentration two hours after his arrest was in excess of 0.10 (above the legal limit). Mata disputed the breath test results because they were obtained “from scien- tific techniques which have not been shown by clear and convincing evidence to be reliable and relevant”. The expert testified that: Only one measurement was taken Several variables were assumed including weight and whether Mata had eaten anything It was not known if Mata engaged in “normal” drinking patterns The jury convicted Mata of driving while intoxicated and Mata was sentenced to 90 days in jail, probated for two years, and a fine of $400. 24  §1.5. SCIENCE AND COMMUNICATION Question 1.5.1 Make a list of potential people (or groups of people) involved in commu- nication in such a case. Question 1.5.2 Choose one person (or group) from the list and identify their communica- tion needs and preferences. End of Case Study 1: Please raise your right hand. 25  §1.6. SCIENCE AND PROGRAMMING 1.6 Science and Programming Computation is important when formulating and applying models, particu- larly when dealing with complex phenomena. In this course we will write programs to model some phenomena. Programming requires technical skill, experience and creativity. The programming sessions and activities will help you to learn these aspects of the course. Refer to Appendix A for the Python programming instruction manual. We can write a short Python program to calculate blood alcohol concentration based on the Widmark formula. Question 1.6.1 In the code below, see if you can relate the calculation that is performed back to the variables used in the Widmark formula.  Program 1.1: Blood Alcohol Concentration  1 # C a l c u l a t e s blood a l c o h o l c o n c e n t r a t i o n (BAC) f o r a p e r s o n 2 3 # Input v a l u e s 4 mass_person = f l o a t ( i n p u t ( "What i s t h e p e r s o n ’ s mass ( i n kg ? ) " ) ) 5 sex_person = i n t ( i n p u t ( " Enter 1 f o r female , 2 f o r male : " ) ) 6 no_drinks = i n t ( i n p u t ( "Number o f s t a n d a r d d r i n k s ? " ) ) 7 drink_time = f l o a t ( i n p u t ( "How many hours ago ? " ) ) 8 9 # Convert d r i n k s t o mass (1 0 g a l c o h o l p er s t a n d a r d d r i n k ) 10 mass_alcohol = 10 ∗ no_drinks 11 12 # Mass o f water ( i n grams ) f o r th e p e r s o n 13 i f sex_person == 1 : 14 mass_water = 1000 ∗ mass_person ∗ 0. 6 15 else : 16 mass_water = 1000 ∗ mass_person ∗ 0. 7 17 18 # M e t a b o l i c r a t e o f removal (%/hour ) 19 removal_rate = 0. 0 1 5 20 21 # Apply Widmark f o r m u l a 22 bac = ( mass_alcohol / mass_water ) ∗ 100 − removal_rate ∗ drink_time 23 p r i n t ( "The p e r s o n ’ s blood a l c o h o l c o n c e n t r a t i o n i s " , round ( bac , 3 ) , "%" )   26  §1.6. SCIENCE AND PROGRAMMING Writing, testing and debugging Most newly written programs include errors, and it is important to adopt a systematic approach to minimising the number of errors, then identifying and fixing any that occur. This process is called testing and debugging. Some programming errors are relatively easy to find (such as missing brack- ets), some will result in runtime error messages (for example, trying to di- vide by zero), but in other cases a program may produce incorrect output without an error message. To find such errors, you will need to test your program with different input values, and check the output “by hand”. When writing programs, make sure that you: Think about the clearest and most logical way to solve the problem; Write your program in an organised, systematic manner. Include useful comments in the program. Test your programs on a range of data; Check some output carefully to make sure it is correct; and Pay attention to any error messages! 27  Chapter 2: Thinking and communicating Workshop 2: Making sense in science Learning objectives ✓ Understand the need for clear communication of scientific information ✓ Evaluate quantitative information presented by the media ✓ Interpret binary classification test tables Scientific examples ✓ Breast and prostate cancer ✓ Birth control and thrombosis ✓ HIV screening Maths skills ✓ Evaluate probabilities, percentages and ratios Key literature ✓ Helping Doctors and Patients Make Sense of Health Statistics Preparation ✓ Read through the preparation section below 2.1 Workshop Preparation In this course you will develop your skills in scientific communication. Some key guiding principles of scientific communi- cation are: be clear, know your purpose, know your audience, and identify your key messages. Appendix B “Communication in Science” provides detailed advice and ex- amples, and we will work through these Image 2.1: The Persistence of Memory concepts throughout the course. (1931), Salvador Dalí Here we consider the communication of scientific knowledge, particularly in the context of effectively communicating and interpreting quantitative data 28  §2.1. WORKSHOP PREPARATION Workshop 2: Making sense in science in a medical context. We will explore the fundamental skills and concepts that will help you with effective scientific analysis and communication. The goal is to motivate why clear scientific communication is so important. We are all producers and consumers of quantitative scientific information, in the form of scientific papers, assignments, media reports, the internet, and professional communications such as doctor/patient discussions. As a producer of such information, we should aspire to be accurate, honest, logical, unambiguous, concise, precise, not excessively technical, and always mindful of the intended audience. Specifically, our aim when communicating in science is to be precise, clear, and concise (see Appendix B). To ensure this, we will follow four principles: Being clear Knowing your audience Knowing your purpose Identifying key messages As a consumer, we should aspire to be thoughtful, reflective, sceptical, log- ical and analytical, while at the same time open-minded and accepting of evidence that may differ from our preconceptions or opinions. The media and internet provide a continual bombardment of facts, reports, summaries, interpretations and opinions, often covering sophisticated concepts but writ- ten and read by non-experts. In many cases there are errors (or deliberate falsities) in such communications. You should form the habit of critically evaluating information, data and (claimed) conclusions. A useful approach when checking your own work, or the work of others, is rough estimation, which is the process of calculating approximate values. It involves building rough, conceptual models, and then evaluating them ‘for sense’. Estimating ‘gives an idea’ whether a particular value is plausible. Often, we aim to find an approximate value within an order of magnitude of the correct value (that is, within a factor of 10 of the correct value). Before attending the workshop, ensure that you have developed a working knowledge of the key principles of communication, and of making rough es- timates using simplified models. 