1. Originally Posted by Speakpigeon
Originally Posted by lpetrich
Some philosophies of mathematics affect which kinds of proof are valid, and thus affect which mathematical results are valid.

...
When Georg Cantor published his theory of transfinite numbers, one of his colleagues, Leopold Kronecker, rather strongly opposed it on finitist grounds. That theory remained controversial until the early 20th cy.
I don't think this is any problem at all. Different mathematicians start with different assumptions, so it's only proper that they should end up with different conclusions.

Again, this just shows that either mathematics is an empiric science or it's not a science at all, but if you want to say it's not a science then nothing is. Your choice.
EB
That is not a problem, because mathematics is purely deductive, and because every deduction starts with some assumptions or premises, and tries to find conclusions from them. Some mathematicians have had notions of what are valid proofs that are different from mainstream mathematicians' notions, but those notions are also premises.

2. Originally Posted by lpetrich
Originally Posted by Speakpigeon
Originally Posted by lpetrich
Some philosophies of mathematics affect which kinds of proof are valid, and thus affect which mathematical results are valid.

...
When Georg Cantor published his theory of transfinite numbers, one of his colleagues, Leopold Kronecker, rather strongly opposed it on finitist grounds. That theory remained controversial until the early 20th cy.
I don't think this is any problem at all. Different mathematicians start with different assumptions, so it's only proper that they should end up with different conclusions.

Again, this just shows that either mathematics is an empiric science or it's not a science at all, but if you want to say it's not a science then nothing is. Your choice.
EB
That is not a problem, because mathematics is purely deductive, and because every deduction starts with some assumptions or premises, and tries to find conclusions from them. Some mathematicians have had notions of what are valid proofs that are different from mainstream mathematicians' notions, but those notions are also premises.
They are premises that delineate subfields within mathematics, like playing poker or playing chess. As long as they don't mix up everything or tramp on each other's territories with their big shoes, they'll be safe.

So, is mathematics a science, then?
EB

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