The abstract science of number, quantity, and space. Mathematics may be studied in its own right ( pure mathematics ), or as it is applied to other disciplines such as physics and engineering ( applied mathematics ).
INTRODUCTION
The Mathematical Paradigm Proofs and conjectures Beauty, Elegance, Intuition Mathematics and Certainty Area of Knowledge:
Mathematics T.O.K Axioms Deductive Reasoning Theorems Mathematics as empirical Mathematics as analytic
Option 3: Mathematics as synthetic as priori Discovered or invented? Criticisms of Platonism Non-Euclidean Geometry & The problem of Consistancy Riemannian Geometry The Problem of Consistency Gödel's Incompleteness Theorem Applied Mathematics
Conclusion Question(1): 1+2=?
“Mathematics is the language with which God has written the universe.”
~Galileo Galilei Math Literacy is required for any branch of a science to be sucessful! “This was one of the great events of my life, as dazzling as first love. I had not imagined that there was anything so delicious in the world”
~Bertrand Russell "If you can't express something in mathematical symbols then it has no intellectual value" A worldview underlying the theories and methodology of a particular scientific subject.
Question(2): Define Mathematics.
" 'The science of rigorous proof' "
Infinite Regress WE CAN'T PROVE EVERYTHING!!
Independent Fruitful Simple Consistent
If you deduce opposing ideas from same set of axioms then it's not consistent Begin with the smallest number of axioms. Deducing an axiom from an axiom is not allowed (then it would be called a theorem) YO! KEPT IT SIMPLE AND STUPID! Many proofs, minimum amount of axioms We did this, REMEMBER!
C'mon, What is Deductive Reasoning?
We use our Axioms to get theroms r C= 2(pi)r π.
That was the famous Beethoven fifth symphony Proofs Conjectures -A theorem is shown to follow logically from the relevent axioms.
-Trying different techniques in order to test if it works in all cases.
-A hypothesis that seems to work, but has not been shown to be necessarily TRUE when you TEST only a small proportion and thinking it works for all. When solving a math problem, you are simply unpacking the truth'.
Difficult to memorize the whole method when solving a problem. Thus, short memory = better in maths.
Mathematical truths do not seem to be trivially true in a matter that "All bachelors are unmarried" since math problems are said to be a puzzle than true definition.
False by definition, certain words in maths might not be applicable in the English language.rather
Math is based on an EXPERIENCE and OBSERVATION .
Maths can be known independent of experience We are able to discover truth about the nature of reality Truths give us a priori insight into the structure of reality Some shapes, such as circles, do not exist in the real word. Are not wheels and coins circles? Can’t YOU draw a circle? NO!! = What is a circle?
impossible to “mathematically” draw a perfect circle, or any other geometrical object. A line is defined as having a length but no breadth, again impossible! The set of all points in a plane that is equidistant from a given point. Some mathematical entities are named as “idealisations” "It is impossible to prove that a formal mathematics system is free from contradiction" All it takes is one baby mistake to destory a mathematical rule Plato believed that mathematical objects are more real than physical objects Robert Downey JR. Empty space with atoms.
Exists briefly, then turns into dust VS Shapes Clear Does not turn into dust Platonism is committed to the view that there are literally an infinite number of mathematical entities 'out there'. If maths has a weird ideal existence then how can physically embodied beings, such as, Robert Downey JR get to know about them? Example:
In what sense does chess exist? Does it still exist if no one is playing it or even thinking of it? What if Robert Downey JR decided to destroy the chess rule books, would it still be there? Some say that if no one played chess for a year there would still be certain statements about it that are true and others that are false.
However, if we disappeared somehow it would not make sense to say that the game of chess still existed.
Applied Mathematics According to Einstein mathematical systems are INVENTED, but it is a matter of DISCOVERY which of the variuos systems apply to reality. Even if Maths is a game, the rules for the most interesting games may be suggested to us by reality.
You know, like monopoly! Although Mathematics cannot give us absolute certainty, it continues to play a key role in a wide variety of subjects. There is something surprising and mysterious about its usefulness there is no reason to believe that it is the only or the best tool for making sense of reality.