29  §2.2. LOSING PATIENTS WITH MATHEMATICS? Workshop 2: Making sense in science 2.2 Losing patients with mathematics? Sometimes, particularly in a medical context, critically evaluating quantita- tive information is a matter of life and death. A paper from 2007 presents the following key findings : Many people (doctors, patients, journalists and politicians) do not un- derstand health statistics. Lack of understanding is due both to lack of knowledge, and intentional misrepresentation of information. The following paragraph is a quote from the paper: “Statistical literacy is a necessary precondition for an educated citi- zenship in a technological democracy. Understanding risks and ask- ing critical questions can also shape the emotional climate in a society so that hopes and anxieties are no longer as easily manipulated from outside... ” Question 2.2.1 In , researchers asked 450 American adults (aged 35-70; 320 had at- tended college; 62 had a postgraduate degree) for answers to the following questions: “1. A person taking Drug A has a 1% chance of having an allergic reaction. If 1,000 people take Drug A, how many would you expect to have an allergic reaction? 2. A person taking Drug B has a 1 in 1,000 chance of an aller- gic reaction. What percent of people taking Drug B will have an allergic reaction? 3. Imagine that I flip a coin 1,000 times. What is your best guess about how many times the coin would come up heads in 1,000 flips?” 30  §2.2. LOSING PATIENTS WITH MATHEMATICS? Workshop 2: Making sense in science Question 2.2.1 (continued) (a) What are the answers to the above three questions? (b) What are the ramifications of getting these answers incorrect for doc- tors, journalists and politicians? Question 2.2.2 In 1995, an emergency announcement in the UK warned that third- generation oral contraceptive pills doubled the risk of potentially life- threatening blood clots (thrombosis). The announcement led to widespread concern and fear, and many women ceased using the contracep- tives. Reports estimate that in the following year there were an additional 13,000 abortions and 13,000 births, with 800 additional pregnancies in girls under 16 years of age. The announcement omitted the following relevant information: young women have an absolute risk of spontaneous thrombosis of 1 in 10,000. the absolute risk of thrombosis when taking second-generation oral 31  §2.2. LOSING PATIENTS WITH MATHEMATICS? Case Study 2: Cancer Question 2.2.2 (continued) contraceptive pills is about 1 in 7000. the relative risk of thrombosis increases by a factor of 4 to 8 during a Caesarean birth. the relative risk of thrombosis during and after pregnancy increases by a factor of around 4. the absolute risk of dying from thrombosis during or after an abortion is around 1.1 in 10,000. (a) Define the terms “absolute risk” and “relative risk”. (b) The Australian Medical Association (AMA) website states that: “... in order to support and enhance the collaborative nature of the doctor-patient relationship, patients must be able to make informed choices regarding their health care. An informed choice is dependent on receiving reliable, balanced health information, free from the influence of commercial considerations, that is communicated in a manner easily understood by patients.” Write a public health announcement that would better advise the public on the risks of the third-generation contraceptive pill. 32  §2.2. LOSING PATIENTS WITH MATHEMATICS? Case Study 2: Cancer Case Study 2: Cancer Cancer is the name for a large group of diseases affecting many different parts of the body. It arises from the uncontrolled, rapid growth of abnormal cells that interfere with the usual bodily functions. Treatments of cancer include chemotherapy, radiation therapy and surgery. These can all have minor to major side effects, including fatigue, nausea, mouth ulcers, hair loss, cognitive problems, infection, anaemia, infertility, graft-versus-host disease, burns, cancer (!) or death. Two commonly reported medical statistics are: the 5-year survival rate, which is the percentage of people who are still alive five years after being diagnosed with a condition; and the annual mortality rate, which is the number of people dying from a given condition each year, often expressed as a rate per 100,000 people. Question 2.2.3 (a) The 5-year survival rate for prostate cancer in American men is 98%; for British men it is 71%. (i) Assume that 1,000 British men and 1,000 American men receive a diagnosis of prostate cancer (at the same time). After 5 years how many men in each country are expected to have died? (ii) Considering only the given data, which country has the ‘better’ health system, and why? 33  §2.2. LOSING PATIENTS WITH MATHEMATICS? Case Study 2: Cancer Question 2.2.3 (continued) (b) The annual mortality rate for prostate cancer in American men is 26 deaths per 100,000; for British men it is 27 per 100,000. Considering only these data, which country has the ‘better’ health system, and why? (c) The medical information given in Parts (a) and (b) is all correct. Ex- plain how the (apparent) discrepancies could occur. (d) Treatment for prostate cancer is invasive with many substantial side effects, including incontinence and impotence. Considering only prostate cancer, which country has the ‘better’ health system, and why? 34  §2.3. BINARY CLASSIFICATION Case Study 2: Cancer 2.3 Binary classification A binary classification test aims to classify objects, people or things into one of two groups. Examples include many medical tests, such as determining whether or not an individual has (or is likely to have) cancer. Binary classification test The table below illustrates how we can represent the possible outcomes of such as a test in tabular form. Disease Yes No Test + Test – Most binary classification tests are imperfect: results can be true positives, false positives, true negatives or false negatives. Identify where these terms would each sit in the table above. We can also represent this information in a probability flowchart which we can use to identify the number of true/false positives/negatives for a given test in a population. This is created as follows: This chart starts at the top with the total number, N , who are tested. This is then divided, on the next line, into those who have the disease (according to some ‘gold standard’ test) and those who don’t have the disease. These are then each further subdivided on the last line into those that test positive according to the binary classification test (true positive, false positive) and those that test negative (true negative and false negative). 