Math: Intro to Math (Lec 3)

Means ‘to learn’ in Greek

Mathema

• study of nos

• solving equation

• collection fo formulas
  
  Big 3: 3 L’s

• language of science

• lens of nature

• labor of thinking

Nature of Mathematics

• optical illusion & justaposed

• contradicts but LOGICAL

Paradox

before an object can travel a certain distance, it must first travel half the distance, and before that, half of the remaining distance, and so on

Zeno’s Paradox

surface area NOT FINITE

“a container that ine can fill but not paint”

Gabriel’s Horn Paradox

Philosophy of Mathematics (3)

1. Platonism

2. Formalism

3. Passive Effectiveness

• math b4 humans

• the followers were called: DISCOVERERS

• math existed THEN discovered

• math is ABSTRACT

Platonism

Supporters (2 GH) & Critics (2 FB) of Platonism

1. Galileo & Hardy

2. Field & Benacerraf

• math existed AFTER humans

• math objs r creation of MIND

• INVENTED

• Concrete - tied to phys interpretation = NEEDS APPLICATIONS

Formalism (by David Hilbert)

Mathematics is a game played according to certain simple rules with meaningless marks on paper. - LIKE CHESSS

Dvid Hilbert

Supporters (2 KE) & Critics (2 GR) of Formalism

1. Kronecker & Einstein

2. Godel & Russel

*n this proposition that 1 + 1 = 2
- from scratch
- nakailang pages na yun pa lang napprove

Principia Mathematica - Whitehead Russell (1910)

invented notions that appear to have no applications but years after are discovered to be useful in the natural world — GREY AREA (development)

• old theory maaapply rin pla in the future (developed theories)

Passive effectiveness

Mathematics is the language with which God wrote the universe.

Galileo

u need an instrument to see what is in front of you

• I believe that mathematical reality lies outside of us, and that our function is to discover, or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations” are simply notes on our observations.

Hardy

Mathematical truth is NOT RELIABLE

• depends on interpretation

Field

abstract object are INACTIVE

• not interacting w/ e/o

• no causal relations

Benacerraf

God created NATURAL NO.s

• all else work of man

Kronecker

Math is NOT PERFECT & u made it to FIT into situations

• How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?

Einstein

INCONSISTENCY of systems

• math is not just w/in its own system

Godel

FAILURE in no. application

• principia mathematica

Russell

Developed Theories (Passive Effectiveness)

1. Grp Theory → Building Blocks of Matter

2. Knot Theory → String Theory