The Goal of Science To create a set of models that describe the measurable universe. These models must – Fit previous verified measurements applicable to the model; – Make testable predictions that can be validated by future measurements; and – Have no more, and preferably less, free parameters than any other equivalent model.

Methods of Verification In order to verify a new measurement, scientists can: Compare the new measurement against a previously accepted value; Compare two measurements that should overlap or differ by a known amount; or Compare the measurement against a predicted result.

The Role of Mathematics Mathematics plays several roles as a tool of scientists. It is used to: Provide information on the statistical significance of a measurement; Describe a functional relationship between a measured quantity and the free parameters associated with it; and

The Role of Mathematics Provide a layer of abstraction. This allows seemingly disparate experiments to be tied together in such a way as to reveal underlying relationships. These relationships are usually more fundamental in nature. It is this last role that we will focus on, and which separates physics from the other natural sciences.

Example: Exponential Decay An RC circuit is charged up and then disconnected from the source. The current in the circuit is then seen to drop to zero according to the formula Similarly, the activity of an I 131 sample varies as

Example: Exponential Decay While these two experiments do not appear to be connected, the equations describing their time evolution show they both undergo an exponential decay. This is indicative of an underlying similarity between the systems.

Types of Numbers Counting Numbers (N): Whole positive numbers and zero, i.e., 0, 1, 2, 3, … Integer Numbers (Z): Whole positive and negative numbers, including zero, i.e., 0, ±1, ±2, ±3, … Rational Numbers (Q): Any number that can be expressed as a ratio of integer numbers p/q, where p and q are integers.

Types of Numbers Real Numbers (R): The set of rational and irrational numbers, excluding those which include a multiple of Complex Numbers (C): The set of rational and irrational numbers, including those which include a multiple of Note that

Zeno’s Paradoxes The Dichotomy Motion is impossible, because whatever moves must reach the middle of its course before it reaches the end; but before it has reached the middle it must have reached the quarter mark, and so on, indefinitely. Hence the motion can never even start.

Zeno’s Paradoxes The Achilles Achilles running to overtake a crawling tortoise ahead of him can never overtake it, because he must first reach the place from which the tortoise started; when Achilles reaches that place, the tortoise has departed and so is still ahead. Repeating the argument we easily see that the tortoise will always be ahead.

Zeno’s Paradoxes The Arrow A moving arrow at any instant is either at rest or not at rest, that is, moving. If the instant is indivisible, the arrow cannot move, for if it did the instant would be immediately divided. But time is made up of instants. As the arrow cannot move in any one instant, it cannot move in any time. Hence it always remains at rest.

Zeno’s Paradoxes The Stadium To prove that half the time may be equal to double the time, consider three rows of bodies: First PositionSecond Position (A) (B)0 0 (C)

Zeno’s Paradoxes (A) is at rest while the other two, (B) and (C), are moving with equal velocities in opposite directions. By the time they are all in the same part of the course (B) will have passed twice as many of the bodies in (C) as in (A). Therefore the time which it takes to pass (A) is twice as long as the time it takes to pass (C). But the time which (B) and (C) take to reach the position of (A) is the same. Therefore double the time is equal to half the time.

Mathematical Disciplines Algebra: The mathematics of generalized arithmetical operations. Geometry: The mathematics of points, lines, curves, and surfaces. Trigonometry: The mathematics of triangles and the functions that describe the relationships between the angles and lengths of the triangle.

Mathematical Disciplines Calculus: The branch of mathematics that is concerned with limits and with the differentiation and integration of functions. Linear Algebra: The part of algebra that deals with the theory of linear equations and linear transformations. Differential Equations: The branch of mathematics that is concerned with the solution of equations containing differentials of a function.

Mathematical Disciplines Topology: The branch of mathematics that deals with the properties of a figure X that hold for every figure into which X can be transformed via a one-to-one correspondence that is continuous in both directions. Probability: The branch of mathematics that is concerned with the the measure of how likely it is that some event will occur and the functional descriptions associated with these measures.

Mathematical Disciplines Complex Variables: The mathematics of complex numbers, including their algebraic, geometrical descriptions and the calculus associated with such numbers. Group Theory: The branch of mathematics dealing with groups. Number Theory: The branch of mathematics dealing with the properties of number sets.

Mathematical Disciplines Knot Theory: The branch of mathematics dealing with discontinuities and singularities associated with surfaces. Fractal Geometry: The mathematics of fractal, or self-similar, shapes that are defined via recursive algorithms.