35  §2.3. BINARY CLASSIFICATION Case Study 2: Cancer Accuracy, sensitivity, specificity and prevalence The terms accuracy, sensitivity, specificity and prevalence are defined mathematically below. Explain what each term means and why it is im- portant. Disease Yes No Test + A B N =A+B+C +D Test – C D A+D Accuracy = N A Sensitivity = A+C D Specificity = B+D A+C Prevalence = N 36  §2.3. BINARY CLASSIFICATION Case Study 2: Cancer Question 2.3.1 A paper studied the effectiveness of combined mammography and ultrasound imaging to screen for breast cancer. A total of 203 patients returned “suspicious or malignant” test results, of whom 138 were later found to have cancer (via biopsy testing). A total of 2811 patients returned “normal or probably benign” test results, of whom 12 were later found to have cancer. Find the accuracy, sensitivity, specificity and prevalence of the combined procedures. 37  §2.3. BINARY CLASSIFICATION Case Study 2: Cancer Question 2.3.2 Most binary classification tests are not 100% accurate. Clearly it is better to have more true positives and true negatives than false positives and false negatives. However, it is often the case that it is not possible to simultaneously optimise both the sensitivity and the specificity. (a) Identify some negative impacts of false positive or false negative cancer test results. (b) When might a higher false positive rate be tolerated? When might a higher false negative rate be tolerated? (c) Are false positive results ‘better’ or ‘worse’ than false negative results? 38  §2.3. BINARY CLASSIFICATION Case Study 2: Cancer Question 2.3.3 A paper quotes an example in which 160 gynaecologists were asked: “Assume you conduct breast cancer screening using mammography... You know the following information about the women in this region: The probability that a woman has breast cancer is 1% (prevalence) If a woman has breast cancer, the probability that she tests positive is 90% (sensitivity) If a woman does not have breast cancer, the probability that she nev- ertheless tests positive is 9% (false-positive rate) A woman tests positive. She wants to know whether that means that she has breast cancer for sure, or what the chances are. What is the best answer? A. The probability that she has breast cancer is about 81%. B. Out of 10 women who test positive, about 9 have breast cancer. C. Out of 10 women who test positive, about 1 has breast cancer. D. The probability that she has breast cancer is about 1%.” (a) Without doing detailed calculations, what is your answer? 39  §2.3. BINARY CLASSIFICATION Question 2.3.3 (continued) (b) Investigate the answer to the question using a probability flowchart and a group of 1000 ‘typical’ women. (c) Repeat Part (b), instead using a binary classification table. (d) What proportion of gynaecologists do you think could answer Part (a) correctly? What are the implications for you and/or your female relatives? End of Case Study 2: Cancer. 40  §2.3. BINARY CLASSIFICATION Question 2.3.4 In the 1980s, blood screening in Florida found that 22 people who had donated blood tested positive for HIV. Once notified of the test results, seven of these donors died by suicide. (At that time, HIV was not well known, and people were not regularly tested. Screening donors for the disease commenced after the discovery that transmission of HIV occurred through contact with infected blood.) The HIV test has a very high sensitivity [percentage of infected individuals who correctly test positive] of about 99.9% and specificity [percentage of non-infected individuals who correctly test negative] of about 99.99%. The prevalence, or rate of infection, for heterosexual men with low-risk behaviour, is around 1 in 10,000. What is the (approximate) probability that someone who tests positive for HIV is infected? 41  §2.3. BINARY CLASSIFICATION Question 2.3.5 To investigate the quality of HIV counselling for heterosexual men with low-risk behaviour, an undercover client visited 20 public health centres in Germany, undergoing 20 HIV tests. The client was explicit about belonging to a low risk group, as do the ma- jority of people who take HIV tests. In the mandatory pre-test counselling session, the client asked: ‘Could I possibly test positive if I do not have the virus? And if so, how often does this happen?’ The answers from the medical practitioners were: No, certainly not False positives never happen Absolutely impossible With absolute certainty, no With absolute certainty, no With absolute certainty, no No, absolutely not Definitely not... extremely rare Never Absolutely not... 99.7% specificity Absolutely impossible Absolutely not... 99.9% specificity Absolutely impossible More than 99% specificity With absolute certainty, no More than 99.9% specificity The test is absolutely certain 99.9% specificity No, only in France, not here Don’t worry, trust me (a) How would you answer the question? 42  §2.3. BINARY CLASSIFICATION Question 2.3.5 (continued) (b) Recall that the Australian Medical Association (AMA) website states: “... in order to support and enhance the collaborative nature of the doctor-patient relationship, patients must be able to make informed choices regarding their health care. An informed choice is dependent on receiving reliable, balanced health information, free from the influence of commercial considerations, that is communicated in a manner easily understood by patients.” Two key aspects of communication are being clear and knowing your audience. Do you think any of the answers from the German doctors demonstrate these principles? 43  Chapter 3: Scientific reasoning - Part 1 Workshop 3: Philosophy - Part 1 Learning objectives ✓ Clarify the module question, ‘How is science rational?’ ✓ Realise that some important questions are not able to be answered by experiment (e.g. how is science rational?) ✓ Develop clarity and precision in language (e.g. being clear about what we mean by the terms ‘theory’, ‘law’, ‘proof’ or ‘model’) ✓ Consider how models are created and used in science ✓ Examine how induction works ✓ Recognise the proper role subjectivity has in science ✓ Consider challenges to the idea that induction is a rational process, including that of justifying induction; formulate a version of the principle of induction that addresses some of these challenges ✓ Understand the problem of induction ✓ Recognise what special challenges models pose for the inductivist view of science Image 3.1: Émilie Du Châtelet (1706-1749), mathematician, physicist and philosopher of the French Englightenment, portrait by Latour 44 §3.1. INTRODUCTION: SCIENCE AND THE ASSUMPTION OF RATIONALITY Workshop 3: Philosophy - Part 1 3.1 Introduction: science and the assumption of rationality News Headlines: “No Rational Person Can Deny Human-Induced Global Warming” What does this mean? What, if anything, is scientific “rationality”? Many people believe that science is rational and that this is because there is such a thing as the scientific method by means of which we are able to make reliable claims about the natural world. Results arrived at by means of the scientific method have a special status – scientific knowledge is reliable knowledge, unlike claims made on the basis of common sense. This is not to say that common sense never leads to truth or science always does, but rather that our grounds for accepting claims made in the name of science are stronger than those for accepting claims made on the basis of common sense – this by virtue of the method used for arriving at the claims made. To illustrate the point, consider the conflict between the common sense view that the sun revolves around the earth (the earth feels stationary, the sun seems to move across the sky, etc.) and the scientific view that, in fact, the earth revolves around the sun. This conflict is typically resolved in favour of the scientific view. In general, where science and common sense conflict, common sense gives way. So consider two public speakers A and B. A stands and proclaims that human-induced global warming is not occurring, B claims it is. When asked, A admits that he claims no global warming due to human activities is occurring because common sense suggests that any fluctuation is more likely to be part of a natural cycle of climate change. B, on the other hand, admits that she claims human-induced global warming is occurring because scientific evidence all points to the fact. Now, regardless of who is in fact right, who do we have more reason to believe? B would commonly be said to be the more credible of the two by virtue of the means employed for arriving at the claim – B’s method is more reliable, more rational, than A’s. The general view underlying this resolution of the conflict seems to be, again, that scientific claims to knowledge have some kind of merit not shared by common sense claims. But if merit attaches to scientific knowledge then why? What makes science rational? The common answer, again, is that science employs a method which is rational and is the means by which scientific knowledge is arrived at. This is what we will be discussing in these workshops. 45  §3.2. GETTING PHILOSOPHICAL Workshop 3: Philosophy - Part 1 3.2 Getting Philosophical Of course, at this point, in discussing the general nature of science, we are engaged in an activity other than science itself. No amount of scientific experimentation will tell us whether or how science is rational. Just think about it for a minute. Asking questions about what science is and how it works is not something that we can do in a lab. White coats, bunsen burners and experiments won’t help. When we question our beliefs about science we step out of science itself to a more abstract level of discussion. We are engaged in philosophical debate and argument about the nature of science. The following workshops are directed at introducing you to some of the philosophical issues that arise in attempting to explain the apparent rationality of science and scientific method. 3.3 Some Preliminaries 3.3.1 What is Science? What is science? As we shall see, it is not a body of facts. Despite common views to the contrary, science is not in the business to putting forward proven facts. Scientific laws and theories are continually being overthrown in the face of problems, or anomalies, that the laws or theories fail to adequately account for – they are always provisional to some degree, as are the specific scientific claims that depend on them. Newton’s Laws, for example, were never scientific facts. They were conjectures that were eventually overthrown by “better” laws, those of relativity theory. Science is a way or method of thinking: thinking critically about the empirical world using evidence to try to justify hypotheses, laws and theories (collections of laws about some set of phenomena), put forward as conjectures that are subject to further critical testing against ever-increasing bodies of evidence. Key Point: Science is not a body of facts. It is a way of thinking critically about the empirical world using evidence to make general conjectures. Just think about the hypothesis of human-induced global warming. We appeal to evidence about humans increasing carbon dioxide levels in the atmosphere and their effect on the heat-retaining capacity of the atmosphere (the so-called “green house effect”), and go on to conjecture that we are thus the cause of a warmer climate. This conjecture is further tested by increasing bodies of evidence about past climate variation and its possible causes. And so it goes. 46  §3.3. SOME PRELIMINARIES Workshop 3: Philosophy - Part 1 This conjecture is, of course, a hypothesis in the applied field of climate science. As applied science, it depends on a large number of even more fundamental physical and chemical laws, as well as associated mathematical principles that enable modelling of exponential growth of gas concentrations and summative effects of gas concentrations as a result of chemical reactions, etc. And these currently accepted laws and mathematical principles themselves are taken as (currently) justified. So how do we justify these? What justifies these fundamental laws and principles as acceptable? For obvious reasons, we shall limit our discussion to scientific theory and set aside the many (interesting) issues surrounding the acceptability of mathematics used in science. The philosophy of mathematics is yet another area of philosophical enquiry with a heritage stretching back to the ancient Greek philosopher Pythagoras and beyond. The notion of mathematical truth, in particular, has been the subject of considerable study – most notably in the late 19th and early 20th centuries with key players like Bertrand Russell, David Hilbert and Kurt Gödel. Our focus, though, is squarely on scientific hypotheses, laws and theories. 