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Priscilla Stone
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2 MATH 195 Part 1 Science

3 What is science? Formal Sciences Natural Sciences (physical sciences and life sciences) Social Sciences

5 HISTORY Aristotle: Deductive Logic ( BC)

6 The problem with deductive logic is that the conclusion of a deductively valid argument cannot say more than is implicit in the premises. although where the argument is complex we may find the conclusion surprising just because we hadn t noticed that it was already implicit in the premises

7 HISTORY Francis Bacon: Induction Or Confirmation ( )

8 Inductive reasoning, or induction, is the name given to various kinds of deductively invalid but allegedly good arguments. The idea is to reach the truth by gathering a mass of information about particular states of affairs and building from them step by step to reach a general conclusion.

9 Empirical observation, data gathering, experiments Experiments: test all possible controlled cases (e.g., by elimination, trial and error). Experiments allow us to ask what would happen if? Experiments should be repeatable, so that others can check the results.

10 Experiments in vivo: done inside the living organism in vitro: done outside the living organism in situ: done where organisms are found in nature in silico: done on computer (computer simulation)

11 Theoretical vs Empirical Empirical observation and experiment is a neutral foundation for scientific knowledge, or at least for the testing of scientific theories.

12 Induction is justified by the theory of probability. Frequentist and Bayesian statistics Scientific knowledge is objective means that it is not the product of individual whim, and it deserves to be believed by everyone, regardless of their other beliefs and values. Observation without any bias whatsoever is impossible.

13 Major Problem of Induction Bertrand Russell ( ) famously argued in the Problems of Philosophy, sometimes inductive reasoning may be no more sophisticated than that of a turkey who believes that it will be fed every day because it has been fed every day of its life so far, until one day it is not fed but eaten. Problem of Extrapolation: in the past the future has been like the past doesn t mean that, in the future, the future will be like the past. It is for this reason that the philosopher C.D. Broad ( ) called induction the glory of science and the scandal of philosophy.

14 But well Inference to the best explanation, which is sometimes called abduction, is the mode of reasoning that we employ when we infer something on the grounds that it is the best explanation of the facts we already know. In science, hypotheses are often adopted because of their explanatory power.

15 Good about science is an attitude of scepticism towards the traditional dogma Question Most people use induction and those that don t are mad and you can t reason with them. What makes you think you re the sane one?

16 HISTORY Karl Popper: Falsification ( )

17 deciding whether something is scientific or not will often be a decision with significant consequences for people s lives It is just too easy to accumulate positive instances which support some theory, especially when the theory is so general in its claims that its seems not to rule anything out Falsificationism: grounds for the demarcation of science from non-science (fringe science, pseudoscience and junk science)

18 Popper came to the view that it is not confirmation but falsification that is at the heart of the scientific method. No matter how many positive instances of a generalisation are observed it is still possible that the next instance will falsify it. The inference from a falsifying instance to the falsity of a theory is purely deductive.

19 Once a hypothesis has been developed, predictions must be deduced from it so that it can be subjected to experimental testing. If it is falsified then it is abandoned, but if it is not falsified this just means it ought to be subjected to ever more stringent tests and ingenious attempts to falsify it. Note: there is no distinction between scientific theories, laws, principles, etc. Popper was particularly impressed by the experimental confirmation of Einstein s general theory of relativity in 1917.

20 Method of conjectures and refutations this is the way in which we can learn from our mistakes; and that in finding that our conjecture was false we shall have learnt much about the truth, and shall have got nearer to the truth Popper theories that in their time were considered highly confirmed and which enjoyed a huge amount of empirical success, have been shown to be quite mistaken in certain domains

21 So can we ignore non-scientific beliefs? It is perfectly possible for someone with religious faith and beliefs to accept a definite demarcation between science and religion (this may be the case with many scientists). But non-scientific beliefs can be a source of discovery!

22 The context of discovery Generation of scientific theories is not, in general, a mechanical procedure, but a creative activity Scientists may draw upon diverse sources of inspiration, such as metaphysical beliefs, dreams, religious teachings and so on, when they are trying to formulate a theory. The kind of speculation and imagination that scientists need to employ cannot be formalised or reduced to a set of rules. In a way this makes the sciences closer to the arts than they might otherwise seem. On the other hand, the sciences differ from the arts in being subject to testing by experience and this must be the final arbiter of any scientific dispute.

23 Studying history of Science In Popper s view then, there are two contexts in which we might investigate the history of science and the story of how certain theories come to be developed and accepted, namely the context of discovery and the context of justification.

24 Some legitimate parts of science seem not to be falsifiable Statements that assert the existence of something cannot be falsified by one s failure to find them Sometimes we need to visit the past or need to investigate evolutionary timescales Falsificationism is not itself falsifiable: it is a philosophical or logical theory of the scientific method, and not itself a scientific theory

25 What could happen Instead of abandoning a theory, scientists thought up modifications or extra assumptions to save it. Where scientists have a successful theory, the existence of falsifying observations will not be sufficient to cause the abandonment of the theory in the absence of a better alternative.

26 Bacon + Popper: Science is about confirmation as well as falsification Science is cumulative. In other words, scientists build on the achievements of their predecessors, and the progress of science is a steady growth in our knowledge of the world. This feature of science is sharply contrasted with other activities, such as art, literature and philosophy, which are progressive in a much looser and controversial sense. Science is unified in the sense that there is a single set of fundamental methods for all the sciences, and in the sense that the natural sciences at least are all ultimately reducible to physics (Reductionism). Scientific terms have fixed and precise meanings.

