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Key quote: “Mathematics is neither physical nor mental, it’s social” – Reuben Hersh
Reflection (upon reading the introduction):
I would disagree with the quote by Reuben Hersh. The only social aspect of mathematics is when it is being taught to another person. And even that is misleading because one cannot teach mathematics, the closest thing to teaching mathematics is guiding someone through the logical thinking that is math. Neither does math even need society, it can exist entirely by itself without any social interaction or any humans at all. Even a moderately intelligent computer can perform mathematics. However, the quote is true in saying that mathematics is not physical in nature. The mathematical principles may govern the physical world but the physical world is of no consequence to mathematics. Math can exist almost entirely independent of any physical matter but the same cannot be said of matter. Mathematica is neither physical nor mental, but it it isn’t social either. It exists independently in it’s own category and we simply strive to comprehend it the best we can with our limited brains.
Mathematics offers a certainty in the world. In math everything is logical and there is one true way to go about doing things. Everything interacts in one true way.
But it is not an innate human characteristic to naturally gain mathematical reasoning skills as part of common sense. One tribe in the Amazonian rainforest has no numbers. They are limited to “one”, “two”, and “many”.
The mathematical paradigm
Euclidean geometry – field of geometry explored by Euclid. What we learn in middle school. Rigorous geometric proofs.
Formal system – the model of reasoning created by Euclid. (Axiomes) -> (Deductive reasoning) -> (Theorems) -> more theorems
Axiom – a starting point for reasoning that does not need proof.
Theorem- a statement that can be proven true through axioms or other theorems
The staring points of a deductive or inductive endeavor. For most of history the axioms of math were thought to be self-evident truths.
A good set of Axioms must be:
Consistent- one should not be able to deduce anything that they want from those axioms.
Independent- the axioms cannot be derived from each other
Simple- clear and simple
Fruitful- must yield something, facilitated deductive reasoning
Example of the axioms that Euclid used to come up with his geometry:
Premise #1 : I went to school on 11/22/16
Premise #2 : Today Is 11/22/16
Conclusion : therefore I went to school today
A posteriori knowledge- knowledge that can only be proven via experience.
A priori knowledge – knowledge that can be justified without experience
Analytic proposition- a proposition that is true by definition
Conjecture- a hypothesis that seems to work but has not been proven yet
Deductive reasoning – reasoning from the particular to the general
Empiricism - school of thought that claims that all knowledge is ultimately based on sense perception.
Formalism – the belief that mathematical proofs are true by definition
Godel’s incompleteness theorem – a theorem that suggests it is impossible to prove that a formal mathematical system is free from contradiction.
Goldbach’s conjecture – Hypothesis that every even number is the sum of two primes. This appears to be true but there has not been a proof for it yet.
Platonism – the view that mathematical truths give us a priori truth into the structure of the universe.
Synthetic proposition - all propositions that are not analytic