3.3.2 Hypotheses, Laws, Theories and Models Science is made up of many hypotheses and laws, and groups of them that work together to make up scientific theories – evolutionary theory, electromagnetic theory, and so on. Let’s just stop for a minute to get clear on our terms here. As potential scientists you will come across the terms ‘hypothesis’, ‘law’ and ‘theory’ a lot, and the way they are used in science is sometimes different to how they get used “on the street”, so some clarification may help here. — A Hypothesis: a (scientifically testable) claim used to predict or explain some par- ticular phenomenon or event. E.g. hypothesising that, since a ball began to move, it was acted on by some force. The hypothesis is that the ball was acted on by some force. — A Law: a (scientifically testable) claim describing a general regularity in nature used to predict or explain some particular phenomenon or event. E.g. All bodies remain at rest unless acted upon by some force. — A Theory: a set of interconnected laws and principles working together to form a model (typically involving significant idealisation) used to explain the general regularities themselves. E.g. Einstein’s general theory of relativity (including E = mc2 , etc.) is a model that explains gravitational laws. Darwin’s theory of evolution (including principles of natural selection, etc.) is a model that explains biological diversity. 47  §3.3. SOME PRELIMINARIES Workshop 3: Philosophy - Part 1 Thus theories explain how the world works in general and why it works as it does by providing a model of the system in question. Hypotheses and laws are simply used to tell us why some particular thing happened but might themselves stand in need of explanation. NB: Later in these workshops we’ll pin down, more exactly, what we mean by “scientifically testable”. For now, the above distinctions should be sufficiently clear. Some more examples: 1. “Gravity is caused by undetectable particles exerting forces” may appear to be a hypoth- esis but is not (it is not testable – as we shall see later when discussing testability). 2. “The postie is sick” is a hypothesis (that one might offer to explain the lack of mail), but is not a law (it is not general). 3. “F = ma” is a law (describing a general regularity: when a force acts on an object it is caused to accelerate by an amount which when multiplied by the mass of the object, equals the force applied), but is not a theory. The relationship between the three quantities remains unexplained. 4. “Einstein’s theory” explains general features of the observable world expressed by grav- itational laws, etc. Common Misuses and Abuses Note that these key terms are not always used this way outside of science. “Theory”: People sometimes mean a mere guess or speculation lacking any support. (For example, “That is just a theory”.) The former US President Ronald Reagan was famously reported as saying “Evolution is a theory, a scientific theory only... ”. He meant it was just a guess, arguing that it was no more reasonable to believe than creationism (the view that the world was created by God rather than evolved). “Law”: People sometimes mean a general regularity in nature that has been proven. (For example, “But that is a law. It cannot be wrong.”) Scientists generally use the terms ‘theory’ and ‘law’ in a way that is neutral concerning whether they have no support, some support, or very strong support. More generally, the terms ‘hypothesis’, ‘law’ and ‘theory’ do not indicate a difference in how well established or proven a scientific claim is; they indicate the kind of claims in question – a specific claim that is general (law) or not, or an explanatory set of claims (theory). When we want to indicate that we are speaking of the currently supported or accepted view, we often speak 48  §3.3. SOME PRELIMINARIES Workshop 3: Philosophy - Part 1 of the theory of evolution, or the law of gravity. (Here we indicate that one from among many candidates is accepted.) Don’t forget about models! — Models: idealised, i.e., simplified or distorted, representations of portions of the world. Models can be theory driven, that is, arrived at by the application of theory. Models can also be data driven, that is arrived at on the basis of data. Often, models are theory and data driven. E.g. The Keeling Curve is a data model of the accumulation of Carbon Dioxide in the Earth’s atmosphere. It is partly because models are idealised that they are so useful. Data would be unmanageable if we did not simplify/distort it, e.g., by modelling it with a simple function. Theories would often be hard, if not impossible, to test or apply if they were not simplified in the process of their application. The idealised nature of models, therefore, enables them to connect our theories, hypotheses and laws – our ‘ideas’ – with data – the world – and thus enables testing our ideas. Models are, as the philosophers of science Mary Morgan and Margaret Morrison neatly put it, mediators. Image 3.2: Margaret Morrison and Mary S. Morgan 3.3.3 The Task Ahead Scientists typically believe that their way of thinking about the world involves some method, “the scientific method”, and this rational method gives a way of justifying hypotheses, laws and theories (the results of scientific activity) as scientific knowledge. So what is this “method” and how does it produce scientific knowledge? 49  §3.4. SCIENCE AND INDUCTIVE REASONING Workshop 3: Philosophy - Part 1 3.4 Science and Inductive Reasoning A popular view of scientific method is that science begins with particular observations of natural phenomena. From observation one logically arrives at general principles – scientific laws – by an inference known as induction.1 For example, imagine Boyle (1627-1691) studying the behaviour of a gas at constant tem- perature. He observes the following numerical measures of its volume at different pressures, in appropriate units: Pressure Volume 1 12 2 6 3 4 4 3 From an examination of these few measures he infers the general law that the product of the pressure and volume is constant (given constant temperature) – Boyle’s Law. Justification of scientific laws is thus by way of a special kind of generalising argument – using “inductive reasoning” to infer the law-like structure of the universe. Our belief in the laws of science is therefore rational since based on logical argument from evidence... or so the story goes according to the inductivist. In the same way, Avicenna (the 10-11th century Arabic philosopher, scientist, physician and mathematician) gave the following example of inductive inference, referring to the purgative effects of scammony2 (not to be tried at home!): It is observed that ingestion of scammony is followed by the discharge of red bile This observation is repeated under circumstances in which other possible causes of the discharge of red bile have been excluded Hence, all scammony according to its nature withdraws red bile. We seem to use this kind of inference all the time in daily life. Why expect that the fire will burn me? Because it has numerous times in the past. Why think that food will nourish me? Because it has numerous times in the past. The main principle underlying this use of “inductive inference” is the idea that what occurs frequently does not do so by chance. 