Mathematics and Knowledge An investigation

What is special about mathematics? Certain Conclusive consistent Deductive Reason

Are mathematicians able to prove their theorems because they’re true Are those theorems true only because mathematicians can prove them?

Slavery is wrong. Did we discover that slavery is wrong? Or Did we come to agree that it is wrong?

Are mathematicians able to prove their theorems because they’re true Are those theorems true only because mathematicians can prove them?

How do we gain knowledge? A priori Known independent of experience A posteriori As a result of experience

What makes a proposition true? Analytic True by definition Synthetic Any proposition which is not analytic. Through relation to the world, to what is “out there”.

Analytic Synthetic A priori True by definition, known without experience. Trivial truth **Proposition are true in relation to the world but are knowable without experience. ** A posteriori x Proposition are true in relation to the world out there, only knowable through experience

Synthetic a posteriori – maths as empirical 3+1=4

Analytic a priori – maths as rational True by definition, Known without experience. 2 defined as 1+1 4 defined as 1+1+1+1 Therefore (1+1)+(1+1)=1+1+1+1 So 2+2=4

Formalism – analytic a priori ‘A higher form of chess’ Imagine the truths of chess if no one played it?

Paradoxes “This sentence is false” (Liar’s paradox) “The set/class of all sets/classes not members of themselves.” (Russell’s paradox) “the class of all classes that has more than five members” is a member of itself. “the class of all men” is not a member of itself. If P is true, then P is false… so P has got to be false. If P is false then P is true… so P has got to be true Therefore P is true and P is false

Formalist response to paradox David Hilbert (German Mathematician, early C20th) response was to FOMALISE. That means make sure mathematics is arranged according to a system of logic which is consistent. At this time formalism was predominant.

Kurt Gӧdel Recluse Isolated Strong intuitions about maths against formalism. Boundary between philosophy and mathematics.

Gӧdel’s Incompleteness theorems 1st theorem: any formal system of mathematics, rich enough to express arithmetic, is either going to be inconsistent – meaning you can prove a contradiction in it - or it is going to be incomplete – meaning that there are certain truths that you can’t be able to prove in the formal system. 2nd theorem: You cannot prove your formal system is complete from within the formal system itself. https://www.nchum.org/video/godels-incompleteness-theorems - 1 hour.

Platonism – synthetic a priori That is that what is knowable is knowable independent of experience, but is more than a trivial truth, it is a truth about the world, about what is ‘out there’. In answer to our question, Are mathematicians able to prove their theorems because they’re true or are those theorems true only because mathematicians can prove them?, Platonism says that mathematicians can prove theorems because they are true. Mathematical Platonism claims that mathematics can through its proposition provide us with knowledge of trans-empirical truths- truths which are ‘out there’ but are not accessible through experience.

Other responses to Gӧdel A mathematical- revisionist approach This seems mathematical knowledge as mechanical, maths is achievable by machines, our minds are versions of machines and mathematics of infinite domains is illegitimate. Mathematicians are fooling themselves with delusions of grandeur. A postmodern response Mathematics like other disciplines which aim at knowledge is unable to achieve certainty. It is not a special discipline. Often linked with an idea that knowledge and truth is relative

Sources used: Prof Rebecca Goldstein