27 HISTORY Thomas Kuhn: Paradigm ( )

28 Kuhn s paradigm If a new paradigm is adopted by the scientific community, then a revolution or paradigm shift has occurred. Kuhn also emphasises the role of psychological and sociological factors in disposing scientists to adopt or a reject a particular paradigm. Some people are inherently more conservative than others, while some enjoy being a lone voice in the wilderness; some people are risk takers and others are risk averse, and so on.

29 MISUNDERSTANDING? Terms and concepts of scientific theories in different paradigms are not mutually intertranslatable; this is called meaning incommensurability (lack of common measure). A simple form of epistemic relativism would say that, for example, a particular theory in physics or biology might be counted as knowledge just because it was believed by those with status and influence within the community of physicists or biologists.

30 Kuhn s five core values common to all paradigms A theory should be empirically accurate within its domain. A theory should be consistent with other accepted theories. A theory should be wide in scope and not just accommodate the facts it was designed to explain. A theory should be as simple as possible. A theory should be fruitful in the sense of providing a framework for ongoing research.

31 MATH 195 Part 2 Mathematics

32 Problem of Reductionism (my own point-of-view) Philosophy Logic Math Math Computers Computer Science Philosophy Math (Language & Queen of Sciences) Reality Science Physics Matter Chemistry Physics Chemistry (the Central Science) Life Biology Biology Behavior & Mind Psychology (note: brain is not equal to mind) Psychology Human interaction Social Science Social Science Wealth Economics Science Technology Engineering (a bit of everything?)

33 Note: this is just a crude representation of the branches of science. The distinctions between branches are vague in the modern perspective. Philosophy Logic Math Math Computers Computer Science Philosophy Math (Language & Queen of Sciences) Reality Science Physics Matter Chemistry Physics Chemistry (the Central Science) Life Biology Biology Behavior & Mind Psychology (note: brain is not equal to mind) Psychology Human interaction Social Science Social Science Wealth Economics Science Technology Engineering (a bit of everything?)

34 What is maths? Theoretical or Pure Math Applied Math But there is no clear line separating the two

35 Maths Study of patterns Mathematicians seek truth, beauty and elegance (Math Institute, Oxford U)

37 Science of patterns (Sir Cortez) ================================ From friend Wiki: Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof (e.g., by deductive logic). When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature.

38 Abstraction The first abstraction was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members. wiki Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. wiki

39 From other colleagues: Sir Egarguin: anything that can be learned (coming from the original Greek word μάθημα or máthe ma which means knowledge, study, learning ) Sir Palines: a language that only few dare to understand Sir Marasigan: course ko Maam D. Cuaresma: greatest discovery of all times (used implicitly or explicitly in all disciplines)

40 Summary of keywords Patterns to Conjectures to Theorems Proof, Deductive logic Abstraction Elegant Language Model of real phenomena, Tool in problem solving

41 The worlds of Mathematics: Axiomatic Systems An axiomatic system consists of some undefined terms (primitive terms) and a list of statements, called axioms or postulates, concerning the undefined terms. One obtains a mathematical theory by proving new statements, called theorems, using only the axioms (postulates), logic system, and previous theorems. Definitions are made in the process in order to be more concise. Axioms in traditional thought were self-evident truths, but that conception is problematic. -wiki

42 The worlds of Mathematics: Axiomatic Systems It is crucially important in a proof to use only the axioms and the theorems which have been derived from them and not depend on any preconceived idea or picture.

43 The worlds of Mathematics: Axiomatic Systems An axiomatic system is consistent if there is no statement such that both the statement and its negation are axioms or theorems of the axiomatic system. (No Contradictions) Contradictions in the real world are impossible. If we exhibit an abstract model where the axioms of the first system are theorems of the second system, then we say the first axiomatic system is relatively consistent.

44 The worlds of Mathematics: Axiomatic Systems An axiom is independent if it is not a theorem that follows from the other axioms. Independence is not a necessary requirement for an axiomatic system; whereas, consistency is necessary.

45 The worlds of Mathematics: Axiomatic Systems An axiomatic system is complete if for every statement, either itself or its negation, is derivable in that system. Kurt Gödel ( ) with his Incompleteness Theorem demonstrated that even in elementary parts of arithmetic there exist propositions which cannot be proved or disproved within the system.

46 Theorem Environment The concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental - wiki

47 Theoretical Math ( study for the sake of math ) A naïve way of classifying maths: Foundations of Math (e.g., Logic, Set Theory) Algebra and Combinatorics ( structure ; e.g., Abstract Algebra, Combinatorics, Number Theory) Analysis ( change ; Calculus, Real Analysis, Complex Analysis, Functional Analysis, Differential Equations, Dynamical Systems) Geometry ( space ; e.g., Trigonometry, Synthetic and Analytic Geometries, Euclidean and Non-Euclidean Geometries, Projective Geometry, Discrete Geometry, Topology, Manifold Theory)

48 Applied Math ( math intended for use in other disciplines ) Examples (some here are still considered theoretical math): Probability and Statistical Theory Actuarial Science Computational and Numerical Mathematics Theoretical Computer Science (including Automata Theory, Coding Theory & Cryptography, Machine Learning, etc.) Mathematical (Physics, Biology, Chemistry, Social Science, Finance, Economics, etc.); Mathematics of Complex Systems Operations Research, Optimization and Control Theory Recreational Mathematics

49 Theoretical and Applied Maths differ on their FOCUS but there is no demarcation line between them Spectrum of Applied Math Theoretical Math Application Area

The roots of computability theory. September 5, 2016

The roots of computability theory September 5, 2016 Algorithms An algorithm for a task or problem is a procedure that, if followed step by step and without any ingenuity, leads to the desired result/solution.