1 “Induction” here is not to be confused with the mathematical inference known as “mathematical induction”. 2 A twining plant having a stout taproot, Convolvulus scammonia, found in Syria, Asia Minor, Greece, etc. The dried milky juice was used as a medicine from ancient times. 50  §3.4. SCIENCE AND INDUCTIVE REASONING Workshop 3: Philosophy - Part 1 In this way, from observations of lunar eclipses and other phenomena concerning light one inductively infers general principles or laws – e.g. that light travels in straight lines; that opaque bodies cast shadows; and that certain configurations of opaque and luminous bodies place one opaque body in the shadow of another. (See the left side of Figure 3.1.) These general principles or laws themselves can then serve as assumptions, along with some other facts concerning particular conditions etc., enabling one to deduce statements about phenomena. Continuing with the example above: the laws concerning opaque bodies and light, along with the fact that the earth and the moon are opaque bodies and the fact that on such-and-such a date the earth will pass across the line between the luminous sun and moon, can be used to deduce that there will be a lunar eclipse on that date. In this way we can make predictions about events not yet observed. (See the right side of Figure 3.1.) Alternatively, by showing how an observation can be deduced from the laws one can progress from an already observed fact that the lunar surface has darkened to an understanding of why this took place. In this way, deducing the observation from laws about the nature of light and opaque bodies, along with particular facts, we can provide an explanation of events already observed. Laws Induction Deduction Observations Predictions or Observations Figure 3.1: Induction and deduction. On this account, “induction” is the crucial process used to arrive at laws. It allows us to infer some general law-like property or relation from a number of particular, observed cases or events. Our experience of some novel scientific phenomenon (for example, the effect of some drug on the human nervous system) is usually very undiscriminating; we do not initially see the general principles at work – we begin with a confused mass. However, over time, with sufficiently many repeated occurrences of the phenomenon, we are able to infer the general principles underlying the phenomenon – we reason our way to a universal law-like feature of the universe. By examining many cases we can inductively infer a formal pattern. Key Point: Inductivist models of scientific thinking claim scientific laws are justified by induction from observations of a particular phenomenon. 51  §3.5. THE RENAISSANCE: EXPERIMENTATION AND MATHEMATICS Workshop 3: Philosophy - Part 1 3.5 The Renaissance: experimentation and mathematics Notice the use made of an actively pursued set of experimental results in the second premise of Avicenna’s inductive inference. Later, in the 16th and early 17th century, the Renaissance development of instruments like the telescope, microscope and accurate clocks meant that much more sophisticated experiments could be undertaken (e.g. in the microscopic realm and the astronomical realm). And more extensive use was made of increasing amounts of experimental evidence that was becoming available. The emphasis on active experimenta- tion to acquire new and relevant data for theorising became increasingly significant. Another significant Renaissance development was the increasing use of mathematics and mathematical modelling in scientific theorising.3 In the Renaissance the idea of a “clock- work universe” developed – a conception of the universe as like a giant mechanical clock (typically with God the great Watchmaker who set it all in motion) governed by laws amenable to precise mathematical modelling, and there was a shift from qualitative anal- ysis to quantitative analysis and measurement. Scientists began to theorise about motion with idealised mathematical models invoking frictionless planes, perfectly spherical bodies, etc. This reached its greatest expression in Newton’s Laws of Motion, whose mathematical simplicity and wide applicability confirmed a view of the cosmos as essentially mathematical in nature. Image 3.3: Galileo, looking a touch haggard A century before this great Newtonian triumph, Galileo had put the point clearly: Philosophy [i.e. science] is written in that vast book which stands forever open before our eyes, I mean the universe; but it cannot be read until we have learnt 3 It seems hard to imagine science without sophisticated mathematics involved but, remember, the calculus of infinitesimals – developed by Gottfried Leibnitz and Isaac Newton, and necessary for the modelling of motion, acceleration, population growth, etc. – and probability theory – developed by Blaise Pascal – were not developed until the 17th century. 52  §3.5. THE RENAISSANCE: EXPERIMENTATION AND MATHEMATICS Workshop 3: Philosophy - Part 1 the language and become familiar with the characters in which it is written. It is written in the mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.4 The book of nature is written in the language of mathematics, and that is why you need to get some mathematical skills under your belt before getting anywhere in science. If you don’t like the maths, blame the Renaissance!5 Key Points: In the Renaissance, technological innovation led to more active experimen- tation and an emphasis on the pursuit of observational evidence. Renaissance science also emphasised the mathematical structure of the scientific universe. QUESTION: Could we do science purely qualitatively – i.e. using notions like strong, weak, hot, cold, near, far, fast, slow, etc. – instead of the Renaissance push to quantitative measurement? If not, why not? Notes: 4 Il Saggiatore (The Assayer). Quoted in A.C. Crombie’s Grosseteste and Experimental Science, Oxford (1953) p. 285. 5 For some examples of this increasingly mathematical approach to science see: H. Kearney, Science and Change 1500- 1700, Weidenfeld & Nicolson (1971), pp. 66-7; A.C. Crombie, Science, Optics and Music in Medieval and Early Modern Thought, Hambledon Press (1990), p. 325. 