The Mathematical Paradigm Proofs and conjectures Beauty, Elegance, Intuition Mathematics and Certainty Area of Knowledge:
Mathematics T.O.K Axioms Deductive Reasoning Theorems Mathematics as empirical Mathematics as analytic
Option 3: Mathematics as synthetic as priori Discovered or invented? Criticisms of Platonism Non-Euclidean Geometry & The problem of Consistancy Riemannian Geometry The Problem of Consistency Gödel's Incompleteness Theorem Applied Mathematics
Conclusion Question(1): 1+2=?
“Mathematics is the language with which God has written the universe.”
~Galileo Galilei Math Literacy is required for any branch of a science to be sucessful! “This was one of the great events of my life, as dazzling as first love. I had not imagined that there was anything so delicious in the world”
~Bertrand Russell "If you can't express something in mathematical symbols then it has no intellectual value" A worldview underlying the theories and methodology of a particular scientific subject.
Question(2): Define Mathematics.
" 'The science of rigorous proof' "
Infinite Regress WE CAN'T PROVE EVERYTHING!!
Independent Fruitful Simple Consistent
If you deduce opposing ideas from same set of axioms then it's not consistent Begin with the smallest number of axioms. Deducing an axiom from an axiom is not allowed (then it would be called a theorem) YO! KEPT IT SIMPLE AND STUPID! Many proofs, minimum amount of axioms We did this, REMEMBER!
C'mon, What is Deductive Reasoning?
We use our Axioms to get theroms r C= 2(pi)r π.
That was the famous Beethoven fifth symphony Proofs Conjectures -A theorem is shown to follow logically from the relevent axioms.
-Trying different techniques in order to test if it works in all cases.
-A hypothesis that seems to work, but has not been shown to be necessarily TRUE when you TEST only a small proportion and thinking it works for all. When solving a math problem, you are simply unpacking the truth'.
Difficult to memorize the whole method when solving a problem. Thus, short memory = better in maths.
Mathematical truths do not seem to be trivially true in a matter that "All bachelors are unmarried" since math problems are said to be a puzzle than true definition.
False by definition, certain words in maths might not be applicable in the English language.rather
Math is based on an EXPERIENCE and OBSERVATION .
Maths can be known independent of experience We are able to discover truth about the nature of reality Truths give us a priori insight into the structure of reality Some shapes, such as circles, do not exist in the real word. Are not wheels and coins circles? Can’t YOU draw a circle? NO!! = What is a circle?
impossible to “mathematically” draw a perfect circle, or any other geometrical object. A line is defined as having a length but no breadth, again impossible! The set of all points in a plane that is equidistant from a given point. Some mathematical entities are named as “idealisations” "It is impossible to prove that a formal mathematics system is free from contradiction" All it takes is one baby mistake to destory a mathematical rule Plato believed that mathematical objects are more real than physical objects Robert Downey JR. Empty space with atoms.
Exists briefly, then turns into dust VS Shapes Clear Does not turn into dust Platonism is committed to the view that there are literally an infinite number of mathematical entities 'out there'. If maths has a weird ideal existence then how can physically embodied beings, such as, Robert Downey JR get to know about them? Example:
In what sense does chess exist? Does it still exist if no one is playing it or even thinking of it? What if Robert Downey JR decided to destroy the chess rule books, would it still be there? Some say that if no one played chess for a year there would still be certain statements about it that are true and others that are false.
However, if we disappeared somehow it would not make sense to say that the game of chess still existed.
Applied Mathematics According to Einstein mathematical systems are INVENTED, but it is a matter of DISCOVERY which of the variuos systems apply to reality. Even if Maths is a game, the rules for the most interesting games may be suggested to us by reality.
You know, like monopoly! Although Mathematics cannot give us absolute certainty, it continues to play a key role in a wide variety of subjects. There is something surprising and mysterious about its usefulness there is no reason to believe that it is the only or the best tool for making sense of reality.
- Imitation Game- Where a genius mathematician is castrated because he is gay. His entire achievement because of which the wwii was shortened by 2 years was overlooked. The Turing test is a test of a machine's ability to exhibit intelligent behavior equivalent to, or indistinguishable from, that of a human. Alan Turing proposed that a human evaluator would judge natural language conversations between a human and a machine that is designed to generate human-like responses. The evaluator would be aware that one of the two partners in conversation is a machine, and all participants would be separated from one another.
- Temple Grandin- an autistic woman(ironic) creates a mathematical design that helps cow die painlessly(ironic:a person who is devoid of feeling) and economically for the ranches. Her faith is that cow will die painlessly when a hot iron rod is jammed into their brain(I am sure it is painless).