Axioms How would mathematics differ if we abandoned the axiomatic approach due to questions of reliability? Can we apply the precise knowledge that the mathematical method gives us, to other areas of knowledge? In science, it would barely work as sciences are based on real-world situations that will always have unexplainable or unaccountable factors.

In ethics, there will always be a conflict of interest and the 'answer' depends on the chosen moral premise. Often times it is hard to say who is 'right or wrong', unlike proofs in mathematics.

Sounds like the axiomatic system of Mathematics will stay in Mathematics, despite our attempts to change this. The Reliability and Use of Mathematics Using proofs instead of experiments in Science What is 12+1? Commutative Property of Multiplication Induction Banach-Tarski Paradox So, are axioms reliable? Axioms and Proofs in Ethics Scientific methods in Mathematics So, can we start doing experiments in Maths? Assumptions that we make in Mathematics to then start creating proofs, i.e, new knowledge. Example: Euclid's Postulates. http://www.math.hmc.edu/funfacts/figures/30001.1-3-8.1.gif Bibliography: Yount, D. J. 2003. Empiricism vs. Rationalism. [online] Available at: < > [Accessed 26 November 2012].

Su, F. E. ?. Banach-Tarski Paradox. [online] Available at: <http://www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml> [Accessed 29 November 2012].

Markie, P. 2004. Rationalism vs. Empiricism. [online] Available at: <http://plato.stanford.edu/entries/rationalism-empiricism/> [Accessed 2 December 2012].

Quartz Hill School of Theology, ?. Rationalism vs. Empiricism. [online] Available at: < http://www.theology.edu/logic/logic4.htm> [Accessed 20 November 2012].

e^ipi+1=0, 2010. Axiom of Choice. [online] Available at: < http://xkcdsw.com/3097> [Accessed 1 December 2012]. We know that 3 x 2 is the same as 2 x 3: this is called the commutative property of multiplication: an axiom we have assumed is true. Is this truly reliable though? No, because consider the concept of matrices. Here, AB is not necessarily equal

to BA. Axioms have paradoxes

Axioms can vary

Axioms deal with a 'closed system'

Axioms are often based on reasoning, however may sometimes be initiated by perception in the case of the commutative property. The axiom of induction is used to prove patterns: sequences and series.

It has two principles. Suppose we have a proposition, Pn, then:

If P1 is true

And Pk+1 is true whenever Pk is true

Then Pn is true. Is this completely reliable? Mathematicians think so, however consider the dominos in real life: can we be always sure that they will fall? For example, Another axiom is the axiom of choice. It can be best explained through the idea of football teams. As seemingly obvious and non-consequential this axiom may seem, it does possess an extremely far-fetched paradox: the Banach-Tarski paradox. In essence, this paradox states that you can cut up a sphere into parts, and rearrange it to form two new identically sized spheres. This is of course impossible in real life. Wow, Maths seems so unreliable, why do I study this in school? - Empiricism is Simpler

- The advance of Science - Verifying Empiricism is Impractical

- All Rationalists do not agree about innate knowledge - More reliable/precise

- More practical Empiricism believes that some ideas or concepts are independent of experience and that truth must be established by reference to experience alone. - Complicated

- Disagreement on the same 'innate knowledge' Rationalism believes that some ideas or concepts are independent of experience and that some truth is known by reason alone. It would be unwise to abandon the Axiomatic approach for now, however this could change in the future as there is also valid justification for the use of Empiricism. Can we apply the axiomatic approach then? So, the answer is not always: it depends (unfortunately). However we have no other alternative beyond the axiomatic approach... or do we? Formal ethics:

(Prescriptivity) — "Practice what you preach"

(Universalizability) — "Make similar evaluations about similar cases"

(Rationality) — "Be consistent"

(Ends-Means) — "To achieve an end, do the necessary means" Unfortunately, these are insufficient. 'Answers' in ethics depends on what moral premise is used: egoism, altruism, utilitarianism, moral duty, and so on. Because the world would crumble to pieces without the advances in Math throughout history.

While axioms have their problems, noone can deny Mathematics is a unique Area of Knowledge that provides a novel way of knowing. We would like to try implementing this elsewhere, however as we have seen, this has proved too difficult.

But atleast now you know that 1+1 doesn't have to be 2! Thank you for listening! By Aseel Babhair, Balaji Ullal Bhat and Naval Handa How is Mathematics reliable when it is based on axioms we ourselves create?

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# ToK Mathematics Presentation

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