53  §3.6. A COMMON VIEW OF SCIENCE Workshop 3: Philosophy - Part 1 3.6 A Common View of Science Consistent with the account of scientific method described earlier, a popular view of science is described by Alan Chalmers as follows: Scientific knowledge is proven knowledge. Scientific theories are derived in some rigorous way from the facts of experience acquired by observation and experiment. Science is based on what we can see and hear and touch, etc. Personal opinion or preferences and speculative imaginings have no place in science. Science is objective. Scientific knowledge is reliable knowledge because it is objectively proven knowledge.6 The notion of “proof” referred to here will, for the inductivist, be proof by inductive infer- ence. On this “inductivist” interpretation of the popular view then, scientific reasoning is inductive and scientific method yields reliable knowledge through the application of induc- tive reasoning from the “facts of experience”. There are problems here though: (i) The idea that scientific knowledge is proven knowledge can mislead us to think of such knowledge as certain. It is not. (ii) The idea that science is objective and that “personal opinion or preferences and specu- lative imaginings have no place in science” is misguided. (iii) The reliability of induction as a form of proof is difficult to justify. Let us turn firstly to the problems concerning (ii). 3.6.1 Discovery versus Justification With a little thought it seems obvious that there is a distinction to be made, first clearly drawn by John Herschel in the 1830s, between the means by which scientific theories are discovered – the context of discovery – and the means by which they are to be justified – the context of justification. How we discover a theory – i.e. the means by which we come to have the theory in our minds – is one thing, whereas establishing a theory as rationally acceptable – i.e. justifying it – is another thing. The common view described above seems to wrongly characterise science by ruling as illegitimate that highly imaginative and creative aspect of science whereby practitioners “cook up” theories for consideration and testing. Who ever came up with the idea that there could be “dark matter” or “dark energy” and what in heaven’s name were they on 6 A. Chalmers, What Is This Thing Called Science?, (1976) p. 1. 54  §3.6. A COMMON VIEW OF SCIENCE Workshop 3: Philosophy - Part 1 when they thought up the idea? Who cares? What is relevant is simply whether or not such an idea (indeed, whether any scientific law, hypothesis or theory) can be justified. For, irrespective of how someone came up with the idea, what matters from the point of view of understanding the universe is whether or not such an idea can be justified. With this in mind, we do not need to give any account of the process whereby we discover scientific theories. We do not need to claim that the process is in any way rational or reliable, let alone suggest that it proceeds by way of induction. An account of scientific method as essentially inductive then need only be committed to the view that scientific laws are justified by induction. Personal opinion and preferences, and speculative imaginings play a key role in science. Science is, in this sense, clearly subjective. Imaginative theorising by individual subjects puts hypotheses on the table for consideration which might otherwise never have been considered (these hypotheses are not “objectively given” to us), and some of the greatest honours in science go to those who have used their subjective imagination in ways that are ingenious, and which have produced theories which have subsequently come to be seen as justified. But how are they justified? This, according to those advocating an inductive scientific method, is by way of inductive inference. And so to problem (i). Key Point: The discovery of scientific hypotheses – a process entirely separate from justification – is a creative process that infuses science with subjectivity. 3.6.2 Inductive Inference and Fallibilism What exactly is the logic of inductive inference? Consistent with Avicenna’s example, inductive inference, according to one of its most famous advocates, the philosopher J.S. Mill (1806-1873), consists in inferring from a finite number of observed instances of a phenomenon, that it occurs in all instances of a certain class that resemble the observed instances in certain ways.7 For example, from the fact that ingestion of scammony is observed on a number of occasions to produce red bile we infer that its ingestion always produces red bile. Similarly, from the fact that Nicole, Bianca, etc. are all mortal we infer, by induction, that all humans are mortal. Let’s look at this in more detail. Observational (singular) statements The simple inductivist account claims that science starts with observation. The scientist, with normal, unimpaired senses records what she sees without prejudice. Impartial reports as to how the world operates are justified by the use of the senses. Statements reporting these particular facts – often referred to as observational statements – serve as the basis for the derivation of scientific laws. Examples of some simple observation statements are: At midnight on Jan. 1 1975, Mars appeared at position x in the sky. Mrs Smith struck her husband. 7 J.S. Mill, A System Of Logic, Vol. I, p. 354. 55  §3.6. A COMMON VIEW OF SCIENCE Workshop 3: Philosophy - Part 1 The water boiled at 100 degrees Celsius at sea level (at place p and time t). Observing what is the case will establish such statements as true or false at a particular place and time. Such statements are singular statements; they describe a particular event or state of affairs at a particular place and time. Universal (general) statements Scientific statements however describe general patterns in nature. For example: Planets move in ellipses around their sun. (Astronomy) Animals in general have an inherent need for some kind of aggressive outlet. (Psychology) Water always boils at 100 degrees Celsius at sea level. (Physics) These statements refer to all events of a particular kind at any place and time; they are not about particular events or states of affairs but suitably general. The laws and theories of science involve general statements of this kind; they are universal statements. The problem then for those who think that science starts with particular observation state- ments is to explain how one can justifiably arrive at universal statements from particular ones. For example: just because water boiled when heated to 100◦ C at sea level by person x1 at place p1 and time t1 , and by person x2 at place p2 and time t2 , and by person x3 at place p3 and time t3 , and... and by person xn at place pn and time tn , how can we thereby justifiably infer that the result holds in general, for all future times, places and persons? How can the general be justified on the basis of the particular? How can a set of observations about how the universe is now justify claims as to how the universe is in general? Inductive reasoning The inductivist reply is that under certain conditions we can generalise. So long as the following conditions are met, we may legitimately generalise to an appropriate universal statement: 1. The number of observation statements forming the basis of a generalisation is large. 