Symbols. How do you unpack it? What do you need to know in order to do so? How could it hurt, rather than help, understanding?

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Corey Eaton
5 years ago
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1 Inner Issues

2 Symbols Mathematical symbols are like natural language in that they grow and change. Symbols gain precision but lose some fidelity to the original problem. Some are very powerful. Whitehead: notation sets the brain free to concentrate on more advanced problems.[and] increases the mental power of the race.

3 Symbols What does 1 T A A 1 T mean? How do you unpack it? What do you need to know in order to do so? How could it hurt, rather than help, understanding?

4 Symbols ( x,, x n ) Let 1 be a formula with all free variables shown; let M be a c.t.m. for ZFC, P be a p.o. set in M,,, 1, n M and p ; then p (,, ) iff, M 1 Of course, is the set of all P names in M, n M[ G] G G is -generic over M p G val( 1, G,, val( n, G. M val,g is defined by transfinite induction on τ, and G is P generic over M if G is a filter on P and for all dense. D, D M G D 0

5 Abstraction Abstraction as idealization: The straight line; the square; the sphere. Assumptions: String is perfectly flexible, constant linear density, constant tension, motion confined to xy plane, each point moves perpendicularly to x axis, displacement is small compared to the length of the string, and the angle between the string and x axis is also small. Finally, no external forces act on the string. (Theoretical Physics meets the Dairy, again) Idealized, Platonic world

6 Abstraction Abstraction as extraction What is essential in this? (and for what purposes?) Groups Graphs How is a doughnut like a coffee cocoa cup?

7 Generalization If it works in 2 and 3 dimensions, maybe it will work in 4 and 5 and n. The Law of Cosines is a generalization of the Pythagorean Theorem. Advantage: A consolidation of information into one neat package.

8 Formalization The process by which mathematics is adapted for mechanical processing. Everything is taken down so that there is only (in theory) the need to process a handful of rules, and manipulate the symbols. Meaning becomes irrelevant.

9 Formalization Axiomatic development is a necessary (but not sufficient) condition for formalization Set Theory Russell and Whitehead 3 volume work. an attempt to derive all mathematical truths from a well defined set of axioms and inference rules in symbolic logic.

10 Formalization Page 379

11 Formalization It enables us to construct a mathematical theory about mathematics a logical analysis of mathematics. This was the original purpose for much of the mathematical logic developed in the 20 th century.

12 Existence of Objects and Structures Objects in modern math, are usually some kinds of sets. Structures also sets, together with other sets which are the relations or special elements. Frechét Ultrafilters and Unicorns

13 Existence of Objects and Structures Like that of the unicorn, there is no single notion of existence of mathematical objects. Existence is intimately related to setting, to demands, to function.

14 Proof Of Thales proof that a diameter divides a circle into two equal parts: The genius of the act was to understand that a proof is possible and necessary. They keep using that word (necessary). I do not think that it means what they think it means.

15 Proof Notice that the language of the proof has a formal and severely restricted quality about it.... this is language that has been sharpened and refined so as to serve the precise needs of a precise but limited intellectual goal.

16 Proof Proof serves many purposes simultaneously. Subjected to constant process of criticism and revalidation. In its best instances it increases understanding by revealing the heart of the matter. Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems. Proof is ritual, and a celebration of the power of pure reason.

17 Infinity How is infinity handled before calculus? How is infinity handled during calculus? How is infinity handled after calculus?

18 The Stretched String The idea of a straight line is intuitively rooted in the kinesthetic and the visual imaginations. The interplay of these two sense intuitions gives the notion of straight line a solidarity that enables us to handle it mentally as if it were a real physical object that we handle by hand.... may imagine it as an Eternal Form... [or] an aspect of Nature, an abstraction of a common quality

19 The Stretched String Ideas that grow from this embodied cognition have to be formalized and axiomatized when dealt with in modern mathematics: E.g. betweenness.

20 The Stretched String If a point B is between points A and C, B is also between C and A, and there exists a line containing the points A,B,C. If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D. Of any three points situated on a straight line, there is always one and only one which lies between the other two.

21 The Stretched String Or, point B is between A and C if A, B, and C are collinear and AB + BC = AC In which case you have to axiomatize distance, instead.

22 The Stretched String Other things about straight lines are not so intuitively clear (and, what is intuitively obvious changes over time) Infinite divisibility Sizes of subsets the continuum hypothesis. Unproveable

23 The Coin of Tyche The characteristics of a fair coin. No finite amount of information at the beginning of the sequence [of heads and tails] has any effect in computing a probability. Thus, we are led to the silent assumption, which is not part of the formalized theory, but which is essential for applied probability, that convergence [to the theoretical probability] is sufficiently fast that one may judge whether a particular real coin is or is not modeled by the fair coin.

24 The Aesthetic Component Poincaré asserted that the aesthetic rather than the logical is the dominant element in mathematical creativity. The mathematician s patterns, like the painter s or the poet s must be beautiful. (G. H. Hardy) It is more important to have beauty in one s equations that to have them fit the experiment. (Dirac)

25 The Aesthetic Component Aesthetic judgment exists in mathematics, is of importance, can be cultivated, can be passed from generation to generation, from teacher to student, from author to reader. But there is very little formal description of what it is and how it operates.

26 Pattern, Order, and Chaos To some extent, the whole object of mathematics is to crate order where previously chaos seemed to reign, to extract structure and invariance from the midst of disarray and turmoil A mathematician, like a painter or poet, is a master of pattern. (Hardy)

27 Algorithmic vs Dialectic Mathematics Two treatments of finding. Algorithmic typifies math up through calculus, dialectic math post calculus (with some exceptions)

28 The Drive to Generality and Abstraction The Chinese Remainder Theorem

29 Mathematics as Enigma

30 Unity within Diversity

35 Chapter CHAPTER 4: Mathematical Proof

35 Chapter CHAPTER 4: Mathematical Proof 35 Chapter 4 35 CHAPTER 4: Mathematical Proof Faith is different from proof; the one is human, the other is a gift of God. Justus ex fide vivit. It is this faith that God Himself puts into the heart. 21