Articles of philosophy

Why are Euclid axioms of geometry considered 'not sound'?

The five postulates (axioms) are: “To draw a straight line from any point to any point.” “To produce [extend] a finite straight line continuously in a straight line.” “To describe a circle with any centre and distance [radius].” “That all right angles are equal to one another.” “That, if a straight line falling on two […]

Theorems that we can prove only by contradiction

While most of the world is fine with proofs performed by contradicting the thesis, direct proofs are sometimes considered more elegant than indirect ones. Those who prefer intuitionism or structuralism as their philosophy of mathematics may even consider indirect proofs as strictly invalid. Therefore while proofs by contradiction are enough to convince others that a […]

Essays on the real line?

Are there any essays on real numbers (in general?). Specifically I want to learn more about: The history of (the system of) numbers; their philosophical significance through history; any good essays on their use in physics and the problems of modeling a ‘physical’ line. Cheers. I left this vague as google only supplied Dedekind theory […]

Ideas about Proofs

If there are two different proofs for one theorem, at some level are the two proofs the same, or can they be fundamentally different? In other words, if you have two proofs of a theorem, can one show that the two proofs are expressing the same thing in different ways, and then remove the redundancies […]

Can the generalized continuum hypothesis be disguised as a principle of logic?

A cool way to formulate the axiom of choice (AC) is: AC. For all sets $X$ and $Y$ and all predicates $P : X \times Y \rightarrow \rm\{True,False\}$, we have: $$(\forall x:X)(\exists y:Y)P(x,y) \rightarrow (\exists f : X \rightarrow Y)(\forall x:X)P(x,f(x))$$ Note the that converse is a theorem of ZF, modulo certain notational issues. Anyway, […]

How can any statements be proven undecidable?

As I understand it, undecidability means that there exists no proofs or contradictions of a statement. So if you’ve proved $X$ is undecidable then there are no contradictions to $X$, so $X$ always holds, so $X$ is true. Similarly though, if $X$ is undecidable then $\lnot X$ is undecidable. But again, this would mean $\lnot […]

How many different proofs can a theorem have?

I notice some problems has many different proofs, do all theorems have multiple proofs, is there some theorems which has only 1 way to prove it? $n$ ways? infinite?

Exactly who popularized the modern definition of domain and codomain of functions?

In Whitehead and Russell’s Principia, domain is the referents of relation; converse domain is the relata. Modern function in mathematics is just one special case of relation whose referent is unique when the relatum is given. I notice that what is called in modern function “domain” is called “converse domain” in PM, and what is […]

What concepts does math take for granted?

I suspect there must be some concepts that math takes for granted (there has to be a starting point). For example, after spending some time thinking about it yesterday, I wondered whether most of math could be produced from the concepts of Negation Identity Cardinality Ordinality Sethood Concepts Universality It seemed to me that other […]

Are axioms and rules of inference interchangeable?

There is an equivalence between cellular automata and formal systems, you can code one into the other and vice versa. But in the the cellular automata (CA) the rules of inference are fixed and are pretty simple (for instance, the universal two neighbor, two state rule 110). So all the rules of the formal system […]