2. The observations are repeated under a wide variety of conditions. 3. No accepted observation statement conflicts with the derived universal law. Condition (1) helps rule out anomalous cases (e.g. a defective measuring instrument) and stops one jumping to conclusions prematurely. Condition (2) implies that it is not enough to increase our base of singular statements by simply repeating tests on the same subject 56  §3.6. A COMMON VIEW OF SCIENCE Workshop 3: Philosophy - Part 1 under the same conditions. To rule out the possibility of the observed phenomenon being due to some hidden factor we ought to test for the phenomenon under as varied conditions as possible. For example, the claim ‘All liquids contract when frozen’ would seem a justified generalisation if water was not considered; testing liquids under a wide variety of conditions will include testing the liquid water which is unusual in that it expands when frozen. Now obviously if water is observed to expand when frozen then the universal law ‘All liquids contract when frozen’ is not justified. Hence condition (3) is necessary. This kind of reasoning – from a finite list of singular statements to a universal statement, from some to all – is called inductive reasoning and the process of reasoning thus is called induction.8 The simple Inductivist position can be summed up by saying: science is based on inductive inference. Principle of Inductive Inference: If a large number of As have been observed in the past, under a wide variety of conditions, to possess the property B without exception we can infer that all As have the property B. Scientific knowledge is built up from and justified by a secure base of particular observation statements by induction. Science then, on this account, is justified by its use of induction in inferring, from particular observations, general laws and theories; scientific statements can be inductively justified by experience. Scientific statements, based on observational and experimental evidence (i.e. the facts) are contrasted with statements of other kinds – those based on pure logic or mathematics, authority, tradition, prejudice, or any other foundation. Scientific statements are derived in a rigorous and objective manner from objective facts. Science is a body of such knowledge and scientific progress then is the piecemeal addition of laws and theories to that body of knowledge; the accumulation of facts, and new laws and theories arrived at via induction form the ever-growing observational base. This cumulative conception of scientific knowledge is sometimes called ‘the bucket theory’. Fallibilism So the notion of proof that the inductivist relies on is inductive “proof”. But it is important to realise that inductive proof falls short of certainty. Call it “proof” if you want, but it would be wrong to think inductive inference is anything like mathematical proof. When we prove Pythagoras’s Theorem, we justify it as true, and because of the nature of mathematical proof we then take it to be established with certainty. There is no “probably” about it. This 8 NB: there are other forms of inductive reasoning, like inferring from the fact that water has always boiled at 1000 C at sea level in the past that it will do so when I next boil it. The problems we go on to discuss apply equally to these other forms but for simplicity we shall concentrate on the simple form presented here. 57  §3.6. A COMMON VIEW OF SCIENCE Workshop 3: Philosophy - Part 1 is not the case with inductive proof. Inductive inference cannot establish laws as certain. At best, it makes them highly likely. For example, if water has been observed, on numerous occasions, to boil at 100◦ C at sea level without exception, at best that only makes it likely or probable that all water boils at 100◦ C at sea level. It is always left open to further contradictory evidence (evidence that we must recognise may be “out there”, for all we know). We must recognise that our scientific investigations yield results that are clearly fallible. No-one in their right mind these days would claim that a currently “proven” scientific law is established as certain and so beyond revision. Since we cannot assume we have all the relevant data, scientific laws are always to be considered open to revision. In fact, we are constantly revising our scientific understanding of the world on the basis of new evidence and this involves admitting that we didn’t have things “quite right” previously – a polite way of saying we were, in fact, wrong in what we previously thought! If you think about it, everything we claimed to know about the scientific structure of the world in the past has been shown to be wrong. Scientists were once confident, for example, that Newton’s laws of motion were absolutely certain, but new scientific evidence found in the early twentieth century led to developments in physics that resulted in their rejection in favour of more general relativity theory. Of course, you may want to say that Newton’s laws weren’t “wrong”, they were just too general – they are a correct account of motion at low speeds. But that is to admit that, as general laws of motion (what they were put forward as describing), they were wrong. They are not true. More recently, scientists claim to have new evidence that suggests that they cannot account for some 80 percent of matter in the universe! They now speak of “dark matter” and “dark energy” – the stuff they can’t yet detect but suppose is there. What revisions of our scientific laws will this lead to? We’ll see. The point is, we can never discount the possibility of new evidence forcing revisions of our scientific understanding. Science is an activity of constant testing of what we think we know – our fallible claims to knowledge – and constant searching for new information that might further confirm or refute what we think we know. Bearing this in mind, the corresponding Principle of Inductive Inference should reflect this. It now says (and perhaps this what many had in mind all along when it comes to induction): The Weakened Principle of Inductive Inference: If a large number of As have been observed in the past, under a wide variety of conditions, to possess the property B without exception we can infer that all As probably h